- Email: [email protected]
Time-resolved four-level double resonance in the microwave spectrum of formaldehyde
Chemical Physics56 (1981) 231,239
..
Ndrth-Ho&and PublishingCompany
TIME-RESOLVED
FOUR-LEVEL
OF FORMALDEtHYDE -.
DOUBiE
RESONANCE IN THE MICROWAVE SPECTRUM
_
David A. &DREW Department of Physics, Schuster Lobmat&,
Received29 Juiy 1980
The Univerdy. Manchester Ml3 9PL:UK
-
The technique o; tram&t double resonance has been applied to the (2 11 G 2&-(311~ 313)~ system in the microwave s$ectmm of formaldehyde. An effective broadkng parameter 7*/27rp = 22.96 f 0.3 MHz Ton--’ was measured and theory is presented to reldte this to other measured multipole relaxation rates.
1_ Introduction
:. Over the past twenty years, much effort has gone into measuring relaxation times for rotational transitions in polar molecules; These measurements fall into two main cl-s: linf%hape analysis of static spectral profies [1,2] and time-resolved measurements of transient phenomena [3-5]_ Analysis of these experiments has usually been confined to a determination of the conventional TI and T2 population and polarisation relaxation times corresponding to a two-level system; however recent theoretical [6-S] and experimental Work [9,10] has indicated that, in a multi-level system in which collisional transfer of population is important, the different types of experiment can give quite different soits of information about relaxation. proce&s. A promistii t&hnique in this respkt is &a! of *e-resolved dotible resonance (TDR).. Steady+ate do_uble tesonance,pticu@rly in the hands of.Oka and co-workers [ll] has&&n very &cessful in &l&dating
midable technical problems associated with the rapid switching of very Iarge voltages applied to the s+nple. Instead the microwave radiation is itqlf modulated by means of a PIN diode.before it enters the sample cell. The systems most commonly studied by double resonance are three-level systems, in which pump and probe transitions share a common energy level, and the pump field directly affects the probe populatio.ns tid polarisations through this common level. Analysis of the transient response for systems of this sort is complicated by this direct coupling and will form the subject of a following paper [X3]. This present paper is confined to the study of four-level double resonances in which the pump can only affect the probe transition by collisional transfer of population. Because the effects of these crossrelaxation processes are smaller than the direct relaxi-. tion,.typic&y by an order of magnitude, the signals observed aF less.intense than those produced by Stark-switching by this factor. However, the Q b&ch
232
D.A. Andrews/
Resonancein
microwavespectrum
offormaldehyde
1 :
-.
‘. I:..:.-
RQ621 klystrons giving into the cell 100 mW a&d 0.2 mW for the pump and probe respectively. The punm was modulated using a Microwave Associates PIN ’ diode switch with a U ns switching time and >30 dB isolation. A Hewlett-Packard ‘%faffleGron” low-pass tiher and isolator were used to prevent the pump radiation falling on the detector. The isolator in addition to providing a good match for the probe radiation will absorb 90% of incident pump power with * PP
Fig. 1. Schematicd&am of apparatus. Key: A ampiitier. AT attenuator, BA boxcar avenger, CK clock generator,CM capacitancemanometergauge,CP cross@d coupIer,D micrcwoe detector, F low-passfilter,G samplega;trinput. I X-band isolator,Kl Oki30 V12 klystron,K2 EM R9621 kfysnon, S PIh diodeswitch.SC waveguide Stark cell. fl microprocessor-baszdcontrol unit.
