Conversion of Gaussian to 51 Units SI
Gaussian
Fundamental
Ratio·
quantity
SI Symbol
Units
Units
Symbol
Gaussian
10)/41( 4l1:xlO-'
Magnetization
4,M
Permeability of free space
-
Anisotropy Exchange Gyromagnetic ratio Gilbert parameter
K
,A •
-
•
-
Parameter
Defining
Units
Defining
Units
Ratio·
Magnetic flekl H strength Magnetic induction B Flux Energy product Demagnetizatjon N factor Volume
•
,.K
susexptibility Permeability Anisotropy field Stability factor
Wall energy density Characteristic length Wall mobility
m meter cm centimeter
M
Aim
J.lo",4IlX
10- 7
erg/em J K erg/em A (S·Oe)-1 Y
Wb/Am
10- 1 10-)
JIm)
Jim (s-A/ml-l 411'/10 J I
(ormula
formula
IH-dl.4IfNI/1O De
JH,-dl=NI
Aim
IOJ/4J11'
Wblm J Wb
10-· 10-' 10- 1
B_H+41tM
G
B=1JJ.H+M)
.-fB·dA BH/8lf.=-MHfl
M,
.~fB.dA
Jim' -
H. __ NM
crgfcm'
BHf1.=:>JloMHfl
-
H.=-NM
l-dM/dH
-
,.=dM/dH
-
4K
-
lJ.=dB/dH
-
411 X 10- 7
10'/411:
Jl-dB/dH -1+4n:X H, Hr -2K/M Q Q_H./4rtM P
•
lJ_4j.!AK
I
l_a/4nM2
o -Y~ P~ P-w· « K
,
G
,-
second
N number of turns
1/4/f
=JlJI+K)
-Po
Q=H.IM
Aim -
ergfcm2
a=4VA'K
J/m 2
10- 3
om
l=alp oM 2
m
10- 2
m 2 /s·A
4nxlO- l
De
-
cm/s·Oe
A Ampere J Joule
Hr =2KIJloM
~a.= 1~ • K 0, Oersted
G
Wb Weber
M,
1
Gauss Maxwell
• To obtain values in 51 units multiply value in Gaussian units by the respective ratio
The effect of Ul",U- 1 for various U's and for a = X,Y,Z. The bottom line applies for spin! only 1.
I,
1,
U e i91 %
1%
I,oos8-/~sin8
I~
ei(Jl y
/%cos8+I~sin8
I,
/~
ei91~
/% cos 8 -I, sin 8 1. c05(8/2)
1,0058+1% sin 8
I,
e-i(JI~S~
cos 8+1, sin 8 cos8-/% sin 8
1, 1, 000(812) + 1,(25, 5;0(812)) \ • • - 1.(25, ';0(p/2))
.
,
'.' Useful relations for unitary operators U. (U is a function of I, or S, or I and S)
Uj,(J,5lf,U,5W-' = Uf,U,5jU-'Uf,U,5)U-' Uexp[i/U,S)IU- 1 = exp(iU I(J, S}U-lj For spin!:
.
" ". 1 - '",
,;1.
ccMi·(I~B~,=·cos (8/2)
sin
(I~9)
12 - 1 "'-'I
=
2l~ sin
f
,-
-,
.
(8/2)
"a~:z:y:;\''''' -"
:~
,..,
1%111 = t/~ I'. : lL.DltliI.cyclic permutations "7 z r->',_ .. 1, 1% = -'2
,.
..
,
.
..
"
;,
...
:1 ',' -
,
.
.
, ., .
..
1
Springer Series in Solid-State Sciences Edited by Peler Fuldc
c. P. Slichter
Springer Series in Solid-State Sciences Editors: M. Cardona
P. Fuldc
K. von Klitzing
Managing Editor: H. K. V. Lotsch so
l\1ull~ Diffnlial:nms "The EJ«lronir and Slatistical·Mechanical Theory ofsp-8ondcd M~I.kand Allo)' By J. 'h'""r ~'.-
lIyS. Kagosh;m •. H. NagaSllwa. andT. Sambollgi Edilors: S. V. Vonsovsky and M. I. Katsnebon 74 QUlntum Monic Carlu Mdhoch;n Equilibrium and Nonequilibrium Syslems Editor: M. Su~uki
7S
E1«1roni~ SIructU'~ ..d Oplinll I'ropt,Iiro; 01 Seraicondl>etors ByM.L.ColKnandJ R.Cbclit""u)·
16 EJectroaic Properlies oIC_jorpttil Poll'MIS Edilors: H. KlaZmlny. M. M~hrins. and S. R<Mh
Principles of Magnetic Resonance Third Enlarged and Updated Edition
11 FenniSoorfatt Etrec'b
Ed,lors: J. Kondoand A. Yosllimori 78 w
nR W Hclf=Ho+-=Flo+-
\ (b)
lowing the pulse, the excess magnetization will be perpendicular to Ho and will precess at angular frequency "'IHo. As a result, the moments will produce a flux through the coil which will alternate as the spins precess. The resultant induced emf JIlay be observed. What we have suggested so far would indicate that the induced emf would persist indefinitely, but in practice. the interactions of the spins with their surroundings cause a decay. The decay may last in liquids for many milliseconds, but in solids it is more typically 100,IS. Even during that shan time, however, there are many precession periods. The technique we have described of observing the "free induction decay" (that is, decay "free" of HI) is a commonly used technique for observing resonances. It has the great virtue of enabling one to study the resonance signal in the absence of the voltages needed to produce HI. Since oscillators always generate noise. such a scheme may be advantageous. One interesting application of the rotating rererence frame is to prove the following theorem, which is the basis of another technique for producing resonance signals. Suppose we have a magnetic field [-[0 of fixed magnitude whose direction we may vary (no other magnetic field is present). Let the magnetization M be parallel to Ho at t = O. We may describe the changing direction of Flo by an angular velocity w. Then the theorem states that if
,
y It I z [e -i"",LI. ei"l'Il\/"ttJi(O)JdT.
(2.73)
WI.
Muhiplying by U;j2 from Ihe left, integrating over spin space, and utilizing the (2.67)
and lise the fact thul I z and I z are Hermitian, we get
J.p'
(O)e -i""l tlz Izei""l If.0 tJi(O)dT
f.ct that (11[.1~) = (il da
="'(Ii Jv*(O)e -i""l t1 rei"", LI. Ize -i"".t1 ze i""l tI" !P(O)dT
= "'(Ii
!II.I-1)
=0, we get
1
T dt = 1IiHtb(~I[.I-~) (2.68)
7
I
(2.69)
(2.74)
Similarly we get
Ii db
By using (2.55), we can write
30
(2.70)
(2.66)
WI ="'(H]
{,tz(t»
{Jlz(t» = -{,JII(O» sin wit + {Jtz(O» cos wit
,p'(0 ::: a(t)u Ij2 + b(Ou_Ij2
J~'(l)I,,~(l)dT
= "'(It
Substituting in (2.68) we get
-
dt
= 1"Ht a ( -
!II.lt)
Utilizing the fact that {!II:,.,j -!)::: i and {two simultaneous differential equations to find
illzl!) =!
we can solve these 31
+ ib(O) sin (WI t(2.) b(t) :::: ia(O) sin (WI t(2.) + b(O) cos (WI tn)
aCt) "" a(O) cos (WI tn.)
(2.75)
(al
{bl -
(2.76) is the classical precession frequency about HI in the rotating frame. The spinor properly is revealed by considering a 211'" pulse. We recall that such a pulse causes (he expectalion value of the magnetization vector to undergo a rotation about HI which returns it to its initial value. Denoting the pulse duration as t2 .... we have. then Wlt2 ... :::: 211'"
6('2.) = -6(0)
Bark
«
I
p;f
y'
y'
·1
".
~~
y'
. ".
'wisled strip
{dl
X·
6(••) = 6(0)
The t ..isled slrip ....itll its ends joiI'led
(2.78)
These relationships show that after a 2x- rotation the wave function has nbl relurned to its original value but has instead changed sign. For the wave funclion 10 return to its initial value. the pulse length, t..... must lead to a 4x- rotalion. Thus. if WI t w '" 4x-
(2.79)
The general property that a 211'" rotalion produces a sign reversal of ¢ and thai a 4x- rolation is needed 10 gel t/J back to its initial value is referred to as the "spinor" properly of ",. It is shared by wave functions associated with spins of ~. ~, etc. The existence of spinors is well known in group theory. For example. in solid-state physics the property is referred to with the tenn a crystal double group [2.2]. The wave functions of particles wilh spins of 0, 1,2, elC. return to their original value under rotations of 2'11". There is somcthing unselliing aboUl finding that a 2'11" rotation does not return one to one's starting point! It is perhaps comforting in this connection 10 think of a Mobius strip, which is shown in Fig. 2.10. As is explained in the picture, one must go around the strip twice to reach the starting point. Thus. we have a physical manifestation or representation of (2.73). Referring to (2.22) we sec that though the wave function changes sign on a 2"'11" rotation, the expectation v.lllIes of the spin components [;e, [y, I: do not. The question arises whether or not the spinor nature is physically observable. The answer is yes. The first explicit demonstration was done by two groups: by Rauch et al. [2.3], and independently by Werner et al. [2.4]. Methods of providing a test had been proposed earlier by Bernstein {l.5] and by Aharmwv and Susskind [2.6]. The essence of the idea is to produce a spatial separation of the spin-up part of the wave function from the spin-down part so Ihal one can act on the two parts independently. Following a spatial region in which a magnetic field acts on only one component of the wave function, the two components are recombined (0
32
"
(2.77)
which gives. using (2.75).
!.
x'
",
y
where
a(t.... ):::: a(O)
x front
"
Fig.2.IO. A Mobius strip can be enviuged by starting with a strip of pllper ... hose fronl (a) "nd back (b) sides can be di$linguished by minting them black and ....hite respectively. The paper is then twisted (e) and the ends joined (d). Suppose one then starts on the bh.ck !lUnOKe at the point Pa. moves on thu surface to i15 other end (point Qo) llt which point one crosses over to point f\v on the white surface. Ancr going the length of the ....hile surface, one lIrrives at point Qw, adjACent to the original slarling point. Po. Thus, one has been around the strip twice to reach the starting point
study their interference. Thus, if one leaves the spin-up ponion alone. it provides a fixed phase reference for (he spin-down function. When one subjects the spindown function to 211" and 4'11" rOtations by passing it through a region of magnetic field, one finds that the interference intensity is the same for 0 and 4Jf rotations, but different for a 21f rotation. Ingenious NMR experiments demonstrating the spinor propeny have also been perfonned by Stoll and co-workers [2.7-2.9).
2.7 Bloch Equations Both quantum mechanical and classical descriptions of the motion of noninteracting spins have in common a periodic motion of the magnetization in the rotating frame. For example, if ')'Ho:::: wand if the magnetization is parallel to Ihe static field at t "" 0, Ihe magnetization precesses around HI in (he rotating frame, becoming alternately parallel and ami parallel to the direction of the static field. Viewed from the laboratory frame. the magnetization is continuously changing its orientation with respect to the large static field. However, Ihe energy that 33
must be supplied to tum the spins from parallel to antiparallel to the static field is recovered as the spins return to being parallel 10 the static field. Thus there is no cumulative absorption over long times but rather an alternate absorption and recovery. The situation is reminiscent of whal we described in the first chapter prior to introduction of the coupling to the thermal reservoir. (We note that there the system, however, simply equalized populations, whereas our present model predicts an alternating reversal of populations. The two models must therefore be based on differing assumptions.) Without contact to a reservoir, we have no mechanism for the establishment of the magnetization. By analogy to the equation
dn
dt
=
no-n
T1
(2.80)
and recognizing that M: = ,,/lin/2, we expect that it would be reasonable for M: to be established according to the equation
dM:
--= dt
Mo -M: TI
1 - - =" T2= (2.81)
where Mo is the thennal equilibrium magnelization. In terms of the static magnetic susceptibility Xo and the static magnetic field Ho, we have
Mo = xoHo
(2.82)
We combine (2.81) with the equation for the driving of M by the torque to get
dM" di""" =
M, - M. (M H) TI + "/ x :
(2.83)
Funhermore we wish to express the fact that in thermal equilibrium under a static field, the magnetization will wish to be parallel 10 Ho. That is, the x- and y-components must have a tendency to vanish. Thus (2.84)
dM M dt Y = "'(M x If)y - T: We have here introduced the same relaxation time T2 for the x- and y-directions, but have implied that it is different from Tl. That the transverse rate of decay may differ from the longitudinal is reasonable if we recall that, in contrast to the longitudinal decay, the transverse decay conserves energy in the static field. Therefore there is no necessity for transfer of energy to a reservoir for the transverse decay. (This statement is not strictly true and gives rise to important effects when saturating resonances in solids, as has been described by Redfield. We describe Redfield's theory of saturation in Chapter 6, beginning with Sect. 6.5). On the other hand, the postulate of the panicular (exponential) form of relaxation we have assumed must be viewed as being rather arbitrary. It provides
34
a most useful postulate to describe certain important effects, but must not be taken too literally. According to (2.84), under the influence of a static field the transverse components would decay with a simple exponential. (This result is readily seen by transfomling to a frame rotating at ,,/Ho, where the effective field vanishes.) A possible simple mechanism for T2 for a solid in which each nucleus has nearby neighbors arises from the spread in precession rates produced by the magnetic field that one nucleus produces at anDlher. If the nearest neighbor distance is 1', we expect a typical nucleus to experience a local field Hloc"'" Il/r J (due to the neighbors) either aiding or opposing the static field. As a result, if all nuclei were precessing in phase at t = 0, they would get out of step. In a time T such that "'(HlocT ::! I, there would be significant dephasing, and the vector sum of the moments would have thus diminished significantly. Since T must therefore be comparable to T2, a rough estimate for T2 on this model is "'(Hl oc
(2.85)
,,/2ft
often about 100 JLS for nuclei. Equations (2.83,84) were first proposed by Felix Bloch and are commonly referred to as the "Bloch equations". Although they have some limitations, they have nevenheless played a most imponant role in understanding resonance phenomena, since they provide a very simple way of introducing relaxation effects.
2.8 soluiion of the Bloch Equations for Low HI At this stage we shall be interested in the sollllion of the Bloch equations for low values of the alternating field, values low enough to avoid saturation. We immediately transform to the coordinme frame rotating al w: taking HI along the x-axis and denoting H o + (w:h) by 11 0 . Then
dM: _ dt
~1 H
- - --"'(I' Y
1+
Mo ~ M: T[
(2.86,)
(2.86b)
(2.86c)
Since M~ and My must vanish as HI -10, we realize from (2.86a) that in a steady Slate, !vI: differs from Mo to order H~. We therefore replace M: by Mo in (2.86c). The solution is funher facilitated by introducing M+ = M~ + iM y . By adding (2.86b) 10 i rimes (2.86c), we get 35
Mx(t) ==
(2.87)
(x' cos wi + i' sin wt)Hxo
(2.93)
defining the quantities X' and Xii. By using (2.90) and (2.93), we get I
,.
(2.88)
a=-+-y10I
T,
- I
Xo 2
......
X = -WO.L2
Therefore
M+ = Ae-o·t + i-yMoHI
(2.89)
1/T2 + i,ho
If we neglect the transient term and subsulUte Mo = xoHo. and define Wo :: 7Ho. = -w, we get
Wz
(2.90)
II
X =
It is convenient to regard both Mx(t) and Hx(t) as being the real parts of complex functions M~(t) and H~(t). Then, defining the complex susceptibility X by I
Mil "" XO(WO T 2) 1 + (W -Wo )'T' H) 2
(2.93,)
Xo...... I -w,p, 2 1+(w-wo)2Ti
.
X=X -IX
I
(wo - W)T2 1 +(w - wo)2Ti
/I
(2.94)
and writing
Equations (2.90) show thai the magnetization is a constant in the TOtating reference frame. and therefore is TOtating at frequency w in the laboratory. In a typical experimental arrangement we observe the magnetization by studying the emf it induces in a fixed coil in the laboratory. If the coil is oriented with its axis along the X -direction in the laboratory, we can calculate the emf from Icnowledge of the time-dependent component of magnetization
MX along the
(2.95)
Hi(i) = H XOeiWl
we find
M~(t) = xHi(t)
or
MX(t) == Re HxH xoeil.Jt)}
(2.%) (2.%,)
X -direction. y
x
"'
Fig.2.1l. Rotating
axes~,
II relative to lll.borillory llXetl
X, Y
Although (2.92) and (2.96a) were arrived at by considering the Bloch equations, they are in fact quite general. Any resonance is characterized by a complex susceptibility expressing the linear relationship between magnetization and applied field. Ordinarily, if acoil of inductance Lo is filled with a material of susceptibility xo, the inductance is increased to Lo{l + 47fxo), since the flux is increased by the factor I + 47fXo for the same current. In a similar manner the complex susceptibility produces a flux change. The flux is changed not only in magnitude but also in phase. By means of (2.93-96), it is easy to show that the inductance at frequency w is modified to a new value L, given by (2.97)
By referring to Fig. 2.1 I, we can relate the laboratory component Mx to the components M~ and My in the rOlating frame. Thus
Mx = M z cos wt + My sin wt
(2.91)
where X(w) = i(w)-ixl/(w). It is customary in electric circuits to use the symbol j for yCT. However, in order to avoid the confusion of using two symbols for the same quantity, we use only i.3
If we write the magnetic field as being a linear field, Hx(t) = Hxo cos wi
2HI = H xo
(2.92)
then we see that both M z and My are proponional to H xo, and we can write
3 In practice, the sample never complC!lely fills all apace, and we must introduce the ~fil1illg factor~ q. Ita calculation del>ends 011 a knowledge of the spatial variation of the
alternating field. Then (2.97) bccomCl!l L = Loll
+ "... qX(loI)J 37
Denoting the coil resistance in the absence of a sample as Ro, the coil impedance Z becomes
Z :::: iLow(1 + 41Tx' - i41TX") + Ro :::: iLow(l + 41TX/) + L ow41TX" + Ro
(2.105)
(2.98)
The real part of the susceptibility Xl therefore changes the inductance, whereas the imaginary part, XII, modifies the resistance. The fractional change in resistance IJ.RI Ro is
Low. -L1R = - 1TX " Ro
Ro
:::: • 1rX"Q
Utilizing the fact that H xo :::: 2H I this can be rewriuen as
(2.99)
whe~ v = W/21T. Thus, if one knows the Q, the volume of the coil, and the power available from one's oscillator, one can calculate how strong an H xo (or HI = Hxo/2) one can achieve, frequently a very useful quantity to know. In (2.105) the units are [H 1 ]: Gauss; [?]: erg/s; [v]: Hz; [V]: cm 3. Using 1w = 107 erg/s, an alternative expression in mixed units is HI =
lOPQ
(2.106)
vV
where we have introduced the so-called quality factor Q, typically in a range of 50 to 100 for radio frequency coils or 1,000 to 10,000 for microwave cavities. Assuming unifonn magnetic fields occupying a volume V, the peak stored magnetic energy produced by an alternating current, whose peak value is io, is
with [Hd: Gauss; [P]: W; [II]: MHz; [V]: cm J . The particular functions X' and XII, which are solutions of the Bloch equations, are frequently encountered. They are shown in the graph of Fig. 2.12. The tenn Lorentzjan line is often applied to them.
(2.100) The average power dissipated in the nuclei P is
p:::: !iijIJ.R:::: ~iijLow41ri'
(2.101)
By substituting from (2.100), we find
I H'xoX "V P :::: ~w
(2.102)
This equation provides a simple connection ,between the power absorbed, X", and the strength of the alternating field. We shall use it as the basis of a calculation of X" from atomic considerations, since the power absorbed can be computed in tenns of such quantities as transition probabilities. Since X' and XII are always related, as we shall see shortly, a calculation of XII will enable us to compute Xl. Moreover, we recognize that the validity of (2.102) does not depend on the assumption of the Bloch equations. Another useful fonnula can be obtained from (2.100) and (2.101), relating the average power dissipated in the coil resistance, Pc, to the strength of Hxo. Since Pc is half the peak power dissipmcd in the coil, (2.103)
Fig. 2.12. X' and X" rrom the Bloch equations ploued versus z "" (wo - ",,)T2
At this time we should point out that we have computed the magnetization produced in the X -direction by an alternating field applied in the X -direction. Since the magnetization vector rotates about the Z-direction, we see that there will also be magnetization in the Y -direction. To describe such a situtarion, we may consider X to be a lensor, such that C(t)M Ct' -
XCl"Cl'
H (toe i""t
0:
= XYZ , ,
o:l=X,Y,Z
In general we shall be interested in Xxx.
2.9 Spin Echoes
Solving for (~)iij and substituting into (2.100) we get
Hl ov : : PcQ 8,
38
w
(2.104)
Just after finishing graduate studies, Erwin Hahn burst on the world of science with his remarkable discovery, spin echoes [2.10]. His discovery provided the
39
key impetus to the development of pulse methods in NMR. and must therefore be ranked among the most significant contributions to magnetic resonance. What are spin echoes and why are they so remarkable? Suppose one applies a 1fn. pulse to a group of spins to observe the free induction signal which follows turn-off of the pulse. According to the Bloch equations. the free induction signal decays exponentially with a time constant T2' For solids. T2 is a fraction of a millisecond, corresponding to line widths of several Gauss. For liquids, the line widths are typically much narrower. corresponding perhaps to times of several seconds. Such lines are a good deal narrower than the usual magnet homogeneity. As a result, the inhomogeneity-induced spread in precession frequency causes the spins in one portion of the sample to get out of phase with those in other JXlnions. The free induction signal arises from the sum total of all portions of the sample. As the separate portions get out of step, the resultant signal decays. The decay lime is of the order of l/(-y.dH) where tj,H is the spread in static field over the sample. fJalm made the remarkable discovery that if he applied a second 7rn. pulse a time T afler the first pulse, miraculously there appeared another free induction signal at a time 21' after the initial pulse. He named the signal the "spin echo". To produce a signal at the time of the echo, the spins must somehow have gotten back in phase. The great mystery of the spin echo was what made the spins get back in phase again? Was the echo a challenge to basic concepts of irreversibility? Was there a Maxwell demon at work producing the refocusing? fJalm discovered spin echoes experimentally, but was soon able to derive their existence from the Bloch equations. This solution showed that as one varied T. the echo amplitude diminished exponentially with a time constant T2. Thus the echo provided a way of measuring line widths much narrower than the magnet inhomogeneity. Understanding the physical basis of echo formation has led to much deeper insight into resonance phenomena in general and pulse work in particular. The essential physical ideas of refocusing can be most easily seen by considering a pulse sequence in which the first pulse produces a rotation of 7I:n.. the second a rotation of 1f. Such a sequence we denote as a 7rn.-7r pulse sequence. It was invented by Carr [2.11] based on a vector model proposed by Purcell for the 1rn.-7f/2 echo [2.10]. and was first described in a famous paper by Carr and Purcell [2.12). Consider a group of spins initially in thennal equilibrium in a static magnetic field H in the z-direction. The thermal equilibrium magnetization Mo then lies along H as shown in Fig. 2.1301. We assume there is a spread in magnetic fields over the sample, and take the average value of field to be Ho. We first analyze what happens if we can neglect the effect of Tl and T2. We apply a rotating magnetic field HI at t = 0 with frequency w. tuned to resonance at the average field Ho. Thus
w=,Ho 40
(2.107)
,
, M,
}---y x
x
(b)
, M!
----
/
0
,
--" --
I"" 0'
, --,
--,,
.""7'----i-- y / --'
x
x (d)
I'" T+
, FIg. 2.13a- 2T is identical to its development immediately after the first 11'!2 pulse. We give this as a homework problem. This result has a useful e)(perimental consequence. The signal immediately after the 7r!2 pulse gives one the Fourier transform of the line shape function P(ho), as can be seen from the results of Problem 2.3. But the signal immediately follows a strong pulse. In practice this pulse may block the signal amplifiers of the NMR apparatus, making it impossible to observe the signal until the amplifiers recover. On the other hand, the echo occurs later than a pulse by a time T, giving the amplifiers time to recover. Thus, the echo is easier to see. It is therefore fortunate Ihat the echo reproduces the earlier signal. 50
2.11 Relationship Between Transient and Steady-State Response of a System and of the Real and Imaginary Parts of the Susceptibility Suppose, to avoid saturation, we deal with sufficiently small time-dependent magnetic fields. The magnetic system may then be considered linear. That is, the magnetization produced by the sum of two weak fields when applied together is equal to the sum of the magnetization produced by each one alone. (ytIe shall not include the slatic field Ho as one of the fields, but may find it convenient to consider small changes in the static field.) In a similar manner, an ordinary electric circuit is linear, since the current produced by two voltage sources si· multaneously present is the sum of the currents each source would produce if the other voltage were zero. Let us think of the magnetization Ll."f(t) produced at a time t and due to a magnetic field H«() of duration Llt' at an earlier time (see Fig.2.18). As a result of the linearity condition we know that LlM(t)oc H(t'). It is also oc Llt' as long as Llt' 0,
J
dm +
1=0-
J~
dt =
1=0-
~~
d met) = dt (Mstep)
(2.146)
t=O-
(2.150)
Equation (2.150) therefore shows us that knowledge of Mstep(t) enables us to compute met). For example, suppose we discuss the magnetization of a sample following application of a unit magnetic field in the z-direction for a system obeying the Bloch equations. We know from the Bloch equations that
= Mlltep
(2.151)
Therefore, using (2.150),
T,
E(t)dt
Fig. 2.19. Step runctKm
By taking the derivative of (2.149), we find
1=0+
J
(2.149)
0
,
met) = XOe-t/TI 1=0+
j m(r)dr
o
MzCt) = xo{l- e- t / T1 ) (2.145)
To find A, we integrate (2.144) across t = 0 from t = 0- to t = 0+:
t=O+
t')dt' =
_I_H(O_ _
To find the equation for m{t), we recognize that met) obeys this equation when H(l) • '(I). Thus.
j met o
We then have
dM% + M r = XO H(t) dt T1 Tl
,
,
(2.152)
Note that in any real system, Ihe magnetiutlion produced by a step is bounded, so that ~
J
which gives
m(r)dr
m(O+) _ m(O-).
0
As t ..... - 00, this function goes to zero. We compute the limit as 8 ..... O. Thus
M~(t)
•
=
J
(2.160)
c,
The integral vanishes as w ----+ 00, leaving us = x'(oo). It is therefore convenient 10 subtract the o-funclion pan from m(r), which amounts 10 saying that
J
00
X(w) - X'(oo) =
met - t')Hxoc i .." " e"' dt'
-00
= lI xo e'W'c"
J J
met - t')eiw("-')c'{"-l)dt'
00
= Ilxoe(iw+.),
m(r)e-(·+iwjT dr
m(r)e-i,,",T dr
(2.161)
o
•
_00
and
•
.-.
x("') = Jim jm(T)e-(.+'W)Tdr The advantage of this definition is that it has meaning for the case of a ~lossless resonator" (magnetic analogue of an undamped harmonic oscillator), in which a sudden application of a field would excite a transient that would never die out.
54
(1') cos (wr)dr + c\
o
where now m(r) has no a-function part. [Of course no physical system could havc a magnetization that follows the excitation at infinite frequency. However, if one were rather making a theorem about penneabilitY!1-, p(oo) is not zero. We keep x'(oo) to emphasize the manner in which such a case would be treated.] We wish now to prove a theorem relating X' and X", the so-called KramersKronig theorem. To do so, we wish to consider X to be a function of a complex variable z = x + iy. The real part of z will be the frequency w, but we use the symbol x for w 10 make the fonnulas more familiar. Therefore
55
=
x(z) - i(oo) =
=
J o = J
m(r)e- iZT d,
The presence of the tenn exp (yr) tells us that Ix(z) - i(oo)I--+ 0
m(T)eYTe-i:l;T dT
(2.162)
o Since an integral is closely related to a sum, we see that X(z) is essentially a sum of exponentials of z. Since each exponential is an analytic function of z. so is the integral, providing nothing too bizarre results from integration. To prove that X(z} - x'(oo) is an analytic function of z, one may apply the Cauchy derivative test, which says that if x(z) - X/Coo)
== U + iv
(2.163)
and
avail ax = - ay
(2.164)
From (2.162) we have
J= o = J
m(T) cos (xT)e
00
-
·We already know that
Ix(z)-i(oo)I-O as
X--I±OO
Therefore X(z) - i(=) is a function that is analytic for Y:5 0 and goes to zero as Iz I -+ 00 in the lower half of the complex plane. Let us consider a camour integral along the path of Fig. 2.20 of the function
xCi) - x'(oo) z' w
X'(z') ~ X'(oo)
j "--''--'-;---'''-'.:.'-'dz
I
ZI_W
u ""
Y --I
By Cauchy's integral theorem this integral vanishes, since X(z) has no poles inside the contour.
where u and v are real, u and v must satisfy the equations
au au ax = ay
as
=0
(2.167)
C
yT
(2.165)
dT
Since IxIV) - x/(oo)1 goes to zero on the large circle of radius e, that part of the integral gives zero contribution. There remains the contribution on the real axis plus that on the circle Zl - w = Rexp (i¢). Thus
m(T) sin (xT)eyTdT
v = -
o giving
=
+
au = -jm(T)T sin (XT)eYTdT= av
ax
au
-
ay
o
ay
ax
which satisfy the Cauchy relations, provided it is pennissible to take derivatives under the integral sign. There are a variety of circumstances under which one can do this, and we refer the reader to the discussion in Hobson's book [2.20]. For our purposes, the key requirement is that the integrals in both (2.165) and (2.166) must not diverge. This prevents us in general from considering values of y that are too positive. For any reasonable mer) such as that of (2.152), the integrals will be convergent for y:S 0, so that X(z) - X'(oo) will be analytic on the real axis and in the lower half of the complex z-plane. Whenever we use functions mer) that are flOl well behaved, we shall also imply that they are to be taken as the limit of a well-behaved function. (Thus an absorption line that has zero width is physically impossible, but may be thought of as the limit of a very narrow line.)
56
Re1q,
= O= P
j
-=
+=
j
o:.J+R
11"
+00
j m(T)r cos (XT)eyTdr = -a,o
'J
,
X(w) -.X (00) Rieiq,d¢+
(2.166)
= =
j
2. [
,
I
X(w') -l(oo)
wI-w
, )J w + 1T1.[ X(W ') -X(oo
X(w) - X (oo)d'
w
I
w
(2.168)
where the symbol P stands for taking the principal pan of the integral (that is, taking the limit of the sum of the integrals ~~R and J:;+~ as R --+ a simultaneously in the two imegrals).
Fig.2.20. Contour integral 57
x"
Solving for the real and imaginary parts, we find
J
1 +00 1/( ') x(w) - x'(oo) = -p ~c/w' 11" w'-w
-n
(2.169)
-00
+oox" (w ) - X,(00) dw' X"(w) = _2.. p
J
(a)
n
w
I I
n!(
~-n
-00
These are [he famous Kramers-Krollig equations. Similar equations can be worked out for analogous quantities such as the dielectric constant or the electrical susceptibility. The significance of these equations is that there are restrictions placed, for example, on the dispersion by the absorption. One cannot dream up arbitrary X'(w) and X"(w). To phrase alternately, we may say that knowledge of X" for all frequencies enables one to compute the X' at any frequency. Note in particular that for a narrow resonance line, assuming x'(oo) = 0, the static susceptibility Xo is given by I
XO = X(O) = -p •
J _,_w_;onance exp·eriment with the static field in ~he z-direc~ion and the alternating field in the z-direetion, we are discussing Xu. Then x'(O) or (2.170) is x~...(O), whereas xo is usually thought or as relating the total magnetization Alo to the field I/o, which produces it, and is thllS X~r{O). lIowever, a Slnll.lIstatie field IIr in the z-directioll simpl)' rotates M o , giving
I/r M oII,
,
=
x.. (0
ThllS X~...(O) = x~AO)
=Xo.
Mr =
58
)
lIr
We shall now tum to obtaining expressions for the absorption and dispersion in tenns of atomic properties such as the wave functions, matrix elements, and energy levels of the system under study. We shall compute X" directly and obtain X' from the Kramers-Kronig equations. We make the connection between the macroscopic and the microscopic properties by computing the average power P absorbed from an alternating magnetic field HrlJ cos wt From (2.102) we have
P -w"H'V -"2X zO
(2.173)
in a volume V. It will be convenient henceforth to refer everything to a unit volume. (We shall have to remember this fact when we compute the atomic expressions in particular cases). On the other hand, Ihe alternating field couples 10 the magnetic moment Itzk of the klh spin. Therefore. in our Hamiltonian we shall have a time-dependent perturbation 1{t)er~ of 7 To show that 6(z) = 6(-z) we consider the integrals II = I~oo f(z)6(z)dz and =I~::: f(z)6(-z)dz. We have irnmedia~ely that II = j(O).1b eval~ate 12, we chAnge variable to z' = -z. Then we get
I~
+12 = Thus 12
J
--
f( -z')6(z')dz'
=/(0)
= II, and ,s(z) =6(-:r:). 59
'}-{pert = - L!1-3:kH3:0 cos wt k = -!1-1;H1;O cos wt
(2.174)
We can compute the absorption rate P ab, due to transitions between states a and b in tenns of Wah, the probability per second that a transition would be induced from a to b jf the system were entirely in state a initially:
where !1-3: is the x-component of the total magnetic moment ~§L!1-1;k
(2.179) a.175)
k
In the absence of the perturbation, the Hamiltonian will typicnlly consist of the interactions of the spins with the external static field and of the coupling '}-{'k . . J between SpinS J and k. Thus
'H" - L"'kHO+ L'Hjk k
(2.176)
The tenns P(Eb) and P(Ea ) come in because the states la) and Ib) are only fractionally occupied. The calculation of the transition probability Wah is well known from elementary quantum mechanics. Suppose we have a time-dependent perturbation Hpert given by HI>ert = Fe -iw! + Ge iw !
(2.180)
j,k
We shall denote the eigenvalues of energy of this many-spin Hamiltonian as E a , Eb, and so on, with corresponding many-spin wave functions as la) and Ib). See Fig.2.22. Because of the large number of degrees of freedom there will be a quasi-continuum of energy levels.
where F and G are two operators. In order that Hpert will be Hennitian, F and
G must be related so that for all states I(t) or Ib), (aIFlb) " (bIGla)'
(2.181)
Under the action of such a perturbation we can write that Wah is time independent and is given by the fonnula (2.182)
Fig.2.22. Eigenvalues of energy
The states la) and Ib) are eigenstates of the Hamiltonilln. The most general wave function would be a linear combination of such eigenstates: (2.177)
where the ca's are complex constants. The square of the absolute value of Ca gives the probability p(a) of finding the system in the eigenstate a: p(a)"
1,.1 2
If the system is in thennal equilibrium, all stales will be occupied to some extent, the probability of occupation p(a) being given by the Boltzmann factor
e- EA / kT
P(E.) "
Le
E,/kT
h l(a!Flb)1 < T
(2.178)
E.
where the sum E c goes over the entire eigenvalue spectrum. The denominator is just the classical partition function Z, inserted to guarantee that the total probability of finding the system in any of the eigenstates is equal to unity; that is,
60
provided certain conditions are satisfied: We do not ask for details that appear on a time scale shorter than a certain characteristic time T. It must be possible to find such a time, which will satisfy the conditions that 1) the populations change only a small amount in T and 2) the possible states between which absorption can occur must be spread in energy continuously over a range LJ.E such thaI LJ.E» hiT. These conditions are violated if the perturbation matrix element l(aIFlb)1 exceeds the line width, as it does when a very strong alternating field is applied. We can see this point as follows: The quantity LJ.E may be taken as the line width. We have, then, that LJ.E< l(alFlb)l. But under these circumstances one can show that the populations change significantly in a time of order IiII(alFlb)j. Thus to satisfy the condition I that the populations change only a small amount during T, T must be chosen less than hll(aIFlb)l. This gives us
But, by hypothesis,
dE < l(alFlb)1
Therefore
LJ.E < !!. T
which violates condition 2 above. Thus it is not possible to satisfy both conditions, and the transition probability is not independent of time. 61
,
This example shows why we did not gel a simple time-dependent rate process in Sect. 2.6, since for that problem, the energy levels in the absence of HI are peneClly sh"'P (LIE = 0), l(aIFlb)1 > LIE. In our formula for Wah we use the O-funclion. This implies thai we shall evenlUally sum over a quasi-continuum of energy states. In writing the rransition probability, it is preferable to use the o-function (onn rather than the integrated form involving density of slales in order to keep track of quantum numbers of individual stales. By summing over all states with E a > Eb, we find
21f H 2o 2 -'-r,w !P(E,) - p(E,)JI(al",lb)! ,(E, - E, - I>w) 11 4 Ea>Eb
The quanta hw correspond crudely to the energy required to invert a spin in the static field. This energy is usually much smaller than kT. For nuclear moments in strong laboratory fields (...... 10,1 Gauss), T must be as low as 10- 3 K so that fiw will be as large as kT. This fact accounts for the difficulty in producing polarized nuclei. For electrons, kT ...... fiw at about 1 K in a field of 104 Gauss. Therefore we may often approximate (2.188) We may call this the "high-temperature approximation". By using (2.178) and (2.188), we have
L:
P = -
=~i'H;o
= e-Ea/kT[e(Ea-Eb)/kT _ I]
P(E,) - p(E,)
(2.183)
Z =
e- Z (Ea kT- Eb) Ea kT /
(2.189)
Therefore
I
X"(w)
=, L:
[P(Eb) - p(E,)II(a!",lb)I',(E, - E, - hw)
. (2.184)
Ea > Eb
Eb.
As long as Ea > only positive w will give absorption because of the 0function in (2.184). Removal of the restriction E a > Eb extends the meaning of xll(w) fonnally to negative w. Note that since lJ(Eb) - P{Ea ) changes sign when a and b are interchanged, i'(w) is an odd function of w, as described in the preceding section: X"(w)
=, L:
IJKEb) - p(E,)II(al,',lb)I',(E, - E, - I>w)
.
(2.185)
Ea,Eb
Assuming i(oo) = 0 for our system, we can easily compute i(w), since x'(w)=.!.P
+=
Jw
1T
lie
I
XI w)dw ' -w
(2.186)
-00
J ,(E, -,Eo -
+00
=, L:
[P(E,) - P(E,lll(al",lb)I'.1. p
Ea,Eb
1T
W
-00
-
,
hw) dw'
W
or, evaluating the integral, x'(w) =
L:
[P(E,) -
P(E,)JI(a!",lb)1
Ea,Eb
,
1
E _ E _, a
b
(2.187)
lW
By lIsing the fact that a and b are dummy indices, one may also rewrite (2.187) to give x'(w) =
L: p(E,JI(al,',lb)I' [(E, -
E, - nw)-l
Ea,Eb
+ (Ea 62
- Eb
+ rlW)~I]
Substitution of (2.189) into (2.185), together with recognition that E a - E b = hw, owing to the o-functions, gives xl/(w) =
~;; L:
e-Ea/kTI(alfl':rlb)120(Ea - E b - hw)
(2.190)
Ea,Eb
Another expression for XI/(w) is frequently encountered. It is the basis, for example, of Anderson's theory of motional narrowing [2.21]. We discuss it in Appendix B because a proper discussion requires reference to some of the material in Chapters 3 and 5. It is important to comment on the role of the factors exp(-Ea/kT). If one is dealing with water, for example, the proton absorption lines are found to be quite different at different temperatures. Ice, if cold enough, possesses a resonance several kilocycles broad, whereas the width of the proton resonance in liquid water is only about 1 cycle. Clearly the only difference is associated with the relative mobility of the H 20 molecule in the liquid as opposed to the solid. The position coordinates of the protons therefore play an important role in detennining the resonance. Fonnally we should express this fact by including the kinetic and potential energies of the atoms as well as the spin energies in the Hamiltonian. Then the energies E a and Eb contain contriblllions from both spin and positional coordinates. Some states la) correspond to a solid, some to a liquid. The factor exp(-Ea/kT) picks out the type of "lattice" wave functions or states that are representative of the temperature, that is, whether the water molecules are in liquid, solid, or gaseous phase. Commonly the exponential factor is omined from the expression for XII, but the states la) and Ib) are chosen to be representative of the known state. The classic papers of Gutowsky and Pake (2.22], on the effect of hindered molecular motion on the width of resonance, use such a procedure.
(2. 187a)
63
Evaluation of x!' by using (2.190) would require knowledge of the wave functions and energy levels of the system. As we shall see, we rarely have that infonnation, but we shalI be able to use (2.190) to compute the so-called moments of the absorption line. We see that the only frequencies at which strong absorption will occur must correspond to transitions among states between which the magnetic moment has large matrix. elements.
3. Magnetic Dipolar Broadening of Rigid Lattices
3.1 Introduction A number of physical phenomena may contribute to the width of a resonance line. The most prosaic is the lack of homogeneity of the applied static magnetic field. By dint of hard work and clever techniques, this source can be reduced to a few milligauss out of 104 Gauss, although more typically magnet homogeneities are a few tenths of a Gauss. The homogeneity depends on sample size. Typical samples have a volume between 0.1 cc to several cubic centimeters. Of course fields of ultrahigh homogeneity place severe requirements on the frequency stability of the oscillator used to generate the alternating fields. Although these matters are of great technical imponance, we shall not discuss them here. If a nucleus possesses a nonvanishing eleclric quadrupole moment, the degeneracy of the resonance frequencies between different m-values may be lifted, giving rise to either resolved or unresolved splittings. The latter effectively broaden the resonance. The fact thaI T, processes produce an equilibrium population by balancing rates of transitions puts a limit on the Iifctime of the Zeeman stales, which effectively broadens the resonance lines by an energy of the order of lilT,. In this chapter, however, we shall ignore all these effects and concentrate on the contribution of the magnetic dipole coupling between the various nuclei to the width of the Zeeman transition. This approximation is often ex.cellent, particularly when the nuclei have spin! (thus a vanishing quadrupole moment) and a rather long spin-lattice relaxation time. A rough estimate of the effect of the dipolar coupling is easily made. If typical neighboring nuclei are a distance r apan and have magnetic moment lA, they produce a magnetic field Bloc of the order I' Bloc ::; r 3
(3.1)
By using,. ::; 2 A. and J-l ::; 10- 23 erg/Gauss (10- 3 of a Bohr magneton), we find Hloc s::o I Gauss. Since this field may either aid or oppose the static ficld Ho, a spread in the resonance condition results, with significant absorption occurring over a range of B 0 ....., I Gauss. The resonance width on this argument is independent of H 0, but for typical laboratory fields of 104 Gauss, we see there is indeed a sharp resonant line. Since the width is substantially greater than the magnet inhomogeneity, it is possible to study the shape in detail without instrumental limitations. 64
65
Flg.3.t. Relationship between rectangular coordina~es :1:, II, , (describing the position of nucleus 2 rdative to nucleus I) and the polar coordinates r, 8, '"
z
3.2 Basic Interaction The classical interaction energy E between two magnetic moments III and 112
;s (3.2)
k--li-r7-- -"
where r is the radius vector from III to JJ.2. (The expression is unchanged if r is Iaken as the vector from 112 to PI.) For the quanlum mechanical Hamiltonian we simply take (3.2). treating 1-'1 and P2 as operators as usual:
(3.3)
P2""'Y2hI2.
PI ""/lhII
where we have assumed that both the gyromagnetic ratios and spins may be different. The general dipolar contribution to the Hamillonian for N spins then becomes
Hd ""
~
EE
[Ili
2 j=1 1e=1
~PIe
_ 3(pj
r jle
.rjk~(Il/c .rjlIJ]
(3.4)
rjk
where the! is needed. since the sums over j and k would count each pair twice, and where. of course, we exclude tenr.s with j = k. By writing PI aod JJ.2 in component fonn and omitting the subscriplS from r, we see from (3.2) that the dipolar Hamihonian will contain terms such as 2
I
2
zy
/112k [lzh="5
/112h hzI2z-, r
r
r'
/12
(A + B + C + D + E + F)
r:-
(3.6)
y
z
Hz "" -71hRoIII - 72hHo12z corresponds to an interaction with a field of 104 Gauss. It is therefore appropriate to solve the Zeeman problem first and then treat the dipolar term as a small penurbalion. (Actually. for two spins of~, an exact solulion is possible.) To see the significance of the various tenns A, B, C. and so on, we shall consider a simple example of two identical moments. both of spin The Zeeman energy and wave functions can be given in tenns of the individual quantum numben ml and m2. which are the eigenvalues of II' and h,· Then the Zeeman energy is
!.
(3.8) We shall diagram the appropriate matrix elemenlS and energy levels in Fig.3.2. It is convenient to denote a state in which ml = m2 = -~ by the nOlation 1+ -). The twO states 1+ -) and 1- +) are degenerate. and both have B-l = O. The states 1+ +) and 1- -) have, respectively. -hwo and +hwo, where wo = ,Ro as usual. We first ask what pairs of states are connected by the various tenns in the dipolar expression. The tenn At which is proponional to II.zh.z' is clearly completely diagonal: It connects Iml m2) with (mtm21. On the other hand, B, which is proportional to It 12" + II only connects Iml m2) to states (ml + I, m2 - II or (ml - I, m2 + II. A customary parlance is 10 say that B simultaneously flips one spin up and the other down. B therefore can join only the states 1+ -) and 1- +). The states joined by A and B are shown diagrammatically in Fig. 3.3.
+!,
It,
where 2
A"" [1,[2,(1 - 3 cos 8)
B = -t(I{ 12" + I) [i)(I
C=
1/
(3.5)
If we express 1)z and [151 in tenns of the raising and lowering operators and I), respectively, and express the rectangular coordinates z, y, z in tenns of spherical coordinates r, 8, tP (Fig. 3.1), we may write the Hamiltonian in a fonn that is particularly convenient for computing matrix elements: '}-{d = /11'2
I'
- 3 cos 2 8)
-~(It h, + 1),Ii) sin B cos Be-i,p
(3.7)
D = -~(Il h, + IlrI2") sin 8 cos 8e i,p E= -~I{Ii sin 2 8e- 2i ,p . 2 ee 2i ,p F -- -;{'1-1I 2 Sin As we have remarked, (/11'2h2)1r3 corresponds to the interaclion of a nuclear moment with a field of about I Gauss, whereas the Zeeman Hamillonian
66
-----+ +--------++ Fig. 3.2. Energy levels or two
iden~iclIl
o
spins
67
,--A ...
(
Fig. 3.3. States joined by matrix elements A .. nd B. The dashed lines &0 between states thal are joined
)
A
...-........ +- ( ,
-----_ .... B
A ....-.... >
( )
.....
where
is the wave function corrected for the effect of the penurbation 'Hpert
the unpenurbed states of u~, and u~. Qy means of (3.9) we can see that the state 1+ +) will have a small admixture of 1+ -), 1- +), and 1- -). The amount of admixture will depend on (n'I'Hpertln) and En - En" The former will be -y 2h 2Jr3 multiplied by a spin matrix element. Since the spin matrix element is always of order unity, and since Hl oc = -yTl/r 3, we can say (n/l'Hperdn) ':! -yIIHloc . On the other hand, En - En' = Ilwo = 'YfIHO, so that
-+
,..-_ ....A
'+ +
(
Un
and where, of course, the matrix elements (n'j'Hpertln) are computed between
Note that B has no diagonal matrix elements for [he m 1m2 representation, but it has off-diagonal elements between two stales which are degenerate. The fael that off-diagonal elements join the degenerate states 1+ -) and J _ +)
(n'I1ip m2, m3 ... mN for the N spins. Therefore
L,(mllplzlmd:: "(ft L,(mI11lzlml):: 0
~
w
(3.24)
•
-~
I+~
o
1
L(al";la) = 2h T, (";)
f(w)dw:: 2h
Therefore we can compute either (&.;2) directly or compule it from the calculations of (w 2 ) and (w). (We shall do the laneL) .
1
j j(w)dw = 2h 0-
j j(w)dw
(&.,') = (w') _ (w)'
~
.
(3.29)
ml
Therefore the contribulion from tenns j :/: k vanishes. For j :: k, we get, taking j :: I,
1,8')
(3.23,)
(3.30) 73
The matrix element is independent of m2, mJ, and so on, but it is repealed for each combination of the other quantum numbers. Since there are (21 + I) values of m2, (21 + I) values of m3, and so on. we get the matrix for each value of ml repeated (2I+I)N-I times. On the other hand. using Trl to mean a trace only over quantum numbers of spin 1. we have that Trl {llIz} = TTJ {Jll y }. This equation
To compute the average frequency or first moment rigorously,
=
wj(w)dw
(w) =
is most simply proved by first evaluating Tq {Plz}' using eigenfunctions of liz'
Then
+1
Tq {Plz} = "(2 112
L:
m2
(3.31)
In a similar way. Trl {ply} may be evaluated by using eigenfunctions of Ily: +I
L:
j(w)dw
o
m=-1
Trl {PIy} = ..,2h2
J -,=~,----- J
m2
(3.3Ia)
m=-I
Therefore Trl {PIz} =Trl {llI y } =Trl
tlltJ = iTrl {pi}
+=
J
wj(w)dw =0
because the integrand is an odd function of w. We therefore are forced to compute r.'wj(w)dw:
J=o
I
wj(w)dw = ",
.
f(w)dw = 21'l72h2 I(I; 1) N(2I + l)N
I = -,
+=
J
(alpzlbXblpz[aXhw),(E. - E, - hw)d(hw)
L
(all'z[bXblpzla)(E. - E,)
.
(3.34)
h E.>E. (B2)
o We tum now to a calculation of the effect of the dipolar coupling on the average frequency of absorption, (w). The existence of such a shift implies that the local field produced by the neighbors has a preferential orientation with respect to the applied field. Since such an effect must correspond to a Lorentz local field .t!J.H, it must be of general order XIIHo. where XII is the static nuclear susceptibility. X'I is given by the Langevin-Debye fomlUla: XII = N"(2h 2 I(I + 1)l3kT, where N is the number of nuclei per unit volume. If the distance between nearest neighbors is a. N ;:' Iht 3• we have, therefore, that t1H S! (,lda3)(-rliHolkT) S! Hloc(,IiHolkT). Since the nuclear Zeeman energy 7hHo is very small compared with kT. we see t1H is very small compared with the line breadth Hloc and is presumably negligible. Notice that the physical significance of our expression for t1H is that the neighbors have a slight preferential orientation parallel to the static field given by the exponent of the Boltzmann factor ("(IiHolkT). Hl oc has a nonzero average to this extent. Since few) of (3.17) corresponds to infinite temperature, it must lead to a .t!J.H = 0, and (w) = woo 74
L
o,b 0
Since there are N identical terms of j = 1.:, finally we get as our answer
j
(3.33)
-= (3.3Ib)
There are 21 + I diagonal matrix elements of PI, each of magnitude 72h2I(I + 1). Therefore
..,2h2 I(I + I) Trl {Jtlz} = 3 (21 + I)
k
is a bit more difficult than the calculation of f(w)dw. In (3.22) it was convenient to extend the limits of integration to go from -00 to +00. As a result. for every pair of energies E a and Eb' there was some frequency w such that Eo - E" = trw. regardless of whether E a was higher or lower than E". We cannot do the same thing for (w), since
The energies Ea and E" are the sum of dipolar and Zeeman contributions (--yhHoM + En), as we have remarked previously. We shall assume that the dipolar energy changes are always small compared with the changes in Zeeman energy and that the latter correspond to absorption near wo (our earlier discussion of the role of the tenns A. H, ... F shows us this fact). Therefore. since Eo > Eb. we write
Eo
= -"(TiHoM + En
Eb = -"(IiHo(M + 1) + En' Eo-Eo = flWo + En-Er./
(3.35)
By using these relations, we can write (3.34) as
=
J
I
wf(w)dw = 2"
o
L
(MalJ-lzIM + 10")
" '"I ,n,a' X (M + 100'1/lz1M a)(!iwo + EO' - EO',)
(3.36)
We shaJl first discuss the contribution of hwo tenn in the parentheses. It is 75
~o L: it
(MaIJJ;!:jM + la')(M + la'IJJ;!:IMa)
.
(3.37)
= EOI, / u Al'OI,PU,\I01 dr
/11,0',0"
= Eo:,(M'o'jPIMa)
Were it not for the restriction to M + 1, (3.37) could be converted to a trace by means of (3.23a). This restriction can be removed by using the properties of the raising and lowering operators and by noting that
.
(3.42)
Therefore
L
(MoIJJ-jM'a')(M'o'lp+jMo)(Ea - EOt')
/11 ,/If' ,CI',O"
•
L M,/I1',&,«'
Thus
(Moll1i~,~-lIM'o')(M'o'I~+lMa)
(3.43)
.T'{[1i~'~-)I'+) . (3.38)
Since Jl+ connects only states M ' and /11 in a matrix element (M' a'IJJ+JM (X) where M' = Ai + I, we can rewrite (3.37), summing over all values of M' as
A detailed evaluation of this trace shows that it vanishes. Therefore, combining the results of (3.36), (3.39), and (3.43), we get
f 00
wf(w)dw =
~: Tr {pi}
(3.44)
But from (3.24), OOJ
(3.39)
I j(w)dw = 2h T, {~~}
o Therefore 00
J
wj(w)dw
(w)
where we have used the facts that Tr {Jl~} = Tr fJl~} and
.
= / =
u~l'Ot'~uMO'dr
J
uA1'o,PEotlMOI dr
= EO'(M'a'IPIMa:)
.
