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(VO"OI)
=
J vO"oIJF(G)dG ,
(5.1.1 )
tiE
where G is the energy of the incident electron, and AE is the threshold excitation energy. On substituting the Maxwellian distribution function (1.2.1) we obtain VO"
_Kex> ~exp(-G/T)8d8 na~ (Ry)I/2T3/2 '
L
( 01) -
(5.1.2)
217 x 10- 8cm3-1 - . K -- 2.Jili.ao s . m It is convenient to express the energy and temperature (Sect. 3.1.3). On setting u
= (8 -
In
P= DE/T.
AE)/DE,
the scaled units
(5.1.3)
Eq. (5.1.2) can be written as DE) 1/2 ex> ( ) (VO"OI) = K ( -R p3/2e - pp J 0" u2 (u y 0 nao
+ p)e -
pudu
(5.1.4)
with p = AE/DE. The rate of the inverse process (deexcitation) can be expressed through the excitation rate by (1.2.7). As was shown in Sects. 2.3 and 3.1,2 the cross section for the transition ao - al can be written, in the general case, in the form O"aOal
= E [Q~(ao,adO"~(lo, ld + Q~(ao,adO"~(lo, IdJ ,
(5.1.5)
I(
where the quantities O"~ and O"~ depend only on the quantum numbers nolo and nlll of the optical electron. The dependence on the total angular momenta of an atom (J,S,L, ... ) is described by the factors Q~ and Q~. For the cross section summed with respect to J,S,L, ... ) in the case of configurations no/~-nol~-Inl/h we have
O"(no/~,no/~-Inl/t>
= m E[O"~(lo, Id + O"~(lo, II)] .
(5.1.6)
I(
Thus the quantities O"~ and O"~ correspond to the single-electron transition cross sections. The quantity O"~ includes the direct and interference terms ,.1
VI(
=,.
VI(
+ ,.int
VI('
Q'(ao,al ) ex 0SoSI ~ .' I(
(5.1.7)
110
5. Some Problems of Excitation Kinetics
the quantity a~ representing purely the exchange term. The expressions of a~ and a~ in terms of radial integrals are given in Sect. 2.3. These quantities, evidently, can be obtained only by means of numerical calculations. The index K varies in the interval of Kmin to K max , where Kmin
= 1/0 -
Id,
Kmax
= 10
+ II
(5.1.8)
.
In accordance with (2.3.9, 10), a~ -=1= 0 only for K with the same parity as Kmin. We shall use the analytical fitting formulas to represent the results of the numerical calculations. In computer codes one can explore formulas with many adjusted parameters providing very accurate fitting to numerically calculated or measured cross sections. However, good accuracy is based on mutual compensations of essential contributions from nearby terms of opposite signs. Therefore, a small change of cross section results in a much larger change of parameters. In particular, the extrapolation of parameters along an isoelectronic sequence becomes very difficult if possible at all. We use comparatively simple formulas with true asymptotic and 2 or 3 adjusted parameters. The accuracy of these formulas is usually a few percents, but an interpolation and extrapolation of parameters for similar transitions, in particular in isoelectronic sequence, is very simple. Generally we use (5.1.5) and the fitting formulas for a~, a~. For cross sections, summed over J, the factors Q' and Q" differ only in the spin parts: Q~ = QK(Lo,LI)Ao, Ao
= b(So, SI ),
Q~
A2
= QK(Lo,Ld A 2 ,
=
(2S1 + l)j2(2Sp
(5.1.9)
+ 1).
In this case the fitting formula may be applied to the total one-electron cross section (5.1.10) This provides often the better accuracy of fitting since for a direct transition a~ is energy function smoother than a~, which may be negative at small and medium energies. It should be noted, however, that for transitions between fine-structure components the J o - JI orbital parts of Q' and Q" are different and the form of (5.1.10) is not valid. Taking into account properties described in Sects. 3.1-2 the one-electron excitation cross sections are written in the form 2 a~(lo, II) = ~
2/0
+I
(R) --L DE
2 (
EI ) Eo
3/2
. C(]I, (u) u + q> u=---
a"(1 1)= K
0, I
1ta~ 2/0
+1
(Ry)2 (EI)3/2. DE Eo
DE
(5.1.11)
C(]I"(u)
u + q>
where Eo,E I are atomic level energies (from the ionization limit). For a~ the function (]II = (]I' is used. C, q> are the fitting parameters, DE is a scaling factor.
5.1 Rate Coefficients for Elementary Processes in a Plasma
111
The functions depend on the type of transition, and generally can include a third fitting parameter D. If one prefers to use a two-parameter formula D = 0 should be assumed. For the excitation cross sections, we have '(u)
= [1 -
D/(u + Ii] '11(U)
"(u)
= [u +
0.4 +Dr2 '11(U)
(5.1.12)
where '11(U)
= {(U+ p)I/2, Z= 1
(neutrals) ,z>l (ions)
1
p = ,1E/DE
(5.1.13)
The excitation rate coefficient can be written in a similar way:
"
,
(5.1.14)
10- (RY 10- 8 (RY 8
EI)3/2 AG'(f3) . - - e - f3p [cm 3 S-I] , f3 + X
(vu (10 Id) = - - _. 2/0 + 1 DE Eo '" _ (vu"(1 0 I») I - --2/0 + 1 '" f3
EI )3/2 AG"(f3) n [ 3 -I] (5.1.15) e - I'P cm s DE Eo f3 + X '
--. - .
= DE/T.
Here A and X are adjusted parameters, the functions G(f3) can include the third adjusted parameter
= f31/2(f3 + 1 + D)'11 (p-I) , G"(f3) = f31/2(f3 +D)'1I(P-I). G'(f3)
(5.1.16)
For the total one-electron rate coefficient (vu~) =Ao(vu~) +A2(VU~)
(5.1.17)
the function Gt = G' is used. Eqs. (5.1.11-16) can be employed in any of the scales DE given by (3.1.30a, b, c). But in the case of closely spaced levels only DE = lEo I or z2 Ry are appropriate. The functions and G are symmetric with respect to the initial and final states. Therefore, in the case of deexcitation collisions 1 ~ 0, we have
(vu(aJ, ao») =
(vu'(1 I») " I, 0
L [Q~(al' ao)(vu~(ll' 10») + Q~(al' ao)(vu~(lJ, 10»)]
" = ~ (RY . EI)3/2. AG'(f3) 2/1 + 1 DE Eo P+ X
_ (vu""(1 I, I») 0 -
10- 8
--2/0 1
+
(RY
[ 3 -I] cm s ,
EI) 3/2 A G"(P) [cm3 s -I] . DE Eo f3 + X
--. - .
, (5.1.18)
(5.1.19)
112
5. Some Problems of Excitation Kinetics
5.1.2 Ionization Cross sections and rate coefficient of ionization can be described in the way similar to that outlined in Sect. 5.1.1. In this book we do not consider exchange effects in the case of ionization. The ionization cross section summed with respect to the quantum numbers J),L),SI is usually of interest. In this case the angular factor Qj does not depend on K. For the transition (5.1.20) We have (5.1.21) Thus we can write fitting formula for the total one-electron ionization cross section summed over K and II : CTj(lO)=
1ta~ (Ry)2CtP(U) 210 + 1 DE U + cp
10- (RD~ ) 8
(VCTj(lo») = 210 + 1
3/2
.
8-Ez DE
(5.1.22)
U=--
A G(P)
P+ X e - Pp [cm3 S-I]
,
(5.1.23)
P=DE
T'
where Ez is the ionization energy of the level ao, the scale DE is Ez or zZRy. The functions
1
0.0 0.000 0.200
0.2 0.Q15 0.200
0.4 0.034 0.200
0.6 0.057 0.200
0.8 0.084 0.200
y, y,
y'u z=1 z > 1
2.0 0.328 0.328
3.0 0.561 0.561
4.0 0.775 0.775
5.0 0.922 0.922
6.0 1.040 1.040
1.0 0.124 0.200
5.1 Rate Coefficients for Elementary Processes in a Plasma
117
On using (5.1.48) for the cross section the excitation rate coefficient can be written in the fonn
(VO"aoa,) = 10- 8 x 32jaoa,
When
(~~r/2 pl/2exp(-p). p(PHcm3 s-I].(5.1.49)
P~l,
p(P)
~ - ~ Ei(-P)·
The values of the factor p(P) are given in Table 5.2. In various applications also the fonnu1as by Drawin [5.4], Mewe [5.5], and Gryzinsky [5.6, 7] are often used. In fact, all these fonnulas practically do not differ. The analytic fonnulas for the excitation cross sections can be obtained in the Born approximation using the model interaction potential VK(R) ()( RK / (R~+R2)K+l/2 [5.8]. For transitions between the levels with small energy spacing (with the same principal quantum number) such a potential is often fairly close to the real potential. In case of optically allowed transitions (AI = 1, K = 1) (5.1.50) where Zp is the projectile charge, M is the reduced mass of the colliding pair, iaoa, is the oscillator strength for transition ao -+ al (5.1.51) Kn(x) are the modified Bessel functions, Xmin = max
Ro ao
fM( V"RY Is
Y-;;
=f
~) VRY
,
(5.1.52)
(5.1.53)
Table 5.2. Factor p(fJ) for atoms (z fJ
z=1
p, p,
z > 1
p, p,
z=1 z > 1
= 1)
and ions (z > 1)
O.oI 1.160 1.160
0.02 0.956 0.977
0.04 0.758 0.788
0.1 0.493 0.554
0.2 0.331 0.403
0.100 0.214
2 0.063 0.201
4 0.040 0.200
10 0.023 0.200
>10 0.066p-'/2 0.200
fJ
0.4 0.209 0.290
118
5. Some Problems of Excitation Kinetics
In the case of quadrupole transitions 4
Uaoal = 135
(I(;
= 2)
22M Ry , -;cQ2(ao,a l )
7ta oZ p -;;;
x (21
I) (10 2/1)2 (l0Ir21/1)2 1+ 000 R~a~ x [4>2(Xmin) -4>2(Xmax )] ,
(5.1.54) (5.1.55)
00
(l0Ir21/1)
= J Pno/o(r)Pnl/l(r)r2 dr
(5.1.56)
o
iff~AE
When both Xmin
~~
Ro 2 ao
and Xmax > I, 4>1(Xmax ) and 4>2(Xmax ) can be neglected, and
(M) m
1/2 AE Ry
(RY) 1/2 iff
(5.1.57)
This approximation corresponds to the first-order impact parameter approximation with rectilinear trajectories. In this case the functions 4>1 (Xmin) and 4>2 (Xmax ) can be approximated by 4>1(X)
~
exp (-2x) . In (2.25 + 0.681/x) ,
4>2(X) ~ exp (-2x) . (2 + xJ3n/2)2 .
(5.1.58) (5.1.59)
The rate coefficients, averaged over the Maxwellian distribution, can be written in the form 16n l / 2 e 2 2 Q~(aoad (VU,,) = 21(; + 1 h ao [(21(; - 1)!!]2 x(21 I
+
I) (/ 0 I(; 11)2 (/olr"I/I)2 000 R2,,-2a2
o 0 xexp (-AE/T) . (Ry/T)1/2I,,(AE/T, Yo) ,
(5.1.60)
00
I,,(x, y) = exp (x/2)
J dt t2"-IKr(t) exp
(5.1.61)
o
Yo
= Ro (AE) 1/2 aO
RY
For the dipole transitions (AI
=
I) (vu) can be written in the following form
< vu >= 1.7410- 7 faoal (Ry/AE)(Ry/T)I/2 (5.1.62)
5.1 Rate Coefficients for Elementary Processes in a Plasma
119
When the argument of Ko does not exceed 3.0 the integral II (x, y) is approximated by the following asymptotic formula (5.1.63) For transitions with small energy spacing in plasmas usually rff ~ AE. Therefore, the Coulomb attraction of the electrons is not important. However, in cases of neutral atoms and ions with low z the effect of normalization can be very substantial. In such cases the formulas given above can be used only for rff~Eo, EI, i.e., when the Born approximation is valid. For multiply-charged ions (z > 3) and rff > AE they give fairly reasonable values of cross sections for the dipole transitions, and allow to estimate the order of magnitude of the cross section for quadrupole transitions. b) Ionization The well-known classical Thomson formula for the cross section of ionization from the shell nolo corresponds to (5.1.22 and 24) when C = 4m(2/0 + 1), ({J = I, DE = Ez and D = 0: aj
2
(Ry)2
= 1tao • 4m Ez
(u
+u 1)2
(5.1.64)
.
To estimate the rate coefficient of ionization for atoms and ions from the ground state, the Seaton formula [5.9] is often used:
(vai)
=
R )3/2 ( 10- 8 x 4.3m; p-1/2exp(-{3) [cm3 S-I], (5.1.65)
{3
= Ez/T, {3
~ 1,
where E z is the ionization potential of an ion X z • This formula corresponds to (5.1.23,24) when z > 1, X = 0 and A = 4.3· (2/0 + 1). Expression (5.1.65) is valid only for {3 2: 1. Sufficiently universal semiempirical formula was suggested by Lotz [5.10]:
(vai) = 10- 8 • 6m
1~~13/2 {3-1/2exp(-{3)f({3)
[cm3 s-I], (5.1.66)
f({3)
=
-{3exp({3)Ei(-{3).
This formula corresponds approximately to (5.1.23,24) when z > I,X = 0.4, and A = 6(210 + 1). The values of the factor f({3) are given in Table 5.3. Table 5.3. Factor f(fJ) fJ f(fJ)
= 1/4 = 0.34
1
4
8
0.59
0.83
0.90
120
5. Some Problems of Excitation Kinetics
The compact semiempirical fonnula which is also often used is given in [5.4]. Classical fonnulas for the ionization cross sections are given in [5.7, II]. c) Dielectronic Recombination A detailed treatment of dielectronic recombination is given in Sect. 5.2. To exhaust the list of analytic fonnulas for bound-bound and free-bound electronic collisional processes we present here a semiempirical fonnula for the rate coefficient of dielectronic recombination proposed by Burgess [5.12]. This fonnula can be written in a fonn similar to (5.1.28) "d(a)
= 1O- 13 BdP3/ 2exp(-PXd)
[cm 3 S-I],
P= (z + 1)2Ry/T ,
(5.1.67)
with Bd = 480jcxocx
Xd
5.2
(
z2
:~3.4 )
1/2
[I
+ O.105(z + l)X + O.015(z + 1ilrl ,
3 )-1 ,
(
= X I + O.Ol\z ~ 1)2
X = (z + 1)2Ry .
(5.1.68)
Dielectronic Recombination
In this section and the following one we discuss some problems related to dielectronic recombination I and fonnation of dielectronic satellites. We have used cgs units here. We recall also that for ions which are members of the isoelectronic sequence of an atom A the designation [A] is used. For example, the designation [H] is used for a set He+, Li 2 + and so on. 5.2.1
Electron Capture and Under threshold Resonances (Simplified Model)
As noted above, the excitation cross section for positive ions has a nonzero value at threshold due to the long-range Coulomb attraction. This attraction also allows the excitation of an ion Xz+ I at an energy below threshold, the electron being captured on some level nl ofthe ion Xz • For example, at an energy lower than the excitation threshold for the resonance level of the He-like ion 06+, the following process is possible: 06+(li) + e
--t
05+(1s2p nl) .
The doubly excited state which is the result of electron capture is unstable, and may decay either through autoionization or spontaneous emission of the resonance 1 Dielectronic recomb~tion is widely discussed in the literature, see e.g. the review articles [5.13-15]. In [5.15] one can find an excellent historical review and numerous references to original articles.
5.2 Dielectronic Recombination
121
photon 2p - Is. In the latter case, the atom X z is transferred to the stationary state, i.e., recombination occurs. This process is called dielectronic recombination (abbreviated to DR below). Generally, the process of dielectronic recombination of an ion Xz+! via the intermediate doubly excited state of an ion Xz is written in the form (5.2.1)
y = anlLSJ,
y' = a'niL'S'J' .
Below, the LS-coupling scheme is adopted. Besides it is assumed that photon emission occurs due to transition of the "inner" electron a - a', and the state nl of the "outer" electron does not change. Radiative transitions of the "outer" electron cause the complementary satellites (Sect. 5.3), but they do not play an essential role in the total balance of dielectronic recombination. The three most important effects of a DR process are: A. Dielectronic recombination for all ions other than bare nuclei is an additional recombination process. In many actual cases, as shown by Burgess and Seaton [5.16] the rate of DR can considerably exceed the rate of radiative recombination. Therefore, in low density plasmas dielectronic recombination should be necessarily taken into account. B. Satellites to the resonance and other lines of an ion Xz+1 originating from radiative transitions in reaction (5.2.1). C. Complementary excitation of levels a' when the autoionization occurs in reaction (5.2.1) with a' =I- ao. The latter has been considered in detail in Sect. 3.4, the satellites will be discussed in Sect. 5.3. Here we shall consider the intrinsic dielectronic recombination. For simplicity we discuss here only the process (5.2.1) neglecting the secondary ionization of the excited ion Xiy'). Both collisional and radiative secondary ionization in a plasma are considered in [5.17]. In some cases the secondary autoionization of ion Xz(y') is also possible. (For a discussion of the secondary processes see also [5.15].) In Sect. 5.2.1 we confine ourselves to the description of the simplified model making the following assumptions: a) the state of ion Xz is described by quantum numbers ani without specifying the terms LS; b) the value of n is large enough, so that the influence of electron nl on the state a of the core can be neglected; the levels nl can be considered as hydrogenic, and capture cross section can be expressed in terms of excitation cross section for transition ao - a using the correspondence principle; and c) in the process of photon emission, the electron returns to the initial state r:t.o. In this case the formulas for calculation of dielectronic recombination cross
5. Some Problems of Excitation Kinetics
122
sections prove to be sufficiently simple. Discussion of these assumptions will be given in Sect. 5.2.3. The general case will be discussed in Sect. 5.2.2. Within the frame of our simplified model, DR process is written in the form /' X;(aonl)
XZ+I (lXo) + e ---- X;*(a nl) '\.
+ hw (5.2.2)
Xz+1(IXO) + e , the lower branch of the reaction (autoionization) being the competing process. Therefore the cross section for dielectronic recombination via the state IXnl is A(a, IXQ) (I) , Wa IXn
I
ad(anl)
= ad(lXQ, anI) A ( IX) +
(5.2.3 )
where A( a, IXQ) is the probability of a radiative transition IX - lXo in an ion Xz+ I, Wa is the autoionization probability for the level anI of an atom X z , A(IX) = LiXJ A(IX, IXd is the total probability of radiative decay of the levela,a~(IXQ,anl) is the cross section for electron capture to the level nl when the transition IXQ - IX is excited. This cross section is represented by a set of resonances at the energies .to (0
~ LIAE _ z2 R2Y < LIAE,
AE
n
= Ea.a.o = Ea. - Ea.o .
(5.2.4)
It is convenient in this section to use again eGS units. The resonance width equals F = hWa • The cross section averaged over the resonances can be obtained
with the aid of the correspondence principle by extrapolating below the threshold the partial cross section for the excitation ao - IX: 2z2Ry n
a~(IXQ, IXnl)F = a(lXQ, 1X1)--3- .
(5.2.5)
Here a(lXQ, IX/) = EAo a(IXQAo, 1X1), where a(IXQAo, 1X1) is the partial cross section for the transition IXQ - IX in the threshold C = AE; Ao, 1 are the orbital momenta of the outer electron. In accordance with condition (b) of the model we should sum the cross section over total angular momenta LTST. The corresponding formulas are given in Sects. 2.3 and 3.2, in which the sum over A. is to be replaced by one definite value of A. = I. The values of Wa and a~(or a) are related to each other as characteristics of direct and reverse processes. To derive this relation it should be noted that at A = 0, the ratio of the populations of Xz+I(IXQ) and Xz(lXnl) is given by the Saha formula. Using this formula, we obtain (21
+ 1)ga. W,(a IXn 1)
= z2C .I:.
1tnn
goa(lXo,1X1)
3
2
1tao
'
(5.2.6)
where ga. and go are the statistical weights of the states a and ao. The rate coefficient of dielectronic recombination is Kd
= E Kd(a), a.
Kd
(a)
= E vad(aO, IX nl)F:#,( C) , ~
(5.2.7)
5.2 Dielectronic Recombination
123
where r is the resonance width, and ff(C) is the Maxwellian distribution for the energies of the electrons. The value of C is given by (5.2.4). Substituting (5.2.3) and (5.2.5) in (5.2.7) we obtain (5.2.8) where ns and
nl
are determined by the relations
( ns)3 n
= Wa(rxnl, rxo) .
(5.2.9)
A(rx, rxo)
One can see that nl is in fact the minimum value of n, i.e., it determines the lowest level at which the capture of an electron is possible in accordance with (5.2.4 ). We now transfer (5.2.8) into a form more convenient for applications. We substitute in (5.2.8) the explicit expression for ff(C) and use the relation (5.2.6), and the relation between A and the oscillator strength /:
g!J.A(rx, rxo)
=
1 (RY) T (.1E)2 Ry go/!J.o!J. .
(5.2.10)
1373
We write the result in the form
Kd(rx)
=
(z
+ 1)2Ry
=
{3
1O-13Bd(rx){33/2exp(-{3x) ,X =
T
[cm3 s-I], (5.2.11)
.1E (z + 1)2Ry ,
where {33/2exp (-{3X) provides the main temperature dependence of Kd, and Bd(rx) only slightly depends on temperature and is equal to
, z 21 + 1 Bd(rx)=C ·4·/"0!J. I: exp(J{3)I: 1+( / )3 nl n>nl l ns for all possible values of n. We consider now as an illustrative example the recombination of a [Li] ion with excitation of 2s - 2 p transition and recombination of [H] ion with excitation of Is - 2 p transition. In the first case, the energy level distance tJE is small, so that X = tJE/(z + I)Ry '" l/z~l, and u(ao, IXI) is large. That means the factor exp( -PX) ~ 1 in (5.2.11), ns ~n) ~ 1, and a great number oflevels concentrated in very narrow energy band ('" tJE) contribute to "d. Because of the small value of A '" X2 j, the value of led is comparatively small in spite of a great number of levels. In the case of recombination of [H] ion or [He] ion X ~ 3/4, and due to the factor exp (-PX), the rate of DR at small temperatures is negligible. The value of ns isn't large and for z ~ 20, ns is even smaller than n) ~ 2. For this reason, a comparatively small number of levels contribute to Bd in (5.2.12), but the contribution of each one is great because of the high value of A. According to numerical calculations the total values of Bd for the Is - 2 P transition usually exceed those for the 2s - 2 p transition. The value of led, however, is greatly dependent on temperature in the case of the Is - 2 p transition.
5.2.2
General Case
The formulas (5.2.11, 12) obtained above provide a useful method of calculation of the DR rate coefficient within the simplified model. In this section we derive a general expression for the DR rate coefficient without the assumptions of the simplified model. We shall consider the process (5.2.1) again using the detailed balance principle (Sect. 1.2) to derive the general formula for recombination rate coefficient. The total DR rate is (5.2.14)
5.2 Dielectronic Recombination
125
where R(y) is the probability of electron capture into the state y of Xz; and A(y) = E y' A(y, y') and Wa(Y) = E y ' Wa(y, y') are the probabilities of radiative and autoionization decays. Since the latter decay is associated with internal electrostatic interaction, it cannot change the total momenta LSJ. If we suppose that A = 0, and hence the system is in thermodynamical equilibrium, then according to detailed balance principle, we can write
NeNz+lR(y) = Nz(y)Wa(y,tXo) ,
Wa(y,tXo) =
E Wa(y, Yo) , ~
where the ratio Nz+dNz is determined by Saba equation (Sect. 1.2). The value of R, of course, is independent of any assumption concerning the radiative decay associated with an electromagnetic interaction. Therefore we can use the last equation to determine the value of R in the general case, and substituting it in (5.2.14), we obtain
I 8n3/ 2 a3 Kd(Y) = 2go . (z + 1~ p3/2exp (-PX
+ bP)
x gyA(y) Wa(y, tXo) ; bP = AE - EylXo , T A(y) + Wa(Y)
(5.2.15)
the values of P and X being determined by (5.2.11); go and gy are the statistical weights of the states tXo(Xz+1) and y(Xz ), and AE = EalXo and EylXO are the excitation energies of the states IX and y. The difference of these energies bE is equal to the bound energy of the captured electron, (5.2.16) The radiative decay probability A(y) summed over all final states y' does not depend on LSJ and is denoted below by A(IX). The values of P and bP are in fact also independent of LSJ. The autoionization probability in the LS coupling scheme does not depend on J, but essentially depends on LS. Therefore we shall write the rate coefficient of DR in the form (5.2.11), the factor Bd being equal to Bd(lX) = C
E exp(bP)q(y),
y = IXnILS,
niLS
C = 1013
4 3/2 3 n ao 3[c m 3 -1] s. go (z+ 1)
(5.2.17)
Due to the nonlinear dependence of Bd on Wa(y) we cannot explicitly sum over LS in (5.2.17). For this reason, the use of this equation requires a great deal of computation. In most applications an approximate formula is used, in which an averaged value of Wa(y),
Wa(lXnl) =
it
(2L + 1)(2S + 1) 2(21 + 1)( 2L a + 1)(2Sa + 1) Wa(Y) ,
(5.2.18)
126
5. Some Problems of Excitation Kinetics
is substituted in the denominator of (5.2.17). After this substitution we can write (5.2.17) in the form Bd(a.) = CEexp(bP)q(rxn/) , nl
(5.2.19)
(a.nl) = 2(21 + l)gIXA(a.) Wa(a.nl,lXo) . A(a.) + Wa(a.n/) q
Calculations with these formulas are much simpler than (5.2.17). Summing of linear expressions of the type (5.2.18) can be accomplished analytically, and only the sum over nl has to be done numerically. Besides, the expression for Wa(rxn/) is much simpler than that for Wa(Y) (Sect. 5.2.3). Approximation (5.2.19) corresponds to the assumption (a) of the simplified model. If n ~ 1 [assumption (b)], we can use the relation (5.2.6) and substitute the threshold excitation cross section in place of Wa in (5.2.19). Thus we obtain Bd(a.)
, z
(21+1)B'
= C . 4" . I IXfJIX E exp (bP) E B B'( I nl
n>nl
l
!3
1>0'> fI.I
:g
1>0'>
S·
fI.I
a.
e
-
trI
0
fI.I
= ....
1>0'>
g.
E.
.g
~
'"C
VI
5. Some Problems of Excitation Kinetics
148
5.4.4 Hydrogen6k.e Ions Both in the coronal limit and at high density, the temperature at which the ions Xz exist is proportional to the ionization potential Ez. For hydrogenlike ions, Ez = z 2Ry. When T <X z2, the quantities (va) <X z-3. The spontaneous radiative transition probabilities Ann' <X z4. Therefore the reduced density and temperature fie
= Ne/z7, f = T/z2
are convenient when considering the ions. At given fie and f, the quantities ro, rio rx./z, and z 3S do not depend on z. The Saba-Boltzman equation (5.4.11) may be rewritten in the form
NE(n) = where Q density
~ n28~/2 :~ (z2:y) 3/2 exp ( ; )
,
(5.4.22)
= Ne/N(z+I). From (5.4.18), it follows that the reduced population
fI(n) = QN(z)(n)/zll
(5.4.23)
will not depend on z. Table 5.8 gives some values of ro, rio rx./z, and z 3S from data of the work by McWhirter and Hearn [5.38). The population densities are obtained by substituting NE(n) from (5.4.22) into (5.4.18) with N(z+l) instead of N(H+). The more recent quantitative results can be found in [5.40,43). The difference between two sets of coefficients ro, rl given by Tables 5.7 and 5.8 is great at low temperature T ~z2Ry. It can be explained by different threshold behaviour of the cross sections for neutral atom and for ion (the excitation cross Table s.s. Parameters ro(n),rl(n) and coefficients S and thin plasma [5.38]. 4.7--6 denotes 4.7 x 10-6 1/e
0
108
1010
1012
ro(2)
4.7--6
5.9-6 1.2-8
8.8--6 1.2--6
3.6-5 1.2-4
1.6-4 1.2-3
1.3-3
1.6-3 5.5-9
2.5-3 5.5-7
1.5-2 5.5-5
1.0-2
1.4-2 5.1-9
2.2-2 5.0-7
2.9-2
3.9-2 4.3-9
7.9-2
1013
ro(3)
rl(3) ro(4)
rl(4) ro(5)
rl(5) ro(7)
rl(7)
for hydrogenlike ions in the optically lOIS
1016
1018
00
103z 2K. 1.4-3 1.2-2
1.3-2 1.1-1
6.1-2 5.1-1
1.1-1 8.9-1
1.1-1 8.9-1
9.6-2 5.7-4
3.2-1 6.4-3
4.3-1 6.0-2
4.7-1 2.9-1
5.0-1 5.1-1
5.0-1 5.1-1
2.1-1 4.2-5
5.2-1 2.9-4
6.9-1 2.9-3
7.5-1 2.7-2
7.7-1 1.3-1
7.8-1 2.2-1
7.8-1 2.2-1
7.0-2 4.1-7
5.5-1 2.3-5
7.8-1 1.3-4
8.7-1 1.3-3
8.9-1 1.2-2
9.0-1 5.8-2
9.0-1 1.0-1 I
9.0-1 1.0-1
1.1-1 3.4-9
3.2-1 2.7-7
8.8-1 6.3--6
9.5-1 3.0-5
9.7-1 3.0-4
9.8-1 2.8-3
9.8-1 1.4-2
9.8-1 2.4-2
9.8-1 2.4-2
T
rl(2)
IX
=4 x
1014
S ·z3
9.1-26
1.1-25
1.9-25
1.0-24
6.1-24
4.6-23
3.7-22
2.0-21
3.5-21
3.8-21
IXlz
7.9-13
9.2-13
1.4-12
5.2-12
2.0-11
1.1-10
8.6-10
8.0-9
7.6-7
7.6-251/e
5.4 Populations of Excited Levels in a Plasma
149
Table 5.8. (continued) fIe
0
108
1010
1012
ro(2) n(2)
1.1-3
1.3-3 8.9-9
1.5-3 8.9-7
ro(3) rl(3)
1.9-2
2.2-2 4.2-9
2.7-2 4.1-7
3.1-3 8.8-5 6.9-2 4.0-5
ro(4) n(4)
5.9-2
6.7-2 3.5-9
7.5-2 3.4-7
3.3-1 2.6-5
ro(5) rl(5)
1.0-1
1.2-1 3.2-9
6.5-1 1.3-5
ro(7) rl(7)
1.8-1
2.2-1 2.5-9
1.6-1 3.0-7 4.0-1 2.0-7
S ·z3
4.9-17
5.3-17
(lIz
4.8-13
ro(2) n(2)
1013
1014
10 15
1016
1018
2.1-1 6.4-2 7.7-1 1.8-2
6.0-1 1.9-1
7.6-1 2.4-1
8.9-1 5.1-2
9.4-1 6.5-2
7.6-1 2.4-1 9.4-1 6.5-2
9.3-1 5.5-3
9.7-1 1.6-2
9.8-1 2.0-2
9.8-1 2.0-2
00
T=8xI03~K.
