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A model for ground-state and excited-state microwave optical double resonance
JOURNAL OF MOLECULAR
A Model
SPECTROSCOPY
64, 86-97 (1977)
for Ground-State and Excited-State Optical Double Resonance’
Microwave
RICHARD F. WORMSBECHER,DAVID 0. HARRIS, AND BRIAN G. WICKET Quantum Institute and Department of Chemistry, University of California, Santa Barbara, Santa Barbara, Calijornia 93106
A model is developed to describe the physical principles of both ground-state and excitedstate microwave optical double resonance. This model uses a steady-state kinetic analysis to determine the populations of three quantum levels in the presence of two monochromatic radiation fields. The microwave transition rate and the laser transition rate are obtained from separate analyses using the semiclassical two-state transition probability averaged over the effects of collisions. The Doppler effect of the optical transition rate is explicitly included. The competing effects of laser power, microwave power, and pressure on signal intensity and lineshape are described. Calculated lineshapes and relative signal intensities based on this model are in good agreement with previous measurements on BaO and NOz. I. INTRODUCTION The study of small molecules in the gas phase is limited by the trade-off between sensitivity and resolution characteristic of most conventional spectroscopic techniques. Optical spectroscopy, for example, has high sensitivity, but its resolution is normally limited by a Doppler width on the order of 1 GHz. In contrast, microwave spectroscopy has substantially greater resolution of typically 0.1 MHz at 10 mTorr pressure, but its sensitivity is basically low. Hence, the study of molecular rotation, Stark effects, hyperfine coupling, etc., via conventional spectroscopy has been restricted by and large to stable ground-state molecules which can be obtained in substantial concentrations. In the past few years several double resonance techniques have been developed in an attempt to combine the high resolution of radiofrequency and microwave spectroscopy with the sensitivity of optical spectroscopy (1-14). Many of these double resonance techniques share a fundamental basis by which the presence of a small energy resonance is detected via a coupled change in a larger energy resonance. Microwave optical double resonance is one such technique in which a microwave resonance is detected by a change in one of the properties of an associated optical transition. Microwave optical double resonance typically involves the interaction among two adjacent rotational levels in one vibronic state and a particular rotational level in another electronic state. The pair of rotational levels may be in the ground electronic state or an excited electronic state, corresponding to ground-state or excited-state microwave optical double resonance, respectively. In the ground-state case the pair of 1Work supported in part by NSF Grant MPS72-04978. 2 Present address: TRW Systems Group R1/1196, One Space Park, Redondo Beach, California 90278. 86 Copyright Q 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
A MODEL FOR MODR
87
rotational levels are connected by a microwave transition, and one of these levels is connected with a single rotational level in an electronically excited state by a laser induced transition. The presence of microwave resonance between the lower levels is detected as a change in the intensity of the laser induced fluorescence from the excited electronic state. In excited-state microwave optical double resonance, microwave resonance is detected as a change in the intensity of the total rotationally unresolved laser induced fluorescence from both the excited electronic-rotational levels. No microwave radiation detection is used. Hence, rotational transitions in the ground state or an electronically excited state are measured with resolution approaching that of microwave spectroscopy but with sensitivity characteristic of laser induced optical fluorescence. From this description it is clear that the term “microwave optical double resonance” is really a misnomer; a more appropriate name would be “optical detection The key experimental feature is that, although microwave of microwave transitions.” transitions are being studied, the detector is typically a phototube. High resolution is derived from the microwave radiation, high sensitivity from the detection of laser induced fluorescence. This paper presents a simple model for the technique of optical detection of microwave transitions. The principal motivation for the development of this model was the practical need for a systematic guide for applying this technique to new molecular systems. This requires an understanding of intensities and linewidths, and their dependences on the laser power, microwave power, collisions, and spontaneous emission. This work is an extension of a previously presented paper (15). In Section II the model is developed. In Section III calculated intensity profiles and microwave lineshapes are presented and compared with available experimental results from this laboratory (10-14). II. DERIVATION
OF THE MODEL
The model treats the interaction between three quantum levels in the presence of two monochromatic fields and collisions, as shown in Fig. la for ground-state, and Fig. lb for excited-state microwave optical double resonance. Levels 2 and 3 interact directly via a microwave radiation field with a transition rate constant k~. Levels 1 and 2 interact directly via a laser radiation field with a transition rate constant kr,. Kinetic processes which populate the levels via other mechanisms, such as through collisions, molecular drift into the interaction region, or chemical reactions, are indicated by the P’s. The effects resulting from depopulation through collisions and spontaneous radiation and nonradiative decay are indicated by the k”‘s. These kinetic processes include all means by which the levels may be populated or depopulated except by the radiation fields. For ground-state microwave optical double resonance the experimentally observed quantity is the rate of optical fluorescence from level 1 (Fig. la), kF1-N1, where kF, is the spontaneous emission rate constant and Nr is the population of level 1. In excited-state microwave optical double resonance the experimentally detected quantity is kF,-Nz + kF,.Ns (Fig. lb). The final result of the model is a general expression for the populations of the emitting levels, from which the rate of optical fluorescence is obtained. The model is developed in three stages. First, on the basis of the kinetics shown in Fig. 1, the population of each level is obtained from a steady-state kinetic analysis. KMand kL, the rate constants for the microwave and laser
88
WORMSBECHER, HARRIS AND WICKE
a 1
A
kL
b 3
FIG. 1. Schematic diagram indicating the three energy levels and kinetic processes considered in the model for (a) ground-state, and (b) excited-state microwave optical double resonance. Levels 2 and 3 are connected by a microwave transition (rate constant kn), and levels 1 and 2 are connected by a laser induced transition (rate constant KL). Spontaneous emission from level 1 in ground state (a), and levels 2 and 3(b) in excited-state microwave optical double resonance are monitored.
induced transitions, are then replaced transition rate constants appropriately
by the corresponding semiclassical two-state averaged over the effects of collisions. The
velocity dependence of the optical transition, the optical Doppler effect, is explicitly included in the laser induced transition rate constant Kn. Finally, explicit expressions for the rate of optical fluorescence for both ground-state and excited-state microwave optical double resonance are obtained by integrating over the molecular velocity distribution. Although this general framework has been used previously in modeling infrared microwave double resonance (16) and microwave optical double resonance (IO, II), the previous results were obtained using approximations not valid for large laser intensity. The results presented below include these previous calculations as special cases,
89
A MODEL FOR MODR
A. General Development of the Model The steady-state kinetic analysis is straightforward. Within the context schematized in Fig. 1, the rate equations for the changes in the populations of the levels are dnJdt
The steady-state
= kai + kM[nz - n3] - ka”ng
dnJdt
(2)
dnJdt
= k1” + kL[n2 - nl] - kl”m.