Likewise, using the fact that 1{~ is Hennitian,
(M'O"I1t'~PIMa)
= / uA'f'OI,1t'3PI.lMO'dr
=
jerC!JU/lf'cr)· PUMO'dr
J
(3.45)
o
(3.40)
We have so far handled the hwo tenn of (3.36). The technique for handling the tenn E OI - E OI , is very simple. We know that }£~ IMa') = EO', 1M a'). Therefore, for an operator P, we have
(M'a'I~IMa)
=wo
j(w)dw
Tr {JJ;!:JJy -,lyp;!:} = "'(2 h 2 Tr {I;!:Iy - fIJI;!:} = ,2h2iTr {I:} =0
=~
(3.41)
The "average" value.of the frequency is therefore uns~ifted by me broadening as we had expected. To get the local field correction that we mentioned in our qualitative discussion, we should, in fact, have to go back to (3.17) and include the exponential factors that we deleted in going from (3.16). [That this is true follows from me fact that the expression fjH ~ Hloc("'(hHolkT) depends on temperalUre. The only place the temperature enters is in the exponentials.] We can compute the second moment (w 2 ) by similar techniques: 00
/ w 2 f(w)dw
~=~
J
O~
j(w)dw
o
Since we have already evaluated the denominator, all that remains is to compute the numerator:
76 77
00
Jo
2
w !(w)dw
1+00
Jw
:; 2
2
{Llw') =
j(w)dw
-00
1+00
J
=:2
-00
=
2'"
~7' ", I(l + I)~ L
4
Lw'(alp.lb)(blp.la)b(E. - E. - ""')dw
(3.47)
a,b
L...c E • - E.), (alp.lb)(blp.!a)
.,'
jl.:
is independent of j. There are then N equivalent sums, one for each value of j, giving us
3
L
{Llw') = -7' ", I(l + I) 4 I.:
(3.48)
= 'Hz + 'H'~ to get
where in Ihe "cross-Ierm" involving {[1tZ. 1Ja:] and l1t3, J.lz] we have used the basic relation true for any pair of operators A and B:
(3.50)
which is readily proved by applying (3.23a). If the dipolar coupling were zero only the first lenn on the right would survive. and of course the resonance would be a 6"-function at w::;; WOo In this case (w 2 ) = Therefore we sec that the first lenn must contribute w~ 10 (w 2 ). Explicit evaluation in fact verifies lhis result. The second. or '\,TOSS", leon vanishes, since every term involves factors such as Trl {Pb-}' The lasl term, when divided by j(w)dw gives
wfi.
k
~,'i ft2 1(1 + I) (~) L: (I N
- 3 c~s2 8jk )2
i,I.:
Now, by (3.21),
{Llw') = (w') _ {w)' Therefore, since (w)::: WO, we have
78
rjl.:
J
We can get a clearer understanding of (3.52) by considering an example in which all spins are located in equivalent positions, so that
I.:
By using the fact thai 'Hla) = Eala), we see, as in (3.42) and (3.43),
Tr {AD} = Tr{BA}
(3.52)
r~1.:
~ (I - 3 cos 2 6 j l.:)2 L.J ,.6
1 "
We can expand, using H
(I - 3 cos' 9j ,)'
N J,. I.:
(I - 3cos 2 6· )2
6
I'
(3.53)
Tjl.:
Each term is clearly of order (-1H{;,,/ where H~c is the contribution of the kth spin to the local field at spin j. The imponant point about (3.53) is that it gives a precise meaning to the concept of a local field, which enables one to compare a precisely defined meoretical quantity with experimental values. So far we have considered only the second moment for a case where all nuclei are identical. If more than one species is involved. we get a somewhat different answer. The basic difference is in the terms of type B in the dipolar coupling that connect states such as 1+ -) 10 J - +). If the two states are degenerate. as in the case when the spins are identical. B makes a first-order shift in the energy. On the other hand, when the states are nondegenerate, B merely produces second order energy shifts and gives rise to weak. otherwise forbidden transitions. It is therefore appropriate to omit B when the spins are unlike. 2 The interactions between like and unlike nuclear spins may be compared and me second moment readily obtained. If we use the symbol 1 for the species under observation. and S for the other species, the effective dipolar coupling between like nuclei is .. ,0
(rtJ)Il:::
I 2 2~(1-3coS26H)
,("h
L.
x,1
3 rk/
(3[zl.:lz1 ~ II.: ·1/)
(3.54)
In computing the second moment for like spins, the terms II.: • II do not can· tribute, since they commute with 1-'% [see (3.49)]. The coupling between unlike spins is
(3.51) 2 Van Vied: points out thlll omitting these terms for unlike spins lIS well as the terms C, V, E, and F for like sl,ins is crucial in computing (.1",,2). The !'i:lISOn is lhlll in computing (.6",,2), the ralher wellk satellite lines at "" 0 lind"" 2,
+V 1i01/;" = E n 1/;" (4.37) 2m Then we can look on the tenns in (4.36) that involve Ao llS pcrturbing the energies and wave functions. We shall compute perturbed wave functions so that we can compute the effect of the magnetic field on the current density. Of course Ao goes to zero when lIo vanishes, being given typically by 1io = -
AO = ~Ho x r
hqW'V" -
,,'V,,') - LAo"',,
,,(nl1ipcr tl°) .1,
./.' _ ./.
'fIO-'fIo+L.J
Eo
tl
.
Ell
'fin
Bu'
"0
(4.47)
(
'V"o
=
J(.)
('~nM+ (X~Y)'VJ(')
giving .() hq ( " 2 Jo r = 2m XJ - yt)! (,.) 2
=2mkxr!(r)
(4.48)
The current therefore flows in circles whose plane is perpendicular to the zaxis. If we define a velocity vCr) by the equation vCr) = io(r)
q"',,
(4.49)
where q1jJ*'I./J is the charge density, we find
(nIJipcrd O)
96
Ji
hq
Let us define
Eo
+ iY)
X
=
2mc
=
*
(4.46)
(4.40)
we need keep only those parts of the perturbation that are linear in Ho, or by referring to (4.36-38), we take q fipcrl = (po Ao + Ao' p) (4.41)
enO
*
hq
,0(r)=-2'("0'V,,0-,,0'V,,0) nu
(4.39)
2m. me we can compute io(r) correctly to tcnns linear inlIo as a first approximation. To do this, we need,p and 1/;* correct to terms linear in flo for that pan of io(I') in the parentheses, but for the last tenn, we can use for"" the unperturbed function 1fio. Since we always have
(4.45)
is the current that would now when lIo = O. When tile orbital angular momelllllfll is quenched .w that t/Jo is real, we see that J(r) = O. and the current density vanishes at all points ill the molecule in the absence of Ho. lt is the tenn J(r) that gives rise. however. to the magnetic fields at a nucleus originating in the bodily rotation of the molecule. that is, the so-called spin-rotation interactions that are obselVed in molecular beam experiments. It is instructive to compute io(I') for afree atom in the In = +1 j)-$tate. lIo being zero. io(r) then equals J(I'), so that
(4.38)
Although this involves a particular gauge, we can see that in any gnuge, Ao will be proportional to Ho. In the expression for the current io(r),
jo(r) =
m'
En
(4.42)
v(r) =
!:...
k x r
m (x 2 +y2)
(4.50)
97
which is tangent to a circle whose plane is perpendicular 1 1.(r)l- -h m +y2
(0
the z-axis, so that
./:£2
(4.51)
This gives a z angular momentum of (4.52)
in accordance with our semiclassical picture of the electron in an m "" + I state possessing one quantum of angular momentum. We see, therefore, the close relationship in this case of the current density, the "velocity", and our semiclassical picture of quantized orbits. When the states tPo and tPn may be taken as real (quenched orbital angular momentum), we have J(r) = 0, and
11 io(r) = 2 .q l)£nO -£;IO)(tPO'VtP" -ljJn'V.pO) Tnl
n
LAotP6 Inc
(4.53)
For (4.53) to be valid, it is actually necessary only that the ground state possess quenched orbital angular momentum, but for excited states, we have assumed that the real fonn of the wave functions has been chosen. Let us now proceed to look at some examples. We shall consider two cases, an ,,-state and a p-state. It will turn out that the chemical shirts for s-States are very small but that, for p-states, the effect of the magnetic field in unquenching the orbital angular momentum plays the dominant role, giving chemical shifts twO orders of magnitude larger than those typically found for ,,-states. To proceed, we must now choose a particular gauge for A o. It turns out, as we shall see, to be panicularly convenient to take
Ao
= !Ho
xr
= !Hok x
(4.54)
T
although an equally correct one would be Ao
= ,Ho x (r -
(4.55')
R)
where R is a constant vector, or
Ao z =0
,
Ao:>: = Hoy
,
Aoy =0
div Ao =0
--q-[Ao' p+ (,), Ao) + Ao .p] 2mc
98
h
,
~('J
·Ao) = 0
llperl
=-2~eHo·(r-R)XP
(4.59b)
(4.59c)
qf' = --HoL,(R)
2me where Lz(R) is the z-component of angular momentum about the point at R. The choice of gauge therefore specifies the point about which angular momentum is measured in the perturbation. It is, of course, most natural to choose R"" 0, corresponding 10 measurement of angular momentum about the nucleus, since in general the electronic wave functions are classified as linear combinations of .s, p, d (and so on) functions. When the electron orbit extends over several atoms, more than one force center enters the problem. The choice of the best gauge then becomes more complicated. A closely related problem in electron spin resonance involving the g-shift is discussed in Chap. 11. Let us now consider an .s-state. Then the wave function is spherically sym· metric: (4.60) ~.(r) = ~.(c) It is clear that, since
0
L,tP. = 0, (4.61)
.
Therefore tno is zero for all excited states, and the entire current io(r) comes from the last term of (4.44):
(4.56)
io(r) = _LAo!J;5 = --q-Hok x r!J;;(r) (4.62) me 2me The current therefore flows in circles centered on the z-axis. The direction is such as to produce a magnelic moment directed opposite to Ho so thm it produces a diamagnetic moment. We see that the current direction will also produce a field opposed to Ho at the nucleus (see Fig. 4.3). It is interesting to n()(e that there is a current flowing in the s-state. There must certainly be an associated angular momentum, yet we customarily think of
(4.57)
where (p' Ao) means p acts solely on Ao. But since p = Ul/i)'V. (p·Ao) =
qh -2-HoLz me Had we chosen the gauge of (4.55a), we would have
1{pert =
(4.55b)
Then we have, from (4.41),
,.-
--'-(Ho x r)'p 2me (4.59,) = --'-Ho' (r x p) 2me We recognize that r x p is the operator for angular momentum in the absence of Ro. It is convenient in computing matrix elements to use the dimensionless operator (l/i)r x 'V for angular momentum. Denoting this by the symbol L, we can write (4.59a) alternatively as 1{pert =
(nl1tpertl¢,) =
In tenns of the Ao, (4.54), we have
1{......rt =
Then, using (4.54), we have
(4.58)
2
2
99
I
FigA.3. Diamagnetic curnlllt flow in an ,,-state atom, and the magnelic fields produced by the current
y
/I,
y
-q
/I
-I/-k
/I
+q
.,
1
yf(r)
"j
zf(r) xf(r)
+q
-q
s-stales as having zero angular momentum. We are confronted with the paradox: If s-states have zero angular momentum, how can there be electronic angular momentum in a first-order perturbation treatment if the first-order pcourbalion uses the unperturbed wave function? The answer is that the angular momentum operator has changed from r x (Ii/i)\! in the absence of a field to r x [(llIi)\7 (q/c)A] when the field is present. By using the changed operator, the unchanged ,s-state has acquired angular momentum. The angular momentum is imparted to the electron by the electric field associated with turning on the magnetic field, since this electric field produces a torque about the nucleus. There is a corresponding back reaction on the magnet. We note that since A is continuously variable, we can make the angular momentum continuously variable. By using typical numbers for H o and r, one finds the angular momentum much smaller than h. Does this fact violate the idea that angular momentum changes occur in units of II? No, it does not, since the electron is not free but rather is coupled to the magnet. The complete system of magnet plus electron can only change angular momentum by h, but the division of angular momentum between the parts of a coupled system does not have to be in integral units of Ii. We turn now to a ]>-state Xf(I') acted on by the crystalline field such as that discussed in Section 4.3. We duplicate the figures for the reader's convenience (F;g.4.4). The energy levels are then as in Fig.4.5. Let us consider H o to lie along the z-direction. In contrast to the s-state, the p-state has nonvanishing matrix elements to the excited states, corresponding to the tendency of the static field to unquench the angular momentum. For this orientation of Ho, the matrix element to Zf(l-) vanishes. That to yf(r) is (nl1i pert 10) =
J (x'!!'-oy H0 "J ' dr
--'-Ho~ 2mc 1
= +-2 q me
7
,
yf(I')
y..£..-)Xf(1")dT
ox
[yf(')1
(4.63)
iqflHo =---2me where we have used the fact that the function yf(r) is normalized. By using (4.63), we find GnO=
100
(nl1i per dO)
Eo
E"
. qhHo I
=1---
2mc Ll
(4.64)
Fig.4.4. Crystalline field due to charges at:l: =±a, .'1= z =0; -q at y = ± a, :I: = Z = 0
+q
The tenn 1/;0 V't/J"
t/Jo \71/;n
-
- t/Jn \71/;0
t/J1l V't/Jo =
. = ell' is the potential due to the nucleus. As is well known, PI is much smaller than P2 in the nonrelativistic region. One customarily calls P2 the "large component". For s~states in hydrogen, P2 is also much larger than PI, even at the nucleus. One can eliminate !PI (still with no approximations) to obtain a Hamiltonian for rJi2, 'H' such that (4.118) By tedious manipulation one finds that
?-i'= EI
1
I/>
2
+e + me
2(e 2p 2+ e2A 2+2ecA_p_iecdivA
+ eflcu ° '\1 x A)+,
enc
'12
1 2encuo'\1xA E +e'f'+ me "2 1
and
(4.120a)
2
(E/+e« 2mc 2, (4.120a) is exactly the same as (4.98) and goes as 1/1'3. If, however, T is so small that el/>_2mc2 , the answer is modified. By writing el/> = e'lh', multiplying (4.120a) by T on top and bOllom, using e 2 /mc2 = TO (the classical electron radius), and neglecting E', we get for (4.120a):
~)~Uo\7XA ( 21'+1'0 2mc
1.3
r)]
(4.12Ib)
The tenn in the square brackets goes as Ih.'l. For 1':»1'0, (4.12Ib) can be shown to be of order 1'0/1' times (4.120b); therefore, much smaller. However, when T :5 1'0, the radial dependence becomes less strong, going over to the hannless 111'0 near I' = O. The tenn is t/lerefore well behaved. It also has the feature that it does not average [Q zero over angle of s-states. It gives the answer of (4.108) for the magnetic interaction energy. We see that these two tenns are very similar to taking a S-function for sstates and asserting that a finite size of the electron prevents the radial catastrophe of the conventional dipolar coupling of (4.98). For computational convenience we may consider that the dipolar interaction of (4.98) should be multiplied by the function 21'/(21' + 1'0), to provide convergence. We shall now turn to the study of some of the important manifestations of the coupling between nuclei and electron spins, considering first the effects that are first order in the interaction and then effects that arise in second order. Further discussion of first-order effects will be found in Chap. lIon electron spin resonance.
lbe Knight shift is named after Professor Walter Knight, who first observed the phenomenon. What he found was that the resonance frequency of Cu 63 in metallic copper occurred at a frequency 0.23 percent higher than in diamagnetic CuCI, provided both resonances were perfonned at the same value of static field. Since this fractional shift is an order of magnitude larger than the chemical shifts among different diamagnetic compounds, it is reasonable to attribute it to an effect in the metal. Further studies revealed that the phenomenon was common to all metals, the principal experimental facts being four in number. By writing W m for the resonance frequency in the metal, Wd for the resonance frequency in a diamagnetic reference, all at a single value of static field, there is a frequency displacement .dw defined by Will
=Wd +.dw
(4.122)
The four facts are 1.
(4. 120b)
Now we no longer have an infinity from the radial integral in computing H:, and it is clear that the angular average makes (4.120b) to be zero. 112
x ("" x
(4.119)
where E is the electric field due to the nucleus. For our present needs, we focus on two tenns only:
'
1'3
4.7 Knight Shift'
(E +el/>+2me)
x (ieEoA +ieE·p - eu' E x p- eu·E x A)
u. [..':.
.dw is positive (exceptional cases have been found, but we ignore them for the moment).
3 See references to UNuclear Magnelic Resonance in Melals" in the Bil>liography
113
2.
If one varies Wd by choosing different values of static field, the fraclional shift .tJ.W/Wd is unaffected. The fractional shift is very nearly independent of temperature. The fraclional shift increases in geneml with increasing nuclear charge Z.
3. 4.
The fact that metals have a weak spin paramagnetism suggests that the shift may simply represent Ihe pulling of the magnetic flux lines into the piece of metal. However, the susceptibilities are too small (10- 6 cgs units/unit volume) to account for an effect of this size. As we shall see, however, the ordinary computation of internal fields in a solid that involves a spalial average of the local field is not what is wanted, since the nuclear moment occupies a very special place in the lattice - in fact a place at which the electron spends, so to speak, a large amount of time in response to the deep, attractive potential of the nuclear charge. As we shall see, the correct explanation of the Knight shift involves considering the field the nucleus experiences as a result of the interaction with conduction electrons through the s-state hyperfine coupling. If we think of the electrons in a metal as jumping rapidly from a10rn to atom, we see that a given nucleus experiences a mI '"
(5.38)
and, assuming E ks '" E", + E s • we get for W mn
We must now express W mb nl/6' explicitly. To do so, we must specify the interaction V. For metals with a substantial s-character to the wave function at the Fenni surface, the dominant contribution to V comes from the s-state coupling between the nuclear and electron spins:
8. , V'" "3,e,nh 1·50(1')
Imks) '" Im)ls)uk(r)e ik • T
(5.34)
It is a simple matter to compute the matrix element of (5.29):
(mkslVlnk's') '" 81r Iclnh2(mII[n). (sISls')uk(O)uk'(O)
(5.35)
3
which gives us
2:
(mII.ln)(nllo,lm)
(5.36) x o(E", + Eb - En - E k '6') We can substitute this expression into (5.32) CO compute W mn . We are once again faced with a summation over k and k ' of a slowly varying function. As before, we replace the summation by an integral, using the density of stales g(Ek , A) introduced in Sect. 4.7. This gives us
r; -9-'C Inti
'"'
L.
0I,OI'6,S'
(mIIOI[n)(nllo,[m)(sISOI[s')(s'[So'ls)
01,01'6,6'
x
JI' since we have already included the effect in (5.23). Moreover, since both u(Ek ,) and ([uk,(0)1 2 ) Ell are slowly varying functions of E k" we may set them equal to their values when Ek '" Ek" In fact we shall evaluate U(Ek ,), and so on at E ks ' This gives us for the integral in (5.39):
J
(5.83)
which are eigenjimcrions of Ihe Hamiltonian 'H, we
uke-(i/')'H'{](O)e(i/l)'ltfumdr (e(i/l)'HI Uk ) •{](O)eWA)'HIUmdr
(5.84)
By utilizing the fact that HUm:: EmU"" and using the power series ex.pansion of the exponential operator, we get
(5.19)
(5.85)
n
This equation is the well-known starting point for time-dependent perturbation theory. We can use (5.79) to find a differential equation for the matrix elements of the operator P, since
."
= -,i
E [ckc;.(nIHlm) -
(5.80)
(kl'Hln)cnc:"l
for the time-dependent matrix element in lerms of the matrix element of t! at t :: O.
So far we have talked about the density matrix without ever exhibiting explicitly an operator for e. For the sake of concreteness, we shall do so now. We shall take an example of a spin system in thermal equilibrium at a temperalUre T. We shall take as our basis states, Uri, the eigenstates of lhe Hamiltonian of the problem, 'Ho. The populations of the eigenstates are then given by the Boltzmann factors, giving for the diagonal elements of (!:
e- E ... / kT
CmC~ :: =--'0--
Z
i
(5.86)
where, as usual,
= h(kl P1t -1tPlm)
Z = 'Le-E./kT
n
where we have used (5.70) for the last step. We can write (5.80) in operator form
as
dP
ill =
i
hlP, 1t1
If we wrile
(5.81)
This equation looks very similar to that of (2.31) for the time derivative of an observable, except for the sign change. If we perfonn an ensemble average of the various steps of (5.80), assuming 'H to be identical for all members of the ensemble, we find a differential equation for the density matrix. (J. Since the averaging simply replaces P by {!, the equation for (J is (5.82)
160
Cn ::
lcllieia-n
we have that CmC~:: ICmIICnlci(Q'... an)
(5.87)
It is customary in statistical mechanics to assume that the phases all arc statistically independent of the amplitudes Ie" I and that, moreover, 0'", or On have all values with equal probability. This hypothesis, called the "hypothesis of random phases", causes all the off-diagonal elements of (5.87) to vanish. If, for example, we were to compute the average magnetization perpendicular to the static field 161
for a group of noninteracting spins, as we did in (2.88), the vanishing of the off-diagonal elements of !! would make the transverse components of magnetization vanish, as they must, of course, for a system to be in thennal equilibrium. More generally, we see from (5.85) Ihal the off-diagon..l elements of fI oscillate harmonically in time. If they do not vanish, we expect that there will be some observable property of the system which will oscillate in time according to (5.75). BUI we should then not have a true thennal equilibrium, since for thennal equilibrium we mean that all properties are independent of time. Therefore we must assume that all the off-diagonal elements vanish. Note. however, from (5.85) (which applies to the situation in which the basis functions are eigenfunctions of the Hamiltonian) that if the off-diagonal elements vanish at anyone lime, they vanish for all time. We have, therefore, (nlelm) = (omll/ Z )e-En /k'l"
(5.88)
It is worth noting that the operator for f! is on a different fOOling from most other operators such as that for momentum. In the absence of a magnetic field, the latter is always li.\J/i. For a given representation the density matrix may, however, be specified quite arbitrarily, subject only to the conditions that it be Hennitian, thai its diagonal elements be greater than or equal to zero. and lhat they sum to unity. There is therefore no operator known a priori. However, in certain instances the matrix elements (nJelm) can be obtained very simply from a specific operator for fl. When this is possible, we can use operator methods to calculate properties of the system. We now ask what operator will give the matrix elements of (5.88) bearing in mind that the u,,'s, and so 011, are eigenfunctions of 'HO. Using the fact that
e
-"Ho/kT
um=e
-Em/kT
Urn
(5.89)
(which can be proved from the expansion of the exponentials), we can see readily that the explicit fonn of e is
e = ~e-"Ho/kT
(5.90)
Z
We can use this expression now to compute the average value of any physical property. Thus suppose we have an ensemble of single spins with spin I, acted on by a static external field. Then ?to is the Hamiltonian of a single spin:
In the high-temperature approximation we can expand the exponenlial, keeping only the first tenns. By utilizing the fact that Tr {M z } = 0, we have
(M,)
=
~ T'{M,(,- ~+)}
~ ~ Tr{ (~~h::oI;) } Now, in the high-lemperanlre limit, Z = 2I + I. Since Tr {4} = we get
(M z ) =
~~fI2 I(l + 1) H 3kT
(5.93)
j I(I + 1)(21 + 1), (5.94)
0
which we recognize as Curie's law for the magnetization. The density matrix gives, therefore, a convenient and compaCI way of computing thennal equilibrium properties of a syslem. One situation commonly encountered is that of a Hamiltonian consisling of a large time-independent interaction 'Ho, and a much smaller but time-dependent tem H,(t). The equation of motion of the density matrix is then
de i dt = !i[e, 'Ho + Hd
(5.95)
If HI were zero, the SolUlion of (5.95) would be e(t) = e -(i/" )1io/ e(O)e(i/h)1i oi
(5.96)
Let us then define a quantity e* (the siar does not mean complex conjugate) by the equation e(t) = e -(i/")1iot e* (t)e(ijh )"Ho/.
(5.97)
If, in fact, HI were zero, comparison of (5.96) and (5.97) would show that e* would be a constant. (Note, moreover, Ihat at t = 0, e* and e are identical.) For small?tl, then, we should expect e* to change slowly in time. Substituting (5.97) into the left side of (5.95) gives us the differential equation obeyed by e*:
-~[?to, el + e- O/")1iotd: * e(ifh)1iot = ~(e, Ho + ?til t
(5.98)
We note thai the commutator of e with ?to can now be removed from both sides. Then, multiplying from the left by exp (il1l)'Hot) and from the right by exp (-(i/h)Hot), and defining
(5.91) We shall illustrate th~of the density matrix 10 compute the average value of the z-magnetization (1I1 z ). It is
(M z ) =Tr{Mze} =
162
~ Tr{M ze-"Ho/k1'}
(5.92)
Hi = ei1iot/h?tle-(i/h)"Ho/
(5.99)
we get, from (5.98),
de* i * * dt = hie, 1£,(')]
(5.100)
163
Equation (5.100) shows us, as we have already remarked, thai Ihe operator r/ would be constant in time if the perturbation ?i1 were sel equal to zero. The transformation of the operalor?il given by (5.99) is a canonicallTansformation, and Ihe new representation is termed the interaction representation. The relationship of U and (J. is illustrated by considering the expansion of the wave function tP in a form
tP = L:ane-(i/l)E.l un
We can make a closer approximation by an iteralion procedure, using (5.109) to get a bener value of U·(t l ), to put into the integrand of (5.108). Thus we find
·e(l) = ,'(0) +
,'(0) +
.
(5.101)
= ,'(0) +
o
(e(O), "H;(,")Jd," }. "H;(")] d"
0
(,(0). "H;(")]d"
o
n
instead of
2 t tl
+
(5.102)
tP=L:Cnu n
(nll?·lm) = e{i/Ii)(En -E".)t(nll?lm)
((,'(0). "H;(''')J. "Hi(")]d"
d'"
(5.110)
(5.103)
We could continue this iteration procedure. Since each iteration adds a term one power higher in the perturbation 'Hi, Ihe successive iterations are seen to consist of higher and higher perturbalion expansions in the interaction ?i I, For our purposes, we shall not go higher than the second. Actually, we shall find it most convenient to calculate the derivative of (J*. Taking Ihe derivative of (5.110) gives
Since (5.101) and (5.102) give the same tP, we must have
d,;,(1) = *["(0). "H;(I)] +
(5.104) (5.105) Comparison with (5.103) shows that ana~ :::::
(D JJ o 0
n
where the un's and En's are the eigenfunctions and eigenvalues of the Hamiltonian ?io. In the absence of ?iI, Ihe an's would then be constant in time. We shall show Ihat Ihe matrix ana~ is simply (nlu·lm). We note first that replacing I?(O) by U· in (5.83-85) gives that
(nil 1m)
Q.E.D.
(5.106)
There is likewise a simple relalionship between (nl1ii 1m) and (nl1i 1 lm). By an argumem quite identical to that of (5.83-85) we have (nj1iiJm) =
Ju~e(i/l)1tot1ite-(i/l)1totumdT
::::: e(ifA)(E"-E,,,)t(nl1i 1Im)
,'(I) = ,'(0)
+
(5.107)
u"
(5.100). By inte-
,
*J .
[,(I'), "H;(I')]dl'
(5.108)
o
This has not as yet produced a solution, since e·(t') in the integral is unknown. We can make an approximate solmion by replacing e"(t l ) by e"(O), its value at t = O. This gives us
.
'(') = ,'(0) +
,
i J(,'(0). "H;(,')Jdl' o
US J
((,'(0). "Hi(I')]. "Hj(l)]dl'
. (5.111)
o
It is important to note that (5.111) is entirely equivalent 10 ordinary timedependent perturbation theory carried to second order. However, instead of solving for the behaviors of a" and am, we are solving for the behavior of the products ana:"n which are more directly useful for calculating expectation val-
"es.
5.5 The Rotating Coordinate Transformation
.
Now we proceed to solve the equation of motion for grating from t = 0, we have
164
*J[{, *J" , *J
(5.109)
We saw in Chap. 2 that it is often convenient 10 go 10 the rotating frame when a system is acted on by alternating magnetic fields. We explored both a classical treatment and a quanlum mechanical treatment. In the laller we ITansformed the Schrtldinger equation. We now examine how to transform Ihe differential equation for the density matrix.. Let us consider first the case of an isolated spin acted on only by the static field Ho and a rotating field HI(t) given by (5.112) where we have already recognized the sense of rotation for a positive "'( in the negative sign in front of j. In the laboratory reference frame de i - = -(,"H - "H,)
d'
(5.113)
h
165
11. = -")'h[HoIz + HI (Iz cos wt - I y sin wt>]
(5.114)
Utilizing (2.55) we express this as
11. = -")'h[HoIz + Hle+ iwU , Ize- iwU ,]
where 17k expresses the chemical or Knight shifts. We omit a chemical shift correction from H I since HI oll. e;(", -ylJi>;) and My corresponding to magnetization along the x·axis (the HI direction) and the y-axis in the rotating frame given by
(M+(2T» = N
.
The physical significance of this result is that at infinite temperature, where only the a term remains, the thermal equilibrium magnetization vanishes. Therefore, no pulse sequence can produce a signal. We will encounter a similar situation in all pulse sequences stnning from thermal equilibrium. The b term gives
In the high temperature approximation, Z '" 21 + I, giving
(M.(t) = (My(t» =
,
"
In the trace, the term labeled a is readily shown to be zero:
(M+(2T» = (5.156)
Z
(5.160b)
'--v-'
(M+(2T» = (;I + I) k;Tr{r+ X(-./2lI.X- (-./2))
(;lR(O-) '" e- iloltl • e(O-)eilo'll.
(;lR(O-) '"
where
(5.160 O. we have a time-independent Hamiltonian. We will treat it in the same spirit as the Hamiltonian of (5.126) in which we recognize that the spin-spin coupling lenns are perturbations, SO (hat it is appropriate to drop the teons in the spin-spin coupling which do not commute with the Zeeman coupling. We thus have as our Hamiltonian 'H.
=
-ihHol~ + L'Yl~Hou~JJ~1e + ~ L,'H.11c -
'YhlzHroo(t)
, (5.172)
i,k
Ie
where 'H1k is the ponion of 'H.jk which commutes with the mean Zeeman energy
II.. 'H1j l = 0
(5.173)
It is convenient, as in (5.126), to define three quantities
'Hz == 'H.p ==
'YhHoI~
L'YhHou~kI~k +~ "L'HJk
(5.174)
i,k
Ie
H6 = -ilil",HrOO(t) Note from (5.174) that 'H.z and 'Hp commute
['HI'. 'Hz] = 0
.
(5.175)
'Hz is the Zeeman energy, apart from chemical and Knight shift differences. Hp is a much smaller tenn ("P" for perturbation) which gives rise to splittings and 175
line widths. For t > 0, since we have then 'H6 = 0, we have a lime-independenl Hamiltonian 'Hz + 'Hr. Then, using (5.95), for t > 0,
e(t) = .. p (-i(Hz + Hp)t/IiJe(O+)exp(i(Hz + Hp)t/Ii)
.
therefore
(5.185)
(5.176)
To find e(O+)we must solve for the time development under the action of the C-function. Since we seek the linear response, it is appropriate to treat 'H6 as a perturbation, keeping only the fim-order lenn. We therefore turn to (5.11 I), using 'H6 for 'HI> keeping only the first lenn on the right. In this equation
e'(.) = exp (-i(Hz + Hp)t/Ii)e(t)exp (;(Hz + Hp)t/Ii) 'H (t) = exp( -i('Hz + 'Hp)t/h)'H6(t)exP (i('Hz + 'Hp)tlh)
(5.177)
de'(t) = .'.[e'(O-)H;(t) - H;(l)e'(O-)J
(5.178)
6
Then
dt
(5.186) The first tenn, linear in I z , corresponds to a thennal equilibrium magnetization. Therefore, when put in (5.176), it will not lead to any transverse magnetization. To get mn(t), therefore, we keep only the tenn involving I y , utilizing (5.170) (H Tl)mxx(t) = (Mx(t»
h
=,/oT,{Ixe('»
Integrating from t = 0- to t = 0+, we gel
,/oHo
= ,liT, { IxT(1) Z(oo)kT,(Hro)IyT
0+
e'(O+) - e'(O-)
=.'./0 J
[T(I)e(0-)H,(t)T- 1(.)
0'
0-
(5.179)
- T(t)H,(t)e(0-)T-1(t)]dt with
(5.180.)
T := exp (-i('Hz + 1ip)tlh)
and (for future use) Tp := exp (-i'Hpt/li)
.
(5. I 80b)
But over the zero time interval we can neglect the time dependence of everything except the c-function giving
.(0+) = e'(O-) - *,/oHTo[e(O-)Ix - Ixe(O-)J
(5.181)
and indeed
(5.182) Now e(O-) corresponds to Ihennal equilibrium, hence is given by I
1
e(0-) = Z ..p[-(Hz+Hp)/kT] '" Zexp(-1iz/kT)
(5.183)
In the high temperature approximation this gives
(I
+ ,/oHoI,) kT
(5.184)
Following the argument after (5.158), we drop the first tenn in the parenthesis since it contributes zero magnetization, getting 176
(I)
.
}
(5.187) (5.188)
A useful variant on (5.188) can be obtained utilizing the fact that 'Hz and 'Hp commute, and expressing Tz as exp (iwotlz), with wo = "tHo. Then
Tr{IzTIIIT- 1 }
Tz := exp (-i'Hztlh)
e(0-) = _1_ Z(oo)
"'I 2h2 Ho -1 mn(t) = Z(oo)kT "'I Tr {IzT(t)IyT (t)}
-I
= Tr {Til IzTZTpIyTpl} = Tr {e- lwotl• IzeifNOLf'TpIyTpl} = Tr {IzTpJyTp I} cos wot + Tr {Jy T p l yT p t} sin wot
This equation is closely related
10
(5.189)
(B.18) in Appendix B. At t = 0+, Tp =
Tj;l = 1 so that the coefficient of the cos wot tenn vanishes (Tr {IzIy } = 0), and the coefficient of the sin wot term is Tr {J;}, which is nonzero. For times shon compared to h/11ipl, where by l1-lpl we refer to the magnitude of a typical nonzero matri", element of 'Hr, we can slill appro"'imate Tp as 1. Consequently, the magnetization nlzz(t) will start as sin wot, a signal oscillating at the Lannor frequency. Since 'Hp gives rise to spectral width, over times of the order of the inverse of the spectral width, Tp will differ significantly from I, and we may e"'pect both traces to contribute. In Appendix B, Eq. (B.18), we show that if 'Hp contains only spin-spin terms and 7 is chosen such that the Uk'S are all zero, the first trace always vanishes. The effect of the operators Tp is to modulate the coefficients of the oscillations at Wo over times of the order of the inverse of the spectral frequency width, L1w. 177
Thus, for times short compared to the inverse of the spectral frequency width, ~, we may write mzz(t) = 7
721i2 Tr {12} Z(oo)kT Ho sin wot
with
t
- Ho, on for a time 7"0 < I/wo, we see that the field tilts the thermal equilibrium magnetization xoHo to have a component in the positive laboratory y-direction. Since we are seeking a response (Le. an x-y magnetization plane) linear in the driving tenn, H, we keep the angle of rotation, L18, small
mu(O = xowo sin wot
(5.195)
While (5.195) looks relatively simple, involving just a few traces, its actual evaluation may be enonnously difficult since the equation contains all the content of the theory of line shapes! Nevenheless, the equation gives a compact statement for attack by the various fonnal methods which have been developed to deal with the line-shape problem. In principle, this mu(t) will yield both the absorption and dispersion spectra utilizing (2.156)
x~z(w) =
z
X{w~}
Tr l y
+sin (woOTr {IyTplyTj;I}]
(5.190)
~
mu(t)=xowo sinwot
mzz(t) =
5.8 The Response to a ,,/2 Pulse: Fourier Transform NMR We tum now to the topic of Fourier transfonn NMR, a method of doing magnetic resonance which is today all pervasive in the field of high resolution liquid spectra, and which fonns the basis for one of the most important other developments in NMR, so-called two-dimensional Fourier transfonn NMR (which we take up in Chap. 7). In 1957, Lowe and Norberg [5.6] discovered an imponant theoretical result which was confinned experimentally by them in conjunction with Bruce [5.7]. Lowe and Norberg showed theoretically that in solids with dipolar broadening, the Fourier transfonn of the free induction deCa'yfoliowing a -rr/2 pulsegave the shape of the absorption line. They verified their result experimentally with a single crystal of CaF2 by comparing their experimenlal free induction decays with the Fourier transfonn of Bruce's steady-state absorption data taken on the same crystal. In 1966, Ernst and Amfer.fon [5.8] pointed out that there were substantial experimental advantages to the use of pulses over steady-state meth<X1s when dealing with complex spectra. (We discuss advllntagcs later in this section.) Their method, which consists of first recording the free-induction signal, then laking ils Fourier transfonn, is referred to as Fourier transfonn NMR. The method allows on~ to obtain both the absorption spectrum and the dispe-;;ion spectrum'"":", ~how ttlatthe cosine transfonn of the free induction decay gives one whereas the sine transfonn gives the other. (See also the book by Ernst et al. [5.9].)
TheY
179
[ These remarks are reminiscenl of linear response theory discussed in Sect. 2.11 where we found thut X' and i l are given by cosine and sine transforms of the function m(T), (2.156):
J
~
x' =
~
m(T) cos (wi)dT
X" =
o
J
m(T) sin (wT)dT
(5.196)
~n
_1_(1
(0-)= (0-)= U Z(oo)
1C = 'Hz +'Hp +1-{1
with
(5.197)
(5.198)
(5.199) (5.200) expressing the fact that the spin is acted on by a rotating magnetic field which is to generate a 7f(l pulse. During the pulse, we go to the rotating frame where UJl is described by (5.131):
i i ,
= "h(un'H eff - 'HeffUIl)
(5.201)
with 'H~rr given by (5.132) and UIl by (5.116) and (5.119). While the H t is on, we take it to be so large that we can neglect everything else in 'HI giving
'H' = -"(fi.HtI~
.
hH 1 o ,) kT
(5.204)
1
-I)
un(t,) = Z(oo) (lhHO 1+ kT X(wltt>I~X (Wltl)
(5.205)
where X(D)
== e+ i8Jz
(5.206)
Now, ulilizing (2.55),
X(8)I z X-'(8) = I y sin 8+ I~ cos 8
so that UR(tl) =
Z(~) (I + 1::0 [Iv sin Witt + I z cos Wtttl)
(5.207)
(5.208)
The first term on the right will nOt contribute 10 the magnetization, following the argument of (5.160) and (5.161) (i.e. it is allihal remains as T-+oo). Moreover, the term involving I z is, apart from a constant of proponionality, the same as the temperature-dependent contribution to a (! R corresponding to thermal equilibrium. Thus, it contributes nolhing to transverse magnetization. (The validity of neglecting the tenns involving 1 and I z can be verified by direct calculation of (Mz(t)) and (M!I(t)).] We are left, then, with I 1liHo. UR(tl) = Z(oo) Sill (wltdIv
---;;r
(5.209)
Therefore, utilizing (5.120),
e(tl) = RUR(tI)R- I I l'IHo. -I = Z(oo) kT ,m (w",)[R(t,)I,R ('1))
(5.210)
where R is given by (5.200): ROI) = ei""lt J•
(5.211)
To gel (!(t) for t> tl' we introduce the operators T, Tz, and Tp defined as in (5.180),
(5.202)
If the HI is turned on suddenly at t = 0 and turned off at t = tit we have, then [os in (5.147)1. UR(tI) = e+iw]It r. un(O+)e-iwtltJ" = e+iW ]lt 1"un(O-)e-i""t 1tl"
+7
whence, as in (5.150a) and (5.151)
•
where m(i) is the response of Ihe syslem 10 a S-function which we treated in Sect. 5.6. We therefore demonstrale that in the linear response regime al high temperatures, a 1f(l pulse (in faCI, any pulse) is equivalent to a S-funclion excitation, hence Ihat the transforms of the free induction signal give ;( and X". Let us consider, then, Ihe Hamiltonian of (5.124) describing a group of N coupled spins, allowing there 10 be both chemical and Knight shifts. We then have a Hamiltonian 'HI given by keeping only Ihat part of the spin-spin coupling which commutes with the Zeeman coupling as in (5.172)
dUll dt
I
with WI = "(HI. Taking UR(O-) to represenl thermal equilibrium, we have, utilizing (5.155-157) generalized for the Nobody Hamiltonian as in (5.184),
(5.203)
Tz(t) = exp (-i'H.ztl1l) = exp (iwotIz) Tp(t) = exp (-i'H.pt/h)
T(t) = exp (-i(1ip + 1iz)tlh) = TzTp '" TpTz
(5.212)
(Do not confuse the operator T(t) with the temperature T!) 18.
181
Thus, since 1tz + 7-ip arc independent of time, we get from (5.83)
Utilizing (5.196) and (5.219) with Hro = I, we have then
=
e(l) = T(t - tt)e(tl)T-I(t - til =
-yIiHo
Z(oo)kT
sin (Wltl)T(t _ tl)R(t\)lyR-I(tl)T-IU - t\)
T(t - tl)R(tl)
= exp (iwotI:) exp (-i1tpt/fi) exp (iHptdh)exp [i(w - WO)t1 1:] (5.214)
If, now, we neglect the effect of Hp and (w-wo)t I during t\ we can approximate Trl(tl)ei(""-~)tl/. ~ I
(5.223b)
o
= exp [iwo(t - t 1)1:] exp (-i7-ip(t - t I )/1i.) exp (iwt I [:)
p
.'0 J(Mz(r»o sin (wr)dr sm
=
Xl/(w) =
= T(t)T I(tl) exp (j(w - WO)tl 1:)
(5.223.)
Sin 0 0
..
(5.213)
Now
(5.215)
getting
Equation (5.223b) includes the result of Lowe and Norberg (they did not include chemical or Knight shifts). Equation (5.223) includes the result of Ernst and Anderson that the cosine and sine transfonns yield respectively the dispersion and the absorption. [It is useful to keep in mind that the tenns "cosine" and "sine" imply a particular phase convcntion. Ours is defined by (5.221).] The validity of (5.218) rests on the approximation that during the pulse we can neglect the effect of Hp and (w - woH I. Since HI' leads to a spectral width, Llw, this approximation is equivalent to
(5.216) Therefore, the expectation value of the transverse magnetization (Mz ), following (5.158), is
(M.(t)
=
(5.217)
This result is identical to the result of (5.187) except that sin WI tl replaces -yHro. We introduce the notation
(Mz(t»o = -y
2,.2H
t 0 sin (O)Tr{IzTlyT-I} Z(oo)kT
(5.218)
the magnetization following a pulse with wltl
(5.219)
== 0
Then we can write sinO
(Mz(t))e = 7n zz (t)-H ,
TO
(5.220)
Now, (Mz(t»e is just the signal we would observe from a pulse in which -yHltl = 0, produced by a rotating field which at t = 0 lay along Ihe x-axis
according to the convention of (5.114). This is also the rotating field generated by a linearly polarized field along the x-axis given by Hz(t) = Hzo cos wt
(5.221)
where, following (2.83),
Hzo 182
= 2H I
"YHI »Llw
and
(w - WO)tl «I
and
"'fHI» Iw - wol
or
(5.224.) (5.224b)
The classical derivation of the b-function response given in (5.192-194) involved assuming that a magnetic field H satisfy
,1m {g(t)I.}
2h2H O Sin . (Wltl )Tr (I z TIy T- 1) = -y Z(oo)kT
j
(Mz(r»o cos (wr)dr
i(w) = ."1
(5.222)
H»Ho
(5.225)
where the field H lasts for a time ro. This condition signifies that the magnetization is rotated during a time less than a Lannor period in H o. This appears to be a much stricter condition than (5.224b). The reason one can use an HI «Ho is that HI is rolating at a frequency w which is nearly the same frequency as the free precession frequency WOo Thus, while HI is on, the total phase angle about the Ho axis which the spin advances is wtl. If a l5-function had kicked the spin at t = 0, the subsequent precession angle in the same time tl would have been wotl' But since I(wo - w)tll and Llwtl «I we can take wOtl = wtl' Thus, the HI builds up the rransverse magnetization over the time it> but does not in the process inrroduce a phase error relative to a "zero time", or 6-function, creation of transverse magnetization followed by free precession for t I. We note also that this argument explains why there is nothing magical about exciting the free induction decay wilh a 0 = 7T/2 pulse. Equation (5.221) shows that any angle of 0 will do. [The fonnula (5.221) appears to blow up for sin 0 = 0, which simply reflects the vanishing of (Mz(t»o for that case]. There are several very imponant advantages to the Fourier [Tansfonn approach. The first is that since one is recording the NMR signal when the HI is zero, one eliminates noise from the oscillator which drives the HI. For steadystate apparatus, the oscillator noise is reduced by bridge balance, but typically such an approach reduces noise which arises from amplitude modulations of HI but does not reduce noise which arises from frequency modulation. 183
If one is dealing with a complex spectrum which has many absorption lines well separated from one another, as in many high resolution spectra of liquids, there is an additional advantage to the Fourier transfonn method. A steady-state search necessarily spends much of its time looking between the resonance lines of such a spectrum. The search procedure is thus inefficient. By pulse excitation, all the spins are induced to broadcast their presence at their characteristic frequencies, thus one immediately gains in dam collection by eliminating the dead time of a steady-state search. Of course, one must use a broader band system, since the steady-state system need only have the band width necessary to examine a single line whereas the pulse system needs a band width large enough to include the whole spectrum. However, since the noise voltage goes as the square root of the band width, one still is better off with the pulsed approach. An important point is that the Fourier transfonn method is the basis for the enonnously powerful two-dimensional Fourier transfonn techniques which have revolutionized NMR. II is useful at this point to make several remarks about instrumentalion. The signal M:c(r) consists in general of a sum of oscillating signals (corresponding to each resonance frequency which has been excited) which decay as a result of line broadening and relaxation phenomena. In general, all of the spread in oscillations is small compared [Q the Larmor frequency. To record these frequencies takes very high frequency response. It is customary to use a technique called "mixing", which in essence translates the fTequency of oscillation while preserving phase relationships. We will not go into the details of how mixers work, but essentially they work as shown in Fig. 5.6. The mixer has three pairs of terminals, one at which the signal is applied, one to which a reference voltage is applied, and one which is the output, the "mixed" signal.
Mixer Signal voltage input o---+~
Mixer signal output
If Vsig(t) = Vs
cos (wst
+,p)
= Vs(cos ¢ cos wst - sin r/J sin wst)
(5.226,) (5.226b)
Vrcrlj(t')], :>lj(')]dt'
d'
(5.308)
We then use (5.303) to express the time derivative (dldt){laQ'" In fact, since (5.309) we have
+ ei(a-a')! d{l(.ta' dt
(5.313)
We compute the 0'0'1 matrix element. There will be contributions from bOlh tenns on the right. We consider first the contribution of [{I·(O),1ij(t)]:
(alle'(O), :>li(t)lIa') = Dale'(O)IP)(PI:>lj(tl!a') p
- (al:>lj(t)IPJ(Plg'(O)la')
(dg·"')(a'IMz1a)
d{l~a' = i[a _ 0"].
these equations are fewer in number than are the equations of the elements of the density matrix, their solution may be considerably simpler than the original set. This trick will work when the relaxation mechanism and operators are such thai the expectation values of the operators pick out only a small number of the possible nonnal modes. We shall illustrate this use of (5.312) shortly. First, however, we tum to a description of the derivation of Redfield's fundamental equation. Our starring point is the basic equation for the time derivative of fl·, (5.111):
(5.314)
We shall now introduce the idea of an ensemble of ensembles whose density matrix coincides at t = 0 but whose penurbations 1i1(t) are different. (We are therefore flot allowing an applied alternating field to be present. That is, we are describing relaxation in the absence of an alternating field. One could add the effect of the alternating field readily. We discuss it in the next section.) We shall assume that the ensemble average of 1i 1(t) vanishes. This amounts to our assuming that1iI(t) does not produce an average frequency shift.~ Let us discuss this point. In general we expect
(5.310)
(5.315)
This relation enables us to transfoml (5.303). By substituting into (5.303) and utilizing (5.309), we find
where J(q is a function of the spin coordinates, and Hq(t) is independent of spin. For example, we saw that hi (t) had this fonn if it consisted of the coupling of a fluctuating magnetic field with the X-, y-, or z-components of spin. In that case q took on three values corresponding to the three components x, y, and z. If hI (t) represented a dipole-dipole interaction of two spins. there would be six values of q corresponding to the tenns, A, B, ... , F into which we broke the dipolar coupling in Chapter 3. Since we are dealing with stationary penurbations, the ensemble average of 1ij(t) is equivalent to a time average. In general we assume that the time average of Hq(t) vanishes, causing 1ii(t) to have a vanishing ensemble average. As a consequence we shall set
dt
d{lQ'a'
d'
fl aa'
= i(a' - a){laa'
+ L RaQ"
,{3{3' {I{3(J'
P,P'
,
= -,. [I?, 7-lo]Q'Q" +
L
Roo',pp'I?PP'
(5.311)
P,P'
l
We then substitute this expression into (5.308), obtaining
d(Mz ) d'
---=
L:
a,Q",pjJ'
(*[e, :>lo] ••' + n••"pp,gpw)(a'IMz1a)
(5.312)
Although it is not obvious from (5.312), under some circumstances the righthand side is proponional to a linear combination of (Mz(t», (My(t», and (AdAt», giving us a set of differential equations similar to those of Bloch. If 200
(5.316) ~ If there is zero shifl.
a
shifl,
we
T( to eO al t ::: O. II is the first leon in a power-series expansion. In order for the convergence to be good. it must imply that f!ppl at time t nOI be vastly.different from this value al t = ~. This implies thai we can find a range of tImes t such that t:> Tel but sull 1!~1J,(t) ':l!' Upp,(O). This laner condition implies that 1
~--'---
R".. ,~W
:» ,
(5.331)
The important trick now is 10 nOle thai if (5.331) holds true, we can replace l'flIJ'(O) by i!P13,(t) on the right side of (5.329). By this step, we convert (5.329) into a differential equation for [/, which will enable us 10 find [/ by "integration" at times so much later than t Tc
(5.332)
As we shall see in greater detail, the condition Tc < T2 is just that for which the resonance lines are "narrowed" by the "mOlion" that produces the fluctuations in 'HI (t). Because ~oJJfJ = R/J/J,OOl
(5.333)
(that is, the transition probability from 0' to fJ is equal to that from fJ to 0'), the solution of the Redfield equation is an equal distribution among all states. This situation corresponds to an infinite temperature. Clearly the equalions do not describe the approach to an equilibrium at a finite temperature. The reason is immediately apparent - our equation involves the spin variables only, making no mention of a thennal bath. The bath coordinates are needed to enable the spins to "know" the temperature. A rigorous method of correcting for the bath is to consider that the density matrix of (5.313) is for the total system of bath and spins. Since in the absence of HI the spins and lattice are decoupled, we may take the density matrix to consist of a product of that for the spins, a, and that for the lattice, ~L. We take for our basic Hamiltonian '110 the sum of the lattice and the spin Hamiltonians (which, of course, commute). 'HI commutes with neither and induces simultaneous transitions in the lattice and the spin system. Then we have 204
Introducing spin quantum numbers sand !l, and lanice quantum numbers / and /', we replace 0' by sf, and so on. Then we assume that the lattice remains in thermal equilibrium despite the spin relaxation:
ll}1' = Off'
e-"/IkT "/"/kT
Ee
(5.335)
I" We then find the differential equation for
d
L• •
I..d.
r
d/ ll /I'uu ') = o/f'llfl' dtan'
(5.336)
and sum over f. The result, in the high temperature limit, is simply to give a modified version of Redfield's equation, with density matrix for spin a replaced by the difference between u and its value for thennal equilibrium at the lattice temperature q(T). We therefore simply assert that for an intcraction in which the lattice couples to the spins via an intcraction 'HI (which, to the spins, is time dependent),7 the role of the lanice is to modify the Redfield equation to be
du;o-' ~ +i(Ol-Ol'-.B+.8')t.. - d - = L. Roo-',fJ.B,e [ufJ.B' - u.BfJ,(T)]
t
(5.337)
.oft'
where a,a',fJ,' stand for spin quantum numbers, and where u.8.8,(T) is the thennal equilibrium value of ufJ.8':
C"/J/ kT
"~~,(T) = 6~W Le WIlT
(5.338)
~"
That (5.337) should hold true is not surprising in view of our remarks in Chapter I concerning the approach to thennal equilibrium. We note here. however, that our remarks apply not only to the level populalions (the diagonal elements of u) but also to the off-diagonal elements.
