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category is a section that treats the effects of instrumental and environmental noise on the precision and accuracy of spectroscopic absorption measurements. Also largely new is the development of general chromatographic theory from kinetic considerations. The chapter on optical instruments brings together information that was formerly found in chapters on infrared, ultraviolet and visible, and atomic absorption spectroscopy; flame and emission spectroscopy; and fluorescence and Raman spectroscopy. Since the publication of the first edition, a host of modifications of wellestablished instrumental methods have emerged and are now included. Among these new developments are Fourier transform nuclear magnetic resonance and infrared methods, infrared photometers for pollutant measurements, laser modifications of spectroscopic measurements, nuclear magnetic resonance measurem2
Z = JR2
I I
XL ~=ilfCtanR
z· o
:lIE
.JR' +(XL -Xci' (X, - Xci arc tan -R--
FIGURE 2-11 Vector diagrams for series circuits: (a) RC circuit, (b) RL circuit, (c) series RLC circuit.
2-42 and 2-44
1 = V"
, Z
= J(SO)2
0 R
:
Za .JR2+xL2
EXAMPLE
Z
----
z
~
Because the outputs of the capacitor and inductor are 180 deg out of phase, their combined effect is determined from the difference of their reactances.
(2-45)
A similar result is obtained when Equations 2-43 and 2-45 are substituted into Equation 2-41. In one sense, the capacitive and inductive reactances behave in a manner similar to that of a resistor in a circuit-that is, they tend to impede the flow of electrons. They differ in two major ways from resistance, however. First, they are frequency dependent; second, they cause the current and voltage to differ in phase. As a consequence of the latter, the phase angle must always be taken into account in considering the behavior of cir-
XL
(2-46)
For the following circuit, calculate the peak current and voltage drops across the three components.
+X~
and for the RL circuit Z
cuits containing capacitive and inductive elements. A convenient way of visualizing these effects is by means of vector diagrams. Vector Diagrams for Reactive Circuits. Because the voltage lags the current by 90 deg in a pure capacitance, it is convenient to let q, equal -90 deg for a capacitive reactance. The phase angle for a pure inductive circuit is +90 deg .. For a pure resistive circuit, q, will be equal to 0 deg. The relationship ainong Xc, XL' and R can then be represented vectorially as shown in Figure 2-11. As indicated in Figure 2-11c, the impedance of a series circuit containing a resistor, an inductor, and a capacitor is determined by the relationship
+ (40 - 2W
1" = 10 V/53.8
n = 0.186
= 53.8 n A
VR = 0.186 x SO = 9.3 V Vc = 0.186 x 20
= 3.7 V
VL = 0.186 x 40 = 7.4 V
Note that the sum of the three voltages (20.4) in the example exceeds the source voltage. This situation is possible because the voltages are out of phase, with peaks occur-
ring at different times. The sum .of the instantaneous voltages, however, would add up to 10V. High-Pass .nd Low-Pass Fikers. Series RC and RL circuits are often used as filters to attenuate high-frequency signals while passing low-frequency components (a low-pass filter) or, alternatively, to reduce lowfrequency components while passing the high (a high-pass filter). Figure 2-12 shows how series RC and RL circuits can be arranged to give high- and low-pass filters. In each case. the input and output are indicated as the volt~es (Vp)i and (Vp) •. ; In order to employ an RC circuit as a high-pass filter, the' output voltage is taken across the resistor R. The peak current in this circuit is given by Equation 2-38. That is, I _ I' -
(Vp);
JR
2
+ (l/wC)2
Since the voltage drop across the resistor is in phase with the current,
1 = (VI'). ,
R
The ratio of the peak output to the peak input voltage is obtained by dividing the first equation by the second and rearranging. Thus,
A plot of this ratio as a function of frequency is shown in Figure 2-13a; here, a resistance of 1.0 x 10· n and a capacitance of 0.10 pF was employed. Note that frequencies below 50 Hz have been largely removed from the input signal. It should also be noted that because the input and output voltages are out of
resistor -behavior just opposite of that of the RC circuit. Curves similar to Figure 2-13 are obtaincd for RL filters. Low- and high-pass filters are of great importance in the design of electronic cirCUits.
~-----~
Resonant Circuits A resonant or RLC circuit consists of a resistor, a capacitor, and an inductor arranged in series or parallel, as shown in Figure 2-14. Series Resonant Filters. The impedance of the resonant circuit shown in Figure 2-14a is given by Equation 2-46. Note that the impedance of a series inductor and capacitor is the difference between their reactances and that the net reactance of the combination is zero when the two are identical. That is, the FIGURE 2-12 Filter circuits: (a) Highpass RC filter, (b) low-pass RC filter, (c) highpass RL filter, and (d) low-pass RL filter.
~ ~ ~ phase, the peak voltages will occur at different times. For the low-pass filter shown in Figure 2-12b, we may write
::
0." 0.2 0.0 1
100
(al
Substituting Equation 2-42 gives
(Vp). = Ip/(roC)
(VP)i
1
O.B
=
roCJR2
+ (l/roC)2
1
~
0.6
"i
;0-
0.4
= j(roCR)2 + 1 (2-48)
Figure 2-13b was obtained with this equation. As shown in Figure 2-12, RL circuits can also be employed as filters. Note, however, that the high-pass filter employs the potential across the reactive element while the low-pass filter employs the potential across the
1
1.= 21tJfC
FIGURE 2-14 Resonant circuits employed as filters: (a) series circuit and (b) parallel circuit.
0.6
(Vp). = IpXc
(Vp).,
Upon rearrangement, it is found that
(bl
Frequency. Hz
Multiplying by Equation 2-38 and rearranging yields
1 2nf.C = 2nf.L
1.0 O.B
Frequency. Hz (b)
FIGURE 2-13 Frequency response of (a) the high-pass filter shown in Figure 2-118 and (b) the low-pass filter shown in Figure 2-12b; R = 1.0 X 104 nand C = 0.10 JLF.
energy stored in the magnetic field of the inductor during the other. Thus, during a one-half cycle, the energy of the capacitor forces the current through the inductor; in the other half cycle, the reverse is the case. In principle, a current induced in a closed resonant circuit would continue indefinitely except for the loss resulting from resistance in the leads and inductor wire. The resonant frequency I. can be readily calculated from Equations 2-42 and 2-43. That is, whenf=f., XL = Xc, and
only impedance in the circuit is that of the resistor or, in its absence, the resistance of the inductor coil and ·the other wiring. This behavior becomes understandable when it is recalled that in a series RC circuit, the potentials across the capacitor and the inductor are 180 deg out of phase (see the vector diagrams in Figure 2-11). Thus, if Xc and XL are alike, there is no net potential drop across the pair. Electricity continues to flow, however; therefore, the net reactance for the combination is zero. Also, when XL and Xc are not identical, their net reactance is simply the difference between the two. The condition for resonance is XL= Xc
Here, the energy stored in the capacitor during one half cycle is exactly equal to the
The follQwing example will show some of the properties of a series resonant circuit.
i EXAMPLE Assume the following values for the components of the circuit in Figure 2-14a: (Vp)i = 15.0 V (peak voltage), L= 100 mH, R = 20 n, and C = 0200 JLF. (a) Calculate the peak current at the resonance frequency and at a frequency 75 Hz greater than the resonance frequency. (b) Calculate the peak potentials across the resistor, inductor, and capacitor at the resonant frequency. (a) We obtain the resonant frequency with Equation 2-49. Thus, 1
1.;= 2nJl00
x 10
3
x 0.2 x 10
6
= 1125 Hz
By defidition, XL = Xc at!.; therefore, Equation 2-46 becomes
To calculate I, at 1200 Hz, we substitute Equations 2-42 and 2-43 into Equation 2-46. Thus, Z = =
JR
+ [27tjL -
J202
+ [27t x
2
(1/27tfC)j2 1200 x 100 x 10
- 1/(27t x 1200 x 0.200 x lO-ti)j2 = J400
+ (754 -
663)2 = 93 0
and I, = 15.0/93 = 0.16 A. (b) At f. = 1125 Hz, we have found that I,= 0.75 A. The potentials across the resistor, capacitor, and inductor (V,)Il' (V,)L, and (J-;,)c are
(V')ll
= 0.75
x 20
= 15.0 V
= 0.75
1050 Frequency,
(V,)L = I,XL X
27t x 1125 x 100 x 10-3
= 530 V
Hz
1100 . Frequenc:y. Hz
FIGURE 2-15 Frequency response of a series resonant circuit; R = 20Q,L = l00mH, C 0.2 IlF, and (J-;,), 15.0 V.
FIGURE 2-16 Ratio of output to input voltage for the series resonant filter; L = 100 mH, C = 0.20 IlF, and (v,,)i = 15 V.