(3)
populations
kL[kM(k?
are found to be
+ kz” + ki) + k2(kz” + k+‘)] + klO[kM(k$ + ki) + kaikxo]
n3 = kL[kM (kl” + kz” + ks”) + ki’(k10 + k&l 182=
(1)
4 = kzi + kM[na - nz] + kL[nl - nz] - k&z,
+ klO[kia (ki’ + kaO)+ kz”ksol’
kL[kM(kli + ka” + kgi) + ksO(kl” + k$)] + kP[kM(k$
+ k&) + kiiks”]
kL[kM (RI” + kz0 + k30) + ksO(k10+ kx”)] + klO[kM(kz’ + ka”) + kz”ksol’ kL[kM(kl:
(4)
(5)
+ kzi + k2) + kz”(kli + kzi)] + kli[kM(kL’ + k30) + kz”kso]
n1 = kL[kM (RI0 + k%o+ KS”)+ k,O(kxO+ k?)] + klO[kM(KS0+ k2) + kznkso]’
(6)
kL and KM are the laser induced optical transition rate constant and the microwave transition rate constant, respectively. We use the rate constants derived by Lamb and others (17-20). Briefly, two quantum levels a and b are assumed to interact through an electric dipole matrix element involving the electric field vector of the electromagnetic radiation. The transition probability is obtained by exact solution of the time-dependent Schrodinger equation for these two levels in the presence of the time-dependent radiation field. When appropriately averaged over the effects of random dephasing collisions, the transition rate constant is given by (17) k
ran -
t (~aa&~/~)~kab~ (ka#
+ (u - ‘O,#’
(7)
where E” is the electric field strength of the radiation field, lab is the electric dipole matrix element between states a and k connected by the electromagnetic field, and k,b” is the average decay rate constant for the two levels. It should be noted that this transition rate constant contains no saturation term (20); the rate constant increases as the electric field strength of the electromagnetic field increases. The transition rate is given by transition
rate = k,,d(n,
- nb).
(8)
As the electromagnetic field strength increases, krnd increases, but the population difference n, - fib approaches a limiting value, and the transition rate saturates. Power broadening of a transition is not to be attributed to any intrinsic modification of the lineshape or transition rate constant, but instead results from a frequency dependent alteration of the energy level populations (20). Finally, the velocity dependence of the optical transition must be included in the laser induced transition rate constant kL. We assume that the homogeneous broadening
WORMSBECHER,
90
HARRIS
AND WICKE
of the microwave transition is much greater than the inhomogeneous broadening due to the microwave Doppler effeet, and consequently the Doppler effect will be ignored for the microwave transition. Because the molecules have a velocity component in the direction of laser propagation, v, the effective frequency seen by the molecule is b
u’ab
=
&ab[l
-
(9)
(v/c>],
where c is the speed of light. If Eq. (9) is substituted into Eq. (7) for a&,, and if the laser frequency is “tuned” to the maximum of the Doppler profile such that o = &b, one obtains for the laser induced transition rate constant, a (/.426L”/A)2K120 h(v)
=
(10) (h20j2
+
(w~v/c>~’
where kL(v) is written as a function of velocity. Assuming a Boltzmann velocity distribution for the one translational degree of freedom and substituting Eq. (10) into Eqs. (4)-(6), one obtains the number of molecules in the ith level with this one velocity component between v and v + dv, dNi(v) = ni(v) (e-‘Z/y2/y7r+)dv,
(11)
where y = (2kT/m)i. The total population velocity.
of the ith level Ni is then obtained
(12) by integrating
Eq. (11) over
B. Ground-State Microwave Optical Double Resonance In ground-state microwave optical double resonance, the laser is tuned to be resonant with the 2 -+ 1 transition, Fig, la. The intensity of laser induced fluorescence from level 1 is monitored while the microwave frequency is tuned. Since level 1 is an excited electronic state, kli, which describes the rate at which molecules enter level 1 by processes other than laser induced excitation, is usually very close to zero; in the following kli = 0 is used. If the molecular species being studied is produced in an excited electronic state, for example, by a chemical reaction, somewhat different results are obtained. It is also convenient to replace kzi and ksi using the equilibrium concentrations Nz” and NaO in the absence of any radiation fields, kz” = kdvz”,
(134
kgi = kJVs”,
W)
where kb = kzo = kso. Using the above assumptions in Eq. (6) for nl(v) and performing the integration over Eq. (ll), @I), one obtains for the rate of optical fluorescence R(wm) for ground-state microwave optical double resonance R(w”.) = kF,.Nl
A MODEL
FOR
91
MODR
where A’2 1+s=-, N20 o(q) = error function,3
C. Excited-State
2Microwave Optical Double Resonance
In excited-state microwave optical double resonance the detected signal is the total fluorescence intensity originating from both levels 2 and 3. KZoand KS0include both collisional depopulation and spontaneous emission. As in ground-state microwave optical double resonance, kz” and ksi are assumed to be zero. Also k1” has been defined in terms of the equilibrium population of level 1, NlO, in the absence of any radiation fields. With the above assumption used in Eq. (4) and Eq. (5) for ns(v) and n2(zt), and performing the integrations over Eq. (11) (21), one obtains the rate of optical fluorescence for excited-state microwave optical double resonance,
where
km(k1o + kz” + k2’) + k30(kz” + kf’) kl”[km(kzo + ho) + kz”k30] f$(q) = error function. III.
RESULTS
AND
DISCUSSION
The rates of optical fluorescence for ground-state Eq. (14) and excited-state microwave optical double resonance Eq. (15) depend on the laser power through &LO,and on both the microwave frequency wm and microwave power (&L2) through k,. The double resonance signal R(w,) has been written as a function of microwave frequency to emphasize that in microwave optical double resonance the detected signal is the change in the intensity of optical fluorescence as the microwave frequency is swept. It should be noted that in both the ground-state case and the excited-state case, R(w,) remains finite in the limit of large microwave pumping rates and large laser pumping rates; that is, R(w,J shows saturation effects even though the individual rate constants aWe use the convention for the Segun, “Handbook of Mathematical
error function as defined in Ref. (21) and M. Abramowitz and I. Functions,”
p. 29.5, Dover,
New York,
1968.