7 "HI involves both spin Rnd lattice coordinRlcs. If we treat the lattice quantum mechanically. the lattice variables are operators, and 'HI dOClS not involve the lillle explicitly. If we treal the lattice classically, 'H\ ;n\'oln'$ the time c:o:plicity. Thill this must be so is evidcnt, since the coupling must be time dependent to induce spin transitions between spin stales of different energy. However, it is time indcll·cndent when the lattice ml'lkcs a simultaneous transitiOIl that just absorbs the spin energy. 205
5.12 Example of Redfield Theory
where q = x, y, z, and
We tum now to an example to illustrate both the method of Redfield and some simple physical consequences. The example we choose is Ihal of an ensemble of spins which do nOI couple to one another but which couple to an external fluctuating field, different al each spin. The external field possesses X-, y-, and z-componenls. This example possesses many of the features of a system of spins with dipolar coupling. However, it is subslanlially simpler to lTeal; moreover, it can be solved exactly in the limit of very short correlation time. For the case of dipolar coupling, Ihe nUClUations of the dipole field arise from bodily motion of the nuclei, as when self-diffusion occurs. The correlation time corresponds 10 the mean time a given pair of nuclei are near each other before diffusing away_ OUT simple model gives the main qualitative features of the dipolar coupling if we simply consider the correlation time 10 correspond to that for diffusion. In panicular, then. our model will exhibit the imponant phenomenon of motional narrowing, which has been so beautifully explained in the original work of BJoembug~n et al. Before plunging into the analysis. we can remark on cenain simple features that will emerge. At the end of this section we develop these simple arguments funher. showing how to use them for more quantitative results. We may distinguish between the effects of the Z·, y-. and z-eomponents of the fluctuating field. A component Hz causes the precession rate to be faster or slower. It. so to speak. causes a spread in precessions. It will evidently nor contribute to the spin-Ianice relaxation because that requires changes in the com· ponent of magnetization parallel to Ho. but it will contribute to the decay of the transverse magnetization even if the fluctuations are so slow as to be effectively static. In fact, as we shall see, it is Hz that contributes to the rigid-lattice Line breadth. The phenomenon of motional narrowing corresponds to a son of averaging out of the Hz effect when the fluctuations become sufficiently rapid. The z- and y-components of fluctuating field are most simply viewed from the reference frame rotating with the precession. Components fluctuating at the precession frequency in the laboratory frame can proouce quasi-static components in the rotating frame perpendicular to the static field. They can cause changes in componenls of magnetization, either parallel or perpendicular to the static field. The former is a T 1 process; the latter, a T2 process. Clearly the two processes are intimately related, since the magnetization vector of an individual spin is of fixed length. The transverse componentS of fluctuating magnetic fields will be most effective when their Fourier spectmm is rich at the Larmor frequency. For either very slow or very rapid motion, the spectral density at the Lannor frequency is low, but for motions whose correlations time, is of order I/wo, the density is at a maximum. The contribution of H~ and H y to the longitudinal and transverse relaxation rates therefore has a maximum as , is changed. Let us consider, then, an interaction 1t1(t) given by
'H\(t) =
-,"h L:Hq(t)Iq
(5.339)
1to = -"(n/tHol: = -liwolz
(5.340)
where. wo is the Larmor frequency. We characterize the eigenstates by a, the eigenvalues of (5.340). These are wo times the usual m-values of the operator I z . (Herem = I, I-I, ... -I). We shall continue to use thenotalion a,d,fJ,fJ', however, rather than m, in order to keep the equations similar to those we have just developed. The matrix elements (al1tI(t)la') are
(al'H,(')la') =
-,"h L:Hq(')(allqla')
.
(5.341)
q
Then the specrral density functions JOtpo'P'(w) are
We now use the symbol kqq.(w) introduced in the preceding section as +~
kq¢(w) =
4J 'H'q"(")H"q-,T.('--:+-:T">e- iw.. d,
.
(5.343)
-~
Clearly the fluctuation effects, correlation time, and so on are all associated with the kqq"s. For simplicity let us assume that the fluctuations of the three components of field are independent. That is, we shall assume
Hq(t)Hq,(t + r) = 0
if q "" q'
(5.344)
For example, (5.344) will hold true if, for any value of the component H q , the values of H q• occur with equal probability as IRq'1 or -IHq,l. We note that kqq(w) gives the spectral density at frequency w of the q-component of the fluctuating field. With the assumption of (5.344) we have, then, (5.345) We now seek to find the effeci of relaxation on the X-, y-, and z-components of the spins. To do this, we utilize the second technique described in the preceding section, that of finding a differential equation for the expectation value of the spin components. Let us therefore ask for (dldt)(Ir ), r = x,y, or z. By using (5.312), we find
d(I,) - = " L. i h[ ll,1to Joo,,(a' I.. 0'l l+) " L. Roa',f3fJ'llf3J3'(U ' IIrla) dt 0,0' o,o',P,P' (5.346)
q
206
207
"
+1'~
The first tenn on the right, involving 'Ha, can be handled readily:
L , 2.r.Q. 'H'o)Cf(f,(a'IIr 10) h
= _,i Tr {(g1iO
,
0,.
-1toe)Ir}
=
Ii Tr {eHoI, -
=
Ii Tr {e[Ho,!,]}
e I , Ho}
= -i'YnlIo Tr {e[I~, I r ]}
(5.347)
If r = z, this term vanishes. If l' = x, we have
-i1'nHO Tr {ily!?}
= +1'n Ho(Iy )
(5.348)
If r = y, we get --YnHo(l:t'). Thus we have
L 0',0"
*[e, 'h'o]O'O',(c/llrlo)
= 1'n«1) x HO)r
(5.349)
RO'O",(3(3'!?(3/3,(O"IIr la)
.
(5.350)
0',0" ,/3,(3' As we have seen, RCl'O" ,(3(3' is itself the sum of four tenns [see (5.330)]. We shall discuss the first teon. JO'(3O"(3'(O" ~ 13'). By using (5.345), we find
L
~
1; I: L
RO'O', ,(3(3' e(3(3' (ollIr 10)
0',0" ,(3,/3'
1; I: (PIIqlo)(oII,Iq -
= i1'~ku(wo)Tr {I:t'Iye - I:t'ely} =
i-y;ku(wo)Tr {U:t'I y - IyI:t')e}
= --y;k:t':t'(wo)Tr{IzU} = -1'~k:t':t'(wo)(lz)
(5.352a)
The tenn q = y gives, in a similar manner,
-1';kyy (wo)(Iz)
.
(5.353)
RO'O', '/3/3'!?/3/3'('/ 1I z 10) = --y;( k:t':t'(wo) + kyy(wo) J(I z )
. (5.354)
By combining (5.346), (5.349), and (5.354), we have
({3'jlq]O")(o'jlrlq!?I,O')kqq(o' - {1)
(5.351)
where the last step follows from the basic properties of orthogonality and completeness of the wave functions 10), and so on. We are able to "collapse" the indices 0 and .0. but we cannot do the same trick for 0' and p' because they occur not only in the matrix elements but also in the kqq's. In a similar way one can obtain expressions for the other three [enns in RO!O!',(3(3" getting finally
=
DW, 10)(011[/" I,lellPlk,,(o - P) ',p = ,,~[ I: 022, un, U'2t· In the relaxation tenns Ra~ ,/Jll', the only tenns of imponance (as we have seen) involve cr - crl = fJ - If. The only tenns Ihal count are therefore
!
R Il ,22 =
R22,1I
R12,12 = R21,21
== I/TI == - 1/1"2
(5.393)
By assuming 1i2 joins states 1 and 2 only, and denoting (l11i2(t)12) by 1i12(t), we find dUl1
_'_e-1{,/"·
,
dM dt
--=/
d.
dU22
=-dt" =
t!22 - ",1 -
[m(T) -
ell (T)] + ii; ( UI2 ~ (t) 1L21
TI
del2
Ul2
-- = -~
&
i
l
i
+ -(£2 - £1)UI2 + -:-(U11 - U22)1i 12(O h h
~ (t)n )
- Iq2
c:21
(5.394)
(5.395)
If E2 is larger than £1 and '}-{12 oscillates at frequency w, we can solve (5.394) and (5.395) in steady state by assuming that
="21e-i~t
l!12 = r'12ei~t,
.!?21
el1 = rll ,
l!22 = r22
(5.396)
where the roo, 's are complex constants. The details of the solution are left as a problem. but the form of the answer is identical to that of the Bloch equations.
'17
If we write
'H2(t) = V cos wt
(5.397)
6. Spin Temperatnre in Magnetism and in Magnetic Resonance
where V is an operator, and define Wo by the relation E 2 - El == hwo, we have in the limit of small V (that is, no saturation) e-El/ kT
1 -El/kT 1'" = ell(T) = e EI/kT + e E21kT ';::' -e 2
'"
_ _ V"T2 [(T) - 2'n 1·( ) ell +IW WOT2 S! IT2 V I 2W O I + i(w WO)T2 4kT
e22
(1')J
6.1 Introduction (5.398)
We note that 1'12 differs from zero only near to resonance and that T2 characterizes the width of frequency over which 1'12 is nonzero. If the states 1 and 2 are the two Zeeman states of a spin ~ nucleus and a static magnetic field parallel to the z-direction, then the transverse magnetization Alz has matrix elements only between states I and 2, the diagonal elements being zero. Therefore (M:r(t»
=
1'12eiwt(2IMzll) + 1'2Ie-iwt(lIMzI2)
= 2 Re {TI2eiwt(2IMzlI)}
(5.399)
Taking (5.400)
V = -MzHzo and recalling that X is defined as
(5.401)
(Mz(t)} = Re {xHzoeiwt} we see that
wo!(IIM.12)I'
'T2
X(w) = 1 + i(w _
2kT
WO)T2
(5.402)
Now, using the fact that I =~. we have X(w) =
iT2
1 + i(w -
WO)T2
wO'Y 2 1i. 2 ](l+I) -
2
3kT
(5.403)
which agrees with the expression for the Bloch equation derived in Chapter 2. We note that we could detennine both 71 and T2 from first principles by computing R",22 and R I 2,12' Alternatively we could simply treat Tl and 72 as phenomenological constants to be given by experiment. If there are more than two levels to a system, the solution may be carried out analogously by simply setting all off-diagonal elements of gaol equal to zero, except those near resonance with the alternating frequency (Err - Eo, S! liw).
218
In Chapter 5 we employed the concept of spin temperature to discuss relaxation. The idea of spin temperature was introduced by Casimir and du Pre [6.1] to give a thermodynamic treatment of the experiments of Gorter and his students on paramagnetic relaxation. It was Vall Vleck [6.2] who first employed the concept for a detailed statistical mechanical calculation of the relaxation times of paramagnetic ions. 80th in this case, and also in his general statistical mechanical treatment of static propenies of paramagnetic atoms [6.3], he recognized and emphasized the fact that expansion of the partition function Z in powers of lIT enabled one to calculate Z without the necessity of solving for the energies and eigenfunctions of the Hamiltonian. Waller evidently was the first person 10 use this property [6.4]. From the partition function, one can compute all the static properties of the system, such as the specific heat. the entropy. the magnetization, and the energy. For example, the average energy of a system. E, at a temperature T is given by
E = kT' :T In Z
(6.1)
In 1955. Redfield [6.5] showed that the conventional theory of saturation did not properly account for the experimental facts of nuclear resonance in solids. In one of the most important papers ever written on magnetic resonance. he showed that the conventional approach in essence violated the second law of thennodynamics. He went on 10 show that saturation in solids can be described simply by applying the concept of spin temperature 10 the reference frame that rotates in step with the alternating field H \. To understand his ideas, one needs to understand certain concepts which predate the discovery of magnetic resonance - ideas such as adiabatic demagnetization. In this chapter we begin by describing a simple experiment which displays the failing of the pre-Redfield theory of magnetic resonance. Then we tum to a discussion of the use of spin temperature in nonresonance cases to build background for the application of these same ideas in the rotating reference frame. We then discuss the Redfield theory of saturution in solids.
219
6.2 A Prediction from the Bloch Equations Let us consider a simple resonance experiment with a rotating magnelic field of angular frequency w. lI"ansverse to the stalic field B o• tuned exactly 10 resonance w: ,Ho
(6.2)
We discuss it by means of the Bloch equations. It is convenient 10 transfonn to a reference frame rotating at w with HI along the x-axis as is done in Sect 2.8. Exactly at resonance. the Bloch equations become dM~
----;It "" -..,MyH I + dM~
Mo - M~
(6.3a)
Tl
Jkrz
(6.3b)
-;U""-T2
dMy "" -vM HI _
dt
,:
~
(6.3c)
T2
Suppose we now orient M along HI sO that at t "" 0 Mz "" Mo. Jvly "" M: "" O. From (6.3b) we see that M z will decay 10 zero in a time T2 • The low HI steadystate solution of the Bloch equations shows Ihat they describe a Lorenlzian line with a frequency widlh I &:-
.~
T,
,
For solids typically Llt..,)
';!'
'1Hnei,hbor ~ '11l ~ '1.-../1(/ + I)/Z J (tJ
a
6.3 The Concept of Spin Temperature in the Laboratory Frame in the Absence of Alternating Magnetic Fields Let us now tum to a discussion of the application of the concept of spin temperature to magnetic experiments not involving resonance. A typical system we might consider is a group of N spins of spin I, gyromagnetic ratio '1, acted on by an external field Ho, and coupled together by a magnetic dipolar interaction represented by a dipolar Hamihonian Hd' We denote the Zeeman Hamiltonian by Hz. The solutions of the Schmdinger equation are then wave functions tPn of energy Ell of the total system.
'li",,: ('liz + 'lid)"" : E"""
(6.5)
(6.6)
Unfonunately, (6.6) is exceedingly difficult to solve, depending as it does on the coordinates of 10'22 spins. One can assume, however, that if the spin system is in thennal equilibrium with a reservoir of temperature 8, the various states n of the total system would be occupied with fractional probabilities PII given by the Boltzmann factor I'll ""
where H ~gl . I bo r is the nuclear magnetic dipole .field due to neighbors, and a is . . the distance to the Z nearest neighbor. For typical solids Llt..,) IS of order of a few tens of kilocycles (e.g. .t1w S! 211'" X lO kc for AI metal). Funher examination of (6.3a) and (6.3c) shows that they do not involve M~ and that if M and M z are initially zero they would remain zero were it not for the tenn invofving T t . If T} ::> T2, therefore, and for times up to about T I after M has been orienled along H I, we still have My "" M: "" O. Therefore these eqlw.tions predict that wilen M is aligned along HI, it wUl decay to zero in a time T2, typically of order 10- 4 to lO-5 sec. Experimentally this prediction (decay in T2 when M is along H) is found to be correct for liquids, but it is not correct for solids. Rather, for solids it is found that as long as HI is turned on and is sufficiently strong, the decay rate of M~ is much more like T, than a time T2 which characterizes the line width. Redfield first stated this fact on the basis of his steady-state experiments, but he did not actually perfonn lhe experiment we have described. That experiment was first performed by Holton et a!' [6.6]. Why do the Bloch equations fail for 220
solids but not for liquids? Redfield has given an explanation. We will discuss the conditions for validity of the Bloch equations later in this chapter, after we have discussed some important background material on spin temperature in cases where there is no alternating field HI present.
E,,/k8 .!..eZ
(6.7)
where Z is the panition function (6.8)
" Quantities such as the average energy E and magnetization M ~ would then be given by
E:E~&
~~
"
(6.9b)
"
As we have remarked, Van Vleck recognized thai expressions such as (6.9) could be evaluated wilhout solving the Schmdinger equation because they could be expressed as traces. For example we can express the panition function as a lrace as follows. (6.10)
"
"
Since the trace is independent of lhe panicular representation used
10
evaluate it. 221
we can use a convenient representation. For example. we could evaluate (6.10)
in principle by using as basis functions the eigenfunctions of the z-component of spin Izk of all the individual nuclei. However, to do so, we need to lake one more step, expansion of Z in a power series. It is often valid for nuclei and for eleclTons to use the high temperature approximation. We expand the exponential in a power series, keeping only the leading lenns. Then the sums are easy to do. 2 Hl1i N', } Z=Tr { 1-/.;8+2/.;28 2 -'" =(21+1) +2k26 2Tr {'Jt}+··(6.11) where we have used the faci that Tr {1t} = 0, as can be readily verified for both
Hz and 'HdUsing these methods one finds C(H2 + lJ2) 0 L
E=
where
8
N"f2fI 2I(I + 1)
C=
(6.12)
(6.13)
3k
is the Curie conslant, and HL is a quantity we call the local field, which is of the order of the field one nucleus produces at a neighbor (several Gauss) and is defined by (6.14) 1 T< {H'} L- k(2I+l)N d Since the trace in (6.14) can be computed, HL may be considered to be precisely known. One finds H[:: -y 2r!2[(I + I) ~)1/rjk)6 (6.15)
CH' -
j
One can compute the magnetization M and finds it obeys Curie's law
M= CHo 8
q=
E+kBlnZ
(6.17)
8
Evaluating Z and E we get C (H 2 +H 2) ~::Nkln(2I+I)-0 L
2
8'
(6.18)
6.4 Adiabatic and Sudden Changes The significance of these results is more fully realized by considering the behavior of the spin system when the applied field Ho is made a function of time. For simplicity, let us assume the spin system is thennally isolated from the outside world and that it mayor may not be in [hennal equilibrium. The first of these assumptions is satisfied if the experiments we conduct are perfonned on a rime scale short compared to the spin-lauice relaxation time. The assumption implies that the Hamiltonian of the system includes only variables intemal to the system since spin-lanice relaxation results from tenns involving variables both intemal and external 10 the system as illustrated by (5.33). If the system is in internal themlal equilibrium, a spin temperature applies, so that (6.18) holds. We can then consider three cases.
(6.16)
Note that this is a vector equation, so that M and Ho are parallel. A moment's reflection shows that (6.16) is truly remarkable. It states that the vectors M and Ho are parallel no matter what the size of Ho as long as the high temperawre approximation is valid. Suppose Ho is small, comparable to the local field a nucleus experiences from its neighbors. One might then suppose that nuclei would tend to line up along the direction of the local field, not along Ho. It would seem reasonable to suppose that the degree of polarization one could achieve per unit of applied field would be less when Ho «..H L than when Ho:» H L . Equation (6.16) shows Ihat this intuitive argument is incorrect - the degree of polarization per unit of applied field is independent of the size of Ho relative to the local field. Not only is this true of the magnitude of M, but also of the direction as well. 222
Another useful property of (6.16) to note is that when Ho :: 0, M :: O. Thus, suppose Ho were tumed 10 zero so suddenly that M does not have time to change. Immediately after Ho :: 0 we have a case where M f. 0, Ho :: O. But if there..were a temperature, (6.16) shows M must be zero. Therefore we can use (6.16) to conclude that at this instant of time the system is not describable by a temperature. Another quantity of great utility is the entropy ~. We know from statistical mechanics that Ihe entropy measures the degree of order in a system. In a reversible process in which there is no heat flow into or out of a system, the entropy of that system remains constant.
a) The Hamiltonian is independent of time (the applied field is static). In this case, the average energy is constant in time, whether or not the system is describable by a spin temperalure. If the system has many parts which are strongly coupled together, but is not initially in a state of internal equilibrium, we expect that irreversible processes within the system will eventually bring the system into an internal equilibrium describable by a spin temperature B. During that process, the energy is conserved since a time-independent Hamiltonian corresponds 10 a system on which Ihere are no applied forces. Proof of the constancy of the energy is left as a homework problem (Problem 6.5). b) The Hamiltonjan changes slowly jn time. The criterion of slowness is that at all times internal transfers of energy shall be fast enough so that the system is always describable by a single temperature B. Under this circumstance, Ihe changes are reversible, and the entropy of the system remains constant. 223
system. The condition will be satisfied if we make the changes on a rime scale rapid compared to the spin-lattice relaxation time T!. Frequently one has T1's of seconds, and by cooling may achieve Tl'S of hours. Such long limes are practically infinite. The second condition we must satisfy is that after each small change in Ho we must allow a new temperature to be reached before making another small change. This condition is typically that we must change Ho slowly on a time scale defined by the precession period of nuclei in the local field of neighbors (I/-yHL)' This time scale is a few tenths of a millisecond. Between these time intervals there is a readily achievable range for which Ho can be changed adiabatically. For an adiabatic change we have a constant entropy. Thus from (6.18) we get that (HJ+HVI02 remains constant. If we stan in an initial field Hi at temperature 0i and change the field adiabatically to a final value Hr, the temperature Or is given by the relation H.I2 +HI.2 _ H2r +H2). (6.23)
c) The Hamiltonian changes discontinuously in time. Such a change could occur if Ho can be changed quickly. The term "discontinuous" means that the change is so fast that the various spins making up the system do not change direction during the change. Let us now investigate these various cases more fully. I) Time-Independent Hamiltonian
Consider a system not in thermal equilibrium initially. Let us suppose that the parts of the system are coupled together. We then expect that eventually the system will achieve an internal equilibrium described by a "final" temperature If we know the energy of the system at t = 0 (call it Eo), we can compute I~e temperature Or as follows, making use of the fact that for a time-independent Hamiltonian the energy is conserved. _ . Utilizing (6.12) which relates the energy E to the temperature, and applymg conservation of energy, we get that
o.
E=Eo
or
8r -
(6.\9)
C(H6 +Hl) Sf = Eo
A famous result, cooling by adiabatic demagnetization, can be seen in (6.23) by taking and Hl,
(6.20)
eRo
8r H L -=- and goes to zero when IIr goes to :tetO
We have considered a process which is reversible. Now we tum to one where things happCn suddenly, resulting in irreversibility. Suppose we describe the system by a wave function t/J. TIle time-dependent Schrtxl.inger equation is
h
-i
a~ {It = 7i(l)~
11£,
II,
This equation is ploued in Fig.6.2. Note that if we lower the field until it is zero, M f = O. However, since the field changes are reversible (and in fact at all times keep the entropy constant), we can recover Mi by raising Hr from zero back up to its initial value Hi. The recovery of Mr in such a process d~s 1I0t involve spin-lattice relaxation since, as we have postulated, everything is done on a time scale shon compared to Tt! This curve is often a surprise to those of us who first learned magnetism by studying magnetic resonance, since we learned that one needs TJ to produce magnetization from an unmagnetized sample. Actually, if one uses Curie's law and (6.23), one sees that if one stans in zero field with spins which are in thennal equilibrium with the lauice (8 = ()/), the mere act of adiabatically turning on the static field to H o ,. HI. will produce a magnetization. The size is (6.26) where Mo is the magnetization one gets when the spin-lattice relaxation produces 226
= -M·H(l)~+7id'"
(6.27)
where M is the tOlal magnetic moment operator and 1id the dipolar coupling. The time dependence of 1i arises because the applied field H is time dependent. For the case of sudden switching, we take H as independent of time except for t = 0, at which time it jumps discontinuously from one value to another. Denoting by 0- and 0+ times just before and just after t = 0, we then have
~(O+) -
0+
"'(0-) = -
J*7i(l)~(l)dl
=0
or
(6.28)
0-
(6.29) since 1i(t), though discontinuous, is nevenheless never infinite. Thus we have that the wave function just before the switch is identical to its value juS! after. We can utilize (6.29) to see that a sudden change in H produces a change in the expectation value of the energy E. The expectation value of energy at any time t, E(t), is E(l) = (~(l). 7i(l)~(l))
(6.30) 227
Thus
whence we gel
E(O+) = -C H(O-) . H(O+) _ C H~
E(O-) = (,,(0-). 'II(O-),,(O-» = -(M(O-» . H(O-) + ('IId(O-»
8.
(6.32)
(6.33)
But
is conserved. 1Oerefore.
Ukewise
(6.41)
The fact that "'(0-) ::: "'(0+), however, means that the expectation values of both magnelization and dipolar energy are the same at t ::: 0+ as at t ::: 0-. This resuh expresses Ihe physical fact Ihal all spins point the same direction al t ::: 0+ as they did at t ::: 0-. Thus we can write
(6.34) This equation is very useful since in general the system just before Ihe sudden change is assumed 10 be in internal thennal equilibrium with temperature OJ. Thus we can compute the expectation values by the methods of Sect. 6.3. So far we have nor employed the density matrix notation, simply to make Ihis chapler accessible to those readers who have not yet become familiar with it. We can write our results compactly by recognizing that (6.29) is equivalent to saying that lhe density matrix f! obeys the relation
(6.35) Thcn
(6.36) Assuming thermal equilibrium at t ::: 0- at a lemperalUre Bi we get
_
riCO ) =
cxp(-'II(O-YkO,) Z(O )
(6.37)
Therefore
('IId(O-» = Tr {e(O-)'/{d} Tr {'lid cxp (-'II(O-)lkO,») Z(O-)
(6.38)
Manipulation, using the high temperature approximalion plus the definition (6.14), gives
('IId(O-» = 228
(6.40)
As discussed earlier, immediately after switching the field. the spin system is in general not in internal thermal equilibrium even though it was in thennal equilibrium at t = 0-. If we wait a long enough time, we expect a temperature to be achieved. Call that time if and the temperature 8r- We can compute 8 r by recognizing that fOT t > 0 the Hamiltonian is lime independent, hence the energy
where (M(O-» and (1{d(O-» are the expectation values of magnetization and dipolar energy at t = O~. defined as
(M(O-» " (,,(0-). M,,(O-» ('IId(O-» " (,,(0-). 'IId"(O-»
8j
(6.31)
CH'L 0,
(6.39)
E(tr) = _CIH'(O+) + HlJ
0, and E(O+) is given by (6.39). Thus
H(O+)' + H' 8r - (J. " - 'H(O ).H(O+)+Hl .
(6 42) .
The significance of (6.42) can be seen by considering a particular example in which the applied magnetic field is turned suddenly from its initial value. Ho, to zero at t = O. Then H(O-) ::: Ho. H(O+) ::: O. giving
Or = 0,
.
(6.43)
We contrast this with the result of turning the field slowly 10 zero
HL
Or::: OJ Ho
(6.44)
The sudden tum-off leaves lhe temperatu~ unchanged, the slow tum-off produces cooling. The results of sudden changes are summarized in Fig. 6.3. for Ihe case described above in which Ho is lurned suddenly to zero at t ::: O. Adiabatic and sudden changes in Ho have been very useful in magnetic resonance. One of the first uses of adiabalic changes was to measure the magnetic field dependence of the spin-lattice relaxation time. In 1948, Turner el al. [6.7] were studying [he spin-lattice relaxation time TI of protons in insulators. The relaxation times were, in some inslances, many minutes long. To observe Ihe dependence of T 1 on stalic field, they used a field cycling lechnique in which they observed the resonance at a high field. bUI allowed the spins to relax. in lower fields to which they cycled belween their observations. Pourui [6.8] discovered that the nuclear relaxation times in a crystal of LiF were so long thai he could take his sample out of the apparatus into Ihe earth's magnetic field and then return it with only a small loss in magnetization. With Ramsty [6.9] he demonstrated that if this sample were removed from the strong field 10 a small static field (40 Gauss
'29
III
i
/ P 'V!v ~
£1
Fig. 6.3. Magnetic field (11), wave function (l{!), energy (E), spin temperature (0), and rnagnc~ization (M) as functions of lime for the case of a magnelic field turned suddenly to zero
i - - - - - !-
,
, ,,
~N;;O~ _ _ {%
,
I
,
"I
-----'--"-----
or less), subjected to an audio-frequency magnetic field, and then returned to the original magnetic field, he could detect aI the strong field the resonant absorption of energy by the spin system when in the weak field. The experiment we have just described is closely related to the famous experiment on negative temperarures by Pound and Purcell [6.10]. They 100 started with the LiF sample in a strong field, but removed {he sample to a solenoid whose field they suddenly reversed, producing a situation in which the magnetization is anliparallel to the static field. They then returned the sample to the strong field where they observed the relaxation of the magnetiz.1tion component along the static field. They point out that such a circumstance, with the upper energy Zeeman levels more populated than the lower, corresponds to a negative Zeeman temperature. Any system with an upper bound on its energy levels can in principle have a negative temperature. In addition to the original paper by Pound and Purcell, we call the reader's attention to the wonderful account of this experiment and discussion of the negative temperature concept to be found in Van Vleck's [6.11] lecture on the concept of temperature in magnetism. (Professor Van Vleck cautions that there are some incorrect numbers in his paper.) This imponant experiment in which a population inversion was produced is the forerunner of the maser and the laser. Demagnetizing to zero field is the basis for the experimenls of Hebel and Slichter [6.12] and Redfield and Anderson [6.13,14] to measure the nuclear relaxation in superconductors. The problem here is that since superconduclOrs 230
exclude the magnetic field, it is hard to observe a magnetic resonance in a metal in the superconducting state. One starts the field cycle with a magnetic field of sufficient strength to suppress the superconductivity. When one turns the field to zero, one achieves two effects: (1) the nuclear spins are cooled and (2) the sample becomes superconducting. With the field zero, the nuclear spin temperature relaxed towards the lattice temperature so that when the magnet was turned on again after a variable time T, the metal returned to the nOTlllal state and the spins warmed to a holler temperature than they had before the cycle began. The increase in temperature is measured by a swift pass though resonance. By varying T, one could deduce the zero-field spin-lattice relaxation time. The NMR results showed that as one lowered the sample temperature below the superconducting transition temperature, Te , the nuclear relaxation rate was at first faster than it would have been if the metal were normal, then at lower temperature it became slower. The temperature dependence of the nuclear relaxation rate is dramatically different from the temperature dependence of ultrasonic absorption, as was shown by Morse and Boltm [6.15]. They found that ultrasonic absorption dropped rapidly relative to its value in the normal metal on cooling below Te · The contrast between the behavior of two low energy scattering processes, nuclear relaxation and sound absorption, is difficult to understand in a one-electron theory of metals, but finds a natural explanation in the Bardeen, Cooper, and Schrieffer (BCS) theory of superconductivity. These experiments constitute a direct verification of the concept of eleclTon pairs, which is the basis of the BCS theory. (See, for example, Leon Cooper's Nobel Prize lecture [6.16].) Anderson [6.17] working with Redfield, combined field cycling with the application of an audio-frequency magnetic field applied while the static field was zero to heat the spins to plot Ollt the zero-field absorption characteristics of a spin system. Thus they used field cycling to give them the sensitivity of resonance in a strong field to monitor the effects they produced in zero field. Abragam and Proctor [6.18] did further studies establishing the validity of the spin temperature concept, again using LiF. An important result of their experiments was their observation of the transfer of energy between the twO spin systems (Li 7 and F19) even in a static field which greatly exceeded the dipolar fields exerted by the nuclei on their neighbors. We discuss these experiments further in Sect. 7.10.
6,5 Magnetic Resonance and Saturation The analysis of magnetic resonance by Bloembergen et lIl. using standard perturbation theory is given rather compactly in (1.32) - the differential equation for the population difference, n, between the two energy levels of a system of spin particles,
4
dn
no-n
- '" -2W(w)n + - dt T1
(6.45) 231
W(w) is the probability/second that a spin will be flipped by the radio-frequency field H t . Standard perturbation theory shows W(w) = i-y2 Hrg(w)
(6.46)
where g(w) is a funclion nonnalized to unit area having the same dependence on frequency as the absorption line -that is, it expresses the fact that the frequency of HI must be close to resonance for H, to induce transitions. no is the thennal equilibrium population difference, and TI the spin-lattice relaxation time. It is always JX)ssible, at least conceptually, to consider TI infinite, in which case (6.45) is especially simple to solve. n = Ae- 2Wf (6.47) where A is a constant of imegration. It is well to recall the conditions for the validity of (6.46). 1bey are two: i) ii)
The perturbation matrix elements inducing transitions must be small compared to the width of the final state energy levels. This means HI < H L. The wave function must not change much. We note, however, that (6.47) predicts that fl --f 0 as t _ 00.
To satisfy condition (ii) we expect that we must consider times less than 1/1V. Thus, though it is always easy to meet condition (i) by making H, small, no matter how weak HI, if we wait long enough we violate (ii). [Note we are requiring here also that I/W < TI, otherwise the Tl tenn would rescue condition (ii) even for times long compared to I/Wj. We have the interesting problem, therefore, that we do not know how to integrate the equations of motion beyond a time for which n is almost its value at t = O. (See Fig. 6.4). The solution of this problem was found by Redfield [6.19] in a truly remarkable paper, the more so when one recognizes that it was his first work on magnetic resonance. In it he shows that the Bloch equations, when applied to a solid, violate the second law of thennodynamics. The essence of his approach is to note that a resonant time-dependent perturbation, no matter how weak,
".
of l"lliitiity of ~Rrgiolt com'elJliOIl(lI/H:1(llrlJo/io/f /lleor)'
will eventually produce large effects. Whenever a perturbation of small size can produce a big effect, it is dangerous to treat it lightly. He therefore eliminates the time dependence by transfonning to a reference system in which the Hamiltonian..is essentially time independent. The residual time dependence is not of the dangerous variety. For such a transfonned system, the energy is conserved. Moreover, the system is highly complex, consisting of many interacting spins. One can thus predict that after a sufficiently long time the system will be found in a state of internal equilibrium. That is, it will be in one of its most probable states. Phrased alternatively, the energy states will be occupied according to a Boltzmann distribution at some temperature 8. We consider the Hamiltonian 1t, given by
1/ = 1/z(I) + 1/d
where llz(t) is the Zeeman interaction with the static field Ho and the rotating field of amplitude HI and angular frequency w rotating in the sense of the nuclear precession. "The TOtating field makes 'Hz time dependent. We are, of course, considering T I infinite. Utiliz.ing the methods of Sect.2.6, we transfonn to the rotating reference frame, gelling a transfonned Hamiltonian J(:
'It = -~h(Ho-wh)I.+HII.J+1f.: + term oscillating at ± w, ± 2w -Iz sin? + I y cos; = e- i/ .? Iyei !.?
and
(6.50) (6.51)
to transform products such as ei!,w.l IzIye -iI
.w. t
(6.52)
to expressions involving sin Wit and cos Wit. The term 'H~ is that part of the dipolar coupling which commutes with Ii' Physically, it is the part of 1id which is unchanged by rotation about the z-axis. (The two statements are of course equivalent since one can consider Ii as generating rotations). II is the sum of the tenns A and B discussed in Sect. 3.2. The fonn of 1t~ is
o
"·ig.6.4. Conventional 6aturation theory predicts that the I>opuiation JI goes to zero exponentially with lime t. However, the assumptions on which il is based are valid only for the initial PlITt of the curve where n s:' no
(6.49)
In deriving (6.49) we have used relations such as
"(2/i,2
1id = - 4
"
(6.48)
L
j,k
(1-3cos 2 8j1.) :I r jk
(3I:jI z k - Ij'
h)
(6.53)
where 9jk and rjk are coordinates of nucleus j with respect to nucleus k. The term in the square brackets in (6.49) may be considered as the coupling of Ihe spins to an effective static field He as we noted in Chap. 2.
He
=
k(Ho -w/-y)+iH I
(6.54)
2/W 232
233
In the absence of the time-dependent tenns, the energy levels of 'HI would be split by He·and the dipolar couplings, so that we expect typical splittings to be IJE ~ iliJHi + (6.55)
M=C
HJ
(6.56)
Now, in the absence of HJ, 'Hz and 'H3 commute, since the fact that
H3
[Iz , 11:3] = 0 was the definition of1f.3. Under this circumstance 'Hz and would separately be constants of the motion. However, if HI f. 0, ['Hz, H~] f. 0, and the Zeeman and dipolar systems can then exchange energy. Since H is independent of time, the total energy is conserved. Moreover, the system is very complex. Redfield therefore postulates that no matter what the state of the system at t = 0, a long time later it will be in a state of internal equilibriulll described by a Boltzmann distribution. In other words, there will eventually be a temperature 6 which can be assigned to the spins. We can thus say that the density matrix g is
e=
..
e-'H/k9
(6.57) Z with Z = Tr {exp(-H/k8)}, where 'H is the effective Hamiltonian of (6.56). Of course, we expect that after a long enough time Redfield's hypothesis would be fulfilled (unless there were some hidden selection rule which we have overlooked, are perfectly isolated from one llnother if HI such as the fact that 1f.z and is zero). But the really important question becomes, how long does it take to reach equilibrium? The answer to this question clearly depends on the size of HI, since HI is needed to prevent isolation of the dipolar and Zeeman systems. We return to the question later, but for the present consider that the time is short enough to make the establishment of a temperature practical.
H?t
6.6 Redfield Theory Neglecting Lattice Coupling
2
CH,2 _
I T, «'lj0)2) L- k (2I+I)N d
E = _C(H; + H'I> 8 234
(6.58)
(6.60)
(6.61)
Evaluation of the trace of (6.61) gives, for a system with only one species, (6.62)
where {IJH 2 } is the second moment of the resonance line. Following our earlier convemion of omitting primes. we shall now use H L for H~, using the prime only when we wish to distinguish between the local field in the laboratory reference frame and the rotating reference frame. It is important to notice that the Redfield assumption leads to Curie's law and that the vector nature of the law shows that the nuclear magnetization is always parallel to the effective field when the system is describable by a temperature in the rotating frame. Thus, if one is exactly at resonance, where He = iHI' the magnetization is perpendicular to the static field. Since the fonn of (6.58), (6.59) and (6.60) is identical to that of the corresponding equations in the laboratory frame, most of the equations of Sect. 6.4 can be immediately applied to the rotating frame by repl:lcing Ho with He and with
HE
Ht[.
6.6.1 Adiabatic Demagnetization in the Rotating Frame An adiabatic demagnetization in the rotating frame can be perfonned readily, as was demonstrated by Holron [6.20]. Suppose that initially HI = 0, and the sample is magnetized 10 Its thermal equilibrium value kMo. Let us shift Ho far above the resonance value wI'"'(, and tum on HI' C'Ne assume Ho - wI'"'( to be much bigger than HL and HI.) We now have an effective field which is virtually parallel to M. Next we change Ho to approach resonance at 11 rate sufficiently slow to satisfy the criterion for a reversible change. C'Ne are here assuming T l to be infinite, which we achieve in practice by perfonning all experiments in a time shorter than T j ). We then have that
M = Mo The significance of (6.56) can be appreciated by calculating again the energy E, entropy a, and magnetization M. We find easily that
(P
where C is the Curie constant, and where
HE H;,
1f. = -'"'(hI· He + 1f.3 = 1f. z + 1f.~
(6.59)
a=Nkln(2I+I)- C (H;+HII)
HE
where the square root is a convenient way of including the two limiting cases of He>HL and He«H L . The time-dependent tenns will connect states of order iliHo apan in energy. Unless ~ + a very low resonance field indeed, the time-dependent tenns are far from resonance and can be neglected since they are unable to produce transitions. They are not dangerous. We thus obtain a Hamiltonian which we call 1f., omining primes for simplicity of notation.
He 8
He JHi+ H [
(6.63)
Notice that M is parallel to He' as is required by Curie's law. Thus, as one approaches resonance, M changes direction, always pointing along He. In general, the magnitude of M also changes, unless Hi ~ HE. We can experimentally measure M by suddenly turning off HI, leaving M}o precess freely about 235
Ho. The induced voltage immediately after turn-off is proportional 10 M',t.. (Use of a phase sensitive detector enables one to measure Al'1: and 1L1y , but for these experiments My = 0). One can measure M z by noting Ihm though M'I: decays to zero within a time of order I/-yH L, after tuming off HI, M l does not change. One can thus wait till M',t. has decayed, and then apply a 7r/2 pulse which rotates M z into the x-y plane for inspection. The theoretical values of M'I: and M l are HI
HI
MQ
------------ ----- -----------------_.-
(6.64)
M. = M - = MOr~==;; He ./H2 +H2
vel,
ho ho M, = M - = MO'f~~=,;' He I H 2+H2 V el.
(6.65)
where ho is the component of the effective field in the z-direction:
110
== (Ho - w/-y)
(6.66)
Notice that if one does an adiabatic demagnetization exactly to resonance, the value of magnetization ~=~
HI /H2 +H2
V
I
0~
L
will persist indefinitely as long as HI remains on (actually, when relaxation to the lattice is included, it decays, but on a time scale typically of order T I ). We contrast this prediction with our earlier conclusion from the Bloch equations that M:z; decays to zero in T2 , where T2 ~ I/-yH1•. The factlhat M does not shrink as long as He is kept constant, and that M precesses in step with HI is often is comparable 10 M:z; described by the graphic tenn "spin locking". If will be less than Mo. However, the "loss" of magnetization is recoverable. Were one 10 go back off resonance, M would grow back to M o when H;:» Figure 6.5 shows (6.67). When Redfield proposed his theory. the fact Ihat the spins were locked to He was one of the most surprising results. It is, of course, nOlhing but the rotating frame equivalent of the statement that the magnetization in the usual laboratory frame adiabatic demagnelization has a one-to-one correspondence with Ho, as expressed in (6.25). Note that if one pulses on Ho when (Ho -w/-y):»HI and H t • and then changes H o so thai one passes through resonance, continuing until one is far on the other side of the resonance (Ho - w/-y negative), one will have turned M antiparallel to Ho. Moreover, although near to resonance, one might have M < Mo (if HI < Hd; by the time one is far from resonance one would have M '";;£ Mo· This experiment provides a simple means of lllrning over the mngnetization. One can see from (6.64) that if one demagnetizes exactly to resonance, the magnetization will be the full Mo provided 1I I :» HI.. TIle resonance signal is then as big as can be achieved using a 1r/2 pulse. If HI ~ HI,. one will not
lIf
El,
Hl.
236
fl i . fI, )'ig.6.S. The transverse nmgnetization At" produccJ in a demllgnetization experiment in the rotating frame, as a runction of the strength or alternating field If, employed. Ih is assumed to be turned on when If a is we11 off resonance, with the slImple initially at its thermal equilibrium magnetization {\'fa
achieve the full magnetization at resonance. Were one to reduce HI slowly to zero after arriving at resonance, M would shrink to zero. In this manner all of the order represented by Mo prior to demagnetization would have been put into the dipolar system, thnt is, into alignment of spins along their local fields. One could, at a later time, slowly turn H t back on again, thereby recovering the magnetization. 6.6.2 Sudden Pulsing A situation frequently encountered in experiment and interesting to contrast with adiabatic demagnetization is the effect of a sudden change in He. We treat an especially simple case, that of suddenly turning on HI at time t = O. We assume that before we tum on HI, we are off resonance by an amount ho, with the system initially magnetized to Mo along Ho. We utilize the ideas of sudden changes discussed in Sect. 6.4. The sudden tum-on of HI is so fast that the system has the same wave function or density matrix just after turn-on as it had before. The dipolar energy Ed which depends on the relative orientation of spins is thus the same at t = 0+ as at t = 0-. Moreover, since the dipolar Hamiltonian 1t~ is the same in both laboratory and rotating frame.
_ C1I2 H2 Ed = -~ = -Mo.::.L. 6] Ho
(6.68)
where 61 is the lattice temperature. The Zeeman energy is
Ez = -M·He = -Moho
(6.69) 237
The tolal energy. E = Ez + Ed, is then
E = -Mo{ho + HVHo) ~ - Moho
6.7 The Approach to Equilibrium for Weak HI (6.70)
A long lime later a spin temperature will be established together with a magnetizalion M parallel 10 He giving
E=_C(Hi;Hl) =_M(HJ;eHl )
(6.71)
But the lolal energy is conSClVed once HI has been turned on, so that, equating (6.70) and (6.71), we get
M = Mo H~O:~2
(6.72a)
L
o
Hillo
M z = NIo H2
H2
(6.72b)
M: = Mo H2 : H2
(6.72c)
0+
L
,,2 o
L
This equation shows that exactly at resonance, M would vanish. The null is vel)' sharp, M:t varying linearly with ho, so that observation of the null provides a precise method of observing exact resonance. II is interesting to contrast these results with conventional saturation theory. For a spin! system, M: is related to the population difference n by the equation /'fm
~-~~ 2 Saturation theory says that the equilibrium population difference, assuming infinite Tl' is n = O. hence M z = O. Equation (6.72c), on the other hand, says M z will be zero only when saturation is perfonned exactly at resonance (ho = 0). Conventional saturation theory assumes the transverse magnetization vanishes as well as M z . Equation (6.72b) shows that in general, M:I; is large, and may in fact be larger than M z . If (6.72) has a simple geomelTical meaning. After H] is pulsed on. M precesses about He. The component of M parallel to He cannot decay without energy exchange to the lattice. but Ihe componenl perpendicular can decay since Ihe local field gives a spread in precession frequencies. Thus, after several times l/-yHL' M will be parallel to lIe. and will have a magnitude given by the projeclion of the il'lilial Mo on He.
Hi:» Hl.
We saw in Sect. 6.5 that standard perturbation theory predicted that following the turn-on of a weak HI the population difference 11 would go to zero for long times, although we recognized that we could not rigorously apply perturbation theory to times greater than l/W. The requirement of a weak HI was necessary in order that perturbation theory be valid for at least short times. Of course, since M z is proportional to n. this implies M z would go to zero. In Sect. 6.6, however, we saw that M z would, under thesc conditions, go to an equilibrium valuc Mz)equil = Mo , 2
10
il 2 °H 2
+
(6.74)
L
fIr
where we have assumed Hr« 116 and «: HE (although we note that the equilibrium expressions in Sect. 6.6 were not limited to weak H J). It is therefore clear, as we suspected, that for fong limes, perturbation theory does flat give correct predictions. For short time intervals, however, it must be correct. Recalling the proportionality between M. and n, we know that
d~~,
_ -2W(w)M,
(6.75)
for times short compared to I/W(w). How can we describe M z for longer times? The solution to this problem was worked out by Provotorov [6.21] in an elegant paper utilizing powerful techniques. Rather than outlining his analysis, we will give an altemative derivation of his result. We note that in the absence of HI, thc Zeeman interaction in the rotating frame is just
1iz = -Iflhol z
(6.76)
Let us make the assumption that we can assign a tempcrature 0z to this Zeeman Hamiltonian, and 0d to the dipolar This assumption may not be rigorously correct, bUI it is al least simple, and corresponds to the facts at the time HJ is turned on. Immediatcly before turning on H] the dipolar system is at the lauice is the samc in thc rotating or laboratory frame. In temperature 01, since rhe laboratory frame we have
'Hl
1t3
CHo
Mo=-8,
(6.77)
But in the rotating frame
Clio Mo=-8z
(6.78)
Inasmuch as Ho ~ h o , Oz «0,. Thus the Zeeman temperature in the rotating frame, 0z, is very cold compared to the dipolar temperature tJ d . Turning on II] couples the two reservoirs and they approach the final equilibrium value given by the analysis of Sect. 6.6. The coupling of HI produces lTansitions between the
°
238
239
energy levels of the Zeeman and dipolar systems which we assume are governed by simple rate equations for the population of the various slates. (This assumption is similar to our postulates of Sect. 5.2. It is quite common in all cross-relaxation calculations. Provotorov makes it implicit in his work when he evaluates the relaxation times.) Since there are many states, a large number of coupled rate equations similar to (5.13) result. As has been shown by Schumacher [6.22], when the two systems are char· acterized by temperatures, the many equations reduce to twO coupled linear rate equations, one for (1/8z), the other for (I/8d) much as the many equations represented by (5.13) reduce to a single rate equation (5.27). But the conservation of energy gives a relationship between 8z and 8J :
Ch 2
CH2
8z
8d
___ 0 _ ~ ::::: const
(6.79)
Equation (6.79) is a first integral of the coupled equations, so that one of the resultant time constants is infinite, and a single exponential results. Since M, oc 118z, this means M~ relaxes according to a single exponential towards its equilibrium value. Using the fact that M, ::::: Mo initially, we get an equation for M, as a function of time,
M" - M,,)equil::::: (Mo - M~)equil)e-t/T
(6.80)
The only unknown in this equation is 1". We can, however, easily calculate il as follows. Taking the derivative of (6.80), evaluating it at t : : : 0, and comparing with (6.75) (which must be valid initially where perturbation theory is correcl), we get
~ T
Mo
:::::2W(w)
Ala
(6.81)
Mz)equil
Using (6.72c) for M')equil we get
2
2
2
_I ::::: 2W(w) h 0 +HL::::: 7(72 H2 (h 0 +H2) II 9(W) 1" H2 IH2 L
(6.82)
L
This result is the same as that firs! found by Provotorov, as indeed it must be since we have made the same approximations as he. The complete time development of the magnetization is therefore
M,::::: 2Mo 2 ho+HLexp -1r"f HI2(h~+Hf) 2 y(w)t )] ho+H L HL
[2
2 (
2
I
Slope b,OW" I,om f'tnuroutioll tlltory
". Filial rolue gi~fm by Uedfield O,eo,y
, Fig.6.6. M. "eUIl! t during saturation ~ given by the argument in ~he te,,~. The equlltion ror M. 115 Il runction or lime is round by joining lhe initilll region, where dM./dt is known rrom perturba~ion theory, to ~he equilibrium "alue, give" by Redfield theory, using Sc1wlllochu's observation lhat ~he approa the equality of dipolar and Zeeman temperalures in the laboratory reference frame implied inequality in the rotating frame. Spinlattice relaxation will allempl 10 bring the two temperalUres together at the lanice temperature in laboratory frame in a rime TI. On the other hand, the presence of H I attempts to equalize them in rotating frame. The two tendencies are in conflict. The slrength of the tendency is indicated by the corresponding relaxation times (a short relaxation lime means a correspondingly strong tendency for the associated equilibrium). Thus a spin temperature will be established in the rotating frame only if 1" is much less Ihan Tt. Thus we find that a spin temperature is established in the rotating frame and that we muSt employ Redfield's approach provided
T, 1 -:» T
(6.83)
This expression is remarkable since it involves the successful integralion of the equations of motion well beyond the time (l/W) for which perturbation lheory is usually valid (see Fig. 6.6).
240
".
1r"'(2 Hr
or
(h~
;r
l )9(W)T1 ,. I
(6.84)
This is almost exactly the conventional condition for saturation. Note that the longer T I , the smaller the HI which will satisfy (6.84). In particular, frequently (6.84) is satisfied even when HI , and Te , must be expected to push the spin system out of thermal equilibrium in the rotating frame, but the internal couplings of the spin system counteract that trend by producing energy exchanges within the spin system. Changes within the spin system must conserve the total energy of the spins. If then we consider the rate of change of the expectation value of energy, E, we can ignore any changes which simply redistribute energy within the spin system. and keep only changes associated with the lattice. Using the fact that
dt
I = -(M", - M)
(6.9Oa)
Ttl
and for the spin temperature 8
(6.9Ob) (6.87)
where To and T/) and T e are relaxation times corresponding to exchange of energy with the lattice. and where (1t3)/ is the value of (1t3) when the spin temperature is equal to the lattice tcmperature. We have used partial derivative signs in these equations 10 emphasize that they represent the changes in these quantities induced by lauice coupling only. Thus. although Ta is the usual TI. T/) is lIot the usual T2 (::::: Il-yH t ,}. Tb is gcnerally of onler TI, a much longer time. A good analogy to these equations is to think of a Boltzmann equation in.
1 We do nol worry !Iboul Ai, since its relAxlllion does nOl c:hange lhe el1crgy in lhe rol.llling frame (in essenc:e we llSSume At, = 0).
where we have introduced a notation Tie and where
M
_ cq -
MoHerr{hoITa ) (hyTo ) + (H?ITb) + (HlJTc )
(6.910)
"
2 2 I (h H2 +:.:J. H ) _I = .::2. + _, Tl e h~+H?+Hl To Tb Tc
(6.9Ib)
[We have here neglected the term (1t~)1 of (6.87)). Note in particular that Mcq := 0 exactly at resonance, is poslUve when Ito";> 0 (i.e., M is parallel to He). but is negative for 110 < 0 (that is, M is antiparallel to He)' The last case corresponds to a negative spin temperature in I
242
243
the rotating frame. Equation (6.91a) shows that the equilibrium 8 is far from the lattice temperature 8/ and may even be of the opposite sign. Since Mo is inversely proponional 108/, it is still true Ihat 8f detennines 8. even Ihough they are quite different. The negative temperature one sometimes finds is a simple manifestation of the fact that M;: always tends to be positive whether ho is positive or negative. In fact. one can say that the equilibrium is reached as follows: The strong internal coupling of the spin system (which guarantees a spin temperature) keeps M along He, since Curie's law is a vector law. The lattice is anempting (a) to make Ihe z·component of M be Mo. but (b) the xcomponent be zero. (a) would make M bigger than MO so that its projection on the z-axis is Mo, whereas (b) would make M be zero. The lauice is thus fighting itself since (a) and (b) are inconsistent. The equilibrium value of (6.91a) results.
6.10 Spin Locking, T 1g, and Slow Motion As discussed above, and shown in Fig. 6.7, a spin temperature in the rotating frame is established in a rime T from an arbitrary initial condition wilhoUl exchange of energy with the lauice. But this is only a quasi-equilibrium value since. over the subsequent time TIL" the spin temperature changes as energy is exchanged with the lanice to drive M to M equil of (6.91 a) (see Fig. 6.7). During this process M lies along HeW' Thus if one starts at resonance having oriented M along HI, M will not decay in a time characterized by the inverse of the
M
line width, but rather with the time Tt, which requires energy exchange with the lattice. In principle. by going to low enough temperalUres one should be able to make T t , as long as one pleases. If one does this, magnetization along HeW in the..rotating frame. following at time T to establish a spin temperature, will remain without decay for as long as one has chosen to make TIL" This time may be seconds in metals or even hours in insulators. Even though we can lock the magnetization along Herr for a time T, when H] is on, if we remove H, suddenly, M wUl decay 10 zero ill a lime oi order of the inverse line width. That is, the Bloch equations with the usual meaning of T2 give a rough qualitative description of what happens. In contemplating the Redfield theory, it is helpfUl to go back to the SilUalion of the laboratory frame without alternating fields. There we see that smning with a magnetized system we can tum Ho to zero slowly and later tum it back up to its original value. When Ho = O. M = 0, but the full M is recovered when Ho is lUmed back on, all without exchange of energy with the lattice. In zero field the order is manifested by the preferential alignment of nuclear moments along the direction of the local fields of their neighbors. The ex.istence of order in the local fields is the basis of a technique [6.13] for observing motions which are much too slow to be seen as a T, minimum or a line·narrowing. Consider a nucleus #1 with a neighbor nucleus #2. Suppose now that #2 makes a sudden jump. as in the process of diffusion. The duration of the jump is perhaps 10- 12 to 10- 13 , very fast compared to nuclear precession frequencies. Thus the local field at #1 arising from #2 changes suddenly in both magnitude and direction. The orientation of the spin of nucleus #1 is thus somewhat randomized relative to the local field. If the mean time a given nucleus sits between jumps is T m , the alignment of nuclei in the local fields of neighbors, that is the ordered state, can persist only a time of T m . Thus to carry out a full demagnetiz..1tion and remagnetization cycle with full recovery of the initial magnetization, we must remain in zero field for a time less than T m. Of course even were there no jumping, any T] process would change the entropy of the spin system. so that in any event we must complete the cycle in a time less than T,. We can conclude i)
Vatu .. glvclI by (6.12a)
~
ii) Votu .. glv,," by (6.9Ia)
T~,1
Fig.6.7. The hierarehy of times followin& sudden tum-on of If l
• During time T, a spin temperature is established in the rolating frame, the value of which is determined by the expectation value of the ener&y, E. immediately after tUMIing on 11\. Over a longer time T 1I. the spin temperature in the rotatin& frame challSes, changing /If correspondingly in response to ener-gy transfer with the lattice. Both processes require Redfuftr s spin temperature hypothesis to analyze sina: we have M5umed T
!