If the output of the circuit shown in Figure 2-14a is taken across the inductor or capacitor, a potential is obtained which is several times larger than the input potential (see part b of the example). Figure 2-16 shows a plot of the ratio of the peak voltage across the inductor to the peak input voltage as a function of frequency. A similar plot is obtained if the inductor potential is replaced by the capacitor potential. The upper curve in Figure 2-16 was obtained by substituting a 10-0 resistor for the 20-0 resistor employed in the example. Parallel Resonant Filters. Figure 2-14b shows a typical parallel resonant circuit. Again, the condition of resonance is that Xc = XL and the resonant frequency is given by Equation 2-49. The impedance of the parallel circuit is given by
filter (Equation 2-46). Note that the impedance in the latter circuit is a minimum at resonance, when XL = Xc. In contrast, the impedance for the parallel circuit at resonance is a maximum and in principle is infinite. Consequently, a maximum voltage drop across (or a minimum current through) the parallel reactance is found at resonance. With both parallel and series circuits at resonance, a small initial signal causes resonance in which electricity is carried back and forth between the capacitor and the inductor. But current from the source through the reactance is minimal. The parallel circuit, sometimes called a tank circuit, is widely used as for tuning radio or television circuits. Tuning is ordinarily accomplished by adjusting a variable capacitor until resonance is achieved.
=
=
Since XL = Xc at resonance, f
Part (b) of this example shows that, at resonance, the peak voltage across the resistor is the potential of the source; on the other hand, the potentials across the Inductor and capacitor are over 35 times greater than the input potential. It must be realized, however, that these peak potentials do not occur simultaneously. One lags the current by 90 deg and the other leads it by a similar amount. Thus, the instantaneous potentials across the capacitor and inductor cancel, and the potential drop across the resistor is then equal to the input potential. Clearly, in a circuit of; this type, the capacitor and inductor may have to withstand considerably larger potenti;als than the amplitude of the input voltage would seem to indicate. Figure 2-15, which was obtained for the circuit shown in the example, demonstrates that the output of a series resonant circuit is a narrow band of frequencies, the maximum of which depends upon the values chosen for L and C.
Z =
JR2 + (XC-XL XLXC
Behavior of RC )2
(2-50)
It is of interest to compare this equation with the equation for the impedance of the series
Circuits with Pulsed Inputs When a pulsed input is applied to an RC circuit, the voltage outputs across the capacitor and resistor take various forms, depend-
ing upon the relationship between the widtp of the pulse and the time constant for the circuit. These effects are illustrated in Figure 2-17 where the input is a square wave having a pulse width of 1",' seconds. The second column shows the variation in capacitor potential as a function of time, while the third column shows the change in resistor potential at the same times. In the top set of plots (Figure 2-17a~ the time constant of the circuit is much greater than the input pulse width. Under these circumstances, the capacitor can become only partially charged during each pulse. It then discharges as the input potential returns to zero; a sawtooth output results. The output of the resistor under these circumstances rises instantaneously to a maximum value and then decreases essentially linearly during the pulse lifetime. The bottom set of graphs (Figure 2-17c) illustrates the two outputs when the time constant of the circuit is much shorter than the pulse width. Here, the charge on the capacitor rises rapidly and approaches full charge near the end of the pulse. As a consequence, the potential across the resistor rapidly decreases to zero after its initial rise. When V;goes to zero, the capacitor discharges immediately; the output across the resistor peaks in a negative direction and then quickly approaches zero. These various output wave forms find applications in electronic circuitry. The sharply peaked voltage output shown in Figure 2-17c is particularly important in timing and trigger circuits.
SIMPLE ELECTRICAL
MEASUREMENTS
This section describes selected methods for the measurement of current, voltage, and resistance. More sophisticated methods for measuring these and other electrical properties will be considered in Chapter 3 as well as in later chapters.
Meter
The Ayrton Shunt. The Ayrton shunt, shown in Figure 2-19, is often employed to vary the range of a galvanometer. The example that follows demonstrates how the resistors for a shunt can be chosen.
f1o-------AJV:J
'" :\,
'-
..!!..'Aft
~ Shunt
(b)
RCO!!T.
(el RC~T.
T+l .JnLJnLJnL T+ !7l J Vi 0
1..
o
t
Vc 0
1..
--Time-
:Tl iT1
LJ U
0 --T~_
L
_
T+~' 1.._
I
-,
I
V" 0
o
-,
I
-
--Time---"
FIGURE 2-17 Output signals VR and Vc for pulsed input signal J-l; (a) time constant ~ pulse width Tp; (b) time constant ~ pulse width; (c) time co~tant z The common unit of frequency is the hertz Hz, which is equal to one cycle per second.
emls
+< I/J)
(4-3)
where y is the electric force, -1 is the amplitude or maximum value for y, t ~s time, and I/J is the phase angle, a term which has ~n defined earlier (p. 12). The angular velOCIty of the vector, w, is related to the frequency of the radiation v by the equation w = 2ltv
or cosine terms such as those shown in Equation 4-5. For example, the square wave form often employed in electronics can be represented by an equation having the form y = A (sin 2nvt
+ i sin
approximate the shape of a square wave. As shown by the solid line in Figure 4-4b, the resultant more closely approaches a square wave when nine waves are incorporated. . Mathematically, resolution of a complex wave form into its sine or cosine components is tedious and time consuming; modern computers, however, have made it practical to exploit the power of the Fourier transformation on a routine basis. The application of this technique will be considered in the discussion of several types of spectroscopy. Ditrraction. Figure _4-5 is a schematic representation of diffraction, which occurs whenever waves of any type pass by a sharp barrier or through a narrow opening. Diffrac-
67tvt
(4-6)
+! Distance
FIGURE 4-2 radiation.
Effect of change of medium on a monochromatic
Substitution of this relationship tion 4-3 yields
into Equa-
y = A sin(2nvt ~ tfJ)
(4-4)
Superposition of Wa,ts. fhe principle of superposition states that when two or more waves traverse the same space, a displacement occurs which is the sum of the displacements caused by the individual waves. This principle applies to electromagnetic waves, where the displacements involve an electrical force field, as well as to several other types of waves, where atoms or molecules are displaced. For example, when n electromagnetic waves of the same frequency but differing amplitudes and phase angles pass some point in space simultaneously, the principle of superposition and Equation 4-4 permits us to write y = Al sin(2nvt
+ tfJd
+ A'2 sin(2nvt t 4>2) + ... + A. sin(2nvt + 4>.) .
i
(4-S)
where y IS the resultant for~ field. The solid line in Figure 4-3a shows the application of Equation 4-S to two waves of identical frequency but somewhat different amplitude and phase angle. Note that the resultant is a sine wave with the same frequency but different amplitude from the component
beam of
waves. Figure 4-3b differs from 4-3a in that the difference in phase angle is greater; here, the resultant amplitude is smaller than those of the component waves. Clearly, a maximum amplitude for the resultant will occur when the two waves are completely in phase; this situation prevails whenever the phase difference between waves (tfJI - 4>2) is 0 deg, 360 deg, or an integer multiple of 360 deg. Under these circumstances, maximum constructive interference is said to occur. A maximum destructive interference occurs when (tfJI - tfJ2) is equal to 180 deg or 180 deg plus an integer multiple of 360 deg. The property of interference plays an important role in many instrumental methods based on electromagnetic radiation. Figure 4-3c depicts the superposition of two waves with the same amplitude but different frequency. The resulting wave is no longer sinusoidal. It does, however, exhibit a periodicity. Thus, the wave from 0 to B in the figure is identical to that from B to C. An important aspect of superposition is that the process can be reversed by a mathematical operation called a Fourier transformation. Jean Fourier, an early French mathematician (1768-1830~ demonstrated that any wave motion, regardless of complexity, can be described by a sum of simple sine
sin
to nvt + ... + ~ sin 2n7tvt)
A graphical representation of the summation process is shown in Figure 4-4. The solid line in Figure 4-4a is the sum of three sine waves differing in amplitude in the ratio of S : 3 : I and in frequency in the ratio of 1 : 3 : S. Note that the resultant is already beginning to
----Time---~
----Time,----
(al
(b)
(el
4-3 Superposition of sinusoidal waves: (a) Al < A2, (tfJI - tfJ2) = -20°, VI = V2; (b) Al < A2, (4)1 - 4>2) = -200°, VI = V2; (c) Al = A2, 4>1 = 4>2' VI = I.S V2' In each instance, the solid curve is the resultant from combination of the two dashed curves. FIGURE
.t
Superposition of 3 .ine _ y - Al.in 2m.