92
WORMSBECHER,
HARRIS
AND WICKE
k&r and kr, may increase formally without limit. It should be noted that Eqs. (14) and (15) are not appropriate for very large radiation field intensities when multiple-photon transitions are possible. In order to examine the predictions of the model, it is convenient to define a relative microwave optical double resonance signal I(w,,J by R(w,)
- R(&
= 0)
I(wm) = R(&
= 0)
’
(16)
where R(w,) is given by either Eq. (14) or Eq. (1.5) for ground-state or excited-state microwave optical double resonance, and R(& = 0) is the appropriate optical fluorescence rate when the microwave field is off. I(wm) then represents the relative change in the optical fluorescence intensity at microwave frequency (Jo. For purposes of comparison, numerical values of I(w,J have been generated for a variety of choices of laser power, microwave power, microwave frequency, and pressure. Other parameters, such as transition dipole moments, radiative lifetimes, laser beam dimensions, etc., have been chosen as appropriate for the microwave optical doubleresonance study of BaO (10). A. Ground-State
Microwave
Optical Double Resonartce
Figure 2 shows the dependence of the relative fluorescence intensity I&) for groundstate microwave optical double resonance at microwave resonance (wm = ~23) as a function of pressure for several laser powers. The microwave power is constant at 10 mW. For a given laser power, I(w,J increases as the pressure decreases. The slope of the curve depends critically on the other parameters. For sufficiently low pressures, I(o,,J saturates and further reduction in pressure will not increase I(wm). Figure 2 also illustrates the effect of laser power on I@%). For constant pressure, an increase in laser power increases I(w,J. Note, however, that the effect is highly nonlinear for the cases illustrated. At sufficiently high laser power, I(w,J saturates; a further increase in laser power results in no further increase in the relative signal. Note, too, that the total fluorescence intensity is a function of pressure, so that while the relative signal I(wJ may be large at low pressure, the absolute double resonance signal may be small. The relative microwave optical double resonance signal may be positive or negative, and may change sign as a function of pressure and laser power. The value of I&) at saturation depends in a complex way on all the parameters, including Nz” - Na”. For 100%. N3O = NZ”, this saturation value may be as large as approximately Figure 3 illustrates the dependence of the relative microwave optical double resonance intensity and lineshape on microwave power for a particular pressure and laser power. at microwave resonance As the microwave power increases, the relative intensity increases, and then levels off. The linewidth increases with microwave power, illustrating power broadening. The qualitative dependencies of the intensity of the microwave optical double resonance signal on laser power, microwave power, and pressure as predicted by this model agree with observations on BaO (7,X)). The broad lineshapes observed in BaO (7,lO) are also in good quantitative agreement with the model except in the tails.
A MODEL
FOR
93
MODR
PC = L 0
20
60
40
0.001w
60
!OO
(millitorrl
PRESSURE
FIG. 2. Calculated relative ground-state microwave optical double resonance signal intensity I(w,,,) at microwave resonance as a function of pressure for several laser powers. Other parameters are those appropriate to the experimental work on BaO [Ref. (lo)]; they are laser beam diameter = 0.2 mm, microwave field area = 0.5 cm2, lifetime of excited state = 3.50 nsec, collisional deactivation rate constants = 10’ collision see-’ Torr-I, ground-state dipole moment = 8 D.
Experimental increasing increase
microwave microwave agreement
the main features in this model. happening simply level
double
clearly
resonance
show
spectra
microwave
for BaO as a function
power
broadening
of
with
little
for BaO suggests
that
in intensity.
The general
double
optical power
The basis
as follows.
resonance proportional
the model microwave
for the model
The
is the
signal
total
by laser
i-
of the excited
0 +5
-5
from
until
level
at laser
P = 0.2
torr
pc = 0.1
watt
resonance a simple
in ground-state
fluorescence
excitation
i
-5
double
la) provides
monitored
unresolved
and N1 will increase,
and observations optical
(Fig.
being
to the population
1 is populated
will decrease
between
of ground-state
from
are incorporated picture
of what
microwave
excited
level
1, which
level. If the microwave 2. As the laser power
saturation
is
optical is
field is off, increases
Nz c- N1. When
-IT2
a strong
iii
0 +5 MICROWAVE
-5
0 +5
FREQUENCY
-5 (w-q~)
0 t5
-5
0 l5
in MHz
FIG. 3. Calculated ground-state microwave optical double resonance lineshapes as a function of microwave frequency at various microwave powers. Laser power and pressure are indicated. Other parameters are as given in the caption to Fig. 2.
94
WORMSBECHER,
HARRIS
AND
WICKE
microwave field is tuned to the 2-3 transition, the microwave field will modify the populations of levels 2 and 3 in the direction which reduces the difference 1ZVz- NS 1, toward the saturation limit of NZ = Ns. If in the absence of the microwave field Na” > 82, the microwave transition will increase Nz. Because Nz is coupled to N1 by the laser field, Nr will increase, and hence the optical fluorescence intensity will increase [I(c+,) > 0] as a result of the microwave transition. If, however, in the absence of the microwave field Ns” < Nz, then the microwave field will decrease Nz and, as a result, NI will decrease and the optical fluorescence intensity will decrease, making I&J negative. Since in the microwave region Na” = iVz”, the laser transition must be strong enough to substantially reduce the population of level 2 or only a small microwave optical double resonance signal is possible. If the laser is strong enough to saturate the optical transition, and if kF1 is very large, then a large relative microwave optical double resonance signal may result. For KM = 0, for example, and with k~ large enough to saturate the optical transition, the rate of optical fluorescence will be limited by the rate of collisional repopulation of level 2. With the addition of a saturation microwave field, however, fluorescence will be limited by the rates of repopulation of levels 2 and 3, and under these limiting conditions the relative microwave optical double resonance signal I(w,,J = 1ooo/o; that is, the optical fluorescence intensity will double on microwave resonance. B. Excited-State Microwave Optical Double Resonance Figure 4 shows the pressure dependence of the relative excited-state microwave optical double resonance signal I@,) at several laser powers. The microwave frequency is set on resonance wm = ws3, and the microwave power is constant at 10 mW. For any given laser power, I(w,,J increases with pressure, then decreases. As the laser power increases the position of the maximum shifts to higher pressure. The slope and shape of the relative intensity profile depend in a complex way CEq. (1.5)] on all of the parameters. Figure 4 also illustrates the laser power dependence of I(w&. For any given pressure,
Pmw= tOmw
PC= 0.1 w
I ,
I,
20
I1
40
PRESSURE
I
60
I
I
80
I
I
100
(millitorr)
FIG. 4. Calculated excited-state relative microwave optical double resonance signal intensity I(w,) at microwave resonance as a function of pressure for various laser powers. Other parameters are as given in caption to Fig. 2, except excited-state dipole moment = 4 D.