=t
251
++--
++--
The resul/ of using the more general form offunction of (7.8), whether as a result of solving the simpler HamUlonian of (7.1) more exac/ly, or as a result of solving the more general (7.7), is thai, on application of an applied alternating magnetic field, transi/iolls other t!lan those shown in Fig.7.2 become possible. We adopt the convention of calling transitions other than the four in Fig. 7.2 "forbidden transitions" .
--+-
--+-
-+--
In the absence of applied alternating fields, populations of the energy levels of the combined spin system are given by the Boltzmann factors when the system is in thennal equilibrium. As discussed in Chapter I, the achievement of thermal equilibrium can be thought of as resulting from lransitions induced by the coupling to a thennal reservoir in which the thermally induced transition probability We'l,c,y from state 1!"'1) to state Ie'l') is related to the rate Wc"'"e.,
-+-(a) DisrulI/
(h) N"urb)' 'IIICIt-,ts
IIlIdt'('$
Fig. 7 Ja b. The cITed of lhe size of the electron-nucle;o.f coupling II/h relative to the nucle'at ~nallce frequency Col on Lhe appearance of the energy level dillgn"us. For nuclei far from the dectron, lJ$ually iAI < h'.h 110 For nuclei ne~r. to the electron, .rrcquen~ly IAI > b.h/fol. The figu~ ll$IIumes a n~"tive 1'n and A positIVe. What would It look hke for positive "fa?
I·
by
~w~ W34, W 43) and a combined nucleus-electron spin flip (W23. W32) such as one obtains from the Fermi contact interaction in a metal as explained in Sect.5.3. An applied alter· nating field induces transitions of the electrons between levels I of II p"ir of nuclei (posilive .,'5). An IIpplied allernalins field induces lhe transition 1 to 2 and 3 to 4 in which rnJ remains fixed, but ms chanses. The lransition rllte IV ill defined in lhe text
l---.·
numbered 1. 2. 3. and 4. We assume that we are irradiating the S transition. producing a transition probability per second of W between states 1 and 2 and between 3 and 4. To find what this does 10 the [-spin polarization, we calculate (It) given by
,
(It) =
L
(il[:li)')i = ~(Pl + P2 - P3 - P4)
(7.3Ia)
.=1
In a similar way (S:) = !(PI + P3 - P2 - P4)
(7.3lb)
To find the Pi'S. we must solve the four rate equations for the populations Pi (i = I to 4)
dp,
dt = PI WH + P2 W 24
+ P3W34 - P4(W41 + W,n + W 43)
+ 2 and])3 = P'I> (7.31) becomes (7.38)
Utilizing (7.34) and (7.35) we can get an exact expression for F and G. However, in the case of a liquid. the high temperature approximation is valid even for electrons (perhaps liquid He is an exception!) so that Bij = I + (Ej - Ej)/kT
giving =
_I_{ 2kT
(7.39)
(7.40)
1- -) to
1+ -)
(7.41)
As we shall see, W42 = W31, so that .
(7.43)
and recalling that
B = -t(S- I+ + S+ r)(1 - 3 cos 2 8)
E = -i(S+ J+) sin 2
ee- 2i6
~~ (/6) ((1 -
3 cos' 8)'),.(+ -1[+5-1 -
+X- + Ir5+1 + -) (7.44)
to 1+ +) in which the I-spin flips up, W 42 is the transition from 1- -) to 1- +). also a transition in which the I*spin flips up. Finally W32 is the transition from 1+ -) to 1 - +) in which the S-spin flips down and the I-spin flips up. We shall define a new notation which makes these processes more explicit by defining U/l1 where M is the totnl chnnge in mr + ms in the tmnsition:
Ul(W) = W31(W)
k)2 T ;
For dipolar coupling, the 'H.1(t)'S represent the various tenns A. B, C, etc. from (3.7). Defining
Uo =
Examination of Fig.7.7 shows that W41 is llie transition from
+) in which both spins flip up. W31 is the uansition from
2Tc
+ (m
we get that
E
W"I(E4 I ) + W31(EJ - E I ) 2(W41 + W31 + W 42 + W32)
+ W 42(£4 - £2) + W32(£3 - E2)} 2(W"1 + W31 + W 4 2 + W 3 2)
1+
= Gmk(O) h2 I
Ao = "'IJ1sh 2/r 3
IG-F (I,) = (P, - p,) = '2 G + F
(I,)
we get from (5.297)
where «(1 - 3 cos 2 8)2)" .. means the average over the 411" solid angle of (I 3 cos 2 8)2. Thus «1_3cos2 8)2)"
•
=~J(I-3COS28)2sin8d8d¢=4/5 4.
(7.45)
Likewise (sin 2 8 cos 2 (J),,'/I" = 2/15
(7.46)
(sin" (J)o\?r:::: 8/15. In this manner we get Uo =
A~ -.!....
U2 =
h2 5"
(7.42)
Tc
11.2 10 1 +(ws -wr)2rt
A33
Tc
I +(wr +wS)2 Tt
(7.47) 261
The resulting polarization of the I-spins is thus, using (7.40), (7.41), and (7.47),
Now, in thennal equilibrium we have from Pj(T) '" PiO')B ij thai (7.55)
Pj(T) - ]>;(1') = Pi(T)(Bij - I) = Pi(Tkij
(I,)
Therefore, in (7.54) we recognize that since Pi (7.48)
';:!
Pi(T) S!'
t
W 4 1P\e14 S!' W 4 1[P4(T) - Pl(T>]
(7.56)
Substituting into (7.54) we gel Suppose
Iwsl »wl (as for S being an electron spin, in which case ws is negative)
and thai W~7";
«: I. Then
(I ) = ''"'5 (3/5 - 1/10) = ~ "W5 z 41.:1' (3/5 + 1110) 7 41.:1'
(7.50)
which for negative Ws is posltlve. Therefore, dipolar relaxation produces an Overhauser effect of the opposite sign to the conventional Overhauser effect since for dipolar coupling U2 »Uo. We have carried out the solution for the steady-state polarization of the 1· spins. If one is doing pulsed experiments, as is frequently the case for double resonance experiments or for two-dimensional Fourier transform experiments, the pulses disturb the 1-S spin system from thermal equilibrium, following which the thennal processes bring the system back to thermal equilibrium. One then speaks of a transient nuclear Overhauser effect. Thus, in general, one wishes to find the time dependence of the observables (I~(t) or (S~(t) for some sort of initial conditions. Solomon, in [7.7J, calculates the transient response and demonstrates what happens in a set of classic experiments. To do the calculation, one needs to solve for the time dependence of the Pi'S, starting with equations such as (7.32) with W = O. Thus we have
dp4
dt =PJWI4 +P2 W 24 +]J3 W 34 -P'I(W41 + W"2+ W"3)
(7.51)
- Pol -
[PI (T) - 1'4(T)j}
+ W",{)" - P4 - [p,(T) - p4(T)j} + W",{p3 -Pol - U'3(T) - 1',1 (T)]}
(7.49)
which is negative for electrons. On the other hand, if there were a strong Fenni contact tenn, so that Uo »UI or U2, as with the conventional Overhauscr effect,
( I ) = _ lIws z 4k1'
dP4 dt = W ll {PI
with similar equations for dpddt for i = I, 2, and 3. We saw in Chap. I that the fact that Wij oF Wj; is important in producing a thermal equilibrium population. That is the physical significance of (7.56). Clearly the family of equations represented by (7.54) describes a system which will relax to the thermal equilibrium populations p/T). We can then replace the Wij'S by the Um's, taking
Uo
W32 s:' W23 =
W42 s:' W2.1 = VI = W3l S! W I 3 W31 s:' W l 3 = U2
and introducing
Ui
(7.58a)
by Jhe relations
WI2 s:' W21 ==
ui = W34
s:' W"3
EI
cl" =
where
~E4
kT
relationships, one can now show Ihal d dt (Pl +!J2 -1'3 -1',,) = 2UI [(pol -]>2) + (P3 - PI)]
+ 2UO(p3 - 1") + 2U,(P4 - P'> - 2U I [PI (T) - P2(1') + P3(T) - PI (T)] - 2Uo[P,(T) - p,(T)]- 2U I [P,,(T) - p,(T)]
(7.52)
From (7.31) one can show that
Therefore (7.59) can be rewritten utilizing 10 and So to represent the thermal equilibrium values of (I~) and (S~) as
(7.53) = [10 _ (1,)](U0 + 2UI + U,) + [So - (5,)](U, - Uo) dt In a similar manner, we get
d(I,)
dp4 dt = W 41 (PI - P4) + W 42U>2 - p,,) + W 43 (1'3 - P4)
,,,
(7.59)
(7,60)
we get
+ W41PIC14 + W42P2C24 + W43P3C34
(7.58b)
U: differs from UI (7.47) by the subslitution of Ws for WI. Making use of these
Writing Wu = Win-nil> only are lhe only sip;nilicallt lhermlll PI"OCes6e$
7.6 Polarization by Forbidden Transitions: The Solid Effect In order for the Overhauser effect to work, the nuclear relaxation process (such as W13, W24) cannot be allowed to short circuit relaxation in which both an electron and a nucleus flip, such as W23 or W!1. It is not always possible to meet those conditions. The condition which one can, in general, be quite sure will hold is that pure electron spin-flip processes (such as W 12 or W34) are much the fastest W;j's because electrons couple more strongly to tile lanice than do nuclei. Jeffries [7.8] and independently Abragam et al. (7.9) recognized that the so-called forbidden transitions were not strictly forbidden in many useful cases, and that one could use them to good advantage in achieving polarization. In fact, Erb et al. (7.10) independently discovered the effect experimemally. Using this scheme, Jeffries and his colleagues at Berkeley, and experimentalists at Saclay collaborating with Abragam. have obtained proton polarizations of over 70%, and have made a rich variety of applications. Invention of this technique was another major step forward. The phenomenon is often referred to as the solid effect. There are two possible forbidden transitions. They are shown in Figs. 7.8 and 7.9. The transition of Fig. 7.8 can be induced by an alternating field parallel to the static field when the simple isotrOpic electron-nuclear coupling of 0.1) is solved to the next higher order to include the effect of the tenn AUz 5 z + IySy ) in admixing 1+ -) with 1- +). The transitions of both Figs. 7.8 and 7.9 can be induced by alternating fields perpendicular to ]fo when the more general Hamiltonian, (7.6), is solved to adequate precision. For example, the dipole-dipole coupling between a nucleus a distrance r from the electron makes the transition matrix elements of Fig. 7.8 and 7.9 of order 'Yefilr 3 Ho times the matrix element of Fig.7.6 in which only an electron is flipped. The ratio also depends on the angle the static field makes with the axis connecting the nucleus and the electron. (The effect of the dipolar coupling is expressible more precisely in terms of the contribution of the dipoledipole coupling components AZ'Zl, Ay'y" and A,,'z')' 264
FIg. 7.'). A rorbidden transition 1Iltemalive to that or Fig. 7.8, which producl'3 Iloclear polarization or the opposite sign
Even though the transition probability Wen may be small, it is frequently possible to achieve strong enough alternating magnetic fields to make it larger than the thennal transition rates Wij in which a nucleus flips, so that the transition connected by Wen produces effective population equalization. Lei us analyze the case of Fig. 7.9 for which the transition between states I (-IPt = 1+ +)] and 4(IP4 = 1- -)) is saturated. Since we assume Ihe only thermal transitions of consequence are those shown, we immediately can write down 1'1 = P4
(7.62,) (7.62b) (7.62c) (7.63,)
(7.63b)
265
P3 :
~~B::,4",-J~_
(7.630)
2+8 12+ B 43
relative to the donor atom. The solution is a straightforward generaljzation of that of Sect. 7.3, using the N quantum numbers mi, the eigenvalues of fZi' ..E = '1ehHOmS +
Using (7.2Ib),
) 1 812 - B modulation amplitude, sweep excursion, etc., to the optimum. A corollary is when looking for an unknown, be willing to set the equipment to give ~ maximum signal even though you may thereby distort the resonance line (for example, from using too large a modulation, or from panially saturating). After the resonance has becn found you can focus attention on it with parameters adjusted to avoid distonion.
7.10 Cross-Relaxation Double Resonance No mailer what one does to improve one's apparatus, one eventually reaches the limit of the current state of experimental an. What then if the signal is still too weak to see? We tum now to the use of double resonance, assuming there are two resonances which can be excited, one the "weak" one too difficult to see directly, the other a "strong" one observable by conventional means. For example, the weak one might result from a low abundance species with spin S, whereas the strong one might result from an abundant species with spin I. (For simplicity, we use the term "rare" to imply the species whose resonance is weak. Since a low ""( could also make the resonance weak, "low ""(" can be substituted for "rare" in most cases.) If an H I is applied at resonance to the rare species, those nuclei absorb energy and their spin temperature rises. If the rare and abundant spins could exchange energy, the abundant spins would thereby get holter, and their signal amplitude would diminish. If the abundant species is thermally isolated from the outside world (i.e., has a long TI), this temperature rise can be made quite large and thus readily observable merely by silling on the weak resonance long enough. By going to low temperatures, TI of many systems can be made exceedingly long. Thus the crucial question becomes how well can the two species exchange energy. The mixing of the Iwo spin systems was sludied by Abragam and Proctor [7.17]. They employed the Li 7 and FI9 resonances in LiF. Their experiments were part of their studies of the fundamentals of spin temperature and of adiabatic demagnetization. Working in a field of several thousand gauss to observe the resonances, they prepared the system in some nonequilibrium state in the strong field (for example, they inverted the F I9 magnctiz..'ltion), removed the sample to a lower static field, allowed the Lj7 and FI9 spins to mix, then returned the sample to the strong field for inspection of the Li 7 and F19 resonances. In this way they found that in a field of 75 Gauss the spins come to a common spin temperature in a mixing time of 6s, that the time was unobservably shon at 30Gauss, and longer than the TI 's (several minutes) above lOOGauss.
271
A detailed study of cross-relaxation was made by Bloembergen et al. [7.18] and by Pershall [7.19] for both electron spin systems and nuclear spin systems. They show that the crucial problem is the failure of the Zeeman energy to match when two different nuclear species undergo mutual spin Aips. The mismatch in Zeeman energy must be made up by the dipolar coupling between the spins. The simplest sort of process arises when two nuclei have nearly the same 'Y. For example, HI and F I9 have 'Y's which differ by 5%. Owing to the existence of tenns such as S+ I- or S- 1+ in the dipolar coupling [the B tenn of the dipolar Hamiltonian of (3.7)], the dipolar coupling couples states in which the proton spin flips up (down) and the fluorine spin flips down (up). If we could consider the 'Y's as being identical, we would then have a situation such as shown in Fig. 7.12. Because the individual energy levels match, the initial state of the two spins, indicated by the two x's, has the same energy as the final state indicated by the two o's. A system started in the x state will undergo a transition to the o state by means of the S+ I- part of the B term in the dipolar coupling. For HI and F t9 , the fact that the 'Y's differ by 5 % means that in a strong magnetic field such as 10 kGauss, the Zeeman energy difference between the x and the o states would correspond to the energy of FI9 or HI in a field of 500Gauss. Such an energy mismatch would prevent the transition unless there were some other energy reservoir whose energy could change to make up the difference. A possible candidate is the dipolar energy reservoir. For a pair of spins, it has a typical value of 'Yl'YSh2/r3, which, expressed in units of magnetic field, is only a few gauss for reasonable values of r. Thus, it cannot make up for an energy mismatch of 500Gauss. If, however, the static field were much lower, the mismatch would be correspondingly reduced, and the mutual flips might become possible! Returning to LiF in a field of IOkGauss, the Li 7 resonance occurs at 16.547 MHz and the Ft9 resonance occurs at 40.055 MHz, a ratio of 2.420. Clearly the mismatch here is even worse than for H t and F t 9. What Bloembergen and his colleagues recognized [7.18,19] was thai 2.4 is close to 2.0, hence a process in which two U 7 nuclei flip up and one F I9 nucleus flips down comes much closer to satisfying energy conservation than a process in which only one Li7 flips. Now, Li 7 has a spin of ~' but for the sake of argument we are going to pretend it has a spin since the explanation is then simpler. To estimate the rate, one must utilize a fonnula for transition probabilities such as (2.182) (with
!'
Ms
=
M l = -1/2
-1/2
M s = +1/2 Spccies 5
272
Specic" J
]c lhe same for lhe two spc where HL is some sort of local field. Then the I-spins. if exactly al resonance, are quantized along the I-spin x-axis, the direction of (H I )/. and the S-spins are quantized along the z-axis, the direclion of Bo_ The term 'Hd/S • given by (3.55), involves products such as I;k5zp. which have matrix elements which are diagonal in Szp. bUI are off-diagonal in J;J;p- However, owing to the spin-Ianice relaxation time of the S-spins, the diagonal matrix elemenlS of Szp are functions of time with a correlation lime of (Tl)S' Defining
we can set
Ta ! = (T1 )! for the S-spins we will have. in analogy to (7.132),
aMzS Mos - M"s --/)t
HE
Ckp=
"'fnSh2
3
rkp
2
(I-3cos 6kp )
(7.133)
-
with
TIIS
(7.134)
and
(7.135)
oMzs Mzs ~=-TbS
(7.136)
where TbS ...... (TI)S. We will have three dipolar relaxation equations similar to (6.87):
~(7i )_ fJt dfJ -
(7.128)
( 7idll) T,11
(7.137a)
BlQembergefi and Sorokjfl show that
_'_=.2..l: C 2 (TI,,)1
1i 2 k,p
5(5+1)
kp
3
(T,)s
I +wrl(TI)~
(7.137b) (7.129)
The reader should compare this expression for the relaxalion of (Iz ) along (HI)z with the expression for TI of (5.372) which applies to the relaxation of (I,,) along Ho. Physically, the fact that the z-component of the S-spins is fluctuating produces a time-dependent magnetic field on the I-spins along the z-direction. A field of this polarization is ttansverse to the I-spin quantization direction, and can therefore relax the I-spin magnetization along its "static" field in the rotating reference frame. The factor TI/(I +wr,Tls ) gives the spectral density at the frequency WII which can produce flips of the I-spins quanlized along (HI)!. If (HI)s is turned on to a level sufficiently strong that the $-spins obey the Redfield spin-Iemperature condition in their rotating frame (6.84), then the magnitude of the S-spin magnetization, Nls, and of the I-spin magnetization, M" will be given by an argument such as is given in 5ecI.6.9. The calculation we described above for (Tle)l can be viewed as a calculation of Tb' defined by a modified (6.86):
z / = - M:z:d oM-
at
'I
w'lh ,
( 7idlS)
where we have neglected the tenns like (~) of (6.87). If we assume that (Tt)s (HI )e. to the Hartmann~Hahn value. Within a cross-relaxation time, Ihe C I3 will be polarized. Since there are many more hydrogen atoms than C 13 nuclei, in this process the HI spin temperature will hardly change. Therefore
Me
-- GdHlk
o
(7.152)
But
8 = Gu(H1h,
and
(7.153.)
Mil
1'dHde = 111(Hd"
Me = M u Ge Gn
so thai
(I'll) 1'e
(7.153b) (7.154)
Suppose. now, Ihal initially MI-I has ils {hennal equilibrium value in Ho
GIIHo 0, The corresponding MColarizcd with (a) "'5 = and (b) "'s vcrsua I-spin scar t1f
fOI
t
< t 1f
fOI
t
> t>r
(7.173)
These are shown in Fig.7.21. If there were more than one paicof spins I-S, each pair with its characteristic chemical shift, the situation would be more complicated, but the same principles would apply. Let us label each I-S pair by k. (k = I to N if there are N distinct pairs.) We shall assume the I-S coupling is only within a given pair. Then we have N Hamiltonians of the fonn Fig. 7.19. The y·componenl of the I-spin lllflgnetizalion versus time, t. Al t t", the S-spins are in\'erled by " 1f pulsc
=
1ik = -liWOfkIzk - flWOSkSzk
+ AkIzkSzk
(7.174)
so that, defining ak = Akl!l, 301
PhilSe iogln 4'lltl iod 4' l ll)
(7.176)
,
{Mj(O}""" = C;i l)exp [- i(!7 lk - (lk!2)t]
+ exp [- i(!7 lk + ak!2)t] Fig. 7.11. The phllSe
Fret/lIt'IIC)' of Clf 1800 pl,lse mell$lfred ,e/atil'e 10 Clfi) mllill line (k1I~)
• r-'.~12~.,----_----,~~.,---__~4~.,---~ 20
~ 1.'3
'."§ ~
j
• • • • •
......•
•
40
60
•
•
'ooL
.,:-__---,4~.~ ,, ,
..
.... ....
:
'•
.;
.'
•
o Spin Echo
.~ 35
~ 30
,
~
"";;25
~20
§ PoSitiol' of 0,·)
z
,
.~
•
..
•
o
•
0 0
0
~ ••
5
• " ' ,•
.L--'-£-,--L-c--J~~',---'-.-,+c__--'-~J. 1.07
1.08
U)9
mallllllle
---'---
• SEDOR
, :
: \
~ 10
1.10
1.11
1.12
1.13
1.14
1.15
1-10'*'0 (kG / MH%)
---.J
Fig. 7.30. Boyce's observation of SEDOR in • Cu powder containing O.~ at. % Co near the Cuu resonance fT«luence of pure Cu at 9901 Gauss and 1.5 K, showing the satellite due to the 1 .... _ 1 transition of lhe first neighbor. The parameters UBed are T = 250"" 1I is :::::5Ga:" for ~"If pulse of the Cu transition.. IOO echoes 'llere averaged for each point and the dots are larger th.n the S(:a-ller and dnft. Note that t.he ?"uch more abundant Cut3 nuclei a long distance from the Co, the SOotalled Cu malll hne, do nol show up in the Co SEDOR
t .... - t
would continue to dephase during the second interval, producing a smaller echo at 2T. An example of a resonance delected in this manner is shown in Fig. 7.30. Boyce [7.49-51) studied a dilute alloy of Co in Cu. He wished to observe the nuclear resonance of Cu nuclei which are near neighbors of the Co nuclei. For such a dilute alloy, the neighbor resonances are weak in amplitude. frequently hidden under the tail of the resonance of Cu nuclei which are distant from impurities, the so-called "Cu main lines". Boyce was able to reveal the "hidden" resonance of the first neighbor by doing a spin echo double resonance in which he observed the effect on the Co spin-echo amplitude of a 1r pulse applied over a sequence of frequencies close to the Cu main line. Since distant nuclei do not produce much field at the Co, the "main line" has negligible effect on the Co echo height. But the first neighbor has a big effect. In this instance the use of double resonance can be seen as a way of selecting pairs of nuclei which are close in space. Makowka et al. [7.52] utilized pt195_C13 SEDOR to detect the surface layer of Pt nuclei for small metal panicles of Pt on whose surface they had adsorbed a monolayer of CO enriched to 90 % in C 13 . The metal particles. which were tens of angstroms in diameter, were supported on A1203, a typical supported catalyst. Makowka et al. observed Pt'95 spin echoes. Figure 7.31 shows their data. The straighl Pt 195 spin echo gives lhe line shape of the Pt nuclei in the small metal particles. This line is over 3kGauss wide! To observe the Pt195 resonance of 312
, 4.
>-
g 15
•
• •
80
• 45 '2
Fig. 7.31. Measurement ofMaki:Jwla et at (7.521 of the spin·echo (0) and SEDOR (.) line shapes of Pt lU for a sample of small particles of Pt metal supported 011 J\12~ whose surface is coal.ed with C 13 0 molecules. The metal particles have diameters of a few tens of A. The Pt lU spin echo gives the totlll line shape of all the Pt ltS in the particle. The SEDOR data, involving pt 1U _C 13 0 double resonance, and lin add-subtract method, give the NMR line shape of the PllU nuclei in clo5e proximity to the CO molecules, i.e. the surfll.Ce layer of Pt atoms
the surface Pt atoms only, they employed an add-subtract technique in which on alternate spin echo cycles they applied a C 13 'If pulse coincident wilh the Ptt95 'If pulse. For those PI nuclei far from the C t3 , the echo was unaffected by the Ct3 'If pulse, whereas for the surface atoms, the Ptt95 echo was diminished by the C I3 pulse. Thus, subtracting the Ptl95 echoes when the C I3 pulse was applied from those when il was nOl, the signal from the Pt l95 nuclei not bonded to C t3 nuclei vanishes. To analyze the SEDOR signal, we nOte that the precession angle. 8, of the I-spins off resonance by (1I o)J during an interval T is
8 = (")'(ho)1 - ams)T
(7.200)
where (ho)1 represents lhe extent to which the particular I-spin is off resonance due either to magnetic field inhomogeneities or chemical shifts. Suppose, then, that at t = 0 we apply a 1r(2 pulse with (H')I along the I-spin x-axis in their rotating frame. This puts the corresponding magnetization M«ho)J) along the +y-axis (Fig. 7.32a). At time T- just before the I and S 1r pulses, 8 has the value 9, (F;g.7.32b):
8, = 80 =f a-r(2
where
(7.201)
80"" ,/(ho)iT
The I-spin 1r pulse reflects M«ho)J) about the x-axis, producing the situation shown in Fig.7.32c. During the next time interval, T, 8 advances an angle 82 (Fig.7.32d) given by 313
(a) t: O'
(b) t :T-
,
7i(I,5) = -'I"[(l>o),I, + (HI),I.) ~ l's rl[(h o )sS: + (HI)SS~J
,
M
M
e,
(7.205b)
" 7i(I) + 7i(S)
where by 1i(f) [1i(S)] we mean a Hamiltonian which is a function only of the spin components of the I-spins [S-spins]. If this Hamiltonian acts from a time t1 to a time t2, we have
, (e) t resenling a chemica.! shift only. (b) The ~hll.vior of. eqUlVlllent, for! > 21 .. , to that of (a). (e), (d) Replot.1ll olea) and (b) f('$pectively for .. value t~ >~". (e) A family of phue trajectorir$ colTesponding to a family of experiments with different times I. (t~, I~, de.)
321
IbJ
la)
0. ,, I I I
-IO~-bI21
--~-------""1-
-t QI,·bI21
-----------t-II
I
iI
---+------t--
iI
I
-!n I -aI21 ---~--
_ 0,
-ttl1u12 1
I
,, ,
I I
-Q l ~ -Q h -Q~~ Fig. 7.34. (a) The lwo-dimensional plol of frequencies WI and W2, 'lhowing poinL5 in "'I corresponding to the chemical shin WI = -(af ±a/2). (b) A plot similar lo (a) corresponding to two I-spins, each coupled lo a single S-spin with corresponding chemiCl\1 shift!! aid and nf6 l\nd spin-spin couplings a and b
iCO a 9(Wl.W2) = -2-5(wl + n!a){5(W2 + !1-la - 0/2)+ 5(W2 + n la + 0/2)] iCOb
+ T5(w, + n ,b)(5(W2 + n,b - bIl) +6(w, + [h, + WZ)J
17.238)
which would give {he plot of Fig.7.34b. This plot allows us {o go along the WI-axis {O detennine the values of the I -spin chemical shifts, then at each chemical shift position (o observe in the W2 direction splitting of those panicular I -spins by the S-spins to which the)' are coupled. That is, there is a correlation of the WI frequencies with the W2 frequencies. The physical origin of this correlation can be traced back to the phase diagram Fig. 7.35, which shows that the single phase line over the time interval 0 10 2t7: splits into two lines for limes after 2t7:' For t > 2t7:' those
lines extrapolate back to an intersection at t = 2t ... The two lines are diverging from their intersection since the chemical shift frequency fl/ is split into two frequencies. n/ ± a/2.. for times after 2t... The example illustrates the general feature of two-dimensional schemes that there is some initial preparation of the system [e.g. the X t (7f!2) pulse], then the system evolves for a lime interval tl under one effective Hamiltonian (in our case, one in which the spin-spin couplings are zero), then data are collected during a second time interval, t2. for which the Hamihonian lakes on some new effective Conn (in our case the full Hamillonian). 1lle experiment must then be repeated for other values of tl_ These three intervals are often referred 10 as the preparation, the evolution, and the acquisition intervals, respectively. The experiment we have just described is in (act essentially the famous experiment invented by Maller et al. [7.60] to resolve the C l3 chemical shifts and C I3 _H I spin-spin couplings. They demonstrated the technique using n-hexane (Figs. 7.35 and 36). They utilized broad-band (noise) decoupling of the protons instead of applying a "" pulse as in our example, and they decoupled during the second time inlerval. t2. instead of during the first time inlerval, tl' as we did in our hypothetical example_ They point out one could choose to decouple during either interval, and remark that since each point tl requires a separate experiment, one needs less data if the tl spectra is chosen to have the fewer spectral lines, Le. is the time during which the protons are decoupled. In a later paper, they describe the sequence of our example [7.61]' This general method is referred to as J-resolved 20 NMR in the literature. Qoli"CH;Qol;0i;CHrQol3 abccba
'"
.,
H'
". 0'.0
I,
I,
Fig. 7.35. Schematic representation or one ronn of C 13 2D-re;olved Spedroaoopy aner Miller ct al. 17.601. The CI3 nuclei are the I-spins, Iii thfJ S-spins. For the inlerval II the fullUamiltonian acts, but during t, the spin-spin interaction i, lUnled ofT by broad-band decoupling. The situation is similar to S-f1ip-only resonance with the role of the times t l lind t2 inlerdlllllged
322
Fig. 7.36. DlIla of Muller, KUI1t(Jr, and Ernsf using lhe pulse !lCquence of Fig. 7.35 for the CI3 spectrum of n-hexane (Clb-CII,-CII,-CII,-CII,-Clb). The ""I-axis gives the combined chemical shin and spin-spin spliuing, the loo'2-axiS the chemical llhirL5. Thesoe data show how a 20 display enables one to lIl""atly simplify unraveling a complex spectrom. TwentylW() experimenL5 Vooe"" coadded for each of the 64 values of tl between 0 and 35 IllS. The authors state lhat the resolution is seve~ly limited by the 64)(64 dab. matrill: used to represent thfJ 20 Fourier lransform. The absolute values or the Fourier coefficients a"" ploued. The undecoupled I D spe T
=+i
=
(I+(t}}'/2 =
~[(++ Iu lIT" 11T2. The Feher approach to coherence transfer using adiabatic passage can be replaced by a pulse technique, the method used by JUller [7.55J and by Maudsley and Ernst [7.67]. Although superficially (7.279) and (7.280) suggest that one is
334
335
performing an Overhauser effect, in fact, since the method does not use spinlattice relaxation, it is clearly based on different principles. We now show how to use pulses to invert one hyperfine line. Consider an WI tuned exactly to the unsplit I-spin resonance frequency Wo/. (7.281)
WI = WOI
AI t = 0 we apply an XI('If(l.) pulse (Fig.7.42a). The magnetization vectors of the mS and mS lines are shown in Fig.7.42b at t 0+. We label them MI(~) and MI(-!) corresponding to ms = and -~ respectively. They now precess in the left-handed sense at angular frequencies (WOI - amS) in the lab frame, or at -oms in the rotating frame. (This is equivalent to precession in the rotating frame frequency oms in the right-handed sense, see Fig.7.42c.) At a time, to, such that
=!
=-4
=
+!
oto = 1f
(7.282)
they are pointed opposite to one another as shown (Fig.7.42d) along the x-axis. At this time (Fig.7.42e), we apply a 'fro. pulse with (HI)! along the y-axis Ia Y/(1fn) pulse]. This puts the M I (+!) along the negative z-axis, and MI(-!) along. the positive z-axis (Fig. 7.42f). Thus, we have achieved a population inline, while leaving the ms -~ line with a normal version of the ms population difference. This is equivalent to an adiabatic passage across just t~e ms = line. There are two important points to n~te. The ~rst .is that t~e sphtting, 0, is essential since il is responsible for prodUCing the suuatlOn of Fig. 7.42d in which the two components of AI I are pointing opposite to one another. The second point is that the phase of the (H 1)1 pulses (Le. their orientation in the x-y plane) is important. If, for example, the second pulse had been another XI(1fo.), it would have been parallel to the MI vectors, and would have had no effect on their orientation. Likewise an X('lfo.), where X means (H')I pointing along the negative x-axis, could be used to obtain a state in which the MI(-!) would be inverted instead of MI(!). So far we have only inverted the ms = transition, therefore we have achieved the effect of the first of Feher's twO steps. To complete the coherence transfer, we must now do a similar thing for the S transitions_ Thus, with ws tuned to Wos, we can apply the sequence XS(1fo.) followed at a rime delay to = 'If/a by a YS(1fn) pulse to invert the Af, = transition for the S-spins. This would have completed Feher's coherence transfer and produced the populations in the matrix of (J .278). If instead of applying four pulses we apply the sequence
=+!
=
!
!
lal
L [H,.II
lei 11,11/21
Ibl~
Hl ll121
MI I-1I21
+i
"
-I
"
Idl
, ,,
"
t1 1 11/21
~
" (fl
"
(a)
Mil-liZ!
[Hll l
"
"
Fig.7.428-1. The effect of (lid! pulses tuned to resonance (WI = ,",od on the I-spin magnetization M,(ms) for the two components ms = and ms _1. (a) At I 0, (Ildl is applied IIlong the z-lIxis in the rotRting frame giving an x/('II'/h pulse; (b) I'll I = 0+, both At I (t) and M l( lie along the y-axis; (c) the two components M /(mf) precellS in o[.posite diredions at angular frequencies ±a/2; (d) at a time t 11:/0, 111'(2) and MI( - ~) point along the -z- and +z-axes respectively; (e) an (HI)I lying along the y-axis gives a Yl(lr!2) putse; (f) the I-magnetization vcdors after the Yt(,../2), with MI(-i) pointing parallel to the ~-dire pointing antiparatlel
+t
336
L (H,ls
1'1 1 11/2)
-!)
•
(7.283)
the I pulses will have produced a situation in which the M I (!) population is inverted, bUI not Ihe MI(-!). so the situation will look like Fig. 7.4Ia. Therefore the XS(1f(l.) pulse (Fig.7.43a) will produce a silUation in the rotating frame such
1'1[1- 112 1
" lei
X,(.12) ... (./a) ... (y/(./2). Xs(.12)) .. . observe S
/11.1-1121
=
=
Ibl
" Mi l-1I21
" Ms !1I21
(e I
=
"
"
~·Ig. 7.43. (a) (/fils is applied along the :I:-direr++. Now (mj_lg(r++)lml+) is prcx1uced by XS(7C!2), hence from elements of e{r+) which are diagonal in mi. At t = 0-, before the first Xj(wll) pulse, we assume f! is in thennal equilibrium, hence diagonal in both ml and ms. Hence, under the action of the two X j (1f!2) pulses, f! remains diagonal in ms· Thus, (ml-Ie(r++)lmj+) arises solely from elements of e{r+) just before the pulse XS(7C!2) which are diagonal in both ml and ms. These mamx elements can be traced back to elements (mlmsle(r-)Imlms) as well as to elements (=F msle{r-)l± ms) using (5.253):
(mjmsle
X S I (,...fl)T/f}B2 .
(7.331)
The effect of the XI(,...fl) pulse is to rotate I, into l y : XI(1rfl)1,X/I(1rfl) = 111
(7.332)
.
Next we apply TIS' Utilizing Table 7.1 we gel TlS(B 1)Iy T ls)(9,) '" I1/ cos 9)12 - 21z 5, sin BI12
(7.333)
Next we apply Xs(,...fl), a left-handed ,...(1 rotation of spin S, giving XS(,...f2XIlI cos 9 1(1- 2Jz 5, sin 9 1(1)X I (,...f2)
S
= I y cos 9i1l - 2Jz 5 11 sin 9 1f2
(7.334)
We now act on this with TIS(92):
s
T/s(B2)ly T/ 1(B2) cos 9 1(1- 2T/S(92)lzT/-s)(92)TIS(92)5yT'SI(~)sin 8,(1 = (1y cos 9212 - 2lz 5, sin e./.fl) cos 8,(2
- 2(1z cos 82(1 + 2l11 5, sin 9212)(511 cos fhfl- 21,5z sin 82fl) sin 9 1(2 "" 111 cos 92fl cos Blfl- 21z 5, sin 82(1 cos 8 1(1 - 2Iz S11 cos 2 8212 sin 9 1fl + 41z 1,5z cos 82n sin 82n sin 81n - 41y 5,5y sin 82n cos 82n. sin 8,n. + 8Iy 5,I, 5 z sin 2 82fl sin B,n.. (7.335)
We now have a number of tenns involving spin products such as 1a lp or Sa5{J. Utilizing (7.324), we reduce the various teons as follows:
.
(7.336)
The initial density matrix at t::: 0 is given by (7.210). Suppose we consider the ponion of e(O-) given by
~~~I,
Evaluation of Ihe portion involving 5, is left as a homework problem. 348
(7.330)
.
(7.328)
for a spin ~ panicle. Substituting (7.328) into (7.325) yields (7.324). Let us now illustrate how Ihese fonnulas are used. Suppose we analyze the 5-fHp-only double resonance for the case of a 7r{2 5 pulse:
e(O-):::
B2:::at2
•
X
plus cyclic pennutations. To prove (7.324), we nOle thai tenns such as l z l l1 are encounlered in the commutation relation
Xr(7r{2) ...
For simplicity, we consider a case in which il r ::: ils::: 0 (exact resonance for both spins). We apply an X r (7f!2) pulse al t '" 0, and an Xs(Tr/2) pulse at time tl, let the system evolve under TIS from 0 to tl, and after t1 for a time t2. We define B, and fh by
(7.329)
l y 1,5,5:I: =
'2 5 ('2i) 17; (i)
I.S 11 "'-~
When these are substituted into (7.335), the two tenns in 111 57; cancel, and the teons in IzS y combine (cos 2 8212 + sin 2 92fl = 1) giving 349
TI S(8 2)XS(1r(2)TIS (8 1)X1(7r(2)I z Xii (7r(2)T1S1(8 1)X5 1(7r(2)Ti!] (82) = I" cos 82/2 cos 8 1/2 - I z S z 2 sin 82/2 cos 8 1/2 - I z S,,2 sin 8tf2
. (7.337)
Therefore, the contribution of (7.329) to
e is
e(tl + t2) = hWOI (lz cos 82/2 cos 9 1/2 - IzS z sin 92/2 cos 9 1/2
ZkT - I z S,,2 sin 9d2)
(7.338)
These various terms tell us what elements (mlmsl,q(t)lm~ms) are nonzero at time tl +t2' For example, the first term involves I z , hence gives nonvanishing elements (± mslu(tl
(7.339)
+ t2)I~ms)
The term IzS z gives similar elements, weighted with mS:
s
(mlmsllzS z Im~ms) = ms(±IIzl=f)Oms,m Omr,±Om/,:,:
(7.340)
Lastly IzS" givcs nonvanishing terms
(± ±!e(t, + ',li'!' '1')
(7.341,)
(7.342)
Tit"', + ',)1'1' ±)
(7.341b)
However, since two nuclei are involved, with II = IS but £2r i- £25 owing to their chemical shift difference, it is better to think of this as
(±
whose further significance we discuss in Sect. 9.1. In Table 7.2 we list the other useful relationships employed in dealing with spin operators. Table 7.2. Useful relations for unitary operators U. (U is a function of I, or 5, or I and 5)
U/J(1,S)h(I,S)U-l
U/l(1,S)U-IUh(l,S)U
I
Ucxp(i/(I,S)lU- 1 = exp[iU/(I, S)U-I I For spin
!:
cos (1.8) = cos (8/2) sin (1.8) = 2/. sin (8/2)
7.27 The Jeener Shift Correlation (COSY) Experiment Since thc early days of high resolution NMR, an important goal has been to tell which nuclear resonance lines arise from nuclei which are bonded to one another. The existence of bonding manifests itself in liquids through the indirect spin350
spin coupling [7.74,75] (Sect. 4.9). The bonding might be direct, as in a C l3 H I fragment, or remote via an electronic framework as in the H-H splittings in ethyl alcohol (CH3CH20H) between the CH 3 protons and the CH2 protons. In solids, the dipolar coupling proves proximity. In liquids, thc dipolar coupling shows up through thc nuclear Overhauscr effect. One thcn distinguishes between bonding and proximity, an important distinction in large biomolecules [7.76] in which a long molecule may fold back on itself. A variety of methods such as spin echo double resonance, spin tickling [7.71], INDOR [7.65J and selective population transfer [7.66J have been employed, to utilize the fact that if I and S are coupled, perturbing the I-spin resonance will produce an effect on the S-spin resonance. All of those methods involve sitting on one line and point-by·point exploring the other lines. leener's discovery of the two-dimensional Fourier transform method converts the approach to a Fourier transform method with all its advantages. In this section we analyze the original Jeener proposal, which now is commonly referred to as the COSY (correlated spectroscopy) method since it reveals which pairs of chemical shifts are correlated by a spin-spin coupling between them. It involves a single nuclear species (e.g. HI) and a pulse sequence
XI(7r/2), XS(7r/2)··· tl ... X / (7r/2), X S (7r/2) ... acquire I(tz) and
S(t2)
,
(7.343) where a single HI of sufficient strength flips bOlh spins. We have already discussed the basic principles of coherence transfer in Sect. 7.25. There we saw that if we are observing the S-spins in time interval two, the density matrix elements (m/-Ie(tl +t2)lmr+) are responsible for the signal. These elements, however, have passed through both (mlmslfl(t)lmrms) and (=fmsle(t>!±ms) during time interval tl. Thus, the I-spin resonance frequencies modulate the S-spin signal during 12 as a function of tt. We can express these facts colloquially by saying Ihe (=f mslu(t)1 ± ms) matrix elemcnts during tl "feed" the (In, ~ Iu(t)!ml ±) elements during t2. Such a description almost suffices 10 describe the pulse sequence (7.343). All one need add is that at t = 0+ both (=fmsle(t)I±ms) and (ml =f lu(t)lml ±) are excited, so that during 12 (ml-lu(t)lml+) is fed by both (=fmslel±ms) and (ml =f lu(t)lml ±). [Note that (mrmslu(t)lmlms) during II also "feeds" (ml -lu(t)lmr+) during 12, as is shown by (7.291) and (7.290), however, since the diagonal elements of e are independent of time they do not by themselves introduce any I I dependence to e.J We could try to carry out an analysis using U similar to our approach to spin coherence. However, Ihis method rapidly becomes quite cumbersome. Indeed, one finds that all 16 elements of U are excited by this pulse scquence! We turn, therefore, to the method of spin operators. 351
Since the two nuclei are identical, we consider a Hamiltonian in the singly rotaling frame, rotating at the angular frequency w of the alternating field. The Hamiltonian includes a chemical shift difference so that il r and ils are not equal. Rather, Ihey are given by (7.344) with ur and us being the two chemical shifts. The Hamiltonian in the rotating frame is still (7.345) Both nuclei experience the same alternating field, H" so Ihat for any pulse the two spins experience the same phase for HI and the same angle of rotation, e. Thus, for example, if there is an Xr(e) there is also an Xs(e). Since we observe both I and S spins, we seek (7.346) with an initial density matrix prior to application of the first pulse given in the high temperature approximation as
(>{O-) =
~(I+ r.~:O(l,+S,»)
(7.347)
Utilizing the time development operators, T, Tr, Ts, Trs, defined in (7.307), we readily get
, H °T, {(l+ + S+)T(t,)XI (,/2)Xs(,/2)T(tl) Z·T
Expressing T(t" as T(t" = Tr(l])Ts(t])T/S(t])
we get T(t,)[yT-1(t]) = Tr(tt)Trs(tt)ciflsS~ll lye-iflsSztl T } (tl )Tr-'(t,) rf l = Tr(t,)Trs(t,)[yTif} (t,)TI- (tl)
T,{ABCj=T'{CABj
r
(7.353)
s
we pull the T '(t2)T '(t2) from the far right hand end of (7.348), giving for the first pan of the trace
Tsl(t2)Tr~ '(t2)([+ + S+)TI (t2)Ts(i2)TIS(t2)..
(7.354)
Then, utilizing eil,S l+e-ilz(};:: l+e i8
(7.355)
and the fact that I and S commute, we get instead of (7.354) (I+e- in ,11 +S+e- insh )T/s(t2)...
(7.356)
so that finally we have
+ S+(t)
J,
s
x X l(rr!2)XS(rr!2)(I: + S:)X '(rr!2)Xi l (rr/2)T-'(tl)
x X,I('/2)Xi 1('/2JT- 1(,,)}
(7.352)
Utilizing the fact that
(I+(t)
(l+(t) + S+(t» = "k
(7.351)
(7.348)
= "'(hHo Tr {1+e- in / 12 + S+e-inS(2) ZkT x [e-ia/,S~h X r(rr!2)X s (rrl2)e ifl ,I,ll e -iaJ,S,ll ly x e+ ia /. S, t1 e -in/ J, 11 Xs(rr!2)X r(rr!2)eiaJ ,S'11]}
(7.357) The expression on the right can be written as the sum of two temls, one containing the I: part of e(O-), the other containing the 5: part. We define these as (7.349) Explicit examination of the two expressions shows that one can be converted to the other if every place one has an [a one replaces it with Sa (a = x, y, z) and every place one has Sa one replaces it with an [a' Thus if one evaluates the teoo involving [: only, one can get the contribution from Sz by interchanging in the first answer Sa's with la's, [a's with Sa's, fir with fl s , and ils with fl[. Therefore, we evaluate just the [z teOll. Then X s(rr!2)Xr (rr!2)[zXi' (rr!2)X = Xs(rr!2)lyX I(rr(2) = ly
S
352
s'
(rr!2)
(7.350)
We have collected inside the square brackets the set of operators which we need to evaluate using the spin-operator foollalism. We saw in our example (Table 7.1) that the operators TJ> acting either on an la or on a product teoo [aSp, at most double the number of teoos, for example transfonning an [y to an ly cos + Ix sin An operalOr Trs acting on a single lerm la or Sp will double the number of tenns, but acting on a product will quadruple the number of teoos. The operators XJ(1r!2) and Xs(rr!2) will change an lz into an [y, etc., bllt not increase the number of teoos. Therefore TIS acting on ly will give two terms, which T J will double to four terms (two single operators, two operator products), which T/S will then convert to at mosl 2 x 2 + 4 x 4 = 12 tenns. (It turns out we get only 9.) They will be of the fonn la, or S{3, or lal{3 after we have reduced any products such as lalp, or [a[pl-y, etc. to the appropriate la" The only temlS which will eventually contribute to (l+(t)+S+(t» f involve either '
e
e.
353
(7.358) These are diagonal in one spin and off-diagonal in the other. All possible spin operators can be expJessed as linear combinalions of the 15 spin functions In. So. IoSIJ (a'" x, y. z; (J = x, y, z) plus Ihe identilY operator. But of these 16, only the eighl operalors (7.359.)
[:nSz. ["Sz. IzS%. IzS"
(7.359b)
will give matrix elements of the fonn (7.358). In fact. using the symbol 0 10 Sland for these eight funclions, (7.357) will consist of a sum of terms Tr {I+O} or Tr {S+O} which vanish if 0 is bilinear. Thus. only the terms (7.359a) of the form [:n. 1y • Sz and
5y
contribute 10 (/+0) + S+(t».
As a result, of the 12 possible terms we will get from [] in (7.357), many will not be of interest. It is therefore useful. as shown by Van den Ven and Hi/bers [7.72] to construct a table in which onc keeps only the spin products, not all the other factors, to find the tenns one needs eventually. then go back 10 get Ihe coefficients. In so doing, we will designale whal we are doing by wriling a sequence of operalors which aCI sequentially in the manner of Sorensen et al. [7.71J. Thus, to indicale that we wish to calculate the effect of lhe operators in the square brackels of (7.357) on an inilial operator I y • we write (7.360) We put these results into a table (Table 7.3) in which over the dividing line belween columns we lisl the operator which has lransfonned one column inlO the next column. If we make our table with 12 lines we should have all the space we need. Comparing Table 7.3 wilh (7.359), we see thai we will gel contributions 10
(£+(1»
Iz
(7.361)
Sz/4
(7.362)
The contribulions 10 I+ can be seen 10 arise from teons which at all stages are either I y or I z • hence diagonal in IllS. Thus they are not fed by tenns which are ever off-diagonal in ms. Therefore, these lenns do nOI involve lransfer of coherence between lhe I-spins and the S-spins. On the other hand. (7.362) shows lhal the tenn Sz/4, which is off-diagonal in ms but diagonal in m/, evolved earlier from the initial 1% to I y , to l z S% to IySy to -1,5". Hence lhe tenn I y • which is off-diagonal in m1 but diagonal in ms. feeds the lenn Sz/4. which is diagonal in m1 but off diagonal in ms. This tenn involves transfer of I-spin 3S4
------- --- -----
I
1.
1.
3
5
-a1,S,I,
X s (,,/2)
-d,S,11
, • , , ""
XI(,,/2)
0,/,11
1.
1.
-I.
I.
,,
I,
I.
I,
I,
3
1,5, _1.5,
1.5,
1,5,
1.5,
[~~:5J x [-~:/.J~
7 8
1.5,
-1,5.
1,5,
10
1,5, +1,5,1,
=
1,5, /.1,5, -(i/2)1,5. -1,5,5, (i/2lf)5. +1,1,5,5. (i/2 11.5,
= = =
5,,/4
polarization to the S-spins. We see Ihat by applying first XS(x{2). then second. X,(7f(1.). we get from I!lS, 10 -I,5,1 via the state IyS". Thi.s operator has malrix elemenls which are off-diagonal in both ml and mS' If mstead we had firsl applied X,(7f{2), Ihen XS(7f(1.). we would have gone from +ly S, to -1,5, to -I,Sll' Thus the intennediate state -I,S, would have had malrix elements which are completely diagonal. This latter Iype of Slate is the one we encountered in our discussion of coherence transfer (7.286). The fact that we go from IlJS, to -I,SlJ no matter which order we apply Xl(7rI2) and XS(7rI2) is an example of our Iheorem of (7.207) that one can interchange the order of the Xl and Xs pulses. We now need to get the rest of the coefficients of I z and Sz on lines 3 and 10 of Table 7.3. We readily find
(r+(,) + s+(t) I, =
-r::;
+ e-insl12 sin (att/2) sin (nll t )2 sin (atV2)Tr{S+Sz/4)). (7.363)
T,{r+r.) =T'{l;J :::I
L
(mlmsl1;lm, 71l S)
=4xi=1
so that
• ,, , "" 5
8
10
Explicit evaluation gives
and 10 (S+(t»I, from line 10
Line
x (e- in/ I , cos (at)(1.) sin (nltl) cos (atV2)Tr{rt l z }
I. from
line 3
Table 7.3. EffecU! of operators from (1.357) on an initial operator I,
(7.364)
(7.365) 355
In writing expressions such as (7.365), Van den Ven and Hi/bers [7.72) introduce a useful nClation which greatly cuts down on the number of lellers one needs to write down. They point out that using this formalism one encounters trigonometric factors such as cos (att!2), sin (atzf2), sin U2[t2), sin (ilst t )
(7.366,)
They write these as
C~
51
== ==
5~ == sin (at2n.) , == sin (nst,)
cos (attn.)