/
'V' ,
"\
\ \
Su-,tion
\ -
y-Ahin2lh't.tsin8""t
,""'-
of 9 .ine_ ••..• ;!; sin 34,,,,,)
\
, Superposition of 3 sine _ , y -
t sin 81M. t sin 1000l
Alsin 2•.vr •
\
t .in 8•.vr • t sin 10,M)
lb)
FIGURE 4-4 Superposition of sine waves to form a square wave: (a) combination of three sine waves, and (b) combination of three, as in (a), and nine sine waves.
lion is readily observed by generating waves of constant frequency in a tank of water and observing the wave crests before and after they pass through a rectangular opening or slit. When the slit is wide relative to the wavelength of the motion (Figure 4-Sa~ diffraction is slight and difficult to detect. On the other hand, when the wavelength and the slit opening are of the same order of magnitude, as in Figure 4-Sb, diffraction becomes pronounced. Here, the slit or opening behaves as a new source from which waves radiate in a series of nearly 180-deg arcs. Thus, the direction of the wave front appears to bend as a consequence of passin.s the two edges of the slit. - Diff~ion is a consequence of interference. This relationship is most easily understood by considering an experiment, performed first by Thomas Young in 1800, by which the wave nature of light was unambiguously d~monstrated. As shown in Figure 4-6, a paralI4I beam of radiation is allowed to pass through a na~row slit A (or in Young's experiment, a pinhole) whereupon it is diffracted and illuminates more or less equally two closely spaced slits or pinholes Band C; the radiation emerging from these slits is then observed on the screen lying in a plane XY.1f the radiation is monochromatic, a series of darlc and light images perpendicular to the plane of the page is observed. Figure 4-6b shows the intensities of the various bands reaching the screen. If, as in this diagram, the slit widths approach the wavelength of radiation, the band intensities decrease only gradually with increasing distances from the central band. With wider slits, the decrC'ase is much more pronounced. The existence of the central band E, which lies in tht shadow of the opaque material separating the two slits, is readily explained by noting that the paths from B to E and C to E are identical. Thus, constructive interference of the diffracted rays from the two slits occurs, and an intense band is observed. With the aid of Figure 4-6c, the conditions for maximum constructive interference, which result
lbl
FIGURE 4-5 Propagation of waves through a slit. (a) x)' ~ l; (b) x)';:: l.
in the other light bands, are readily derived. The angle of diffraction (J is the angle from the normal formed by the dotted line extending from a point, 0, halfway between the slits, to the point of maximum intensity, D. The solid lines BD and CD represent the light paths from the slits B an
Substitution into Equation 4-7 gives
n.l. - ~
;;;:~ DD
).
/
(4-8) .
DE
Equation 4-8 permits the calculation of.~ wavelength Crom the three measurable quantities.
I-
EXAMPLE Suppose that the screen in Figure 4-6 is 2.00 m Crom the plane of the slits and that the slit spacing is 0.300 mm. What is the wavelength of radiation if the fourth band is located 15.4 mm from the central band? Substituting into Equation 4-8 gives
DiffrllCtion byl slngllsfit
' _ 0.300 )( 15.4 A 2)( 1000 mm .l.- 5.78 )( 10-· mm
4
f I Ii n n
~olLVJJJlliL X-Diltlncl-Y (bl
FIGURE 4-6 slits.
(el
Diffraction of monochromatic
Because BC is so very small compared to DE, FD closely approximates BD, and the distance IT is a good measure of the difference in path lengths of beams BD and CD. For the two beams to be in phase at D, it is necessary that CF correspond to the wavelength of the radiation; that is, .l.z= CF
z=
BC sin 8
Reinforcement would also occur when the additional path length corresponds to 2)., 3)., and so forth. Thus, a more general expression
radiation
by
for the light bands surrounding the central band is n.l.= BC sin 8 (4-7) where D is an integer called the order of interference. The linear displacement DE of the diffracted beam along the plane of the screen is a function of the distance DE between the screen and the plane ofthe slits, as well as the spacing between the slits BC; that is, DE= OD sin 8
or
578 mn
Coherent Wiatioa. In order to produce a diffraction pattern such as that shown in Figure 4-6a, it is ncccssary that the clcctromagnetic waves traveling from slits B and C to any given point on the screen (such as D or E) have sharply defined phase differences that remain entirely constant with time; that is, the radiation from slits Band C must be coherent. The conditions for cohcrcncc are: (1) the two sources of radiation must have identical frequency and wavelength (or sets of frequencies and wavelengths); and (2) the phase relationships between the two beams must remain constant with time. The necessity for these requirements can be demonstrated by illuminating the two slits in Figure 4-6a with individual filament lamps. Under these circumstances, the well-defined light and dark patterns disappear and are replaced by a more or less uniform illumination of the screen. This behavior is a consequence of the incoherent character of filament sources (many other sources of electromagnetic radiation are incoherent" as well). In incoherent sources, light is emitted by individual atoms or molecules, and the resulting beam is the summation of countless indi-
vidual events, each of which lasts on the order of 10-1 second. Thus, a beam of radiation Crom this type of source is not continuous, as is the case in microwave or laser radiation; instead, it is composed of a series of wave trains that arc a few meters in length at most Because the processes that produce a train are random, the phase differences among the trains must also be variable. A wave train from slit B may arrive at a point on the screen in phase with a wave train from C, and constructive interference will occur; an instant later, the trains may be totally out of phase at the same point, and destructive interference will occur. Thus, the radiation at all points on the screen is governed: by the random phase variations lU1,1ongthe wave trains; uniform illumination, which represents an average for the trains, is the result. Sources do exist which produce- electromagnetic radiation in the form of tr~ins with essentially infinite length and consfant frequency. Examples include radio frequency oscillators, microwave sources, optical lasers, and various mechanical sources such as a two-pronged vibrating tapper in a riffle tank. When two of these are employed as sources for the experiment shown in Figure 4-6, a regular diffraction pattern is observed. Diffraction patterns can be obtained from 'random sources, such as a tungsten filament, provided that an arrangement similar to that shown in Figure 4-6a is employed. Here, the very narrow slit A assures that the radiation reaching Band C emanates from the same small region of the source. Under these circumstances, the various wave trail1$ exiting from slits Band C have a constant set of frequencies and phase relationships to one another and are thus coherent. If the slit at A is widened so that a larger part of the source is sampled, the diffraction pattern becomes less pronounced because the two beams arc only partially coherent. If slit A is made sufficiently wide, the incoherence may become great enough to produce only a constant illumination across the screen.
ELECTROMAGNETIC RADIATION
PlU1:icleProperti •• of R_iation Energy of EIedromagnetic Radiation. An understanding of certain interactions between radiation and matter requires that the radiation be treated as packets of energy called photons or quanta. The energy of a photon depends upon the freql,lency of the radiation, and is given by
where h is Planck's cOnstant (6.63 x 10-27 erg see). In terms of w~velength, he E=-
;1
Thus, an X-ray photon (1- 10-8 em) has approximately 10,000 times the energy of a photon emitted by a hot tungsten wire (1- 10-4 em~ TIle Photoelectric ~ffed. The need for a particle model to de¢ribe the behavior of electromagnetic radiation can be seen by consideration of the photoelectric ~fJect. When sufficiently energetic radiation impinges on a metallic surface, electrons are emitted. The energy of the emitted electrons is found to be related to the frequency of the incident radiation by the equation E = hv - w
(4-11)
where w, the work function, is the work required to remove the electron from the metal to a vacuum. While E is directly dependent upon the frequency, it is found to be totally independent of the intensity of the beam; an increase in intensity merely causes an increase in the number of electrons emitted with energy E. Calculations indicate that no single electron could acquire sufficient energy for ejection if the radiation striking the metal were uniformly distributed over the surface; nor could any electron accumulate enough energy for its removal in a reasonable length of time. Thus, it is necessary to assume that the energy is not uniformly distributed over the
beam front, but rather is concentrated at certain points or in particles of energy. The work w required to cause emission of electrons is characteristic of the metal. The alkali metals possess low work functions and emit electrons when exposed to radiation in the visible region. Metals to the right of the alkali metals in the periodic chart have larger work functions and require the more energetic ultraviolet radiation to exhibit the photoelectric effect. As we shall note in later chapters, the photoelectric effect has great practical importance in the detection of radiation using phototubes. Energy Units. The energy of a photon that is absorbed or emitted by a sample of matter can be related to an energy difference between two molecular or atomic states, or to the frequency of a molecular motion of a constituent of the matter. For this reason, it is often convenient to describe radiation in energy units, or alternatively in terms of frequency (Hz) or wavenumber (cm -1), which are directly proportional to energy. On the other hand, the experimental measurement of radiation is most often expressed in terms of the reciprocally related wavelength units such as centimeters, micrometers, or nanometers. The chemist must become adept at interconversion of the various units employed in spectroscopy. Conversion factors commonly encountered for transformations are found inside of the front cover of this text. The electron volt (eV) is the unit ordinarily employed to describe the more energetic X-ray or ultraviolet radiation. The electron volt is the energy acquired by an electron in falling through a potential of one volt. Radiant energy can also be expressed in terms of energy per mole of photons (that is, Avogadro's number of photons). For this purpose, units of kcal/mol or cal/mol are convenient.
kCII/moi
Electron volts. .V
em-'
9.4 X 10'
4.1 X loa
3.3 X 10'·
W••••• ngth. A
--em 3 X 10-11
Frequency.