AIMODEL
9.5
FOR MODR
zx
2 i
1 Q
kF3 =
T
I
I
kF,
_-----------
_------
I
0
I
I
20
I
1
40 PRESSURE
---
60
I
80
I
1
too
(millitorr)
FIG. 5. Calculated excited-state relative microwave optical double resonance signal intensity I&,,,) at microwave resonance as a function of pressure for when kF2 = +kF8 and kF8 = $kFz. Laser power and microwave power are indicated. Other parameters are as given in Fig. 2.
I(w,J
decreases
with
increasing
laser
power.
The
effect
is highly
nonlinear.
I(wm)
Eq. (15), however, increases with increasing laser power. Once saturation of the l-2 transition is achieved, the rate at which the laser depletes level 1 can be made no larger than the rate at which level 1 is collisionally repopulated. Since the collisional repopulation rate is unaffected by the microwave field, I(wJ decreases with increasing laser power. The microwave frequency dependence of the relative microwave optical double resonance signal for the excited-state case has essentially the same shape as for the ground-state case, as shown in Fig. 3. As the microwave power increases the relative intensity increases, then levels off, indicating saturation of the microwave transition. However, in excited-state microwave optical double resonance saturation occurs at a higher microwave power, since levels 2 and 3 can be depopulated not only by collisions as in the ground-state case, but also by spontaneous emission. The linewidth also increases with increasing microwave power, illustrating power broadening. Additional numerical calculations show that the lineshapes exhibit pressure broadening, power broadening, and lifetime broadening features at the appropriate limits. An interesting case arises in excited-state microwave optical double resonance when the spontaneous emission lifetimes of the two excited-state rotational levels are different. This effect offers one explanation for the experimentally observed pressure dependence of the relative excited-state microwave optical double resonance signal study made by is the relative
signal;
the abso2ute signal,
Tanaka et al. (11) on the 808-919 transitions in the highly perturbed 2Bz excited state of pu’02. Figure 5 shows the pressure dependence of the relative signal I(w,J when the fluorescence lifetime of level 2 is greater than that of level 3, and vice versa. When the long-lived level is optically pumped the microwave field transfers molecules from the slow emitter to the fast emitter, making the relative signal very large and positive. However, when the fast emitter is optically pumped, the microwave field transfers molecules to the slow emitting
level, making the relative signal small. At high pressures
96
WORMSBECHER,
HARRIS
AND WICKE
the molecules in level 3, the slow emitting level, are collisionally deactivated faster than they can emit, hence, the relative signal becomes negative. The qualitative dependences of the excited-state microwave optical double resonance signal on microwave power, laser power, and pressure predicted by this model, are in good agreement with the experimental studies on BaO (8, 10) and NOz (II). In excitedstate microwave optical double resonance (Fig. lb), the signal being monitored is the total fluorescence from both levels 2 and 3. In the absence of the microwave field, level 2 is populated by laser excitation from level 1. In the presence of a saturating laser field, the populations of levels 1 and 2 will be approximately equal. In the limit of small fluorescent rate constants such that fluorescence merely “samples” the excited-state populations, when a saturating microwave field is applied, the populations will be shifted to an equal distribution between levels 1, 2, and 3. In this limit I&J = 33%. If, however, the fluorescent rate constants are very large such that fluorescence “depletes” levels 2 and 3, I(w,J = 0. This effect places a lifetime limit on excited-state microwave optical double resonance which is a function of the choice of parameters, i.e., laser beam dimensions, etc. For the choice of parameters used in this paper (IO), this limit is approximately 30 nsec. By monitoring the rotationally resolved fluorescence intensity of level 2 this lifetime effect is eliminated. Although the restrictions on excitedstate microwave optical double resonance are somewhat greater than the ground-state case, under appropriate conditions this technique can be approximately as sensitive as laser induced fluorescence spectroscopy. ACKNOWLEDGMENTS We would like to thank Professor Larry Davis for many helpful discussions. We would also like to thank Mr. Steven Lane for his assistance.
RECEIVED:July 23, 1976 REFERENCES 1. A. M. RONN ANDD. R. LIDE, JR., J. Chem. Phys. 47, 3669 (1967). 2. K. GERMANAND R. N. ZARE, Phys. Rev. Leti. 23, 1207 (1969). 3. S. J. SILVERS,T. H. BERGEUN, ANDWa6. KLEMPERER,J. Chem. Phys. 52, 4385 (1970). 4. T. SHIIUU ANDT. OKA, Phys. Rev. A 2, 1177 (1970). 5. R. W. FIELD ANDT. H. BERGEYAN,J. Chem. Phys. 54, 2936 (1971). 6. S. M. FREUNDANDT. OKA, Appl. Phys. Lett. 21, 60 (1972). 7. R. W. FIELD, R. S. BRADFORD,D. 0. HARRIS, ANDH. P. BROIDA,J. Chem. Phys. 56,4712
(1972).
8. R. W. FIELD, R. S. BRADFORD,H. P. BROIDA,AND D. 0. HARRIS, J. Chem. Phys. 57, 2209 (1972). 9. R. SOLARZANDD. H. LEVY, J. Chew Phys. 58,4026
(1973).
10. R. W. FIELD, A. D. ENGLISH,T. TANAKA, D. 0. HARRIS, ANDD. A. JENNINGS,J. Chem. Phys. 59, 2191 (1973). 11. T. TANAKA, A. D. ENGLISH,R. W. FIELD, D. A. JENMXGS, ANDD. 0. HARRIS, J. Chem. Phys. 59, 5217 (1973). 12. T. TANAKA, R. W. FIELD, ANDD. 0. HARRIS, J. Chem. Phys. 61,340l
(1974).
13. T. TANAKA, R. W. FIELD, ANDD. 0. HARRIS, J. Mol. Speclrosc. 56, 188 (1976). 14. T. TANAKA ANDD. 0. HARRIS, J. Mol. Spectrosc. 59, 413 (1976). 15. B. G. WICKE ANDD. 0. HARRIS,in “Thirtieth Symposium on Molecular Structure and Spectroscopy, 1975, Columbus, Ohio,” TC11.
A MODEL fli. M. TA~AMT ANI) K. S~IIMODA, /a@.