2. Experiment A schematic d&ram of the apparatus is given in fig. 1_The system is based around a conventional Hewlett-Packard X-band Stark cell with, as signal sources, the phase-stabilized OKI 3OVi2 and Ehfl
only a few percent reflected, thus improving the impedance match for pump, as well as supplementing the rejection of *-hefilter. The transient signal from the
J-band detector (el PV) was amplified by 90 dB using wideband amplifiers (risetime ~3 ns overall) and fed to a boxcar signal averager. The minimum gatewidth available on the boxcar was 10 ns and the effective resolution of the camplete system was measured to be 20 ns_ A 1 MB crystal clock divided down to give a 125kHz squarewavewasusedtodrivethe PINdiode modulator and to give a reference signal to trigger the boxcar averager 1 MSbefore the pump was switched Off. The data acquisition system [I91 controlled by a hlC68CKlmicroprocessor, provided a stepped ramp voltage to sweep the time delay of’the boxcar; the
F
DA; Andrews / Resonnnce in microwave spectrum of formnldehyde analogue output from the avetiger was synchronously converted to an 8_bit digital form and accumulated in memory. The results could be dispIayed on an oscilloscope or punched out onto tape for further analysis. Typically a double sweep over 64 channels would take 2 miu witha box& time constant of 300 ms. A typical signal response as the pump power Is switched offisshowninfig.2. The pump and probe frequencies were kept exactlyonresonanceforthe3,,+3,, and2,,+2,2 resonances at 28975 and 14.458 GHz respectively, by phaselocking each to an harmonic of an UHF frequency synthesizer. The pressure in the cell was mot&red by a.capacitan? manometer gauge. Data was taken over a series of pressures from 60 to 160 mTorr and at low and moderate probe powers (70 and 170 ccw)_ To obtain an acc\lrate t&e-scale calibration for the boxcar, the time delay produced by the boxcar was compared with that’produced by an oscilloscope with a delayed timebase; this was in turn caIiirated Using a 100 MHz reference sinewave to generate markers every 10 ns.
3_ Theory The theory of transient double resonance phen&nena has been given by several authors [13-16,20-221, so only a simplified treatment adequate for the pres-
233
ent experiment will be given here,Consid& the fourlevel system .&own in figi 3, assuming for the moment that the levels are non-degenerate. The difference in fractional populations for the pump and probe transitions will be represented by W andM respectively, which in the absence of applied fields will relax back to their thermal equilibrium vahxes w and B with characteristic rates 7~ and ye_ A change in W however will also produce a change in N by cross-relaxation, and this is represented by the ~NW term, coupling the population differences. Thus for the situation of pump field off and finite probe field e(t) = .$ cos wr exactly on resonance w = wg we may write down the simplified Bloch equations [3,7,23] I -8,’ = -2FQ
+ ~&IV - rs, + ~lVw(w - @),
(la)
~=~N+Y~Q, -??=y&v-
(lb) m)+y~&v-B),
(14
where Q is related to the inquadrature component of the polarisation yDiinduced by the probe field E(r) by
where No is the total number density of absorbing molecules and &,b the transition dipole moment. Q may also be related to the observed absorption Q per unit length of the probe radiation by (r = --wNo!&&&
a&
F = .&abl~
-
A couple of comments may be made about these equations. Firstly, the pump polarisation does not appear explicitly in any of these equations and its decay therefore does not contribute.to the observed signal (unlike the three-level case). Secondly, we have as&med that the population sums (n, + nb) and . (n, + na) are *e-independent. This approximafion is particularly good f&r molecules with energy leveLs formi$g closely spaced doublets, such as formaldeHyde, but more generally it has been shown [7], *ht. the coupling of population sum to popiilation difference only contributes to second drder in the. observabl&. Similarly populati&&ansf$r o&o‘f th& syste&.