•
Hz
1021
Type rad. tion
9.4 X 10'
4.1 X 10'
3.3 X 10·
3 X 10-"
Type IpeCtroIcopy
3.3 X loa
3 X 10-'
10"
9.4 X 10'
4.1 X 100
3.3 X 10'
3 X 10-5
10'"
Nucl.lr
f
~
4.1 X 10"
t
G"""""roy emission
10'·
9.4 X 10"
Type quantum transition
t
Gemma I'IY
Electronic
(inner sh.lI)
1
1
V
VK UV. absorption
Yr· 9.4 X 10-'
4.1 X 10-2
3.3 X 102
3 X 10-'
10"
It
9.4 X 10-'
4.1 X 10-4
3.3 X 10·
3 X 10-'
10"
t
9.4 X 10-1
4.1 X 10'"
3.3 X
10-2
3 X 10'
10·
fl'::'r
9.4 X 10-'
4.1 X 10-·
I
_1_-
3.3 X 10"
!
absorption. 1m1n Molecullr vibration
1
f
Molecular rotation
- n:_I~
Micro-
Microwave
porlmlQnellC resonence
•
Nuclear magnetic
J 3 X 10'
t
EllCtronic (outer sh.lI)
1
Radio
FIGURE 4-7 radiation.
r
l
X..-oy Ibsorption omission
U\ }~ion
The Electromegnetlc Spectrum The electromagnetic spectrum covers an immense range of wavelengths or energies. Figure 4-7 depicts qualitatively its major divi-
Wavenumber. a
Energy
101
resonenc.
Magnetically
1 ~
spin states
--
10'
Spectral properties, applications, and interactions of electromagnetic
sions. A logarithmic scale has been employed in this representation; note that the portion to which the human eye is perceptive is very small. Such diverse radiations as gamma rays and radio waves are also electromagnetic radiations, differing from visible light only in the matter of frequency, and hence energy. Figure 4-7 shows the regions of the spectrum that are useful for analytical purposes and the names of the spectroscopic methods associated with these applications. The
molecular or atomic transitions r~ponsible for absorption or emission of radiation in each region are also indicated.
THE INTERACTION OF RADIATION WITH MATTER As radiation passes from a vacuum through the surface of a portion of matter, the electrical vector of the radiation interacts with the
atoms and molecules of the medium. The nature of the interaction depends upon the properties of the matter and may lead to transmission, absorption, or scattering of the radiation.·
Transmlulon of Rlldietlon It is observed experimentally that the rate at which radiation is propagated through a trans- . parent substance is less than its velocity in a vacuum; furthermore, the rate depends upon the kinds and concentrations of atoms, ions, or molecules in the medium. It follows from these observations that the radiation must interact in some way with the matter. Because a frequency change is not observed, however, the interaction CQllIIOt involve a permanent energy transfer. The refractive index of a medium is a measure of its interaction with radiation and is defined by
where IIj is the refractive index at a specified frequency i, Vi is the velocity of the radiation in the medium, and c is its velocity in vacuo. The interaction involved in the transmission process can be ascribed to the alternating electrical field of the radiation, which causes the bound electrons of the particles contained in the medium to oscillate with respect to their heavy (and essentially fixed) nuclei; periodic polarization of the particles thus results. Provided the radiation is not ab- . sorbed, the energy required for polarization is only momentarily retained (10-14 to 10-15 s) by the species and is reemitted without alteration as the substance returns to its original state. Since there is no net energy change in this process, the frequency of the emitted radiation is unchanged, but the rate of its propagation has been slowed by the time required for retention and reemission to occur. Thus, transmission through a medium can be viewed as a stepwise process involving oscil-
Iating atoms, ions, or molecules as intermediates .. One would expect the radiation from each polarized particle in a medium to be emitted 'in all directions. If the particles are smal~ however, it can be shown that destructive interference prevents the propagation of significant amounts in any direction other. than that of the original light path. On the other hand, if the medium contains large particles (such as polymer molecules or colloidal. particles 1this destructive effect is incomplete: and a portion of the beam is scattered as a. consequence of the interaction step. Scatter-' ing is considered in a later section of this' chapter. Dispersion. As we have noted, the velocity' of radiation in matter is frequency-dependent; since c in Equation 4-12 is independent of this parameter, the refractive index of a substance must also change with frequency. The varia'; tion in refractive index of a substance with' frequency or wavelength is called its disper- i sion. The dispersion of a typical substance is: shown in Figure 4-8. Clearly, the relationship' is complex; generally, however, dispersion plots exhibit two types of regions. In the
Frequency, Hz
FIGURE
4-8
Typical dispersion curve.
normal dispersion region, there: is a gradual increase in refractive index with increasing frequency (or decreasing wavelength). Anomalous dispersion regions are those frequency ranges in which a sharp change in refractive index is observed. Anomalous dispersion always occurs at frequencies that correspond to the natural harmonic frequency associated with some part of the molecule, atom, or ion of the substance. At such a frequency, permanent energy transfer from the radiation to the substance occurs and absorption of the beam is observed. Absorption is discussed in a later section. Dispersion curves are important in the choice of materials for the optical components ofinstruments. A substance that exhibits normal dispersion over the wavelength region of interest is most suitable for the manufacture of lenses, for which a high and relatively constant refractive index is desirable. Chromatic aberration is minimized through the choice of such a material. In contrast, a substance with a refractive index that is not only large but also highly frequency-dependent is selected for the fabrication of prisms. The applicable wavelength region for the prism thus approaches the anomalous dispersion region for the material from which it was fabricated. Refraction of Radiation. When radiation passes from one medium to another of differing physical density, an abrupt change in direction of the beam is observed as a consequence of differences in the velocity of the radiation in the two media. This refraction of a beam is illustrated in Figure 4-9. The extent of refraction is given by the relationship
If the medium M1 is vacuum, VI becomes c and nl is unity (Equation 4-12); thus, the refractive index n2 of M2 is simply the ratio of the sines of the two angles.
FIGURE 4-9 Refraction of light in passing from a less dense medium M1 into a more dense medium M2, where its velocity is lower.
Reflection and Scattering of Rlldletlon RellectioL When radiation crosses an interface between media with differing refractive indexes, reftection occurs. The fraction reflected becomes larger with increasing differences in refractive index; for a beam traveling normally to the interface, the fraction reflected is given by I, 10
(n2 - nd2 (n2 + nd2
- = ~-~
(4-14)
where 10 is the intensity of the incident beam and I, is the reflected intensity; nl and "2 are the refractive indexes of the two media. EXAMPLE Calculate the percent loss of intensity due to reflection of a perpendicular beam of yellow light as it passes through a glass cell containing water. Assume that for yellow radiation the refractive index of glass is 1.5, of water is 1.33, and of air is 1.00.
In passing from air to glass, we find
!.!. = (1.50 10
1.00f = 0.040 (1.50 + l.00f
or
4.0%
In passing from glass to water, we find the loss to be 1,
(1.50 - 1.33)2
10 = (1.50 + 1.33)2 =.