7.4-3 8.8-4 2.5-1 3.6-4 6.9-1 1.4-4
3.4-2 8.5-3
9.5-1 3.1-4 9.9-1 5.9-5
9.8-1 2.0-3
9.9-1 5.7-3
9.9-1 7.3-3
9.9-1 7.3-3
9.2-1 3.3--6
8.8-1 5.4-5 9.8-1 9.9--6
1.0 3.8-4
1.0 1.1-3
1.0 1.4-3
1.0 1.4-3
6.8-17
1.7-16
4.1-16
2.2-15
1.6-14
4.5-14
4.5-14
5.1-13
6.1-13
1.2-12
2.5-12
7.6-12
3.7-11
3.8-14 1.7-10
1.0-8
1.0-26,,_
2.2-2
2.4-2 6.8-9
8.6-1 5.7-2
9.4-1 6.3-2
9.4-1 6.3-2
9.3-2
1.0-1 3.2-9
9.7-1 5.8-3
9.8-1 9.9-3
9.9-1 1.1-2
9.9-1 1.1-2
ro(4) n(4)
1.7-1
1.8-1 2.6-9
2.0-1 2.5-7
9.8-1 1.5-3
9.9-1 2.5-3
1.0 2.8-3
1.0 2.8-3
ro(5) rl(5)
2.3-1
2.5-1 2.3-9
2.9-1 2.2-7
T = 1.6 x 104~K. 5.7-2 1.5-1 3.6-2 6.7-5 6.6-4 6.0-3 3.8-1 1.8-1 7.4-1 3.0-5 2.4-4 1.3-3 4.3-1 9.3-1 7.6-1 1.8-5 8.6-5 3.6-4 7.1-1 9.1-1 9.8-1 3.2-5 9.3--6 1.2-4
5.2-1 3.2-2
ro(3) rl(3)
2.6-2 6.8-7 1.1-1 3.2-7
9.9-1 4.9-4
1.0 8.4-4
1.0 9.1-4
1.0 9.1-4
ro(7) rl(7)
3.1-1
3.5-1 1.8-9
4.8-1 1.5-7
9.3-1 2.1--6
9.8-1 5.7--6
1.0 2.1-5
1.0 8.5-5
1.0 1.5-4
1.0 1.6-4
1.0 1.6-4
S
1.3-12
1.4-12
1.5-12
2.4-12
4.2-12
1.2-11
4.3-11
7.2-11
7.8-11
7.8-11
(lIz
2.9-13
3.0-13
3.2-13
4.3-13
6.3-13
3.0-12
7.9-12
3.1-10
3.1-28".
ro(2) rl(2)
1.2-1
1.3-1 5.4-9
6.8-1 2.0-2
9.3-1 3.0-2
9.7-1 3.1-2
9.7-1 3.1-2
ro(3) n(3)
2.5-1
2.6-1 2.6-9
1.3-1 5.4-7 2.7-1 2.5-7
1.2-12 4 = 3.2 x 10 z2K. 1.9-1 3.1-1 4.5-3 5.2-4 5.1-1 8.1-1 1.8-4 8.8-4
9.5-1 2.9-3
9.9-1 4.2-3
1.0 4.4-3
1.0 4.4-3
ro(4) n(4) ro(5) rl(5)
3.3-1
3.5-1 1.9-9 4.1-1 1.7-9
ro(7) rl(7)
4.6-1
5.0-1 1.3-9
S ·z3
2.4-10 1.7-13
2.4-10 1.8-13
'Z3
(lIz
3.9-1
T 1.5-1 5.4-5 3.3-1 2.4-5
6.1-1 2.6-3 8.7-1 8.4-4
3.7-1 1.9-7 4.4-1 1.6-7 5.9-1 1.1-7
5.4-1 1.4-5 7.5-1 7.6--6
8.1-1 6.4-5 9.3-1 2.3-5
9.5-1 2.3-4 9.8-1 7.4-5
9.9-1 6.8-4 1.0 2.2-4
1.0 9.8-4 1.0 3.1-4
1.0 1.0-3 1.0 3.3-4
1.0 1.0-3 1.0 3.3-4
9.4-1 2.0--6
9.9-1 4.1--6
1.0 1.2-5
1.0 3.6-5
1.0 5.1-5
1.0 5.4-5
1.0 5.4-5
2.5-10
3.2-10
4.4-10
8.5-10
2.8-9
2.9-9
2.9-9
1.8-13
1.8-13
2.0-13
2.4-13
2.0-9 3.3-13
5.0-13
2.9-11
2.9-29".
5. Some Problems of Excitation Kinetics
150
Table 5.8. (continued) 1015
1016
10 18
00
5.2-1 3.6--3
7.8-1 1.5-2
9.5-1 2.3-2
9.8-1 2.4-2
9.8-1 2.4-2
6.5-1 1.5--4
8.6--1 7.1--4
9.7-1 2.1-3
9.9-1 2.9-3
1.0 3.1-3
1.0 3.1-3
6.7-1 1.2-5
8.6--1 5.5-5
9.6--1 1.8--4
9.9-1 4.8--4
1.0 6.7--4
1.0 7.0--4
1.0 7.0--4
6.2-1 1.1-7
8.1-1 5.9-6
9.4-1 2.0-5
9.9-1 5.8-5
1.0 1.5--4
1.0 2.1--4
1.0 2.2--4
1.0 2.2--4
6.6--1 9.2-10
7.1-1 7.9-8
9.6--1 1.3-6
9.9-1 3.3-6
1.0 9.7-6
1.0 2.4-5
1.0 3.4-5
1.0 3.5-5
1.0 3.5-5
3.4-9
3.5-9
4.0-9
4.8-9
7.3-9
1.4-8
1.8-8
1.8-8
1.8-8
1.0-13
1.1-13
1.2-13
1.6--13
2.1-13
5.6--12
5.6--30'1e
= 2.56 x
105z2 K.
'1e
0
108
1010
10 12
= 6.4 x
104z2 K.
ro(2) rl(2)
3.1-1
3.5-1 4.5-9
3.6--1 4.5-7
3.8-1 4.4-5
4.2-1 4.3--4
ro(3) rl(3)
4.6--1
4.8-1 2.1-9
4.9-1 2.1-7
5.3-1 1.9-5
ro(4) rl(4)
5.3-1
5.5-1 1.5-9
5.7-1 1.5-7
ro(5) rl(5)
5.7-1
6.0-1 1.2-9
ro(7) rl(7)
6.2-1
s. Z3
3.4-9
1013 T
rx/z
1.0-13
1.0-13
1.0-13
T
10 14
ro(2) n(2)
1.1
1.1 3.5-9
1.1 3.5-7
1.1 3.5-5
1.1 3.4--4
1.1 2.9-8
1.0 1.4-2
9.8-1 2.4-2
9.7-1 2.6--2
9.7-1 2.6--2
ro(3) rl(3) ro(4) rl(4)
1.0
1.0 1.6--9 1.0 1.1-9
1.0 1.6--7 1.0 1.1-7
1.0 1.5-5 1.0 9.2-6
1.0 1.3--4 1.0 5.2-5
1.0 6.7--4 1.0 1.8--4
1.0 2.0-3 1.0 4.5--4
1.0 3.0-3 1.0 6.8--4
1.0 3.2-3 1.0 7.2--4
1.0 3.2-3 1.0 7.2--4
ro(5) rl(5)
9.9-1
1.0 6.0-10
1.0 5.8-8
1.0 3.9-6
1.0 1.8-5
1.0 5.4-5
1.0 1.4--4
1.0 2.1--4
1.0 2.2--4
1.0 2.2--4
ro(7) rl(7)
9.7-1
1.0 4.6--10
1.0 4.1-8
1.0 9.0-7
1.0 3.0-6
1.0 8.5-6
1.0 2.2-5
1.0 3.2-5
1.0 3.2-5
1.0 3.2-5
1.0
section for a neutral atom at the threshold is equal to zero, but, for ions, the threshold value is not zero: see Sect. 3.2). In the range T rv z2Ry, the values of ro, rl from the Tables 5.7 and 5.8 are found to be in reasonable agreement. 5.4.5 Population Densities of Highly Excited Levels at High Density; Steady-Flow Regime The highly excited bound levels with n ~ no are populated and evacuated exclusively by collisions. After an electron has been transferred to a level through three-body recombination it may be either reionized through electron impact, or transferred to another bound level through inelastic or super-elastic (quenching) collision. If n > no, the collisional transition frequency between the bound levels is n2 times larger than the frequency of reionization. The most probable are
5.4 Populations of Excited Levels in a Plasma
151
collisional transitions between adjacent levels, followed by transfer of a small energy amount. The electron wandering between the highly excited levels can be treated as diffusion in the space of quantum numbers which can be described by Focker-Planck equation [5.32-34,44]. Neglecting the radiative processes the electron flux j(n) [cm- 3 s- l ] in the space of quantum numbers can be determined by 00
j(n) =
L:
L:[N(n + k' + l)Wn+k'+I,n-k - N(n - k)Wn-k,n+k,+d, (5.4.24)
k'~Ok~O
the first term in the square brackets representing the electron flux via n directed to the ground state, the second term representing the flux directed to the continuum. Using the ratio N(n)/NE(n) = b(n) which shows the departure of the level population from the Saba-Boltzmann population density, (5.4.24) may be rewritten in the form
j(n) = N(Z+I)-S2 L:[(n + k' k,k'
+ 1iexp (En+k,+dT)b(n + k' + l)Wn+k'+I,n-k
-(n - kiexp (En-k/T) b(n - k)Wn-k,n+k,+d ,
(5.4.25)
where S = z383/2/4n3/2aijNe, 8 = T/z 2Ry. Assuming b(n + k) = b(n) + kob/on, retaining the terms of the order and neglecting the weak functions of n, one can obtain from (5.4.25)
j(n) =
N(Z+I)Ne~exp
(ETn)
~b n2 L: k 2 (vun,n+k)
un
Ikl~1
.
rv
k 2,
(5.4.26)
In n ~ 1 the sum with respect to k in (5.4.26) may be extended to infinity. Let the basic dependence of (VUn,n+k) on temperature and the numbers n and k, which determine the rate coefficient order of magnitude, be written in explicit form (Sect. 3.5) 2
2
4
2 e nao -1/2n (VUn,n+k ) -_ .jn'h--;'38 k 3 cp(n, k, 8) ,
(5.4.27)
where cp(n, k, 8) is a weak function of its arguments. It is convenient to introduce the continuous variable e = 1/n2. Using (5.4.27) we can write the formula for the flux as
'() __ N(Z+I)N232n2e2 58-2-3/2 (~)LOb ] e e z6 Ii, ao e exp 8 oe'
(5428) . .
The factor L = L:k k-1qJ(n = e- I/2 , k,8) is a weak function of e and 8. (In reference [5.20] it is estimated to be ~ 0.2). Using the variable e the diffusion equation can be written
2e3/2 d~~) = q(e) - N(e)fB .
(5.4.29)
Here q(e) [cm- 3 s-I] is the rate for the direct population of the level e, and
152
5. Some Problems of Excitation Kinetics
f e[s -I]
is the total frequency of electron transfer to the continuum and to the lower levels n' with n' < no. The collisional transfer to the highly excited levels is already taken into account in the expressions (5.4.26,28) for the electron flux j(e) and should be excluded from fe. In order to obtain b(e) and j(e), the equation (5.4.29) should be supplemented with appropriate boundary conditions. The boundary condition at I'. = 0 is found easily. In the limit of I'. ~ 0 (n ~ 00), the function b( e) should correspond to the continuum distribution. Wlien Maxwellian velocity distribution is valid, limb(e)
e..... O
=
I .
(5.4.30)
The second boundary condition in general must be chosen by fitting the solution of (5.4.29) to the solution of the set of rate equations (5.4.10) for lower levels where the discrete structure and radiative transitions are of importance. In some cases it is possible to obtain the explicit form of this boundary condition. Somewhat later we shall consider low-temperature recombining plasmas when the condition of total absorption at some value 1'.1 can be treated as the second boundary condition. Without any particular pumping or evacuation of the highly excited levels, q( e) is equal to the rate of three-body recombination to the levell'., and f B is the probability of the inverse process, ionization by electron impact. Assuming the departure of b(e) from the Saha-Boltzmann equilibrium value (equal to unity) to be small, the right-hand part of (5.4.29) may be put equal to zero, because q(e) ~ N(e)fe due to detailed balance. Thus we obtain the constant-flow approximation j(e)
= const. = j
Then solving the equation (5.4.28) together with the boundary condition (5.4.30) yields the relationship b(e)
=
z 6Le 2 e I - j N(Z+llNic[(e')3/2 exp
(
-
1'.') e
de'
(5.4.31)
C = 32n2 aije2 jh .
If
e~e
. z 6Le2
b(e) ~ 1- ] N(z+I>N2Ce
5/2
(5.4.32)
e
The flux value should be determined using the second boundary condition. Now we shall consider low-temperature recombining plasmas. The flux j( e) is positive definite, and b( e) decreases with e. As the second boundary condition for equation (5.4.29), one can use the condition of total absorption of the flux at some value 1'.1, b(eJ)
=0
.
(5.4.33)
5.4 Populations of Excited Levels in a Plasma
153
If we deal only with low temperature e ~ 81, the value of j determined from (5.4.31) and (5.4.33) does not depend on 8t, and the integration with respect to 8' may be extended to infinity. Thus j = N(z+I)Ne a. ,
(5.4.34)
where
N. 27
3/2
2
a. = ~ _7t_ ~a~e-9/2 z6 3 Ii. =
N ez3
4V27t3/ 2 e 10 3 ml/2T9/2L
(5.4.35)
is the recombination coefficient. The order of magnitude and the dependence of the coefficient a. on temperature appear to be in agreement with the results of numerical calculations by Johnson and Hinnov, and Bates et al. given in Tables 5.7 and 5.8.
6 Tables and Formulas for the Estimation of Effective Cross Sections
In this chapter, tables of cross sections and rate coefficients of excitation and
ionization by electron impact and rate coefficients of dielectronic recombination are given. The cross sections and rate coefficients are presented in the form of products of angular and radial factors, the latter being expressed in analytical form containing two or three adjusted parameters. The tabulated parameters are obtained from the results of numerical calculations. The first section contains a description of the contents of the tables and relevant fitting formulas. In the second and third sectins of the chapter the formulas for angular factors are given which are necessary for applying the tables.
6.1 6.1.1
Tables of Numerical Results Methods of Calculations and Survey of the Tables
In this section the results of numerical calculations for the cross sections a, collision rate coefficients (va), and dielectronic recombination rate coefficient /Cd are given. The calculations are made using the Born method (see Sects. 2.3 and 3.1) and its modifications which are described in Sect. 3.2. The atomic wave functions are assumed to be constructed from single electron wave functions in accordance with a specific scheme of angular momenta coupling. A singleconfiguration approximation is used. The radial wave functions for all levels of a given electronic configuration are assumed to be the same. Under these assumptions the cross sections may be expressed in the form (2.3.4,8). In this chapter, however, we use the formulas somewhat different from (2.3.8). For the excitation cross sections of the transitions ao ---+ al we write in the general case a(ao,
ad = a'(ao, ad + a"(ao, ad = I: Q~(ao,ada~(lo, Id K
+ I: Q~(ao,ada~(lo, Id
(6.1.1 )
K
where a' consists of the direct plus interference terms, and a" is the purely exchange contribution to the cross section (Sects. 2.3,3.1,2). A summary of formulas for the Q-factors is given in Sect. 6.2. In order to simplify the use of the tables given below the subsequent subsections comprise the specific formulas for analytical approximation and Q-factors which can be used for specific tables. The exchange cross sections are given either summed over /C or for those cases in which index /C has a single value. Therefore, in most cases the exchange
I. I. Sobel'man et al., Excitation of Atoms and Broadening of Spectral Lines © Springer-Verlag Berlin Heidelberg 1995
6.1 Tables of Numerical Results
155
cross sections can be determined only for transitions between the atomic terms as a whole, so (6.1.2) where index K is not included, and a"(lo, 1\) = ~Ka~(lo, 1\). The calculated quantities a~, a~ have been approximated by means of simple analytic formulas which contain the two or three fitting parameters: C, cp, D for a and A, X, D for (va). The fitting parameters have been found from the results of numerical calculations by the method of least squares. The errors of analytic approximation R are also given in the tables. We pass on to a brief description of the tables and of the approximate methods which have been used for calculations. The Born approximation with normalization has been used for calculation of the cross sections for hydrogen atom summed over 10 and 1\ (Table 6.1). Tables 6.2 and 6.3 contain the fitting parameters for cross sections and rate coefficients calculated in the Born approximation with the Bates-Damgaard approximation for the atomic wave functions. The Bates-Damgaard approximation used for calculations of Tables 6.2 and 6.3 is most valid in cases when the maxima of both radial functions lie outside the atomic core. This condition is usually formulated explicitly as
no = no
Vz2Ryi lEo I >
ni
> nc ,
10 + 1/2,
nr = Vz 2Ry/lEd > 1\ + 1/2,
(6.1.3) (6.1.4)
> nc ,
where nc is the largest of the principal quantum numbers of the electrons of the atomic core. The condition (6.1.4) is sometimes stricter than (6.1.3). However, in many cases when conditions (6.1.3) are fulfilled, but (6.1.4) are not fulfilled, the error does not exceed the factor of 2. Such errors are inherent in the Born method itself. The excitation cross sections for specific atoms and ions (Tables 6.1,4-10), have been normalized with the use of the K matrix. For calculation of radial BOO
t 600 N
= A J ~«(3 + 3) In( 16 + 1/(3) . +X
(6.1.10)
= C (_u_) 1/2 _ _ ,
(6.1.11 )
¥- 1 :
cPK(u)
u+l GK «(3)
= A J~«(3 + 3) +X
u+q> .
(6.1.12)
The transition under consideration is characterized by the assignment of the effective principal quantum numbers of the lower level no and of the upper level nj no
= VZ2RY/IEol, nj = Vz2Ry/IE I I
The quantities Eo and EI are the ionization energies corresponding to a specified state of the atomic core; z is the spectroscopic symbol of an ion. For a neutral atom, z = 1, for a singly charged ion, Z = 2, and so on. The parameters C, q> and A, X for transitions s -+ S, S -+ p, S -+ d, p -+ s, p -+ p, p -+ d, d -+ s, d -+ P are given in Tables 6.2 and 6.3 as functions of no and LIn = nj - no' The spacing with respect to no and LIn adopted in the tables ensures the possibility of linear interpolation almost everywhere. The tables give the order and the mantissa of the number; for example, 24 - 1, 47 - 0, 59+0, 42+ 1, and 12+2 denote respectively 0.024, 0.47, 0.59, 4.2, and 12. The range of approximation for cross sections is 1 :S u :S 128. The cases where the errors of approximation exceed 10 percent are indicated in Table 6.2 by asterisks. These errors, however, do not exceed a factor of 2. The rate coefficients are approximated in the range 1/32 :S (3 :S 4. The asterisks in Tables 6.3 indicate the cases in which errors of approximation are greater than 25 percent. 6.1.4 Normalized Cross Sections for Specific Atoms and Ions (Tables 6.4--8) (i) LIS = 0 For transitions with no change of spin (LIS = 0) the cross sections (J and rate coefficients (v(J) are fitted by
(J' = naij [EI] 3/2 Q~(aoad CcP'(u) z4 Eo 2/0 + 1 u + q> ,
(6.1.13)
204
6. Tables and Formulas for the Estimation of Effective Cross Sections u = (8 - AE)/z'l-Ry ,
(vn) =
1O-8~ [EI] 3/2 Q~(aoad z3 Eo 210 + 1
P=
Ry/T,
z2
AE/Z2
p=
• AG'(P)ex (_ P) P+ X P P ,
cm3 s- 1
(6.1.14)
Ry.
Here, z is the charge of the parent ion (spectroscopic symbol of ion, Z = 1 for a neutral atom, Z = 2 for a singly charged ion, and so on), and Q~(aoad is the angular factor. In fact, in all tables only the states with s core electrons (besides closed shells) are considered. In this case
Lp
Gfo~o p p
= 0,
= 1,
Q'
=m
for excitation from the shells of equivalent electrons 1'0, and Q' = 1 for one electron outside closed electron shells. Since Q' are independent of K we give the values, summed over K, of n and (vn). The formulas for tP'(u) and G'(P) are given by (5.1.12, 13 and 16). (ii) AS = 1 In the case of intercombination transitions n" = naij
z4
[EI]3/2 Q~(aoat>CtP"(u) Eo
210 + 1
(6.1.15)
u + cp ,
u = (8 - AE)/z2 Ry,
(vn") =
1O-8~ [EI]3 /2Q~(aoad.AG"(P) exp(-pp) [cm3 S-I], z3
P=
z2 Ry/T,
Eo
210
+I
P+ X
(6.1.16)
p = AE/z2 Ry.
In the tables for intercombination transitions, as in those for AS = 0 only the states with s core electrons (besides closed shells) are considered. In this case
L
p = 0,
"LnSo" lTL;Sp
= 1,
Q
= mA 2 = m
2S1 + 1 2(2Sp + 1)
(6. l.l 7)
for excitation from the shells of m equivalent electrons. Since Q" are independent of K we give summed over K values of n" and (vn"). The formulas for tP"(u) and G"(P) are given by (5.1.12, 13 and 16). The energy dependence of exchange cross sections varies from one transition to another, and the errors of fitting are rather large. For this reason only the rate coefficients are tabulated in most cases. The set of parameters C, cp,D is adjusted for the range 0.02 < u < 16, and the set A,X,D, for 0.25 < P < 16.