51
sin (nlf2)
(7.366b)
in a notation which is self-evident. One merely has to be careful to remember in the arguments involving a which is not present in those involving l the or ils. Returning to our discussion of (7.365). we now add to this the conuibution from 5~ to e(O-). which we obtain simply by interchanging ill and ils in (7.365). The final result is
"i"
n
(P"(l) + s+(t»
: ~::; {cos (atl/2l cas (at,J2)
(7.367) The various terms in this expression have simple physical meanings. We note fi~t that we have a term involving cos (att/2) cos (atz!2). This term is present even as tt and t2 approach zero. It has the terms
e- inst2 sin (ilstil
and
e-inl12sin (ilstt)+e-insf2sin (illtl)
(7.368a) (7.368b)
which correspond to oscillation near il, during both t I and t2, or oscillation near ils during both tt and t2. Of course the actual eigenfrequencies are ill ± an. and ils±a!2, but if a parallel 10 Ihe uniform field is H~=Ho+z
(OH') fu
(7.373)
so that planes of constant z correspond to planes of constant precession frequency. Thus, a given frequency interval is bounded by two frequencies which correspond to two planes (of constant z). The total NMR intensity in that frequency interval is proportional to the number of nuclei in the sample lying between those planes. Thus the NMR absorption spectrum provides a projection of the sample spin density integrated over planes perpendicular to the gradient direction. From a series of such projections for various gradient directions, one can reconstruct the object. We are considering, then, a uniform field kHo to which there is added a small additional field h(r) which is static in time:
H = kHo + h(r)
(7.3740)
with
h(r) :::: ihz(r) + ihl/(r) + kh~(r)
and
Ihl Ih(r)l, the effects of ih;e and ih" are merely to rotate H slightly without the first oroer changing its magnitude, whereas h~ changes H(r) to firs! order:
H = V(Ho + h~)2 + =
hi + h~
JH~+2Hoh~+h~+I.;+h~
ll~(r)
= h:(O) + x
==
({)~~: )0 + Y (a~~v )0 + Z(aa~: )0 + ...
h:(O) + xG;e +yGl/+zG;z
(7.378)
which defines the components Gz , G", G: of the gradient G of h~(r) at the origin: (7.379) G = iG z + iG y +kG: The reader may wonder whether we can guarantee that it is possible to generate an 11 with the spatial dependence of (7.378). After all, h must obey the laws of physics (V. It = 0, V x It = 0). We explore these aspects in a homework problem. The answer is that we can achieve (7.378), but for there to be a term such as kxG z for example, there must also be terms in the i and/or i directions. However, since as we have seen the transverse components do not affect IHI to first order, we can neglect them. Let us then define two functions, a frequency distribution function, !(w), which gives the NMR intensity at frequency wand is normalized to satisfy
J
f(w)dw = 1
(7.380)
and a spin density function e(r), which gives the number of nuclear spins in a unit volume at point r, and which is likewise normalized so that
J
(7.381)
(,(r)d'r = 1
If, then, the gradient G is oriented in the i-direction,
G=
(7.382)
Z'G .
planes i = constant are planes of constant precession frequency. Since the magnetic field changes by G dz' between the planes al i = constanl and (z' + d.z') = constant, the precession frequency change, dw, is (7.383)
dw=;Gdz' Therefore
!(w)dw = dz'
f'lJ
dx 'd y"",x,y,z I"J I I I)
(7.384)
%':::consl
(7.376)
Neglecting the terms quadratic in the components of h, we get
H = HoJI + 2h~/Ho ~ HoO + h:IHo) = Ho + h~
What spatial forms can h: take? If we simply consider it to be a gen~ral function of r, we can expand it in a power series in z, y, z about a conventent origin. We assume we have shaped the field so that only the lowest terms are needed. Then
(7.377)
The integral in (7.384) is the projection of the spin density funclion e(x', yl, z') on the z'-axis. Thus, utilizing (7.383), "(Gj(w)::::
11
dx1dy'e(:i,y',/)
(7.385)
~/=consl
358
359
Since this projection depends on the direction of G, and since the magnitude of f(w) depends on the magnitude of G, we put a subscript on f(w) :
.."Gfc(w)
=
JJ
dx'dy'e(z', y', z')
(7.386)
:r'=const Since fc(w) can be measured, we can view the integral on the right as known experimentally as a function of z'. We can think of making a variety of measurements 10 delennine the w dependence of fc(w) for a variety of different G directions, thereby getting the projecled spin density for any desired projection axis, z'. It is now easy 10 show that such measurements enable one to reconSmJct e(z, y, z). Firsl we take the experimental dara for a given z' and fonn the Fourier transform of the data:
ek =
J
e-ik:r'dz'
JJ
dx'dy'e(z',y',/)
problem then becomes one of finding a suitable numerical approach for getting the most accurate value of the integral from the data as given. Of course, in the process of selecting the approach, one gets guidance as to whal data to collect. The approach of reconstructing e(r) by Fourier transforming the spin density distributions is one approach to finding e(r) from data for fc(w). Lawerbur [7.79] demonstrated a simple graphical means, and pointed out that Ihe solution to the problem of getting e(r) from spin-density projections was well known from Olher areas of science. Another experimental approach was proposed by Mansfield, based on 0bserving free induction decays. If one applies an X(ll'"{l) pulse to the spins and observes the free induction decay in Ihe reference frame rotating at wo, the complex magnelization is
(M+(tl) = (M.(t)} +i(M,(tl)
(7.387a)
= iMo
where = iMo
k = H'
(7.387b)
Utilizing (7.373) wilh Wo
ek = exp(ikwo!"'(G>
== "'(Ho
exp(-ikw!"'(G)fc(w)dw
.
(7.388)
If we know ek for all k, we can get e(1") by the relalionship 1 l'\r = (2ll'")3 -J )
j eik·.ek. cr"k· .
(7.389)
'. k,
.'11:.7.46. The paths in k-space (k = ki') eXI>lored by applying the gradient C = G i , in a variety or directions i' in a rhUle. The various directions 91, 92, ... ,9,. correspond to different directions i;,
-,
" 360
(M+(t» = iMo
so that (M+(t» =iMo
In practice, since each direction k is explored in a separate experiment, ek. is never known experimentally for k as a continuous variable. For example, if one has a two-dimensional object, the various k's would be chosen to lie in the plane of the object. Then the k values explored could be represented as in Fig.7.46. It is seen that they represent radial lines out to some maximum k, and k max • Since performing an integral such as (7.389) is a numerical operation, the
9,
e(z', y', z')e-iw(z' ,JI,z')ldz'dy' dz' f!{z', y', i)e-i.,.G:r"d:i dy' dz'
(7.39Oa)
which can be rewritten to make ils meaning clearer as
and (7.386) we can express (7.387) as
J
J J
i:;,
J J
JJ JJ
dz/e-i.,.G:r'1
dz'e- iq ''''
dz' dy' f!.z', y'. z')
dz'dy'(!{Z',y',z')
(7.3901»
(7.391)
where
(7.392)
We can rewrite \1.391) to emphasize that on the left we are observing
(M+(t» for a particular gradient direction i', and rewrite the right hand side to bring our its meaning to get
(M.+(t»j;' = je-iq • .,. e(r)d3r
.M,
(7.393)
The expression on the right of (7.393) is identical to that on the right hand side of (7.387a) if one replaces q by k. We will henceforth replace q by k. Thus, in the presence of a static field gradient, the NMR signal following a 7r{l pulse sweeps out in time the Fourier transform (?k' where the direction of k is given by the direction of the field gradient, and each point in time corresponds to a magnitude of wave vector given by (7.392). Clearly, the free induction decay is also producing f:!k along radial lines in k-space as in Fig. 7.46. In his initial paper, Mansfield [7.80] proposed using the free induction decay, giving (7.393). Since his initial idea related to crystallography, he explored the case that the spin density is given by a periodic lauice, and demonstrated the imaging scheme by making a macroscopic layered sample to simulate atomic periodicity.
36.
In Fig.7.46 we display a set of k trajectories for a planar sample. Even for nonplanar samples, it is convenient to collect data from planar slices in me sample, a sequence of slices thus providing the full three-dimensional object The methods of selecting well-defined slices are pan of the present an of imaging. We refer the reader to (7.77] for example. A simple approach to generate a slice perpendicular to the z-direction is to start by applying a gradient in the z-direction. If one applies a pulse at a panicular frequency WI, the spins at coordinate z such that
which is the Fourier transform in the plane z of the spin density. Since f z is fixed while recording (M+(t», the data are for a fixed kz • but for a continuous k y . They may be represented as in Fig. 7.47. They thus sample (}k for the z-slice along parallel lines in (k;r:. kll) space. In a full three-dimensional version, one applies the initial rl2 pulse in the absence of gradients. Then. one applies
(7.394) are perfectly at resonance. An X , (7r!2) pulse will then rotate them into the x-y plane. Spins within a Ll.z given by
(7.396,)
G=iG;r:
fo
makes an angle 26,p wIth the z -" plane. In Fig. 8.2e we see that if the III of the second pulse lies along the -x-axis, the rotation direc:tiOll is reversed, so the _(,.. + 6.,> rOlalion restores the spin to ils orientation at t ::: 0+
t ::: 7 - the spins will have fanned oul in the x - y plane. a typical spin making an angle 8 with the -y-axis (for negative ")'). At t "" T we apply a ow + 6¢ pulse about the +x-axis where 6¢ gives the deviation from being a perfect 'If pulse. For a spin having 8 "" the spin lies along the x-axis. It is not affected by the pulse at t"" T. A spin for which 8"" 0 would rotate as shown in Fig. 8.2c. It will lie an angle 6¢ above the +y-axis in the y - z plane where it will sit until 2T later when the next + b¢ pulse is applied. The result is shown in Fig.8.2d. The spin now makes an angle U¢ with the -y-axis. Each successive imperfect 1'1" pulse adds another 6¢ to the deviation. The cllmulative effect becomes exceedingly serious in praclice because the HI is never uniform throllgh tile sample. COllsequclIlfy thOllgh part o/the sample may have a ow pulse, other part.r do f1ot. These sons of troubles have major implications for all multipulse sequences ~uch as those described in the later portions of this chapter. One simple approach IS to apply the H J for the ow pulses alternating along the +x- and -x-axis. Thus the sense of rotation about the x-axis reverses in alternate pulses. The result is shown in Fig. 8.2e. Instead of getting a cumulative rotation eITor of U¢ as in Fig.8.2d, the cumulative error is zero!
7rn..
7r
370
11le dipole-dipole coupling between neighbouring nuclei provides valuable in-
formation for many purposes, but on occasion it is a problem. For example, it may obscure chemical shifts, and it may cause free induction decays to be soonlived and thus difficult to see. We tum now to several interesting approaches to eliminating or effectively reducing dipolar coupling. For a group of spins an ordinary spin echo [e.g. an X('IT/2) ... T ... X(:IT) sequence] refocuses the dephasing which arises because they are placed in an inhomogeneous magnetic field. Since the magnelic dipolar coupling among neighboring spins is in some way analogous to a magnetic field inhomogeneity, one might ex:pect that the same spin echo pulse sequence would also refocus the dephasing resulting from magnetic dipolar coupling. If the dipolar coupling is between different nuclear species (e.g. HI and C I3 ). such an echo sequence does refocus the dipolar coupling. If, however, the coupling is between like nuclei (e.g. HI with HI), the echo does not work. It is easy to see why. The 'IT pulse inverts all the neighbors, so that a nucleus which was precessing more rapidly than average in the first time interval T, finds its neighbors inverted, and thus precesses more slowly in the second interval. The usual echo works because the 11" pulse converts a precession phase lead into a precession phase lag, of equal size. If at the same time the rate of precession changes, the spins do not come back into phase. Nevertheless, the fact that a well-defined relationship exists between the precession frequency when a neighbor points up versus down makes one feel that there should be some way to undo the dephasing which the dipolar coupling produces. In the following sections, we explore these ideas.
8.5 Solid Echoes Powles and MallSfield [8.7] discovered that the free induction decay of a coupled pulses pair of identical spin ~ nuclei can be refocused by applying a pair of shifted in phase by ow/2 with respect to each other. Such a sequence might be denoted by
7rn.
371
X('rr{2) .. . r . .. Y('II"/2) . .. tt
(8.6)
with the echo occurring when tl := , . This sequence has come to be known as the solid echo. Powles and Mansfield demonstrated it experimentally for CaSO'1 ·2H20, and showed that it followed theoretically. The pulse sequence perfectly refocuses spin nuclei which interact only in pairs, but does not refocus larger groupings of spins perfectly. In this section, we explain the echo for coupled pairs. This same pulse sequence refocuses the first order quadrupole coupling of a spin I nucleus for reasons explained in Appendix H. It is therefore imponam for deuterium and N t4 NMR. The existence of echoes related to quadrupole splittings was first demollStrated by Solomon [8.8] for a case of I := ~. It appears to have been Davis et al. [8.9] who recognized that for the 1:= I case sequence (8.6) gave perfect refocusing. The fact that one needs exactly sequence (8.6) may seem surprising at first glance. Why does one need to phase shift the second pulse by 11:/2? Why does one use a '11"/2 pulse instead of a 'II" pulse? To help give a feel for the answers to these questions, we shall derive the result Ihree ways. The first method is a semigraphical one, utilizing a special value of r. The second derivation generalizes the first for arbitrary r. The third derivation utilizes the spin operator method. We start with the simple physical picture. However, we must present it rather carefully. Consider a sample consisting of pairs of nuclei. That is. each nucleus has one and only one neighbor which is sufficiently close thai only the dipolar coupling to it matters. In the presence of a static field, the nuclei are weakly polarized, there being a slight excess with magnetic moments parallel to the static field, H o. If there are N nuclei and if we denote by p+ and p_ the probabilities for a given nuclear moment to point either parallel (p+) or antiparallel (p_) to H o, we have then that there are N+ and N_ nuclei pointing parallel or antiparallel, respectively, given by
4
(8.7) The population difference, n, then obeys n:=N(p+-p_)
(8.8)
p+ and p_ are related by the Boltzmann factor p+/p_
:=
e"tldfo/kT
(8.9)
so that, if ",(hHo «kT, P+:= ~(1 + "'(fiHo/2kT)
p_
:=
~(1 - "'(hHo/2kT)
(8.10)
Note that the expressions for p+ or p_ do fU)( depend on the orientation of the other nucleus in the pair, because we have assumed that the only field acting on a nucleus is Ho. This assumption should be valid if Ho d> "'(1lIr 3 , where r is the distance between neighbors. Now, it is only the excess of nuclei N+ over those N_ which can give rise to a NMR signal. To follow what happens, we need 372
only follow what happens to the excess, since Ihe remaining nuclei carry no nel polarization. Let us therefore in a thought experiment reach in at random to examine the pairs. We take n/2 pairs in which both nuclei point up, and n{2 pairs in which one nucleus points up, and the other points down. This will give us a net n nuclei pointing up. The olher N ~ 2n nuclei in the sample must then be equally pointing up and down, hence be unpolarized. We neglect them. Now, among the pairs with one nucleus pointing up, Ihe other pointing down, we label the up nucleus blue, the down nucleus red. Among the pairs with both nuclei up, we also label one blue, the other red (it does not matter which of the pair we choose for which color). We thus have n{2 red nuclei with spin up, and n/2 red nuclei with spin down. Then the blue nuclei give the magnetization, since the red nuclei have no net magnetization. We represent the spin-spin coupling by a term 1t'spin-spin of the form (8.11)
!
For pairs of identical particles of spin this is exactly equivalent to dipolar coupling, as explained in Appendix H. We assume that apart from 1t'spin-spin we are exactly at resonance for any pulses we apply. Then, following an X(1r{2) pulse, the blue spins lie along the +y-axis in the rotating frame. Now, although we have flipped both red and blue spins by 1r{2, the red spins at this point have no net polarization and thus their density matrix remains diagonal with the two diagonal elements equal to each other. Thus, we can describe the situation at t := 0+, just after the pulse, as in Fig. 8.3b, in which we represenl the blue spins by vectors I and 3, lyir]g along the y·axis, and the red spins by vectors 2 and 4, one pointing up, the other down, along the z-axis. Spins I and 3 will now precess at rates -a!2 and +a/2 (in the left-handed sense) respectively about the +z-axis. The explanation of how the echo fonns becomes very simple for a particular value of " the value which makes arl2 := 1r12 (Le. a, := 1r). For that time, the spins will be aligned as in Fig.8.3c. At this time, the y('II"!2) pulse will produce the situation of Fig. 8.3d. Now, during the next time interval" spins 1 and 3 will not precess (they lie along the z-axis), but spins 2 and 4 will precess at rates +a/2 and -a/2 respectively so that at t:= 2" 2 and 4 will lie along the +y-axis. An echo therefore occurs at t "" 2,. It is interesting to note that in this process we have transferred the net polarization which was initially in spins 1 and 3 10 spins 2 and 4. However, we have not made an unpolarized system become polarized. Thus, we can remain confident that the other N - 2n spins, which we have said are initially unpolarized, do nOI become polarized by the echo sequence. If one chooses a , other than Ihe one for which arl2 := '11"12, the explanation becomes more complicated. Then one needs to utilize the ideas of Sect. 7.24 in which we discuss the time development of one spin when it is coupled to another spin which is in a mixture of the spin-up and spin-down eigenstates. By choosing a,/2 := 'II"!2 we have made this mixture be in fact pure eigenstates. 373
However, a true echo means thai the refocusing condition is independent of the strength of the source of the dephasing. Thus. if spins are dephased by an inhomogeneous magnet. all the spins get back in phase at the echo. independent of how far the inhomogeneity has shifled their resonance from the average field_ In our case. that means the existence of an echo should be independent of a, hence of the PrOOuC[ aT, hence of T. Let us therefore reexamine the situation for a more general T, using Fig. 8.4. We start (Fig. 8.4a) with the magnetization along the z-axis, and apply an X('K(l) pulse to prOOuce the situation in Fig. 8.4b. A time T later the vectors I and 3 have rotated in the % - Y plane through angles (aT(l) and (-aT(l) to produce Fig.8Ac. Then the Y(1I'/2) pulse produces the situation of Fig.8Ad in which magnetization vectors 1 and 3 are in the y - z plane, making angles -aT(l and arl2 respectively with respect to the y-axis. In Fig. 8.4e we show the projections of the four vectors onto the x - y plane. Spins 2 and 4 lie entirely in the plane. We define their lengths as Mo. Spins I and 3 have projections in the x - y plane of Mocos(aT(2.). Now, using the concepts of Sect. 7.22, we realize that each of the four magnetization vectors is coupled to a spin which is in a mixture of up and down states. Since the quantum states of spins 2 and 4 are equal mixtures of up and down. spins 1 and 3 will each break into two counter-rotating components of equal amplitude (Fig. 8At). Spins I and 3 have z-components of -Mo sin(ar(2.) and +Mo sin(aT(2.) respectively. Thus they have an excess of m = (for spin 1) and m = +i (for spin 3). We need to calculate what fraction of their stales correspond to spin up and spin down. Denoting Ihe occupation probabilily of slate m for spin i as (mIUilm) we have that
h: O'
, y t: 0'
l 4
,
,IT: "
,
Idl 2
-!
y I: '['
,
,
(8.12)
from nonnalization. The z-component of magnetization of spin I,
y
lei
2
,
t :
M" = Mo (~I",I~)
2'[
expressing the fact Ihat if the spin is entirely in respectively. But
I'Ig.S.Ja·e. The fonnation of a solid echo for two pairs of spin!! (I and 2, 3 and 4) for a particular value of the time T" at which the refocusing pulse is applied. (8) The thermal equilibrium magnetization at 1 0- showing the vector sum of the four magnetization veclors. Spins 1 and 3 are the blue sl)ina, spins 2 and 4 are the red spins. (b) Just after the X(Tr/2) pulse (t 0+), the blue spina (1 and 3) lie along the y-axis while the red spins (2 and 4) can be taken to point along t.he positive Il.Ild negative ",-axC!! respeclively. (c) The magnetization vectOrs at times L = T- just before the Y(lr/2) pulse, for the particular T such that (JT/2 lr/2 (or or = lr). (d) The magneti:u.tion vectors at time 1 1+ immediately llfter the Y(lr/2) pulse. (e) The magnetization vectors at time t 2r, showing that there is a net magnetization IIlong the y-axi!! from spin!! 2 llnd 4. Note that initially spill!! I and 3 carried the mllgnetization. At t = 2,., the magnetiZll.tion arises from spins 2 and 4
=
=
=
- Mo (- ~Ied -~)
=
=
M~l
= -Mosin(ar(l)
M~t>
is (8.13)
i or -i. M
z1
is Mo or -Mo (8.14a)
so we get
(~Ied~)
- (-~Ied-~)
=-s;n(aT/2)
(8.I'b)
Solving (8.12) and (8.13) we get Well~) = ~[I - s;n(aT!2)]
(- ~Ied -~)
=
1(1 + s;n(aT!2J]
(8.15)
In a similar way, we get 374
375
,
Fig.8.4 Caption see opposite page
(- ~I",I-~)
lal
, ,
Ibl
,
,
,
2
2 all2
--
,
,
~o [1
,
4
1
11
/
/
,
lei
,
(8.18)
N~O [1
- sin(ar/2)] sin(ar/2)
~O [sin2(or/2) + sin2(ur/2)]
(8.19)
Therefore
L
t~,'
No
4
M y (t+2) = Mo[sin 2(or/2) + cos 2(or/2)] = Mo
,
Igi
(8.20b)
,
M
(os(aTl2)
•
Fig.8.4a·g. Formation of a solid ocho for two pairs of spins (I and 2, 3 and 'i) for a general value of the spin-spin coupling conslant u. (a) The thermal equilibrium magnetization at t = 0- showing the vector sum of the four magnetization vectors. Spins 1 and 3 are the blue SpillS, spills 2 and 'I are the red spins. (b) The magnetization vectors at 1=0+ immediately following the X(1r/2) pul$(!. (e) The magnetization vedors at time t = r-, just before the Y(1r/2) pulse. (d) The magnetization vectors at time t r+, immediately after the Y(tr/2) pulse. (e) The projections of the magnetization vectors on the :l' - Y plane at time t = r+, immediately after the Y(tr/2) pulse. (f) The projection of the magnetization components of a spin 1 on the 2: - Y plane at later times, showing how its Illagnetization breaks into two counter-rotating components whose amplitude is determined by the orientation of spin 2. (g) The projection of the magnetization components of a spin 2 on the z - !I plane at later times, showing how its magnetization vector breah into two counter-rotating components whose amplitude is determined by the z-components of spin 1
=
spin 2
spin
("'T---'
(8.20')
By a similar argument
Ho
-f
~o [1 + sin(or/2)] sin(or/2)
=
=
_-'-;:,-...L__ ,
If I
M y2
_
~--Mo(os(aTl2l--- 3
1
~o cos2(aT/2) + ~o cos2(aT/2)
,
,lTn /
M y1 =
,
,
Idl
- sin(ar!2)]
as shown in Fig. 8.4f. We now add up the y-components, My; (i = 1,2,3,4) of the spins at time -r after the Y(7l"/2) pulse:
"-k'::..o" )3 aT 12
,
(8.17)
while the counterclockwise component of 2 will have amplitude
,
4
,
I,}
~o (1 + si,(arl2)]
3
,
(8.16)
= HI - si,(arl2)]
Thus', as spin 2 precesses, it will break into two counter-rotating components whose amplilUdes are detennined by the relative amounts of spin 1 in the up and down states. That is, the component of 2 which rotates clockwise (looking down on the x - y plane) which comes from the down component of 1 will have amplitude
,
376
W",I~) = ~(I + si,(arl2»)
M. T
l1-sin(aT1211
377
This result is independent of a, therefore corresponds 10 a real echo. Indeed, (8.20) shows that in a powder sample (in which a would take on different values for each crystal orientation if it were representing magnelic dipolar coupling), spins of all the crystallites would refocus at time 2r. This exercise has been long, laborious and exhausling, but it does serve to show how coupled precessing veclors behave in quantum mechanics. Now we lum to a much more elegant approach to analyze the same problem: use of the product operator method. It has the advantage of giving us a simple way of figuring OUI what the second pulse should be in order to produce the echo. (See Seci. 7.26. See also [8.10, lIn. We start with a system in thermal equilibrium and work in the rotating reference frame. The density matrix JUSt before the first pulse (t = 0-) is (8.21)
where A is a constant. Following the pulse X(1r(2), the density matrix becomes
~~=~+~
~~
The density matrix at time t = r- JUSt before the Y(ll"/2) pulse is then e(r-) = e- ia1• S ... e(O+)e+iol.S.T
Now, utilizing (8.23) we have e+iolzS... e(O+)e-iaI,S, .. . = (Iy + Sy)cos(ar/2) + ([ZSl + I z S;l:)2sin(ar/2)
(8.29)
which, using (8.24), is equal to
Re(r-)R- 1
(8.30)
and using (8.23) gives
R[(ly + Sy}cos(ar/2) - (l;l: + S;l:)2 sin(ar/2)]R- 1
(8.31)
Comparing (8.29) with (8.31) we see that R must satisfy the relations
I y + Sy = R(Iy + Sy)R- 1
and
(l;l:SZ + IzS;l:) = -R(lzSz + IzSz)R- 1 We automatically satisfy (8.32a) if we make R be a rotation about the satisfy (8.32b), the rotation must be 1r/2:
(8.32a) (8.32b) y~axis.
To
(8.33)
(8.23a) a result which follows from the relationships (Table 7.1) such as
which, utilizing Table 7.1, gives (8.23b) Now, we wish to apply a pulse which produces some rotalion R which will cause an echo. The effect of R is to change e from its value at t = r- 10 a new value at t = r+ given by (8.24) We want e(r+) to be such that during the next interval r, e returns to its value at t = 0+, just after the first pulse. Thus, the condition for an echo is e(2r) = g(O+) but e(2r) = e- ial• S ... e(r+)e+io1• S , ..
(8.25) (8.26)
If the effect of R on e(r-) were equivalent to changing the sign of a during the interval between the X pulse and R, so that Rg(r-)R-1 = e+ io1,S, .. e(0+)e- ia1 • S... (8.27)
then we would satisfy (8.25) and produce an echo, since putting (8.27) in (8.26) we would have e(2r) = e- ioJ• S e(r+)e+ ia1 ,S,..
Y(1r!2)I y y- I (7r!2) = I y Y(1r/2)I;l:y- 1(71/2) = +Iz Y(1r/2)lzY(7rI2) "" -Iy
and
(8.34) (8.35)
Before leaving the solid echo, we should remark on one more matter. In Fig. 8.3c we show the spins 1, 2, 3, and 4 at a time ar!2 = 1r/2. We want to refocus the spins to their condition in Fig. 8.3b where 1 and 3 lie along the +y-direction. Looking at Fig. 8.3c we note that if we simply reverse vectors 2 and 4, vectors 1 and 3 will reverse their precession directions, and thus refocus as desired a rime r later along the +y-direction. All that is needed is a 7r pulse about the x-axis. So why do we use a Y(7r/2) pulse instead of an X(7r) pulse? Clearly the idea will work if Gr!2 = 7r/2. But to give a true echo, the scheme must work for other values of a, hence of aT. The easiest test is to look at (8.32). An X(1r) pulse will leave I z and S;l: alone, but will reverse I y • S", I l , Sz. It will therefore satisfy (8.32b), but it willllot satisfy (8.32a). Note, however, in (8.29) that when (ar/2) = 7r/2, the coefficient of the lenn (Iy + Sy) vanishes, so we will refocus this particular product ar. We will not, however, refocus for either a general value of a or of r. Thus an X(7r) pulse will not produce a true echo.
= e- iol• S
Re(O+)R-1e+io1,S,T = e-iol.S, .. e+ial. 5... e(O+) e -iol. 5... e+ io1• S ... = ,,(0+)
378
(8.28) 379
The resulting magnetiUltion is shown in Fig. 8.5c. So far, everything we have done is similar to our first steps in discussing the echo using Fig. 8.3. Examination of the four magnetization vectors in Fig. 8.5c shows that their resultant is zero. We appear to have lost the magnetic order. However, that cannot be true since we know that a Y(71/2) pulse will enable us to recover the magnetizalion. The secret of the Jeeller-Broekaen pulse sequence is the recognition that the spins in Fig. 8.5c have spin-spin order. This can be understood by analyzing what happens if one applies a Y(1I"/4) pulse, producing the result of Fig. 8.5d. If now 380
Fig. 8.Sa-e. The Jeener-Broekaert method of creating dipolar order from pairs of sl)ins (I and 2, 3 and 1). (a) The thermal equilibrium magneti~ation at ! = 0- showing the vector sum of the four magneti~ation vectors. Spins I and 3 are the blue spins, spins 2 and 4 the red Sl)ins. (b) The magnetization vectors at t 0+ immediately following the X(1r/2) pulse. (c) The magnetization vectors at time t T- ,just befoN! the Y(1r /4) pulse, for the case (JT/2 = 1r/2(aT = 1r). (d) The magnetizatiOll vedors at t = T+, immediately after the Y(1r/4) pulse. (e) The magnetization vc/4)
can be seen, in the limit T _ 0, to be X(rm ... r
J2
y
J2
J2
2(1~S~
- IzS z ) sin(aTfl)]
We now invoke the argument that couplings to more distant spins. not included in the Hamiltonian (8.11), cause the ofT-diagonal components of ~, which arise from the lenns (Iy + Sy) and IzS z in (8.38), 10 decay. Thus at a later time T we can drop the off-diagonal elements of ~, so that f!{T + T) is then
(J{T + T) ;;
-2AI~S~ sin(a;f2)
.
(8.39)
This Hamiltonian has zero net magnetization, as can be seen easily by explicit evaluation of Ot;;z,y.~ ;;Tr{Iau(,+T)}
(8.40)
However, the average dipolar energy
;; -Tr{haI~S~/?(r+T)} ;; -2AhaTr{I; sin(ar/2)} ;; -Alia 2 sin(aT/2) .
m ... r ... Y(r/4) ... T ... Y(r/4) ... ',
(8.42) 383
arfl ::: attfl = 7C(2
•
(8.50)
Then
'H.::: --yhhoI~ - -yhH) I~ + L Bij(3I~iI~j -1; .lj)
Comparing this signal with that of the free induction decay following an X(7C(2) pulse,
= T,{(r+ + S+)AUv + Sv» = iA n{(Iv + Sv)2}
(8.54b)
i>j
We now define an effective field in the rotating frame
(8.51)
Herr:::iH, +kh o
we see the maximum signal is one half that of a pure X(1rfl) pulse. If there is a distribution of values of a, there is still a signal at tl ::: r since (8.52) sin 2(ar!2) > 0
(8.55)
We shall concern ourselves with cases in which Herr is much larger than the local fields. In Ihal c~se, it is appropriate to quantize the spins along the effective field. Defining this direction to be the Z-direction, we make a coordinate transfonnation (Fig. 8.6) for all spins, getting
Thus, we can create dipolar order by applying X(7C!2) ... T ... Y('ll14) with T chosen such that a T ~ 7Cfl. and wecan inspect the dipolar order by later applying a Y(7C/4) pulse to produce an echo a time T later. The inspection pulse can be used to study the decay of the dipolar order resulting from either conventional spinlauice relaxation mechanisms (e.g. coupli.ng to conduction electrons in a metal), or from motional effects which modulale the strength of the dipolar coupling. We have treated the problem for coupled pairs. JCClicr and Brockacrt treat the general dipolar case in their classic paper [8.12].
Iz ::: I",cos8 + I",sin8 Ix :::I~cos8-Iysin8
Iy = Iv
•
(8.56.)
giving the inverse transfonnations for an individual spin j
Izj::: I Zj cos 8 - I Xj sin 9 IZj::: Ixjcos8+Izjsin8 Iyj::: IYj
(8.56b)
Subsriruting into 'H., we gel
8.7 The Magic Angle in the Rotating FrameThe Lee-Goldburg Experiment
+2
'H.::: -"fhHcrrIZ +
L
A/l1(8)'H/l1
where
(8.57)
/1,1:::-2
Another imponant concept concerning dipolar order is connected with the famous Lee·Goldburg experiment (8.13] 10 which we now turn. Suppose one has a number of spins all with the same precession frequency, coupled together by dipolar coupling. We keep just the secular tenos of the dipolar coupling, giving a Hamiltonian in the rotating frame of
'H.::: -
- "fllH II~ 2 "f2h + :L -,-(1 - 3 cos28ij)(3I~i l~j - Ii' Ij)
'H.o:::
,
"fhhoI~
i>j
(8.53.)
where as usual
r
h,
etc.
ho::: Ho -wl-y
(8.53b)
It is convenient to collect all the radial and angular lenns of (8.53a) in a simple symbol by defining I "f2h2 B;j=zT(I-3cos2 8ij)
Ii' Ij)
(8.58)
z
----
T ij
I~::: r:,I~j.
L Bij(3I~iIzj i>j
,
I I I I HrH ! I I H,
, x
Fig. 8.6. .The errej
1-£+2 =
L
3BijUt It}
,
i>j
1LI
= (1-£+1)", 1-£-2 = (1-£+2r
and
(8.59)
-'0(9) = !(3cos 28-1) (8.60)
-' ± 1(8} = -! sin 9cos 8 -'±2(8) = sin 28
-i
The 1-£/I.1's satisfy a commutation relation
lIz, 1iM]- M1iM
(8.61)
as is easy to show by explicit calculations of matrix elements. From (8.61), or by examination of the explicit form of the 'HM's, we see that only 1-£0 commutes with the Zeeman interactions of the effective field. In the limit of large effective field, we can then as a first approximation drop all terms but the term involving 1-£0. We get then a truncated Hamiltonian 1-£ = -yhHefflz + !(3 cos 28 - I) Bij(3IziIZj - Ii' Ij}. (8.62)
L
i>j We can compare this with the secular part of the Hamiltonian of the spins in the lab frame,1tIl\b' in the absence of an alternating field, H,: 'H'l"b = -"'('iHolz +
L
Bij(3IziI:j - Ii' Ij) (8.63)
Equations (8.62) and (8.63) are identical in form, except in the rotating frame the dipolar term has been multiplied by the factor 3,os 29 _ I 2
(8.64)
['This is juSt the tenn -'0(8)]. 8 is delennined by the relative size of ho, the amount one is off resonance, with HI, the strength of the rotating field. Lee and Goldburg noted that if they chose HI and ho properly, they could make cos 28 (8.65)
=!
in which case the dipolar term in (8.62) vanished. In this manner they could effectively eliminate the dipolar broadening. An effective field for which (8.65) is salisfied is said to be at the magic allgle. We can rephrase their result by saying that if the effective field is at the magic angle, a spin will precess in the rotating frame solely under the influence of Neff, without suffering a dephasing and consequent decay of the components 386
(8.66) The failure of the dashed theoretical curve to intercept the origin arises because of slight inhomogeneity of H1. BarntUJl and Low [8.14] made an extensive study of the systems in which HI was exactly at resonance. In this case 3cos29 - I (8.67) 2 They did experiments in which they turned on HI for a time T, then observed the free induction decay a time t after H 1 is turned off. They solved the problem of an interacting pair exactly. For an HI along the x-axis they found My(t) = M(O)
i> j
:= 1-£Zeemlln + 'H'dipolllr
of magnetization perpendicular to Herr as usually occurs when there is dipolar coupling. To make a precise test of this concept, they did pulse experiments in which they observed the decay of the magnetization with time following a sudden tumon of H t at a frequency w somewhat off resonance. They varied the angle 8 and the strength of the effective field. To measure the effective dipolar strength, they determined the second moment from the transfonn of the decay curves. They studied the F ID resonance of CaF2. Figure 8.7a shows their measurements of the square root of the second moment versus (3cos 28- l)n.. AI the magic angle, the second moment should vanish. It does not quite do so owing to the nonsecular terms involving 'H± I and H±2. The effect of these tenns should vanish in the limit of infinite Herr. In Fig.8.7b we show their measurements of the faJl-off of second moment, nonnalized to its value in the lab frame, versus llw; where
~ sin(flT)cos [ (3:,: 2 ) (t + T!2)]
(8.68)
where
n' -_ (3BI2)' - - +w,,
"
4~
(8.69)
In the limit of large HI, this expression agrees with the result of the truncated Hamiltonian: My(t) = M(O) sin(wl T) cos [ (3:,:2 ) (t + T!2)]
(8.70)
A striking feature of (8.68) is Ihal the oscillation after the tum-off of HI is identical to what it would be following an X(7r/2) pulse except for (1) a slighl amplitude correction and (2) a change of T in the apparent zero of time t. In a beautiful set of experiments, Barnaaf and Lowe showed that for CaS04 ·2H20, CaF2, and ice the first zero crossing of the free induction decay moves 10 later times as T is increased. The delay in zero crossing is T/2. for values of T which are up to about half of the nonnal free induction decay zero crossing.
387
I
(oF,IU doptlll H.1I111lJ
.. ~ 0.6
>J A
."
0.'
v
0.'
~
"-
s
A
."
., ~
pulse a time T later as suddenly lransfonning the Hamiltonian to be the negative of the actual Hamiltonian. Then, over the next time interval T, the time development of the magnetization unwinds, returning the magnetization to its value just after the initial pulse. In discussing spin temperature (Chap. 6), we talked about irreversible processes connected with the complexity of a system of many coupled dipoles. We now tum 10 a remarkable discovery which shows how one can run backwards the dephasing produced by dipolar coupling, thus showing that it is possible, even after the free induction decay is over, to recover the initial magnetization. This experiment shows that the spin temperature approach does not in all cases accurately describe the evolution of a spin system. In the process an echo is fanned. This sort of echo has become known as a magic echo. The first intimation that one could refocus dephasing arising from dipolar coupling was noticed by Rhim and Kessemeier [8.15,16], who discovered the effect experimentally, and showed theoretically by an approximate method Ihat an echo was fanned in which the loss of signal owing to dipolar dephasing was recovered. Following this work, done at the University of North Carolina, Rhim went to MlT where, in collaboration wilh Pines and Waugh, he extended and perfected the experimental techniques and the theoretical analysis [8.17]. In order to refocus spins which are defocused by dipolar coupling, one needs 10 be able [0 reverse the sign of the dipolar Hamiltonian. Then, as we have seen, we can effectively run the system backwards in time to undo Ihe dephasing. The trick is closely related to Ihe Lee-Goldburg experiment. The critical equation is (8.62), the truncated Hamiltonian in the rotating frame:
H, : 5.1!O.2 Gauss
/"
0.'
/:
'"
0.2 0.1
/'
~OO~5;:;-0::;.';:;-0:::J~-0:::.2~-0:;.1---'~0'--;0:;.1--:0::.2:-:0::J:-:0~.'---'0:':5,---J0.' ]cos'e~l
lal
2
• ';
-v., , -.,
.10. 1
//
,
/
•
/ /'//
A
~
/
/
2
A
/
•/0
//
~
v
/
/
/
(oF1lU doped I
Ho lll110l 2 /3(05 2
&.,/
~
0.01
11. = -",lill wIz + ,ell
0:'0--;:-;-----;;7'-----:':c--.J 0.1
Ibl
0.2
0.3
0.4
Il&.ll'lwli....:
F1g.8.7a,b. The ~ Ilnd ?oldburs experimental results for F" relIOnance in CaF, sins1e ct)'st.I~. (a~ The normllh~ second moment. of the Fit resonance for flo par.. llel to the ~lll J dued,.on, as a function of (3 COlI" - 1)/2, where B specifiCil the orientation of H
c:r
III the rotatmg rr.~ (see Fig. 8.6). (b) The normalized second moment iLS II function the .second moment III the lab frame < (4....') > LAn divided by",,' where .... = 11 e' e " elf110 1$ parallel to the crystal (1101 direetion
8.8 Magic Echoes In Secrs. 2.10.and 5.~ we saw thai one way of understanding how a spin echo comes ~bo~t IS to view me first pulse as initialing a time development of the
(3cos 26-1)
,,,
'"' B·1)·(3[z1z· I J) -1·I - I·) )
2~.
(8.62)
In this equation, () is the angle between the static field Ho and the effective field, Heff, and the Z-direction is the direction of the effeclive field. If Heff is nearly parallel to Ho,cos8 = 1 and the angular factor, (3cos 2() - l)Il, in front of the dipolar tenn is + I. However, if () = as when H t is exactly at resonance, the angular factor becomes 3cos 2(} _ 1 I (8.71) 2 =-1:
trn,
Thus, it has a negative sign. The magic echo makes use of Ihis negative sign to unwind the dipolar dephasing. Let us then rewrite (8.62), the truncated Hamiltonian, exactly at resonance, using the coordinates x, y, z in which HI lies along the x-axis: 1 1i = -"'thHII~ - 2" BjJ.:(3J~jI~k - I j .IJ,;) (8.72) ;>k
L
In addition to Ihese tenns, there is the nonsecular tenn
magneUZ3tJon under the influence of the Hamiltonian, and to view the second 388
389
1{nonsec =
~
L
3D jk
(lt rt + I j- Ii:)
,
(8.73)
j> k
which we are for the moment neglecting. How can we utilize the negative sign to unwind dipolar dephasing? For the moment, consider an experiment in which we produce some transverse magnetization [for example, by a Y(nl2) pUlse]. Let it dephase under the action of the dipolar system, for a time T, then turn on H I to produce a negative dipolar coupling. Will this refocus the dipolar dephasing? If the density matrix (in the rotating frame for our whole discussion) is initially (t = 0-) given by (8.74) where A is some constant which we set equal to I for convenience, the pulse sequence would produce at a time t a density matrix e(t)=ex p [
L
~1{zzT) Y(-Ir/2)Iz {inv}
(8.76) Bjk(3IzjIzk - Ij' h)
j>k
e{r+2T
)
=ex p [
x exp
-* (r!lHII~-~1{XZ)TI] [-
X exp ( =exp
X exp ( 390
*(-~!lHI[~ *
-
~1{~z)
*
2T
T ]
(8.77)
(8.80)
where tl is the observation period. Picking T ' = r will cause the signal at tl = 0 to correspond to the full magnetization. It is easy to extend the discussion to show Ihat if one were to hold T I fixed and reduce T, the dipolar refocusing would occur at a time tl given by
tt +r=T'
(8.81)
so that if r goes to zero, there would be a dipolar echo at t I = r ' . Note that if T goes to zero, the Y(n!2) and y-I(lT!2) pulses just undo one another, so both could be omitted. This is exactly what Rhim and Kessemeier did in their first experiments. From (8.78), it is clear that
= ex p ( I
')]
1iZZ T) Y(-Ir!2)Iz {inv}
(8.79)
Clearly when T I = T, the dipolar dephasing has vanished. Our pulse sequence is thus (reading left to right)
y- 1 (n/2) ... r,/!]
1{zzr) Y(-Ir/2)[z {inv}
[-1ii( - 2'it..
1{;z2T')] y- I (1T/2)
Y( n /2) ... T... Y -I( n /2) ... TIIl I ... TI-III ... Y(lTl2) ... tl
This expression has two dipolar temlS with opposite signs in front, but one is 1{~~, the other 1{zz, so they are not the negatives of each other. Moreover, there is also a term involving Ih which induces precession around the x-axis, whereas the initial dipolar term involves only 1{zz, the dipolar dephasing. It is easy to get rid of the precession, as was shown by Solomoll [8.18] in his paper describing the rotary echo. He showed that he could get rid of dephasing from precessing in an inhomogenous HI by suddenly reversing the phase of HI so that the spins precess in the opposite sense. We employ the same idea here. We keep HI on for a time TI then reverse it for an equal time r'. Then we get I
*(_
(8.75)
Bjk(3I~jI~k - Ij' h)
L
I e{T+2T ) = Y(1T!2)exp [_
=exp(*1i zz T')ex p ( -*1izzT)Y(lT/2)Iz{inV}
j>k
1{zz:=
Therefore, we add two pulses, a Y(1T/2) and a y-I (1T!2), where the latter is simply a Y( -1T/2) before and after the time interval 2r' giving
xex p ( -*1i zz r)Y(1T!2)Iz {inV}
where {inv} is the inverse of the operators to the left of I z and where we define 1{~~ ==
(8.78)
-i( -'''HII:2 _
iy.: y2 - 2izy
(8.83) 393
'.
Suppose now we consider two coordinate systems x, y, z and x', y, Zl, with corresponding angles 0, tP and 8', ¢I defined by (8.84). The Yim(O, tP)'s form a complete set for a given I, which means that we can always express }'i fi (8', tP') in tenns of the Yim(O, ¢)'s: }'jfi(e', tP') =
Lafim l'im(O, tP)
and
(8.86a)
m
Y,m(B,~) = L:em"Y,,(B',?,)
,
lal
(8.86b)
It is then slTaightforward, utilizing the onhogonality propenies of the l'im's when integrated over 4lf solid angle, to show that
x'
z'
".
I
Ibl f f / l / /
//
f y/
'. 1(1 f
I
I
Idl f
'.
L: Y,;.(e;., ~o)Y,m(B', ~')
Yo /'
/'\
I
(8.89)
01
m
(This theorem can be derived by making an expansion of a 5-function lying along the z-axis. One utilizes (8.86), the fact that the 5-function is axially symmetric about the z-axis, and equates the expansion in the lI",(D.?)'s to that in the
Yim(8', tP')'s] . We now wish to consider a spinning sample. To do this. we wish to define some axes (Fig. 8.8). We define first the laboratory z-axis. ZL, along the static field, and an axis zit about which we will eventually rolate the sample. The angle between ZL and Zit is 00 (Fig.8.8a). We can then, without loss in generality, define the laboratory axis XL to be perpendicular to both zL and Zit (Fig. 8.8b). We define XIt to be coincident with x I, (Fig.8.8c). The axes xn.. !JR, and Zit are fixed in the laboratory frame (Fig. 8.8d). We then define axes xs, YS, and zs which are fixed in the sample with zs coincident with Zit at all times, and Xs and YS coincidem with Xlt and Yit respectively at time t = 0, but making an angle nt at" later times (Fig. 8.8e). n is the angular velocity of sample spinning. Utilizing these definitions, we can specify the orientation of Ho in the coordinate system Zit' YIt, Zit (at angle 00R, 4toRl and in the system zs' !IS' zs (at angle Dos, ¢os)·
/
"
(8.88)
They can also be found using the formalism of the Wigner rotation matrices (see for example the text [8.23]). Let us now specify the orientation of the z-axis as being at an angle (9'0' ¢o) in the primed system. Then there is a famous theorem, the addition theorem for spherical harmonics, which tells us that
/
/
/
"
'. "
= b31 x + b32Y + b33Z
Y,o(O,O)Y,o(B,~) =
'.
/' /
)//
=b II X+bI2!J+b 13 Z
y' = b:ll x + b22!J + b23Z
'.
"
'.
'. "
(8.87)
Noting the forms of the Yi", 's in rectangular coordinates, we realize that the coefficients all'" or Cnll' could be found by explicit substitution of the relationship between the primed and the unprimed coordinates:
"
I
'. --
I
J../
/
Yo y,
'. '."
Flg.8.8a-e. Axes imporh.nt ror sample spinning. (a) The laboratory z-axis, ZL, is chO$en along the static field /lo. The axis ZR makes an angle 80. Together. ZL and ZR define a. plane. (b) The lab axis %L is defined as lying perpendICular to the (ZL, ZR) plane. (e) The axis %R ill coincident with %L. (d) The axes lIL And YR lie in the (:R,zd plane. Af{ the axes %L.1IL,rt. and %R,YR,zR are fixed in t~e laboratory rrame. (e) The axes %s,VS. and Zs fixed In the sampl~ and rotate with il, Zs coincidins chosen to be coincident with ZL With ZR. At t = 0,Z5 and ZR, so that it makes an angle Ot with them at later times
" u:e " "
I'
From the figure we see °OR=OO,
Oos = nOR = 00 '
(8.90,) ¢os =
tPOIt -
nt = 7rn - nt
(8.9Ob)
We now consider the interaction between a pair of spins j, k which we have previou.sl~ w~tten as proportional to 3cOS 2 0jk - I (3.7). We now, however. must dlstlOgUish between coordinate systems. hence we replace (Jjk by OLjk> writing the coupling 2 3COS 0Ljk - 1 = JI67r/5Y20«(JLjk, tPLjk) (8.91) Then, we can utilize (8.89) to get Y20(O, O)Y20(OLjk, ¢I,jk) = LY2"m(00s, ¢OS)Y2m(OSjk> tPsjk)
(8.92)
m
395
where 0Sjk, ¢Sjk give Ihe orienlation of the internuclear vector Tk in Ihe coordinate system xS' Ys. zs which is fixed in the sample and where Oos and ¢os are given by (8.90). Substituling the explicit expressions for the Ylm'S from Table 8.1, we get
1
3 COS 20Ljk _
= (3COS
2~OS -
1)
We turn now to a discussion of the effect of spinning on a pulse experiment in which we observe the free induction decay of a spinning sample following a -;r/2 pulse. Then we might express the dipolllr Hamiltonian, 11.d as
(3cOS20Sjk _ I)
j
02
"
1/n where
Tjt"" I/.6.wd
(8.103.) (8.103b)
These can be rewriuen to give (8.97)
3 2
{»
'" (b)
..
I (I's/
""
f!
,I
3
P
"
"" ii'G~
2
f"'
__'_ r'W t F(11(') = exp ( -
Y2.,O(O,O)
(8.115)
where (a, fJ) gives the orientation in spherical coordinates of the spinning axis
[(I(LO-O'LO) (3Ca"a-l) 2 {1+(I(-'i7)+ P)] (3CO"Bo-I)} ' _O'TR)(,inaCOS2 + (} 2 2 .
1£timeindep= ~'YfIHo[,
(8.116)
(This expression can be checked in the limiting case that 80 = O. hence the spin axis lies along Ho. Then the spinning has no effect and the time-independent tenn is the entire Hamiltonian. Then 0' = 0 or 0' = rr(2, {J = 0 or 1r(2 correspond to.flo lying along the three principal directions. The result is that for the three cases we get shifts respectively 1(3 - 0'3. 1(O)
(8.119)
and recalling that T = 27f/n. we see that when
Y;,_,(a,p) °0,-2::
%s in the Xp. yp. %p (principal axis) coordinate system. The result is
\TR
"1JI
Utilizing the fact that
and the spherical hannonic addition theorem (8.89)
Y;o(a,p)
+(I( -
. [ + A,(P,S) 2il ( sm 2ilt - r2(p,S)] + sin r 2(p,S)} )
(8.114)
C_',o(P,S) = a;,_,(S,P)
ao,o =
([1
. [ + AI(P,S)( n smm-r 1 (P,S)]+sinrl (p,S)}
(8.87) = a;,o(S,P)
(8.117)
where AI(P.S) and A2(P.S) are coefficients, and rl(p,S) and r2(P.S) are phase angles, all of which in general will depend on the orientation of Zs in the xp, yp, zp system. Utilizing this fonn, we get
Thus, it is identical to the fonn taken in a liquid in which the anisotropic shift contributions average to zero. If 80 is nO[ at the magic angle we might wish to know the coefficients CO,o(P, S), C2,O(P, S), and C-2,O(P, S). These we get from the inverse relalions
CO,o(P,S)
[nt - rl(P.S)]
+ A,(P,S)co, [W. - r,(p,S)))
(8.112)
The coefficients C 2,0 (P, S) and C- 2 ,0 (P, S) can be obtained (if desired) as we explain below. But even before that, we nOie thai when 80. the angle between the rotation axis and Ho. is 3t the magic angle. we gel 'Htime indep =
erties of the Y2ms 's (Table 8.1) we see that 1£ is a sum of tenns which are either independent of time. or vary as exp(int), exp(-inl). exp(2int). exp(-2int). Keeping in mind that the Hamiltonian must be real. and that we are at the magic angle,.we can thus write
t=nT
•
n=O,I,2 ...
'!>(I) = exp ( - i'YHo1,(1 + [( -
(8.120.)
a)') ,!>(O)
(8.12Ob)
just the result we would have if the time-dependent terms were missing. Note that this result says that at the magic angle the time development over integral multiples of the rotation period is independent of the orientation of the crystal axes relative to Ho. Thus, if we applied a 7f{l. pulse to a spinning sample. the free induction decay at times given by (8.120a) would act as though there were no anisotropy to the shift tensor. If one had a powder sample which was not spinning, the spread in precession frequency arising from shift anisotropy. 6.wshif~' would cause the free induction to decay in a time 1/6.wshif~' If, however, the sample is spinning we can see from (8.120b) that the signals are rephased at times given by (8.120a). thereby producing a string of echoes. Notice that we have not placed any requirement as yet on how fast the spinning must be, in contrast to the previous siwalion concerning narrowing of dipolar line broadening. For the dipolar case we compared the rigid lauice line width ~d with n. What frequency should we compare with w in the present case? Dimensionally, there is only one relevant frequency. the total excursion
403
of the precession frequency over the spinning cycle. Note that all our equations so far imply there is a single crystal being rotated since there is a well-defined orientation of zs in the xp, yp, zp coordinate system. However, for a powder, all possible orientations occur. For the rest of our discussion we will focus on a powder. Then, the maximum frequency excursion one can have in a rotation cycle is less than 1'Ho[(J(zI'Zp - qzpzl') - [(I{:cI':C1' - q:cp:cp)]
== .6wshifL
n '"
404
•
(8.121)
where we are defining the zp and xp axes as having the maximum and minimum shifts respectively. So we ask, what happens when we vary the relative sizes of fl and .6wshift.? Suppose, first, that the powder sample is not spinning. Then, as we have noted, following a rr/2 pulse, the transverse magnetization will decay to zero in a time :::::: l16.wshift. The Fourier transform of this free induction decay would give us the line shape. If we then applied a 11" pulse at time tll" we could refocus the magnetization into an echo. If t1l':» 1/.0.wshifL' the echo and the free induction decay would be well separated in time. Indeed, we could apply a string of such 11" pulses (a Carr-Purcell sequence) pnxlucing a string of echoes. The Fourier transform of anyone echo will give us the powder line shape in frequency space. If, instead, we took a Fourier transform of the sIring of echoes, we would now have introduced a periodicity. The resultant J1ansform differs from that of a single echo in the same way that a Fourier series differs from a Fourier integral. The transform of the string of echoes would consist of 5-functions spaced apart in frequency by the angular frequency l/t rep where t rep is the time between successive rr pulses. The Fourier transform of a single echo would be the envelope function of the spikes. If now one shortened t rep , one would get to a point where t rep :::::: lI.6wshifL' Then the echo would not have decayed completely to zero when the 11" pulse is applied, so that over a single period one no longer would have the complete shape of a single free induction decay. The Fourier transform of the sequence of echoes would still have a center line plus sidebands spaced in angular frequency by 1/t rep , but the envelope of these sidebands would not be the powdeI line shape (the rransform of a single free induction decay). Stejskal et al. [8.27] realized that the same situation would arise from magic angle spinning. When n« .6wshifL, the NMR signal following an initial 11"/2 pulse decays rapidly compared to the period of rotation, hence is essentially identical to what it would be with O. However, after one full rotation there is an echo, as described by (8.120). Indeed, a train of "spinning echoes" will be formed, analogous to a Carr~Purcell train. The Fourier transform of any echo will give the 0 powder line shape. The Fourier transform of the train of echoes will give o-function spikes spaced apart in angular frequency by with an intensity envelope versus frequency identical to the nonspinning powder patlern. This situation is illustrated beautifully by Fig. 8.10 showing the data of Herzfeld et al. for a p13 compound [8.28,29].
n '"
Proton decoupled p3l spectra at 119.05 MHz of barium diethyJ phosphate spinning aL the magic angle. The figure illustrates how, in the limit of slow spinning, the spinning sidebands reproduce the shape or the Ilonspinning spectrum [8.28] .'ig.8.IO.