0004 or
4. O. %
The same losses would occur as the beam passes from the water and through the glass. Thus, total percent loss = 2(4.0
+ 0.4) = 8.8
It will become evident in later chapters that losses such as those shown in this example are of considerable significance in various optical instruments. Reftective losses at a polished glass or quartz surface increase only slightly as the angle of the incident beam increases up to about 60 deg. Beyond this figure, however, the percentage of radiation that is reflected increases rapidly and approaches 100% at 90 deg. Scattering. As noted earlier, the transmission of radiation in matter can be pictured as a momentary retention of the radiant energy, which causes a brief polarization of the ions, atoms, or molecules present. Polarization is followed by reemission of radiation in all directions as the particles return to their original state. When the particles are small with respect to the wavelength of the radiation, destructive interference removes nearly all of the reemitted radiation except that which travels in the original direction of the beam; the path of the beam appears to be unaltered as a consequence of the interaction. Careful observation, however, reveals that a very small fraction of the radiation is transmitted at all angles from the original path and that the intensity of this scattered radiation increases with particle size. With particles of colloidal dimensions, scattering becomes sufficiently
intense to be seen by the naked eye (the Tyndall effect~ Scattering by molecules or aggregates of molecules with dimensions significantly smaller than the wavelength of the radiation is caIIcd Rayleigh scattering; its intensity is readily related to wavelength (an inverse fourth-power effect~ the dimensions of the scattering particles, and their polarizability. An everyday manifestation of scattering is the blueness of the sky, which results from the greater scattering of the shorter wavelengths of the visible spectrum. Scattering by larger particles, as in a colloidal suspension, is much more difficult to treat theoretically; the intensity varies roughly as the inverse square of wavelength. Measurements of scattered radiation can be used to determine the size and shape of polymer molecules and colloidal particles. The phenomenon is also I1tilized in nephelometry, an analytical method considered in Chapter 10. Raman Scattering. The Raman effect differs from ordinary scattering in that part of the scattered radiation suffers quantized frequency changes. These changes are the result of vibrational energy level transitions occurring in the molecule as a consequence of the polarization process. Raman spectroscopy is discussed in Chapter 9.
Polarization of Radiation Plane PoiarizatiolL Ordinary radiation can be visualized as a bundle of electromagnetic waves in which the amplitude of vibrations is equally distributed among a series of planes centered along the path of the beam. Viewed end-on, the electrical vectors would then appear as shown in Figure 4-lOa. The vector in anyone plane, say X Y, can be resolved into two mutually perpendicular components AB and CD as shown in Figure 4-10b. If the two components for each plane are combined, the resultant has the appearance shown in Figure 4-1Oc. Note that Figure 4-1Oc has a
FIGURE
4-10 (a) A few of the electrical vectors of a beam traveling perpendicular to the page. (b) The resolution of a vector in plane XY into two mutually perpendicular components. (c) The resultant when all vectors are resolved (not to scale).
different scale from Figure 4-lOa or 4-lOb to keep its size within reason. Removal of one of the two resultant planes of vibration in Figure 4-IOc produces a beam that is plane polarized. The vibration of the electrical vector of a plane-polarized beam, then, occupies a single plane in space. Plane-polarized electromagnetic radiation is produced by certain radiant energy sources. For example. the radio waves emanating from an antenna commonly have this characteristic. Presumably, the radiation from a single atom or molecule is also polarized; however, since common light sources contain large numbers of these particles in all orientations, the resultant is a beam that vibrates equally in all directions around the axis of travel. The absorption of radiation by certain types of matter is dependent upon the plane of polarization of the radiation. For example. when properly oriented to a beam, anisotropic crystals selectively absorb radiations vibrating in one plane but not the other. Thus, a layer of anisotropic crystals absorbs all of the components of, say, CD in Figure 4-IOc and transmits nearly completely the
components of AB. A polarizing sheet, regardless of orientation, removes approximately half of the radiation from an unpolarized beam, and transmits the other half as a plane-polarized beam. The plane of polarization of the transmitted beam is dependent upon the orien~ation of the sheet with respect to the incident beam. When two polarizing sheets, oriented at 90 deg to one another, are placed perpendicular to the beam path, essentially no ~radiation is transmitted. Rotation of one results in a continuous increase in transmission until a maximum is reached when molecules of the two sheets have the same ltlignment. The way in which radIation is reflected, scattered, transmitted, and refracted by certain substances is also dependent upon direction of polarization. As a consequence, a group of analytical methods have been developed whose selectivity. is based upon the interaction of these substances with light of a particular polarization. These methods are considered in detail in Chapter 13.
Absorption of Radiation When radiation passes through a transparent layer of a solid, liquid, or gas, certain frequencies may be selectively removed by the process of absorption. Here. electromagnetic energy is transferred to the atoms or molecules constituting the sample; as a result, these particles are promoted from a lowerenergy state to higher-energy states, or excited states. At room temperature, most substances are in thei~ lowest energy or ground state. Absorption then ordinarily involves a transition from the ground state to higher-energy states. 1 Atoms, molecules, or ions have only a . limited number of discrete, quantized energy levels; for absorption of radiation to occur, the energy of the exciting photon must exactly match the energy difference between the ground state and one of the excited states of the absorbing species. Since these energy
a molecule is given by difTerences arc unique for each species, a study of the frequencies of absorbed radiation provides a means of characterizing the constituents of a sample of matter. For this purpose, a plot of absorbance as a function of wavelength or frequency is experimentally derived (absorbance, a measure of the decrease in radiant power, is defined on p. 149). Typical plots of this kind, called absorption spectra, arc shown in Figure 4-11. The general appearance of an absorption spectrum will depend upon the complexity, the physical state, and the environment of the absorbing species. It is convenient to recognize two types of spectra, namely, those associated with atomic absorption and those resulting from molecular absorption. Atomic Absorptia The passage of polychromatic ultraviolet or visible radiation through a medium consisting of monatomic particles, such as gaseous mercury or sodium, results in the absorption of but a few welldefined frequencies (see Figure 4-11d ~ The relative simplicity of such spectra is due to the small number of possible energy states for the particles. Excitation can occur only by an electronic process in which one or more of the electrons of the atom is raised to a higherCDerIYlevel. Thus, with sodium, excitation or the 35 electron to the 3p state requires energy corresponding to a wavenumber of 1.697 x 1()4em - I. As a result, sodium vapor exhibits a sharp absorption peak at 589.3 nm (yellow light). Several other narrow absorption lines, corresponding to other permitted electronic transitions, are also observed (see Figure 4-11d). Ultraviolet and visible radiation have sufficient energy to cause transitions of the outermost or bonding electrons only. X-ray frequencies, on the other hand, are several orders of magnitude more energetic and are capable of interacting with electrons closest to the nuclei of atoms. Absorption peaks corresponding to electronic transitions of these innermost electrons are thus observed in the X-ray region.
E - £.••••••••• 1. + Eribra'lanal
+ E"""IIOIIoI
.
the number of possible energy levels for a molecule Is much greater than for an atomic particle. Figure 4-12 is a graphical representation of the energies associated with Ii few of the electronic and vibrational states"'of a molecule. The heavy line labeled Eo represents the electronic energy of the molecule in its ground state (its state of lowest electronic energy); the lines labeled E1 and Ez represent the energies of two excited electronic states. Several vibrational energy levels (eo, eh .,., e.) are shown for each of these electronic states.
(4-15)
where E•••et •••••l• describes the electronic energy of the molecule, and EribraiiOlUlI refers to the energy of the molecule resulting from various atomic vibrations. The third term in Equation 4-15 accounts for the energy associated with the rotation of the molecule around its center of gravity. For each electronic energy state of the molecule, there normally exist several possible vibrational states, and for each of these, in turn, numerous rotational states, As a consequence,
I i
t
,
£,
.
o 220
240
260 280 Wovelength, nm
300
·3 ••
Cd)
FIGURE
4-11 Some typical ultraviolet absorption spectra.
~r"'"---~t
-
L
~
.") J
I
eo'
1
energy
Vi.lbl.
Regardless of the wavelength region involved, atomic absorption spectra typically consist of a limited number of narrow peaks. Spectra of this type are discussed in connection with X-ray absorption (Chapter 15) and atomic absorption spectroscopy (Chapter 11). Molecular Absorption. Absorption by polyatomic molecules, particularly in the condensed state, is a considerably more complex process because the number of energy states is greatly enhanced. Here, the total energy of
..,_)1' •
J
• •
1
2
f I
•
fa)
FIGURE 4-12
E,
Excited electronic state t
f~:ae:'::..-;""'"
~
. ".
t -. g£~ ''.i'~lv
Nonparallel radiation
'X-i- ~ transmitting mirror
.
.
Schematic representation
processes with lifetimes in the range of 10-9 to 10-12 s, the detection and determination of extremely small concentrations of species in the atmosphere, and the induction of is0topically selective reactions.3 In addition, laser sources have become important in several routine analytical methods, including Raman spectroscopy (Chapter 91 emission spectroscopy (Chapter 12), and Fourier transform infrared spectroscopy (Chapter 8). The term laser is an acronym for light amplification by stimulated emission of radiation. As a consequence of their lightamplifying properties, lasers produce spatially narrow, extremely intense beams of radiation. The process of stimulated emission produces a beam of highly monochromatic (bandwidths of 0.01 nm or less) and remarkably coherent (p. 99) radiation. Because of these unique properties, lasers have become important sources for use in the ultraviolet, visible, and infrared regions of the spectrum. A limitation of early lasers was that the radiation from a given source was restricted to a relatively few discrete wavelengths or lines. Recently, however, dye lasers have become available; tuning of these sources provides a
1
• For a review or some or Ihese applicalions. see: S. R. Leone, J. Chem. Educ.,53, 13 (1976~
•
=s --.