6.1 Tables of Numerical Results
205
Table 6.4. Normalized Born and Coulomb-Born excitation cross sections. Transitions with no change of spin (LIS = 0) Atom
HI
He I
C 3.46 70.32 2.33 38.11 1.81 2.06 31.58 2.07 0.03
0.67 4.27 0.68 3.64 1.13 0.69 3.49 1.29 0.61
D 0.00 0.00 0.00 0.00 0.30 0.00 0.00 0.20 0.40
R 0.04 0.03 0.01 0.05 0.01 0.01 0.05 0.Q1 0.01
A 7.52 24.12 5.72 18.59 2.03 5.24 16.77 2.40 0.06
X 12.13 0.34 2.94 0.28 0.68 2.53 0.32 0.65 0.87
D 9.90 0.50 1.40 0.00 0.20 1.00 0.00 0.10 0.00
R 0.03 0.02 0.02 0.02 0.02 0.02 0.Q1 0.02 0.02
2s-3s 2s-3p 2s-3d 2s-4s 2s-4p 2s-4d 2s-4f 2p-3s 2p-3p 2p-3d 2p-4s 2p-4p 2p-4d 2p-4f
18.27 139.93 57.52 8.34 59.82 16.73 8.94 10.51 60.96 1014.92 4.89 27.75 313.54 38.55
0.26 0.60 0.18 0.22 0.60 0.12 0.13 0.02 0.41 1.59 0.Q1 0.24 1.05 0.35
0.00 0.90 0.00 0.00 0.90 0.00 0.00 0.90 0.00 0.20 0.90 0.00 0.00 0.00
0.02 0.53 0.02 0.02 0.58 0.01 0.03 0.57 0.06 0.06 0.44 0.02 0.11 0.08
20.15 15.20 89.18 13.57 7.72 46.84 21.24 8.73 42.94 147.18 5.80 42.45 122.06 41.11
1.64 0.53 1.14 0.74 0.68 10.17 4.10 0.26 1.75 0.67 0.54 0.84 0.80 1.74
2.10 9.90 0.60 0.00 9.90 6.70 2.60 0.00 3.90 6.70 0.00 0.20 2.70 2.30
0.02 0.16 0.02 0.02 0.12 0.02 0.02 0.08 0.02 0.02 0.17 0.02 0.02 0.02
3s-4s 3s-4p 3s-4d 3s-4f 3p-4s 3p-4p 3p-4d 3p-4f 3d-4s 3d-4p 3d-4d 3d-4f 11S-2 1S 11S-ip 11S-3 1S 11S_3 1P 11S-3 1D 11S-4 1S 11S-4 1p 11S-41D 11S-41F
55.53 0.16 359.31 0.26 114.84 0.11 56.65 0.03 61.83 -0.01 197.65 0.27 1644.09 0.51 394.47 0.13 0.05 6.43 52.52 0.02 251.01 0.17 5485.46 0.73 1.26 1.85 8.22 26.71 1.43 1.32 21.53 7.78 2.90 0.36 1.34 1.32 21.77 8.99 0.44 2.91 0.00 1.48
0.00 0.90 0.00 0.00 1.00 0.00 0.60 0.00 0.10 0.70 0.50 0.70 0.00 0.20 0.00 0.20 0.20 0.00 0.10 0.20 0.30
0.07 0.25 0.12 0.01 0.44 0.07 0.16 0.10 0.05 0.25 0.04 0.14 0.Q1 0.02 0.01 0.03 0.02 0.01 0.02 0.02 0.02
40.45 44.43 111.48 121.39 25.01 88.72 228.43 332.30 10.84 49.05 84.22 502.57 4.13 8.09 3.43 7.55 0.33 3.25 7.35 0.42 0.00
3.07 0.57 4.54 0.99 0.63 2.46 0.74 4.44 1.29 0.50 1.83 0.38 1.61 0.25 1.23 0.25 0.52 1.25 0.27 0.54 0.97
7.70 9.90 8.70 0.00 2.90 9.90 9.90 9.90 0.70 0.30 9.90 9.90 0.40 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.20
0.02 0.22 0.02 0.02 0.02 0.02 0.13 0.02 0.Q1 0.02 0.02 0.31 0.02 0.02 0.02 0.01 0.02 0.Q1 0.01 0.02 0.02
1.00 0.00
0.15 0.02
72.85 17.95
1.32 1.32
9.90 1.30
0.07 0.02
Transition Is-2s Is-2p Is-3s ls-3p Is-3d ls--4s Is-4p Is-4d Is-4f
2 1S-21p is-3 1S
287.51 14.84
qJ
0.00 0.28
206
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.4. (continued) Atom Transition He I
Li I
LiII
D
R
0.46 0.79 0.79 0.53 12.25 2.77
7.20 0.10 0.00 7.60 9.90 1.50
0.02 0.02 0.00 D.03 0.02 0.02
5.55 42.80 147.66 3.59 42.42 121.92 40.64
0.73 1.75 0.67 0.67 0.84 0.81 1.74
9.60 3.90 6.60 3.90 0.20 2.70 2.30
0.04 0.02 0.02 0.03 0.02 0.02 0.02
0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.07 53.34 0.03 15.61 0.65 8.05 0.01 48.95 0.01 10.51 0.53 5.23 0.01 34.76 0.04 5.97
1.13 0.94 0.22 0.71 0.85 0.16 0.93 1.51
8.60 0.50 0.00 0.00 0.00 0.10 0.00 0.40
0.03 0.02 0.13 0.01 0.01 0.09 0.02 0.02
0.34 0.43 1.81 0.34 0.24 1.21
0.90 0.00 0.10 0.90 0.00 0.00
0.31 0.04 0.05 0.42 0.03 0.11
9.16 41.32 137.32 3.45 40.94 112.13 30.57
0.80 1.60 0.66 0.93 0.72 0.77 1.70
9.90 3.30 5.70 7.80 0.00 2.70 2.30
0.07 0.02 0.02 0.06 0.01 0.02 0.02
242.74 9.92 6.02 24.23 129.92 57.91 917.36 1261.32 75.59 2994.65
1.14 0.26 0.12 0.37 1.30 0.42 1.79 0.13 0.06 0.10
0.10 0.20 0.10 0.00 0.00 0.00 0.10 0.90 0.00 0.20
0.06 58.31 0.04 14.43 0.06 20.23 0.02 36.32 0.30 26.90 0.04 46.49 0.05 148.53 0.15 232.48 0.04 115.65 0.231495.79
0.86 0.66 7.03 0.70 0.86 1.22 0.63 0.93 2.17 2.48
5.20 0.00 3.40 0.00 6.00 2.10 5.40 9.90 2.10 9.10
0.02 0.01 0.02 0.01 0.03 0.02 0.02 0.07 0.02 0.02
1.87 43.32 1.37 31.48 0.79
0.78 7.77 0.80 7.63 3.46
0.00 0.20 0.00 0.20 0.40
0.06 0.02 0.08 0.02 0.02
1.30 0.21 1.41 0.22 0.51
0.00 0.20 0.00 0.10 0.80
0.01 0.02 0.01 0.02 0.02
({J
D
R
62.58 45.04 7.12 34.69 18.89 5.27
3.01 0.23 0.25 2.21 0.18 0.17
0.20 0.00 0.00 0.50 0.00 0.00
0.11 5.35 0.01 70.67 0.01 12.06 0.09 3.39 0.01 43.96 0.03 12.09
ip-3 l S 21P-3 1P 21p-3 1D 21p-4 IS 2lp-4lp iP-4 ID iP-4 IF
44.98 60.81 1010.04 16.10 27.69 313.07 38.15
0.97 0.41 1.59 1.73 0.24 1.05 0.36
0.60 0.00 0.20 0.00 0.00 0.00 0.00
0.21 0.06 0.06 0.09 0.02 0.11 0.08
23S_23P 23S-3 3S 23S_3 3P 23S-3 3D 23S-43S 23S-43P 23 S-4 3 D 23S-43F
255.67 11.64 11.93 31.92 5.91 10.22 18.13 2.72
0.83 0.31 0.60 0.30 0.28 1.02 0.27 0.23
23P_3 3 S 23 p_3 3 p 23p_3 3 D 23 P-4 3 S 23 P-4 3 P 23 P-4 3 D 23 P-4 3 F
60.27 55.97 865.03 16.22 25.80 309.41
2 1S-3 1P is-3 ID 2 IS-4 IS 2 1S-4lp 2 IS-4ID is-4 IF
2s-2p 2s-3s 2s-3p 2s-3d 2p-3s 2p-3p 2p-3d 3s-3p ·3s-3d 3p-3d lis-is I Is-ip 11S-3 1S 11S-3 1P 11S-3 1D
C
A
5.62 11.31 4.64 9.38 0.39
X
207
6.1 Tables of Numerical Results Table 6.4. (continued) Atom Transition
C
cp
D
R
A
X
D
R
2 l s-ip is-3 IS 2 1S-3 1P is-3 ID ip-3 l S 21P-3 1P 21p-3 1D
309.65 14.84 93.49 50.10 32.36 59.47 892.85
0.03 0.19 1.66 0.32 2.31 0.21 1.12
0.70 0.00 0.60 0.00 0.00 0.00 0.00
0.15 211.50 0.02 27.82 8.41 0.15 0.03 61.23 0.07 4.48 0.01 93.50 0.16292.64
1.33 1.02 0.53 1.18 0.54 1.21 0.62
3.10 0.20 9.10 1.10 5.20 0.70 2.60
0.02 0.02 0.02 0.02 0.02 0.02 0.02
23S-23p 23S-33S 23S-3 3p 23S-3 3D 23P_3 3S 23p_3 3p 23p-3 3D
249.92 13.55 57.24 41.93 43.37 54.68 780.14
0.03 0.23 1.87 0.41 2.15 0.22 1.21
0.80 0.00 0.70 0.00 0.10 0.00 0.00
0.18 164.06 0.01 23.92 0.15 4.15 0.02 48.10 0.17 5.70 0.01 87.89 0.16268.62
1.11 1.12 0.45 1.12 0.60 1.09 0.51
2.60 0.40 9.90 1.10 6.20 0.50 1.90
0.02 0.02 0.02 0.02 0.03 0.02 0.02
Be I
2S-2P
151.61
2.11
0.00
222.45 12.24 29.67 35.05 70.98 54.88 790.48 1517.48 68.55
0.02 0.26 4.26 0.44 1.72 0.21 1.20 0.03 0.03
0.90 0.00 0.00 0.10 0.00 0.00 0.00 0.80 0.10
0.92 1.11 0.30 1.08 0.65 1.10 0.51 1.51 1.11
0.90 2.20 0.40 3.00 1.10 4.80 0.50 1.90 6.60 0.30
0.02
2s-2p 2s-3s 2s-3p 2s-3d 2p--3s 2p--3p 2p--3d 3s-3p 3s-3d
58.21 141.37 21.28 3.48 38.34 13.17 89.02 273.34 615.05 129.97
0.40
Be II
0.09 0.24 0.01 0.03 0.02 0.10 0.01 0.16 0.15 0.02
0.02 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.02
CI
2p2 3p-2p3s 3p 18.47 2p21D-2p3s Ip 20.37 2p2 IS-2p3s IP 30.78 2 1s-ip 209.19
2.92 2.36 2.09
0.50 0.50 0.20
0.08 0.10 0.08
4.14 4.97 9.70
0.41 0.43 0.46
2.00 2.00 1.50
0.03 0.03 0.02
0.04
0.70
0.25 193.41
0.53
0.40
0.02
223.10 303.97 27.94 10.46 19.82 1392.49 254.78 70.07 85.49
0.03 1.37 0.34 0.33 0.11 1.44 0.92 0.38 0.45
0.70 0.10 0.00 0.00 0.60 0.30 0.00 0.00 0.00
0.23 0.06 0.01 0.04 0.09 0.05 0.11 0.10 0.07
71.46 1286.57
0.05 0.25
0.00 0.80
209.07 67.57 42.88 14.91 31.17 159.32 59.15 53.67 55.35 676.14 0.03 109.74 0.14215.54
0.54 0.76 0.72 0.65 0.74 0.68 0.97 1.39 1.36 1.62 2.24 0.88
0.40 4.80 0.00 0.00 0.00 9.10 6.40 2.70 3.10 9.90 2.20 9.90
0.02 0.02 0.01 0.02 0.02 0.02 0.03 0.02 0.01 0.04 0.02 0.06
Lill
OV F VI Na I
is-2IP 3s-3p 3s-3d 3s-4s 3s-4p 3p--3d 3p-4s 3p-4p 3p-4p 3d-4p 4s-3d 4s-4p
Na VIII
is-ip
244.83
0.02
0.70
0.20 225.78
0.62
0.60
0.02
MgI
3 1S-3 1P 3 IS-4 I S
214.17 5.29
2.32 0.54
0.00 0.10
0.06 62.81 0.03 6.87
0.45 -0.61
1.60 0.00
0.020.01
208
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.4. (continued) Atom Transition MgI
C
qJ
D
R
A
X
D
R
26.04 25.34 281.84 1140.87 84.65
0.44 0.79 0.84 1.06 0.18
0.00 0.00 0.00 0.50 0.00
om
3 1S-4lp 3 I P-4 I S 3 1P-3 1D 3 1P-4lp
45.35 27.74 68.56 130.28 127.20
0.85 0.55 1.02 0.68 1.18
0.00 0.00 6.50 9.90 0.70
0.01 0.02 0.02 0.01 0.02
33 P-4 3 S 33 P-43 P 33 p-3 3 D
103.99 298.92 706.60
1.56 2.62 2.22
0.10 0.00 0.10
0.18 23.65 0.51 68.46 0.05 141.69
0.73 0.15 0.53
4.30 0.60 3.30
0.03 0.03 0.02
406.66 15.14 51.07 208.84 1604.38 48.25
0.49 0.21 0.32 0.74 0.79 0.00
0.00 0.00 0.00 0.00 0.00 0.00
0.22 198.56 0.02 25.99 0.03 63.40 0.21 66.65 0.17475.20 0.01 104.53
0.87 1.33 1.20 0.76 0.81 0.99
2.70 0.70 1.10 3.70 4.30 0.00
0.02 0.02 0.02 0.02 0.02 0.00
3 1S-3 1D
0.13 0.10 0.07 0.02
Mg II
3s-3p 3s-4s 3s-3d 3p-4s 3p-3d 4s-3d
Mg IX
2S-2P
253.98
0.02
0.70
0.19 235.97
0.63
0.60
0.02
MgX
2s-2p 2s-3s 2s-3p 2s-3d 2p-3s 2p-3p 2p-3d
319.36 16.14 119.68 49.81 23.35 56.80 745.42
om 0.16 2.26 0.10 1.69 0.15 0.68
0.70 0.00 0.00 0.60 0.00 0.00 0.00
0.14278.24 0.02 34.07 0.11 22.37 0.06 68.92 0.17 5.40 0.01 119.17 0.29439.66
0.90 0.96 0.38 0.65 0.39 0.95 0.40
1.30 0.00 2.70 0.10 2.30 0.00 0.60
0.02 0.01 0.02 0.01 0.02 0.01 0.02
Al I
3p-4s
77.72
1.68
0.20
0.13
18.76
0.66
3.50
0.02
329.94
0.81
0.00
0.19 157.76
0.57
1.50
0.02
390.91 12.79 40.96 23.27 294.73 1522.11 80.13 33.26 1586.55 858.63
1.02 0.30 0.22 0.13 0.95 0.98 0.36 0.02 0.14 0.53
0.30 0.00 0.00 0.50 0.00 0.50 0.00 0.00 0.90 0.00
0.08 0.02 0.01 0.08 0.13 0.06 0.13 0.06 0.16 0.11
73.65 16.82 70.01 35.02 61.19 171.45 59.54 45.52 281.97 196.56
0.83 0.92 0.79 0.70 0.99 0.64 1.50 3.70 0.90 1.34
6.80 0.50 0.00 0.00 7.30 9.90 3.10 5.00 9.90 9.90
0.02 0.02 0.00 0.01 0.03 0.05 0.02 0.02 0.07 0.03
AlII
3 1S-3 1P
KI
4s-4p 4s-5s 4s-3d 4s-5p 4p-5s 4p-3d 4p-5p 5s-3d 5s-5p 3d-5p
Ca I
4 1S-3 1D 4 1S-4lp 3 1O-4 l p 43 p-3 3 D
12.19 302.55 304.09 283.09
0.16 1.82 0.19 0.59
0.00 0.10 0.20 0.00
0.02 0.06 0.10 0.09
25.12 68.44 116.87 69.84
0.99 0.62 1.95 1.40
0.00 3.50 9.10 9.40
0.02 0.02 0.02 0.02
Ca II
4s-3d 4s-4p 4s-5s 3d-4p 3d-5s 4p-5s
10.89 596.95 20.11 369.95 1.75 294.60
0.03 0.03 0.17 0.02 0.06 0.68
0.00 0.90 0.00 0.80 0.40 0.00
0.03 24.65 0.27 178.03 om 33.95 0.15 236.50 2.88 0.03 0.22 85.36
1.05 0.99 1.54 1.23 0.73 0.82
0.00 3.70 1.00 3.00 0.00 4.60
0.00 0.02 0.02 0.02 0.01 0.02
Table 6.4. (continued) Atom Transition
C
cp
D
R
A
X
D
R
Cu I
4s--4p
167.62
1.71
0.00
0.08
58.81
0.51
1.60
0.02
Zn I
4 1S--41P 4 1S-5 1S 4 1S--4 1D 4 1S_5 1P 41P_5 1S 41p--41D 41p_5 1p
163.76 3.66 15.12 30.46 240.58 1082.46 66.48
2.58 0.77 0.67 1.22 0.83 1.73 0.24
0.00 0.00 0.00 0.00 0.00 0.20 0.00
0.06 0.04 0.00 0.13 0.10 0.04 0.03
56.38 4.67 23.90 27.88 62.92 152.82 88.16
0.38 0.62 0.79 0.48 1.02 0.63 1.24
0.90 0.00 0.00 0.00 6.00 6.40 1.00
0.02 0.00 0.Ql 0.02 0.02 0.02 0.02
4 3P_5 3S 43p_5 3p 43P--4 3D
88.10 753.48 470.12
1.62 5.43 2.25
0.00 0.00 0.10
0.10 22.88 0.47 85.75 0.07 114.73
0.71 0.08 0.44
3.50 1.00 2.10
0.02 0.02 0.02
315.14 10.72 35.49 13.28 4.17 177.20 1253.98 243.28 71.52
0.62 0.24 0.39 0.19 0.38 0.80 0.96 0.69 0.15
0.00 0.00 0.20 0.30 0.00 0.00 0.00 0.00 0.60
0.21 0.02 0.02 0.05 0.05 0.20 0.17 0.67 0.03
156.02 20.05 38.25 24.07 6.70 57.89 388.52 133.56 69.26
0.78 0.84 1.05 0.83 1.41 0.74 0.70 0.26 1.19
2.20 0.00 1.10 0.00 0.90 3.40 3.30 0.20 1.70
0.02 0.02 0.02 0.00 0.02 0.02 0.02 0.03 0.02
Zn II
4s--4p 4s-5s 4s--4d 4s-5p 4s--4f 4p-5s 4p--4d 4p-5p 4p--4f
Ga I
4p-5s 4p-5p 4p--4d 4p-6s 5s-5p 5s--4d 5s--6s 4 1S--4 1p 4 1S-5 1S 4 1S--4 1D 41P_5 1S 41p--41D 43P_5 3S 4 3P--4 3D
78.66 22.83 479.96 21.00 700.64 57.69 21.33
1.83 0.00 2.64 1.47 0.69 0.11 0.24
0.00 1.00 0.00 0.30 0.50 0.00 0.00
0.09 0.04 0.05 0.12 0.11 0.04 0.06
20.93 30.47 123.45 8.21 105.03 96.59 23.53
0.66 0.70 0.42 0.54 0.88 0.99 1.52
3.00 0.20 1.70 1.50 9.90 0.30 1.90
0.02 0.02 0.02 0.03 0.02 0.02 0.02
299.06 7.91 40.57 230.70 1285.02 124.23 986.67
0.94 0.26 0.48 0.63 0.74 1.23 1.20
0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.18 138.94 0.04 16.22 0.02 48.84 0.23 90.02 0.17417.23 0.15 33.28 0.15 336.24
0.54 0.93 1.07 0.76 0.83 0.67 0.52
1.40 0.00 0.90 3.00 4.00 3.50 2.00
0.02 0.Ql 0.02 0.02 0.02 0.02 0.02
5s-5p 5s--4d 5s--6s 5s--6p 5s-5d 5p--4d 5p-6s 5p-6p 5p-5d 4d--6s 4d--6p 4d-5d
421.39 39.60 13.38 29.20 8.52 1330.95 338.26 85.10 42.03 22.90 395.52 194.77
1.03 0.19 0.30 0.12 0.11 0.66 0.92 0.34 0.03 0.02 0.55 0.03
0.30 0.00 0.00 0.60 0.00 0.60 0.00 0.00 0.80 0.00 0.00 0.90
0.08 0.01 0.02 0.11 0.01 0.07 0.11 0.13 0.37 0.04 0.15 0.17
0.83 0.82 1.03 0.60 5.07 0.71 1.01 1.52 0.41 2.89 1.20 0.79
7.00 0.00 0.70 0.00 2.30 9.90 7.70 3.20 0.00 2.90 7.60 3.40
0.02 0.00 0.02 0.00 0.01 0.06 0.03 0.02 0.04 0.02 0.03 0.02
Ga II
RbI
76.70 70.04 17.21 38.59 27.98 174.03 68.96 62.76 42.62 37.65 103.37 83.95
210
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.4. (continued) Atom Transition Sr I
SI8-4 1D SiS-Sip SIS-6 IS SIS-6 l p 4 10-S1P 41D-(j IS 4 1D-(jl P S3P-43D S3P-63S S3P-63P 4 3D-(j3S 43D-(j3p
A
D
R
0.93 0.63 1.01 3.58 1.81 0.82 0.46
0.00 4.10 0.80 1.90 9.90 0.00 1.60
0.01 0.02 0.02 0.01 0.03 0.01 0.02
0.13 112.4S 0.24 33.94 62.69 0.03 11.S9 12.21
1.32 0.79 1.38 0.86 0.31
9.90 6.60 1.70 0.00 0.40
0.03 0.03 0.02 0.00 0.01
cp
D
R
21.03 338.S3
0.16 1.66
0.00 0.20
8.23 SS2.67 6.46 31.88
0.20 0.29 0.10 0.73
0.00 0.00 0.00 0.00
0.02 41.28 0.07 70.01 11.6S 0.01 21.27 0.08 181.86 0.02 11.79 0.32 13.04
448.12 181.20
0.00 0.89
1.00 O.SO
6.11
0.09
0.00
C
X
Sr II
Ss-4d Ss-Sp Ss-6s Ss-Sd Ss-6p Ss-4f Sp-6s Sp-Sd Sp-6p Sp-4f 4d-Sp 4d-6s 4d-Sd 4d-Sd 4d-Sd 4d-6p 4d-4f
18.13 726.97 21.9S 33.20 2S.6S 20.16 S03.13 2168.30 128.12 281.14 489.30 2.98 108.S2 SO.72 SO.72 22.17 978.39
0.03 0.S1 O.1S 0.16 0.11 0.16 0.03 0.69 0.11 0.12 0.03 0.06 0.13 0.01 0.01 0.07 1.02
0.00 0.00 0.00 0.50 0.30 0.00 0.90 0.00 0.10 0.30 0.80 0.40 0.10 0.70 0.70 0.40 0.00
0.01 39.07 0.22 288.11 0.01 36.83 O.OS 29.87 0.04 43.48 0.03 38.S2 0.32 199.26 0.19 SS8.87 0.02 183.3S 0.04 308.03 0.19277.21 0.04 4.S6 0.02 162.86 0.8S 82.76 0.8S 82.76 O.OS 3S.77 0.18280.00
0.99 0.90 1.61 1.32 0.77 0.86 0.93 0.88 1.70 1.88 1.21 0.96 1.36 0.81 0.81 0.74 0.66
0.00 3.70 1.10 2.20 0.00 0.00 4.30 S.70 1.60 2.70 3.S0 0.40 1.00 0.00 0.00 0.00 3.40
0.01 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.02 0.23. 0.23 0.00 0.02
Ag I
Ss-Sp Ss-6s Ss-6p Ss-Sd Sp-6s Sp-6p Sp-Sd 6s-6p 6s-Sd
189.97 6.10 20.24 13.98 162.11 48.7S 820.00 986.91 77.S6
1.82 0.47 0.62 0.56 1.06 0.38 1.89 0.39 0.08
0.00 0.00 0.00 0.10 0.00 0.10 0.10 0.70 0.00
0.07 0.03 0.12 0.02 0.10 0.04 O.OS 0.13 0.06
62.12 9.29 26.49 19.52 42.21 43.26 140.91 lS6.80 119.28
0.50 0.73 0.6S 0.67 0.87 1.21 0.61 0.86 1.84
1.70 0.00 0.00 0.00 4.80 1.80 4.80 9.90 1.60
0.02 0.00 0.03 0.01 0.03 0.02 0.02 0.04 0.02
Cd I
SiS-Sip S IS-6 IS SiS-SiD S IS-6 l p Slp-6 IS Sip_SiD Slp-6 l p
194.S9 4.44 17.94 29.17 269.74 1162.16 76.03
2.81 0.66 0.69 1.20 0.87 I.S4 0.19
0.00 0.10 0.00 0.00 0.00 0.30 0.00
O.OS 0.02 0.01 0.12 0.09 0.06 0.03
S7.54 S.78 26.70 26.20 66.26 lS4.64 114.68
0.42 0.63 0.74 0.46 1.01 0.63 1.12
1.30 0.00 0.00 0.00 6.30 7.10 0.60
0.01 0.01 0.01 0.01 0.02 0.02 0.02
6.1 Tables of Numerical Results Table 6.4. (continued) Atom Transition
C
D
R
0.71 0.17 0.45
3.90 0.50 2.40
0.02 0.03 0.02
0.67 0.85 0.42
4.90 0.50 2.40
0.03 0.02 0.03
0.18 162.58 0.02 17.44
0.54 1.08
1.50 0.30
0.02 0.02
0.00
0.16 42.80
0.68
3.70
0.02
1.03 0.13 0.29 0.12 0.19 0.15 0.00 0.92 0.33 0.87 0.21 0.03 0.02 0.03 1.12
0.30 0.00 0.00 0.60 0.30 0.00 1.00 0.00 0.00 0.60 0.00 0.00 0.90 0.90 0.30
0,07 0.02 0.02 0.11 0.02 0.02 0.15 0.14 0.13 0.16 0.04 0.02 0.35 0.11 0.09
80.69 55.13 18.03 41.66 6.68 15.99 191.12 72.16 66.24 57.17 170.92 20.70 21.00 63.48 169.13
0.85 0.87 1.11 0.55 0.66 2.13 1.19 1.03 1.57 0.74 1.60 0.90 0.77 0.90 0.80
7.90 0.00 0.90 0.00 0.00 0.90 9.90 8.20 3.40 9.90 1.70 0.00 2.40 3.20 9.20
0.02 0.00 0.02 0,0} 0.01 0.02 0.04 0.03 0.02 0.03 0.02 0.01 0.02 0.02 0.02
3 Transition 11 S-21 S 11 S-21 P 11 S-3 1 S
C 3.31 62.45 2.25
qJ
1.43 5.56 1.64
D 0.10 0.50 0.20
R 0.02 0.04 0.01
A
X
D
R
3.77 10.57 2.38
0.60 0.31 0.56
0.10 1.70 0.10
0.02 0.01 0.02
6.1 Tables of Numerical Results Table 6.6. (continued) Transition
C
qJ
D
R
R
41.37 1.70 2.01 36.13 1.96 0.02
6.90 3.76 1.78 7.33 3.99 1.69
0.50 0.70 0.20 0.50 0.70 0.00
0.04 0.07 0.Ql 0.06 0.08 0.33
6.37 0.31 2.05 5.46 0.35 0.02
X 0.32 0.44 0.54 0.32 0.43 0.55
D
IIS-3 IP IIS-3 ID IIS-4 IS IIS-4IP IIS-4ID IIS-4IF
1.80 3.20 0.10 1.80 3.10 0.00
0.Ql 0.02 0.02 0.02 0.02 0.08
zIS-2lp 21S-3 1S 2 1S-3 1P 2 1S-3 1D zIS-4 IS 2 1S-41P 2 1S-41D zIS-4IF 21P-3 1S 21P-3 1P 21p-3 1D 21P-4 1S 21p-41p 21p-41D zIP-4 IF
409.42 17.37 166.30 55.29 8.14 70.82 17.46 8.20 21.23 58.14 836.51 8.47 27.18 291.36 38.25
0.03 0.20 2.46 0.20 0.28 2.16 0.32 0.00 2.84 0.18 0.89 2.56 0.24 0.95 0.51
0.50 0.00 0.00 0.50 0.00 0.20 0.40 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.30
0.10 0.02 0.09 0.07 0.03 0.12 0.04 0.08 0.04 0.02 0.23 0.08 0.02 0.19 0.07
290.52 29.40 23.32 47.55 12.86 11.51 15.60 6.44 2.59 103.91 373.36 1.54 48.39 155.65 23.51
1.99 1.63 0.50 1.08 1.76 0.55 1.03 1.83 0.38 0.87 0.48 0.36 0.79 0.48 0.95
4.80 1.10 4.70 1.80 1.40 4.50 1.50 4.20 4.30 0.10 1.30 2.60 0.00 0.90 2.30
0.02 0.02 0.02 0.02 0.02 0.03 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02
3 1S-4 1S 3 1S-41P 31S-41D 3 1S-41F 31P-4 1S 3 1P-41P 3 1P-41D 31P-41F 3 1D-4 1S 3 1D-41p 3 1D-41D 3 1D-41F
5l.98 387.56 116.56 56.59 99.72 190.96 1463.19 380.73 6.80 50.17 247.32 4214.
0.06 1.41 0.03 0.10 1.41 0.06 0.56 0.90 0.13 0.02 0.05 0.02
0.00 0.00 0.80 0.10 0.00 0.10 0.00 0.30 0.20 0.90 0.00 0.90
0.04 0.11 0.22 0.16 0.27 0.04 0.31 0.19 0.27 0.32 0.02 0.35
102.46 41.23 68.13 64.90 10.43 354.90 427.74 341.30 5.48 25.18 505.88 2237.
0.89 0.65 2.91 2.85 0.61 0.92 0.74 2.51 2.54 0.90 0.93 0.64
0.00 9.90 8.90 4.40 9.90 0.10 4.50 5.10 5.80 3.20 0.00 2.00
0.01 0.02 0.02 0.02 0.05 0.02 0.02 0.01 0.02 0.02 0.Ql 0.02
23S_23P 23S-3 3S 23S_3 3P 23S-3 3D 23S-43S 23S_43P 23S-43D 23S-43F 23P_3 3S
361.31 16.59 146.76 51.27 7.82 68.41 16.82 7.36 16.66
0.Ql 0.19 2.41 0.09 0.26 2.43 0.10 0.05 1.27
0.70 0.00 0.00 0.70 0.00 0.00 0.70 0.80 0.00
0.10 0.02 0.08 0.09 0.01 0.08 0.10 0.11 0.37
270.49 29.62 22.56 54.73 13.75 14.29 22.59 10.68 4.91
1.48 1.22 0.46 0.82 1.12 0.39 0.56 0.81 0.27
3.10 0.50 3.90 0.80 0.40 2.30 0.00 0.30 1.40
0.Ql 0.02 0.02 0.01 0.02 0.03 0.Ql 0.02 0.01
A
213
214
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.6. (continued) Transition 23p_3 3p 23p_3 3D 23P-43S 23p_4 3p 23p_43D 23p-43F 33S-43S 33S_43P 33S-43D 33S-43F 33P-43S 33p_43p 33p_4 3D 33p_4 3F 33D-43S 33D_4 3p 33D-43D 33D-43F
C
D
qJ
A
R
X
D
R
0.00 0.02 0.03 0.01 0.02 0.02 0.01 0.02 0.02 0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.00 0.02
56.23 772.12 5.64 26.31 269.86 30.56
0.13 0.79 0.73 0.11 0.75 0.03
0.10 0.00 0.00 0.30 0.00 0.90
0.02 0.27 0.54 0.02 0.26 0.22
114.22 390.74 3.40 58.49 181.43 33.07
0.92 0.42 0.21 1.04 0.37 0.58
0.00 0.90 0.10 0.00 0.30 0.40
50.54 350.20 106.33 56.00 98.32 187.59 1351.28 391.17 7.65 53.06 250.60 4199.
0.06 1.41 0.02 0.06 1.15 0.05 0.55 0.04 0.01 0.00 0.04 0.02
0.00 0.00 0.80 0.30 0.00 0.10 0.00 0.60 0.70 0.90 0.10 0.90
0.04 0.13 0.22 0.14 0.37 0.03 0.34 0.15 0.20 0.35 0.03 0.35
100.15 37.00 66.88 76.69 11.50 356.10 403.69 371.17 9.49 37.24 522.72 2302.