I'AQT- OkHz
n,
"OOT - 0.94 kHz b
I'OOT"
2.92 kHz
d
16
,
a
0
-
a
, -16
Freq (kHz)
n
Once becomes comparable to or larger than .6wshifL> the satellites occur outside the envelope of the nonspinning line. As can be seen fIOm (8.117), the coefficients A\(P,S) and A2(P,S) ale multiplied by lin, hence become progressively smaller than the lalger n. This is part of a general theorem discussed by Andrew et al. [8.22] that the second moment is invariant under rotation. Thus, the intensity of lines spaced by multiples of must fall off as l/n 2 in the limit of large n. When there are many spinning sidebands, how can one tell the central line? The easiest technique is to change n. One line will be undisplaced - the cenrral line.
n
405
The beauty of the slow spinning regime of Stejskal, Schaefer. and McKay is that it gives one simultaneously a high precision determination of the isotropic average shift, as well as a picture of the associated powder paHem. It also removes the difficulty of spinning at a rate faster than ~Wlihirt., a genuine difficulty in the strong fields of superconducting magnets when one has large shift anisotTOpies. A common use of the slow spinning regime is to slUdy low abundance nuclei (e.g. C 13 ). so that there is no like-nuclei dipolar coupling with which to contend, combined with standard decoupling to any high abundance nuclei like HI whose mutual spin-flips would otherwise broaden the spinning side bands. In some instances it is desirable to remove sidebands in order to simplify the spectra. This can be done by applying 1( pulses synchronously with the rotation as shown by Dixon [8.30) and by Raleigh et al. [8.31].
H( 1 r[5 ) =
406
S
M[ - -3-
r lS
r lS
(8.122)
We consider M[ to lie along the k-direction, and consider three cases in which S lies at a distance a respectively along the X-, y-, and z·axes (see Fig.8.ll). Application of the formula shows that when S is on the z·axis it experiences a field 2M, HI(O, 0, a) • - 3 (8.123) a whereas when S is on the x· or y-axes HI(a,O,O)' HI(O,a,O)' - M.,I (8.124) a If, for some reason, S were to jump rapidly among the three positions, spending equaJ limes in each on the average, it would experience a time-averaged field < H/ > given by
8.10 The Relation of Spin.F1ip Narrowing to Motional Narrowing The strong spin-spin coupling characteristic of solids was essential for achieving the high sensitivity of double resonance. Its disadvantage is that it produces line broadening, and thus may obscure details of the resonance line such as the existence of anisotropic couplings (chemical shift, Knight shift, or quadrupole interactions). Within the past few years, a varielY of clever techniques have been introduced which utilize strong rf pulses to eliminate much of the dipolar broadening. We now lum to the principles on which the ideas are based. The highly ingenious concepts owe their development to several groups. The pioneering experimenlS were performed by two groups, one headed by John Waugh [8.4], Ihe other headed by Peter MallSfitld [8.S]. Subsequently Robert Vaughan [8.6] and his colleagues have contributed imponantly, as have many of the scientists [8.32,33] who worked with Waugh, Mansfield or Vaughan. The essence of these schemes is to apply a repetitive sel of rf pulses [8.3439) which produce large spin rotations and which, by a process akin to motional narrowing, cause the dipolar coupling to average to zero. Each cycle consists of a small number of pulses (8 pulses per cycle is common). To achieve the effect, Ihere need to be many cycles within the normal dephasing time of Ihe rigid lattice line width. Since the rf pulses must produce large spin rotations (90" pulses are typically used), and since the rotations should occur within a small fraction of a cycle of pulses, one needs H ,'s which are large compared to the rigid lauice Ijne breadth. We have remarked on the similarity between the multiple pulse schemes and motional narrowing. We stan by making the analogy explicit. For convenience we shall name the multiple pulse schemes "spin-flip narrowing". Consider two spins, I and S. Let 1" [5 be the vector from spin I to S. The magnetic field H[ at spin S due to I is then
3(M/-rIS)r/5
(8.125)
, ,..---... , ,
,,
,,
,
/
, s (,j
,
, I
x
s
"F----
(b)
s
\, ,I
, ,,
IF----
(c)
Ag. 8.lla-e. The magnetic field or [ III S for lhree localions of S all the nme distance, a, from ,pin I. The dashed line indicates II. magnetic line of force
407
This result is the essence of motional narrowing. Though for our case we picked only discrete locations for spin S, the same result is found for continuously variable sites for which the strength of the interaction goes with angular position 81 s of rlS with respect to the z-axis as 3cos 2 81S - I. This averages to zero over a sphere. Suppose now we consider a variation in the above picture. Suppose we position S at (0,0, a), Le. on the z-axis, and consider the orientation of both M I and Ms (Fig. 8.12). In tenns of the orientation of the vector r I S with respect to MI' it is useful to introduce some names. Referring to Fig. 8.11, we denote the (c) configuration as the on-axis position of S, we denote the position of S in (a) and (b) as the side position. Note that for the on-axis position HIS is parallel to MI, whereas for the side position HIS is anti-parallel to MI and half of its on-axis magnitude. Referring now to Fig. 8.12, we see that in (a) and (b) S occupies a side position, whereas in (c) it occupies an on-axis position. Moreover, since we have taken M , and Ms parallel in all parts, for configurations (a) and (b) the magnetic energy E llIag = -Ms' H,s is:
a quick rotation of both spins to the side position of Fig. 8.12b. Wait another time 'T. What is the average magnetic energy < Em > over the interval 3'T? Using (8.126) and (8.127) we get
- .!...(-2MI MS'T MIMs'T ll'hMST)_o + + a3 a3 a3
< E m> - 3T
.
(8.128)
That is, over such a cycle in which, at stated times, we apply selected 7fn. rotations to both spins, we can make the magnetic energy average to zero. Thus by spin-flips we can cause the dipolar coupling to vanish just as we could for position jumps in Fig. 8.11. This is the principle of spin-flip line narrowing. We make the dipolar energy vanish by flipping the spins among selected on-axis and side positions, spending twice as long in the side positions as in the on-axis ones. All of the complicated pulse cycles are based on exactly this principle.
8.11 The Formal Description of Spin-Flip Narrowing
(8.126) whereas for (c) Emag)e =
-Ms' HIS =
-2NhMs
(8.127)
a3
Suppose, then, we began with the configuration of Fig. 8.12c, the on-axis arrangement with MI and Ms parallel. After time 'T let us quickly rotate M I and Ms by 7fn. about the y-axis to the side position of Fig. 8.12a. After a 'T give
,
(,)
S
~x
, (
(b)
S
(c)
S
"
" (
FIg.8.I2. The two spill!l Ai, and M. are oriented parallel to one another. In (a), (b), lind (c) their magnetic moments are parallelrespcctive!.y to the Z-, Ih and z-directiolls
408
From the previous section we see that it is possible to cause the dipolar interaction 10 average 10 zero if we cause the spins to flip between the "on-axis" and "side" arrangements. For the conventional motional narrowing, narrowing occurs when the correlation time Tc and the rigid lauice line breadth (in frequency) DwRL satisfy TcDwRL $ I
(8.129)
Withoul jumping, a set of spins precessing in phase initially get out of step in a lime ..... (l/OwRL)' The condition expresses the faci that for motional narrowing to occur, the jumping must occur before the dephasing can take place. In our example of motional narrowing, the correlation time would be on the order of the mean time spent in one of the three configurations of Fig.8.11. If T is the time in anyone orientation, roughly Tc
:::::: 3T
.
(8.130)
In a similar way we expect that spin-flip narrowing will worle only if the spins are flipped among the needed configurations before dephasing has occurred. Thus we get a condition on T:
(8.131) The better the inequality is satisfied, the longer the spins will precess in phase. We expect, therefore, that we shall wish to flip the spins again and again through lhe configuration of Fig. 8.12. This we can do by applying a given cycle of spin-flips repetitively. The basic cycle will bring Ihe spins back to their starting point at the end of each cycle. 409
A fonnal description of what takes place begins with a specification of the Hamiltonian. We shall use it to compute the development of the wave function in time. We write lhe Hamiltonian as 1i(t) = 1irr{O + 1iint
where
(8.133) (8.134)
1t3 = 2: Bij(rjj)(Ii .Ij -
(8.135)
'Ho includes the chemical and Knight shifts qUi."H~ is the secular part of the dipolar coupling, the coefficients Bij(f"ij) including the distance and angular factors [see (8.54a»). The objecl ofspin-jlip narrowing is to cause ~ to vanish while maintaining 1to nonzero. To illustrate the principles of spin-flip narrowing, it is useful to idealize the situation. Correclions to the idealization are important as a practical matter. We return to them in Sec!. 8.13. The idealization is to consider that 11rf is zero except for very short times during which it is so large that 'Hint can be neglected in comparison. This approximation enables us to say that 'Hrr produces spin rotations at the time a pulse is applied, but that between pulses the wave function ¢(t) develops under the action of the time independent Hamiltonian 'Hint. Thus between pulses the wave function at lime t can be related to its value at an earlier lime t I by (2.49) ¢(t) = exp{ - (ilh)'Hint(t - tt»)¢(t t)
(8.136)
which defines Uillt. The effect of pulse of amplitude HI and duration t w along the a-axis (a = x, y, z) at time tt is to produce a transfonnation by the unitary operator Pj for the ith pulse, 410
Pi = e i(1rj2)lo
(8.138)
if the pulse produces a 1r/2. rotation about the a(= x, y,or z) axis. Let us consider a three-pulse cycle for simplicity and concreteness. We always begin an experiment by having a sample which has reached thermal equilibrium in the magnet. We then tilt the magnetization into the :t - Y plane at t = O. We call that pulse the preparation pulse. After a time TO we begin applying the repetitive pulse cycles. Choice of TO and the phase of the preparation pulse is made in tcnns of producing a signal at some point in the pulse cycle which is convenient for observation (at a time which is called "the window" in the spin-flip literature). Let .p(tn)be the wave function after n cycles [Le., just before pulse PI of the (n + I)th cycle]. Then we can follow the wave function in time with Table 8.2. Table 8.2. Varialion of the wave function with time for. three-pulse sequence Time
3I::iI::j)
ij
= Uint(t - tlhb(t)
(8.131)
where, for example, "'{Htt w is chosen as 1ffl giving
(8.132)
We shall work in the rotating reference frame with the z-axis defined as the direction of the stalic field. 1irr{t) is the coupling to the applied rf pulses used to apply the spin rotations. It is time dependent because the pulses are switched on for very short intervals, two In principle the corresponding H) can be oriented along the x-, y-, or z-axes. Selection of the x- versus the y-axis is a question of the phase of the rf pulse. Though pulses along the z-axis can be applied in principle, in practice they are not used since they would require addilion of ,mother coil 10 the rig. However, a z-axis rotation can be achieved by two successive rotations about the x- and y-axes. (Show how this is done!) The tenn 1tin~ consists of two tenns, 1tint = "Ho+~
¢(tt) = ei..,lI,tw l o .p(ti)
I; (just before PI) (just after ~)
It t.
+ T,-
(just before 1'\)
+ Tt (just .ner P,) I. + Tl + Ti" (just before 1\)
,,= "('.)
.p=P,¢(I,,)
'" =exp( -i1ti•• 1"1 j")~ ¢(t.. ) =UI.,(TdPt.p(I.)
t.
¢ =
I.. +TI +T1+T3-
tP = Ui•• (T1)1'\ Ul •• (Tl) PI ,,(t.) ¢(I,,+d = UihtCr~)/"JUi ... (T1)P:!Ulnl(Tl)Pl\/.,(t,,) == UT ¢(! .. ) defining UT
Dust before (II
+ l)th cycle]
~Ui .. (TdP,
tP(t.)
We can thus write (8.139) where UT is independent of n. Since UT is a product of unitary operators, it is itself unitary. After N pulses, we have
,p(tN) = u!f,p(to)
(8.140)
so that the problem can be considered solved if the effect of UT can be deduced. Let us examine one cycle, and introduce the unitaty operators p j - I which are the inverses of the Pi'S. For a unitary operator Pi-I = P/
(8.141)
where the" stands for complex conjugate. 411
Then we write UT as UT :::: Uill~(T3)P.JUin~(T2)P2Uin~(Tt )Pt :::: P3 P2 PdP\-t p 2- t P3- 1Uinl(T3)P3P2 PIHP.-l p 2- t UinL(T2)P2Ptl
X [P1-tUinl(T)Pd
(8.142)
Now we showed in our example of Sect. 8.10 that a complele cycle of spinflips should get us back to the starting point so that we can repetitively flip the spins among "on-axis" and "side" configurations. Therefore the cycle PI, P2, PJ should get us back where we started. Hence (8.143) giving us
(8.144a)
:: Uinl(TJ)[PI- 1P2- 1Uinl(T2)P2PI][PI-IUinL(TI)Pl]
(8.144b)
The meaning of the individual tenus can be made evident by several transformations. Erst, consider a unitary operator P and a Hamiltonian 1t, and the corre· sponding U: P-IU(t - to)P:: p-l e -(i/l)1{(t- 1o) P
.
(8.146)
so that the effeci of p- 1 UP is 10 cause U to develop in time under a transfonned Hamiltonian. If we were to evaluate the expression in the exponent of Ihe transfonned Uin~(T2)'
Pit P2-I1tinL(T2)P2Pj
(8.147)
we could first do the trunsfonnation p2-t'HinlP2, then sandwich the result between p t- 1 and PI and evaluate that, using the appropriate expressions for exponential operatOrs. 111is order of application of the opemtors is the rever.fe of the order in time of application of Ihe spin rQlation operators. Why is that? Consider a SchrOdinger equation
_':.iN=1i'" (8.148) i 'f' We can transfonn this equation with a unitary operator R, which is independent of time, to the problem
at
412
(8.150)
as the 7t corresponding to the rQlation P- I • We can consider the following two descriptions of the effect of 1t~ following 3. 1fn. rotation: i) Use ~ untransfonned acting on a
t/J which
is rotated
+wn.
ii) Leave the spin function alone bUI rolate the spin coordinates in ~ by
-wn., the inverse of lhe spin rotation in (i). Equation (8.144) corresponds to (ii).
(8.145)
By expanding the exponential, insening p-l PC:::: 1) between factors, and regrouping, we find that p-I U(t - to)P = e.p[ - (Uh)(P-'1iPXt - to»)
which leaves the problem lhe .fame. Thus, if R were a spin rotation operator giving a +1r{2 rotarion of the spins about some axis, R1tR- 1 musr transfonn the coordinates of Ihe Hamiltonian corresponding to the same rotation. If 1t:: -..,hHoIt and the spin is in a Slate corresponding to the spin-up state, a transformation, R, which rotates Ihe spin function into the up spin state along the +y-direclion requires rotating 1t Ihe same way, which is done by replacing I: by I" in 1t. We can therefore interpret p- l 1tP:: (P- I )1t(p- I )-1
UT :::: (P1- 1P2- 1P;'Uinl(T3)P:! P2P j][P1- 1P2-IUinL(T2)P2Pd X (PI-IUinl(TI)Pd
(8.149)
If a rolalion is made up of several rolations in succession, the inverse consists of lhe inverse rotations perfonned in the opposite order. Thus, if the spin is flipped by P2P\, the inverse transfonnmion Q is (P2PI)-I, so the 1t transformed by the inverse is
(8.151) The prescription for finding the transfonned 1t conesponding to the ith interval Ti of an n pulse sequence is to take the spin rotations PI, P2 Pi, which preceded the interval and transform the coordinates in the Hamiltonian by applying rhe inverse rotations in the reverse sequence (e.g., first p;-t, then Pj-=.ll , ••••... , lastly PI-I). We now define the three rransfonned Hamiltonians 11.", 'HB, and 'He as 1t" :::: PI-I1tintPI 1tB :: P l- l P2-I1tinLP2PI
(8.152)
11.e :::: 1tint On expanding the exponentials we get
413
('H~)B : 'L Bij(Ii ·Ij - 3lzi l zj )
UT =ex p ( -*1iCT3)eXp( -*1iBT2)eXp(-*1i"'11) = 1-
i<j
i
(~>c
1i (1iCTJ+1iB1'2+1-£",Tt)
• L: B(I; . I j -
3I,;I,j)
(8.158)
i<j
and
- (*Y(1iCT31iB1'2 + 1tC7'J1-£", 11 +1iB1'21iA11)
T) :
7"2 : TJ.
(8.159)
2 2 +'H. 2 7"22 + 'H.",7"I) 22 + ... +-l(i)2 - (1-£C7"3 B 2 I,
(8.153)
If the 7"'S are short, (8.154) where by 1(I'HA,B,clhl)1 we mean a quantity of the magnitude of typical matrix elements of the transformed Hamiltonians. Under these circumstances, UT is well approximated by keeping only the leading two teons on the right of (8.153). The condition on 7"i is similar to the requirement on the correlation time 7" for there to be motional narrowing. Introducing the period of a cycle. t c = Tl + 7"2 +"f'J, we get
(T'J
7"2 7") ) UT = I - -i 1ic- +'H.B-+'H.Atc II tc te te
i __ = I -1i'Hinttc ';:!
ex p ( -*'H.inttc)
(8.160)
Setting w : wo for simplicity. we would then get
Ho :
liwo"" -3 ~ azzi(Izi + I y ; + lzi)
,
(8.155b) (8.161) (8.155c)
'H.in~ : -h(wo - w)Iz - liwQ 'Laui1zi + 'H~ (8.156)
The trick then is to choose the pulse cycle (PI> P2, P3, etc.) so that we eliminate the dipolar coupling but maintain Ihe chemical shift and Knight shift information: (8.157.)
-h(wQ - w)lz - hWQ 'Lazzi1zii-Q
(1io )", : -h(wo -w)III-llwo 'Lazzi1yi
(8.155.)
where (b) defines 'H.in~' the average 1iinh and (c) serves to remind us that over a cycle the system develops to a good approximation as though 1iint were replaced by its average 1iint' defined above. Now we expand. 1iin~ into its elements
" 110 + 11~
These results could be achieved if p,-I were equivalem to a 1r/2 rotation about the x-axis which would transform Izi imo [yi and if P 1- 1p-I were a 1r12 rotation about the y-axis which would transform Izi and Izi. 2 These pulses would also transform 'Ho
(8.157b)
(8.162) Equation (8.162) shows that the chemical shift and Knight shift are reduced by v'3 (i.e., multiplied by 1/Jj) from their value without averaging. We have succeeded in making the dipolar coupling vanish without eliminating the chemical shift. To the eXlent that the term in (w - wo) is included (it represems frequency offset or field inhomogeneity). it is also reduced in the same proportion. According to (8.139) and (8.155c), ¢(to) and f/J(t) are given by
,,(t) . ul( ,,(to) : exp( - (i/li)1tinl(t - to»)¢(to)
(8.163)
provided
t = to + Nt c
(8.164)
How abour ¢(t) at other times? Consider a time t in the Nih interval such rhar
Thus if
(~)A : 'L Bij(li .Ij - 3111iIlIj)
t:tO+Ntc+tl
(8.165)
i<j 414
415
The time t) could fall into anyone of the time intervals T), T2 • ••• Tj which make up the basic cycle t c . Over anyone complete cycle there is no large change in ¢ if (8.152) is true. However, each large pulse produces a sudden big spin change _ typically a 1fo, rotation. These large pulses might, for example. progressively cycle the spin along the X-. y-. and z-axis in the rotating frame. If one always observes during the ith interval. however. the effect of the big pulses in successive cycles will always have returned the nuclear magnetization to the same direction in the rotating frame. Thus we can write ,,(t) = e/dt during turn-on and dtPldt during turn-off equal and opposite. Making l[tPldt zero is not practical, but the balancing during turn-on and tum·off is. Among the various pulse sequences used or referred to in the literature, we shall list three. They are known as WAHUHA, HW·8, and REV-8. These mysterious symbols are made up from initials of their inventors plus numbers which tell how many pulses there are per cycle. The WAHUHA is named after J.S. Waugh, L.M. Huber, and U. Haeberlen
[8.34J. \( is X, T(T,X, T. Y,2T, Y,T,X, T)
This sequence can be sampled at the times corresponding to just before the parenthesis by measuring M lI • The name WAHUHA refers to the contents of the parentheses. The preparation pulse can be chosen to give convenient sets of timing. The panicular sequences shown have the property that the X,T, at the start are in fact like the last pulse and interval of the cycle; hence no special timing intervals are needed at the stan. That is, the preparing pulse in this case is pan of the cycle. For the preparing pulse shown, the magnetization is initially turned to the y--direction. Thus, if one detectS My at the inverval between the X and -X pulses, one will get a signal. We can write the HW-8 [8.35] as (T,X,2T,X, T, Y,2T, Y, T, Y,2T, Y,T,X, 2T,X, T)
There are a number of equivalent variations. The MREV-8 [8.38] sequence is
X, T(T, X, T, Y,2T, Y, T,X,2T,X, T, Y,2T, Y, T,X, T) This can be viewed as pair of pulses X, Y separated by T followed at 2T later by their inverse y, X. MREV demonstrate the highest resolution obtained with their MREV-8.
8.14 Analysis of and More Uses for Pulse Sequence The importance and utility of Illultipulse techniques goes far beyond spin· flip line narrowing. We turn now to several examples, as well as to illustrations of some calculations of the effects of the pulse techniques. Before beginning our discussion, it is useful to introduce a notation and 10 demonstrate some useful relationships. We shall be following the effect of pulses on individual components of spins such as I:r:, I y • or I: and on dipolar tenns. For the dipolar tenn we introduce the notation 'Hon(a = x, y. z) defined as
423
1i&& =
L
(8.191)
Bik(/j ·lk - 3I&jI&k)
i>k Explicit calculation then shows that
1iu +1iIlIl +1iu =O
~4~J
.
(8.192)
In general. the effect of pulses Pi will be rotations. To discuss the effects of nonzero pulse lenglh:i. one must treat the Pi 's as time-dependent operators such as (8.193) where t is the time variable during the pulse. For our present purposes. however. we shall Ireat the pulses as negligible duration (so-called 6-function pulses) and moreover treat them as all being 1tll pulses. We then have to evaluate expressions such as
P1- 1I: P,
(8.194a)
P,-t HuPt
(8.194b)
But once we know what (8.194a) is. it is easy to get (8.194b). For example. let
Pt = X
~.I~
using right-handed rotations. For a pair of pulses PI followed by P2. then the lefl-handed rotations
PI-tP2-t1&P2PI
(8.I99b)
would be replaced by the right-handed rotations Pt.l'2. etc. p)
P21&P2-J p)-t
(8.199wI
(9.30)
We assume that W3 -"'2
4-
"'2 -WI
(9.31)
not e32). and that we have previously excited 1!21 (hence e12), but not 1!23 (hence Fig.9.2. in spins of pair a for shown is system a such of ion An actual realizat Assuming lhey have a coupling of the form (9.32) rand 5, and are identical except for a chemical shift difference between spins ed in discuss we as cies, frequen t differen at occur the four allowed transitions Chaptet 7. Let us rewrite (9.24) by multiplying both sides by cxp(iw jlt). Then .I t del ./ i · (9.33) ~ :: h ~ [ejke+lI ·Ik/IVkl (t) - Vjk(t)e + "-'J1- I!I./] with the This is actualJy just what we would have gOllen if we had started (9.21)] ing [follow equation for I! in the interaction representation, with Vkl(t) :: Vkl(t)ci"-'k/t
etc. Then, 438
(9.34)
....
,
I•• )
2
...., h -' h ex""'---· E, E,,.. labeled 1, '1 , 3 wilh ener&ies E, " levels lergy :>-- 10.01
V
1111
--(1119 100 1
11,01
r--_ \
11019100)
"-
\
-.;1
10,0)
11,11---
Fig, 9.4. (a) An alternating potential joining the states J10) and lit) can couple (lotf?IOO) to (t1If?IOO). If we wanted to insp«tan element (tllf?IOO) whidl is itself not directly observable, we a~ uJ\$uccessfulsince we can only pump it into an element (tOI(lIOO), which is also unobservable. (b) If we produced (toteIOO), a furlher IIpplieation of \' tuned to the JI- t) -110) trllnsition will produce (t -lleIOO) which is still nOl observable 442
=
--yhhOFz + hahrh: _ -yh( HI; H2 )Fz
--yl{ HI; H2 )(lb - hz)
(9.49)
The tenn Itz - hz now connects singlet to triplet states, as can be seen by evaluating matrix elements between Ill) and Joo) using the spin-up and spin· down functions it and fJ fO express the IFM F) Slates:
III) = ,,(1),,(2) .nd
=
For the two spins, we have F = 0, M F 0 for the singlet, F I, M F = I, 0, -I for the triplet. We now demonstrate a type of matrix element of t! which we cannot probe. Since
101
1ieff = --yllhoFz +haIlzl2z --yh(HIIlz +H2I 2z)
(9.50.)
I
1(0) = j2[,,(I)P(2) - P(I),,(2)]
(9.50b)
We find that
1
(II Ill. - 12' 1(0) = - j2
(9.5Oc)
Thus, if we had initially a nonzero malrix clement (10IU' ' = 2e- iS7t2 cos [a(t} - r - t2)/2] cos (ntd sin (f}r)
the conversion of the initial z magnetization to one-quantum coherence for the preparation period, the fact that zero-, one-. and two-quantum coherences have been brought into existence for time interval tl' and the conversion back to single-quantum coherence for observation. We have seen that the pulse sequence X(-rr/2) ... r ... XC1r/2) will produce zenr, one-, and two-quantum coherence. As we remarked earlier. if we study column 10 of Table 9.1, we note that the zero- and two-quantum coherence terms (lines I and 3) are proportional 10 cos f}r, whereas the one-quantum terms (lines Therefore. if = 0 (exact resonance) we 2 and 4) are proportional to sin would not produce one-quantum coherence. The question, how, when we are tuned exactly to resonance. can we produce one-quantum coherence with a pair of spins. turns into the question. how do we produce odd-order quantum coherence for an arbitrary number of spins. when tuned exactly to resonance? We turn now to a simple approach. The disappearance of the one-quantum coherence for a pair of spins can be traced back to column 5 of Table 9.1 where we see the effect of the spinspin coupling alone [i.e. before the resonance offset term f}r(1z + Sz) acts]. It produces (ly + Sy)C; on line I and -(lz:Sz + IzSz:)C:; on line 3. These tenos, if acted on by X(1r/2), produce (!z + Sz)C; (a zero-quantum coherence) and -(!z:Sy + I y S z )2S:; (two-quantum coherence). To get the one-quantum terms we allowed f}T(lz + Sz) 10 turn I z into -Iy • I y into I z , then we applied an X(1r!2) pulse. Therefore, if we have f} = 0, so that I z and I y are not rotated by 1r/2, we should get one-quantum terms by rotating the pulse from being an X(1r!2) to being a Y(1r/2) pulse. Indeed, we see that a y(7I'"!2) pulse would leave (Iy + Sy)C; alone as a one-quantum coherence. but tum -(IzSz + I z S z )2S; into (IzSz + I z S z )2S:;, a one-quantum term. In general, of course, f} =f O. Then the points we have just discussed show that production of a particular order of quantum coherence depends on the amount one is off resonance. That is frequently a disadvantageous situation. It can be remedied by making f} effectively zero by means of a spin echo, then choosing the phase of the second pulse to get the desired order of coherence. Thus. we can use a pulse sequence
(9.76)
which, in complex fonn, gives
(F+)I = ~[e-i(n-aj2)12(ei(!Haj2)tl +e-iC l?-aj2)1 1)
x (eiCS7+aj2)T +e- i(S7-a/2)T) + e-i(J?+a/2)t2(ei(l?-aj2)t1 + e- i(l?+aj2)t 1 ) x (e i(S7+a/2)T +e-iCS7-a/2)T)]
nr.
For two-quantum coherence we get
(F+)2 = _2ie- int2 sin (at2/2) cOS(2f}tl) cos (f}r) sin (ar(2) = ~ [e-icn+a/2)t2 _ e- i(l?-a j 2)t2]
8
x (e2int1 + e-2int1 )(ei(S7+a/2)T _ eicn-aj2)T +e-iCS7-a/2)T _ e-icn+a/2)T)
(9.77)
Referring back to (9.62), we recognize thac the elements of the density matrix will have time variations
e+ i(l?±a/2)t
(9.78.)
for the states M - M' = +1, will have e- i(l?±aj2)t
(9.78b)
for M - M/ = -1, will have e 2int
(9.78k 1 + It 4i II 2
p= +2
0'
.!-I+ r 4i 1 2
p=o
0'
1 4i II Ii
p= -2
where the fl represents being off resonance and 1-l88 and 1-ljk'S are the secular part of the spin-spin couplings. That is
[1-ljk' IzTJ =0
,
(9.95b)
[1-l 88 ,I:Tl =0
For example, for dipolar coupling
1-l j k = B j k(3I:jlzk - Ij' h)
(9.91)
(9.96.)
= B j k(2Iz jlzk - IZjIzk ~ IyjIyd
Once one has made such a table, one can work out the result in one's head since an Izk or an Iyk gives ± I, so a product of two such operators gives p values of 1 + I = 2, 1 - 1 = 0, -I + 1 = 0, -I - I = -2. We see that the three spins give us p values ranging from +3 to -3, as we proved earlier. Now, however, we can see how long T must be to produce Ihe coherence. Note that if a12 ~ GI3, that the three-quantum coherence will develop in a time when sin (aI2T/2) ~ 1 or UI2'/2 ~ 7r/2. If al3 «: al2, then we must wait until a13T/2 ~ 1r/2. In the process, as time develops, sin (aI2T/2) will have been through several oscillations. Thus, we must wait for the weakest coupling. Suppose, however, we had three spins in a line so that spins 1 and 3, on opposite ends of the chain, are weakly coupled. Then we expect (9.92) Then our expression (9.90) would imply GI3 would control how long it would take to generote a three-quantum coherence. Thai conclusion would, however, be wrong since, had we analyzed the time dependence of I z 2 instead of I zl ' we would have gouen a three-quantum coherence dependent on 458
(9.96b)
As in (9.7b), we then have quantum numbers Al and
IzTIMo:) = MIMo:)
and
0:
where
1-l ss IMo:) = E(> IMo:)
(9.97)
Then, the time development up to just before the second X(1r:I2) pulse is given
by {?(r-) = exp (
-*
1-lr) X(1rI2)I:TX(rr/2) exp
(*
=ex p ( -k1-lT) IyT exp (*1-lr) = exp (ifhIz) exp ( -*1-l88 T)IYT exp
1-lr) (9.98)
(*
1-l88 T) exp (-iflr I:)
The spin-spin exponential can be expanded using the theorem (see for example [9.16]) eABe- A = 1 +[A,BJ+~[A,[A,BlJ+
(9.99)
to give 459
e(r-) = e+
+
iI7r1
• { fYT - *r['Hss, IyT)
(.!.)2 ..!..-[1i,u, [1i [yT]J + ... }e tl 2! 88
•
(9.100)
-inrI,
Consider now the first commutator
[H.., IyTI =
L ,
(9.101)
[Hjk,!,,]
i>k
Equation (9.107) is, however, the condition for one-quantum coherence. In a similar way, one can show that every tenn in the series (9.101) generates onequantum coherence, a fact to which we will return. Returning to the two-spin Cllse, (9.100 and 103), we can now apply the Zeeman operators exp(inTI.:T) to get for the density matrix at time Te(T-) "" [yT cos DT + [zT sin DT -
~ fl I2[Izl [.:2 cos DT
- [yI I t 2 sin Dr+I.: I [z2 cos Dr-IdIY2 sin Dr]
If 1 is different from both j and k. the commutator vanishes. Thus, we have typical lenns such as (9.102)
(9.108)
Then, the X(1r!2) pulse will produce e(T+) "" - I.:T cos DT + IzT sin DT
3.
- 2'flI2[(IxIIy2 + I y 1Iz 2) cos DT
which are just whm one would have for .a pair of coupled spins. It is straightforward to evaluate the commutation for a specific choice of H 12. Thus, for dipolar coupling, we get
[1i12,(IyJ + [y2)] = BI2 [2Izl I z2 -
1:0:1/2:2 - [yl[y2,(1y1
+ [y2)]
= B I 2[2(Izl,!yl)Iz2 - Uzl,Iy d I :t2
+ r)J 2
(9.103)
From (9.103) we see that this commutalOf at this stage has one-quantum coherence. Indeed. we can see this in general by calculating a matrix element of
[H u ' I yT ] (Ma,!,H s8 I y'l'
"" L:
-
IyTHuIM'a')
(9.104)
0"
(9.105)
so that
I.)It2] sin Dr} ..
(9.110)
8S '
['H s8 , IyTll
(9.111)
Here we will have teons in the double commutation '
(9.112)
[Hjk> ['Hlm,(IYIII + I yl )]]
(9.113)
Now, if neither j nor k is 1 or m, the outer commutator will vanish. Thus, we get either two-spin terms such as
Thus M and M t are states which are joined by IyT' But IT)
- Ii) +ui -
DT
but since either for m must equal n, this will be of the foon
I
"" L:(M al'HsslMa")(M a"IIyTIM 0")
~i([,t -
II Ii) cos
We see clearly the one- and two-quantum coherences. The fact that the B I2 tenn is proportional to T is analogous to the small r behavior of the tcnn sin (aT!2) of lines 3 and 4 of column 10 of Table 9.1. Clearly, we can use a similar approach to the next Icon in the series of (9.100)
[Hjk, [Hln!' [ynn
which, using (9.97), which says 'H S8 is diagonal in AI, gives
Iy'f ""
+ [I.: lui
'12 (.I,: )' [H
(Mal'Hs8IM"a")CM"a"IIyT1M'a')
/If'' ,a"
- (M alIyTIM' al/)(M'a" IH8IM' a')
Expressing the [z and I y operators in tenns of raising and lowering operalOrs, (or using Table 9.2) we have
3i T{ + + ~ 2'Bt22i (II I 2 -
= -3iUz1 l z2 + 1:1 [z2)BI2
3.
(9.109)
e(T+) "" - ItT cos DT + Ix'!' sin DT
+2(Iz2.Iy2)Iz1 - (lzz,!y2)Iz d =-"2BI2[(I\+ +[1- )Iz2+ Jzl (1+ 2
+ (Itl I y2 + I y1 I z 2) sin Dr] ...
(9.106)
['H 12' ['H 12,Uy l + I y 2)ll
(9.114)
or three-spin terms such as
M""M'±l '60
(9.107)
(9.115) 461
In this mannerlwe see that each higher order or T in the series (9.IOO) adds one more possible spin to the couplings. We can now see another userul point by looking in detail at (9.103). AI· though lhe top line involves products of three spin operators, the commulation step reduces the prodUCI to two operators (e.g. ZI with [22)' In a similar way, one finds that the three·spin tenn (9.115) involves three spin operators, such as [zll1/2Iz]. In general each higher tenn in the series involves PrOOtiCts of the previous tenn Wilh a pair of spin operators, bUI lhen the commutator reduces the num~r by one. Thus, each successive lenn has one more spin operator in the product. Since it takes three spin products to get a three-quantum coherence, the T 2 lenn is the first one in the series which can give three-quanlUm coherence. The T] tenn is the first one which could give four-quantum coherence. Note that we say "could", implying that il does nO! necessarily do so. Thus, if one has two spin-j nuclei, the highest-order coherence one can produce is second, hence the T 2 , T , and higher tenns cannOI produce more lhan second-order coherence (double-quantum coherence) in this case. To find what coherences are aClually realized, one mUSI look in detail at the lenns or use the general rules about the maximum coherence 2NI of a group or N nuclei of spin I. Suppose in (9.115) one has a three-spin prodUCI involving coordinates of spins 1,2, and 3. Expressing the 11:1: and lyA: 's in tenns of raising and lowering operators, one might ask, will there be tenns in ()(T-) Uust berore the second. X(1fn) pulse] such as
I
ItriI:
?
(9.116)
The answer must be "no" since this is a three-quantum tenn whereas we have said each tenn in the infinite series prior to the second X(7fn) pulse has one-quanmm coherence. Permissible terms then would be of the general fonn
IiliIt
or
(9.117a)
It 1:2[,]
(9.l17b)
These may be thought of as grouping a raising (or lowering) operalor (e.g. with either zero-quantum operators or with paired raising and lowering operators (e.g. Ii It)· When these are operated on by the Zeeman operators. they simply multiply or I; by a phase exp(inT) or exp(-Wr) respeclively, or leave the the I:k's alone. Thus. the second X(7rn) pulse still acts either on expressions such as (9.117a or b).
Ii)
I:k
r:
Now
xIi X: X(l~t + i/y 1)X: 1",1 X/ 1X: /"'1 + il,l
so
il:1
X(It Ii !jJX = (1z1 - iI: I )(lz2 + il:2)(I",3
+ i/", I 1:2 /",] + 1"'11:21:] - iI: 11",2 1 z3 - 1:1 1z2 1:] + J:t /:21~3 - il:II:21:3
.= 1",1/",2/",3 - i/",t 1",21:3
(9.118)
When we express [zl> 1",2, and /",3 with raising and lowering operators, we see explicitly how the three-quanlUm coherence arises. TIle lenn (9.ll7b) will give
XIi h:IJ:X
: (lzt - iI,t)ly21yJ : [.1;\ I Y2I y] - il: 1l y2 / Y:1
(9.119)
TIle tenn l"'t1Y21Y3 contains three-quantum coherence. From our analysis of the power series expansion, using (9.9a), our con. c1usion is. then. that the pair or pulses X(1:(l) ... T •.• X(7(fl) will generate all orders of quamum coherence permitted. While it is a question of proper choice of T to achieve lhe desired orders, in general, all allowed orders will be presem. We thus have IwO problems yet to consider. (I) How can we detecl a particular desired order? (2) Is it possible to generate a particular order without generating other orders? We tum to these IOpics in the next two sections. 9.3.5 Selecting the Signal of a Particular Order of Coherence We turn now to a discussion of how [0 select the signal arising from a panicular order of coherence. We have already mentioned one approach at lhe end of Sect. 9.3.3, where we noted that if the signal is Fourier analyzed with respect to the evolution time t], the coherence or order p gives rise to lines at frequency pil, where is lhe amoum one is off resonance. Thus. introducing a deliberlarge compared to the spin-spin splillings ate resonance offset by an amount produces groups of lines in which the spectra of the different orders are well separaled. This approach, and a variant or it (lime proportional phase incrementation, TPPI) in which the offset is actually zero but is made 10 appear nonzero, has been widely employed. especially by the Pines group [9.5,7J. We discuss TPPI at the end of this section. AnO!her powerful method was imroduced by WotOIUi and Ernst [9.I7J. We turn to il now. Their method is based on a recognition that the phase of the multiple quantum coherence depends on the phase of the exciting pulse in a manner which depends on the order of the multiple quantum coherence. Their concept also underlies the scheme for production of a selected order of coherence which we take up in the next section. The basic Wokaun-Emst scheme may be described as follows. Let us denote by Xf>' a 7(/2 pulse applied about an x'-axis in the rolating frame where the Xl. axis lies in the x-y plane. making an angle ¢ with the x-direction according to lhe equations
n
x' "" x cos ¢ - y sin ¢ 462
- il:])
n
(9.120.)
(9. 120b)
I;,:I = I;,: cos ¢ - I y sin ¢ = eil.t/J I;,:e-il.t/J
(9.12Oc)
Note Ihal Xl is rotmed from x in a left-handed sense about I z by an amount Wokaun and Ernst apply a sequence Xt/J ... r ... X",,,.tl ... X ... t2
1J.
(9.121)
recording the complex signal 5(T, 1>,tl, t2) as a function of t2, for fixed values of the parameters T and tl, and for a succession of appropriately chosen phases 1>. Let us write the signal for ¢ = 0 as a sum of contributions G p from the various orders of coherence p (p = - N 10 +N)
1 I
Then, utilizing the fact thm X = eiJrTff/2 we have Xt/J = exp [i!(l;,:T cos ¢ - I lIT sin
L
G"
-!...
G =
h
P
J
5W,e-; pod.
.
(9.123)
o
Since in practice Ihere are 2N + I values of p, measurement of 5(1)) for 2N + 1 values of 1> should suffice, as we show below. To understand the Wokaun-EmSI theorem, we need to
1. 2.
examine Ihe general form of the contributions of a given order of coherence, examine the effect of a phase shifl on the differenl orders of coherence,
,nd 3.
show how to ulilize Ihis knowledge to pick OUI the contributions G p of the particular order of coherence. We consider a system characterized by a Hamiltonian in the rotating frame 1{ =
-
E hDkI:k + E k
1{jk
==
'Hz + 'H u
(9.124a)
X", = exp(iRI;,:TR- I 1r!2) = ReilrT1r/2 R- 1
= eil.Tt/J Xe-il.Tt/J
where for the spin-spin interactions 'Hjk we keep only the secular terms so Ihal (9. 124b)
Then, we have the quantum numbers NI, a such that (9.1240)
'HuIMa) = EnlMa) Note Ihat (9.124a) allows for a variety of chemical shifls. 464
(9.125d)
We now wish to express the ¢ dependence of the signal, 5. Abbreviating 5(T, 1>,tt> i2) as 5(t/J), we have S(t/J) = Tr {
If [e -(i/h)'H"J2 X e -(if h)'H"tl J£:I(T+, t/J)
[inverse]}
(9.126)
where (I(T+,t/J), defined as f!{,,+, t/J)
== Xt/Je-(i!h)'HT X",IzTX;le(i/h)'HT X;I
(9.127)
is the density matrix jusl after the second 1r/2 pulse, for the case that both 11' pulses have phase shifts t/J. The tP dependence of S(t/J) therefore arises in £:I(T+, ¢). Expressing Xt/J utilizing (9.125d) and utilizing the faci Ihal I zT ' and thus exp (iIzT¢), commutes with H, we get f!{T+, t/J) = eil,Tt/J Xe-(i/h)'HT X IzX-leCi/h)'HT X-le-i/,Tt/J
= eil.Tt/J £:I(T +, O)e-il.Tt/!
(9.128)
We recalllhm (1(,,+,0) is the density matrix just after the second X pulse, hence it contains all the multiple quantum coherence. In general, for a given spin system, £:1(,,+,0) consists of a number of terms representing not only the different possible orders of coherence, bUI also the various ways of generating a one can generate p = 0, given order. Thus, for a three-spin system, of spins ± 1, ± 2, ± 3. Consider p = I. It could arise from a variety of operators such as
!'
k>j
['Hjk> IzT] =0
(9.125c)
we gel
p=-N
,.
(9. 125b)
R=ei/.Tt/J
(9.122)
and then consider 5(1J) to be known as a continuous function of 1J. The essence of the Wokaun-Ernst method is then that the Gp's can be found as transfonns of 5(.)
1»]
Defining
+N 5(.=0)=
(9.125')
It
It12:
It h.hz
ItIiIi
(9.129)
plus all the operators one can get by permuting the labels 1,2, and 3. Utilizing a symbol f3 to designate all such various ways of generating a given order, we can clearly break £:I(r+ ,0) into a set of components Yp{J of given p
"T+,O) = Lgpp p,p
(9.130)
465
Now, a given order, p, is distinguished by having p more raising operators than lowering operators. (If p is negative, there are more lowering operators than raising operators.) We can then easily evaluate g(r+,4J) as i ' ,1'.;P9p13e -i/,T.;P (9.131) e< r +, f) "" eiT'TtI> g( r+, O)e -it ,T.;P =
l.:e
P.P
But,
ei/'T h±eif,.;p ""
h
±e±i.;p
eif'T Izke-i/'T "" Izk
by p, and in which the coefficients of the hannonics are the Gp's. Therefore, if one knew S(tjJ), one could deduce the Gp's by taking the Fourier transfonn of 5(
J5(
where
(9. 136a)
p=-N
G
p"" Tr{ r.t
[e-(i/h)"Ht 2xe-(i/h)1"f.t l ]
(~yp.B)
[inverse]}
. (9. 136b)
The individual tenns 9p/3 depend on r, on the ilk'S and the strength of the spinspin couplings as illustrated in the tables we have worked out for g(r+) for the two- and three-spin cases. Examination of (9.136a) shows that it says that S(tjJ) can be represented by a Fourier series in tjJ in which the harmonics are labeled
466
Je;(p-p')Odp [i(p - p')1d k,p " = L: G p L: "p [i(p - p')kh/(2N + I)J
5(0)+5('11") = L:Gp(eipO+eip1r)
=
p
p
(9.142)
p
=
Now, defining
2N
=
1.=0
1.=0
1,,=0
=
=
(
L:I'
)
L:
I' I
f2N+l
-1-1
G 1(1 +ei1r/2 +ebr +e i31r / 2)
if
1. 1
(9.144)
1,,=0
Examination of (9.144) shows that as long as p 1= pi the exponent in neither zero nor a multiple of 21l', hence
f
f
is
=F 1. But
(9.145)
f2N+l = e i (p-pl)211" = 1
I
p
=
L. '1'1. e -;P'¢' ( - -I - )"'5(")
2N+l
with
,
(9.146)
k=0,1, .. .,2N
Therefore, measurement of S(¢k) for these 2N + t values of
k
Then X-I ex p (
-~1tzzr)x
= exp [
_~(X-l1tHX)T]
= ex p (
~~1tyyr)
(9.160)
coherence?" In fact, as we shall see. we can find a way to generate coherence of order pk. where k = 0,1 •.... By special means, one can eliminate k = O. Then. the means to distinguish among the other k values is by the duration of the e?tcitation period. How then do we limit generation to an order pk? We will find that there are two parts to the task. The first part is concerned with finding a way to limit the order, and the second part. with making the amount generated large. In order to keep the discussion sufficiently general, we will therefore not as yet specify the Hamiltonian. However. we suppose that as with 1t yy it can generate various orders of coherence, p. That is
so that
'Ii =
e(r+) = exp (
-~1tyyr ) I z exp (~1tYYT)
L
(9.166)
p
(9.161)
where. for example. if 1t zz were given by (9.159),
1tyy =
I: 'lip
BjkU~jI~k + IzjIzk - 2Iyj I yd
(9.162)
j>k
We now want to see how to limit the orders generated. We already have a clue from the ideas of Wokml1l and Ernst; we should utilize the phase shift properties. Let us think about what happens to H if we shift the phase of the rf pulses which generate it. For the case of two rr/2 pulses so that the coherence is generated by
In this formulation, explicit mention of the pulses has disappeared, replaced by a transformed Hamiltonian. 1t yy , which starts to act at r = O. Now, e(r+) possesses. as we have seen, multiple quantum matrix elements. That is
xex p ( -*1t.ur)x = ex p ( -*1tyy r)
(9.167)
we have seen thai (9.163) X
However, since the density matrix at t = O. I z , has no matrix elements between different stales M and M', we may say that 1t yy acting over time r generates the multiple quantum coherence. Indeed, expanding the exponentials we have exp ( -*1t yy r ) = 1 _ i
1t~yr + (~y 1tyy ~yyr2
+...
=
(9.164)
x
From (9.162), substituting raising and lowering operalors (9.159).
1tyy =
I::
(9.165)
which therefore contains zero- and two-quantum coherences. We have secn numerous examples of ways in which the ingenious resonator can effectively modify a Hamiltonian: spin echoes. in which magnetic field inhomogeneities are effectively reversed; magic echoes, in which dipolar dephasing is undone by creating the negative of the dipolar Hamiltonian; and spin-flip line narrowing, in which the dipolar Hamiltonian ?i zz is averaged to zero by being made to jump to 1t~~ and H yy . Therefore, we may rephrase our question "can we generate one and only one desired order of coherence" to "can we find a way to achieve a Hamiltonian which will generate one and only one desired order of 472
eXP(*IZT~)X ex p ( -*IZT~)
=
eXP(*IzT~)X ex p ( -*1t zz r)x ex p ( -*IZT~)
=
eXP(*IZT~) exp ( -*Hyyr) ex p ( -*IZT~)
Bjk(t(ItIt" + IjIi;) + IzjI zk
j>k - tut Ii; + Ij It»)
eXP(*IZT~ )X ex p ( -*IZT~) ex p ( -*1tztr)
(9.168)
Let us then think about a general Hamiltonian, which for zero phase. ~, we call
1to. We decompose it into various orders of coherence (9.169a)
Then 1tt/> == e i1•T =
I:: eil.T¢J1tpe -il.Tt/>
(9.169b)
p
In the spirit then of (9.133), we then get
473
L 'Hpeipo), 3(2P//>o) ... 0>0 -1)(2·/Po) This cycle lasts for a time t c given by the number of phases n limes the duration of each phase
=nLl, =PoL\T
(9.197)
However, in order for the average Hamiltonian theory to hold. we must be able to approximate exp (
-i1itfl,o_tLl,')
eXP[-*(1iI/>,O_1 +
ex p ( -i1ioLh') 1
+ 1iO)LlT
]
by (9.198)
This condition is satisfied if
l1ioiLlT' there is some loss of order. Let us therefore next examine the general result (9.209) for the case that we eliminate such losses by making t I = O. Then, going back to the last line of (9.207), we get (lJ/')'I=O as (1:\1)(1 =0 = Tr (Iz{VU IzU- 1V-I)}
(9.211)
Comparing this result with (9.21Ob), which gives us the maximum possible (Iy ), we see that if we make
VU = I
(9.212.)
Le.
v=U- 1
(9.212b)
we will guarantee the maximum signal. Physically this condition states that what U does. V undoes. This statement is clearly a general kind of condition analogous to fonning the conventional echo or the magic echo. We can compare this resull with what we would get if we used U to create the coherence. but only a single 7I:n. pulse for the mixing. Since as we see in (9.207) we have already explicitly included a final pulse X, this situation corresponds to having V = I:
(I,)"
=0
=T'(I,(UI,U-')j
(9.213)
says that both V-I and U "rotate" I z • Unless V-I and U produce the same "rotation" the signal is diminished from its maximum value. In general. the bigger the relative "rotation", the smaller the net result. In particular, suppose U generates panicularly large p's, but Y- t does not. Then U will put the order of the system into large p, but Y will not be able to bring it back down to zero order to pennit the X pulse to conven it to transverse magnetization. The Pines group has made extensive use of sequences satisfying (9.212) V=U- I to generate and then observe large-order coherences. It is the author's opinion that the Warren. Weitekamp, Pines melh<Xi of selective excitation of muhiple quantum coherence can be thought of as a kind of graduation exercise in spin Hamiltonian manipulation. It requires that one be completely at home with the idea that pulses manipulate the Hamiltonian, with average Hamiltonian theory, possess knowledge of the specific kinds of Hamiltonian one can generate and how to generate them. It illustrates then how once one has these details in mind, one can advance to a higher level using the ideas as building blocks, much as in electronics one thinks of oscillators. amplifiers, mixers, pulses and so on, without then being overwhelmed by the details of each circuit.
How does this compare with the result when Y = U-I? U is a unitary transformation. This expression reminds us of the dot product between a vector (1z) and the same vector rotated (by U). It should be less than Tr {I;} or for that matter than Tr{(Ulz U- 1)2}. Indeed
T,{(UI,U-')'j =T,{UI,U-'UI,U-'j =T,{UI;U-'j =T'{I;U-'Uj =T'{I;j
(9.214)
Consider then Tr {lIz - (U I z U-I)]2}. Since Ihis is the trace of the square of an Hennitian operator, it must be positive. Thus
oST, {(l, -
(U I,U-'))'} = T, {I; j + Tt «U I,U-')'} - T, {I,UI,U-'} - T, {UI,U-' I,}
(9.215) Pennuting the order of the operators of the last trace, and utilizing (9.214), we thus get that (9.216) In general, as in the Schwarz inequality, we expect the inequality to hold unless U is the identity operator. Thus. the signal of (9.213) will be less than the maximum. We can think of the signal for tl = 0 along these lines: the expression
(9.217) 482
483
10. Electric Quadrupole Effects
10.1 Introduction So far we have considered only the magnetic interactions of the nucleus with its surroundings. To be sure, by implicalion we have considered Ihe effect of Ihe nuclear charge, since it detennines the electron orbits and where the nucleus sits in a molecule. However, we have not considered any electrical effects on Ihe energy required to reorient the nucleus. That such effects do exist can be seen by considering a nonspherical nucleus. Suppose it is somewhat elongated and is acted on by the charges shown in Fig. to. I. We see that Fig.1O.1b will correspond to a lower energy, since it has put Ihe tips of the positive nuclear charge closer to the negative external charges. There is, therefore, an electrostatic energy that varies with the nuclear orientation. Of course,1 turning the nucleus end for end does not affect the electrostatic energy. Consequently, for spin ~ nuclei the electrostatic energy does not split the ml degeneracy.
-q
+q
(a)
-q
+q
-q
+q
(b)
+q -q
Fig.IO.I. (a) A cigar-shaped nucleus in the field of four charges, +q on the z-axis; -q on the y-axis. The configuration of (b) is energetically more favorable because it puts the positive charge of the ends of the cigar closer to the negative charges -q
1 See references to "Quadrupole Effects" in the Bibliography and the articles by Cohen and Rei! and by Das and lfalin under "Books, Monographs, or Review Articles" in the Bibliography.
•
485
10.2 Quadrupole Hamiltonian - Part 1
(We note that sometimes Poisson's equation applies instead. Some care must then be exercised because we are, of course, interested only in the orientation depen-
To develop a more quantitative theory, we begin with a description in tenns of the classical charge density of the nucleus, (!. We shall obtain a quantum mechanical answer by replacing the classical (! by its quantum mechanical operator. Classically, the interaction energy E of a charge distribution of density (! with a potential V due to external sources is (10.1)
E = j g(r)V(r)dr We expand V(r) in a Taylor's series about the origin:
w)
V(r)=V(O)+LX aO 0' Xa
\
r=O
a'v +21 LXaX/3a a . 0',/3 X(J/ X/3
)
+ ...