Active I~ing mediu'!'
JZ:"
Radiation ----Pumping
..•.._ .•..,._A'
of a typical
narrow band of radiation at any chosen wavelength within the range of the source. Figure 5-3 is a schematic representation showing the components of a typical laser source. The heart of the device is a lasing medium. It may be a solid crystal, such as ruby, a semiconductor such as gallium arsenide. a solution of an organic dye, or a gas. The lasing material can be activated or pumped by radiation from an external source so that a few photons of proper energy will trigger the formation of a cascade of photons of the same energy. Pumping can also be carried out by an electrical current or by an electrical discharge. Thus, gas lasers usually do not have the external radiation source shown in Figure 5-3; instead, the power supply is connected to a pair of electrodes contained in a cell filled with the gas. A laser normally functions as an oscillator, in the sense that the radiation produced from the laser mechanism is caused to pass back and forth through the medium numerous times by means of a pair of mirrors as shown in Figure 5-3. Additional photons are generated with each passage, thus leading to enormous amplification. The repeated passage also produces a beam that is highly parallel because nonparallel radiation escapes from the sides of the medium after being reflected a few times (see Figure 5-3~ In order to obtain a usable laser beam, one of the mirrors is coated with a sufficiently
thin layer of reflecting material so that a fraction of the beam is trilnsmitted rather than reflected (see Figure 5-3). . Laser action can be understood by considering the four processes depicted in Figure 5-4, namely, (a) pumping, (b) spontaneous emission (fluorescence1 (c) stimulated emission, and (d) absorption. For purposes of illustration, we will focus on just .two of several electronic energy levels that wil( exist in the atoms, ions, or molecules making up the laser material; as shown in the figure, the two electronic levels have energies E, and Ex. Note that the higher electronic state is shown as having several slightly different vibrational energy levels, E" E~, E;, and so forth. We have not shown additional levels for the lower electronic state, although such often exist. Pumping. Pumping, which is necessary for laser action, is a process by which the active species of a laser is excited by means of an electrical discharge, passage of an electrical current, exposure to an intense radiant source, or interaction with a chemical species. During pumping, several of the higher electronic and vibrational energy levels of the active species will be populated. In diagram a-I of Figure 5-4, one atom or molecule is shown as being promoted to an energy state E;; the second is excited to the slightly higher vibrational level The lifetime of excited vibrational states is brief, however; after 10-13 to 10-14 S, relaxation to the lowest excited vibrational level (E, in diagram a-3) occurs with the production of an undetectable quantity of heat. Some excited electronic states of laser materials have lifetimes considerably longer (often I ms or more) than their excited vibrational counterparts; longlived states are sometimes termed metastable as a consequence. Spontaneous Emission. As was pointed QUt in the discussion of fluorescence, a species in an excited electronic state may lose all or part of its excess energy by spontaneous emission of radiation. This process is depicted in the
E;.
diagrams shown in Figure 5-4b. Note that the wavelength of the fluorescent radiation is directly related to the energy difference between the two electronic states, E, - Ex. It is also important to note that the instant at which the photon is emitted, and its direction, will vary from excited electron to excited electron. That is, spontaneous emission is a random process; thus, as shown in Figure 5-4, the fluorescent radiation produced by one of the particles in diagram b-I differs in direction and phase from that produced by the second particle (diagram b-2). Spon.taneous emission, therefore, yields incoherent radiation. Stimulated Emission. Stimulated emission, which is the basis of laser behavior, is depicted in Figure 5-4c. Here, the' excited laser particles are struck by externally produced photons having precisely the same energies (E,- Ex) as the photons produced by'spontaneous emission. Collisions; of this type cause the excited species to r~lax immediately to the lower energy statei and to emit simultaneously a photon of exactly the same energy as the photon that stimulated the process. More important, and more remarkable, the emitted photon is precisely in phase with the photon that triggered the event. That is, the stimulated emission is totally coherent with the incoming radiation. . Absorption. The absorption process, which competes with stimulated emission. is depicted in Figure 5-4d. Here, two photons with energies exactly equal to E, - Ex are absorbed to produce the metastable excited state shown in diagram d-3; note that the state shown in diagram d-3 is identical to that attained by pumping diagram a-3. Population Inversion a'" Light Amplification. In order to have light amplification in a laser, it is necessary that the number of~hotons produced by stimulated emission exceed the number lost by absorption. This condition will prevail only when the number of particles in the higher energy state exceeds the number in the lower; that is, a population inversion from the normal distribution of energy states must exist.
(1)
-_
EM
=+=~. .,- -r· Pumping
energy
I I
-+-
I
E•
:±: ~
Heat
I
+E.
-----
~~
----
__=cvu-
FIGURE 5-4 Four processes important in laser action: (a) pumping (excitation by electrical. radiant. or chemical energy). (b) spontaneous emission. (c) stimulated emission. and (d) absorption.
system. In contrast, it is only necessary to pump sufficiently to make the number of particles in the E, energy level exceed the number in Ex of a four-level system. The lifetime of a particle in the Ex state is brief, however, because the transition to Eo is fast; thus, the number in the Ex state will generally be negligible with respect to Eo and also (with a modest input of pumping energy) with respect to E,. That is, the four-level laser usually achieves a population inversion with a smaller expenditure of pumping energy. Some Examples of Useful Lasers. The first successful laser, and one that still finds widespread use, was a three-level device in which a ruby crystal was the active medium. The ruby was machined into a rod about 4 em in length and 0.5 em in diameter. A ftash tube was coiled around the cylinder to produce intense ftashes of light. Because the pumping was discontinuous, a pulsed laser beam was produced. Continuous wave ruby sources are now available. A variety of gas lasers are sold commercially. An important example is the argon ion
Population inversions are brought about by pumping. Figure 5-5 contrasts the effect of incoming radiation on a noninverted population with an inverted one. 11Iree- •••• Four-Level LIser Systems. Figure 5-6 shows simplified energy diagrams for the two common types of laser systems. In the three-level system, the transition responsible for laser radiation is between an excited state E, and the ground state Eo; in a four-level system, on the other hand, radiation is generated by a transition from E, to a state Ex that has a greater energy than the ground state. Furthermore, it is necessary that transitions between Ex and the ground state be rapid. The advantage of the four-level system is that the population inversions necessary for laser action are more readily achieved. To understand this, note that at room temperature a large majority of the laser particles will be in the ground-state energy level Eo'in both systems. Sufficient energy must thus be provided to convert more than 50% of the lasing species to the E, level of a three-level
AbIofption "',
Ey
rvv-
rv\ri
I E, lal Ey
V\r Stimulated / emission
'Ulr
Light amplification
by stimulated emission
E,
E.
E. "
____
•FISttransition nonradiative Ey
/--
Fast transition
FIGURE 6-6 Energy level diagrams for two types of laser systems.
laser, which produces intense lines in the green (514.5 om) and blue (488.0 nm) regions. This laser is a four-level device in which argon ions are formed by an electrical or radio frequency discharge. The input energy is sufficient to excite the ions from their ground state, with a principal Quantum number of 3, to various 4p states. Laser activity then involves transitions to the 4s state. Dye lasers4 have become important radiation sources in chemistry because they are tunable over a range of 20 to 50 nm; that is, dye lasers provide bands of radiation of a chosen wavelength with widths of a few hundredths of a nanometer or less. The active materials in dye lasers are solutions of organic compounds capable.()fftuorescing in the ultraviolet. visible. or infrared regions. In contrast to ruby or gas lasers. however, the lower energy level for laser action (Ex in Figure 5-6) is not a single energy but a band of energies arising from the superposition of a large number of small vibrational and rotational energy states upon the base electronic energy state. Electrons in E). may then undergo transitions to any of these states, thus producing photons of slightly different energies.
(b)
FIGURE 6-6 Passage of radiation through (a) a noninverted population and (b) an inverted population.
:rE~ 1E,
• For rurther inrormation, see: R. B. Green, - Dye Laser Instrumentation," J. Chem. Educ.• 54 (9) A365, (10) A407, (1977~
Tuning of dye lasers can be readily accomplished by replacing the nontransmitting mirror shown in Figure 5-3 with a monochromator equipped with a reftection grating or a Littrow-type prism (p. 126) which will reftect only a narrow bandwidth of radiation into the laser medium; the peak wavelength can be varied by rotatioQ of the grating or prism. Emission is then stirllulated for only part of the fluorescent spec(rum, namely. the wavelength reflected from the monochromator.