FOR MODR
97
J. Appl. Phys. 11, 1648 (1972).
17. W. LAMB, Phys. Rev. A 134, 1429 (1969). 18. A. JAVAN, Phys. Rev. 107, 1579 (1957). 19. C. H. TOWNESANDA. L. SCHAWLOW,“Microwave
Spectroscopy,”
McGraw-Hill,
New York, 195.5.
20. R. KARPLUSANDJ. SCHWINGER,Phys. Rev. 73, 1020 (1948).
21. I. S. GRADSHTEYNAND I. W. RYZHIK, “Table of Integrals, Series and Products,” 3.466, 1 arltl 2, Academic
Press, New York,
1965.
A Model
SPECTROSCOPY
64, 86-97 (1977)
for Ground-State and Excited-State Optical Double Resonance’
Microwave
RICHARD F. WORMSBECHER,DAVID 0. HARRIS, AND BRIAN G. WICKET Quantum Institute and Department of Chemistry, University of California, Santa Barbara, Santa Barbara, Calijornia 93106
A model is developed to describe the physical principles of both ground-state and excitedstate microwave optical double resonance. This model uses a steady-state kinetic analysis to determine the populations of three quantum levels in the presence of two monochromatic radiation fields. The microwave transition rate and the laser transition rate are obtained from separate analyses using the semiclassical two-state transition probability averaged over the effects of collisions. The Doppler effect of the optical transition rate is explicitly included. The competing effects of laser power, microwave power, and pressure on signal intensity and lineshape are described. Calculated lineshapes and relative signal intensities based on this model are in good agreement with previous measurements on BaO and NOz. I. INTRODUCTION The study of small molecules in the gas phase is limited by the trade-off between sensitivity and resolution characteristic of most conventional spectroscopic techniques. Optical spectroscopy, for example, has high sensitivity, but its resolution is normally limited by a Doppler width on the order of 1 GHz. In contrast, microwave spectroscopy has substantially greater resolution of typically 0.1 MHz at 10 mTorr pressure, but its sensitivity is basically low. Hence, the study of molecular rotation, Stark effects, hyperfine coupling, etc., via conventional spectroscopy has been restricted by and large to stable ground-state molecules which can be obtained in substantial concentrations. In the past few years several double resonance techniques have been developed in an attempt to combine the high resolution of radiofrequency and microwave spectroscopy with the sensitivity of optical spectroscopy (1-14). Many of these double resonance techniques share a fundamental basis by which the presence of a small energy resonance is detected via a coupled change in a larger energy resonance. Microwave optical double resonance is one such technique in which a microwave resonance is detected by a change in one of the properties of an associated optical transition. Microwave optical double resonance typically involves the interaction among two adjacent rotational levels in one vibronic state and a particular rotational level in another electronic state. The pair of rotational levels may be in the ground electronic state or an excited electronic state, corresponding to ground-state or excited-state microwave optical double resonance, respectively. In the ground-state case the pair of 1Work supported in part by NSF Grant MPS72-04978. 2 Present address: TRW Systems Group R1/1196, One Space Park, Redondo Beach, California 90278. 86 Copyright Q 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
A MODEL FOR MODR
87
rotational levels are connected by a microwave transition, and one of these levels is connected with a single rotational level in an electronically excited state by a laser induced transition. The presence of microwave resonance between the lower levels is detected as a change in the intensity of the laser induced fluorescence from the excited electronic state. In excited-state microwave optical double resonance, microwave resonance is detected as a change in the intensity of the total rotationally unresolved laser induced fluorescence from both the excited electronic-rotational levels. No microwave radiation detection is used. Hence, rotational transitions in the ground state or an electronically excited state are measured with resolution approaching that of microwave spectroscopy but with sensitivity characteristic of laser induced optical fluorescence. From this description it is clear that the term “microwave optical double resonance” is really a misnomer; a more appropriate name would be “optical detection The key experimental feature is that, although microwave of microwave transitions.” transitions are being studied, the detector is typically a phototube. High resolution is derived from the microwave radiation, high sensitivity from the detection of laser induced fluorescence. This paper presents a simple model for the technique of optical detection of microwave transitions. The principal motivation for the development of this model was the practical need for a systematic guide for applying this technique to new molecular systems. This requires an understanding of intensities and linewidths, and their dependences on the laser power, microwave power, collisions, and spontaneous emission. This work is an extension of a previously presented paper (15). In Section II the model is developed. In Section III calculated intensity profiles and microwave lineshapes are presented and compared with available experimental results from this laboratory (10-14). II. DERIVATION
OF THE MODEL
The model treats the interaction between three quantum levels in the presence of two monochromatic fields and collisions, as shown in Fig. la for ground-state, and Fig. lb for excited-state microwave optical double resonance. Levels 2 and 3 interact directly via a microwave radiation field with a transition rate constant k~. Levels 1 and 2 interact directly via a laser radiation field with a transition rate constant kr,. Kinetic processes which populate the levels via other mechanisms, such as through collisions, molecular drift into the interaction region, or chemical reactions, are indicated by the P’s. The effects resulting from depopulation through collisions and spontaneous radiation and nonradiative decay are indicated by the k”‘s. These kinetic processes include all means by which the levels may be populated or depopulated except by the radiation fields. For ground-state microwave optical double resonance the experimentally observed quantity is the rate of optical fluorescence from level 1 (Fig. la), kF1-N1, where kF, is the spontaneous emission rate constant and Nr is the population of level 1. In excited-state microwave optical double resonance the experimentally detected quantity is kF,-Nz + kF,.Ns (Fig. lb). The final result of the model is a general expression for the populations of the emitting levels, from which the rate of optical fluorescence is obtained. The model is developed in three stages. First, on the basis of the kinetics shown in Fig. 1, the population of each level is obtained from a steady-state kinetic analysis. KMand kL, the rate constants for the microwave and laser
88
WORMSBECHER, HARRIS AND WICKE
a 1
A
kL
b 3
FIG. 1. Schematic diagram indicating the three energy levels and kinetic processes considered in the model for (a) ground-state, and (b) excited-state microwave optical double resonance. Levels 2 and 3 are connected by a microwave transition (rate constant kn), and levels 1 and 2 are connected by a laser induced transition (rate constant KL). Spontaneous emission from level 1 in ground state (a), and levels 2 and 3(b) in excited-state microwave optical double resonance are monitored.
induced transitions, are then replaced transition rate constants appropriately
by the corresponding semiclassical two-state averaged over the effects of collisions. The
velocity dependence of the optical transition, the optical Doppler effect, is explicitly included in the laser induced transition rate constant Kn. Finally, explicit expressions for the rate of optical fluorescence for both ground-state and excited-state microwave optical double resonance are obtained by integrating over the molecular velocity distribution. Although this general framework has been used previously in modeling infrared microwave double resonance (16) and microwave optical double resonance (IO, II), the previous results were obtained using approximations not valid for large laser intensity. The results presented below include these previous calculations as special cases,
89
A MODEL FOR MODR
A. General Development of the Model The steady-state kinetic analysis is straightforward. Within the context schematized in Fig. 1, the rate equations for the changes in the populations of the levels are dnJdt
The steady-state
= kai + kM[nz - n3] - ka”ng
dnJdt
(2)
dnJdt
= k1” + kL[n2 - nl] - kl”m.