(10.2)
r=O
- aw) Xa ~
Va /3
r=O
= aa'v a ) X(J/ :r:/3
we have
E = V(O) j
dr + L Vo' j XO'g dr + a
VO'/3 =0 if
r=O
(10.5)
Moreover, V must satisfy Laplace's equation:
\7'V = 0
(10.6)
This equation, evaluated at the origin, gives us
LVcrO' = 0
" 486
which, combined with (10.7), makes all three derivatives zero. The quadrupole coupling then vanishes. This situation arises, for example, with Na 23 in Na metal. The face-centered cubic crystal stnlclure puts each nucleus al a site of cubic symmetry. It is convenient to consider the quantities QO'(j defined by the equation 2
(10.9)
Qa/3 = j(3xcrx/3 - ocrpr )gdr
(10.10)
j x a x/3gdr = !(Qa/3 + j 8cr /3r gdr)
As we shall see, the introduction of the QO'p'S amounts to our subtracting from the left side of (10.10) a term that does not depend on the orienllltion of the nucleus. We have, then, for the quadrupole energy E(2),
;!
a f. {J
(10.8)
(cubic symmetry)
2
(10.3)
L VO'/3 j XO'x/3g dr... . (10.4) cr ,/3 Choosing the origin at the mass center of the nucleus. we have for the first tenn the electrostatic energy of the nucleus taken as a point charge. The second tenn involves the electrical dipole moment of the nucleus. It vanishes, since the center of mass and center of charge coincide. That they do coincide can be proved if the nuclear states possess a definite parity. All experimental evidence supports the contention that nuclei do have definite parity. Moreover, a nucleus in equilibrium experiences zero average electric field Va' It is interesting to note that even if the dipole moment were not zero, the tendency of a nucleus to be at a point of vanishing electric field would make the dipole tenn hard to see. In fact it was for just this reason that Purcell, Ramsey, and Smith [10.1] looked for signs of a possible nuclear electrical dipole moment in neutrons rather than in charged nuclei. The third tenn is the so-called electrical quadrupole tenn. We note at this point that one can always find principal axes of the potential V such that (!
Vu = Vyy = V.;z
In tenns of the QcrP's, we have
where Xo' (a=1,2,3) stands for x, y, or z, respectively. Defining
V."
dent part of the potential, and must therefore subtract the spherically symmetric parts.) If one has a nucleus at a site of cubic symmelry,
(10.7)
E(2) =
~L
V a/3
j xo'x/3(! dr
",~
=
~ L(Va/3Qa/3 + VO'/38a !3 j
2
r gdr)
(10.\1)
",~
Since V satisfies Laplace's equalion, the second tenn on the right of (10.11) vanishes, giving us
E(2) =
~ L: V"~Q"~
(10.12)
",~
Even if this tenn were not zero, we note that it would be independent of nuclear orientation. 2 2 If there is an electronic charge at the nucleus, we must usc Poisson's equation. Thcn
LV.... =
_'hrel,p(O)1
2
" 2 where 1,p(0)1 is the electronic probability density at the nucleus. The orientation independent term, A.E, of (IO.lt) becomes LJ.E
= ~ EV.... j
r20dT
=_4;e J,p(O)J2 j
r 2qdT
"
This LJ.E wi\.l...bC"different for two nuclei of the same c1l11rge but different charge distribution~'iliOi:opcs), or for two nuclei of the same mass and charge but different nuclear ~a.tes'(iSomers).In an electronic transition between an 8 and a p-slate, LJ.E will make a contribution that will in general be different for different isotopes or isomers. Effects also show up in nuclear transition [10.21. 487
To obtain a quantum mechanical expression for the quadrupole coupling, we simply replace the classical e by its quantum mechanical operator e(op), given by e(OP)(r) '"
1:: qAo e(r -
•
rAo)
,
(10.13)
where the sum runs over the nuclear particles, 1,2, ... k . .. N, of charge qAo' Since the neutrons have zero charge, and the protons a charge e, we can simply sum over the protons: e(Op)(r) '"
e
1::
S(r - rk)
By substituting (10.14) into the classical expression for QOlf3' we obtain the
1::
J(3:t a zfJ - 6a fJ r2 )S(r - rk)dT
protOll!l
1::
'" e
(10.17)
where' C is a constant, different for each set of the quantum numbers I and '1' In order to justify (10.17), we need to digress to discuss the Clebsch-Gordan coefficients, the so-called irreducible tensor operators TLM' and the WignerEckart theorem.
10.3 Clebsch-Gordan Coefficients, Irreducible Tensor Operators, and the Wigncr-Eckart Theorem
Q~;>:
Q~1) '" JO%et%f3 - Detpr 2 )e(oP)('r)dT '" e
(lm"IQ~;)IIm'") = C(lml~(loIp + Iplo ) - 'opI'IIm')
(10.14)
protons
quadrupole operator
These can be shown to obey the equation
(3ZakZ,Bk - Set,Br:).
(10.15)
protonll
We have, then, a quadrupole term for the Hamiltonian 1tQ' given by Q(oP) 'liQ = (51 '" L..- Vop .p
o,p
(10 16)
.
The expressions of (10.15) and (10.16) look exceedingly me~sy to handle because they involve all the nuclear panicles. They appear to reqUlr: us t~ m:at the nucleus as a many-panicle system, a complication we have aVOIded In diScussing the magnetic couplings. Actually a similar problem i.s involv~ in ~h magnetic dipole and electric quadrupole cases, but we have Simply aVOIded diScussion in the magnetic case. The quadrupole interaction represenled by (10.15) enables us to treat problems of much greater complexity than those we encounter in a discussion of resonance phenomena. When performing resonances, we are in general concerned only with Ihe ground stale of a nucleus, or perhaps with an excited state when the excited state is sufficiently long-lived. The eigenslates of the nucleus are characterized by the tolal angular momentum I of each state, 21 + I values of a component of angular momentum, and a set of other quantum num~rs '1, which we shall not bother to specify. Since we shall be concerned only With the spatial reorientation of the nucleus for a given nuclear energy state, we shall be concerned only with matrix elements diagonal in both I and '1' Thus we shall need only matrix elements of the quadrupole operator, such as
The Wigner-Eckart theorem is one of the most useful theorems in quantum mechanics. In order to state it, we must introduce the Clebsch-Gordan coefficients C{LJ'J;MMJ,MJ), and the irreducible tensor operators TLM' We shall first state the Wigner-Eckart theorem and then define the Clebsch-Gordan coefficienls. Next we shall discuss the irreducible tensor operators, and lasdy we shall indicate the derivation of the Wigner-Eckan theorem. We consider a sel.0f wave func.lions characterized by quantum numbers J and J' for the total angular momentum, M J or My for the z-eomponenl of angular momentum, and as many other quantum numbers '1 or 1'/ as are needed to specify the state. We are then concerned with calculating the matrix elements of the operators TLM' using these functions as the basis functions. The WignerEckart theorem states that all such matrix elements are related to the appropriate Clebsch-Gordan coefficients through a set of quantities (J'111 T L II J'1'/) that depend on J. J', '1. '1' and L but which are independent of M J' M J" and M. Stated mathematically, the Wigner-Eckart theorem is (JMJ"ITLMIJ'MJ") = C(J'LJ; MJ'M MJ)(J" II T L II 1'"')
.
(10.18)
Lei us now define the Clebsch-Gordan coefficienls. They are encounlered when one discusses the addition of two angular momenta to form a resultant. We therefore consider a system made up of two pans. Let us describe one part of the system by the quantum numbers L and A!, (Q describe the tOial angular momentum of that part and its z-component. Let us use the quantum numbers J' and MJI correspondingly for the second part of the system. For the system as a whole we introduce quantum numbers J and M J . We have, then. wave functions 'l/JLM and ,pJlM ,,10 describe the two parts, and tPJMJ for the whole system. The function tPJA:J can be expressed as a linear combination of product functions of Ihe two parts, since such products form a complete set:
tPJMJ '"
L
C(J'LJ;MJlMMJ),pJ'MJ,tPLM
(10.19)
J'MJ,;LM
(Im'1IQ::)IIm''1) 488
489
The coefficients C(J'LJ; M)'MM J ) are called the Clebsch·Gordtln coefficicms. Certain of their properties are very well known. Forexample, C(J' LJ; M)'M M J ) vanishes unless MJ "" M + M)'. A second property, often called the triangle rule, is that C(J'LJ;M;MMJ) vanishes unless J equals one of the values J' + L. J' + L - I, .. . IJ - Ll, a fact widely used in atomic physics. Let us now define the irreducible tensor operators Tu". Suppose we have a system whose angular momentum operators have components Jz, J". and J:. We define the raising and lowering operators J+ and J- as usual by the relations
J+ == Jz +iJlI
J- == Jz - iJ"
(10,20)
One can construct functions G of the operators of the system and examine the commutators such as [J+, G], [J- ,G], and [Jl' G]. It is often possible to define a family of 2L + I operators (L is an integer) labeled by an integer M(M ::: L, L - 1, _.. - L) which we shall term irreducible tensor operators T,.M, which obey the commutation rules
[J±. T LM ) = JL(L+ I) [J•. Tur)
M(M ± I)TLM ± 1
We shall wish to compute matrix elements of the TLM'S. We are familiar with the fact that it is possible to derive expressions for the matrix elements of angular momentum from the commutation rules among the components. It is possible to compute the matrix elements of the TLM'S by means of (10.21) in a similar manner. Let us illustrate. We have in mind a set of commuting operators J 2 , J., plus others, with eigenvalues J, MJI and '1. We use ." to stand for all other quantum numbers needed. We wish to compute matrix elements such as
(10.21)
(10.25) By means of the commutation rule
[Jz,Tu·,]::: MTUf
(10.26)
we have (10.27.) B",
.
Tlo::: J l
- (JMJ "ITLAI J.IJ'MJ ,.,,') , .
.
(10.22)
2
::: (MJ - M)')(JMJ1JITLMIJ'M)',,')
Another example of a Tl/Ir can be constructed for an atom with spin and orbital angular momentum operators sand " respectively, and total angular momentum J. Then we define the operators
1+:::/z+ilu
r:::lz-il u
One can then verify the operators T tM defined by 1 1 Til ::: - ..j2'+ TJO::: I: TI_l ::: ,jir
v
I
An example of such a set for L ::: 1 is -I + Til ::: ..j2J
.
(JMJ."I[Jz,TI.MlIJ'MJ,,,') ::: (JMJ"IJ.TutlJ'MJ ,,,')
= MTuf
(10.23)
(10.24)
obey (10.21). (Actually the operators of (10.24) form components of an irreducible tensor TIA'f with respect to the operators 1+, 1-, and Il as well as J+ J-, and Jz.) We may write the Tl/Ifs of (10.22) as TI/",(J), to signify that they are functions of the components Jz , J", and J. of J. The Tl/If'S of (10.24) are in a similar manner signified as T1J\f(l). It is helpful to have a more physic"l feeling for the definition of the operators TLAf by the commutation rules of (10.21). We realize that angular momentum operators can be used 10 generate rOiations, as discussed in Chapter 2. It is nOi surprising, therefore, that (10.21) can be shown to guarantee that TLM transforms under rotations of the coordinate axes into linear combinations TUf ', in exactly the same way that the spherical harmonics YL/lf trnnsform into linear combinations of YUI ' 's. This theorem is shown in Chapter 5 of Rose's excellent
Equation (10.27b) shows that
(JMJl1ITUI1J'M)'11')=0
unless
MJ-M),:::A{
(10.28)
I
book [10.3]. 490
In a similar way we may find conditions on the matrix elements of the other terms of (10.21). Thus (J MJ.I(J±. Tl.AII1J' Ml"') = JL(L + I) - M(M
± 1)(JMJ.ITl.Af ± i1J' M J ,"')
(10.29)
(J MJ"IJ±Tu"IJ'M J 'l1')
= (JMJ.IJ±IJ MJ 'f I.)(JMJ 'f I"ITLMIJ' Ml"") =
J J(J + I) -
(MJ 'f I)MJ(JMJ 'f I.ITLMIJ' Ml"')
(10.30)
By combining (10.29) and (10.30), we obtain the other recursion relations: 491
trix elements of Tr,M 's are related. The relationship is called the \Vig"er-Eckart
../J(J + I)
(MJ" I)M;{JMJ" hlITwIJ' M J,"')
- (J MJ'1ITLA1IJ' MJ'
=../L(L+1)
± 1'1')';JI(Jf + I)
theorem:
MJ'(MJ' ± 1)
M(M±I)(JMJ '/ITuf±dJ'MJ , " ) .
(10.37)
(10.31)
We note thal the only nonvanishing ICnTlS must satisfy (10.27b). However, if any one term in (10.31) satisfies this relation, all do. Equation (lO.27b) and (10.31) constitute a set of recursion relations relating matrix elements for TLM to one another and to those of TL/l1" 111cse equations lurn OullO be sufficient to enable one to solve for all TLM matrix elements for given J, )', lJ,1/ in ternlS of any onc matrix element. A further insight inlQ the significance of the recursion relations is shown by returning to the Clebsch-Gordan coefficients. In so doing, we shall sketch the proof of the Wigner-Eckart theorem. As is shown by Rose, the C's obey recursion relations identical to those of the TL/lf'S. We shall derive one: the selection rule on M. M;. and MJ'.
where the notation (JI'/II TL II J',./) stands for a quanlity that is a constant for a given J,£,J',I'/,I/ independent of M;. MJ'. and M. As we can see specifically from (10.22) and (10.24), for a given L and AI there may be a variety of functions, all of which are Tr,M'S. The Clebsch-Gordan coefficient is the same for all functions TUf that have the same Land M, but the constant (JI'/II T/, II J',/) will depend on what variable is used to construct the TLAI·S. To illustrate this point further, let us consider a particle with spin 8 and orbital angular momentum I and position r. The total angular momentum J is given by J::: 8+1
Consider the operator
J: ::= L: + J~
1.
(10.32)
where
(10.33) J~¢J'MJ' = MJ'¢J'M J, TIlen. using (10.19). consider the following matrix element of the operator J:: (tPLMtPJ' M J,' J:tJiJ MJ) = MJ(!/;LMtPJ' AlJ,' 1Ji; (HJ) =MJC(J'LJ;MJ'MMJ )
(10.34)
where we have let J: operate to the righl. But. writing J: as L:+J~ and operating on the functions to the left. we get (wur¢;'M J,' Jzwn.r) = (M + AJJ')C(J'LJ; M;,M M;)
(10.35)
By equating (10.34) and (10.35), we find (M + MJ' - MJ)CU ' LJ; MJ'M MJ) =
a
(10.36)
This equation is quite analogous to (1O.27b). provided we replace 1
(IMjIJIT/,flIIJ J\!!J"/)
By
~(z~ -%~) i ax 8z
I,
~(%~ -y~)
=
1
fJy
(10.39)
ax
We shall now list two T2M 's: one a function of the angular momentum J; the other, of the coordinate r. One can verify that the functions of Table 10.1, which we shall call Tu,r(J) and T 2M(r), indeed obey the commutation rules of (10.21) with respect to J+, J-, and J:. We have used the notation T2M (r) as shorthand for a TU,f constructed from the components z, y, and z of r. There is an obvious similarity between T2 Af(J) and T2M(r): Replacement of J+ by (z+iy), J- by (x-iy), Jz by z will conven TU1(J) into T2M(r). This similarity is a direct consequence of the similarity of the commutation relalions of components of J and r with J7.' J II , and J z : Table 10.1
by
T"
.
One can proceed in a similar manner to compute matrix elements of the raising and lowering operators, 10 get equations similar to (10.31). In fact the C(J'LJ;M;,/",fM;)'s obey recursion relalions identical to those of the (J M;I'/ITLAI IJ' M;,,/)'s. As a result, one can say that Ihe C's and the ma492
Bz
1 = 11
L:'PLM = MtPr,M
-
(10.38)
~ (y~ z~) I
J:oJi;M J = M;oJi;MJ
C(J'LJ;MJ,MMJ)
=
where
1'21
T" T'_I 1'2_2
J+'
-(J.;+ +J+;.) # ( 3 ) : ' _ ;') J.;- +;- J. J-'
493
[Jz.y] = iz
(10.40.)
[Jz • J y ] = iJ,
I
etc.
(1O.40b)
where (10.403) can be verified by means of (10.38) and (10.39). It is clear Ibat any function G(x. y. z) of x. Y. z, constructed from a function G(Jz , J y, J,) of J z , J1/' J, by direct substitution of x for J z , and SO on, will obey the same commutation rules with respect 10 J z • Jv' and Jz. Thus, if a function of J ZI JVI J, is known to be a TuJ, the same will be true of the function fonned by replacing J z • J 1I • J, by X. Y, z. respectively. The only caution we note in procedures such as this is that we must remember that the components of some operators do not commute among themselves; so that for example in T21 (J) we have the symmetrized product J+ J, + J,J+, not 2J+ J,. The method of direct replacement will work for olher variables as long as they obey commutation relations such as those of (10.40). For an excellent review anicle including tables of T/,M's of various Land M. sec [10.4]. Returning now to (10.37), let us consider two Tu",s, one a function of variables q and the other a function of variables p. Then (10.37) tells us thai 1
Iz =
, 1 (hili TL
.
(10.53)
.,/J
II is interesting (0 note lhat of the nine components of Q~o;>. only one nuclear constant, eQ, is needed. The reason is as follows: The faci Ihal the nucleus is in a state of definite angular momentum is equivalent to the classical statement Ihal the charge has cylindrical symmeuy. Taking z as the symmetry axis, the energy change on reorientation depends. then, only on the difference between the charge distribution parallel and transverse to z: / z2(}dT
and
/ z2(}dT
.
This gives us the critical quantity /(z2 _ z2)(}dT
"'! /(2z 2 _ x 2 _ y2){ldT '" ! /(3z2 _ r 2)gdT
(10.54)
The last integral, we see, is the classical equivalent of our eQ. The effective quadrupole interaction of (10.53) applies for an arbitrary orientation of the rectangular coordinates a '" z. y, z. The tensor coupling to the symmetric (in z, y, z) tensor VO{j tan be simplified by choice of a set of principal axes relative to which Vo{j '" 0 for a oF {3. In terms of these axes. we have 1iQ '"
6I(;~- I) [Vu(3I; -
2 2 1 ) + VIIII (3l; - 1 )
+ Vu (3I"2 - l 2 )]
(10.55)
This expression can be rewritten, using Laplace's equation 1iQ '"
4I(;~- 1)[Vu (3l; -
Eo V
1 ) + (V~z - Vyy)(I; - I;)j 2
CfO
= U, to give (10.56)
Equation (10.56) shows that only two parameters are needed to characterize the derivatives of the potential; Vu and V:r:z - Vyy . IE is customary to define two 496
eq'" V" '1'", Vu - VIIy
'Q
'Q 1) L.. " V.pl,U.lp J 'liQ' 61(2l + lpl.) - 5.pl2 )
symbols, '1 and q. called the asymmetry parameter and the field gradient. by the equalions
(10.57)
V" The case of axial symmetry. often a good approximation. is handled by taking the axis to be the z-direction. giving '} '" O. Since we have seen that the raising and lowering operators often provide panicularly convenient selection rules. it is useful to write (10.53) in tenns of I+, I-. and I, for an arbitrary (that is. nonprincipal) set of axes. By defining
Vo
",Vu
V± 1
'"
V±2 '"
V,z
± iV,~
(10.58)
levu - VlIlI}±iVzlI
we find by straightforward algebraic manipulation that
'Q(22 -1iQ'" 41(21 _ I} Vo(3!., - I ) + V+1(l I, + I,I )
+ V_I (.P" I" + I"I+) + V+2(r)2 + V_2([+)2)
(10.59)
Equation (10.59) gives a form of the quadrupole coupling that is particularly useful when considering relaxation for which the principal axes are not fixed in space but rather are functions of time. An anempt to use principal axes would then be exceedingly cumbersome. We shall not attempt to describe nuclear relaxation by the quadrupolac coupling, although it is a very imponant mechanism in insulating crystals, often dominant at room temperature.
10.5 Examples at Strong and Weak Magnetic Fields In order to illustrate the use of the effective quadrupolar interaction. we shall make the simplifying assumption of a field with axial symmetry (or any other symmetry such that Vu '" Vyy for a set of principal axes). Let us then consider a magnetic field applied along the z'-axis where in general the z- and z'-axes differ. Then we have, for our Hamiltonian.
1i '" -'YnliHoI", +
4It22;~ I) (31; -
2 1 )
(10.60)
First we shall consider what happens when the quadrupole coupling is weak compared 10 the magnetic interaction. In Ihis case we can consider the spin quantized along the zl·axis. We proceed 10 U'Cat the quadrupolar coupling by 497
J. E SzHz) (11.19) z z z y By combining this result with the Zeeman teon, 211H· 5, we obtain a spin Hamiltonian for the ground orbital state
508
509
H = {3(gzzH z S z + gyyHySy + gz: HzSz) where 9u = 2
9yy = 2 ( 1 - E
,
z
AEz)
We may employ the dyadic notation
9 10
(11.20)
g, defined by
= igzzi + i9 yyi + kgzzk
(11.22)
write the interaction as
'Ii=pH·
Ii
(11.23)
·5
in place of
'Ii = 2pH· 5
(11.24)
Comparison of (11.23) and (11.24) shows that the combined effect of the spin~ orbit coupling and orbital Zeeman energy is as though the real field H were replaced by an effective field Herr, given by
H
- H·
err-
2
9
-'H 9zz + 'H 9yy + kH 9zz Z 2 J y 2 z 2
-t
(11.25)
with the resonance given by
'Ii=2PH,rr. 5
.
(11.26)
Since gzz, gyy, and gzz are in general different, the effective field differs from the actual field in both magnitude and direction. If we denote by z" the direction of the effective field, it is clear that a coordinate transfonnation will put (11.26) into the fonn
'Ii = 2PH,rrS."
(11.27)
where Herr is the magnitude of Helf' The resonant frequency wo therefore satisfies the condition ~-,------:--:---:--;:
H;giz + H~9~y + H;g;z
= {3H
0~9iz + 0~9~y + o~g~z
(OI1t
(11.28)
where 01, 02 and 03 are the cosines of the angle between H and the x-, y-, and z-axes. Often one writes (11.28) as tlWo = g{3H
'H perl = AL· 5+ {3H· L
(11.29)
new
10') =
L /I
510
222222 gzzoJ + 9yy Cf 2 + 9z:03
(OI'li",wIO') =
Eo
(11.32)
E tl
L [(01.\£' 5In)("IPH· £10') + (OIPH· £1")("1.1£·510') n
+ (11.30)
(OIHperdn)(nl1t per d.L· 5 on the state formed from spin functions and the three p-states xf(r), yJ(r), and ZJ(I·). The sum of the angular momenta Land S is the total angular momentum J:
J::L+5
.
(11.35)
By squaring, J, we have
~L·5= ~(J' -
2 with eigenvalues
Esc
~
L' - 5')
L(L + I) - 5(S + I»)
;:;;
L
Clm ,.Il'jm
(11.39)
I,m
where the YIIlI'S are spherical hannonics and the Clm'S are conslants. If the potential is due 10 the charges of Fig. 11.1, it vanishes on the z·axis and changes sign if we replace z by y and y by -z (a coordinate rotation). It is a maximum on Ihe x-axis and a minimum on the y-axis for a given distance from the origin. The lowest I in the series of (11.39) is clearly I:: 2. Of the five 1:: 2 functions, .:z:y, xz, yz, 3z 2 - r 2, .:z:2 - y2, only the last is needed. We have, therefore, as an approximation, insofar as teons for 1> 2 are not required, (11.40) where A is a constant. (We shall see shortly that no higher terms are needed for an exact treatment.) We have, then, to consider the effect of VI on the states of Fig. 11.3. Two sorts of matrix elements will be important: those entirely within a given J, such as (JM; IVtlJMJ), and those connecting the different J states. The fonner will be the more important because they connect degenerate states. We can compute the matrix elements internal to a given J by means of the Wigner-Eckan theorem, for we notice that, with respect to L z , L y , and L: (hence with respect to J z , JIJ , and J:), F I is a linear combination of Tu,,'s. That is, the commutation relations of VI with J:, JT. ± iJy show VJ 10 be a linear combination of T 2M 'so (In fact Vt is proportional to T22 + T2-2.) Thus we have
(11.36)
(JMJlVtlJM~)
=A(JMJlz' -
y'IJM'/)
:: CJ(JMJIJ; - J~IJMJ) (11.37)
Since we are concerned with states all of which are characterized by an orbital quantum number L and spin quantum number 5, the possible values of J are £+5, £+5-1, ... IL- 51. For our example, L:: 1, 5::::t!, so that J:: ~ or The J :: ~ and states are therefore split apart by an energy spacing
!
LIE = ~~
FI
given by
£So = 2:[J(J + 1) -
!.
We consider next the effect of Fl' Here il becomes convenient to assume a specific fonn. Assuming that the potential arises from charges external to the atom, Ihe potential in the region of the atom can be expressed as a sum of the fonn
(11.41)
This is equivalent to our replacing VI by the operator 1ft.
Hl :: CJ(J; - J;)
(11.42)
as long as we only compute matrix elements diagonal in J. As an allernative, had the potential VI been B(3.: 2 - r 2), we should have used
(11.38)
(11.43)
(More generally, the stale J is AJ above the state J - 1.) The energy levels are shown in Fig. 11.3 for a positive A.
This case of what is called the "axial field" is frequently encountered. We describe the calculation of CJ and CJ below.
512
513
We have still to consider the effect of Ihe magnetic field terms. Again lei us consider only the malrix elements dia{:onal in J. This means that we wish to have
(JMJIPH· L +2pH· SIJMj) = pH· (JMJIL +2SIJMj)
(J MJIL + 2SIJMj) = 9J(J M J IJIJMj)
(11.46)
Equation (11.45) is equivalenl (as far as malrix elements diagonal in J are concerned) to our having a term 'Hz replacing lhe two magnetic terms, where 'Hz is given by (11.47)
.
By combining (11.42) and (11.47) with (11.36), we obtain an effective spin Hamiltonian 1te fr' which describes our problem accurately within a given J: 'Hefr = C;(J; - J;)
~/2 =
~
(11.49)
The tWO tenns on the right of (11.48) and (11.49) lift the (2J + I)-fold degeneracy of each J Slate. We shall not discuss the details of handling this problem other than 10 remark Ihal clearly it is formally equivalent to the solution of the problem of a nucleus posscssing a quadrupole moment actcd on by an electric field gradient and a static magnctic field. So far we have not computed the constants C; or Cj. We lum now 10 Ihat task, illustrating the method by computing C We have. using (11.43)
(11.50)
By choosing /If; = Mj = J, we have
B(JJI3z' - r'IJJ) = Cj[3J' - J(J + 1)1
c'J =
J(2JB _I)(JJ I 3,' -r 'I JJ)
Now for J = 514
4, all matrix elements of 3J; -
(11.50.) (11.51)
J2 vanish (analogous to the faci
I 2 + y2)f2(r)(3z 2 - r 2)dr 2"(x
(11.53)
2B--r' 15
(11.53,)
!.
A(LML! z 2 - y2JLM~) = CL(LMLlL; - L~ILM~) 2 B(LMLl3z - r 2 JLM}) = GL(LMd3L; - L2ILML)
(11.54)
These matrix elements are equivalent to those we should have had by replacing VI by an equivalent Hamiltonian 1t1:
1t1 = GL(L; - L;) or 1t) = C'L(3L; - L 2)
(11.55)
In terms of 1tI, the effective Hamiltonian 1te rr. which will give all matrix elements fonned from the three p-states xf(r), yf(r), and zf(r), is
1te rr=>.L'S+C L(L;-L;)+PH.(25+L)
J.
B(J M;13z 2 - r21JMj) = Cj(JMJ I3J; - J2lJMj)
BJ
'3
i
or, for the axial field,
1tcfr=Cj(3J;-J 2)+9;pH.J
( 11.52)
where r 2 is the average value of r 2 for the p-states. The ~p.in. Hamiltonians of (11.48) and (11.49) do nOI include any malrix elements JO,"lng states of J = wilh those of J = If we wish to include such effects, we can actually apply the Wigner-Eckart theorem in an ailemative manner. All matrix elements of VI are between states of L = I. Thai is, we are concerned. only with (LMr,lVdLMJ). However, the commutators of L z , L y • and ~, with VI show that VI is a linear combination of T 2M 's. Therefore all matnx elements are of the form
(11.48)
+ g;pH· J
.
Th~ angular ~nions of the integral can be carried through, using spherical coordinates, to gIVe '""3/2::
J(J + 1) + S(S + 1) - L(L + I) g; = 1 + 2J(J + I)
~
where we have denoted Ihe spin function S~:: +~ by ul/2' Therefore we find
(11.45)
where 9; is a conSlanl for a given J, independent of M; or M J. We recognize this problem to be the same as that of compuling Ihe Zeeman effect of free atoms, and the constant 9; is therefore the familiar Lande 9-factor:
"/iz=gJPH·J
1
levels). For J :: 3, we
IJJ) = J2(%+'Y)
(11.44)
We can apply the Wigner-Eckart theorem to the malrix element, which, with respect 10 J, is a linear combination of TtM's. We can therefore write
i
thai a quadrupole coupling cannot split a pair of spin have
or
1teff :: >.(L· S) + Ci, are given in Sect. 11.4.] Clearly the operator Sr.' will have matrix elements which join either C1'U_ or OU+ to both fJv+ and {iv-. So far we have treated the electron as being quantized along the direction of the static field Ho. There are two other refinemenls worth mentioning. The first is concemed with anisotropy in the electron g-faclor. If the electron g-factor is anisotropic, the electron will actually be quantized along the direction of the effective field acting on ii, H g • given by
523
(11.91)
This refinement will slightly modify the treatment of (I 1.81-84). The other refinement worth mentioning is to include the fact that the nucleus exerts a magnetic field on the electron. II therefore should playa role in the clec(ron spin quantization. Thus, if the nucleus is flipped, the electron quantization direction changes slighdy. This change is ordinarily small and can frequently be neglected. However, there are some situations in which it is important. Waugh and Stichter [11.2] have called this slight change in electron spin quantization "wobble," They point out thai it gives the electron a degree of freedom much less costly in energy than flipping between the stales 0' and p. They point out that it may playa role in nuclear relaxation by paramagnetic ions at very low temperatures since wobble can be excited by the small thermal energies, but electron spin flips between states 0' and (J cannot.
11.4 Electron Spin Echoes The use of pulse techniques, in particular of spin echoes and stimulated echoes, has become important in electron spin resonance. We turn now to some special features that arise for this case which are not encountered for spin echoes in nuclear magnetic resonance. As we have seen, if we have two coupled nuclei belonging to two different species (e.g. 13C and IH), the spin-spin splitting of one species produced by the other does not affect the size of the echo. Indeed, it acts just like a static magnetic field inhomogeneity. Therefore, although it affects the free induction decay, it is refocused perfectly at time 21" by application of the second pulse of the echo sequence. In order to observe the spin-spin coupling on an echo, one must do a double resonance experiment (Sects.7.19-21). For an electron (spin 5) coupled to a nucleus (spin I) the situation is different owing to the fact, which we have just discussed, that in many cases the nuclear quantization direction depends on the electron spin orientation. As we shall see, the consequence of this coupling is that the amplitude of the electron spin echo is modulated as a function of the time 1" between pulses by the 1- S spin-spin coupling. The modulation frequencies are the splitting frequencies of the nuclear magnetic resonance. These are the frequencies one observes in ENDOR. Therefore, the electron echo envelope in effect reveals the ENDOR frequencies. It is this special feature which has made spin echoes such an important technique for electron spin resonance. This phenomenon was first reported by Rowan et al. [11.3] and subsequently elucidated in greater generality by Mims [11.4 1. Before providing a formal description, let us examine physically what is happening. In particular, why is the electron echo case different from the nu-
clear echo case? For a pair of coupled nuclei of spins I and S respectively, the quantization of one nucleus is to an excellent approximation independent of the orientation of the other nucleus. Consider, then, application of a 1r/2 ... r ... 1r puls~ sequence to the 5-spins. If the I-spin has a particular orientalion, it maintains this orientation despite the S-spin pulses. Therefore, whether the I - S coupling enhances or diminishes the 5-spin precession rate does not matter. The dephasing produced thereby between the first and second pulse is refocused between the second pulse and the echo. Now consider the electron case. Suppose the electron (5) is initially pointing up. Let the nucleus (I) be pointing up along the effective field. Thm is, the nucleus is in a stale u+ of (11.90) and the total system is in the state 1+ +). The 1r/2 pulse now tilts the electron. As a result, the nucleus suddenly finds the effective ficld acting on it tilted. The nucleus thus begins to precess about the new field direction. Consequently the field it produces on the electron changes in time. Clearly the magnetic field acting on the electron during the interval 0 to T will, in general, differ from Ihal between T and 2T. Therefore. the echo will not refocus the spin-spin coupling completcly. If, however, T corresponds exactly to a 2'/T precession of the nucleus, then the spin-spin coupling will have the s:lme time average during the two intervals, hence the echo will be perfec!. Thus, we expect Ihe echo envelope to oscillate with T such that maxima in the echo envelope correspond to r's that are integral multiples of the nuclear precession period. We will see in our fonnal treatment of the problem that the description we have given is exact. In that connection, we can note that 10 think about lhe quantum treatment of the effect of a '/T/2 pulse on the electron system, the general concepts developed in Sect. 7.24 are helpful. In particular, they help us deal with the problem that there is not just one nuclear precession frequency. Rather, there are twO, depending on whether the electron is in its spin-up or spin-down state. We begin by expressing the reduced Hamiltonian (11.83) ill different notation. First we define the frequencies a, b, We, and W n :
rl2a :: Ax sin:! () + A z sin:! () h2b :: (A'l; - Az)sin () cos () W
n
::inHO
(11.92)
hWe :: 2(JHo Then 'Hu'<J ::
h[we5 z' - Will..., + (a1z1 + b1'1:' )25z/]
(11.93)
We now !Tansfonn this Hamiltonian to the electron spin rotating frame at frequency W
De:: We
-
(11.94)
W
using the operator exp(iw5z d) 10 !Tansfonn the wave function
t/J1::e-iwlS.,t/J
with
(11.95) 525
524
1i~ed '" fl[neS~, -wnl~, + (al~, + bl:t:,)2Sz']
(11.96)
To save in writing symbols, we now drop the primes from Xl and z' in (11.96), understanding that from now on the new unprimed axes are really the old primed axes. To solve for the eigenstates we note first that Sz commutes with 1i~ed' Therefore, the eigenstates, 1/J1 , of 1i:.w can be labeled by ms, the eigenvalue of Sz. Thus
1fred ¢'(ms) "'1i(OemS -wnl: + (al: + bIz)2ms]¢'(ms) = ~il.ms + 'li/(msl] .p'(ms)
For ms ::::
(11.98)
-! (11.100)
We now define two vectors w+ and w_ by
w+",j(-b)+k(wn-a)
(11.101.)
w_=ib+k(wn+a)
They enable us to write (1I.IOIb)
The vectors w+ and w_ are shown in Fig. 11.9. Their magnitudes are given by
z+
which give for the components of I along w+ and w_
,
1z _
'"
k_ . I
(11.105)
These enable us to write
::::-flW+·1=-/lW+1:+
1i,( -
i):::: -llw_· I'" -llw_lz _
I
Wn -
x
I
1z+u+ '" il.l+
1:+1.1_ '" -41.1-
1:_v+ :::: !v+
1,_1}_ '" -iv-
(11.107)
we have that the u's and v's are the eigenfunctions of 1t,(ms) as follows:
'liIWU+ 'li/
'" -/1(W+!2)I1+
'" 1I(w+!2)I.1_
'iiI ( - 1)"+ '" -li(w_!2)u+ 'iiI ( - 1)"_ '" -li(w_!2)u_
(11.108)
1++) =:(tu+
1+
1-
l- -) " pv_
+)
"pu+
-) =: (tIl_ (11.109)
that they are eigenfunctions of 1t~ed with eigenvalues
I
I
L_.J.:__
I (J
W+
h(ilo - w+)/2 , h(-ilo -w_)/2
I
1__ -"---_8+ -L-
_ x
FIg. 11.9. The two angular momentum vectors '"'+ and '"'_ in the :t: - : plane, defining the angles 8+ and 8_ made by '"'+ and '"'_ respectively with respect to the ~-axis. Note the sign conventioll (indicated by the arrows) for positive 8+ and 8_. The axes:+ and ,_ lie along '"'+ and '"'_ respectively 526
(11.104)
Then, returning to (11.97) to express 1i~ed' we find that when 1t~ed operates on any of the four functions
z
z
-b-
We then define the unit vectors k+ and k_ as
Then if we define the eigenfunctions 1.1+. u_, v+. v_ by the equations
'iiI ( -!) = -h[(wo + a)I. + bI.]
+a 6_
(lU03)
wn+a
(11.106) (11.99)
'li/m = -"(w" - a)I. - bI.]
Wn
b tao 6_ '" - -
Wn-a
1i,(!)
+! we have
and for ms '"
b tao 8+ '" - -
1z+ :::: k+ . I
where 1i/(ms) is an operator in the components of I, given by
1i/(ms) '" -1i(Wn1: - (a1: + b1:t:)2ms]
The directions with respect to the z-axis are shown in Fig. 11.9. The angles 6+ and ~_ shown in the figure obey the equations
(11.97a) (l1.97b)
•
(11.102)
Ii(nc +w+)12 , h( - ilo +w_)/2
(J 1.110)
respectively. The four functions of (11.109) therefore solve the reduced Hamil~ tonian. Utilizing the definitions of 6+ and 6_ given by (11.102) and shown in Fig. 11.9. we can calculate the angle 60 between z+ and z_. shown in Fig. 11.10. (11.111) 527
•
[ z
J.'ig.1I.10. The angle 80 between the angular momentum vectors \+ and ",_. With 0+ and (L both positive in the sense of thO) arrows, (}o = 8+ + 9_
where for simplicity we do not write out the contents of the inverse brackets and where the subscripts I and 2 on the brackets serve to distinguish the contents of the brackets. E.xplicit evaluation of X S1(1T)T(t - r)XS(rr) gives us X
s
1
(1t)T(T)X 5(11")
= exp{i[ .RcSz + (alz + bI>;:)2Sz + Wnl:] (t - r)}
(11.117)
We now note that we can commute the DeS: factors, to obtain
(S+(I) = ""--
~
Te{ S+ [Xs(.)"p{ i[D,(1 -
r) - D,r] S,}
x exp{ i [(aI, + bI.)2S, - woI,] (I - r) 1
x
xexp{ -i[(aI,+bI.)2S.-woI,]rl],SY[ Now, owing to the spread in De due
Referring to (11.89) and Fig. 11.8, we have that
¢ = -80
(11.112)
so thaI
10
(11.113)
We are now ready to consider the electron echo signaL We begin by assuming that Ho has some inhomogeneity. This may either resuil from actual inhomogeneity in the magnet or from some other source such as hyperfine cou-
pling to nuclei other than the spin I. This inhomogeneity would cause the free induction signal of the electron spin 10 decay, and thus causes us to go to a
magnelic field inhomogeneity, the
tenn exp{i[fle(t ~ ,) - fleT1S z} makes the signal {5+(0) be small at values of t. However, we note that when t '" 2t the factor
XsC1r)
(11.114)
Then defining the lime development operation T(t), in the electron spin rotating frame we have T(l)
=ex =
p(
maSI
Therefore, at t = 2. the signal is independent of fie, and an echo forms. We now confine our attention to this time. We next factor out X S (7r) from the left-hand side of [ h and from the right-hand side of [ J31, and utilize the fact that Tr{ABC} = Tr{ CAB} 10 produce X
1
(IU20a)
S (1T)S+ Xs(rr)
which we rewrite as
spin echo to eliminate the effects of inhomogeneity. We therefore analyze for the signal at time t = 2r for a pulse sequence T ...
(11.118)
(11.119)
v+ = u+ cos(80 /2) - u_ sin(8o/2) v_ = u+ sin(8o/2) + u_ c05(80/2)
XS(1rI2) ...
13'}.
(11.120b) Moreover, we get a more compact notation if we define the unitary operators R(Sz) and R(-Sz) by
R(S,) " exp{ -i[(aI, + bI.)2S, -woI,]r}
(11.121)
R(-S,) " exp{i[(aI. + bI.)2S, +w"I,)r}
(11.122)
,ad
-*1{~edt)
«p{ - i [D,S, + (aI, + bI.)2S, - woI,] I)
(11.115)
where R(-Sz} is obtained by replacing 5 z of R(S:) by -5:. Then we get
We take the density matrix e(O-) prior to the first pulse as ,q(O-) = Sz. Then the signal at time t (we take t to be after the 11" pulse), (S+(t», is given by
(S+(I)) = Te{S+ e(l)) = Te{ S+ [T(I -
r)X s (·)T(r)Xs(·/2)] , S,[
= Te{ S+[Xs(.)Xs'(.)T(1
We now evaluate the (face using a complete set of basis function. For the complete set we pick initially the two electron States a and fJ with quantum number
1,' 1
- r)Xs(.)T(r)],S,[
I,'} (11.116)
528
ms and the two nuclear states u+ and u_. which we denote by the quantum numThus, using the fact that (msllIS_lmsp./) = 0 ber p.. which can be +! or unless p. = l, and so on,
-!.
529
L:
(S+(2T) =
(msIIIS_lmSII)(msPIR( -S:)R(S:)lmSIt')
Then, for any function j expressible in powers of the components of I,
"'S,I'
...5·.. ',1''' X
R1/2fR_I/2=exp(iw+Tlz)jexp(-iw+rlz)=j
(msJ/ lS'y Im~Jt') (lnsp'IR-1 (Sz)R- 1(-S:)lm~plI) (11.123)
In order to have the sums be diagonal sums, we get that m~ = ms and p" = JJ. Now the matrix element of S_ requires that
ms=mS-I.
(11.124)
ms
-t '
=
+!
(11.125)
,. ,
and therefore (mspIS-lm's II) = 6", ",' -I' Then we get for Ihe matrix elemen! of Sy (ms/IS,lms/) = WS,I-!) = ;i WS+I-!) = ;i
so that
(S+(2T»
=
(11.126)
;i L (!,'IR(-S.)R(S.)IV) x (- !/IR-1(S.)n-'(-S.)I-
!")
(11.127)
Now, lcuing (he operator S: of R(Sz) and R(-S:) operate, we replace S: by ms (and -Sz by -ms) in the matrix elements. Then, defining the operators
R- 1/2 =exp{i(alz+bl.l"+wnI:)T} , = exp{i( - aI: - bI", +wnlz)T}
we get
(S+(2T»
=
,
~iTrJ{R_I/2RI/2R=:/2RI>~}
530
.
(S+(2T»
I I = 2iT~}
(q IR_ I / 2 Iqll ) (0.1 IR 1/ 2 1qll) (u"in: :/210'1/1) (alII IR~/~ 10")
",17" =
T
(aIR 1/ 2 Iqll ) e-iUItW_T (qIlIR1/2 Iu)
eiW_T( YI> zl. %2. Y2, and %2. the z-axis lying along the bond. The functions %1 + %2 and %1 - %2 are shown schematically in Fig. 11.14. A study of the figure shows that %1 - %2 corresponds 10 a lower energy than does %1 + %2. since it has fewer nodes and lends to concenlrate the electronic density between lhe two atoms where it can share their altractive potential. The Slates are in fact referred to as bonding (ZI - Z2) and antibonding (ZI +Z2). In a similar manner it turns out that XI + X2 and YI + Y2 are bonding, and XI - X2 and YI - Y2 are amibonding. (The z-states are the so-called q-states and the X or y states are the lI'·states.) The energy levels of these states are shown schematically in Fig. 11.15. Actually the states Xl + X2 and VI + Y2, which are degenerate in the free molecules, are not degenerate in a crystal, but we neglect that splitting. Since there are 6 orbital functions, there is room for 12 p-electrons. The VA: center, which has only II, therefore has a hole in the ZI + Z2 state. That is, there is an unpaired electron in that state. We have introduced, in Fig. 11.15, a labeling u or g (ungerade or gerade) that describes the parity of the orbital state. One may expect to observe an optical absorption due to VA: centers. Since electrical dipole transitions are allowed only for transitions u to g or g to u, the optical absorption will arise from transitions of an electron in either the states ZI - Z2, Xl - x2, or YI - Y2 to the empty %1 + %2. The strongest optical transition
538
"1
t
" - '2 (u,)
,
(r,,)
(11.154)
m ,
In the absence of an applied magnetic field this expression is correct but in the presence of a magnetic field described by a vector potential A we must modify it to oblain a gauge invariant result: 1ioo ""
2~~c2S'
[E x (p+ ~A)]
(11.155)
where -e is the charge of the electron. Equation (11.155) follows directly from the Dirac equation but it is also intuitively obvious because. as we have discussed earlier, we always replace p by p - (qlc)A in the presence of a magnetic field where, in our case q -e. The orbital Zeeman interaction 1ioz is
=
2mc
2mc
(11.156)
We can consider that 1iOZ and 'HSO together constitute a penurbation 1iperL :
(11'",) "y, --• "J Y2
"y, + "2
eli.
'liSO=-2 "S·(Exp)
" A2 1ioz "" -2( p ' A + A· p) + __ 2
'. + .1:2 (u,,) ,,
which has quenched orbital angular momentum. In our earlier discussion of the g·shifl for a problem with only one force center we saw that the g-shift arose from the interplay between the spin-orbit coupling and the slight unquenching of orbital angular momentum produced by the orbital Zeeman energy. When more than one force center is present there is no unique point about which to measure angular momentum and it is natural to return to the more basic fonn of the spin-orbit coupling:
1iperL ""
Fig. IUS. Molecular orbiUlIs ronned rrom ,..sta~ in .. halogen molecule ion. The allowed optica.ltransitions into the unfilled
All = A2 = tHo x (r - Rz)
Eo
c.c.}
E"
eZh { (0.I H .o,(1)lvoo') " - 2 2 3('1 5 1")' (OlE x "71I vo)
mc
c.c.}
+ _1_ L;,(OIE x vln)(nlv' "71 + "71· vivo) + 2m" Eo-En =0 (11.176)
(11.171)
The integral 1, defined as
1==
Jtb~(p·V,p+V¢·V)fJOdT
(11.1n)
can be transfonned, making use of Ihe fact that the wave funClions are real and utilizing panial integrations, to be
(11.172)
To prove the Iheorem of (11.170). we stan wilh (11.161). We express the matrix elements involving A in tenns of the u's and v's. and neglect overlap tenns. For example, 542
vln)(v"lv' "71 + "71' vivo) +
To derive our theorem, (11.170). we must show Ihal the lenns involving ¢ add to zero. By making use of the faci Ihat we are neglecling overlap [enns, our proof is equivalenl to showing thai the quantity (Oul1ll1g(¢)lvou/). defined below. vanishes:
(they
The beauty of (11.170) is that il allows us to choose the veclor polential A' used to evaluate the integrals with Ihe u's independently of the veclor potential All used for integrals involving v's. C'Ne shall discuss handling the matrix elemems (OlE x pIn) shonly.) In panicular, we shall see, if the IWO nuclei are at R I and R z, respeclively. we can evaluate the matrix elements readily by choosing
A' = Al == tHo
X
(11.175)
2~zhc3 (qJ5Iq')· {(UoIE x A'luo) + (volE x A"I1l1J)
==
(11.174)
defining Ihe funclion ,p. (That (11.174) is simply a gauge transfonnation follows, of course, from Ihe faci that it satisfies the requirement V x A' = V X A"). We then substitute A' for A in integrals involving u's. and All + V,p for A in integrals involving v's. By collecting tenns. we gel
(OaIH.ogIOo-')
(uIlIAI.p+p.A/luo)
(11.173)
AI = A'I + V?
We shall neglect all contribulions 10 matrix elements involving a produci of a u and a \/. This approximation is often good, but can lead to errors in some cases. We have, then, as our theorem Ihat the combined effeci of the spin-orbit and orbital Zeeman coupling is to give a g-shift characterized by
where A' and A" are any vector potentials Ihal give the static field differ therefore, at most by a gauge transfonnalion), and where
.
Then we introduce two vector polenlials. AI and All, which differ by a gauge transfonnation:
(11.169)
=
=(volE x Alvo) + (volE x Alvo)
II
1= ih
J
is simple
10
(vo¢V 2 1/Jn -1/Jn,pV 2 vo)d,.
(11.178)
re-express Ihe firsl lenn on Ihe righi, since
2 2m V tbn = /;2(V - En).pn
(11.179)
where V is the potential aCling on the electron. We evaluate the second tenn by 543
nOling that, neglecting overlap,
J
Wnt/J\l2 vo dr ""
J
v,,4> V2v odr =
!
2 vn4>V ¢odT
(IU80)
By utilizing (11.179) in (11.180) and again neglecting overlap, we obtain finally 2m (11.181) (nip' \74> + \74>' vIvo) = -:-. - , (Eo - En) 1/Jn¢vodT
j
r,
, r.
Therefore. when the upper sign applies, the two tenns in the square brackets of (11.185) cancel, and (DIE x pIn) vanishes. On the other hand, for the lower sign, the tenns add, giving twice either one. Thus the state (Xl + x2)/.,fi does not contribute to the g-shift, but the state (XI - X2)/,fi does. Of course a similar argument shows that (Y1 + Y2)1.,fi also makes no contribution although (YI Y2)1V2 does. The states involved in the g-shift are shown in Fig. 11.16 by the solid arrow.
We can substitute this expression into (11.176) and collect tenns to obtain
,'I.
[
(Oo'l'HLlg(t/J)lvoql) = 2m 2 c3 (uISI0"') (DIE x 'V¢lvo)
- 2 L'(OIE
x ''In)(nl''lvo)]
(11.182)
n
The prime can be removed from the summa{ion, since the diagonal spinorbit mauix: elements vanish, giving
(Oul'H"g(,,)lvou') = =
e. 2 tt
--(uISlu'). [(DIE x"" 2m2 c3
2,.
.
2(E
-~(O"ISI17I). jE x V(¢va)dr 2
x ""lvo)J (11.183)
2m cwhere (E x \7) signifies thai E x \l is to operate on all functions to ils right, that is, on both'" and VQ. But the integral can be shown to vanish. utilizing the fact that V x E = 0 and transfonning the integral J \l x (E¢wfi)dr into a surface integral. We omit these details since they are quite standard. Our theorem is thus proved. We have not as yet said anything about the spin-orbit matrix elements to excited states. By utilizing the fact that the electric field E is large only near the nuclei, we are always able to neglect overlap when evaluating spin-orbit matrix clements. Thus (DIE x pin) = (uolE x plu,,) + (volE x plv,,)
(11.184)
To see the full import of (11.184) as well as to illustrate our theorem (11.170) concretely, we now tum to the evaluation of the specific problem of a molecular complex in which only one electron occupies the Vk center orbitals. The ground state is therefore given by (11.153), t/;o = (1/V2)(zl - Z2). We are concerned with excited states such as (l/V2)(Xl ± X2). We have. using (11.184),
(DIE x pin) = (
Z1-Z2
.j2 IE x pi
= WztlE
.j2
(kI2~~"S'EXPll) =A(kIL·Sll)
(11.185)
(11.186)
(11.187)
where k and I denote free-atom states associated with the particular A. For our example. XI, YJ, and Z1 are being taken as free-atom p-states. Therefore we can write
'I., , (0-1510-') . (z11E x plxd = A(uISlo-')· (zIILllxl)
2m ,
(11.188)
where IiLl is the angular momentum about the nucleus of the first a1om, and where A is the spin-orbit coupling constant appropriate to the (np) electron configuration of the outer electron. Evaluation of the matrix element (zJlLJlx n ) proceeds as in (11.17) and (11.18). We now tum to evaluation of the matrix elements (unIAI. p+p .A'luo) of (11.170). We have that Un = xl/..fi or Yd..fi. uo = zd..fi. By utilizing (11.163) and the fact that the u's are real, we have (unIAI·p+p·AJluo) = IiHo'
J
UnLluodr
= -2-' (:1:1ILdzl)
Since the two atoms are identical and since E is large only near a nucleus,
544
If we had a free atom, the spin-orbit matrix elements could be expressed in tenns of the free-atom spin-orbit coupling constant A according to the equation
hHo
X1±X2)
x plxt) Of (z,IE x plx,)J
I.jg.ll.I6. The solid line indicates the states joined to the ground state Zl - Z2 by the spin-orbit coupling
or
But. by the symmetry of the atoms, (unlA1 'p + P ·Alluo) = (V,dA2' P + p' A2lvo)
(11.190)
so that we have. neglecting the first-order tenns such as given by (11.170), 545
(o.I1i",IO.') = 2PA(.ISI.'). [(Z1l
L 1I 2 (a "hole"
-!)
!)
-04)
-!)