WAVELENGTH SELECTION; MONOCHROMATORS With few minor exceptions, the analytical methods considered in the following seven chapters require dispersal of polychromatic radiation into bands that encompass a restricted wavelength region. The most common way of producing such bands is with a device called a monochromator.'
j
• For a more complete d~ussion or monochromators, see: R. A. Bauman, AbJO,ption Spect,OM:OPY.New York: Wiley, 1962; F. A. Jenkins and H. E. White, FUMamentats of Optia. 3d ed. New York: McGraw-HiI~ 1957; E. J. Meehan, in T,etIliM 011 A""lytiClJI CMmUt'Y. cds. I. M. KolthofTand P. J. Elving. New York: Wiley, 1964. Part t, vol. 5. Chapter 55: and J. F. James and R. S. Sternberg, The Design of Optical Spectromners. London: Chapman and Hall. 1969,
Focal plane
~~7B
••••... A
A2/Exit /
sfit
A
(bl
FIGURE 5-7 Two types of monochromators: (a) Bunsen prism monochromator, and (b) Czerney-Tumer grating monochromator. (In both instances, 11 > 12,)
Mooochromators for ultraviolet, visible, and infrared radiation are all similar in mechanical construction in the sense that they employ slits, lenses, mirrors, windows, and prisms or gratings. To be sure, the materials from which these components are fabricated will depend upon the wavelength region of intended use (see Figure 5-2b and c). Components of • Monochromator All monochromators contain an entrance slit, a collimating lens or mirror to produce a parallel beam of radiation, a prism or grating as a dispersing element, 6 and a focusing element
• Much less gencraIIy employed are interference wedges. These devices are described in the next section (p. 136).
grating, angular dispersion results. from diffraction, which occurs at the reflective surface. In both ~esigns, the dispersed radiation is focused on the focal plane AB where it appears as two "images of the entrance slit (one for each wavelength).
which projects a series of rectangular images of the entrance slit upon a plane surface (the focal plane). In addition, most monochromators have entrance and exit windows, which are designed to protect the components from dust and corrosive laboratory fumes. Figure 5-7 shows the optical design of two typical monochromators, one employing a prism for dispersal of radiation and the other a grating. A source of radiation containing but two wavelengths, 11 and 12, is shown for purposes of illustration. This radiation enters the monochromators via a narrow rectangular opening or slit, is collimated, and then strikes the surface of the dispersing element at an angle. In the prism monochromator, refraction at the two faces results in angular dispersal of the radiation, as shown; for the
Types of Instruments Employing Monochromators The method of detecting the radiation dispersed by a monochromator varies. In a spectroscope, detection is accomplished visually with a movable eyepiece located along the focal plane. The wavelength is then determined by measurement of the angle between the incident and dispersed beams. In a spectrograph, a photographic film or plate is mounted along the focal plane; the series of darkened images of the slit along the length of the developed film or plate is called a spectrogram. The monochromators shown in Figure 5-7 are also utilized for dispersing elements in spectr~ters. Such instruments have a fixed exit slit located in the focal plane. With a continuous source, the wavelength emitted from this slit can be varied continuously by rotation of the dispersing element. We need to define two other types of instruments to complete this discussion. A spectrophotometer is a spectrometer which contains a photoelectric device for determining the power of the radiation exiting from the slit. A photometer also employs a photoelectric detector but contains no monochromator; instead, filters are used to provide radiation bands encompassing limited wavelength regions. A photometer is incapable of providing a continuously variable band of radiation. Prism Monochromators Prisms can be used to disperse ultraviolet, visible, and infrared radiation. The material used for their construction will differ depending upon the wavelength region.
Construedoa ~terials. Clearly, the materials employed for fabricating the windows lenses, and prisms of a monochromator must transmit radiation in the frequency range of interest •. ideally, the transmittance should approach tOO%, although it is sometimes necessary to use substances with transmittances as low as 20%. The refractive index of materials used in the construction of windows and prisms should be low in order to reduce reflective losses (see Equation 4-14, p. 103). However, for lenses which function by bending rays of radiation, a material with a high refractive index is desirable in order to reduce focal lengths. Ideal construction materials for both lenses and windows should exhibit little change in refractive index with frequency, thus reducing chromatic aberrations. In contrast, just the opposite property is sought for prisms, where1he extent of dispersion depends upon the rate of change of refractive index with frequency. In addition to the foregoing optical properties, it is desirable that monochromator components be resistant to mechanical abrasion and attack by atmospheric components and laboratory fumes. Needless to say, no single substance meets all of these criteria and trade-offs are thus required in monochromator design. The choices here depend strongly upon the wavelength region to be used. Table 5-1 lists the properties of the common substances employed for fabricating monochromator components. Clearly, no single material is suitable for the entire wavelength region. For the ultraviolet, visible, and near-infrared regions (up to about 3000 nm~ a quartz prism is often employed; it should be noted, however, that glass provides better resolution for the same size prism for wavelengths between 350 and 2000 nm. As shown in column 5 of Table 5-1, several prisms are required fo cover the entire infrared region. Types of Prisms and Prism Monochromators. Figure 5-8 shows the two most common types of prism configurations. The first is a 6O-deg design, which is ordinarily fabricated
FIGURE 5-8 Dispersion by a prism: (a) quartz Cornu type, and (b) Littrow type.
from a single block of material. When crystalline (but not fused) quartz is employed, however, the prism is usually formed by cementing two 3().deg prisms together, as shown in Figure 5-8a; one is constructed from right-handed quartz and the second from left-handed quartz. In this way, the opticaUyactive quartz causes no net polarization of the emitted radiation; this type of prism is
TABLE 1·1 OPTICS
Material Fused silica Quartz Flint glass Calcium fluoride ~fluorite) Lit ium fluoride Sodium chloride Potassium bromide Cesium iodide KRS-5 (TlBr-TlI)
CONSTRUCTION
called a Cornu prism. Figure 5-7a shows a Bunsen monochromator, which employs a 6O-deg prism, likewise often made of quartz. Figure 5-8b depicts a Littrow prism, which is a 3().deg prism with a mirrored back. It is seen that refraction occurs twice at the same interface; the performance characteristics of the Littrow prism are thus similar to those of the 6O-deg prism. The Littrow prism permits
MATERIAL
FOR SPECTROPHOTOMETER
=w
Uleful 1tuIe fer Prisms, IJI11
HarUelsDd CIIemicaI Resistance
0.18-2.7 0.20-2.7 0.35-2 5-9.4
Excellent Excellent Excellent Good
0.29 x 10-4
2.7-5.5
Poor
1.54
0.94 x 10-4
8-16
Poor
0.3-29
1.56
1.45 x 10-4
15-28
Poor
0.3-70
1.79
15-55
Poor
1-40
2.63
24-40
Good
Tl'lIDSIIlittance Range, pm
Refractive
0.18-3.3 0.20-3.3 0.35-2.2 0.12-12
1.46 1.54 1.66 1.43
0.52 x 0.63 x 1.70 x 0.33 x
0.12-6
1.39
0.3-17
• Radians/ l'1li at 0.589 pm
Index
10-4 10-4 10-4 10-4
somewhat more compact monochromator designs. In addition, when quartz is employed, polarization is canceled by the reversal of the radiation path. A typical Littrow-type monochromator' is shown in Figure 7-10. Angular DispersioR of ~ The angular dispersion of a prism is the rate of change of the angle 8 in Figure 5-8 as a function of wavelength; that is, d8/dl. The spectral purity of the radiation exiting from a prism monochromator is dependent upon this quantity. The angular dispersion of a prism can be resolved into two parts d8 d)'
d8 dn
= dn . d)'
(5-1)
where d8/dn represents the change in 8 as a function of the refractive index n of the prism material and dn/d)' expresses the variation of the refractive index with wavelength (that is, the dispersion of the substance from which the prism is fabricated). The magnitude of dO/dn is determined by the geometry of the prism and the angle of incidence i (Figure 5-Sa~ In order to avoid problems of astigmatism (double images), this angle should be such that the path of the beam through the prism is within a few degrees of being parallel to the base of the prism. Under these circumstances. dO/dll depends only upon the prism angle ClC (Figure 5-Sa) and increases rapidly with this quantity. Reflection losses, however. impose an upper limit of about 60 deg for IX. For a prism in which ClC = 60 deg, it can be shown that 2
dO/dn = (I - n /4,-t
(5-2)
The term dll/d)' in Equation 5-1 is related to the dispersion of the substance from which the prism is constructed. We have noted (p. 102) that the greatest dispersion for a given material is near its anomalous dispersion region, which in turn is close to a region of absorption. The dispersion of some substances employed for construction of prisms is shown in Figure 5-9. Note that the rapid
rise in the refractive index for glass below 400 Dm corresponds to the sharp increase in absorption that prevents the use of this material below 350 Dm. in the region of 350 to 2000 DID, however, glass is greatly superior to quartz for prism fabrication because of its larger change in refractive index with wavelength (dll/d).). Focal-Plane DiSpersion of Prism Monochromators. The focal-plane dispersion of a monochromator refers to the variation in wavelength as a function of y, the linear distance along the line AB of the focal plane of the instrument (see Figure 5-7). That is, the focal-plane dispersion is given by dy/d)'. Figure 5-10 shows the focal-plane dispersion of two prism monochromators and one grating instrument. Note that the dispersion for the two prism monochromators is highly nonlinear, the longer wavelengths being bunched together over a relatively small distance. The two prism monochromators were Littrow types. cach having a height of 57 mm. Notc the much largcr dispersion exhibited by the monochromator with a glass prism, in the region of 350 to 800 nm.