0.90 0.64 2.72 2.15 0.65 0.86 0.72 2.26 1.44 0.50 0.97 0.61
0.00 9.90 7.70 2.40 9.90 0.00 4.30 4.00 1.40 0.90 0.00 1.80
Table 6.7. Normalized cross sections for intercombination transitions (AS = 1) Atom
Transition
He I
lIs-is IIS-2IP IIS-3 IS IIS-3 IP IIS-3 ID IIS-4 IS IIS-4IP IIS-4ID IIS-4IF 23S-2 1S
23 S-ip 23S-3 1S 23S-3 1P 23S-23D
is-23p is-3 3S
is-3 3p is-3 3 D 23p-2 1 p 23P-3 1S
C
qJ
D
R
22.66
0.69
2.60
0.31
27.63
0.37
3.70
0.38
28.54
0.34
3.90
0.38
X
D
R
1.10 9.92 1.91 12.18 0.75 2.14 12.94 0.98 0.Q3
2.25 1.88 3.68 2.37 0.73 4.06 2.52 0.74 0.93
0.20 0.20 0.20 0.20 0.00 0.20 0.20 0.00 0.00
0.03 0.04 0.Q3 0.04 0.07 0.Q3 0.04 0.07 0.14
9.38 22.20 13.66 15.76 88.17 13.13 4.93 19.10 108.73 19.58 17.22
1.77 1.78 3.89 2.16 4.80 1.69 2.59 4.26 5.50 1.04 3.70
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.04 0.02 0.02 0.02 0.01 0.04 0.04 0.03 0.01 0.05 0.02
A
6.1 Tables of Numerical Results
215
Table 6.7. (continued) Atom
Lill
Transition 23p_3 1p 23p_3 1D 21P_3 3S 21p_3 3p 21p_3 3D 11S-23S 11S-23P 11S-3 3S 11S_3 3P 11S-3 3D 23S-2 1S 23S-21P 23S-3 1S 23S_3 1P 23S-3 1D is-23p 2 1S-3 3S is-3 3p is-3 3D 23p_2 1p 23P-3 1S 23p_3 1p 23p_3 1D 21P-3 3S 21p_3 3p 21p_3 3D
Mg IX
2 1S-23P is-23p 2 1S-23P 2 1S_23P is-23p 2 1S-23P 3 1S-3 3P 31S-43S 3 1S_43P 31S-3 3D 33p_3 1p 33P-4 1S 33p_3 1D 33p_4 1p 2 1S_23P
AlII
3 1S-3 3P
Be I BII CIII OV F VI Na VIII MgI
C
D
R
1l.43
l.20
l.20
0.42
1l.64
l.23
1.40
0.48
X
D
R
36.07 166.02 14.15 40.87 159.60
3.89 5.17 3.22 4.57 4.81
0.10 0.00 0.00 0.10 0.00
0.05 0.04 0.01 0.06 0.03
0.04 7.20 0.04 6.35 l.53
0.87 0.45 0.85 0.40 0.90
9.90 0.00 9.90 0.00 0.00
0.33 0.05 0.34 0.06 0.02
12.95 25.83 8.42 9.80 52.71 14.93 6.10 6.54 50.03 41.35 9.70 22.67 107.21 14.98 19.92 123.15
3.64 3.60 5.78 3.20 4.79 3.27 4.59 2.50 4.77 l.87 3.28 l.64 3.63 4.25 l.69 3.94
0.00 0.00 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.04 0.03 0.08 0.04 0.03 0.03 0.04 0.02 0.02 0.02 0.03 0.05 0.01 0.02 0.05 0.01
10.55 14.67 10.16 6.83 7.55 7.05
0.80 l.76 2.37 2.05 2.09 2.15
0.00 0.00 0.00 0.00 0.00 0.00
0.04 0.01 0.04 0.02 0.02 0.02
15.38 2.36 11.54 64.55 38.63 5.74 185.98 37.61
0.95 0.92 1.31 2.36 l.66 l.59 4.44 3.46
0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.10
0.05 0.08 0.02 0.02 0.03 0.04 0.06 0.05
7.01
2.21
0.00
0.02
35.75
3.87
0.00
0.02
A
216
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.7. (continued) Atom Ca I
Zn I
Ga II
Sr I
Cd I
In II
Ba I
Transition 4 1S_43P 4 1S-3 3D 4 3p-4 1p 43p-3 1D 4 1S_43P 4 1S-5 3S 4 1S_53P 4 1S-43D 43p-4 1p 43P-5 1S 43p-41D 43p-5 1D 4 1S_43P 4 1S-53S 4 1S-43D 43p-4 1p 43P-5 1S 43p-41D 51S_5 3P 5 1S-43D 51S-63S 51S_63P 53p-41D 53P-5 1P 53P-6 1S 53p-6 1p 43D-5 1P 51S-5 3P 51S-63S 51S_63P 51S-5 3D 53P-5 1P 53P-6 1S 53p-5 1D 53P-6 1P 51S-53P 51S-63S 53P-5 1P 53P-6 1S 6 1S-5 3D 6 1S_63P 6 1S-73S
C
D
R
X
D
R
20.73 34.57 37.l9 31.84
1.54 2.12 1.78 1.42
0.00 0.00 0.00 0.00
0.04 0.02 0.00 0.Q1
13.08 1.64 10.63 36.60 32.85 8.37 222.55 35.37
0.68 0.73 1.04 1.90 1.40 3.75 6.75 3.27
0.00 0.00 0.00 0.00 0.00 0.10 0.10 0.10
0.06 0.03 0.02 0.02 0.04 0.08 0.06 0.04
37.95 7.66 62.19 72.04 16.86 127.73
3.53 4.37 4.24 3.02 4.45 2.81
0.00 0.00 0.00 0.00 0.00 0.00
0.02 0.04 0.03 0.Q1 0.Q1 0.Q1
25.27 43.30 8.71 23.66 36.50 39.25 8.87 29.75 22.87
1.99 2.38 3.08 3.l3 1.47 2.07 2.14 2.89 1.40
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.03 0.Q1 0.03 0.02 0.02 0.Q1 0.05 0.02 0.07
15.11 2.l3 11.86 45.80 35.l7 8.47 241.55 33.72
0.80 0.91 1.27 2.32 1.49 3.70 7.05 3.41
0.00 0.00 0.00 0.00 0.00 0.l0 0.l0 0.10
0.08 0.09 0.02 0.02 0.03 0.06 0.08 0.05
40.08 6.08 89.74 20.63
4.32 4.49 4.11 5.93
0.00 0.00 0.00 0.00
0.03 0.04 0.Q1 0.02
12.56 22.01 5.62
1.30 1.71 2.48
0.00 0.00 0.00
0.04 0.03 0.03
A
6.1 Tables of Numerical Results
217
Table 6.7. (continued) Atom
Hg I
A
X
D
R
53 D-6 1P 53 D-7 1 S 5 1D_63 p
16.62 6.85 11.02
1.12 1.90 0.65
0.00 0.00 0.00
0.08 0.05 0.13
61 S_63 p 6 1 S-73 S 63 p-6 1P 63 P-7 1 S 6 1P_7 3 S
19.83 3.84 37.16 12.30 21.52 13.63
1.17 1.34 1.61 4.18 4.36 3.94
0.00 0.00 0.00 0.10 0.00 0.00
0.08 0.12 0.01 0.06 0.04 0.01
Transition
C
D
R
73 S-is
Table 6.S. Excitation of multipy charged ions. Normalized exchange cross section for heliumlike ion Na X. Intercombination transitions (LIS = 1). Parameters C, q>, D and A, X, D can be used for any heliumlike ion with z > 3. Transition
C
q>
D
R
A
X
D
R
IS-2S IS-2P IS-3S IS-3P IS-3D IS-4S IS-4P IS-4D IS-4F 2S-3S 2S-3P 2S-3D 2S-4S 2S-4P 2S-4D 2S-4F 2P-3S 2P-3P 2P-3D 2P-4S 2P-4P 2P-4D 2P-4F 3S-4S 3S-4S 3S-4P 3S-4D 3S-4F 3P-4S
2.45 4.61 2.27 4.50 0.36 2.30 4.35 0.46 0.01
1.31 0.81 1.52 0.84 0.52 1.37 0.90 0.57 0.36
1.00 0.50 1.00 0.60 0.30 1.10 0.60 0.30 0.10
0.09 0.11 0.08 0.10 0.21 0.08 0.10 0.21 0.35
1.59 10.64 1.49 9.57 2.29 1.53 9.13 2.83 0.17 2.10 5.42 16.26 1.92 4.90 11.33 8.76 4.02 26.37 65.05 3.65 23.65 54.02 22.88 2.01 2.01 4.54 8.38 20.05 3.87
0.69 1.12 0.69 1.09 1.60 0.70 1.08 1.57 2.14 3.33 3.46 3.47 3.35 3.44 3.23 4.51 3.06 3.64 4.42 3.05 3.65 4.25 5.53 6.90 6.90 7.02 7.17 6.76 6.44
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.02 0.03 0.02 0.03 0.02 0.02 0.03 0.02 0.02 0.03 0.03 0.03 0.04 0.03 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.03
218
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.8. (continued) Transition 3P-4P 3P-4D 3P-4F 3D-4S 3D-4P 3D-4D 3D-4F
C
D
qJ
R
A
17.28 32.97 58.74 6.48 24.22 50.84 121.00
X
6.79 8.34 7.55 6.77 7.53 7.86 8.86
D 0.00 0.00 0.00 0.00 0.00 0.00 0.00
R
0.02 0.01 0.02 0.02 0.02 0.02 0.02
6.1.5 Transitions between Closely Spaced Levels (Tables 6.9-10)
In the case of transitions between closely spaced levels under the conditions AE ~ Eo, Eland AE ~ Iff, the dependence of multipole and exchange cross sections calculated by means of the first-order methods on AE is almost absent. For the optically allowed transitions (AI = 1) a weak logarithmic dependence exists. The calculations for multiply-charged ions have been made using the Coulomb-Born approximation for transitions with no change of spin (AS = 0) and using the orthogonalized functions method for intercombination transitions. The data of Tables 6.9 and 6.10 were obtained for a set of values of AE and can be applied to arbitrary multiply-charged ions with z > 3. For quadrupole and intercombination transitions the value of AE is not important. For dipole transitions one has to interpolate data for particular values of AE. The fitting formulas and the range of analytic approximation are quite the same as in Sect. 6.1.4. 6.1.6 Ionization Cross Sections (Table 6.11 and 6.12)
The ionization cross sections have been calculated in accordance with (3.1.38) in the partial wave representation. In cases of ions the Coulomb-Born approximation has been used:
O'i(aO)
= QiO'(lO),
(vO'i(ao»)
u
= Qi(VO'i(lO»)
= (Iff - EO)/z2
P= z2
DE
Ry/T,
= z2
Ry ,
.
(6.1.18) (6.1.19)
Ry,
p = EO/z2 Ry.
The fitting formulas and the angular factors Qi are given by (5.1.21-25). For the total cross section of ionization from a shell 1'0,
Qi=m,
(6.1.20)
where m is a number of equivalent electrons. The set of parameters C, cp,D is adjusted for the range 0.0625 < u < 64, and the set A,X,D, for 0.125 < P < 8.
Table 6.9. Transitions between the closely spaced levels with no change of spin (LIS Coulomb-Born-exchange cross sections for multiply charged ions Transition
1:l.E/z2
C
cp
R
D
A
[cm- I ]
= 0)
The
X
D
R
0.96 1.72 2.06 2.16 1.96 1.72 1.52 1.42 1.66 2.10 2.24 2.19 1.89 1.55 1.47 1.68 2.11 2.24 2.15 1.84 1.55 1.46
1.60 4.00 4.80 4.30 3.20 2.30 1.70 1.40 4.10 5.20 4.60 3.70 2.60 1.70 1.40 4.10 5.20 4.60 3.60 2.50 1.70 1.40
0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.01 0.02 0.02 0.02 0.Q2 0.02 0.02 0.01 0.02 0.Q2 0.02
1.15 1.14 1.69 0.85 0.84 0.87 0.87 0.85 0.93
0.70 0.70 1.20 0.00 0.00 0.00 0.00 0.00 0.10
0.02 0.02 0.02 0.Q2 0.02 0.Q1 0.02 0.02 0.02
Dipole transitions 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 2s-2p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3s-3p 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d 3p-3d
1480. 740. 370. 185. 93. 46. 23. 12. 856. 428. 214. 107. 54. 27. 13. 856. 428. 214. 107. 54. 27. 13.
2p-2p 2p-2p 3s-3d 3s-3d 3s-3d 3p-3p 3p-3p 3d-3d 3d-3d
100. 10. 1000. 100. 10. 100. 10. 100. 10.
321. 389. 433. 491. 531. 589. 647. 704. 1849. 2163. 2439. 2762. 3092. 3436. 3677. 2356. 2738. 3080. 3482. 3892. 4319. 4617. 13.53 13.51 51.26 51.64 51.53 91.07 91.17 75.06 75.09
0.Q1 0.01 0.02 0.Q1 0.01 0.01 0.01 0.00 0.Q1 0.03 0.02 0.Q1 0.Q1 0.01 0.01 0.01 0.03 0.02 0.Q1 0.01 0.01 0.Q1
0.70 0.15 0.08 0.70 0.50 0.09 0.50 0.06 0.08 0.40 0.06 0.40 0.40 0.05 0.40 0.04 0.12 0.70 0.50 0.10 0.40 0.10 0.08 0.40 0.40 0.06 0.05 0.40 0.30 0.07 0.70 0.11 0.50 0.10 0.40 0.10 0.40 0.08 0.40 0.06 0.40 0.05 0.07 0.30 Quadrupole transitions 0.00 0.30 0.03 0.00 0.04 0.30 0.Q1 0.20 0.05 0.00 0.20 0.04 0.00 0.20 0.04 0.00 0.10 0.03 0.00 0.10 0.03 0.00 0.03 0.10 0.00 0.10 0.03
265. 269. 315. 406. 524. 650. 773. 883. 1232. 1492. 2024. 2671. 3403. 4173. 4890. 1588. 1904. 2565. 3387. 4304. 5242. 6132. 20.46 20.37 86.06 96.24 95.54 178.25 178.31 144.24 143.78
Table 6.10. Intercombination transitions between the closely spaced levels (LIS = I). The summed over K exchange rate coefficients for multiply charged ions Transition 2s-2s 2s-2p 2p-2p 3s-3s 3s-3p 3s-3d 3p-3p 3p-3d
1:l.E/z2 200. 747. 200. 20. 210. 252. 20. 42.
[cm- I ]
A 6.00 7.84 62.32 6.08 9.00 15.36 41.08 47.72
X 3.07 2.83 3.51 6.62 7.80 8.35 7.15 8.59
D
R
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.03 0.02 0.03 0.02 0.02 0.01 0.02 0.01
220
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.11. Ionization cross sections for atoms and ions in Coulomb-Born approximation Atom Level A D R C D R cp X Is 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 5g
9.581 32.6 86.9 66.9 175.0 294.0 106.0 283.0 483.0 680.0 145.0 396.0 674.0 956.0 1296.0
2.37 0.13 0.33 0.03 0.11 0.08 0.00 0.01 0.00 -0.01 -0.01 -0.02 -0.03 -0.03 -0.03
0.60 0.00 -0.60 -0.50 -0.80 -0.80 -0.70 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80 -0.80
0.07 0.11 0.08 0.11 0.10 0.12 0.13 0.11 0.12 0.21 0.16 0.12 0.13 0.19 0.28
7.371 198.5 430.0 746.3 1961.0 3462.0 2084.0 5919.0 10416.0 17566.0 4445.0 13595.0 24815.0 35980.0 57240.0
0.12 -0.70 0.01 21.45 5.10 0.05 5.25 1.00 0.02 15.23 1.60 0.03 9.00 0.60 0.02 7.11 0.20 0.01 19.66 1.00 0.02 15.61 0.50 0.02 14.02 0.30 0.02 14.32 0.10 0.00 0.80 0.02 26.75 25.52 0.50 0.02 25.57 0.40 0.02 22.74 0.20 0.01 23.95 0.00 0.01
lIS 23S 23p 21S
5.986 19.67 88.1 25.3 88.0
4.50 0.21 0.44 0.16 0.34
3.50 0.20 -0.60 0.10 -0.60
0.05 0.11 0.07 0.12 0.08
1.583 86.14 366.8 128.4 426.8
0.16 -0.10 0.02 17.47 5.60 0.06 5.65 1.50 0.03 18.37 5.10 0.06 5.42 1.10 0.02
Is
2s 2p 3s 3p 3d
9.570 32.45 85.1 66.2 176.0 287.0
2.41 0.22 0.40 0.07 0.09 0.11
1.10 0.00 -0.60 -0.50 -0.70 -0.80
0.07 0.12 0.05 0.12 0.09 0.03
4.500 89.60 295.6 406.4 1388.0 2548.0
0.25 -0.10 0.04 13.12 7.20 0.06 3.81 1.10 0.02 1.60 0.03 8.30 6.37 0.60 0.02 5.65 0.30 O.oI
Li I
2s 2p 3s 3p 3d
13.80 88.7 34.1 179.0 294.0
0.24 0.40 0.08 0.13 0.08
0.00 -0.60 -0.70 -0.80 -0.80
0.06 0.08 0.03 0.12 0.12
77.41 389.9 386.5 1913.0 3458.0
16.35 5.45 9.60 9.65 7.12
4.10 1.30 0.70 0.80 0.20
0.04 0.02 0.02 0.02 0.01
Li II
lIS 23S 21S 23p 21p
8.046 25.00 29.41 85.3 85.7
2.68 0.30 1.33 0.49 0.41
4.00 0.20 -0.80 -0.60 -0.60
0.07 0.11 0.09 0.07 0.05
1.872 44.27 60.58 259.3 294.2
0.20 7.56 9.83 3.89 3.95
0.30 6.00 6.80 1.40 1.20
0.02 0.07 0.07 0.03 0.02
Bel
2 1S 2~P 21p
BI
2p
8.56 64.51 88.5 53.08
0.73 1.05 0.56 1.70
0.60 0.10 -0.50 0.60
0.08 0.09 0.08 0.08
11.87 77.73 285.4 36.56
0.17 -0.80 0.09 0.13 -0.80 O.oI 7.85 3.30 0.04 0.29 -0.30 0.04
CI
2s 2p3p 2plD 2p l S
4.529 36.88 41.11 47.58
3.25 2.60 2.30 1.88
1.10 1.70 1.30 0.90
0.05 0.10 0.08 0.07
HI
He I
ip He II
2.349 12.47 16.79 25.70
0.18 -0.40 0.01 0.36 0.50 0.02 0.36 0.30 0.02 0.30 -0.10 0.02
6.1 Tables of Numerical Results
221
Table 6.11. (continued) Atom
cn CIll
C IV CV
NI 01
on o III o IV OV
o VI FI Ne I Na I
MgI
MgIX
Level 2p 2 1S 23 p 21p 2s 2p lIS 23 S 23 p 2p 28 2p 2s 2p 2s 2p 2s 2p 21S 23 p 21p 2S 2p 2p 38 3p 3d 31S 33 p 31P
is 23 p
ip MgX
2s 2p
Mg XI ArI KI
lIS
Ca I
3p 48 1 4 S 43 p 33 D 3 1D
C
A
D
R
ffJ
D
R
67.48
1.01
0.00
0.11
74.04
0.23 -0.60 0.04
21.63 76.3 84.6 25.86 84.4
0.31 0.40 0.34 0.23 0.42
0.50 -0.10 -0.30 0.30 -0.60
0.10 0.09 0.09 0.11 0.08
31.72 165.9 239.8 53.42 302.0
0.17 -0.80 0.05 11.48 8.20 0.05 6.21 3.00 0.04 9.73 6.70 0.07 4.89 1.60 0.03
9.203 29.04 83.4 26.43 2.644 24.92 8.321 50.62 13.67 66.66 19.41 75.1
1.42 0.19 0.40 3.78 3.66 3.64 2.56 1.88 0.92 0.74 0.40 0.41
3.80 0.20 -0.60 3.00 8.00 2.70 1.50 0.80 0.80 0.20 0.70 0.00
0.05 0.11 0.07 0.10 0.04 0.06 0.12 0.08 0.11 0.11 0.11 0.10
3.519 0.21 -0.10 0.02 76.64 12.60 7.00 0.06 4.89 312.3 1.50 0.03 0.25 0.70 0.02 5.18 0.485 -0.11 -0.10 0.01 0.19 5.45 0.30 0.01 0.45 2.540 1.10 0.03 0.54 0.90 0.03 21.90 0.23 -0.50 0.04 11.63 0.11 -0.80 0.03 80.48 24.52 0.13 -0.80 0.02 9.42 7.40 0.05 143.3
23.95 79.1 83.9 28.03 17.07 13.17 13.06 85.2 296.0
0.26 0.31 0.33 0.19 5.88 5.82 0.42 0.26 0.09
0.50 -0.20 -0.40 0.30 5.00 6.50 -0.60 -0.70 -0.80
0.11 0.08 0.09 0.10 0.06 0.03 0.02 0.11 0.11
37.80 226.1 273.7 66.22 1.98 1.39 83.04 553.7 3362.0
5.43 7.11 5.95 11.34 0.10 0.09 8.29 5.03 6.89
9.21 68.4 77.3
0.68 0.57 0.32
-0.40 -0.40 -0.70
0.03 0.08 0.10
39.92 209.8 464.8
9.37 6.26 4.99
27.47 80.4 83.1 29.94 83.7
0.20 0.32 0.35 0.17 0.29
0.40 -0.40 -0.50 0.30 -0.50
0.10 0.08 0.08 0.10 0.07
57.88 272.0 304.3 80.34 331.4
9.80 5.79 5.79 12.65 5.37
9.477 3Q.62 16.36
1.23 1.95 0.36
3.20 1.70 -0.60
0.06 0.41 0.05
0.71 0.53 0.39 0.39
-0.40 -0.50 0.60 0.40
0.11 0.14 0.14 0.15
12.31 88.3 174.1 188.4
4.417 18.03 92.98 34.93 285.1 200.9 234.6
X
4.40 3.50 2.40 6.90 0.40 0.30 1.50 0.50 0.20
0.08 0.04 0.03 0.07 0.01 0.01 0.02 0.02 0.02
2.90 0.03 2.60 0.03 0.60 0.02 6.50 2.20 2.00 6.80 1.60
0.07 0.03 0.03 0.06 0.03
0.23 -0.20 0.02 0.73 0.70 0.06 6.23 1.10 0.02 2.40 0.03 5.37 4.35 1.40 0.03 0.12 -0.80 0.02 0.13 -0.80 0.03
222
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.11. (continued) Atom
Cu I Krl Rb I
Sr I
Sr II Ag I Xe I Cs I
Ba I
Ba II
Hg I
Level
C
41p
99.4
0.42
-0.60
0.09
405.2
4s 4p 5s 5p 4d SiS 53 p 43D 41D Sip
10.27 39.98 16.32 95.7 306.0 13.22 99.7 236.3 39.77 117.9 25.0 325.5 11.28 46.68 19.2 125.3 297.0
0.75 2.47 0.42 0.23 0.30 0.59 0.54 0.46 1.59 0.36 0.18 0.22 0.71 1.87 0.38 0.22 0.23
-0.30 1.70 -0.60 -0.70 -0.80 -0.50 -0.50 0.70 1.40 0.20 0.20 0.60 0.60 1.60 0.10 0.10 0.40
0.04 0.31 0.05 0.05 0.13 0.04 0.11 0.14 0.09 0.13 0.03 0.16 0.04 0.18 0.05 0.08 0.16
37.25 16.19 77.12 624.6 1926.0 56.38 331.0 393.7 36.03 585.7 144.2 785.6 40.13 35.78 122.0 869.6 1128.0
13.17 5.40 0.04 0.60 1.00 0.05 5.69 1.30 0.02 6.07 0.80 0.02 7.56 1.30 0.03 8.94 2.90 0.03 7.97 3.20 0.04 6.67 5.40 0.07 0.20 -0.60 0.03 9.49 2.50 0.04 5.44 0.90 0.02 8.18 4.90 0.05 18.90 8.30 0.04 0.34 -0.30 0.03 6.53 1.00 0.02 6.49 0.80 0.02 4.20 0.05 1l.l0
14.52 153.3 163.3 96.6 27.3 352.1 127.0
0.51 0.45 0.38 0.43 0.14 0.21 0.16
0.20 2.70 2.60 0.20 0.20 0.60 0.00
0.05 0.09 0.09 l.l0 0.02 0.16 0.04
73.18 100.5 118.8 399.1 171.0 883.3 1018.0
6.39 1.40 0.02 0.22 -0.40 0.01 0.20 -0.50 0.01 4.96 1.20 0.03 5.58 0.80 0.02 6.90 3.80 0.05 5.52 0.40 0.02
9.67 89.2 114.9
0.24 0.22 0.85
2.60 1.80 0.10
0.12 0.13 0.12
18.60 129.9 413.2
12.23 8.50 0.16 0.18 -0.80 0.12 12.41 4.90 0.05
5s 4d 5s 5p 6s 6p 5d 6 1S 53 D SiD 63 p 6s 5d 6p 6 1S 63 p 6 1P
D
qJ
A
R
Table 6.12. Rate coefficients of dielectronic recombination in Coulomb-Born-exchange approximation. Parameters and Xd.
He II BeN
C VI
o VIII NeX Mg XII SiXN
S XVI
IXo (XI H-like ions Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p Is - 2p
31.18 36.69 31.34 25.78 21.06 17.20 14.08 11.56
0.74 0.73 0.71 0.69 0.67 0.66 0.64 0.63
A.I
X
6.91
D
R
2.10 0.03
6.1 Tables of Numerical Results
223
Table 6.12. (continued) Xz+1
IXo
(XI
A.i
CaXX Fe XVI
Is Is He-like lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS lIS -
2p 2p ions 21p
7.828 4.420
0.61 0.60
17.43 22.21 0.024 18.21 0.078 14.37 0.170 11.16 0.309 8.659 0.495 6.738 0.714 4.144 1.151 2.138 1.500
1.13 0.85 0.90 0.79 0.85 0.75 0.82 0.72 0.80 0.69 0.78 0.68 0.75 0.66 0.71 0.64 0.65
10.81 6.405 3.706 2.400 1.690 1.182 .9387 .6229 .5012
0.07 0.04 0.02 0.02 0.01 0.01 0.01 0.01 0.01
21.07 11.62 7.010 5.665 4.663 1.003
0.10 0.06 0.04 0.03 0.03 0.01
Li II CV CV o VII o VII NeIX Ne IX Mg XI Mg XI Si XIII Si XIII S XV SXV CaXIX Ca XIX Fe XXV Fe XXV Be II C IV o VI Ne VIII MgX Si XII S XIV Ca XVIII Fe XXIV CIII
ov
Ne VII Na VIII Mg IX Fe XXIII
ip 23 p
ip 23p
ip 23p
ip 23 p
ip 23p
ip 23 p
ip 23 p
ip
23 p Li-like ions 2s - 2p 2s - 2p 2s 2p 2s - 2p 2s - 2p 2s 2p 2s - 2p 2s - 2p 2s - 2p Be-like ions 21p 2 1S
is - ip 21S - ip is ip 21S - ip ip 2 1S
Xd
6.1.7 Dielectronic Recombination Rate Coefficients (Table 6.12) The methods of calculations of the rate coefficients for the dielectronic recombination process,
Xz+\(IXo) + e
--+
Xz(lX\nl)
--+
Xz(lXonl)
+ fut),
224
6. Tables and Formulas for the Estimation of Effective Cross Sections
are described in Sect. 5.2. The simplified model (5.2.12) with its modification (5.2.29) for s - p transitions was used. The excitation cross section for the transition IXo - 0(1 of an ion XZ+I has been calculated in the Coulomb-Born approximation with exchange whenever it has been substantial. The rate coefficient for dielectronic recombination connected with the transition IXo - IXI is expressed in the form
f3
=
(z
+ 1)2Ry T
.
(6.1.21) Parameters Ad and Xd for the most important actual cases are given in Table 6.12 and the angular factors Qd for these cases are given by
Qd(nolO', nol:;'-lnI1d = m, Qd(noltnllf', nol:-lnllf'+I)
=N
(1- 2(21~+ 1»)·
(6.1.22)
In the case of heliumlike ions the total rate coefficient for dielectronic recombination is the sum of contributions from excitation of both singlet and triplet P levels.
6.2 Formulas Defining the Angular Factors 6.2.1
Rules for the Addition of Cross Sections
In various applications, cross sections are required for transitions between separate levels, between two groups of closely spaced levels, for transitions from a given level to a group of levels, and for transitions from the whole group of levels to a given level. For example, one may be interested in transitions between separate fine structure components LoSoJo - LISIJI of two terms or in the transition between the terms LoSo - LISI as a whole. The cross section for transition from a given level a of the group A to the group B of levels b is, clearly, a(Aa, B)
= L a(Aa, Bb) ,
(6.2.1 )
b
where a(Aa, Bb) is the cross section for the transition a-b. If every level a of the group A is populated proportionally to its statistical weight, then the cross section for the transition A - Bb is defined by a(A, Bb)
=
1 g(A) ~g(a)a(Aa, Bb),
(6.2.2)
and the cross section for the transition A - B, by 1 a(A, B) = g(A) ~g(a)a(Aa, Bb) .
(6.2.3)
6.2 Formulas Defining the Angular Factors
225
La
Here g(a) is the statistical weight of level a, and g(A) = g(a) is the statistical weight of the group of levels A. The tabulated cross sections are given by formulas (6.1.1,2) where the dependence of effective cross sections on angular momenta is determined by the factors Q~ and Q~. Therefore the summation of the cross sections over the finestructure components of terms and over the terms belonging to a single electronic configuration is equivalent to the summation of these angular factors. The next subsections give a summary of formulas defining the factors Q~ and Q~ for the cases which can be met when using the tables of cross sections given in Sect. 6.1. 6.2.2
LS-Coupling; Q" for transitions between levels LSI
In this and the following subsections we give general formulas for Q-factors in the LS-coupling. The derivation for transitions not involving the shells of equivalent electrons was given in Sect. 2.3. We consider also some most important particular cases. Q~P) means Q~ or Q~. To simplify the notation, we denote by y the whole set of quantum numbers defining the term, specifying if necessary the spin S P and orbital angular momentum L p of the atomic core, the orbital momentum of an electron I, the total spin S, and the total orbital momentum of an atom L. The unnecessary quantum numbers will be omitted in formulas. The multipole order " in general can vary between "min
= 1/0 -
Id,
"max
= 10 + II
,
and Q~ is not zero only if " = "min, "min + 2, ... ,10 + II ,
For transitions between LS.! levels Q-factor can be written as follows, compare (2.3.3-5) 2/0 + 1
Q~p)(LoSoJo, LI SIJd = 2Jo + 1 FvB~lCv(J)C(P)(q), C'(q)
= 2b(q),
C/(q)
= ~[qf ,
Bq1u(J) = BqlC(SL)MqlCv(SLJ)
(6.2.4)
(6.2.5) (6.2.6)
The factor M according to (2.2.23) is equal to
(6.2.7)
BqlC(SL) == BqiSoLo, SILd does not depend on J, but depends on the type of transition. It is discussed in the next subsection.
226
6. Tables and Formulas for the Estimation of Effective Cross Sections
The sum over fine-structure components J) is independent of J o:
Q£p)(LoSoJo. L)S)
= Q£p)(LoSo• L)S) (6.2.8)
= ~ " B2 (SL) C(p) (q) [LoSoF Lq' qK • QK averaged over Jo of the initial levels is Q£p)(LoSo. L)S)JI)
[J ]2
= [L)~)F Q£p)(LoSo• L)SI) •
(6.2.9)
i.e .• it is proportional to the upper level statistical weight.
6.2.3 LS-Coupling; Q" for transitions between Terms LS Q-factors for transition LoSo-LIS) are defined by (6.2.8). On substitution of C(p) we obtain I 2[/0]2 2 QK(LoSo• L) S) = [LoSoF BOK (SoLo. S)L) •
(6.2.10)
2[/of " 2 2 QK(LoSo. L) SI) = [LoSoF Lq' BqK (SoLo. SI LI) [q] . /I
a) Transitions not involving the shells of equivalent electrons ao = [LpSp]loLoSo. BqK(SL)
a) = [LpSp]I)L)SI.
(6.2.11)
= MOqq(SpsS)MoKK(LpIL)[SpLp] .