(3)
populations
kL[kM(k?
are found to be
+ kz” + ki) + k2(kz” + k+‘)] + klO[kM(k$ + ki) + kaikxo]
n3 = kL[kM (kl” + kz” + ks”) + ki’(k10 + k&l 182=
(1)
4 = kzi + kM[na - nz] + kL[nl - nz] - k&z,
+ klO[kia (ki’ + kaO)+ kz”ksol’
kL[kM(kli + ka” + kgi) + ksO(kl” + k$)] + kP[kM(k$
+ k&) + kiiks”]
kL[kM (RI” + kz0 + k30) + ksO(k10+ kx”)] + klO[kM(kz’ + ka”) + kz”ksol’ kL[kM(kl:
(4)
(5)
+ kzi + k2) + kz”(kli + kzi)] + kli[kM(kL’ + k30) + kz”kso]
n1 = kL[kM (RI0 + k%o+ KS”)+ k,O(kxO+ k?)] + klO[kM(KS0+ k2) + kznkso]’
(6)
kL and KM are the laser induced optical transition rate constant and the microwave transition rate constant, respectively. We use the rate constants derived by Lamb and others (17-20). Briefly, two quantum levels a and b are assumed to interact through an electric dipole matrix element involving the electric field vector of the electromagnetic radiation. The transition probability is obtained by exact solution of the time-dependent Schrodinger equation for these two levels in the presence of the time-dependent radiation field. When appropriately averaged over the effects of random dephasing collisions, the transition rate constant is given by (17) k
ran -
t (~aa&~/~)~kab~ (ka#
+ (u - ‘O,#’
(7)
where E” is the electric field strength of the radiation field, lab is the electric dipole matrix element between states a and k connected by the electromagnetic field, and k,b” is the average decay rate constant for the two levels. It should be noted that this transition rate constant contains no saturation term (20); the rate constant increases as the electric field strength of the electromagnetic field increases. The transition rate is given by transition
rate = k,,d(n,
- nb).
(8)
As the electromagnetic field strength increases, krnd increases, but the population difference n, - fib approaches a limiting value, and the transition rate saturates. Power broadening of a transition is not to be attributed to any intrinsic modification of the lineshape or transition rate constant, but instead results from a frequency dependent alteration of the energy level populations (20). Finally, the velocity dependence of the optical transition must be included in the laser induced transition rate constant kL. We assume that the homogeneous broadening
WORMSBECHER,
90
HARRIS
AND WICKE
of the microwave transition is much greater than the inhomogeneous broadening due to the microwave Doppler effeet, and consequently the Doppler effect will be ignored for the microwave transition. Because the molecules have a velocity component in the direction of laser propagation, v, the effective frequency seen by the molecule is b
u’ab
=
&ab[l
-
(9)
(v/c>],
where c is the speed of light. If Eq. (9) is substituted into Eq. (7) for a&,, and if the laser frequency is “tuned” to the maximum of the Doppler profile such that o = &b, one obtains for the laser induced transition rate constant, a (/.426L”/A)2K120 h(v)
=
(10) (h20j2
+
(w~v/c>~’
where kL(v) is written as a function of velocity. Assuming a Boltzmann velocity distribution for the one translational degree of freedom and substituting Eq. (10) into Eqs. (4)-(6), one obtains the number of molecules in the ith level with this one velocity component between v and v + dv, dNi(v) = ni(v) (e-‘Z/y2/y7r+)dv,
(11)
where y = (2kT/m)i. The total population velocity.
of the ith level Ni is then obtained
(12) by integrating
Eq. (11) over
B. Ground-State Microwave Optical Double Resonance In ground-state microwave optical double resonance, the laser is tuned to be resonant with the 2 -+ 1 transition, Fig, la. The intensity of laser induced fluorescence from level 1 is monitored while the microwave frequency is tuned. Since level 1 is an excited electronic state, kli, which describes the rate at which molecules enter level 1 by processes other than laser induced excitation, is usually very close to zero; in the following kli = 0 is used. If the molecular species being studied is produced in an excited electronic state, for example, by a chemical reaction, somewhat different results are obtained. It is also convenient to replace kzi and ksi using the equilibrium concentrations Nz” and NaO in the absence of any radiation fields, kz” = kdvz”,
(134
kgi = kJVs”,
W)
where kb = kzo = kso. Using the above assumptions in Eq. (6) for nl(v) and performing the integration over Eq. (ll), @I), one obtains for the rate of optical fluorescence R(wm) for ground-state microwave optical double resonance R(w”.) = kF,.Nl
A MODEL
FOR
91
MODR
where A’2 1+s=-, N20 o(q) = error function,3
C. Excited-State
2Microwave Optical Double Resonance
In excited-state microwave optical double resonance the detected signal is the total fluorescence intensity originating from both levels 2 and 3. KZoand KS0include both collisional depopulation and spontaneous emission. As in ground-state microwave optical double resonance, kz” and ksi are assumed to be zero. Also k1” has been defined in terms of the equilibrium population of level 1, NlO, in the absence of any radiation fields. With the above assumption used in Eq. (4) and Eq. (5) for ns(v) and n2(zt), and performing the integrations over Eq. (11) (21), one obtains the rate of optical fluorescence for excited-state microwave optical double resonance,
where
km(k1o + kz” + k2’) + k30(kz” + kf’) kl”[km(kzo + ho) + kz”k30] f$(q) = error function. III.