K
~ .S
1.60
-----------
.~
~OCk
~ ~
salt
.
1.50
1.40
o
200
400
600
Wavelength.
FIGURE 5-9 materials.
800
1000
1200
nm
Dispersion of several optical
reflection from a reflection grating. A transmission grating consists of a series of parallel and closely spaced grooves ruled on a piece of glass or other transparent material. A grating suitable for use in the ultraviolet and visible region has between 2000 and 6000 lines per millimeter. An infrared grating requires considerably fewer lines; thus, for the far infrared region, gratingS with 20 to 30 lines per millimeter may suffice. It is vital that these lines be equally spaced. throughout the several centimeters in length of the typical grating. Such gratings require elaborate apparatus for their production and are consequently expensive. Replica gratings are less costly. They are manufactured I;>yemploying a master grating as a mold for' the production of numerous plastic replicas; the products of this process, while inferior in performance to an original grating, are adequate for many applications. When a transmission grating is illuminated from a slit, ~ch groove scatters radiation and thus effedtively becomes opaque. The nonruled portions then behave as a series of closely spaced slits, each of which acts as a new radiation source; interference among the multitude of beams results in diffraction of
Figure 5-10 illustrates one ofthe important advantages of grating mopochromatorslinear dispersion along the focal plane. Resolving Power of Prism Monocbromators. The resolving .power R of a prism gives the limit of its ability to separate adjacent images having slightly different wavelengths. Mathematically, this quantity is defined as
where dl represents the wavelength difference that can just be resolved and l is the average wavelength of the two images. It can be shown that the resolving power of a prism is directly proportional to the length of the prism base b (Figure 5-8) and the dispersion of its construction material. That is,
Gl'IIting Monochromatora for Wavelength Selection Dispersion of ultraviolet, visible, and infrared radiation can be brought about by passage of a beam through a transmission grating or by
200 ~. om
I
Gl'lting 500
. 300
I
I
Glass prism
350
400
450
500
A-,j\il9£~;;; -.-,
I
I
I
200 X,om
600 800
I I II
Quartz prism
A,nm
200
250
300
I
I
I
350 400
I
A
'!
,
500600800
I
II 8
I
0
5.0
10.0 15.0 Diltance y along focal plane, em
20.0
25.0
FIGURE 6-10 Dispersion for three types ofmonochromators. The points A and B on the scale correspond to the points shown in Figure 5-7.
Monochromatic
beam It incident ongtel .
. FIGURE 6-11 Schematic diagram illustrating the mechanism of diffraction from an echellette-type grating. (From R. P. Bauman, Absorption Spectroscopy. New York, Wiley, 1962, p. 65. With permission.)
the radiation as shown in Figure 4-00 (p. 98). The angle of diffraction, of course, depends upon the wavelength. Reflection gratings are used more extensively in instrument construction than their transmitting counterparts. Such gratings are made by ruling a polished metal surface or by evaporating a thin film of aluminum onto the surface of a replica grating. As shown in figure 5-11, the incident radiation is reflected from one of the faces of the groove, which then acts as a new radiation source. Interference results in radiation of differing wavelengths being reflected at different angles r. Diffraction by a Grating. figure 5-11 is a schematic representation of an echellette-type grating, which is grooved or blazed such that it has relatively broad faces from which reflection occurs and narrow unused faces. This geometry provides highly efficient diffraction of radiation. Each of the broad faces can be considered to be a point source of radiation; thus interference among the
reflected beams 1,2, and 3 can occur. In order for constructive interference to occur between adjacent beams, it is necessary that their path lengths differ by an integral multiple n of the wavelength l of the incident beam. In Figure 5-11, parallel beams of monochromatic radiation 1 and 2 are shown striking the grating at an incident angle i to the grating normal. Maximum constructive interference is shown as occurring at the reflected angle r. It is evident that beam 2 travels a greater distance than beam 1 and that this difference is equal to (CD - AB). For constructive interference to occur, this difference must equal n).. That is, M = (CD - AB) Note, however, that angle CAD is equal to angle i and that angle BDA is identical with angle r. Therefore, from simple trigonometry, we may write
where d is the spacing between the reflecting surfaces. It is also seen that AB= -dsin
r
The minus sign, by convention, indicates that reflection has occurred. The angle r, then, is negative when it lies on the opposite side of the grating normal from the angle i (as in Figure 5-1I~ and is positive when it is on the same side. Substitution of the last two expressions into the first gives the condition for constructive interference. Thus, III =
d(sin i + sin r)
(5-5)
Equation 5-5 suggests that several values of.t exist for a given diffraction angle r. Thus, if a first-order line (D = I) of 800 nm is found at r, second-order (400 nm) and third-order (267 nm) lines also appear at this angle. Ordinarily, the first-order line is the most intense; indeed, it is possible to design gratings that concentrate as much as 90% of the incident intensity in this order. The higherorder lines can generally be removed by filters. For example, glass, which absorbs radiation below 350 nm, eliminates the higherorder spectra associated with first-order radiation in most of the visible region. The example which follows illustrates these points. EXAMPLE An echellette grating containing 2000 blazes per millimeter was irradiated with a polychromatic beam at an incident angle 48 deg to the grating normal. Calculate the wavelengths of radiation that would appear at an angle of reflection of +20, + 10,0, and -10 deg (angle r, Figure 5-11). To obtain d in Equation 5-5, we write Imm nm nm d=---x 106=500-2000 blazes mm blaze When r in Figure 5-11 equals + 20 deg , SOO(. 48 . 2) 542.6 A=sm +sm 0 =-D
n
and the wavelengths for the first-, secondo, and thUd-order reflections are 543, 271, and 181 nm, respectively. Similarly, when r = -10 deg (here r lies to the right of the grating n0"ttal)
RfIOIviDg Power of a Gntiac- It can be shown 7 that the resolving power R of a grating is given by the very simple expression
.t = SOO[sin 48 + sine-10)] = 284.7
where n is the diffraction order and N is the number of lines illuminated by the radiation from the entrance slit. Thus, as with a prism (Equation 5-4~ the resolving power depends upon the physical size of the dispersing element.
.t
R = .u = aN
(5-7)
For the grating, we employ Equation 5-7
.t -=aN .u For the first-order spectrum (D = I) N
n
n
Wavelength (nm) for r
20 10 0 -10
n=1 543 458 372 285
n=2 271 229 186 142
n=3 181 153 124 95
A concave grating can be produced by ruling a spherical reflecting surface. Such a diffracting element serves also to focus the radiation on the exit slit and eliminates the need for a lens. Dispersion fJl GntiDp. The angular dispersion of a grating can be obtained by differentiating Equation 5-5 while holding i constant; thus, at any given angle of incidence
EXAMPLE ~mpare the size of (I) a 6O-deg fused silica' prism; (2) a 6O-deg glass prism; and (3) a grating with 2000 lines/mm, that would be required to resolve two lithium emission lines at 460.20 and 460.30 nm. Average values for tlje dispersion (dn/d.t) of fused silica and glass ~n the region of interest are 1.3 x 10- 4 and 3.6 x 10-4 nm-I, respectively. For the two prisms, we employ Equation 5-4
.t
.u =b-d)'
.t
I
.u
dn/d.t
b-- x-dr D -=-d)' dcosr
(5-6)
Note that the dispersion increases as the distance d between rulings decreases or as the number of lines per millimeter increases. Over short wavelength ranges, cos r does not change greatly with .t, so that the dispersion of a grating is nearly linear. By proper design of the optics of a grating monochromator, it is possible to produce an instrument that for all practical purposes has a linear dispersion of radiation along the focal plane of the exit slit. Figure 5-10 shows the contrast between a grating and a prism monochromator in this regard.
dn
R=-
460.25
I
------
460.30 - 460.20
x--
dn/d)'
For fused silica, b = 460.25 nm x I x 10-7 em 0.10 nm 1.3 x 10-4 nm-I om = 3.5 em and f