Therefore Q~(LoSo. L)S)
= QK(L o• L I ) • Ao. Q~(LoSo. LISI ) = QK(L o• L I ) • A2 •
Ao = !5(So. SI) • A2
= (2S1 + l)j2(2Sp + 1)
(6.2.12)
where
22
QK(Lo. L I) = [LpSp] MOKK(LpIL) = [loLo]
2{LoLl }2 II 10Lp K
(6.2.13 )
The sum over L) and average over Lo are independent on Lo:
QK(Lo. LplI)
= QK(Lplo. LplI) =
(6.2.14)
1•
QK(Lplo. LI) = [Ld 2j[Lpld 2 .
(6.2.15)
Similar sums take place for spin factors 1
2[S ]2 p
2
l: [So] Ao = ~
l:Ao = l:A2 = 1. ~
~
1
--2
2[Sp]
l: [So] 2A2 = So
A2 . (6.2.16)
6.2 Formulas Defining the Angular Factors
227
b) Transitions from the shell of equivalent electrons ao = I[J'LoSo, BqK(SL)
=
al
1[J'-I[LpSp]IILISI ,
= v'mG£o~o . MOqq(SpsS)MoKK(LplL)[SpLp] , p p
(6.2.17)
where G£Ip ~Ip is the fractional parentage coefficient (Sect. 6.2.5). Q-factors are defined by (6.2.12) with (6.2.18) We see that QK(Lo, LI) depends in fact on So through G. The sums over LI and LpSp are QK(aO,
l~-I [LpSp]ld = m (G£~~~)2
QK(l~Lo, I~-Ild
= QK(l~,
I~-Ill)
,.
(6.2.19)
= m.
(6.2.20)
c) Transitions between shells of equivalent electrons ao = 1~lrLoSo,
=
al
1~-llr+1 [LpSp]LISI
N
= 410 +2
We assume here for simplicity that l~ is the closed shell. Here, LpSp are momenta of the shell Ir+l. For such transition one can use (6.2.18) for the transition I~-m - /f-m-I/ o, M = 2(2/1 + 1). (6.2.21 ) and corresponding substitutions into (6.2.19,20). We note also that for any value of n ::; 4/0 + 2 r-llm+l) = n(M - m) QK ( Inlm 0 I' 0 1M·
(6.2.22)
We used here for the total statistical weight the expression g(lm)
=
(41 + 2)! m!(M -m)!
(6.2.23)
d) Transitions inside the shell of equivalent electrons ao = rSoLo, BqK(SL)
al
=
ImSIL I ,
= (lmSoLolwqKll/mSILd
= L: v'mG£os£ . Moqq(SpsS)MoKK(LplL)[SpLp]. SpLp
p p
(6.2.24)
228
6. Tables and Formulas for the Estimation of Effective Cross Sections
Table 6.13. Reduced matrix elements (p3LoSollU 2l1 p3LISJ)
(p2LoSolIU2l1p2 LISJ) 3p
IS IS
0 0
3p ID
ID 2p 2D
0
- 1
2/-.13
4S
2/-.13
0
..jff3
0
4S
2p
2D
0 0 0
0 0
--.13
-.13
0 0
Thus we obtain
Q"'( ao, a\ )
=
2[1]2 (l mSoLo II U "II ImS\Ld 2 [SoLoF
Q~ (ao, a\) = ~Q~ (ao, ad + [:0[~~~2(lmSoLollV\"lllmS\Ld
(6.2.25)
where
(l mSoLoIIU"1I1mS\Ld =
G£o~o G£I~I
E
p p
p P
[Lp]
fllP7P~}, OL\K
LpSp LoSo (ImSoLollvq"lllmS\L\) = "L.J GLpSp .
LpSp
x
GLlsl LpSp
[L P SP ]
{:P:P~} . f7P7P~}. SoS\q
10L\K
(6.2.26)
The reduced matrix elements (aoIIU"llad and (aollV q"lIad were defined and partly tabulated in [6.1]. More detailed tables are given in [6.3]. For K = 0
(lmSoLoIlUOlllmS\Ld = b(Lo, Ldm[Lo]/[l]
(6.2.27)
Therefore, transitions with a change of L are possible only with K ~ 2 or due to exchange. At AL = 0, AS # 0 also exchange is necessary; at AL = AS = 0 there is no transition. In the Table 6.13 reduced elements for K = 2 are given for m = 2,3; for m > M/2 one can use relation: (6.2.28)
6.2.4 j/ Coupling
Below we shall give formulas for two cases: (i) both initial and final levels are described by jl coupling; (ii) the initial term LoSo is described by LS coupling, and the final term K is described by j I coupling.
6.2 Formulas Defining the Angular Factors
229
(i) For transitions between the fine structure components
Jo = Ko ±
!,
J, = K, ±
!,
we have
Q" (KoJo, K, J, ) = (2Ko + 1)(2J,
1}2 Q" (Ko, K,) .
KoJo K, ~
+ 1) { J,
(6.2.29)
Summation over J, gives
Q,,(KoJo, K,)
= Q" (Ko, K,) .
(6.2.30)
Equation (6.2.29), being averaged over Jo, yields
2J, + 1 Q" (Ko, K,J,) = 2(2K, + 1) Q" (Ko, K,) .
(6.2.31 )
The j I-coupling scheme cannot be used for a shell with equivalent electrons. Therefore only the case of transitions which do not involve groups of equivalent electrons should be considered. For transitions between the terms Yo = [LpSpj]/oKo, y, = [LpSpj]/,K, as a whole the factor Q" is
Q,,(yo, y,) = ( 2/ 0 + 1)(2K,
/o
+ 1) { K,
KOj}2 I,
(6.2.32)
K
Summing (6.2.32) over K, gives
Q" ([LpSpj] 10Ko, [LpSpj] Id
= Qd/o,
Id
= 1,
(6.2.33)
and averaging over Ko provides
Q (I "
IK )_ 0,
,
,
-
2K, + 1 (2j+ 1)(2/, + 1)
(6.2.34)
(ii) For transitions from the level LoSoJo described by LS coupling to the level [LpSpj]l,K,J, described by the jl-coupling scheme, we have
(6.2.35)
= (2/0
+ 1)(2J, + 1)(2So + 1)(2Lo + 1)(2j + 1)(2K, + 1)
LOSPK [ 10 Joj
K,]2 !
Lp I, SoJ, The definition of the 12j symbol used here is given by (6.3.23). Averaging over J o and summing after that over J, gives for transition between the terms,
Q,,(Yoyd = ( 2/ 0 + 1) (2j
+ 1)(2K, + 1) I)2r + 1) {Lp I, r 2Sp + 1 r K Lo/o
}2 {LpK,Sp} I, ~}2
(6.2.36)
230
6. Tables and Formulas for the Estimation of Effective Cross Sections
Summing (6.2.36) over K I , we obtain Q (
KYO,
[L S .] 1 ) p p}
I
=
(2Lp
2j + 1 + 1)(2Sp + I)
.
(6.2.37)
By summing further over j, one has (6.2.38) Averaging (6.2.36) over Lo, we have _ 2KI + 1 Q ([L S ] I S, K P pO 0,Yd-(2L p +l)(2S p +I)(2/ 1 +1)
(6.2.39)
For transitions from the shell of equivalent electrons I~ -1;-1/ 1, the formulas )2. (6.2.35-38) should be multiplied by m(G£o~o p p
6.3 3nj Symbols and Fractional Parentage Coefficients Formulas for the angular factors Q~ and Q" contain 6j symbols, 9j symbols, 12j symbols, and the fractional parentage coefficients G£sp sp . The detailed description of their invariance properties, and formulas, sum rules, and numerical values can be found in [6.2-7]. Here we give only those which are necessary for this book. I) The 6j symbol {:: :~::} obeys the following symmetry relations: it remains invariant under any permutation of its columns and also on transposing the lower and upper arguments in each of any two columns. For example,
The 6j symbol is nonzero if the following triangular conditions are fulfilled: LI(al a2 a3),
LI(al b2 b3),
LI(b l a2 b3),
LI(b l b2a3) .
The triangular condition LI (a be) means that the sum of any two arguments is greater than or equal to the third argument and the modulus of the difference of any two arguments is less than or equal to the third one. The 9j symbol remains invariant under any permutation of its rows or columns and also under transposition (change of rows to columns). Triangular conditions are fulfilled for every row and column. Here we give summation formulas for j-symbols, including some formulas that are missing in most books. {abc} means ''OJ-symbol'': the set of triangle rules LI(abc) and condition that a + b + c is
6.3 3nj Symbols and Fractional Parentage Coefficients
231
integer. Sums of one j-symbol.
L [x]z {hjlx} = [jzjIl Z
(6.3.1a)
x
(6.3.1b) x
~ [x]z {j:j~~3} = (-I)Zh{hhh}
{f: {:~3} = L [x]z {f: {~{~ } = [lIl-zJ(jz, h 13x L t [x]z {f: {~{:} = hx
~ (-It [xf
(6.3.2a) (6.3.2b)
(-1 )-h -II [h IIlJ(h, 0)
(6.3.3a)
II)
x
(-1 iI2+h-/1 [jIl-zJ(jI, I z )
(-1
x
(6.3.3b)
13
Sums of two j-symbols.
(6.3.4a) (6.3.4b) (6.3.4c)
(6.3.5)
(6.3.6a)
L (-lt2 [XIXZ]Z XlX2
f{:{~{~}J{:~~{t} = (-lih+/3-/~ {{~{~{~} ~lxzk3 ~lxzk3
(6.3.6b)
j{ l{k3
Sums of three j-symbols.
{~l/zI3} {kl~zk3} = {{:{~{~}
L(-I)2x[xf {hhh} x 13 k3x Jzxkz
xJI/I
klkz k3
(6.3.7a)
232
6. Tables and Formulas for the Estimation of Effective Cross Sections
(6.3.8)
Sums of four j-symbol.
(6.3.9a)
(6.3.9b)
(6.3.10)
For what follows it is convenient to adopt the designations: {a
bk} = {a.'. b.k.} './' , a = (a2alao) i p,q, ,
pq /
(6.3.11) Then
L [k2kd 2f(abk)f(pqk) {a bk} P qI
kJk2
=
t:
[/of f(aql) f(pbl) {;
2
:i}
{a qbk}/ P
I
(6.3.12) 0
6.3 3nj Symbols and Fractional Parentage Coefficients
233
using the definition of the M by (2.2.23) this equation can be written as " Mk(a,b)Mk(P,q) {abk} k7k2 pq/ z {abk} pql ; [kzkd z/[ko] z q I} =EMq(l,a)Mp(b,l) {a qbkI} =EMI(a,q)MI(p,b) {abk POlo P 10
(6.3.13) . 0
We have also
E [kzk\ko]z f(abk)f(pqk) {a bk} {a bk} {a bk} {pqk} P q I z p q I \ U V s o u vt 0
to k\ k2
= E [lof f(aql) f(pbl) 10
P}
t s ub a qI V
{
(6.3.14)
, 0
. {a' b'k'} {a' b'k'} {a bk} {a bk} fJ(k' k)fJ(O' 2) p' q' I' z p' q' I' \ P q I \ P q I 0 ' ,
(6.3.16)
= E[lz]z f(a'q'I')f(p'b'I')f(aql)f(pbl)fJ(O', 2) 12
fJ(O', 2)
= fJ(a~b~p~q~/~,
azbzpzqzlz).
Sums including 3jm-symbols.
~(_I)}-m (~~m~') E(-I)}-m[j]Z }m
=
t
= [j]fJ(j'm', 00),
(6.3.17)
(h h (h ,h ) ) j) m\ mzm -m\ -mz-m
(-I)h-m\fJ(m\mz, m~mi), (_I)h- m\-h- m2 [j]z
m\m2
= (-I)}-mfJ(jm,j'm'),
(6.3.18)
(h hj) (h h j ,) m\ mzm
-m\ -m2- m
(6.3.19)
234
6. Tables and Formulas for the Estimation of Effective Cross Sections
(6.3.20)
In particular, if ml = m2 = m; =
E(-IY[X]2 x
= (-1 i
m~ =
0,
{x1 hh} (x0 jlh) (x0 011/2) 11/2 0 0 0
(I h h) (I 000
II h ) 000 .
(6.3.21 )
A summary of formulas for 6j symbols in which one of the arguments does not exceed unity is given in Table 6.14. II) 9j symbols and 12j symbols are defined in terms of the 6j symbols in the following way:
~;} = E( -lfX(2x + I) {afrx bC}{dbxq ef}{pxqr}, {~P qr ad x
Table 6.14. Formulas for 6} symbols s=a+b+c
{ o~
b c} = (-I),(2b+ 1)(2c+ IW I/ 2 c b
b c} { a 1/2 c - 1/2 b + 1/2
s [ (a + c - b )(a + b - c + I)] 1)2 (2b + 1)(2b + 2)2c(2c + I)
= (-I)
b c} s[(a+b+c+ 1)(b+c-a)]-1/2 { a 1/2c-I/2b-I/2 =(-1) 2b(2b+I)2c(2c+l) abc 2[a(a+I)-b(b+I)-c(c+I)] 1 c b =(-I'[2b(2b+I)(2b+2)2c(2c+I)(2c+2)p/2 {a
b
c}
1 c-I b-I
(I)S[ =
-
s(s+I)(s-2a-I)(s-2a) ]1/2 (2b-I)2b(2b+I)(2c-I)2c(2c+l)
C}_(_I,[ 2b(2b 2(s+ 1)(s-2a)(s-2b)(s-2c+ I) ]1/2 + 1)(2b + 2 )(2c - I) 2c (2c + I)
{a b 1c- 1b -
c }_(_I)S[(S-2b)(S-2b-I)(S-2C+I)(S-2C+2)]1/2 {a b 1 c- 1b+ 1 (2b + 1)(2b + 2)(2b + 3)(2c - 1)2c(2c + I)
(6.3.22)
6.3 3nj Symbols and Fractional Parentage Coefficients
[
235
a)a 2 a 3 a 4 ] b) b2 b3 b4
c) C2 C3 C4
= (-1 )bl-b2-b3+b4 L:(2x + 1) {C) C2
x} {C3 )b2x} {C) bx} {C3 bx} .
a3 a )b)
x
C4
a3 a
C2
C4
a4 a 2 3
a4a2 4
(6.3.23 ) III) Our notation for the fractional parentage coefficients, G£sp s p agrees with that of [6.2]. The Racah notation is related to this by
The values of fractional parentage coefficients for electron configurations p'" with m = 3, 4,5 are given by Tables 6.15-17. For configurations s2, p2, p6, the fractional parentage coefficients are equal to unity.
Table 6.15. Fractional parentage coefficients (p2[L pSp]pLS} p 3LS) p3 2p 4S
l
IS
Ji
0
ID
0
3 I
3p
I
-Vi
-{{s
0
Table 6.16. Fractional parentage coefficients (p3[L pSp]pLS} p4LS) p4 3p IS p3 4S
0
2p 2D
0
2D
I
-J3
Vi I -Vi
ID
0
I 2
2
H,
~
236
6. Tables and Formulas for the Estimation of Effective Cross Sections
p4 2p
p5 IS
3p ID
fFs fs If
7 Broadening of Spectral Lines
Various phenomena of spectral line broadening connected with the most interesting applicati0!ls of atomic spectroscopy to plasma diagnostics, astrophysics, laser physics, and other areas are considered in this chapter. The presentation of the general theory of impact broadening is based on the density-matrix and quantum kinetic equation methods. These methods permit not only the line shape to be described in the case of spontaneous emission or linear absorption, but also allow nonlinear effects arising in laser spectroscopy to be considered. There are many books and review articles discussing the progress in theoretical and experimental work on the problem of spectral lines broadening [7.1-16]. For an extensive bibliography on line shapes see [7.17-19]. For a brief review of recent developments in the theory with stress to applications to nonlinear laser spectroscopy see [7.20].
7.1
Model of a Classical Oscillator
7.1.1
Formulation of the Problem
The theory of spectral line broadening caused by the interaction of an atom with surrounding particles is closely connected with the general theory of atomic collisions. Moreover in the region of not very high pressure, when the impact approximation is valid, the calculation of the profile of a spectral line includes calculation of the scattering amplitudes or scattering phases. Nevertheless it is useful to begin the study of pressure effects by considering a model simplified to the maximum extent. We shall make the following assumptions: i) the relative motion of the atom and the perturbing particle is quasi-classical, which enables one to use the concept of the trajectory of the perturbing particle; ii) this trajectory is rectilinear; iii) interactions with the nearest perturbing particle (binary interactions) play the principal role in the broadening, therefore multiparticle interactions can be neglected; iv) the perturbation is adiabatic, i.e., does not induce transitions between different states of the atom. Within these assumptions, the picture of broadening is outlined as follows. The perturbing particle produces an external field V(R) = V
[J p2 + v2(t -
to)2] ,
I. I. Sobel'man et al., Excitation of Atoms and Broadening of Spectral Lines © Springer-Verlag Berlin Heidelberg 1995
(7.1.1)
7. Broadening of Spectral Lines
238
where R is the distance between the atom and perturbing particle at a given time t, p is the impact parameter, to is the time of nearest approach, and v is the relative velocity. As a result the energy levels of the atom and, consequently, the frequency of oscillations of the atomic oscillator vary in time. Therefore the oscillation of the atomic oscillator can be described in the form t
J
= exp [iwot + i
f(t)
(7.1.2)
K(t') dt'] ,
-00
where Wo is the unperturbed frequency and K(t) is the frequency shift due to the interaction. Perturbation of the monochromaticity of the oscillations leads to broadening of the corresponding spectral line. The line shape is given by the expansion of the function f(t) in a Fourier integral, 1 I(w) = lim 1 ~ T-oo v 2nT
T/2
J
f(t)exp(-iwt)dt
12
-T/2
(7.1.3)
= lim - 12 1 T/2 J exp [-i(w nT
T-oo
-T/2
wo)t + i11(t)] dt
12
,
t
11(t)
J K(t') dt'
=
(7.1.4)
,
-00
where 11(t) is the phase of the oscillation caused by interaction. If the frequency w is measured from the unperturbed frequency Wo, then the exponent exp (iwot) must be omitted. In this case, I(w)
1 1 T/2 exp [-iwt 2nT -T/2
J
= lim T-oo
+ i11(t)] dt 12
(7.1.5)
In the theory of spectral line broadening, conditions are usually considered when gas pressure and temperature, state of ionization, and so on, do not vary with time. This means that the functions ,,(t) and f(t) = exp [i11(t)] are stationary random processes, and (7.1.3) can be rewritten in the following way:
*
I(w)
=
4>(r)
= lim
(7.1.6)
Re{7 4>(r) exp (-iwr) dr} ,
T-oo
1 -T
T/2
J
f*(t)f(t
+ r)dt =
f*(t)f(t
+ r) ,
(7.1.7)
-T/2
where 4>( r) is the correlation function. Time averaging can be replaced by averaging over the statistical assembly of quantities defining the function f(t). We shall denote such averaging by angle brackets, 4>(r)
=
(f*(O)f(r») .
(7.1.8)
7.1 Model of a Classical Oscillator
= exp [i1](t)], we have (,) = exp {i[1](t + ,) -1](t)]} =
239
For f(t)
7.1.2
(exp [i1](')]) .
(7.1.9)
Impact Broadening
We shall consider in this section an approximation which is called the impact approximation. This approximation is based on the assumption that the decisive factor in the broadening of a line is the disruption of the coherence of the oscillations of an atomic oscillator during collisions. In other words if the duration of collision is small as compared with the mean time between collisions, then one can neglect radiation during collisions and consider the collisions to be instantaneous. Therefore the collisions are manifested only in phase shifts 1]. Using this assumption of instantaneous collision, it is possible to calculate the correlation function (,) in the following way [7.21]. In accordance with (7.1.9) the difference .,1 = (, + .,1,) - (,) can be written in the form .,1
=
(exp [i1](, + .,1,)]) - (exp [i1](')])
= (exp [i1](')] exp (iL11]») - (exp [i1](')]) ,
where .,11] is the phase shift produced by collisions during time interval .dr. Since collisions are instantaneous, the phase shift .,11] does not depend on 1](,). Therefore 1](,) and .,11] are statistically independent, and consequently .,1
= (exp [i1](')])[ (exp (iL11]») -
1]
= -(,)(1 -
exp (iL11]») .
(7.1.10)
We shall denote the number of collisions per second with parameters p and v as P(p, v)dpdv. The number of collisions during time interval .,1, is equal P(p,v)dpdvL1,. Therefore (l-exp(iLl1]»)
= '!9L1"
'!9
=
f[1-exp (i1])] P(p, v)dpdv ,
(7.1.11)
where 1] is the phase shift produced by collision with parameters p, v. If the density of perturbing particles is N and their distribution over velocities v is given by the distribution function ~(v), then 00
00
o
0
() = N J v~(v)dv2n J pdp[1
- exp(i1])] .
(7.1.12)
Denoting 00
(i'
= 2n J(1 o
- cos 1]) pdp,
(7.1.13)
00
(i"
= 2n
J sin 1] pdp o
,
(7.1.14)
240
7. Broadening of Spectral Lines
we have () = N(v (a' - ia")) .
(7.1.15)
From (7.1.10,11) it follows that
= -()tfJ
dtfJ dr:
(7.1.16)
'
tfJ = exp (-()r:) .
(7.1.17)
By substituting (7.1.17) into (7.1.6), we obtain
y I(w)
= 2n . (w _
y = 2N (va');
1 LI)2 + (y/2)2 '
LI = N (va") .
(7.1.18) (7.1.19)
The spectral distribution given by (7.1.18) is usually called the Lorentzian distribution. The width of the distribution (the distance between symmetrical points WI and W2, for which I(wJ) = I(w2) = Imax/2) is y. The shift of the line peak from Wo is LI. The quantities a' and a" are called the width and shift effective cross sections. Let us assume that the perturbing particle at a distance R produces the frequency shift K = CnR-n. Then (7.1.20) (7.1.21)
r(~) rtn =
r
Vi
G) .
(7.1.22)
For n = 2, 3, 4, 5, 6, we have rtn = n, 2, n/2, 4/3, 3n/8. Substituting (7.1.21) in (7.1.13,14), it is not difficult to obtain the following formulas for y and L1 which we shall use below in estimations of the width and shift:
LI _
v'3
-TY'
n
=6
LI
~
Y ~ 8.16 C;/5 (V 3/5)N ,
0.36 y .
(7.1.23)
7.1 Model of a Classical Oscillator
241
It is not difficult to show that the main contribution in (1' is given by the strong collisions for which '1 ;::: 1 and P < Po, where Po is defined by condition
'1(Po)
=
1:
cx.nCn) I/(n-l) Po= ( - .
(7.1.24)
v
The impact parameter Po is usually called the Weisskopf radius. Therefore to an order to magnitude, (1'
~ np~
(7.1.25)
The shift cross section (1", see (7.1.14), is determined by more distant collisions P ;::: Po. In the case of n = 2, the phase '1(p) ex: P-I. Thus (1' diverges as In Pm and (1" diverges as Pm, where Pm is the upper limit of integration in (7.1.l3) and (7.1.14). The divergence of the integrals (7.1.13) and (7.1.14) means that the approximation of binary collision is not valid. It is evident that in this case broadening is determined by distant (weak) collisions with P > Po. 7.1.3 Quasi-Static Broadening If the external field varies sufficiently slowly, i.e., if it is quasi-static, it is pos-
sible to assume that I( w )dw is simply proportional to the statistical weight of the configuration of perturbing particles for which the frequency of the atomic oscillator is included in the interval w, w + dw. In the binary approximation the frequency shift is produced by the nearest particle. Consequently, to calculate I( w), it is necessary to find the probability W(R)dR of the nearest particle being within the range of distance (R,R + dR) from the atom. For R much larger than the atomic dimensions the interaction potential could be neglected and this probability is W(R)dR
= 4nR2N exp ( -
~n NR 3) dR = exp [- (:0
y] (:J 3, d
(7.1.26) where Ro = (3/4nN)1/3. Substituting R = (Cn/K)lln = [Cn/(w - wo)]lln in (7.1.26), we obtain the probability distribution for a frequency shift of an atomic oscillator. In accordance with the basic assumption of the quasi-static approximation, the shape of the spectral line is also determined by this distribution. If the notation .dw = CnR{jn is introduced it follows from (7.1.26) that 4n I(w)dw=-NC~/n(w_wo)-(3+n)/nexp n
/ [ - ( -.dw - - ) 3 n] dw.
(7.1.27)
w-~
This distribution is valid only for sufficiently large values of w - Wo for which
242
7. Broadening of Spectral Lines
R = C~/n(w - wo)-I/n ~Ro For R :oo nT k I
(7.1.33)
= lim.!.. L T--->oo
T
k
(dK)-1 . dt Ik
It is easy to see that Lk(dK/dt);/dw is the time during which K(t) is included in the interval w - Wo, w - Wo + dw. Since d!k and dw in Fig. 7.1 are connected by the relation (dK/dt)lkd!k = dw, (7.1.33) gives the quasi-static intensity distribution W(w - wo)dw. We shall replace the summation in (7.1.33) by integration. The number of particles incident on the annular element 2np dp in the time T is 2npdp NvT, where N is the density of perturbing particles. Taking into account that each collision with p :::: PAw = (Cn/AW)I/n, Kmax = Cnp-n~Aw gives two points tk and tk+l (Fig. 7.1), we obtain
/(w)dw
= dw
P-1w
{4np
(dK)-1 dt Nvdp
4n
dw
= -;;Nc~/n Aw 1+ 3/n
'
(7.1.34)
i.e., the quasi-statical distribution in the wing of the line.
/(
w~~--~+-~~--------------
Fig. 7.1 Instantaneous frequency shift K(t)
1 It is assumed that the phases ak will be discussed below.
=
['1(tk) + (w - Wo )ttJ are independent. This assumption
244
7. Broadening of Spectral Lines
If a small neighbourhood around the instant of closest approach is not considered, then
dJ( C"v dt ~ p,,+l'
d 2 J( c"v2 dt2 ~ pn+2 '
(7.1.35)
and relation (7.1.32) takes the form
C" vp"-
(7.1.36)
--I~I.
Only collisions with p ~ P,1QJ = (C"/Aw)l/,, give points (7.1.36) can be rewritten in another form
tk
and
tk+1.
Therefore
v"/(,,-l)
Aw~ C~/("-l) = D .
(7.1.37)
According to condition (7.1.37) the quasi-static distribution is valid for large Aw, i.e., in the wing of a line. We shall now consider (7.1.3) in the limiting case of small Aw. If Aw is so small that l/Aw is much greater than the duration of the collision -
I
Aw
~
p v
- ,
(7.1.38)
the change of phase in the collision can be considered to be instantaneous. Hence it follows that the impact approximation can be used. The main contribution in the impact broadening of a line is given by collisions with p '" Po = (a" C,,/V)l/(,,-l). Substituting Po in (7.1.38), we obtain a relation opposite to (7.1.37): v"/(,,-l)
Aw~ C~/("-l) = D . Thus in the center of a line, Aw~D, the impact (Lorentzian) distribution of intensity is valid. For high values of Aw, Aw ~ D, the impact distribution is replaced by the quasi-static one. The quasi-static wing can appear both on the long-wave and on the short-wave side depending on the direction of shift of the terms. If D considerably exceeds the impact width y, then the greater part of the integral intensity of a line is concentrated in the impact region. Taking into account that
y ~ 21tpt,Nv = 21CNv(aC,,/vi/(,,-I) , we obtain
21tpt,Nv~D = .!:. ; Po
whence (7.1.39) where the dimensionless parameter h determines the number of perturbers in the sphere of the Weisskopf radius. Thus, for low pressures and high velocities, so
7.1 Model of a Classical Oscillator
245
long as the inequality (7.1.39) is fulfilled, the impact mechanism of broadening plays a decisive role. A relatively negligible part of the total intensity is concentrated in the quasi-static wing. At high pressures and low velocities, when h
= p~N
'" I ,
(7.1.40)
the impact approximation is inapplicable even to the inner part of a line. Let us note that if condition (7.1.39) is not fulfilled then the binary approximation is violated. In fact relation (7.1.40) means that the effective radius Po is approximately equal to the mean distance between perturbers. Although when p~ ~ I the quasi-static distribution is applicable practically to the whole profile of a line, the expressions (7.1.27,28) obtained above in the approximation of binary interactions are valid only in the wing of a line. The assumption of the independence of the phases (1.k was made above in the derivation of the formula (7.1.33). Since only strong collisions for which '1 ~ C,.lp" • plv'$> I are responsible for the quasi-static wing, the difference (1.k+l(1.k'$> I. In a nonpublished work [7.4] Anderson and Talman investigated in detail the limiting expressions for /(00) valid for the central part of the line and for the wings, and obtained also an interpolation expression for the intermediate part. The same problem is discussed also in [7.22,23]. 7.1.5
Doppler Effect
The frequency observation is amount 000 vie. defined by the
of an oscillator whose velocity component in the direction of v is displaced in accordance with the Doppler principle by an Let the distribution of the radiating atoms with respect to v be function W(v). Then 00 = 000 + ooovle, v = e(oo - (00)/000, and
00-(00) e /(oo)dw = W ( e - - - doo . 000
.