RESULTS
AND
DISCUSSION
The rates of optical fluorescence for ground-state Eq. (14) and excited-state microwave optical double resonance Eq. (15) depend on the laser power through &LO,and on both the microwave frequency wm and microwave power (&L2) through k,. The double resonance signal R(w,) has been written as a function of microwave frequency to emphasize that in microwave optical double resonance the detected signal is the change in the intensity of optical fluorescence as the microwave frequency is swept. It should be noted that in both the ground-state case and the excited-state case, R(w,) remains finite in the limit of large microwave pumping rates and large laser pumping rates; that is, R(w,J shows saturation effects even though the individual rate constants aWe use the convention for the Segun, “Handbook of Mathematical
error function as defined in Ref. (21) and M. Abramowitz and I. Functions,”
p. 29.5, Dover,
New York,
1968.
92
WORMSBECHER,
HARRIS
AND WICKE
k&r and kr, may increase formally without limit. It should be noted that Eqs. (14) and (15) are not appropriate for very large radiation field intensities when multiple-photon transitions are possible. In order to examine the predictions of the model, it is convenient to define a relative microwave optical double resonance signal I(w,,J by R(w,)
- R(&
= 0)
I(wm) = R(&
= 0)
’
(16)
where R(w,) is given by either Eq. (14) or Eq. (1.5) for ground-state or excited-state microwave optical double resonance, and R(& = 0) is the appropriate optical fluorescence rate when the microwave field is off. I(wm) then represents the relative change in the optical fluorescence intensity at microwave frequency (Jo. For purposes of comparison, numerical values of I(w,J have been generated for a variety of choices of laser power, microwave power, microwave frequency, and pressure. Other parameters, such as transition dipole moments, radiative lifetimes, laser beam dimensions, etc., have been chosen as appropriate for the microwave optical doubleresonance study of BaO (10). A. Ground-State
Microwave
Optical Double Resonartce
Figure 2 shows the dependence of the relative fluorescence intensity I&) for groundstate microwave optical double resonance at microwave resonance (wm = ~23) as a function of pressure for several laser powers. The microwave power is constant at 10 mW. For a given laser power, I(w,J increases as the pressure decreases. The slope of the curve depends critically on the other parameters. For sufficiently low pressures, I(o,,J saturates and further reduction in pressure will not increase I(wm). Figure 2 also illustrates the effect of laser power on I@%). For constant pressure, an increase in laser power increases I(w,J. Note, however, that the effect is highly nonlinear for the cases illustrated. At sufficiently high laser power, I(w,J saturates; a further increase in laser power results in no further increase in the relative signal. Note, too, that the total fluorescence intensity is a function of pressure, so that while the relative signal I(wJ may be large at low pressure, the absolute double resonance signal may be small. The relative microwave optical double resonance signal may be positive or negative, and may change sign as a function of pressure and laser power. The value of I&) at saturation depends in a complex way on all the parameters, including Nz” - Na”. For 100%. N3O = NZ”, this saturation value may be as large as approximately Figure 3 illustrates the dependence of the relative microwave optical double resonance intensity and lineshape on microwave power for a particular pressure and laser power. at microwave resonance As the microwave power increases, the relative intensity increases, and then levels off. The linewidth increases with microwave power, illustrating power broadening. The qualitative dependencies of the intensity of the microwave optical double resonance signal on laser power, microwave power, and pressure as predicted by this model agree with observations on BaO (7,X)). The broad lineshapes observed in BaO (7,lO) are also in good quantitative agreement with the model except in the tails.
A MODEL
FOR
93
MODR
PC = L 0
20
60
40
0.001w
60
!OO
(millitorrl
PRESSURE
FIG. 2. Calculated relative ground-state microwave optical double resonance signal intensity I(w,,,) at microwave resonance as a function of pressure for several laser powers. Other parameters are those appropriate to the experimental work on BaO [Ref. (lo)]; they are laser beam diameter = 0.2 mm, microwave field area = 0.5 cm2, lifetime of excited state = 3.50 nsec, collisional deactivation rate constants = 10’ collision see-’ Torr-I, ground-state dipole moment = 8 D.
Experimental increasing increase
microwave microwave agreement
the main features in this model. happening simply level
double
clearly
resonance
show
spectra
microwave
for BaO as a function
power
broadening
of
with
little
for BaO suggests
that
in intensity.
The general
double
optical power
The basis
as follows.
resonance proportional
the model microwave
for the model
The
is the
signal
total
by laser
i-
of the excited
0 +5
-5
from
until
level
at laser
P = 0.2
torr
pc = 0.1
watt
resonance a simple
in ground-state
fluorescence
excitation
i
-5
double
la) provides
monitored
unresolved
and N1 will increase,
and observations optical
(Fig.
being
to the population
1 is populated
will decrease
between
of ground-state
from
are incorporated picture
of what
microwave
excited
level
1, which
level. If the microwave 2. As the laser power
saturation
is
optical is
field is off, increases
Nz c- N1. When
-IT2
a strong
iii
0 +5 MICROWAVE
-5
0 +5
FREQUENCY
-5 (w-q~)
0 t5
-5
0 l5
in MHz
FIG. 3. Calculated ground-state microwave optical double resonance lineshapes as a function of microwave frequency at various microwave powers. Laser power and pressure are indicated. Other parameters are as given in the caption to Fig. 2.
94
WORMSBECHER,
HARRIS
AND
WICKE
microwave field is tuned to the 2-3 transition, the microwave field will modify the populations of levels 2 and 3 in the direction which reduces the difference 1ZVz- NS 1, toward the saturation limit of NZ = Ns. If in the absence of the microwave field Na” > 82, the microwave transition will increase Nz. Because Nz is coupled to N1 by the laser field, Nr will increase, and hence the optical fluorescence intensity will increase [I(c+,) > 0] as a result of the microwave transition. If, however, in the absence of the microwave field Ns” < Nz, then the microwave field will decrease Nz and, as a result, NI will decrease and the optical fluorescence intensity will decrease, making I&J negative. Since in the microwave region Na” = iVz”, the laser transition must be strong enough to substantially reduce the population of level 2 or only a small microwave optical double resonance signal is possible. If the laser is strong enough to saturate the optical transition, and if kF1 is very large, then a large relative microwave optical double resonance signal may result. For KM = 0, for example, and with k~ large enough to saturate the optical transition, the rate of optical fluorescence will be limited by the rate of collisional repopulation of level 2. With the addition of a saturation microwave field, however, fluorescence will be limited by the rates of repopulation of levels 2 and 3, and under these limiting conditions the relative microwave optical double resonance signal I(w,,J = 1ooo/o; that is, the optical fluorescence intensity will double on microwave resonance. B. Excited-State Microwave Optical Double Resonance Figure 4 shows the pressure dependence of the relative excited-state microwave optical double resonance signal I@,) at several laser powers. The microwave frequency is set on resonance wm = ws3, and the microwave power is constant at 10 mW. For any given laser power, I(w,,J increases with pressure, then decreases. As the laser power increases the position of the maximum shifts to higher pressure. The slope and shape of the relative intensity profile depend in a complex way CEq. (1.5)] on all of the parameters. Figure 4 also illustrates the laser power dependence of I(w&. For any given pressure,
Pmw= tOmw
PC= 0.1 w
I ,
I,
20
I1
40
PRESSURE
I
60
I
I
80
I
I
100
(millitorr)
FIG. 4. Calculated excited-state relative microwave optical double resonance signal intensity I(w,) at microwave resonance as a function of pressure for various laser powers. Other parameters are as given in caption to Fig. 2, except excited-state dipole moment = 4 D.