(11.215)
If N ;: I, clearly A::: (. If N represents a shell that has only one missing electron, A ::: -(. Since ( is always positive, we obtain in this way the faci that holes have negative A. We can utilize the free-atom ('s 10 evaluate (11.213) since it enables us to write
f------552
ZI -
Z2
!o'ig.lI.2t. Filled states when the V center orbitals contain only five electrons
553
shift). The close proximilY of these two states would make l.dyul > 1.d9zzl, and 9"y, of course, is Slj]] 2, since Ihe peninenl matrix elements vanish. It is clear that Ihe 12 states are less Ihan half full, yel the predominant g-shift is that of a "hole". We see, Iherefore, Ihat we must use extreme caution in characlerizing centers as "eleclron" or "hole" centers simply from the g-shift data. We should also comment that we have assumed very simple functions with no overlap between atoms, to compute the matrix elemenls. In general, we would need to make correclions both for overlap and for the possibilily thai the funClions :1:), Yl, ZI> and so on are linear combinations of alomic orbilals, as we did in discussing the hyperfine coupling. However, these corrections do nOI aller Ihe principles, although they do complicate the numerical calculations.
12. Summary
We have considered a variety of effects - line widths, chemical shiflS, Knight shifts, hyperfine splillings - a bewildering array of seemingly special cases. As we look back, we see some effeclS that occur in first-order penumation theory, others thai require a higher order. Since we have discussed the phenomena one by one, it is appropriate to summarize by wriling a single Hamiltonian that includes everything. As we coOlemplale it. we should remind ourselves of the significance of each tenn. We write below Ihe Hamiltonian describing a nucleus interacling with an electron in the presence of a magnelk field Ho. We define the vector potenlials Ao. associated with the field Ho. and An' associated with the field at the electron owing to the nuclear moment (An = P X rlr 3 nonnally). We also define the quanlity
.
,
1f=7~+-Ao 1
(12.1)
C
Then we have Ihe following Hamillonian: 2
n2 'l.J=_Ji I~ 2mv
.1"",,,,,, m~
k;~'lio
+V0+ V~rrl\
.leo'_ """.Iial '.'/' ;~po'.~';,,1 'h. n.w ._". d... ,~ Io•••ad l' .U'....... IOi4t ,h.
.1...1t... 01, oth
1.. ".....
.._
.lec,roo .pi. loc",...
.1"",,,,,, .pi._hi' cOIIpli••
, (p + 2mc .oupli••
..o,u
"
• Ao + Ao • p)+ 2mc2 A~
~r
.I.. ,,~. O'bital ",o'io. '0 Ho
couplio. or uolu. m"",.o' '0 .1""00 O'bi"llDo,io~
"'''''''0'
.....pli•• of uol... .. i,b .1.." .....pio momo" for r-IIol..
couplio. or ~uol..r mo",.01 .. i'b .1"'1'S, (ho)s = (Hds .for mJ = ! and mi = Show that the energy eigenvalues of (7.190) are given by
x {[(ho)s ~(AmdYsh)12 + (Hd~}'/2
aod
b)
+ k[(ho)s - (ArnII"Ysh»)
E(mJ, ms') = -1"JhHomJ -,stuns'
dl
.'dt!.( (5,) -
Show that for a given mI, the S-spins see an effective magnetic field (H(mI»eff given by
Show that the I echo, M/(2r), varies with
l'
according to the equation
M/(2r) = Mo cos (21"/h/S1')
7.8 Consider the energy levels shown in Fig. 7.4. Suppose one uses an adiabatic passage at the electron frequency (E 1 - E2)/;1 to interchange the populations of levels I and 2 and then quickly observes nuclear resonance. a)
b)
Show that the nuclear resonance transition (E2 - E 4 )/h has an increased absorption rate and compute the ratio of the rate of energy absorption to its normal value. Show that the nuclear resonance transition (E: ~ Ea)/li has a stimulated emission, and compute tre ratio of the rate of energy emission to the nonnal rate of energy absorption of this transition.
7.9 The density matrix of a pair of spins [ and S in thermal equilibrium at temperature 8 is, in the high temperature approximation, ) 11'B_
d!.
(A.5)
By utilizing (A.4) and multiplying from the left by exp(-..\B)exp(-..\A), we can rewrite (A.5) as
e-~Be-~A Bc>'Ae>'BG
_ BG = dG
d!.
(A.6)
We can evaluate the expression e-~A Be>.A
== R(..\)
(A.7)
as follows: by taking the derivative of both sides of (A.6) with respect to ..\, we find e~>'A(BA _ AB )eM = dR
d!.
- C=
(A.8)
579
since AB - BA =: C commutes with A. Integrating (A.g) we have R(>.) = -C>'+constant
t
-
-=r
(A.9)
.
We can evaluate the constant by setting>. '" 0 and by noting from (A.7) that R(O) = B. Thererore R(~l.-C~+B
h we get
J
~/I(w) = _w_ r:.e-E./kT 2kTZ -co a,b
(A. 10)
By substituting (A. 10) into (A.6) and using the fact that C commutes with
B, we get dG
-~CG=
(A. II)
d~
x (al".lb)(bl".la)exp {(ice. - E,ltVhjexp (-iwt)dt
(8.5)
We can use the fact that the states la) and Ib) are eigenfunctions of the Hamiltonian 1i to express the expression more compactly as Xl/(w) ==
which can be interpreted to give
~ 2kTZ
J ~(ale-1t/kTe
+~
~,
i7iL / Ait
-co a,
X (blpzla)e-ilJldt
(A.12)
C '" exp { - (>.2Cfl + const»)
(B.4)
-
•
e- i1tt / Alb)
.
(B.6)
But the summation over a and b is clearly just a trace, so that
The constant must be zero because, from (A.4). C(O) = I. Therefore
e A +B = e Ae Be- C/ 2 . Q.E.D.
(A.13)
X"(w) == 2k;Z
J
Tr {e- 7i / kT ei1it / l itze-illt/A J1r }e-ilolldt
(B.7)
-~
In the high-temperalUre approximation, we replace exp (-'H/kT) by unity. If we then define the operator Pz(t) by
B. Some Further Expressions for the Susceptibility
flz(t) == ei1tI / Altze-i1tt/h
(This appendix requires familiarity with Chaps. 2, 3, and 5.) Equation (2.190) gives an expression for XII. Another expression is frequently encountered in the literature. It provides an alternative derivation for the moments of the shape function. It can be obtained from (2.190).
(B.8)
we can also express (8.7) as +~
XI/(w) == 2k;Z
J
(B.9)
Tr{Pr(t)JJz}e-ilJ1dt
-~
X" = ; ;
Le-E./.T!(al".lbll'6(E. - E, -
hwl
(B.I)
.~
by use of the integral representation of the 6-fuoction:
1 +co
.
~)=hJe-~rh.
(B.~
-~
By substituting into (B.I), we obtain
J~e
xl/(w) 1 jew) = -w- == 2kTZ
+~
hw w =-x"() 2kTZ
t...-
-co E ., E•
+J~
.
Tr {/lz(t)/lz }e-llJ1dt
.
(B.IO)
-~
-E /kT
•
x (alJ1rlb)(bIJlzla)exp [i(Ea - Eb - llw)TJdr
We can prove another interesting theorem by taking Fourier transform of (B.10): ,
(8.3)
and substituting for the variable r a new variable t that has the dimensions of time, 580
The quantity Tr {pz(t)J1z} is a form of correlation function. and (8.9) states that i/(w) is given by the Fourier transform of that correlation function. By using this expression for Xl/(w), it is easy to show that omission of the dipolar terms C. D. E. and F of (3.7) gives one absorption at the Lannor frequency only, but that their inclusion gives absorption at 0 and 2wo. We get also a very compact expression for the shape function jew):
J ..
+~
I
I 2kTZ Tr{l-lr(t)/lr} == 21f
j{w)ellJ dw
(B. II)
-~
581
We see that, setting t t
=0, I
2kTZ T,{",(O)",). 2.
+=
J
(B.12)
f(w)dw
-~
By taking the nth derivative or(B.I1) with respect to t, and evaluating at t = 0, we find
I
1 tr' (1)" 2kTZ dtn Tr {llz(t)llz} t=o = 211"
+~ "
J
(B.13)
w f(w)dJ..J
-~
We get, therefore, a compact expression for the nth moment of the shape function few): _
J
(w") -_
wnf(w)dw
-~
+J~
__ (i)-"(tF/dt")Tr (JlzO)llz}lt T, (",(O)",)
0
(B. 14)
f(w)dw
-~
As an illustration let us derive an expression for the second moment (w 2 ). Taking the derivative of Ilz(t) gives
~Tr(ei'Hj/l" e-i'Ht/l" J dt 2 ,..z ,..z
.G)'
T, {.m'I'[1i. [1i. ",)).m'I' ",}
= - hl2 Tr {ei'Ht/ll (1i'llz)e -i'Ht/lI ('H:,llzl} (B. 15)
Therefore
(w') •
_2.. T, ((1i, ".)'j h2
Tr {Il;}
(B.16)
This formalism provides a very simple way of generating expressions for the higher moments. Note, however, that the odd moments all vanish, since few) is an even function of w. So far, apart from assuming the high-temperature approximation, we have left the specification of the Hamiltonian completely general. We can proceed further if we assume that it consists of the sum of a Zeeman term Hz and a tenn 'H:p , often a perturbation, which commutes with Hz. A typical H p is the terms A and B of the dipolar coupling. Then, since H p and 1{z commute,
Tr {e i('H p/ll)1 /lye -i('H p/ll)t Jlz} = Tr {ei('H~/lI)l(- Jjy')e -i('H~/1t)tJlz' } = _ Tr {ei('H~/")1Jly'e(i/ll)'H~1 /J z '}
Tr {Ilz(t)pzl = cos WOt Tr {ei{'Hp/l)t lJze-(i/l)'HptJjz)
(B.20)
Since this is the Fourier transform of the shape function f(w), we see that the transient behavior consists of a term cos wot multiplied by an envelope function. If we define Il;(t) as (8.21)
p;(t) '" e(i/l)'Hplllze-(i/l)'Hpi
we can say that the envelope function is Tr {Il;(t)pz}. By writing the cos wot as cos wot = !(eiWOI +e- iWOI )
(8.22)
we can say that the two exponentials correspond to lines at +wo and wish to discuss only the line at +wo, f+(w), we can therefore write f+(w) =
4k~Z
-woo If we
+~
J
Tr {p;(t)/Jz }e+ilotole-ilotldt
(8.23)
-~
(B.24)
This can be rewritten as
J
1 +~ 4kTZ
'" e(ilh )11' pte -iwoJztJlze""O / z!e -(i/h )1t pt
(B.t9)
But the last trace is clearly the same as the first. Therefore the trace is equal to ilS own negative and must vanish. We have, then, that the correlation function Tr {llz(t)llz} is given as
_1_ Tr {1l;(t)Il'z} = -2
IlzO) :: eOlh )(1l z +'Hp)tIlze -(ilh)(1tz +'Hp)t
= e+(i!h)'H pt(llz cos wot + Jly sin wot)e-(i/h)'Hpt
In general, if 1ip is invariant under a rotation of 180° about the x- or yaxes (as is usually the case), the second term vanishes. This can be shown by evaluating the trace, using coordinates x' '" x, y' '" -y, Zl = -z, which differ only oy a 1800 rotation about x. Then 'H. p = H~ by our postulate, so that
71'
!+(w)ei(w-loI(J)fdw
(B.25)
-~
(B.17)
By laking derivatives as before, we now get +~
J
where we have used (2.55). We have, then,
(w - wo)f+(w)Jw
Tr {Jlz(t)llz} = cos wot Tr {e(i!h)'Hpt Ilze-(i/")'Hptllz} + sin wot Tr {e(i!h)'H pt Il y e -0/" )1t pt Ilz}
58'
(B.18)
(d"ldt") Tr {[Il;(t)pz]/ oj • Tr {1l:(O)J'z}
(i")=-~~~::--- +~
J
f+(w)dw
=i"{(w-wo)"}' (B.26)
-~
583
This, then, gives the nth moment with respect to the frequency wo . This fonnalism has supressed the line at -we, which would, of course, make an inordinately large contribution to wo)"} were it included! By following steps similar to those of (8.15), we now find
«w -
« _ w We also
wo
),)=-.:.. Tr{[Hp,p.)'} h2
Tr {pi}
(8.21)
see that
(w - wo) =
I
Ii Tr {[Hp, P,Jp,}
(8.28)
which can be shown 10 vanish if 1ip consists of lhe dipolar tenns A and B, as shown in Chapter 3.
C. Derivation of the Correlation Function for a Field That Jumps Randomly Between ± h o We shall assume the field jumps randomly between the two values we shall label as states I and 2. We shall call
+ho = HI
- ho = H2
± ho, which
,
ddP2 = W(PI - P2)
dT
.
(C.S)
T
This is a "nonnal modes" problem, with solutions obtained by adding or subtracting: PI(r)
+ neT) = conSI. (= 1 from normalization)
PI(r) - n(r) = Ce- 2WT
(C.6)
where C = PI(O) - 1'2(0) = PI(O). Since nCO) vanishes and since PI(O) = I, C=1. By making use of (C. I), (C.4), and (C.6), we have H(O)H(T) = HI (HIPI(T)
+ P2(T)H2]
= h~e-2I1'T
(C.?)
An identical answer is found for H(O)H(T) if the field is assumed 10 be H2 at T = O. We must weigh these equally (Ihat is, average the answers over the initial fields) 10 gel Ihe final ensemble average. We denote this by a double bar, 10 indicate the fact that we have averaged over an ensemble of initial conditions as well as a variety of histories for a given initial condition: H(O)H(r) = hije- 2WT = G(T) . This is the correlation time assumed in Chapter 5, with 2W
';;!
llro·
(C. I)
Then we wish to know the correlation function G(r): G(T) = H(t)H(l + T)
dpi = W(P, _ PI)
D. A Theorem from Perturbation Theory (C.2)
where the bar indicates an ensemble average. If the field is HI at time t = 0, then we can write for a single member of the ensemble: (C.3)
where PI(r) and P2(T) are either zero or one, depending on whether at time the field is HI or H2' We now perfonn an ensemble average of (C.3) over the various histories. This replaces quantities PI (r), P 2(r) by their ensemble averages PI(T) and l>:2(r), which are the probabilities that in an ensemble in which the field was HI at r = 0, it will be HI or H 2 at lime T. Thus we have
In this appendix we shall derive from perturbation theory a theorem that has wide utility in magnetic resonance. It is closely related to second-order perturbation theory but gives the results in a fonn particularly useful when there is degeneracy. A typical situation in which the theorem has great use is illustrated by the g-shift calculation of Section 11.2. We may divide (he Hamiltonian into three teoos;
1i = 1£0 + 1£1 + 1i2
where
(D. I)
T
p'
1£o=-+VO+VI
2m
HI = 2pH· S
(D.2)
H, = AL·S+pH·L
(CA)
This equation, of course, assumes thai at T = 0, H(r) = HI, so that as r --+ 0, PI(T) --+ I, n(r) --+0. Equally likely is the situation that the field is H2 at T = 0, which will give a similar equation except thai I and 2 are interchanged. We shall assume the behavior of PI and 1'2 as a function of r 10 be given by a rale equation: 584
Since 1£0 does not depend on spin, its eigenstates may be taken as products of an orbital and a spin function. We denote the orbital quantum numbers by 1 and the spin quantum numbers by a. Then
Hoi/a) = Ed/a)
(D.3)
The states 110') are degenerate for a given 1 because of the spin quantum numbers. 585
The tenn 1{t lifts the spin degeneracy. Since it depends on spin only. it has no matrix elements between different orbital states: (D.4)
In general the matrix elements of 1{l between staies 110) and Ila') where a #a' will be nonzero. Therefore the presence of 1{1 still leaves us a group of submathese trices (lal1{llla') to diagonalize. For our example, since the spin was submatrices are only 2 x 2 and are easily handled. The presence of the term 1{2 spoils things, since 1{2 joins states of different I. However. as a result of the quenching of the orbital angular momentum. the matrix elements of 1{2 that are diagonal in l vanish:
(D.7)
where dr and drs represent integration over spatial and spin variables, respectively.. By utilizing (0.6) and the Heonitian property of 5, we have
J
¢iO/1{¢I,O',dr drs =
!.
I)
~~~
",
I] Il
I,
{---1---, ---~---·~---i--'::
{
,,
,,
:
:
----,--, ,
,__ ---'-----,--, , , I
o
'HllIO/l'Zt'rU
0'H2/l
01l ' Uro
I_::..l:i:_:_::'--i--_-l.i-_--.J-_-_--'--l--_-
.-
'
"
= (IO'le-iS1{eiSll'O")
(0.8a) (0.8b)
where we have used the notation 110') for matrix elements calculated using the 1/J's. We may interpret (0.8) as saying that we can look either for transformed functions, rPIO/. or a transfonned Hamiltonian, (0.8b). If we define ?i' as (D.9)
We may schematize things as shown in Fig. 0.1, where the Hamiltonian matrix is illustrated and where we have labeled which tenns 1{1 or 1{2 have nonvanishing matrix elements. Il
1/;iO/e -iS1{e iS 1/J/IO/ldr drs
(D.5)
(laIH,lla') = 0
I,
J
.
Fig.D.1. Hamiltonian matrix. The regions in which nonvanishing clements of?i l or 1i2
we may state as our goal the detennination of a Hermitian operator 5 that generates a transformed Hamiltonian ?i' such that ?-t has no matrix elements between states of different I. Presumably S must be small, since the original Hamihonian 11. has small matrix elements off-diagonal in 1. Therefore we may approximate by expanding the exponentials in (0.9): ?i' = e-iS?ie iS = (l-iS-
~; + ... )?i(I+iS- ~; + ... )
. S' H - -2HS') = H + .[H, SJ + ( SHS - T = H + i[H, S] - ~[[H, SI, SI
may be found are labeled by the shading. The quantum numbers II, 12 lind b designate different eigenvalues of 1io
Writing ?i =?io +?il + ?i2. we wish to choose S to eliminate 11.2· By writing out (0.10), we have
The technique that we shall describe below in essence provides a transformation which reduces the size of the matrix elements of 11.2 joining states of different 1. In the process, new elements are added which are diagonal in 1. In this way states of different I are. so to speak, uncoupled, and we are once again faced with diagonalizing only the smaller submatrices diagonal in I. The basic technique may be thought of fonnally as follows. The set of basis functions 1/J10/ fonns a complete set, but has the troublesome 11.2 matrix elements between states of different I. We seek a rransfonned set of funclions ¢/O/ given by • _e iS •I• (D.6) '1'/0/ '1'/0/
1{1 :: 1io + 1{t + 11.2 + i[?io +?i h Sj + i[1t2' 5j - ![5. [S,?ill
where S is a Hennitian operator that reduces the size of the troublesome matrix elements. In tenns of the ¢'s, the Hamiltonian matrix elements are
(D. 10)
(D. II)
We can eliminate the third teon on the right by choosing ?i2 + i[1{o + ?iI, S] =
a
(D.12)
Then we have 1{1 = 1{o + 1{1 + i[11.2. 5]
i2
i2
+ 2 [[Ho + HI, S], sl + 2 [[H" 5], S]
(D.13)
If ?i2 were zero, S would vanish. Therefore, we expect 5 will be of order 1{2. and the last teon of order (?i2)J. Neglecting it, and utilizing (0.12), we have i 1t'=Ho+HI+ [H"SJ . (D.14)
Z
586
587
Equmion (0.12) may be put in matrix fonn to obtain an explicit matrix for S. Using the facts that 1i1 has no malTix elements between states of different 1 and that 1i2 has none diagonal in 1, we have
L:
(f0'1 1i211'0'1) + i
(fO'I1-£'jfO'I)
i
=
= E/ona , + (lal 1t llla
+~ L
[(lO'I1io + 1i III"0'")(1/1 alii S II'0'/)
I"a"
.
(0.15) =
0"
-(lO'ISI/' 0''')(1' a"I'H JlI'0'1)] = 0
(0.16)
If I f.l l , we may neglect the tenns in 'HI! as being small compared with those
involving E, - E". Then
=.; (lal1i2 1110") (E" - E,)
(0.17)
1
If 1= 1' , we have, for (0.16) L:(lO'I1ill/all)(la"lSlla') = L(faISlla")(la"IHllla')
(0.18)
0"
0"
This is readily satisfied by choosing (0.18a): ll (laISlla ) = 0 .
~(lcrll'H2' Sill, cr')
=
~ L
L
(10' JH2
I" ,0"
EI-E/"
E. The High Temperature Approximation In severnl places in the text we make use of the high temperature approximation. For example, on page 63 in Chapter 3 we replace the exponentials by unity in the expression for X"(w):
[(lal'H2I t 'a")(I"a//ISll'a')
i'(w) =
- (la ISIlII 0 11 )(11/ a ll l'H211' a')]
~
W' 0'//)(1'/ a" JH2IIa')
and neglecting the coupling between states of different I.
2 I"a"
=
lll EIOaa l + (fal1i IlIa') + L (lal'H',d a")(11/ a"I H 211a') I",a" EI EI" (0.21)
(0.19)
since 'Ho and 'HI are diagonal in 1. Thus,
(lal1i'll'a')
[(laIH21///al/)(I//a Ii ISI/....l )
If 0' = ai, we recognize that the tenns in 'H2 give the familiar expression for the energy shift in second-order penurbation theory. However, our expression also includes matrix elements for 0' f. 0". In this connection we wish to emphasize that in degenerate penurbation theory, ordinarily one must find zero-order functions that have vanishing off-diagonal elements. The method we have described places no such restriction on the basis functions 110'). If the quantum numbers 0' lead to elements (lO'IH'lla') between states of different 0', it means merely that we must still diagonalize the matrix (laIH'lla') of (0.21). We conclude that the presence of a tenn 1-£2 is to a good approximation equivalent to adding to the Hamiltonian Ho + 'H I matrix elements diagonal in I of
(0.18,)
Therefore S does not join states of the same I. [Equation (0.17) and (0.18a) enable one to verify that S is Hennitian; that is, that (l0'151/'al ) = (l'a'1511a)*, where the star indicates a complex conjugate.] By using (0.17) in (0.14), we may find the new matrix elements between 1/0') and 11'a'). First we note that the states off-diagonal in 1 are
(IQI'H'II' cr') =
)
- (falSII" 0'1/)(1// 0"11 'h2 Ila')J
(l0'11i211' a') + i(EI - E/I )(fa lSI t 0'/) + i L [(10' l'h I I/a l/)(1O''' IS\l' 0 ' )
(l0'ISI1'(;/)
l
/1J,a"
-(loISII"0")(1'1 O'"I'Ho + 'HIlt 0")] = 0
Th"
(lcrl'Ho + 'HI + 2 ['H2, SlIlcr')
where
L (laIH21/"a//)(f"a//I'h21/'a')
:~~ L::e- E,,/k7'l(alll x lb)1 2o(Ea ",_
Eb - tiw)
(E.I)
1"&" X
[Ell
~ EI" + EI ~ EIII ]
(0.20)
The off-diagonal matrix elements are therefore reduced in the ratio of 1i2 10 the difference between eigenvalues of 'Ho, and the states of different 1 are "uncoupled". The matrix elements diagonal in 1 are modified, too. They become, using (0.14) aod (0.17), 588
is the partition function. Since the energies E a are energies of the N-panicle system, they may range from -N'YhHo1 to +N'YhHo1 as a result of the Zeeman energy alone. Of course, the energy -N'YhHoI would occur only if all N spins were in the state m = I, and is thus quite unlikely on a statislical basis. However, we expect to find typical values of lEal';::' .../N'YhHoI combining the m values 589
of the N spins at random. Since N ';;t 1023 in a typical sample, how can we approximate Ea/kT . (E.?) becomes
ml,m2, ...
j:::::l
mj. we can write
exp (liwoMlkT) ;;: exp (llwom/kT)exp (tlWomjlkT)
N
Eo. ::::: -'YlIHo
},f -
L
exp(m;1lWo/kT) = (2J + I)
(E.lI)
m; giving
L
Z = (2l + I)
exp(mhwolkT) = (2l + I)Z(N _ I)
"II ,m2, ... ,fIlj-l,m j+ I ...
Then
where ZeN - I) is the panition function of N - I panicles. The sum
l(al"zlb)I'
L
(al 2:>zjlb)(bl LI'z.la)
=
j
•
L(al"zjlbXbl"z.la) .
=
(E.5)
j,k
Since the 1J.zj's involve the coordinates of only one nucleus. we see from (E.3) that we only get tenns in which j = k so that 2 l(al1J.:elb)12 = l(ml,m2 •...• mi' .. ·11J.:ej!m\,m2•... ,mi. ... )1
L
(E.6)
j
giving us
exp(+mhwo/kT)
now factors oul of the numerator of (E.9) and out of Z giving
"()_ 1r"W ~ X w - (21 + l)kT ~
tlW X (w)= kTZ 'If
L
590
. (E. 12)
mNTlIj
= (2[ + I)N-I
L l(mjl"zjlmj)I'
(E. 13)
mj,mj giving
2
j
x 'lhwo(mj - mj) - hw J
mj) -1iw]
L
exp(tlWoMlkT)
l(m\.m2, ... • mi' .. ·1p:zjlmJ, m2 .... • mi .... )1
2
We wish to re-express this in tenns of the states la) and [b). To do so we note that J(m"m2• ... mj. ···IJ~",jlm"m2, ... ,mj, ... )1 2
ml,m2 •... ,mj •...•mN,mj
x
I
L.J, l(mjllt",jlmj)1 6(llwo(mj -
)_Imj,fflj
fill ,11I2, ... ,fIlj". /I
'"
1/
(E.7)
_
lI"nw
X (w) - (2l + I)N kT
'" 2 L, l(al"zlb)1 ,(E" -
0,'
E, - tlW)
(E. 14)
591
But this is just the result we would have had had we replaced all exponentials in (E.I) by unity. We see that we have never asserted thac lEal
2:::: R3
(F. 17)
where Z is a small numerical factor representing the fact Ihat there is more than a single neighbor. The process of self-diffusion enables the neighbor atoms to move, so that one neighbor is replaced by another. In the process, the resonance frequency of the nucleus undcr observation may change depending upon whether a neighbor is replaced by a nucleus whose moment is oriented in the same direction or in the opposite direction from the magnetic moment it replaced. If T m is the mean lime a neighbor sits before jumping, we surmise T:::: T m
(F. 18)
At low temperatures, then, we expect a nuclear resonance line width order 6w/2 given by (F.17). As we increase the temperature, we expect T m to become shorter. When (F. 19)
we expect the nuclear resonance line width to begin to narrow. Exactly such effects are seen, as is discussed in Chapter 5. Chemical exchange can be studied by its effect on the nuclear resonance absorption lines. For example, the spin-spin coupling in a molecule gives rise to structure in liquids, as illustrated in Fig. 4. I I. In liquid CH3CH20H, the OH protOn splits the CH2 proton resonance structure if the OH proton remains on the molecule. But if there is rapid chemical exchange of the OH proton with other protons in the liquid, the structure may be washed oul [F.9].
G. Diffusion in an Inhomogeneous Magnetic Field
1
2
o Ow " I h"n' I' h resolution .'ig. F J The avcnging by molecular reoricn~ll~ion of Achcmlca S l In a ug" ( d NMR ~pec~rum" It is assumed lhlll mu~ual reorientAlion oecurs belw~n ~wo Sll~ ~ a~.... b) which arC equally populaled and which hAve resonancc frequencIes separat y radians in lhe absenec of thc exchangc
596
In Section 2.9 we explained how diffusion in an inhomogeneous static magnetic field causes a spin echo to decay as the time between pulses is increased. The cause of the decay is the possibility for a nucleus to change its precession frequency by diffusing to a different point in the sample at which the static field is different owing to the inhomogeneity in the magnetic field. Such a situation is closely related to that treated in Appendix F in which the nucleus had two
597
possible natural precession frequencies. For the case of diffusion in a magnetic field, there is a continuum of magnetic fields. For simplicity, we assume the magnetic field, though inhomogeneous, has axial symmetry so that
8H
H(x,y,z,)=Ho+z az
Mz. and My arising from precession. If we transform to the reference frame rocating at the precession frequency, we eliminate the precession and can write in this frame
8~::
(G.l)
In Appendix F we described the process whereby spins switch between sites of different precession frequency by the nile term (F.12a) and (F.l2b). When diffusion can take place, the possible precession frequencies form a continuum as expressed by (G. I). The usual way to describe diffusion is by means of a diffusion equation. The use of a diffusion equation in conjunction with the Bloch equations was introduced by Torrey [G. I ]. We shall follow a rreatment which is a slight simplification of his. Suppose we had a homogeneous static field in the z-direction, had no applied alternating field, and by some means had produced a nonuniform M z , as in Fig. G.Ia. Let us suppose T t is infinite. TIlen the total z-component of magnetization cannot change, but as a result of diffusion the region of magnetization will spread as in Fig. G.Ib, eventually leading to a uniform M z throughout the sample (Fig. G.lc). (Recall that the symbol M z denotes the magnetization density.) The process of Fig. G.I is described by the equation 8Mz = DV 2 M (0.2) at ' where D is the diffusion constant If the magnetic field were uniform, and there were initially also :t- and ycomponents to the magnetization density, there would also be diffusional effects for Mz. and My. But since the static field causes a precession, two effects are present: (1) changes in Mz. and My arising from diffusion, and (2) changes in
'" M DV2
(0.3,)
z
a;:y = DV2My or
(O.3b)
8M+
--=DV 2M+ at
(0.4)
using the definition M+ == Mz. + iMtI • . Since M+ describes a two-dimensional effect, (GA) treats vector effects. It IS analogous to (F. 12), if we bear in mind that M+ is a function of position. We now proceed as in Appendix F to recognize that when the static field is inhomogeneous, we must now include the effect of the spread in precession frequency. We therefore add the precession driving terms giving
aM+(~ y, %, t)
= _ i-yh(x, y, z)M+(x, y, z, t) _ M+(:t. y, %, t)
T, (0.5)
where
h(:t, y, z) = H(:::. y, z) - H o
(0.6)
and Ho is the spatial average field over the sample. Substituting (G.I) for h(:t, y, z~, w~ ~btain an equation describing free precession in a static magnetic field which IS mhomogeneous but possesses axial symmetry:
8M+
(8H) M+ - -M+ + DV 2M+ T2
(0.7)
- - '" -i-yz i)t 8z (.)
M,
M,
M,
o
L
(,)
(b)
o
L
o
L
Fig. C.lao/(t)u/e- /
wet)
(H. 19)
I
Since the coefficients a/ are independent of time if H p = 0, the foml (H.~9) is the interaction representation. Substitution of t/J(t) in SchrOdinger's equauon leads in a straightforward way [see (5.78,79)} to
da" dt
604
=.!. 2::a/(l'I1-£p l1)e-iW,t 1l /
i
.
datO
dt =
- Wto)t]
i
.
- 'h{a ll (lOI1-£ppl)exp[l(w lO -wlllt)
+ al _I (lOI'HpP - I)exp [i(wlo - WI dal_l
i
_I )t]
.
~ = -'halo(1 - tI'HpllO)exp[ -1(WI_I-wlOlt]
These equations are identical for all three cases provided one notes the relation (H.17) between the three coupling constants; a is equivalenl to 3b or to 6c
for lhe three cases. Table n.4
(H.2t)
(H.16)
where Fz = [z + Sz for cases (a) and (b). We have already shown that Fz has no matrix elements joining the F I and F 0 states of cases (I) and (2). Thus, the only lransitions induced are between Ill) and [10) and between [10) and'p -I). We can use the tables of energy levels 10 calculate these two freq~encles. See Table H.5. These frequencies correspond to a pair of lines separated In frequency
L\w=3b
.
The other aFAI's obey the equations
da
1-£1(t) = -"(llH zo F z cos wt
,
aoo = constant
ll dt = -'halO(l l[1-£pjlO)exp [l(Wll
producing a time-dependent perturbalion, 1-£1(t), given by
.6w=a
Since (l'J1-£pjl) vanishes between F = 1 and F = 0 stales, for cases (I) and (2) of coupled pairs of spins we get. using the notation aFAI
.
Therefore there are no stalic or dynamic observations one could make which would enable one to tell which of the Ihree systems one had. One may ask. what if one had chosen to use inSlead the Im/ms) representation? Clearly the physical result should not change. All that will change is thai one now has four instead of three "allowed" transitions (all' to ap or po'. and fJP to afJ or fJa). However. if one were to do the problem explicitly one would need to be mindful of the exact degeneracy of the afJ and fJa stales for case (I), which guarantees coherent effects. If one uses the product operator formalism all these things are automatically taken care of since we do not specify a representation when we use operator algebra. One could. of course, start with the representation lm/ms). formally express the states in terms of the states IFM), solve the problem using the IFNI) states. then transform back [0 jm/ms). An impor1ant practical consequence of the equivalence we have proved is that if an X(?r(2) ... T ••• Y(?r(2) pulse sequence refocuses the coupling aI~S~, it will also refocus the true dipolar coupling. As is shown by (10.59), the effect of a general quadrupole coupling to first order is always of lhe fonn of ?le. Thus, for a spin I nucleus, 'He describes the first·order quadrupole effects. By analogy to the echoes described above, we can refocus the first-order quadrupole coupling of a spin I nucleus using lhe same X(1fn.) .. . T . .• Y(1J'fl) pulse sequence.
I. Powder Patterns
(H.20)
In Chapter 4 we saw thai, in general. Knighl shifts are anisotropic. The same thing is true for g-shifts encountered in electron spin resonance and discussed in 605
Chapte r 11. These features give rise to a resonan ce frequency in (4.187), goes as
W
which, as shown
0.1) to the cryswhere (8, q,) specify the orientation of the static field Ho with respect giving rise ly, random present are (8,,p) tal axes. For powder sample s, all angles singularieristic charact quite r, howeve to a spread in resonant frequencies with, powder called are spectra Such . ties from which wo • wb. and We can be deduced We [1.1]' d Rowlan and ergen B/oemb pattern s. This problem was first treated by we so, doing In s. powder tum here to calcula ting the intensity pallems for such some or crystals liquid are omillin g discuss ion of fascinating systems such as crystals and form of oriented polyme rs which are concep tually in between single random powder s. that the We express the random orientation of the crystallites by saying by given is probabi lity dP that Ho lies in any infinitesimal solid angle dfJ (I.2)
dP "" dfJ "" sin 8 d8 d.#47r = d(- cos 8)d#41f
4,
Introdu cing the variable z defined as (1.3.)
z=-oo s8 we have that dP is
dP = d~dz
(l.3b)
4~
The advanta ge of the variable s (q" z) is that equal areas in the spond to equal probabilities. Inttodu cing fJ defined as fJ
==
W -
rlrz
plane corre--
(1.4)
Wo
we have
n "" W(I sin 2 8 cos2 ,p +wb sin2 (J sin 2 q, + We cos2 8 2 = W a + (wb - w a ) sin
,p + [(we
2 - w a ) - (Wb - w a ) sin q,]z2
(1.5)
frequency, fJ. This express ion can be thought of as detenni ning lines of constan t have we in the rlrz plane. Solving for z2 2 -wa - (Wb -wa )sin q, 2 ~~ z= 2 (We - wa) - (wb - W(I) sin 4J q, at constant SO that it is straight forward to calcula te the curves of z versus er, then, frequency in the ,p-z plane. We illustrate such curves with Fig. 1.1. Consid lity of probabi the (1.3), of result a As dfJ. + fJ nand two curves at frequencies area the just is range cy frequen this within cy frequen finding a nucleus whose the also is lity probabi the But curves. two L1A in the rlrz plane between these have we Thus J. 1(fJ)df nonnali zed NMR line intensity
n
606
06
0.'
0.2 l'lSlJ.
-2:,j L~.,-o~o'---~20I-'--,,,:o--,,~-,:;--f-;;:;;--:-::--:~. 160 le.O 80
90 100
120
'"
¢lldrgrr ul pJ F'ig.l.I. Curves of constan t frequene Y .In the..t ane for" general anillOtropic chemic,,' '1'": or Knight shift
l(fI)Ll fi = LlA
~
~
. This relation ship shows that if we divide the total frequenc y range Into equal be frequency interval th s. e area tween successive curves of constan t fre uenc . . P~portlofnalto (he in~ensity at that frequency range, giving us a simple ~Phrc~: picture 0 where the Intensity is high, where it is low. We case now describ e how to calcula te len). We first treat the simplest . I ' case. general the do then try, symme axla the same Axial symme try means that two of the principal frequencies are . . easIest ' IS (e.g,w(J =w,,~w orw =w t The ca Icu Iatlon et) if we pick Wb, c.. C (I c' . th 8 _ 0 aXIs to be the symme try axis (W(I = Wb) since then axial sYrnme try e .. . h gIving means that the q,..dependent teons of (1 .5) vams
n=
W(J
+ (we -
w(I)z2
0.8)
q,-z
plane are straight Jines at Therefo re, cu.rves of constan t frequency in the ~n~~ti: t~~~' 1.2). The area, L1A', between two curves differing in frequency
LlA' = 211"Llz
(1.9) vertical the whe~ the 21f ~mes from the total range of ¢J, and where Liz is spaclOg approp nate for the range Lin :
Llz =
(afla,) -
Ll
fl.
Such a strip occurs both for
(1.10)
O f en> wb) we have
I(y) = -
~' "" Ly sin 2 ~
I rbm.x
setting z = 0 (1.22):
If ¢mu = the integral turns out to be a complete elliplic integral of the first kind. GradshU!yn and Ryzhik [1.2] show thai if O; j=1
625
true
for the Hamiltonian of (8.158). Then, since
[H"HtI:O
.
Alas. this does nOI vanish. How can we construct a pulse sequence which vanishes? Haeberlen and Waugh point out that one can readily find a six-pulse sequence which vanishes. Introducing the notation
[H;.H,I " i. k
(K,21)
S = 6,5 + 6,4 + 6,3 + 6,2 +6, 1 +5,4+5,3+5,2+5,1 +4,3+4,2+4,1 +3,2+3,1 + 2,1 Now, previously we had 11. 1 + 11.2 + 'H 3 = 0
(K,22) (K.23)
Suppose we keep this condition, and add to it the condition
J>k
N
as was
(K.19)
we have
2([e. b] + [c. al + [b. al) : --.2 ([1t'C.'HB]rC TB
(K.18)
(K.20)
we get that the correction tenn of (K.6)
'H(I) =
For a three-pulse sequence this can be written
1 = - - , (['Hs, 1{",)TATS
'
(K.17)
we can write
Now. identifying a. b. and c as
Tj
[H;.H,I .
1-£1+1-£2+1-£3=0
: _ _i_({'H8, 'H",lT8TA + ['Hc, 1t'",]TCTA + (1{c, 1(8)TC1'"S)
i a= --",11. AT",
(K.16)
Now suppose
Jd', Jd'J!H(.,).H(tI)] 2tch
Jill) : - - ' -
"
L
S: [H,.H,I + [H,.HtI + [H,.HtI .
i'H(1lt c
[H;.H,)
j>k
We therefore get
-
L
2Th j >k
S:
0
= [1i8' 'HA)T8 TA
,
H(I): -~
so that the vanishing of 11.(1) arises from the vanishing of the sum S defined as
Jd', Jd', [H(.,).H('I)] : Jd', Jd'I[HB.HA]
I....
.
.
0
(K.Ilb)
'H4+'HS+11.6=O
.
(K.24)
Then, adding the columns of (K.22) venically, and insening items such as 4,4 in the second column, etc. we get that the second, third, and fifth columns add to zero, leaving solely contributions from the first and founh columns: 627
s =6, 5 + 3,2 =['Ii"
Selected Bibliography
'Ii,] + ['Ii" 'li2]
It is now obvious that if we pick
1i6 '" 11.2
1£5 '" 113
the commutators will be each other's negative. with the result that S will vanish. Thus. a cycle in which (K.23) is [flIe and in addition ~=1iA=~
~=1iB=~
~=~=1il
will have a vanishing 1-£(1).
There is an ex.tensive literature on methods of finding such cycles. See for example [K.4,5J.
The problem of preparin& .. complete bibliO&~.lIphy of magnetic re!lOlilInce is hopeless, there have been so ffiilny papers. Such .. bibliography would not even be usdul for 11 student, since he would not know whcre to begin. Therdore, a short list of articles has b~n selected which touches on II number of the most important ideas in resonl'lncc. In some instances, papers were chosen because they are basic references; others because they were representative of a dl\SS of papers. In some CMCll 1\11 aLtempt hIlS been made Lo augment the treatment of the text. An important technique for searn. Phys. Rev. 69,127 (1946) F. Bloch, W.W. Hansen, M. Paebro: The nuc:lu.r inductH>n experiment. PhYll. Rev. 70, 474-485 (1946) F. Bloch: Nuc:lear induction. Phys. Rev. 70, 460-474 (1946) N. Bloembergen, E.M. Purcell, R.V. Pound: Relaxation effects in Iluc:lear magnetic res0nance absorption. Phys. Rev. 73, 679-712 (1948) Boo~s,
Monographs, and Review Articles
G.E. Pake: -Nuelear Magnetic Resonance·, in Solid Slale PlIpics, Vol. 2, ed. by F. Seitz, D. Thmbull (Academic, New York 1956) pp. 1-91 Collected artic:les. Nuovo cimento Suppl., Vol. VI, Ser X, p 808 fr. (1957) E.R. Andrew: Nuclear MagMI~ ReSOMfll:e (Cambridge University Press, Cambridge 1955) J.A. Pople, W.G. Schneider, II.J. Bernstein: lligh·Resolulitm Nudear Magll£lic ResoNlfll:e (McGraw-llill, New York 1959) C.J. Gorter raratllagnel~RtI='ion(Elsevier, New York 1947) O.J .E. Ingram: Spectroscopy al Radio and Microwave Frequencies (Butterworth, London 1955) A.K. Saha, T.P. Oas: Theory and A-pplit:alions of Nudear Induclion (Saha Inslilule of Nuclear Phyllics, Calcutta 1957) M.II. Cohen. F. Reif: ~uadrtJpole ER'ccl.s in Nuclear Magnd.ic Resonance Sludies of Solids", in Solid Stole Physics, Vol. 5, ed. by F. Seitz, D. Turnbull (Academic, New York 1957) pp. 321-438 T .1'. Oas, E.L. lIahn: "Nuclear Quadrupole Resonance Spedroscopy· , in SoIidSkilt Physics. Supplement 1, ed. by F. Seitz, D. Thrnbull (Academic, New York 1958) T.J. Rowland: Nuclear magnetic resonance in metals. Prog. Maltr. Sci. 9,1-91 (1961)
628
828
II .12 Coupling
R.V. Pound: Prog.Nucl. Phys. 2, 21-50 (1952) . William Low: ~Paramagnclic Resonance in Solids", in Solid Start PhysICS, Supplement 2, ed. by F. !kit~, D. Turnbull (Academic, New York .1960). . . J.S. Griffith: TM Thea? ofTr(lllSilUm·Mefalloru (Cambridge Umvennly Press, Cambndge 1961) N.F. Ramsey: NucltoT MonunlS (Wiley, New York 1953) N.F. Ramsey: Moiteular Beams (Clarendon, Odord 1956) A. Abragam: The Principles ofNudear Magnefism (Clarendon, Ox£O«I 1961) N. Bloembergen: NuclearMag~f~Re/arJJlUm(W.A. ~njamin, New York 1961) J.D. RoberLs: Nuclear Magntl~ ReSDfl(JllCe (McGraw-lilli, ~ew .York 195~) R.T. Schumacher: IntroducliOfl 10 Magntlic RtSOfI(J)lU (Bel\lamm-Cummmgs, Menlo Park, CA 1970) I{ .A. McLauchll\ll: Magnetic Resonance (Clllrcndon, Oxford 1972) . . G.E. Pake, T.L. Estle: TM Physical PrillCiples of Electron Paramllgnttic ResontJflCe (BcnJammCummings, Menlo Park, CA 1973) .. A. Abragam, B. Sleane)': Elecfron Paramagntlk Resonat1Ce o/TrarulflQn Ions (Clarendon, Odord 1970) . Maurice Goldman: Spill Tentplral/Ut tlfId Nuelear Magntlk RtuJNUlCe in Solids (Clarendon, Oxford 1970) . ' C. Kittel: lnlroductiOfiIO Solid SIo4 PhyslU, 5th ed. (WIley, New Y~rk 1976) Chap. 16 C.P. Poole, Jr., II.A. Faraeh: TM TMory c{Magntlic RUOIVUtCe (WIley, New York 1972)
11.5. Gutowsl:y, D.W. McCall, C.P. Slichler: Nuclear magnelic fCIIOnance mulliplels in liquids. J. Chem. Phys. 21, 279-292 (1953) E.L. Hahn, D.E. Maxwell: Spin echo measUT(!menls of nuclear spin coupling in molecules. Phys. Rev. 88, 107G-108-I (1952) N.F. Ramsey, E.M. Purttll: InteractiOn5 beh.:ccn nuclear spins in molecules. Phys. Rev. 85,1>\3-1>\4 (1952) (Idter) N.F. Ramsey: Electron coupled illleracl;ons between nuclear spins in molecules. I'hy•. Rev. 91, 303-307 (1953) N. Bloembergell, T.J. Rowland: Nudearspin exchange in solids: 1'1103 and 1'1 105 lllll.gnetic resonance in thalliu.,n and ll~allic oxide. Phys. Rev. 97,1679-1698 (1955) M.A. Ruderman, C. I{,ttel: Indirect exchange coupling of nuclear magnetic momenLS by conduclion eleclrons. Ph)·s. Rev. 96, 99-102 (1954) K. Yosida: ~lagnetic properties of CIl-Mn alloys. Phys. Rev. 106, 893-898 (1957) H.M. McC~lllnell, A.D. McLean, C.A. Reill)': t\nalysis of spin-spin multiplet.s in nude"r magnetic reI:IOnance speclra. J. Chem. Phys. 23, 1152-1159 (1955) II.M. McConnell: Molecular orbilal approximll.lion lo e1eclron coupled inlerAction between nuclear spins. J. Chem. Phy•. 14, 46G-467 (1956) W.A. Anderson: Nuclear magnelic resonance speclra of some hydrocarbons. I'hy•. !lev. 102,151-167 (1956)
General Theory or Resonance
Pulse Methods
D. Pines, C.P. Slichler: RelaxaLion limtlJ in magnetic resonance. Phys. Rev. 100, 10141020 (1955) f1.C. Torrey: Bloch equations wilh diffusion terms. Ph)'s. Rev. 104, 563-565 (1956) R. Kubo, K. Tomila: A general theory of magnetic resonance absorption. J. Phy!. Soc. Jpn. 9,888-919 (1954) P.\V. Anderson. P.R. Weiss: Exchange narrowing in paramagnetic resonance. Rev. Mod. Phys. 2S, 209·276 (1953) . ' P. W. Anderson: A mathematical model for lhe narrowmg of spectral hnes by exchange or motion. J. Phys. Soc. Jpn. 9, 316-339 (195)\) .. n.K. Wangness, F. Bloch: The dynamical theory of nuclear mductlon. Phy•. Rev. 89, 72S-739 (1953) F. Bloch: Dynamical theory of nuclear induction II. Phy•. Rev. 102, 104-135 (1956) A.G. Redfield: On the theory of relaxation processes. IBM J. I, 19-31 (1957) II.C. Torrey: Nuclear spin relaxat;on by translational diffusion. Phys. Rev. 91, 962-969 (1953)
E.L. f1ahn: Spin echoCll. Phys. Rev. 80, 58G-59>\ (1950) II.Y. Carr, E.M. Purcell: Effects of diffusion on free precession in nude..r magnelic resonance experiments. Phys. Rev. 94, 63G-638 (1954)
Nuclear Magnetic Resonance in Metals C.II. Townes, C. Herring, W.O. Knight: The effect of electronic paramagnetism on nuclear magnetic resonance frequencies in metals. Phy•. Rev. 77, ~52-853 (1950.) (ldter). . W.O. Knighl: ~Eleclron Paramagnelism and Nude...r Magnetic Rt$Qnance ill Mdall ,In SolidS/alt Physics, Vol. 2, ed. by F. Seilz, D. Thrnbull (Academic, New York 1956) pp. 93-136 J. Korringa: Nudear magnelic relaxation and resonance line shifl in metals. Physical6, 601-610 (1950) D.F. Holcomb, R.E. Norberg: Nudear spin relaxation in alkali metals. Phys. Rev. 98, 1074-1091 (1955) . G. Benedek, T. Kushida: The pressure dependence of the Knighl shirt in the alkah metal. and copper. J. Phys. Chern. Solids S, 241 (1958)
630
Second Moment L.J.F. Broer: On the tl~oory of param~gnelic relaxation. Physiea 10, 801-816 (1943) J.II. Van Vleck: The dipolar broadenlllg of magnetic resonance lines in crystals. Phys. Rev. 74, ]]68-1183 (1948) G.E. Pake: Nuclear resonance absorption in hydrated cry.Lals: Fine struclure of lhe proton line. J. Chem. Ph)'s. J6, 327_336 (1948) 11.5. GUlowsky, G.E. Pake: Nudear magnetism in studielll of molecular struc:tUrt and rotatx)fl in solids: ammonium sal~. J. Chem. Ph)'s. J6, 1l64-1165 (l!H8) (letter) E.R. Andrew, R.G. Eades: A nuclear magnelic resonance investigation of lhree IIOlid benzel1t$. Proc. R. Soc. London A118, 537-552 (1953) H.S. G~towsky, G.E. Pake: Structural investigat;ons by means of nudear magnetism II - IImdered rotalion in solids. J. Chem. Phys. 18, 162-170 (1950)
Nuclear Polarization A.W. Overhauser: Polari;/;alion of nudei in metals. Phys. Rcv. 91, 411-415 (1953) T.R. Carver, C.P. Slichlcr: Experimental verification of lhe O"erhauser nudear po[ari;/;alion elfOX:l. Phys. Rev. 101,975-980 (1956) A. Abraga.m: Overh.au8l?r effecl in n.onmetals. Ph)'s. Rev. 98, 1729·1735 (1955) C.D. Jeffnes: PolarlzallOn of nuclei by resonance saturation in l:>aramagnetic crystals. Phys. Rev. 106, 164-165 (1957) (Ieller) J. U.e~feld, J.L. MOl.chane, E.Erb: Augmentation de II. polarisation nucleaire da", les hqmdes el gu adsorbes sur un charbon. Extension aux solidc8 conlenanl des impuretes paramagnetiques. J. Ph)'ll. Radium 19, 843-84'4 (1958) A. Abr~am, W.G. Proct.or: Une nouvelle melhode de polarisation dynamiquedes noyaux atomique dan. Its 8Olides. C.R. Acad. Sci 146, 2253-2256 (1958)
631
C.D. Jeffries: "Dynamic Nuelear Polarization- in Progrusin Cryo~nia (Heywood, London 1961) G.R. Khulsishvili: The Overhauser effed and related phenomena. Soviet Phys. - Usp. 3, 285-319 (1960) R.H. \Vebb: Steady-state nuclear polarizations via electronic transitions. Am. J. Phys. 19,428-444 (1961)
Quadrupole Effects R. V. Pound: Nuclear electric quadrupole inleractions in crystals. Phys. Rev. 79, 68:>-702 ( 1950) N. B1ocmber8en: "Nuclear Magnetic Resonance in Imperfect Crystals", in Report of the Bristol Conference on Defects in Crystalline Solids (Physical Society, London 1955) l>p. 1-32 T.J. Rowland: Nuclear magnetic resonance in copper alloys. Electron distribution around mlute atoms. Phys. Rev. 119, 900-912 (1960) W. Kohn, 5.11. Voslco: Theory or nuclear rCllOnance intensity in dilute alloys. Phy!l. Rev. 119,912-918 (1960) T.P. Das. M. Pomeralllz: Nuclear quadrupole interaction in pure metals. Phys. Rev. 113, 2070 (1961) T. Kushida, G. Benedek, N. Bloernbergen: Dependence of pure quadrupole resonance frequency on pressure and temperalure. Phys. Rev. 1G4, 1364 (1956)
Chemical Shifts W.G. Proctor, f.C. Yu: The dependence of a nuclear magnetic resonance frequency upon chemical compound. Phys. Rev. 77 , 717 (1950) W.C. Dickinson: Dependence of the piS nuclear resonance position on chemical compound. Rev. 77, 736 (1950) 11.5. Gutowsky, C.J. 1I0ffmann: Chemical shins in the mllgnetic I'ClIOnance of F1S. Phys. Rev. 80, 1I0-lll (1950) (letter) N.F. Ramsey: Magnetic: shielding of nuclei in molecules. Phys. Rev. 78, 699-703 (1950) N.F. Ramsey: Chernic:a1 effect.s in nuclear rnasnetic resonance and in diamasnetic susceptibility. Phy!l. Rev. 86, 243-246 (1952) A. Saika, C.P. Slichter: A note on the nuorine I'ClIOnance shifts. J. Chern. Phya. 22, 26-28 (1954) J.A. Pople: The theory of chernielll llhms in nuclear magnetic resonance I - Induced current denllities. Proc. R. Soc. London A139, 541-549 (1957) J .A. Pople: The thoory of chemicalshifb in nuclear magnetic resonance II -Interpretation of proton shiftll. Prot. R. Soc. London A1J9, 550-556 (1957) II.M. McConnell: Theory of nuclear magnetic shielding in molecules 1- Long range dipolar shielding or protons. J. Chern. Phys. 17, 226-229 (1957) R. ~man, G. MUrTay, R. Richards; Cobah nuclear resonance llpect.... Proe. R. Soc. London 141A, 455 (1957)
Spin Temperature N. Bloembergen: On the interaction of nuclear apins in a crystalline lattice. Phyaica 15, 386-426 (19-49) E.M. Purcell, R. V. Pound: A nuclear spin llyalem at nqative lemperatu~. Phy•. Rev. 81,279-280 (1951) (Iella-) A. Abragarn, W.G. Proctor: Experiment!l on spin temperature. Phys. Rev. 106, 160-161 (1957) (letter) A. Abragam, W.G. Proctor: Spin temperatUl"e. PhYll. Rev. 109, 1441-1458 (1958)
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