(7.1.41)
Wo
With a Maxwellian distribution (7.1.42)
where Vo = J2k Tim, we obtain I
/(00 )
[ ( 00A-ooD000 ) 2]
doo = yin exp -
doo AWn'
AooD
= 000 -Voe
•
(7.1.43)
The intensity distribution (7.1.43) is symmetrical. The magnitude of the broadening is defined by the parameter AWn. The width of the line, which we shall denote by ~, and the peak density /(wo) are expressed in terms of the parameter
246
7. Broadening of Spectral Lines
Amo:
() = 2Jln 2 Amo l(wo) = 1/v'n Amo
(7.1.44) (7.1.45)
Here () is defined as a difference between the symmetrical frequencies WI and W2 for which lewd = l(w2) = l(wo)/2. The parameter Amo is usually called the Doppler width of a line. When deriving (7.1.41,43), it is assumed that there is only one frequency wo(1 + vic) in the spectrum of the oscillator with velocity v. This assumption is valid if v does not vary in time or remains a constant quantity during a sufficiently long time. If velocity is constant only during time interval 't, then this interval contributes to the intensity of radiation in a spectral interval with width l/'t around the frequency Wo + Wovle. Formula (7.1.41) is valid if wovle~ 1/'t. Substituting for 't the free path time 'to = Llv, where L is the mean free path, we have (7.1.46) In the general case, the Doppler broadening is determined by Fourier transform of the function f(t)
WO] = exp [i-x(t) e
,
x(t)
=
Jt
(7.1.47)
v(t')dt'
-00
Substituting (7.1.47) in (7.1.6,7), we have l(w)
= ~ Re
{?
4>('t) exp (-iw't) d't}, 4>('t)
= (exp [i
:0
x('t)]). (7.1.48)
The function ('t) = (exp[ik· r('t)]), r('t) = Jv(t)dt ,
(7.1.49)
o and introduce the distribution function f(r, v,t) for the oscillator coordinate r and velocity v. This distribution function satisfies the Boltzmann equation
of -+v·Vf=
ot
(Of) -
ot
coil.
- Gf ( Of) ot coil. -
,
(7.1.50)
and the initial condition f(r,v,O) = W(v) {)(r) .
(7.1.51)
Here (oflot)coll. is the collisional integral or collisional term, G is the linear operator of collisions, and W(v) is the distribution function for v. The correlation
7.1 Model of a Classical Oscillator
247
function O. We find J(w)
.
e
W(v)dv
= hmJ -n ( w- kv)2 +e2 =
J J(w - kv) W(v)dv
(7.1.57) We shall consider now the influence of collisions assuming the model of Brownian motion [7.25J. This model can be used in the case of so-called weak collisions. In the framework of the model of Brownian motion, the collisional term in (7.1.50) has the form
a
( Of) t colI.
. = v dlV.(vf) + 2v~ v L1./ .
(7.1.58)
The effective frequency of collision v is assumed not to depend on velocity and v~ = 2kT/m. Solving (7.1.50) with the collisional term (7.1.58) and taking into
248
7. Broadening of Spectral Lines
account the initial condition (7.1.51), it is possible to obtain tP(t) = exp [/(w) =
~1 (vt - 1 + e-
vt )]
,
..1eoo = kvo ,
(7.1.59)
.! Re {A Wo2 2v 2.IVW tP (1,1 + ..12WV1' - i~;V ..121)} v
,
(7.1.60)
1[
where tP( a, 1; z) is a confluent hypergeometric function az a( a + 1) z2 a( a + 1)( a + 2) z3 tP(a,1;z) = 1 + 1I! + 1(1+ 1)2! + 1(1+ 1)(1+ 2 )3! + ....
For v = 0, (7.1.60) gives the usual Doppler distribution. When v v ~ AWn, for the central part of a profile W = 0 and for the far wing we have, respectively, 1
/(0) = Ji..1eoo
(
2
v)
1 + 3Ji ..1wo
'
v
1 ..1w~ /(w) ~ Ji..1eoo 2Jiw4 .
(7.1.61 )
#
0, but
W ~ ..1wo,
(7.1.62)
Thus due to collisions with v ~ ..1wo the intensity in the central part of a profile is increased and the wing with intensity distribution ()( w- 4 appears. In the limiting case of high densities when v ~ ..1eoo, 1 /(w) ~ 1[
2 W
Yd
•.2
+ Yd
Vd
1 ..1w~ = ---. 2 v
(7.1.63)
The central part of a line is described by the Lorentzian distribution (7.1.64)
with width 2Yd = ..1~/v. Since v ~ vo/L, 2Yd = ..1eoo 21tL/). i.e., the width decreases ()( L with increase of density. This result was first obtained by Dicke [7.26]. For w~v, (7.1.63) coincides with the intensity distribution in the wing from (7.1.62). The qualitative picture of modification of the Doppler distribution due to collisions does not depend on the specific model of Brownian motion used above. We shall consider now the model of strong collisions assuming that after every collision the distribution of velocities does not depend on the velocity before collision and is Maxwellian. In this case, the collisional term in (7.1.50) can be written in the form
( f)f)If) t
=-vf+vW(v)jf(r,v',t)dv'.
(7.1.65)
coil.
For (Gf)k in (7.1.56), we have
(Gf)k = -vF(v,k) + vW(v)j F(v',k)dv' .
(7.1.66)
7.1 Model of a Classical Oscillator
249
Substituting (7.1.66) in (7.1.56) we obtain
F(v,k) =
vW(v) " .( k )JF(v,k)dv V+IW- ·V
W(v)
+ V+IW.( k·V)
(7.1.67)
Integrating the right-hand side and left-hand side of this equation over v we have
J F(v,k)dv
=J
W.~V~dV) (v J F(v',k)dv' + 1) ·V
V+IW
and after replacing in the right-hand side v' by v
W(V)dV) W(v)dv JF(v,k)dv ( I-vJ V+IW-'V .( k ) =J V+IW-'V .( k ) It is possible now to find F(v,k). Then using (7.1.54) we obtain
_ { I-v. kJ v+:~v2~. v) W(v)dv
1(w)-Re
J v + i(W -
}
(7.1.68)
.
k . v)
When V~LlWD, the second term in the denominator of (7.1.68) has the order of magnitude of V/LlWD. If this term is neglected, (7.1.68) gives the usual Doppler distribution. In the general case, the intensity distribution (7.1.68) is similar to that given by (7.1.60). Instead of (7.1.62), it follows from (7.1.68) for 1(0) and I ( n-2 v ) 1(0) ~ y'nLlWD 1+ y'n LlWD '
1(w)
1 vLlWb ~ --=:--- y'nLlWD 2y'nw4
(7.1.69)
In the limiting case v ~ LI mo, (7.1. 68) leads to a Lorentzian distribution in the central part of a line with width LlwMv. In the region of high frequencies W > v, the Lorentzian distribution is replaced by a wing 1(w) <X w- 4 • Thus in both cases considered above (weak and strong collisions), with an increase of the density a narrowing of the central part of the line occurs. At high densities, when L < A/2n, the narrowing of a line is proportional to L <X N- 1, and in the limiting case of L ~ A/2n, the central part of a line is described by a Lorentzian distribution with width 2nL/A times less than the usual Doppler broadening LlWD. Such a narrowing of the Doppler profile caused by collisions can be observed only in cases when there is no broadening due to interaction with perturbers. In the general case there are no grounds for separating the effect of interaction and the Doppler effect. The same collisions can produce both a phase shift and a change of velocity of the atom. This means the statistical dependence of both effects. It must be noted that Doppler broadening is usually of interest just under the condition L > Aj2n. In fact, the condition LlWD ;;:: y, where y is the impact width can be rewritten in the form 2nv/A ;;:: Nva' = Nvaa'/a = va'/La, where a is the gas-kinetic effective cross section of the atom. As a rule, a' ;;:: a and, consequently, Limo ;;:: y when L > A/2n.
250
7. Broadening of Spectral Lines
Nevertheless the statistical dependence of Doppler and impact broadening in some cases must be taken into account. This problem will be considered below in the framework of the quantum theory of broadening. A bibliography on Dicke narrowing may be found in [7.27].
7.1.6 Convolution of the Doppler and Lorentzian Distributions If L ~ A/2n, the combined treatment of impact and Doppler broadenings (statistically independent) leads to the convolution of Doppler and Lorentzian distributions. The Lorentzian intensity distribution with width y and shift L1, corresponding to the atom with velocity component v in the direction of observation, is given by
y I ------::----=2n (w - L1 - wov/c)2 + (y/2)2
(7.1.70)
/v(w) = -
To obtain the intensity distribution for an assembly of atoms, it is necessary to average (7.1.70) over the velocity distribution W(v). Thus /(w)
=l
2n
J
W(V)d~
(w - L1 - wov/c)
+ (y/2)
2 •
(7.1.71)
For a Maxwellian distribution
/(w) = l _ l _ exp [-(v/voil dv 2n y'nvo J (w - L1 - wov/c)2 + (y/2)2
(7.1.72)
When L1wo~y/2 the term wov/c can be neglected in the denominator in (7.1.72), after which the integration over v gives a Lorentzian distribution with width y. Consequently, when L1wo ~y/2 Doppler broadening can be neglected. When L1roo~y/2 a significant contribution to the integral (7.1.72) can be given by two ranges of values of v: v '" 0 and v'" c(w - L1)/wo. In the first of those ranges, the term Wov/c in the denominator can be neglected and in the second v can be replaced in the numerator by c( w - L1 )/wo. After this it is easy to obtain two approximate expressions for / ( w) valid for the center of a line w - L1 ~.oo and for the wing w - L1 ~ .00 , where .00 is determined by the relation
Dt, =
L1w~ In [2n3/2 L1~ (L1'!: Yl
.
(7.1. 73)
In the center of a line w - L1 ~ .00 , / (w) coincides with the usual Doppler distribution. In the wing of a line, /(w) ex y/2nw2. Thus for any relation between L1wo and y/2 at sufficiently high values of w, the Doppler distribution is replaced by the Lorentzian wing. We shall write (7.1.72) in the form I o Re {(W-L1) /(w) = y'n L1w W L1roo '
(7.1.74)
7.2 General Theory of Impact Broadening i 00 exp (-t 2)dt 1 00 W(x,y)=-! . = r.;!exp[-z2+i(x+iy)z]dz, 1t -00 x + ly - t v 1t 0
251
(7.1.75)
where x = (w - A)/Awo, y = y/2Awo, t = (v/c)(wo/Awo). The function W(x,y) can be expressed in terms of the probability integral with complex argument!
1.
2 i(x+iy) W(x,y) = exp[-(x+iyi] [ 1- y'n { exp (_t 2) dt
(7.1.76)
The intensity distribution I(w) for any relation between parameters y and Awo can be calculated using (7.1.76).
7.2 General Theory of Impact Broadening 7.2.1
Density Matrix Method in the Quasi-Classical Approximation
In the quasi-classical approximation the interaction of the atom with the surrounding particles can be described by the time-dependent perturbation V(t). In this case the coordinates of the perturbing particles can be considered not as dynamic variables but as assigned functions of time, which enables one to introduce the perturbation V(t) instead of the perturbation V(R). It will be shown in this section how the shape of a line is calculated when an atom undergoes an arbitrary perturbation V(t). From the theory of the interaction of a quantum system with electromagnetic radiation we know that for dipole transition 0( --+ p [7.2] J(w) ex
I! pocp(i) exp (-iwt) dtl 2 ,
(7.2.1)
where P ocp(t) is the matrix element of the dipole moment of an atom calculated by means of the perturbed wave functions tpoc(t) and tp p( t). These functions are the solutions of the Schrodinger equation for the Hamiltonian H
= Ho + V(t)
(7.2.2)
.
Formula (7.2.1) is the natural generalization of the classical formula (7.1.3).·It is helpful to write this formula in a form similar to (7.1.6), I(w)
=
~ Re
{1
(-r) = E(Pa.p(-r)P:p(O») . a.p
Equations (7.2.6,7) are easily generalized to the case when a line is formed by a set of transitions between two groups of closely spaced levels. We shall indicate by the indices 0( states belonging to initial levels and by the indices p, those belonging to final levels, and we shall denote by Wa. the population of the state 0(, Ea. Wa. = 1. Then IJ>(-r)
= E Wa.Pa.p(t + -r)Ppa.(t) = E Wa. (Pa.p(-r) Ppa.(O»). a.p
a.p
(7.2.8)
The perturbed functions 'l'a.< t) and 'l' p(t) can be expanded in terms of timeindependent functions of the isolated atom 'l'a.(t)
= ~aa.la.(t)'l'a./exp( -*Ea.d) ,
'l'p(t) = f,aplp(t)'l'p/exp( -*Eplt) ,
where p~~~? = (a:/a.ap'p) is the density matrix of an atom, the matrix elements Pa.lpl and PPa. do not depend on t. The upper indices (O(P) define the initial conditions p~~~?(O) = ba.a.lbpp" The evolution of the density matrix with time is given by the following equation dp i dt = ",(Hp - pH),
H = Ho
+ V(t) .
(7.2.9)
7.2 General Theory of Impact Broadening
253
For correlation function kov):
f
I ~ k 2 v2 -:-:-_ _ _=_=_ 411: i(w-kov)+Nvii iw+Nvii - -3-(iw+Nvii)3 .
dO"
For frequencies w4.Nvii (7.2.59) gives
II}
/(W)=.!.Re{fW(V)dV 11: iw + _k2V2(Nvii)-1 3
We shall introduce the notation D = v/3Nii. Then
/()
w =
2
f
-
()d k Re{D} W
v
V
11:
1
0
(w + k2Im{D})2
+ (k2Re{D})2
(7.2.62)
If 1m = In, and ii coincides with the elastic cross section (1, then Im{D} = 0 and Re{D} = D is the diffusion coefficient depending on v instead of (v). In the general case of complex ii and D, the real part of D determines the width and the imaginary part of D determines the shift. The width 2k2Re{D} is proportional to N- 1. Thus, at Nvii 't> k v, the Doppler distribution narrows due to collisions. The 0
7.2 General Theory of Impact Broadening
267
resulting line profile has the form of a superposition of Lorentzian distributions with widths 2k 2 Re{D} and shifts k2 Im{D}. This profile is asymmetric. If (J' and (J" are less than Ii but are not equal to zero, then on increasing N, there is first a narrowing of the Doppler contour to a width Nva', and then a broadening. The results of the calculation of the spectrum J (OJ) by means of the quantum kinetic equation contain a number of new elements, the most interesting of which are the following. Even in the treatment of the simplest example, purely impact broadening, qualitative differences from the formulas usually used arise. Only in the case of broadening by light particles, such as electrons, does a single Lorentzian contour arise, with width 2N(vp(J') and shift N(vp(J"), where vp is the velocity of the electrons and the angle brackets denote averaging over vp. In the general case of mp m, after averaging over the velocities of the perturbing particles vp, the cross sections (J' (v) and (J" (v) contained in the collisional term retain their dependence on the velocity v of the atom. As a result, the following intensity distribution arises: r-..J
r-..J
J(OJ) = /w(v)dvNv(J'(v) 1C
I [OJ - NV(J"(v)]2
+ [Nv(J'(v)F
.
(7.2.63)
This distribution is asymmetric. The greatest difference arises in the case of scattering of a light atom by heavy (almost at rest) perturbing particles. The perturbations due to different perturbing particles combine in completely different ways, depending on the masses of these particles. If the perturbing particles of type 1 and type 2 are light, then the sum of the corresponding widths and shifts arises:
~
- iii
= NI (VI (JD + N2 (V2(J~) + i NI (VI (J~') + i N2 (V2(J~) .
But if the perturbation is created by heavy particles (type 1) and electrons (type 2), then (at Ii = 0)
w(v)dv
}
All this is a reflection of the statistical dependence of the Doppler and impact broadenings. The second characteristic feature is the fact that the cross section Ii responsible for the collisional compensation of the Doppler broadening is complex. Let us recall that Ii =f:. 0 if both the scattering amplitudes 1m and In are nonzero. It is not difficult to show that in the examples treated, asymmetry arises for two reasons - the dependence on the atomic velocity of the parameters of the equation, and the fact that the cross section Ii is complex.
268
7. Broadening of Spectral Lines
In calculation of the width of the resulting spectrum, all characteristic features of collisional broadening connected with the effect of the statistical dependence of the Doppler and impact broadenings are usually not very important. Nevertheless they can be of interest for some other problems, for example, those arising in the theory of nonlinear resonances in the spectra of gas lasers [7.38,39] (see also the bibliography given in [7.20]. 7.2.5
Absorption Spectrum
The energy absorbed in one second by a system of electric charges interacting with an electric field tI(R, t)
= ~{tlo exp [i(wt -
k·R) +
tlo exp [-i(wt -
k·R)]}
is (in the electric dipole approximation)
Q = -tl.J = Re {iwtlo ·dro exp(ik.R)} , where d = dro(R) exp (iwt)
+ d~(R) exp (-iwt)
is the electric dipole moment induced by the field tI(R,t). The quantum generalization of this expression for Q has the form
Q = Re [iw
tlo Trace {d exp (ik.R) p(w)}] .
(7.2.65)
Here d is the electric dipole operator, R is the coordinate of the center of mass of the atom, k is the wave vector of the photon, and p( w) is the Fourier component of the density matrix, satisfying the equation dp i "i dt - h (Hop - pHo) - Gp = h (tI·dp - ptl·d).
(7.2.66)
This equation contains an additional term describing the interaction of the atom with the electric field. Solving this equation by the method of successive approximation, it is possible by means of (7.2.65) to determine the absorption (or emission) power Q. If a set of transitions IX --+ P are perturbed by the field tI (the frequencies wa.p ~ w), then using (7.2.65) we obtain
Q = Re {iw tloEda.pJ dp PPp,a.p+k(W)} , a.p
(7.2.67)
where p is the momentum of the atom. We shall neglect the collisional term simplicity and solve (7.2.66) in the linear approximation according to the field. We shall substitute in the right-hand side of (7.2.66) the zeroth-order density matrix, which is diagonal in indices IX, p and
7.2 General Theory of Impact Broadening
269
p,p':
p~ipp = NpW(p),
P~o;+k,rxP+k = NrxW(p + k) ~ NrxW(p) ,
where Np and Nrx are respectively the populations of the states p and IX, and W(p) is the distribution function for momentum p. Since k~p, we can assume that W(p + k) ~ W(p). In this approximation we have [cf. (7.1.57)]
i 1 Ppp,rxp+k = 2f/o·dpiNp-Nrx)W(P)i(w_wo+hp.k/m) '
1(w)
= !Re {lim J. n
6-+0
W(p)dp } hp·k/m) + e
I(W - Wo -
Therefore the absorbed power Q is proportional to the difference of the populations (Np - N r,.) and the function 1(w), which describes the usual Doppler distribution in the spectrum of spontaneous emission. In the general case, when (7.2.66) contains the collisional term, the expression for Q remains the same, but the function 1(w) has a more complex form describing the Doppler and impact broadenings. All results obtained above for the spectrum of spontaneous emission can be obtained also from (7.2.65, 66). By solving (7.2.65) in the next approximations according to the field, it is possible to calculate the power of nonlinear absorption [7.38]. 7.2.6
Interference Effects: Narrowing of Spectral Lines
In cases when the frequencies of some atomic transitions coincide or are so closely spaced that the corresponding spectral lines overlap, specific interference effects can arise [7.40]. In some particular cases, these interference effects are so important that they alter the entire picture of the broadening. We shall illustrate this by considering as an example the four-level system shown in Fig. 7.2. We shall assume that the transition frequencies Wkt = WI and Wmn = W2 are almost
~---I.--n
Fig. 7.2. Levels scheme and the radiative transitions which are considered
270
7. Broadening of Spectral Lines
the same, co, ~ CO2, but that all other transition frequencies differ very much from CO, and CO2. Therefore in (7.2.16) and (7.2.15), the indices a.,a.',a." stand for k,m and the indices p,P',P" stand for I, n. We shall denote the pair of indices k,1 by 1 and the pair of indices m, n by 2. We shall use the notation Wk = W, and Wm = W2. By solving the system of equations (7.2.15), we obtain J(co) = !Re {W,IP,1 2G22
+ W21P212G l1 -
W2 P ,Pi G'2 - W,pjP2G2'}, G" G22 - G2' G'2 (7.2.68)
11:
where Gl1 = i( co - cod + J (1 - SaSIl) P( v) dv , G22 = i(co - CO2) + J(1 - S;mSnn)P(v)dv, G'2
(7.2.69)
= - JS;kSnlP(v)dv ,
G2' = - JS;"'SlnP(v)dv. We shall assume that S matrix elements obey the relations
= SIl, Smm = Snn, Skm = Sin, Ski = Smn = Skn = Sml = 0 .
Sa
Smk
= Snl ,
(7.2.70)
This means that collisions produce only the mutual perturbation of the states I and n and also the states k and m. It will be shown that such a situation can arise in a number of systems. Using (7.2.70) and also the unitarity properties of the S matrix, L:ISab12
=
1,
b
we have S;kSnl = ISmk 12 = 1 - ISmm 12 , S;"'Sln
= ISkm 12 = 1 -
(7.2.71 )
ISa 12 •
From the general definition of the inelastic cross section u, it follows that (7.2.72) where N is the density of the perturbing particles, v is the relative velocity, and the angle brackets denote averaging over velocities. In Boltzmann equilibrium the level populations W" W2 and transition frequencies are connected by the relations N(vUkm)
=
i,
N(vumk)
Thus G" = i(co - co,) + y1/2, G'2
= -(WI/W2)yI/2,
=
i,
W,y,
= W2Y2
.
G22 = i(co - CO2) + (WI/W2)yI/2, G2'
= -y1/2 .
(7.2.73)
(7.2.74)
7.2 General Theory of Impact Broadening
271
We shall introduce the notation COl
= COo -
15,
CO2
= COo + 15,
215
= L1 .
(7.2.75)
After this (7.2.68) gives
I(co)
=~
. (WIYI + W2Y2)I(co - t5)PI - (co + t5)P21 2 2n [(co + I5)(co - 15)]2 + [YI(CO - 15) + Y2(CO + 15)]2
(7.2.76)
We shall consider now the limiting cases of small and high values of y. In this first case, when YI,2/t5~ 1, the second term in the denominator of (7.2.76) is small, and the function I(co) has two sharp maxima at co = ±t5. Equation (7.2.76) can be rewritten in the form
I(co)
~
.!. ( n
WIy!lP I 12 (co+t5)2+YI
+
W2Y2!P2 12
(CO-t5)2+y~
) .
(7.2.77)
In the other limiting case, when YI, 2/15 ~ 1, (7.2.76) has one sharp maximum at
co ~ COM = YI - Y2 t5 . YI +Y2 In the vicinity of this frequency, (7.2.76) gives
.!.
I(co) '" (WIYI + W2Y2)IY2 P I + YI P21 2 r 2YIY2(YI + Y2) n (co - COM)2 + r 2 '
(7.2.78)
where
r =
4y IY2 152 (YI + Y2)3 .
Equation (7.2.78) describes a Lorentzian distribution with width rex N- I . Therefore at low pressures, when Y\,2/t5 ~ 1, the two components of a line are independent and their widths YI and Y2 are proportional to N. On further increase of N when the components of the line begin to overlap, the picture of the broadening alters completely, and in the limiting case of YI, 2/15 ~ 1 a single Lorentzian distribution with width proportional to N- I arises. In the far wing of a line co~Y\' Y2, 15, in accordance with (7.2.76), the spectrum I(co) has the form
I(co)
~
WIYI ~ W2Y21w(P I -P2) :/(PI +P2)i2
(7.2.79)
Note that in the particular case PI = P2, I(w) ex co- 4 . All qualitative features of the example considered above are connected with the conditions (7.2.70) for the S matrix. There are a number of systems for which the S matrix obeys such conditions. We shall consider two subsystems I and II with the levels A, B and a, b, respectively, shown in Fig. 7.3a. Let us assume that subsystem I is perturbed by collisions, but they do not act on subsystem II.
272
7. Broadening of Spectral Lines 4 (8b) J(Ab)
8--
2 (8a)
A--
f(Aa)
b
a
Fig. 7.3. (a) Levels scheme of subsystems I and II (b) Levels scheme of the whole system.
The system as a whole has four levels: I(Aa), 2(Ba), 3(Ab), and 4(Bb), shown in Fig. 7.3. Two transitions 3 --+ 1 and 4 --+ 2 correspond to the tr~sition a --+ b of subsystem II. If the interaction between subsystems I and II is not great, the interaction produces a splitting of the frequencies W31 - W42 = A I- 0, but does not influence the probability of the collisional transition A-B in subsystem I. In this case the S matrix has the form SII [ S21 S31
SI2
SI3
S14]
S22 S32
S23 S33
S24 S34
S41
S42
S43
S44
_ -
SAB SBB
00 0] 0
0
0
SM
SAB
0
0
SBA
SBB
[SAA SBA
'
(7.2.80)
i.e., is in full agreement with the conditions (7.2.70). Thus the example considered above describes the rather typical situation when the relaxation processes in subsystem I perturb the spectrum of subsystem II. We assume of course that subsystems I and II interact. If this interaction, and consequently also the frequency splitting A, is great enough, i.e., YIA ~ 1, then relaxation processes in subsystem I produce broadening of the line, corresponding to the radiative transition is subsystem II. In the other limiting case of weak interaction (y I A --+ 00), the spectrum of subsystem II is not sensitive to the relaxation processes in subsystem I. For a given value of A, the inequalities YIA ~ I and YI A ~ 1 correspond respectively to low and high densities N. Therefore, on increase of N, there is first a broadening of the spectrum of subsystem II to a width rv y, and then a narrowing proportional to N- I . The spectrum narrowing with increase of N is due to the interference of the amplitudes of the radiative transitions when the corresponding line components begin to overlap. Let us consider for example the splitting of the term 2P in a strong magnetic field H [Ref. 7.30, Sect. 8.2]: (7.2.81 )
7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma
273
where J4J is the Bohr magneton, A is the fine structure constant, and ML,Ms are the magnetic quantum numbers of the orbital and spin angular momenta. We shall which gives three consider the magnetic dipole transition Ms = --+ M~ = components of the line corresponding to the three possible values of ML = 0, ± 1. The frequency splitting of these components is of the order of A. The cross section of the spin reorientation (is usually is less than the cross section of the orbital angular momentum reorientation (iL. Therefore one can assume '
!
-!,
If Y ~A, collisional reorientation of the orbital angular momentum produces broad(ML = 0, ± I). If density N is so ening of the transitions Ms = --+ M~ = high that y > A, then the relaxation transitions ML --+ M{ begins to be ineffective in broadening the spectrum of the transition Ms = M~ = Moreover, on increase of N, a narrowing of the spectra must be observed. In the limiting case of N(V(iL) ~A, the width of the spectrum is less than the initial splitting A. It must be noted that the condition N(V(iL) ~A can be fulfilled only for light atoms for which the fine splitting is not too great. For example, the fine splitting of the ground level of Li atom is 0.34 cm -I. As will be shown in Sect. 7.3 the interference narrowing of a line can take place in the spectra of highly excited hydrogen atoms.
!
-!
!-
-!.
7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma Preliminary Estimates
7.3.1
The main contribution to the broadening of lines of the hydrogen spectrum in a plasma is due to the linear Stark effect in the fields of electrons and ions. The perturbing particle with charge Ze produces the electric field S = ZeR - 2 • Using the well-known formula for the linear Stark effect [7.30] Am = 3/2n (nl-n2)ea oS/Ii, where n,nl,n2 are the principal and parabolic quantum numbers, we can assume Am = C2/R2. The constant C2 for the level with principal quantum number n has the order of magnitude Zn(n -1 )e2ao/1i ~ Zn(n -1) [cm2 s- I ]. We shall estimate the magnitude of the dimensionless parameters (7.1.39) h
rv
e
2 )3
N. (n(n - l)e ao e
liVe
k '
rv I
N (Zn(n - l)e2ao I IiVi
)3
(7.3.1 )
where Ve and Vi are the velocities of electrons and ions, respectively. The range of temperatures and densities for which hi ~ 1 and he ~ 1 is usually of greatest interest. This means that the field of the ions is quasi-static and the electrons cause impact broadening.
274
7. Broadening of Spectral Lines
7.3.2 Ion Broadening: Holtsmark Theory For hi ~ I, number of ions in the sphere of Weisskopf radius is large and the binary approximation is inapplicable. Thus the main problem which arises in considering ion broadening is to find the quasi-static intensity distribution taking into account the simultaneous action on the atom of a large number of ions. We shall consider the component ex-p of a line and denote the shift of this component in the field C by [Ref. 7.30, Sect. 7.2] (7.3.2) where ex and p are the set of parabolic quantum numbers nln2m and intensity distribution at a given field C is given by
n~n~m'.
The
(7.3.3 ) Averaging this expression by means of the distribution function W(C), we obtain (7.3.4 ) The resulting ionic field C is equal to the vector sum over all ions,
The function W(C) determines the probability of a given magnitude of the absolute value of C. This function was calculated by Holtsmark in the ideal gas approximation. (A detailed discussion is given in [7.25]). In this approximation, one assumes that each of the ions can with equal probability be located at any point of the volume independently of how all the other ions are located. Therefore the function W(C) can be calculated in the following way:
WH(C) dC
=
(15
(c -Ze I: R:) dC) k=IR k
= JdR I JdR2 ... 15 (c-zeI: R:) dC V V k=IR k = (2
1 1t
dR I dR2
.
NV
(.zep.R k )
)3 J - J - ... J dp exp (Ip· C) ~ exp -I V
V
k=l
R3
dC .
k
Here we use the well-known representation of the t5-function, introducing the additional integration over p. By changing the order of integration over R and
7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma p, we obtain in the limit V
WH(C)
275
--> 00
=
(2~)3 J dp exp (ip
=
(2~)3 J dp exp (ip
0
0
C) {I -
~J dRk [1 _ exp (-ize~tk )]}
N V
C) exp { -N J dR [I - exp ( _/e;;R) ] }
Integration over R gives
J dR [I - exp ( _i ze;3oR )]
=
1~(2nZep)3/2 .