AIMODEL
9.5
FOR MODR
zx
2 i
1 Q
kF3 =
T
I
I
kF,
_-----------
_------
I
0
I
I
20
I
1
40 PRESSURE
---
60
I
80
I
1
too
(millitorr)
FIG. 5. Calculated excited-state relative microwave optical double resonance signal intensity I&,,,) at microwave resonance as a function of pressure for when kF2 = +kF8 and kF8 = $kFz. Laser power and microwave power are indicated. Other parameters are as given in Fig. 2.
I(w,J
decreases
with
increasing
laser
power.
The
effect
is highly
nonlinear.
I(wm)
Eq. (15), however, increases with increasing laser power. Once saturation of the l-2 transition is achieved, the rate at which the laser depletes level 1 can be made no larger than the rate at which level 1 is collisionally repopulated. Since the collisional repopulation rate is unaffected by the microwave field, I(wJ decreases with increasing laser power. The microwave frequency dependence of the relative microwave optical double resonance signal for the excited-state case has essentially the same shape as for the ground-state case, as shown in Fig. 3. As the microwave power increases the relative intensity increases, then levels off, indicating saturation of the microwave transition. However, in excited-state microwave optical double resonance saturation occurs at a higher microwave power, since levels 2 and 3 can be depopulated not only by collisions as in the ground-state case, but also by spontaneous emission. The linewidth also increases with increasing microwave power, illustrating power broadening. Additional numerical calculations show that the lineshapes exhibit pressure broadening, power broadening, and lifetime broadening features at the appropriate limits. An interesting case arises in excited-state microwave optical double resonance when the spontaneous emission lifetimes of the two excited-state rotational levels are different. This effect offers one explanation for the experimentally observed pressure dependence of the relative excited-state microwave optical double resonance signal study made by is the relative
signal;
the abso2ute signal,
Tanaka et al. (11) on the 808-919 transitions in the highly perturbed 2Bz excited state of pu’02. Figure 5 shows the pressure dependence of the relative signal I(w,J when the fluorescence lifetime of level 2 is greater than that of level 3, and vice versa. When the long-lived level is optically pumped the microwave field transfers molecules from the slow emitter to the fast emitter, making the relative signal very large and positive. However, when the fast emitter is optically pumped, the microwave field transfers molecules to the slow emitting
level, making the relative signal small. At high pressures
96
WORMSBECHER,
HARRIS
AND WICKE
the molecules in level 3, the slow emitting level, are collisionally deactivated faster than they can emit, hence, the relative signal becomes negative. The qualitative dependences of the excited-state microwave optical double resonance signal on microwave power, laser power, and pressure predicted by this model, are in good agreement with the experimental studies on BaO (8, 10) and NOz (II). In excitedstate microwave optical double resonance (Fig. lb), the signal being monitored is the total fluorescence from both levels 2 and 3. In the absence of the microwave field, level 2 is populated by laser excitation from level 1. In the presence of a saturating laser field, the populations of levels 1 and 2 will be approximately equal. In the limit of small fluorescent rate constants such that fluorescence merely “samples” the excited-state populations, when a saturating microwave field is applied, the populations will be shifted to an equal distribution between levels 1, 2, and 3. In this limit I&J = 33%. If, however, the fluorescent rate constants are very large such that fluorescence “depletes” levels 2 and 3, I(w,J = 0. This effect places a lifetime limit on excited-state microwave optical double resonance which is a function of the choice of parameters, i.e., laser beam dimensions, etc. For the choice of parameters used in this paper (IO), this limit is approximately 30 nsec. By monitoring the rotationally resolved fluorescence intensity of level 2 this lifetime effect is eliminated. Although the restrictions on excitedstate microwave optical double resonance are somewhat greater than the ground-state case, under appropriate conditions this technique can be approximately as sensitive as laser induced fluorescence spectroscopy. ACKNOWLEDGMENTS We would like to thank Professor Larry Davis for many helpful discussions. We would also like to thank Mr. Steven Lane for his assistance.
RECEIVED:July 23, 1976 REFERENCES 1. A. M. RONN ANDD. R. LIDE, JR., J. Chem. Phys. 47, 3669 (1967). 2. K. GERMANAND R. N. ZARE, Phys. Rev. Leti. 23, 1207 (1969). 3. S. J. SILVERS,T. H. BERGEUN, ANDWa6. KLEMPERER,J. Chem. Phys. 52, 4385 (1970). 4. T. SHIIUU ANDT. OKA, Phys. Rev. A 2, 1177 (1970). 5. R. W. FIELD ANDT. H. BERGEYAN,J. Chem. Phys. 54, 2936 (1971). 6. S. M. FREUNDANDT. OKA, Appl. Phys. Lett. 21, 60 (1972). 7. R. W. FIELD, R. S. BRADFORD,D. 0. HARRIS, ANDH. P. BROIDA,J. Chem. Phys. 56,4712
(1972).
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(1973).
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(1974).
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A MODEL fli. M. TA~AMT ANI) K. S~IIMODA, /a@.
FOR MODR
97
J. Appl. Phys. 11, 1648 (1972).
17. W. LAMB, Phys. Rev. A 134, 1429 (1969). 18. A. JAVAN, Phys. Rev. 107, 1579 (1957). 19. C. H. TOWNESANDA. L. SCHAWLOW,“Microwave
Spectroscopy,”
McGraw-Hill,
New York, 195.5.
20. R. KARPLUSANDJ. SCHWINGER,Phys. Rev. 73, 1020 (1948).
21. I. S. GRADSHTEYNAND I. W. RYZHIK, “Table of Integrals, Series and Products,” 3.466, 1 arltl 2, Academic
Press, New York,
1965.