Then it is possible to carry out the integration over the angular variables of the vector p. As a result we have
WH(C)dC
= d: .1t (~) ,
(7.3.5)
where
.1t(f3)
Co
=
~f3J:xsinx exp 4
= 2n ( 15
[-
(~r/2l
(7.3.6)
dx,
)2/3ZeN 2/3 = 2.6031 ZeN 2/3 .
(7.3.7)
Values of the function .1t(f3) for a wide range of values of the parameter f3 are given in Table 7.1. In addition the function .1t(f3) is shown in Fig. 7.4. The maximum of the function .1t(f3) corresponds to the point f3 = 1.607. In the two limiting cases, high and low values of f3, the function .1t(f3) can be approximated by the series 1.496 f3-S /2( 1 + 5.107 f3- 3/2
.1t(f3)
~
{
tn
f32 (1- 0.463f32
+ 14.93f3- 3 + ... )
+ 0.1227f34 + ... )
(f3~ 1).
(f3 ~>1),
(7.3.8) (7.3.9)
If in the expression for .1t(f3), the field Co is redefined by putting Co = ZeR(;2, where Ro = (3/4nN)I/3, then instead of (7.3.8) we have .1t(f3) ~ l.5f3- S/2 , which coincides with the binary distribution (7.1.26). We note that from the practical point of view the difference between the two definitions of Co is unimportant. In accordance with (7.3.8), in the wing of the line,
I(w) ~ (w - wO)-5/2l.5D~p(B~p)3/2C~/2 ,
(7.3.10)
~p
in full agreement with the binary distribution (7.1.28). This is due to the fact that the strongest fields are created mainly by the nearest ion. It must be noted that the distribution function of the binary approximation is fairly close to .1t(f3) everywhere, with the exception of the range of low values of f3. Weak fields,
276
7. Broadening of Spectral Lines
Table 7.1. Holtsmark distribution function
p
Jf(P)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2
0.00000 0.0042245 0.016665 0.036643 0.063082 0.094596 0.129587 0.166360 0.203233 0.238641 0.271221 0.299870 0.323782 0.342461 0.355702 0.363566 0.366334 0.364456 0.358502 0.349109 0.336939 0.306821
Jf(P) 0.272746 0.238221 0.205563 0.176063 0.150242 0.128118 0.109422 0.093753 0.080674 0.069765 0.060654 0.053023 0.046604 0.041180 0.036573 0.032640 0.029263 0.026349 0.023822 0.021619 0.019690 0.017993
P 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6
P 6.8 7.0 7.2 7.4 7.6 7.8 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 17.0 20.0 24.0 28.0 33.0 38.0 43.0 48.0 53.0
Jf(P)
0.016494 0.015165 0.013981 0.012922 0.011974 0.011120 0.010350 0.007438 0.005561 0.004289 0.003392 0.002739 0.002249 0.001875 0.001351 0.0008856 0.0005537 0.0003733 0.0002457 0.0001718 0.0001256 0.0000952 0.0000741
0.4
/ '~
OJ 0.2
/ o
/
~
/
"~ .......
2
p-
3
r---5
4
Fig. 7.4. Holtsmark distribution Jf(P)
obviously, are produced by a large number of comparatively distant ions. The calculations of Holtsmark intensity distribution for a number of hydrogen spectral lines were carried out in [7.41]. It is convenient to rewrite (7.3.10) introducing the effective Stark-effect constant B for a line as a whole [7.42]
I(w) ~ 101.5(w - wO)-5/2(B80)3/2,
10
= L/~p, ~p
(7.3.11)
7.3 Broadening of Lines of the Hydrogen Spectrum in a Plasma
277
where in accordance with (7.3.2) B
3/2 -_10-I",
L.. I~p
(e)3/2
h
~p
(Z~~
- Zpp)
3/2
(7.3.12)
.
Here Z is the coordinate of the atomic electron. Comparison with the results of accurate numerical calculations shows that for a hydrogenlike ion with nuclear charge ~e, the constant B can be approximated by the expression B
= (~)2/3 _h_(n2
_ n'2 ),
(7.3.13 )
~me
8
where n and n' are the principal quantum numbers of the initial and final levels. Similarly for the contour of the line 1(OJ) = E~p/~p( OJ), one can also use the approximate expression [7.42] (7.3.14 ) The dependence of TH(fJ) on [3 is given in Table 7.2. At high values of [3, TH ([3) ---+ 1.5 [3-5/2. Since the contour of the line (7.3.4), and also (7.3.14), is symmetrical with respect to OJo, the Holtsmark width of a line LlOJH is approximately equal to 8BSo. Using (7.3.13), we obtain for the hydrogen spectrum, (7.3.15) Formula (7.3.14) describes sufficiently well the contour of the line everywhere apart from the central region. In order to improve the Holtsmark theory one must take into consideration the mutual correlation of ion positions. In the Holtsmark theory, the exponential factor exp[-V(RI' R 2 ,
•••
)jkT]
in the expression for probability of an ion configuration R 1, R 2 , ••• with potential V is neglected. Thus, the relative probability of such configurations to which high positive values of V correspond are overestimated. In particular, the Holtsmark theory overestimates the probabilities of large frequency shifts /C, i.e., of high values of S, and underestimates the probabilities of low /c. The simplest way of introducing the corresponding corrections to the Holtsmark theory is to take the Debye-Hiickel screening into account. The field of the ion surrounded by a cloud of other ions and electrons of the plasma decreases at large distances as
Table 7.2. Function TH(P)
P TH(P)
0 0.1
0.5 0.1
1 0.098
2 0.086
3
5
7
10
15
0.070
0.039
0.02
0.0072
0.0023
20 0.00099
278
7. Broadening of Spectral Lines
0.6,-------,----,----,---...,-----, 0.5 t-----.~---+----+---_::__+--___l 0.41------h~+_'oO'(n, n + k) are the effective cross sections of inelastic scattering. Using the quasi-classical calculations of cross sections for highly excited level described in Sect. 3.5, one can write for the quantity (vO'(n)} the following approximate formula: (vO'e(n)}
~
4
10- 8 z3~1/2 CP(x) f( 8) [cm3 s- I ],
(7.3.42)
where 8 = T/~Ry,
x = En/T = l/n2 8 , CP(x) = 2.18{0, 82q>(x) + 1.47[1 - xq>(x)]}, 1 + 1.4yx ] q>(x) = -exp(x)Ei( -x) ~ In [ 1 + y.x.(l + 1.4x) ,
f(8) = In
[1 + z(1 +nvle vie ]1n(1 + nvle/z) 2.5/z 8)
(7.3.43) Y = 1.78
(7.3.44)
7. Broadening of Spectral Lines
286
when z -+ 00 or 8 -+ 00,/(8) -+ 1. If perturbing particles are protons, the temperature T in (7.3.42-44) should be replaced by the quantity Tm/M, where m is the mass of electron and M is the reduced mass of the colliding particles. Stark broadening of the lines in the far-infrared solar spectrum corresponding to the transitions between highly excited hydrogenlike states is of interest for diagnosing the structure of the solar atomosphere [7.82].
7.4 Line Broadening of Nonhydrogenlike Spectra in a Plasma 7.4.1
Preliminary Estimates
The spectral lines of nonhydrogenlike atoms in the presence of a constant and homogeneous electric field undergo a shift and also a splitting proportional to @"2-the quadratic Stark effect. We shall assume that the field @" = QR- 2 , produced by a charge Q, varies little for atomic dimensions (this is valid for sufficiently large values of R). Then in (7.1.20) for the shift of oscillator frequency, n = 4 and K = C4R- 4 . The constant C4 for a transition n -+ k is defined as C4
= (C4 )n -
(C4 )k; (C4 )n
= Q2/h 2: 1(~~nmI2 m
,
nm
where (Dz )nm are the matrix elements of the z component of the electric dipole operator, .1Enm = En - Em (see [1.1]). The parameters he (electron broadening) and hI (ion broadening) are he = N
(~ C4) 2
, hi = N
Ve
(~ C4 ) 2
.
(7.4.1 )
Vi
The quadratic Stark-effect constants C4, as a rule, have the order of magnitude 10- 12 -10- 15 cm4 s-I, although values of C4 < 10- 15 and C4 '" 10- 11 - 10- 10 are also encountered. For C4 = 10- 12 - 1O- IS,ve = 5.107 cm s-I and Vi = 2.105cm S-I, we have he = 3.(10- 19 - 1O- 22 )N, hi = 0.75(10- 17 - 1O- 2o )N. At not very high values of the density of charged particles N < 1015 cm -3, he ~ I, and hi ~ 1. This means that both electrons and ions produce impact broadening. According to (7.1.23) Y4,.1 4 ex: vl / 3. Thus electrons play the principal role in the broadening of a line. The interaction with ions only slightly increases the impact width and shift of a line, by approximately 15-20%, because (Ve/Vi)I/3 ~ (M/m )1/6 ~ 5-6. Since K ex: Q2 the sign of the shift of a line is the same for electrons and for ions.
7.4.2 Electron Broadening We shall describe electron broadening of lines of a nonhydrogenlike atom in the framework of quasi-classical theory discussed in Sect. 7.2. In the case of the
7.4 Line Broadening of Nonhydrogenlike Spectra in a Plasma
287
isolated spectral line the spectrum I (w) is described by Lorentzian distribution (7.2.17). The width and shift are given by (7.2.18) and (7.2.20). The elements of the S matrix averaged over M components of atomic levels must be substituted in these equations. As a rule the main contribution to the broadening is given by collisions with relatively large values of the impact parameter p. Therefore in calculating the S matrix we can restrict ourselves to the first terms of the expansion in powers of rand 1'/. Therefore one can average over M directly the quantities r and 1'/. In the case of the dipole interaction between a neutral atom and a charged particle V = -d·iff the linear term in (7.2.25) for the S matrix S(v), being averaged over M (or over directions of the vectors p and v), is equal to zero. Only the next term containing C\(t) C\( (I) in the integrand is nonvanishing. We shall assume now that the perturbation of one of the levels (initial or final) can be neglected. Then for the radiative transition n-k, assuming that the level k is not perturbed, we can obtain (7.4.2)
r
= ~'rs = ~'2 s
s
(.!!...)2 (Ins Ry ) ~A (W ns P ) mv JEns P
.
(7.4.3 )
V
Here JEns = Es - En, Ins is the oscillator strength of the transition n -+ s (see [Ref. 1.1, Sect. 9.2]), and the sum over s extends over all atomic levels, for which Ins =I O. The functions A and B coincid~ with the functions A+ and B+ defined by (7.3.29, 30). Values of the functions A and B are given in Table 7.4. As a rule the principal contribution to (7.4.2,3) is provided by the nearest perturbing levels and, in some cases, by only one of them. In the approximation of one perturbing level, y
= 2N (v)(J~( (v))J' (f3)
,
(7.4.4 )
J
= N(v)(J~( (v) )J"(f3) ,
(7.4.5)
Table 7.4. Functions A(z) and B(z). z
A(z)
B(z)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
1.000 1.035 0.962 0.829 0.680 0.540 0.418 0.318 0.239
0.000 0.160 0.359 0.498 0.576 0.606 0.603 0.580 0.546
z
A(z)
B(z)
1.8 2.0 2.4 2.8 3.2 3.6 4.0 5.0
0.177 0.130 0.0688 0.0355 0.0181 0.0090 0.0045 0.00075 nze- 2z
0.507 0.467 0.393 0.331 0.283 0.245 0.216 0.166 n/4z
~oo
7. Broadening of Spectral Lines
288
where (J~ and (J~ are the width and shift cross sections defined by (7.1.13,14),
(J~ = (~r/3 r (~ ) C~/3 (v) -2/3 ':::!. 5.7C~/3 (v) -2/3 , (7.4.6)
(v) is the mean value of the electron velocity, and the constant C4 is defined as C 4
= 2t?a~f(Ry)2
ft
,1E
The functions J'(P) and J"(P) depend on the dimensionless parameter
P =lfRyll/2 I,1EI2 . ,1E m(v) As
P---- 00, J', J"
(7.4.7 )
---- 0.97. This case corresponds to adiabatic perturbation when
r = 0, and the broadening is determined by the phase shift '1.
For
P~ 1, on the
contrary, inelastic collisions play the main role. In this case,
,
Ry
(v(J ) = 4nf ,1E (v
(ft) 2In (mv2 1 Ry 11/2) ) , mv 21,1EI f ,1E
(7.4.8)
(7.4.9) The results of numerical calculations of the functions J'(P) and J"(P) are given in Table 7.5. We shall now consider to what extent the results obtained above can be generalized to the case of several perturbing levels. This problem obviously arises
Table 7.5. Factors J'(fJ) andJ"(fJ).
fJ
J'
J"
64 32 16 8 4 2 I 0.5 0.25 0.125 0.625 x 10- 1 0.312 x 10- 1
0.97 0.97 1.02 1.03 1.06 1.12 1.17 1.20 1.15 1.09 0.927 0.764
0.97 0.97 0.97 0.96 0.94 0.90 0.861 0.746 0.604 0.455 0.326 0.223
fJ 0.156 0.78 0.39 0.195 0.97 0.48 0.24 0.12 0.61 0.305 0.15
xlO 1 X 10- 2 X 10- 2 x 10- 2 x10- 3 X 10- 3 X 10- 3 X 10- 3 x 10- 4 x 10- 4 X 10- 4
J'
J"
0.594 0.451 0.334 0.239 0.171 0.119 0.0824 0.056 0.038 0.024 0.017
0.151 0.094 0.063 0.0405 0.0245 0.0167 0.0103 0.0065 0.004 0.0026 0.0016
7.5 Broadening by Uncharged Particles
289
only in the case when, for one or several perturbing levels, the parameter P is of the order of or less than unity. If P~ 1 for all perturbing levels, then the perturbation is adiabatic and y and LI are expressed by means of (7.1.23) in terms of the quadratic Stark effect constant C4 for a given line. The magnitude of this constant is determined by the total perturbing effect of all atomic levels. If for the nearest perturbing levels which give the principal contribution to '1 and r in (7.4.2,3), the parameters P~ 1, then the width y can be obtained by summing (7.4.4):
y = 2N(v)LI(J~S«V))J/(PS)'
(7.4.10)
Such an approximation is valid because in this case the broadening is caused by inelastic collisions, and the partial widths corresponding to different collisional transitions are additive. The shift of a line cannot be calculated by summing (7.4.5) even if for all perturbing levels Ps ~ 1. In the general case when both levels nand k (initial and final) are perturbed, r
= rn + rk,
'1
= '1n - 17k,
where r n, rk, '1n and 17k must be calculated by means of (7.4.2, 3). An extensive bibliography on numerical calculations of shapes of the nonhydrogenlike spectral lines in plasma can be found in [7.6]. Results of numerical calculations of widths and shifts for a large number of nonhydrogenlike spectral lines of different atoms and ions also are given in [7.6]. The experimental data on Stark broadening in nonhydrogenlike spectra may be found in [7.12.13]. The very accurate experimental data on the lines of neutral He are given in [7.83].
7.5
Broadening by Uncharged Particles
7.5.1
Perturbation by Foreign Gas Atoms (Van der Waals Interaction)
The interaction of neutral atoms at large distances has the form V(R) ex: R- 6
.
Therefore usually one assumes K
= C6R-6.
(7.5.1 )
The crude estimate of the constant C6 is given by C6 ~
e2(r2)IX.
- f j , - ' IX. ~
4
(Ry)2 ao (rp)2,
"3 m T
(7.5.2)
where (r2) ~ 5n*4a~/2 is the mean value of r2 for the excited state of the
290
7. Broadening of Spectral Lines
radiative atom, (r~) is that for the perturber, n* is the effective principal quantum number, IX is the polarizability of the perturbing atom, I is its ionization potential, and m is the number of equivalent electrons. The constant C6 has the order of magnitude 10- 30 n*4. Thus for v rv 105 cm s-I, we have h = (387t
~6 )
3/5
N
~ 10- 21 N
.
(7.5.3 )
This indicates that at not very high pressures, of the order of few atmospheres or less, line broadening can be described in the impact approximation. We shall also compare the quantities Q = V6/5C;:I/5 and Awo. As Q ~ 10 12 S-I and Awo ~ IO IO s- l , we have Q~Awo. Consequently the region of impact broadening extends far beyond the limits of the Doppler width. In accordance with (7.1.23) the width and shift of a line can be estimated using the relations (7.5.4 ) The typical values of yare y rv 10- 8N. In order to treat impact broadening by uncharged particles more accurately it is necessary to take into consideration that at small distances the interaction VCR) has a more complicated form than VCR) ex R- 6 . Depending on the type and states of interacting atoms both attraction and repulsion can take place at large distances R. At small distances the potential VCR) is repulsive. In some cases atom and perturber can form a quasi-stable molecule. Moreover in the general case, the interaction V is dependent not only of R but also of the angular variables. The results of calculations in which a more realistic interaction than V(R) ex R- 6 is used cannot be described by a simple Lorentzian distribution with width and shift as in (7.5.4). In particular, the intensity distribution depends on the type of the transition j-j'. A detailed treatment of the foreign gas broadening is given in [7.4,10,11,1414-16]. The repulsive part of the interaction is usually taken into account in the form of the Lennard-Jones potential V(R) = CI2R-12_C6R-6. The line shift A and the ratio ylA are especially sensitive to the form of the potential VCR). Experimental data on line broadening in the spectra of alkali atoms obtained at low values of foreign gas pressure, less than 10 atm, are in qualitative agreement with the impact theory. The broadening and shift of the lines are proportional to the concentration of perturbing particles. For the initial members of a principal series perturbed by different foreign atoms (He, Ne, Ar, Kr, Xe, H2, N2, and so on), as a rule, a red shift is observed, the ratio ylA being close to 2.8. In some cases (usually for the higher members of a principal series), a blue shift instead of a red one is observed. The sign of the shift of one and the same line can be different for different perturbing particles. The dimensionless parameter h reaches values of order unity only when N > 1021 , i.e., at pressures of about tens of atmospheres. In this case the mean distance between atoms has the same order of magnitude as atomic dimensions and consequently the simplest expression VCR) ex R- 6 is not valid. The experi-
7.5 Broadening by Uncharged Particles
291
mental data on line shapes are usually used to obtain information about the form of potential V(R) at small distances. Specific features of molecular-lines broadening have been described in [7.4,
8,9]. 7.5.2
Self-Broadening
We shall consider now the single-component gas. With an increase of density of such a gas, resonance lines broaden considerably more than on the addition of a foreign gas. This is due to the fact that in the case of collision of two identical atoms, one of which is excited, a resonance transfer of the excitation energy is possible, the effective cross sections of such collisions being extremely large. They can exceed considerably (by several orders) the gas kinetic cross sections. The effective cross sections of resonance energy transfer (J were calculated in Sect. 4.2. For electric dipole transition, the energy transfer is caused by the dipole-dipole interaction V ex: R- 3 • The cross section (J and corresponding line width yare of the order of magnitude
e2 y'" -IN,
mwo
(7.5.5)
where S is the line strength and I is the oscillator strength. Assuming that I ~ 1 and Wo ~ lOIS, we have y 1O- 7N. In the case of foreign-gas broadening, typical values of widths are y rv (l0-9-10-8)N. The effective cross section of energy transfer can be relatively large not only under conditions of exact resonance but also in the case of a collision of two atoms with close energy levels. Thus, when calculating the width of the component 2P1/2-2S1/2 of the resonance doublet of an alkali atom, it is necessary to take into account not only energy transfer 2P I /2 -+2 SI/2 (radiating atom), 2S1/2 -+2P 1/2 (perturbing atom), but also the excitation 2S1/2 -+2P3/2 of the perturbing atom. The cross section of the energy transfer 2P I /2 -+2S1/2 , 2S1/2 -+2P3/2 has the same order of magnitude as the cross section of the resonance energy transfer 2P I/2 -+~\/2' 2SI/2 -+P3/2, if the following condition is fulfilled (Sect. 4.2): V3/2C:;I/2 = V3/2(t? l/mcoO)-1/2 is also proportional to (CO-COO)-2 as in the impact approximation. However, the intensity /(co) in the quasistatic wings is somewhat different from that in the impact approximation [7.84,85,90] /(co)
= a (JoJd
2nt? 1 -lOIN ( )2 . mcoo co-coo
(7.5.7)
The factors a(JoJd are also given in Table 7.6. In the case of Li resonance doublet Jo = 0, JI = 1 should be assumed in (7.5.7). The calculations taking into account the accurate adiabatic potential curves [7.90] show a slight asymmetry of blue and red wings. The direct experimental studies of the resonance broadening as a rule encounter very serious difficulty connected with extremely large optical depth in the center of the lines [7.91]. The lines corresponding to transitions between the resonance level and other excited levels also undergo broadening due to the resonance interaction. The resonance contribution to the widths y of such lines can be evaluated to be equal to the linewidth of the resonance line.
TabIe7.6. Parameters describing the resonance broadening Jo
0
112
112
1
JI
1
112
3/2
1
1.042
0.903
1.039
0.092
-0.031
0.050
0.698
1.047
0.805
A(JoJl)
2
Aly
a (JoJl)
0.983 -0.01
7.6 Spectroscopic Methods oflnvestigating Elastic Scattering of Slow Electrons
293
7.6 Spectroscopic Methods of Investigating Elastic Scattering of Slow Electrons 7.6.1 Perturbation of Highly Excited States The broadening of a line corresponding to a transition between the ground state and a state with a large value of the principal quantum number n is completely determined by the perturbation of the upper level. For sufficiently large values of n, the mean distance of the valence electron from the nucleus rv aon2 is so large that the neutral perturbing particle either interacts with the electron and does not interact with the atomic core, or interacts only with the atomic core. In this case, the broadening is caused by the scattering of the atomic electron by the perturbing particles and by the scattering of the perturbing particles by the atomic core. These two mechanisms of the broadening are statistically independent. We shall first consider the interactions of the first type. If only one level is perturbed, then in accordance with (7.2.39) we have
2n
(I'
= kIm {/(O)},
2n
(I"
= -k Re {/(O)}
,
(7.6.1)
where 1(0) is the amplitude of forward scattering of the perturbing particle by the atom, and hk is the momentum of the perturbing particle (we assume for simplicity that the mass of the atom is large as compared with the mass of the perturbing particle). If aon2~Peff. ~ (nlX/4)1/3(e'2/hve)I/3, where Peff. is the effective radius of the interaction between the electron and the perturbing particle and IX is the polarizability of the perturbing particle, then in the volume of interaction the field produced by the atomic core and consequently the electron velocity Ve are practically constant. In the state with principal quantum number n, Ve is of the order of magnitude vo/n, where Vo is the atomic unit of velocity. If the velocity of the perturbing particle vp = hk/M is less than Ve rv vo/n, then the scattering amplitude 1(0) in (7.6.1) can be expressed in terms of forward scattering amplitude 1;(0) of a free electron with momentum hq by the perturbing particle [7.92] for the derivation of this, and subsequent formulas of this section):
1(0) = M
m
J I G(nlmlq)1 2/;(0)dq.
(7.6.2)
Here m is the electron mass, and G(nlmlq) are the coefficients of the expansion of the atomic function I/!nlm in plane waves: (7.6.3) States with large values of the principal quantum number n are hydrogenlike. It is therefore possible to use as the expansion coefficients G(nlmlq) the well-known expressions for hydrogen functions in the momentum. representation in terms of Gegenbauer polynomials [7.93].
294
7. Broadening of Spectral Lines
By substituting (7.6.2) in (7.6.1), integrating over the angular variables, and averaging over all possible orientations of the perturbing particle angular momentum, we have
41t1i Y = N-;;- dq W(q) 1m {Jq(O)} ,
(7.6.4)
21t1i A = -N -;;- dq W(q)Re {fq(O)} ,
(7.6.5)
= q2 IGn,(qW, J W(q)dq = 1 .
(7.6.6)
J
J
W(q)
As already noted above these formulas describe the width and shift caused by scattering of the atomic electron in the highly excited state by perturbing particle if the following conditions are fulfilled: (7.6.7) (7.6.8) We shall now consider interaction of perturbing particle with the atomic core. As the charge of the atomic core is e, this interaction has the form (7.6.9) It produces the polarization of the perturbing particles by the atomic core and leads to a shift of the frequency of the atomic oscillator, IX 2 1 K(t) = - 21i e [R(t)]4 .
(7.6.10)
Broadening due to interactions of this type was examined in Sect. 7.4.- If
hi=PfN~l'Pi=C:)
1/3 (
~
) 1/3
livp
,
(7.6.11)
where N and vp are the density and velocity of the perturbing particles, then the central part of the line lro-rool ~D = (2Iiv:/lXe 2 )1/3 is described by the Lorentzian distribution with width l' and shift A' given by the following formulas: ~
y'= 114 . (~) 21i
2/3
;;:;
1/3N. Vp '
A' = _~ 2 Y, .
LJ
(7.6.12)
It can be shown that when hi ~ 1, the corresponding quasi-static distribution has width of the order of lON4/ 3 ~ Iii. The case hi ~ 1 corresponding to pressures of the order of or less than an atmosphere is the most interesting. In this case, one can calculate l' and A' with sufficiently good accuracy. Subtracting the cal-
7.6 Spectroscopic Methods ofInvestigating Elastic Scattering of Slow Electrons
295
culated values y' and ,1' from experimental values of the width and shift, one can determine y and ,1 from (7.6.4,5). Expressing in these equations the amplitude 1;(0) in terms of the phase of scattering of the electron by the perturbing particle 111, we also have
N~m J
y=
,1
=
[4n E(21 + l)sin2111] W(q)dq q 1
-N -h J m
= N~ Jqa(q) W(q)dq, m
[n- E(21 + l)sm2111 . ] W(q)dq. q
(7.6.13)
(7.6.l4)
1
Here a( q) is the effective cross section for elastic scattering of the electron with momentum hq by the perturber. If exchange interaction is also taken into account, then the following substitution must be made in (7.6.l3, 14): (7.6.l5) sin 2111 -+ C+ sin 211~ +)
+ c- sin 211~ -) ,
(7.6.l6)
where 11~ +) and 11~ -) are the scattering phases calculated taking exchange into account for states of the system perturbing particle plus electron with given value of the total spin S = Sp ± Ij2,Sp being the spin of the perturbing particle; and C+
=
Sp
+ 1,
C-
2Sp +1
=
(7.6.l7)
Sp
2Sp +1
We shall now consider the resonance transitions no atoms. For np state [7.93] W(q) = 2
(~2
n n - 1
) (_1_)2 [(n nqao
-(n - l)sin(n
+ l)sin(n -
+ l)qJ]2 ao ,
s-np(n~
1) of the alkali
l)qJ
(7.6.l8) (7.6.19)
The function W(q) has n peaks, nj2 peaks being located in the range 0 < q < Ijnao. In this range, the envelope behaves approximately as (1 + n2q2a~)-2. For q~ 1jnao, the function W(q) decreases monotonically: 27 (n 2
W(q)C::'3
1 )n3
_
n
(
1 ) 8 -. nqao
(7.6.20)
Thus the principal contribution to the integral over q in (7.6.13, 14) is given by the range 0 < q < Ijnao.
296
7. Broadening of Spectral Lines
If n is so large that the principal contribution to the sum with respect to 1 is given by the term 1 = 0 (s scattering) and in addition q-l sin 21'/1 differs little from its limiting value
1. (1. ) 1m
q-+O
- sm 21'/1 q
=
-I1'1'//-I00 ~ -0"(0) , n
(7.6.21 )
then
fL t=:7i\\. 1'/0
.1 = -;,;;vna(O)i%TN.
(7.6.22)
Here a(O) is the limiting expression for the elastic scattering cross section at q - t O. By 1'/0 is understood that part of the phase which after subtraction of pn where p is integer, lies in the interval -nI2, n12. Since as q - t 0, qa(q) - t 0, in the range of applicability of (7.6.22), Y~I.1I. Some additional effects of broadening are discussed in [7.94-96]. Experimental data on broadening of Rydberg levels can be found in [7.97-100].
7.6.2 Fermi Formula Equation (7.6.22) has been obtained by Fermi [7.4]. In accordance with this equation it is possible using the experimental value of .1 to determine the elastic scattering cross section for extremely slow electrons (in the limit q - t 0). Thus the cross section a(O) for the atoms He, Ne, Ar, Kr, and Xe have been found by the shift of the absorption lines of Cs in an atmosphere of noble gases. Some other gases have also been investigated by the same method (see [7.4] and also [7.101]). The Fermi method enables one to obtain information on elastic scattering of electrons at very small energies, i.e., in the range most difficult to investigate by other experimental methods. It must be noted that shift .1 is sensitive not only to the magnitude but also to the sign of the phase 1'/0. In the general case, when several terms of the sum over 1 contribute to y and .1, it is not possible to determine the scattering phases from known values of y and .1. Knowledge of these quantities, however, enables one to control the quality of approximate calculations of the scattering phases [7.102, 103].
References
Chapter 1 1.1 1.2
1.1. Sobel'man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoU2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) M. Venugopalan (ed.): Reactions under Plasma Conditions (Wiley-Interscience, New York 1971) VoU
Chapter 2 N.F. Mott, H.S.F. Massey: The Theory of Atomic Collisions (Pergamon, Oxford 1965) 2.2 M.L. Goldberger, K.M. Watson: Collision Theory (Wiley, New York 1964) 2.3 ChJ. Joachain: Quantum Collision Theory (North-Holland, Amsterdam 1975) 2.4 M.R.H. Rudge: Rev. Mod. Phys. 40, 564 (1968) 2.5 R.P. Peterkop: Teoriya Ionizatsii Atomov Elektronnym Udarom (Theory of Ionization Atoms by Electron Impact, in Russian) (Zinatne, Riga 1975) 2.6 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vol.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) 2.7 L.A. Vainshtein, 1.1. Sobel'man: Zh. Eksp. Teor. Fiz. 39, 767 (1960) 2.8 M.J. Seaton: Proc. Phys. Soc. 77, 184 (1961) 2.9 MJ. Seaton: Adv. Atom. Molec. Phys. 11,83 (1975) 2.10 H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody: Electronic and Ionic Impact Phenomena VoU (Clarendon, Oxford 1969) 2.11 R. Courant, D. Hilbert: Methoden der Mathematischen Physik (Springer, Berlin 1931) VoU 2.12 L.A. Vainshtein: Phys. Scripta 33,336 (1986) 2.1
Chapter 3 3.1 3.2 3.3 3.4
3.5 3.6
L.A. Vainshtein, 1.1. Sobel/man: Zh. Eksp. Teor. Fiz. 39, 767 (1960) B.L. Moiseiwitsch: Rep. Prog. Phys. 40,843 (1977) K. Smith: The Calculation of Atomic Collision Processes (Wiley-Interscience, New York 1971) 1.1. Sobel'man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vol.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) M. Inokuti: Rev. Mod. Phys. 43, 297 (1971) K. Omidvar: Phys. Rev. 188, 140 (1969)
298
3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.l9 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27
3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40
References
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Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14
4.15 4.16 4.17 4.18 4.19 4.20
4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33
N.F. Mott, H.S.F. Massey: The Theory of Atomic Collisions (Pergamon, Oxford 1965) H.S.W. Massey, E.H.S. Burhop, H.B. Gilbody: Electronic and Ionic Impact Phenomena (Clarendon, Oxford 1969) VoU H.S.W. Massey, H.B. Gilbody: Electronic and Ionic Impact Phenomena, Vol.4 (Pergamon, Oxford 1974) E.E. Nikitin, S.Ya. Umanskii: Theory of Slow Atomic Collisions (Springer, Berlin, Heidelberg 1984) R.K. Janev, L.P. Presnyakov, V.P. Shevelko: Physics of Highly Charged Ions (Springer, Berlin, Heidelberg 1985) B.N. Bransden, M.R.C. McDowell: Charge Exchange and the Theory of IonAtom Collisions (Clarenden, Oxford 1992) H.B. Gilbody: Adv. Atom. Molec. Phys. 22, 143 (1986) L.D. Landau, E.M. Lifshitz: Quantum Mechanics (Pergamon, Oxford 1965) L.A. Vainshtein, 1.1. Sobelman, L.P. Presnyakov: Zh. Eksp. Teor. Fiz. 43, 518 (1962) [English transl. SOy. Phys. -JETP 16, 370 (1962)] D.R. Bates: Proc. Phys. Soc. A 73,227 (1959) M.R. Flannery: Phys. Rev. 183,241 (1969) M.R. Flannery: J. Phys. B 2,909 (1969) J.C. Gay, A. Omont: J. de Phys. 35, 9 (1974) 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, Vo1.12 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoU (Springer, Berlin, Heidelberg 1979) K. Alder, A. Bohr, T. Huus, B. Mottelson, A. Winther: Rev. Mod. Phys. 28, 432 (1956) D.R. Bates, R. McCarrol: Adv. Phys. 11,39 (1962) M.R.C. McDowell, J.P. Coleman: Introduction to the Theory of Ion-Atom Collisions (North-Holland, Amsterdam 1970) O.B. Firsov: Zh. Eksp. Teor. Fiz. 21,1001 (1951) H.C. Brinkman, H.A. Kramers: Proc. Acad. Sci. Amsterdam 33,973 (1930) B.M. Smirnov: Asimptoticheskii Metod v Teorii Atomnykh Stolknovenii (The Asymptotic Method in the Theory of Atomic Collisions, in Russian) (Nauka, Moscow 1973) R.A. Mapleton: Theory of Charge Exchange (Wiley-Interscience, New York 1972) D.S.F. Crothers, N.R. Todd: J. Phys. B 13,2277 (1980) V.P. Shevelko: Z. Phys. A 287,18 (1978) A.M. Brodskii, V.S. Potapov, V.V. Tolmachev: Zh. Eksp. Teor. Fiz. 58, 264 (1970) [English trans!.: SOy. Phys. -JETP 31,144 (1970)] V.V. Afrosimov, R.N. Ilyin, E.S. Solovyev: Zh. Techn. Phys. 30, 705 (1960) V. Schryber: Helv. Phys. Acta 40, 1023 (1967) A.V. Vinogradov, L.P. Presnyakov, V.P. Shevelko: JETP Lett. 8,449 (1968) H.D. Betz: Rev. Mod. Phys. 44, 465 (1972) N.V. Fedorenko: JTP 15, 1947 (1972) A. Salop, R.E. Olson: Phys. Rev. A 13, 1312 (1976) A. Salop: Phys. Rev. A 13, 1321 (1976) G. Harel, A. Salin: J. Phys. B 10, 3511 (1977) J. Vaaben, J .S. Briggs: J. Phys. B 10, L521 (1977)
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Chapter 5 5.1 5.2 5.3 5.4
5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15
5.16 5.17 5.18 5.19 5.20 5.21 5.22
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5.23 W. Lochte-Holtgeven (ed.): Plasma Diagnostics (North-Holland, Amsterdam 1968) 5.24 W. Neumann: Spectroscopic methods of plasma diagnostic, in Progress in Plasmas and Gas Electronics, VoLl, ed. by R. Rompe, M. Steenbeck (Akademie, Berlin 1975) p.3 5.25 H.W. Drawin: Validity conditions for local thermodynamic equilibrium, in Progress in Plasmas and Gas Electronics, VoLl, ed. by R. Rompe, M. Steenbeck (Akademie, Berlin 1975) p.593 5.26 M.J. Seaton: Mon. Not. Astron. Soc. 119,90 (1959) 5.27 I.L. Beigman, E.D. Mikhalchi: J. Quant. Spectrosc. Radiat. Transl. 9, 1365 (1969) 5.28 G. Ecker, W. Weizel: Ann. Phys. 17, 126 (1956/57) 5.29 G. Ecker, W. Kroll: Phys. Fluids 6, 62 (1963) 5.30 H.R. Griem: Phys. Rev. 128, 997 (1962) 5.31 H.W. Drawin: Ann. Phys. (Leipzig) 14, 262 (1964) 5.32 S.T. Belyaev, G.1. Budker: Mnogokvantovaya Rekombinatsiya v Ionizovannom Gaze, in Fizika Polasmy i Problema Upravlyaemykh Termoaydernykh Reaktsii (Plasma Physics and the problems of Controlled Nuclear Fusion) Edition of Academy of Sciences, Moscow 1958) Vol.3, p.41 5.33 L.P. Pitaevskii: Zh. Eksp Teor. Fiz. 42, 1326 (1962) [English transl.: SOy. Phys. -JETP 15, 919 (1962)] 5.34 A.V. Gurevich, L.P. Pitaevskii: Zh. Eksp Teor. Fiz 46, 1281 (1964) [English transl.: SOY. Phys. - JETP 19, 870 (1964)] 5.35 M. Cacciatore, M. Capitelli, H.W. Drawin: Physica C 84,267 (1976) 5.36 D.R. Bates, A.E. Kingston, R.W.P. McWhirter: Proc. R. Soc. London A 267, 297 (1962) 5.37 D.R. Bates, A.E. Kingston, R.W.P. McWhirter: Proc. R. Soc. London A 270, 155 (1962) 5.38 R.W.P. McWhirter, A.G. Hearn: Proc. Phys. Soc. London 82,641 (1963) 5.39 L.C. Johnson, E. Hinnov: J. Quant. Spectrosc. Radiat. Transf. 13,333 (1973) 5.40 H.W. Drawin, F. Emard: Physica C 85,333 (1977) 5.41 M. Venugopalan (ed.): Reactions under Plasma Conditions (Wiley-Interscience, New York 1971) VoLl 5.42 R. Hess, F. Burrell: J. Quant. Spectrosc. Radiat. Transf. 21, 23 (1979) 5.43 H.W. Drawin, F. Emard: Physica C 94, 134 (1978) 5.44 H. Risken: The Fokker- Planck Equation, 2nd edn., Springer Ser. Syn., VoLl8 (Springer, Berlin, Heidelberg 1989) Chapter 6 6.1 6.2
6.3 6.4
L.A. Vainshtein: Trudy FIAN 15, 3 (1961) 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoLl2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoLl (Springer, Berlin, Heidelberg 1979) D.A. Varshalovich, A.N. Moskalyv, V.K. Khersonske: Quantum Theory of Angular Moment (World Scientific, Singapore 1988) A.R. Edmonds: Angular Momentum in Quantum Mechanics (Princeton Press, Princeton, NJ 1957)
302 6.5 6.6 6.7
References M. Rotenberg, R. Bivius, N. Metropolis, J.K. Wooten, Jr.: The 3j and 6j Symbols (MIT Press, Cambridge, MA 1959) H. Appel: Numerical Tables for 3j, 6j, 9j Symbols, Landolt-Bomstein (Group I), Vol.3 (Springer, Berlin, Heidelberg 1968) A.P. Jucys, A.J. Savukynas: Mathematical Foundations of the Atomic Theory (Mintys, Vilnus 1973)
Chapter 7
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19
7.20 7.21 7.22 7.23 7.24 7.25 7.26 7.27 7.28
R.G. Breene Jr.: The Shift and Shape of Spectral Lines (Pergamon, New York 1961) M. Baranger: In Atomic and Molecular Processes, ed. by D.R. Bates (Academic, New York 1962) G. Traving: In Plasma Diagnostics, ed. by W. Lochte-Holtgreven (NorthHolland, Amsterdam 1968) S. Chen, M. Takeo: Rev. Mod. Phys. 29, 20 (1957) H.R. Griem: Plasma Spectroscopy (McGraw-Hill, New York 1964) H.R. Griem: Spectral Line Broadening by Plasmas (Academic, New York 1974) J. Cooper: Rev. Mod. Phys. 39,167 (1967) C.J. Tsao, B. Curnutte: J. Quant. Spectrosc. Radiat. Transf. 2, 41 (1962) H. Rabitz: Ann. Rev. Phys. Chern. 25, 155 (1974) W.R. Hindmarsh, J.M. Farr: in Progress in Quantum Electronics, Vol.2 (Pergamon, Oxford 1972) p.141 F. Schuller, W. Behmenburg: Phys. Rpt. 12,273 (1974) N. Konjevic, J.D. Roberts: J. Phys. Chern. Ref. Data 5,209,259 (1976) N. Ksonjevic, M.S. Dimitrijevic, W.L. Wiese: J. Phys. Chern. Ref. Data 13, 619 (1984) E.L. Lewis: Phys. Repts. 58,1 (1980) G. Peach: Advances in Physics 30,367 (1981) N. Allard, J. Kielkopf: Rev. Mod. Phys. 54,1103 (1982) J.R. Fuhr, L.J. Roszman, W.L. Wiese: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366 (19720; Suppl.1 (1974) J.R. Fuhr, G.A. Martin, B.J. Specht: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366, Suppl.2 (1975) J.R. Fuhr, B.J. Miller, G.A. Martin: Bibliography on Atomic Line Shapes and Shifts, NBS Spec. Publ. 366, Suppl.3 (1978) J.R. Fuhr, A. Lesage: Ibid., Suppl.4 (1993) P.R. Berman: Appl. Phys. 6, 283 (1975) P. Anderson: Phys. Rev. 76, 647 (1949) J. Szudy: Acta Phys. Polon. A 40,361 (1971) J. Szudy, W.E. Baylis: J. Quant. Spectrosc. Radiat. Transf. 15,641 (1975) S.G. Rautian, 1.1. Sobleman: Usp. Fiz. Nauk 90, 209 (1966) [English trans!': Sov. Phys. - Uspekhi 9, 701 (1967)] S. Chandrasekhar: Rev. Mod. Phys. 15, 1 (1943) R.H. Dicke: Phys. Rev. 89, 472 (1953) D.R.A. McMahon: Austral. J. Phys. 34, 639 (1981) V.N. Faddeyeva, N.M. Terentyev: Tables of the Probability Integralfor Complex Argument (Pergamon, Oxford 1961)
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7.29 M. Abramowitz, LA. Stegun (eds.): Handbook of Mathematical Functions (NBS Math. Ser., Washington 1964) 7.30 1.1. Sobel/man: Atomic Spectra and Radiative Transitions, 2nd edn., Springer Ser. Atoms Plasmas, VoLl2 (Springer, Berlin, Heidelberg 1992); 1st edn., Springer Ser. Chern. Phys., VoLl (Springer, Berlin, Heidelberg 1979) 7.31 L.D. Landau, E.M. Lifshitz: Quantum Mechanics (Pergamon, Oxford 1965) 7.32 1.1. Sobelman: Opt. Spectrosc. 1,617 (1956) 7.33 M. Baranger: Phys. Rev. 111,481,494 (1958); ibid. 112,855 (1958) 7.34 P.R. Berman, W.E. Lamb, Jr.: Phys. Rev. A 2, 2435 (1970); ibid. A4, 319 (1971) 7.35 E.W. Smith, J. Cooper, W.R. Chappel, T.D. Dillon: J. Quant. Spectrosc. Radiat. Transf. 11, 1547, 1567 (1971) 7.36 W.R. Chappeli, J. Cooper, E.W. Smith, T.D. Dillon: J. Stat. Phys. 3, 401 (1971) 7.37 V.A. Alekseyev, T.L. Andreyeva, 1.1. Sobleman: Zh. Eksp. Teor. Fiz. 62, 614 (1972) [English trans!': SOy. Phys. -JETP 35,325 (1972)] 7.38 V.A. Alekseyev, T.L. Andreyeva, I.I. Sobelman: Zh. Eksp. Teor. Fiz. 64, 813 (1973) [English trans!.: SOy. Phys. -JETP 37, 413 (1973(] 7.39 V.S. Letokhov, V.P. Chebotayev: Nonlinear Laser Spectroscopy (Springer, Berlin, Heidelberg 1977) 7.40 V.A. Aleseyev, 1.1. Sobelman: Zh. Eksp. Teor. Fiz. 55, 1874 (1968) [English trans!.: SOY. Phys. -JETP 28, 991 (1968) 7.41 A.B. Underhill, J. Waddell: NBS Circular No.603 (1959) 7.42 H.R. Griem: Astrophys. J. 132,883 (1960) 7.43 C.F. Hooper, Jr.: Phys. Rev. 165,215 (1968) 7.44 H.K. Wimmel: J. Quant. Spectrosc. Radiat. Transf. 1, 1 (1961) 7.45 G.V. Sholin, V.S. Lisita, V.1. Kogan: Zh. Eksp. Teor. Fiz. 59, 1390 (1970) [English trans!.: SOY. Phys. - JETP 32, 758 (1970)] 7.46 W.L. Wiese, D.E. Kelleher, V. Helbig: Phys. Rev. A 11, 1854 (1975) 7.47 R.L. Green: J. Quant. Spectrosc. Radiat. Transf. 27, 639 (1982) 7.48 R.L. Green, D.H. Oza, D.E. Kelleher: in Spectral Line Shapes Vo!.5, ed. by J. Szudy (Ossolineum, Wroclaw 1989) p.127 7.49 J. Seidel; Z. Naturforsch. 32a, 1195, 1207 (1977) 7.50 J.W. Dufty: in Spectral Line Shapes, VoLl, ed. by B. Wende (de Gryter, New York 1981) p.41 7.51 D.B. Boercker, C.A. Iglesias, J.W. Dufty: Phys. Rev. A 36,2254 (1987) 7.52 D.B. Boercker: in Spectral Line Shapes, Vo!.5, ed. by J. Szudy (Ossolineum, Wroclaw 1989) p.73 7.53 V.S. Lisitsa: Usp. Fiz. Nauk 122, 449 (1977) [English trans!.: SOY. Phys. Uspekhi 20, 603 (1977)] 7.54 C. Deutsch, L. Herman, H.-W. Drawin: Phys. Rev. 178,261 (1968) 7.55 R.L. Green, J. Cooper, E.W. Smith: J. Quant. Spectrosc. Radiat. Transf. 15, 1025, 1037, 1045 (1975) 7.56 H.R. Griem, A.C. Kolb, K.Y. Shen: Phys. Rev. 116,4(1959) 7.57 H. Pfennig: Z. Naturforsch. 26a, 1071 (1971); ibid. J. Quant. Spectrosc. Radiat. Transf. 12, 821 (1972) 7.58 R.L. Green, J. Cooper: J. Quant. Spectrosc. Radiat. Transf. 15, 1490 (1975) 7.59 M. Lewis: Phys. Rev. 121,501 (1961) 7.60 Nguen-Hoe, H.-W. Drawin, L. Herman: J. Quant. Spectrosc. Radiat. Transf. 4, 847 (1964)
304
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7.61 M. Caby-Eyrand, G. Gouland, Nguen-Hoe: J. Quant. Spectrosc. Radiat. Transf. 15, 593 (1975) 7.62 D. Voslamber: Z. Naturforsch. 21a, 1458 (1969); ibid. 27a, 1783 (1972); ibid Phys. Lett. A 42,469 (1973) 7.63 H. Van Regemorter: Phys. Lett. A 30,365 (1969) 7.64 N. Tran-Minh, H. Van Regemorter: J. Phys. B 5,903 (1972) 7.65 V.S. Lisitsa, G.V. Sholin: Zh. Eksp. Teor. Fiz. 61, 912 (1971) [English transl.: Sov. Phys. -JETP 34, 484 (1971)] 7.66 P. Kepple, H.R. Griem: Phys. Rev. 173,317 (1968) 7.67 C.R. Vidal, J. Cooper, E.W. Smith: J. Quant. Spectrosc. Radiat. Transf. 10, 1011 (1970); ibid. 11,263 (1971) 7.68 C.R. Vidal, J. Cooper, E.W. Smith: Astrophys. J. 214, Suppl.25, 37 (1973) 7.69 N. Tran-Minh, N. Feautrier, H. Van Regemorter: J. Phys. B 8, 1810 (1975); ibid. B 9, 1871 (1976); J. Quant. Spectrosc. Radiat. Transf. 16, 849 (1976) 7.70 W.L. Wiese, D.E. Kelleher, D.R. Paquette: Phys. Rev. A 6,1132 (1972) 7.71 G. Boldt, W.B. Cooper: Z. Naturforsch. 19a, 968 (1964) 7.72 R.C. Elton, H.R. Griem: Phys. Rev. 135, 1550 (1964) 7.73 D.E. Kelleher, W.L. Wiese: Phys. Rev. Lett. 31, 1431 (1973) 7.74 K. Grutzmacher, B. Wende: Phys. Rev. A 16, 243 (1977) 7.75 K. Grutzmacher, B. Wende: Phys. Rev. A 18, 2140 (1978) 7.76 R. Stamm, E.W. Smith, B. Talin: Phys. Rev. A 30, 2039 (1984) 7.77 R. Stamm, B. Talin, E. Pollock, C. Iglesias: Phys. Rev. A 34,4144 (1986) 7.78 U. Frisch, A. Brissaud: J. Quant. Spectrosc. Radiat. Transf. 11, 1753 (1971) 7.79 A. Brissaud, U. Frisch: J. Quant. Spectrosc. Radiat. Transf. 11, 1767 (1971) 7.80 H.R. Griem: Astrophys. J. 148,547 (1967) 7.81 L.A. Minaeva, 1.1. Sobelman: J. Quant. Spectrosc. Radiat. Transf. 8, 783 (1968) 7.82 H. Van Regemorter, D. Hoang-Binh: Astron. Astrophys. 277, 623 (1993) 7.83 D.E. Kelleher: J. Quant. Spectrosc. Radiat. Transf. 25,191 (1981) 7.84 Yu.A. Vdovin, V.M. Galitskii: Zh. Eksp. Teor. Fiz. 52, 1345 (1967) [English transl.: Sov. Phys. -JETP 25,894 (1967)] 7.85 Yu.A. Vdovin, N.N. Dobrodeyev: Zh. Eksp. Teor. Fiz. 55, 1047 (1968) [English transl.: Sov. Phys. -JETP 28,544 (1968)] 7.86 A.W. Ali, H.R. Griem: Phys. Rev. A 140, 1044 (1965); ibid. A 144, 366 (1966) 7.87 D.N. Stacey, J. Cooper: Phys. Lett. A 30, 49 (1969) 7.88 C.G. Carrington, D.N. Stacey, J. Cooper: J. Phys. B 6,417 (1973) 7.89 J. Cooper, D.N. Stacey: Phys. Lett. A 46,299 (1973) 7.90 M. Movre, G. Pichler: J. Phys. B 13, 697 (1980) 7.91 R.J. Exton: J. Quant. Spectrosc. Radiat. Transf. 15, 1141 (1975) 7.92 V.A. Alekseyev, 1.1. Sobel'man: Zh. Eksp. Teor. Fiz. 49, 1274 (1965) [English transl.: Sov. Phys. - JETP 22,882 (1965)] 7.93 H. Bethe, E. Salpeter: Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, GOttingen 1957) 7.94 L.P. Presnyakov: Phys. Rev. A 2,1720 (1970) 7.95 A. Omont: J. Physique 38,1343 (1977) 7.96 M. Matsuzawa: J. Phys. B 10,1543 (1977) 7.97 B.P. Stoicheff, E. Weinberger: Phys. Rev. Lett. 44, 733 (1980) 7.98 R. Kachru, TW. Mossberg, S.R. Hartmann: Phys. Rev. A 21, 1124 (1980) 7.99 R. Kachru, TJ. Chen, T.W. Mossberg, S.R. Hartmann: Phys. Rev. A 25, 1546 (1982) 7.100 H. Heinke, J. Lawrenz, K. Niemax, K.H. Weber: Z. Phys. A 312,329 (1983)
References
305
7.101 H. Massey, E. Burhop: Electronic and Ionic Impact Phenomena (Clarendon, Oxford 1952) 7.102 MA Mazing, M.A. Vrubleyskaya: Zh. Eksp. Teor. Fiz. 50, 343 (1966) [English trans!.: SOY. Phys. -JETP 23, 228 (1966)] 7.103 M.A. Mazing, P.D. Serapinas: Zh. Eksp. Teor. Fiz. 60, 541 (1971) [English trans!': SOY. Phys. -JETP 33, 294 (1971)]
List of Symbols
Constants
Bohr radius Velocity of light e Elementary charge If = h/21t Planck's constant divided by 21t m Mass of electron Ry = me 4 /2lf 2 Rydberg unit of energy a o = If/me 2
c
Quantum numbers j J JT
Electron angular momentum Atomic angular momentum Total angular momentum of a system including atom and outer (scattered) electron / Electron orbital momentum L Atomic orbital momentum Lp Orbital momentum of atomic core (of parent ion) LT Total orbital momentum of a system including atom and scattered electron m, M Magnetic quantum numbers n Principal quantum number s Electron spin momentum S Atomic spin momentum Sp Spin momentum of atomic core (of parent ion) ST Total spin momentum of a system including atom and scattered electron A. Orbital momentum of outer (scattered) electron Basic Notations
ao
a, a 1 A Aij
Set of quantum numbers for initial state of an atom Set of quantum numbers for final state Fitting parameter for approximation of rate coefficients (vu) Einstein coefficient for spontaneous emission (radiative transition probability) [S-I]
C
Fitting parameter for approximation of cross sections u D Fitting parameter for analytical approximation of calculated cross sections and rate coefficients DE Energy scaling parameter E. Energy of bound electron in state a .to,.t Initial and final energies of free electron lij Oscillator strength 1(8) Scattering amplitude F)., F;" FJ, Radial functions of scattered electron in various representations g(a) Statistical weight of level a G:;L.=(/n-l[SpL p]/SL} /nSL) Coefficient of fractional parentage [see Ref. 1.1] G (P) Function of analytical approximation for rate coefficients (vu) Gr(r,r') Green's function jx(z) Spherical Bessel function Jf (P) Holtsmark distribution function n* = Z2 Ry/I E I Effective principal quantum number N Number density of particles [cm - 3] PI (cos 8) Legendre polynomials P" Pn/ , P"I (r) = r Rnl (r), where Rnl (r) is radial function for bound electron Principal value of integral P Q, QK Angular factor defining the dependence of cross sections on angular momenta for transitions with no change of spin Q", Q; Angular factor for exchange cross sections S, Sit Scattering matrix T Temperature in energy units 7;k Transition matrix u = .t/AE = (.to - AE)/AE Electron energy in threshold units v Velocity of particles (vu) Rate coefficient averaged over Maxwellian velocity distribution [cm 3 S-I] Dimensionless transition probability, frequency of collisional transitions [S-I]
J
J
w.o.
307
List of symbols
w. (c)
Autoionization probability for atomic state c [S-I] z Charge of atomic core (of parent ion) Z = z -1 Charge of ion !E Charge of nucleus y Line width (full width) in Chapter 7, set of quantum numbers for atomic term in Sect. 6.2 y = aM lmm 3 Set of quantum numbers for a system including atom and scattered electron r = aA1/2L T ST Set of quantum numbers for a system including atom and scattered electron in representation of total momenta LI Line shift in Chapter 7 LIE = E j - Ek Energy difference for levels i and k K Multipole order Kd Rate coefficient of dielectronic recombination [em 3 s - 1] K, Rate coefficient for three-body recombination [cm 6 s- 1] K. Rate coefficient of radiative recombination [cm 3 s- 1j
o
Solid angle Impact parameter, density matrix P.u Spherical components of density matrix uaoa,u(ao, a) Effective cross section u(aoAo,aA) Partial cross section cp Fitting parameter for approximation of cross sections q. (u) Functions of analytical approximation for cross sections X Fitting parameter for approximation of rate coefficients Ud2" ·jn] = J(2jl + 1)(2j2 +1) ... (2jn+ 1) (ml m2lsu) Klebsch-Gordan coefficients (abbreviated notation)
p
Wigner's 3j symbol
6j symbol
~U} (aoll
Til
9j symbol a)
Reduced matrix element
Subject Index
Action function 80 Analytic approximate formulas for cross sections 78,85,86,111,112,117, 118, 119 for rate coefficients 79, 86, 111, 112, 113, 114, 116, 117, 118, 120, 123, 130, 285 for recombination coefficient 153 Angular factors 30-35, 225-231 Atomic core 1 Autoionization 1, probability of 126, 127 Bethe formula 40 Bohr quantization condition 80 Boltzmann distribution 4 Born approximation 36, 74 Born formula 37 Born-Oppenheimer approximation 31 Brinkman-Kramers formula 102, 104 Broadening by neutral particles 289-296 Doppler 245-251,263 impact theory of 239, 251-263 of highly excited (Rydberg) states 284-286 Cascade matrix 137, 138 Clesch-Gordan coefficient 257 Collision strength 23 Collisional-radiative model 136 Coronal model 136 Correlation function 238 Coulomb-Born approximation 31,51 Cross section differential 9 normalization of 54-56 of broadening 239, 240, 254 of shift 239, 240, 254 partial 21 Debye radius 278 Density matrix method in broadening theory 251-273 Detailed balance, principle of 5 Dicke narrowing of spectral lines 250
Dielectronic recombination cross section of 122 rate coefficient of 122, 123 semiempirical formula for the rate coefficient of 130 simplified model of 120, 121, 127 Dielectronic satellites 130 -136 Differential cross section 9 Distorted wave approximation 31 Doppler broadening 245-251,263 Effective principal quantum number 159, 203 Exchange radial integral 31, 32, 51 Fractional parentage coefficients 39, 235, 236 Green's function free electron 61 Coulomb 67,68,75 Holtsmark distribution 276 Hooper distribution 278 Impact parameter 14,84 Intercombination transitions 50 Intermediate coupling 32 Ionization cross sections 45-47 Ionization equilibrium 7, 8 OJ symbol 231 3j symbol, Wigner's 20 6j symbol 231 9j symbol 21, 236 12j symbol 236 Klein-Rosseland formula 5 Kramers approximation for radiative recombination 115, 116 for radiative transitions 139
309
Subject Index Landau-Zener formula 91 Line strength 99 Local thermodynamic equilibrium 136 LS coupling 20, 29 Matrix, density 252 K 22 S 22 T 22 Maxwell distribution 4 Milne formula 6 Normalization of cross sections 54 Ochkur approximation 51, 52 Optical electron 1, 17 Optical theorem 11 Orthogonalized functions approximation 51 Oscillator strength 40 Partial cross section 21 Partial wave expansion 10 Partition function 4 principle of detailed balance 5
Radial integrals, direct 31 exchange 31 Recombination collisional-radiative 143 dielectronic 1 radiative 1 three-body 1, 150-153 Reduced matrix element 229 Saha-Boltzmann equation 141 Saha distribution 4 Satellites, dielectronic 130-136 Scattering amplitude 10 Scattering channel 15, 69 Scattering matrix 22 Spherical Bessel, Hankel, and Neumann functions 27 Sum rules for cross sections 225 for 3nj symbols 232-235 Thermodynamic equilibrium 3 Thomson formula 119 Three-body recombination 1, 150-153 T matrix 22 Triangular condition 231 Unitarity of S matrix 16
Q-factors 30-35, 225-231 Quadratic Stark effect constant 286 Quantum theory of broadening
Weisskopf radius 241 Wigner-Eckart theorem 38 Wigner's 3j symbol 20
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