DA. Androvs
234
/ Remncttce
in microwave spectrum of fomialdehyde and the startingvalues Q(O) and N(O) have been
7~w(W - @) in eq. (la)_ This approximation essentially neglects the effect of the probe in transferring population to the pumped transition, thus changing its reaction on the probeThe solution of this set of equations is most elegantly performed using Laplace transform techniques [24], which when applied to eqs. (I) give
exponent&& and damped sinusoids driven by the probe field (probe nutations), and agrees with the form given by Glorieux $201 if his approximation 7N = 7rv = yQ = ,y is made:
0 = p& - N(0) + ‘yiy(R - R/p)
Q(t) = QW - QX e-”
*r&9
@a)
- W/P) - 2@!,
0=&j-
Q(O)+7&
o=pW--
W(O)‘7”@_
eliminated by solutions of-the steady-state equations. The form of eq. (4)is seen to be a combination of
CZ!Ffi, Rip),
(2b) (2c)
where the transformed variables fi, Q and i? are functions of the dummy variable p only. Solving for D and rii in terms of R and substituting into eq. (2a) gives
This equation may be converted by an inverse Laplace transform to give Q(f), which has two different forms depending on whether 4F* is greater or less than &rN - yQi* _ Taking the case of large probe power (4F= >$7~-7Q~2)theSOhtiOUhasthefO~Otig
Eq. (4) is a complicated expression to use for the analysis of the experimental data, especially.since an average has to be taken over the non-uniform power distribution in the cell_ At high pressures however, the transient response is dominated by relaxation processes and it is worthwhile to obtain a simpler expression to fust order in the probe power. TO this end consider the case of small probe power (4F2 < 7’) by returning to eq. (3) and putting 4F2 = 0 in the denominators This is essentially equivalent to deleting the fit term on the rhs of eq. (la), which would normally produce saturation effects for the probe transition, and are negligible at low probe payer. inverting the Laplace transform of the thus modified eq- (3) and =ummg h’ - YQ 1, fTN - TQi * rQ > such that we can neglect second order terms Irw - -ye I* compared to 7’ , the exponentials may be expanded to give Q(t)= Q(m) - QA[l +7*t
(6)
+:r”t’]exp(-7*t),
fern
where 7* = tk
YQ *%7,
= -w,YNi+’
+ TN - 2rQ), [w(o)
-
(7)
WhQrN-
It may be noted_that the pumping conditions only appear in eq. (6) through the coefficient QA, so the forrnof equation is true even for non-uniform power distriiution across a waveguide. This present form differs somewhat unexpectedly from that obtained .,.. by Brown [15]., namely
&j-= Q(->7 QA(1 .* 70
ai&-&
:--.
1 (8)
D-4. &drews/
Resonance in mictottave spectrum offormaldehyde
235
direct transfer of population from pumped to probed levels’is absent. The correctness of eq. (6j may be confirmed by taking the expression (5) and making ari expansion in powers of F*/7*, obtaining Q(r) = Q(i) - QA emn [l f 7t f 4 y* tz
which agrees with eq. (6> in the limit F*/yZ + 0, if the y’s are equal. Throughout this section the assumption has been made that the levels are non-degenerate even though eachJ state in fact has (25 + 1) fold degeneracy. It has been shown however that this degeneracy may be elegantly taken into account by use of a tensor pperator formalism [23,25,26]. In this approach the populations and polarisations are represented by a series of spherical tensors 7$) whose symmetry properties greatly simplify the analysis of complex multi-level systems. As shown in the appendix the analysis of this section may be readily carried through for the multidegenerate case to yield an equation of exactly the same form aseq. (7), but with a _ different interpretation of 7*, namely 7* = 7& + 4-{$(7fl+ + fi’(7,z
7fl-
27Qx)
+ 7$ - 27Q1))@’ -Pfi*>-’ ,
(10)
where 7p, 7rrp, etc. are the relaxation parameters for the individual spherical tensors p, No, etc., and the coefficients fi are given by eq. (A-6) in the appendix. The effective relaxation parameter is seen to be a weighted means of-the relaxation,parameters associated with the different multipole tensors contributing to ihe signal.
4. D%ti kalysis and discussion
..
-’
The data from each individual run was leastsquares fit to the 4 parameter formula: v(t) =Ai +A3 exp(-r)(l
+ r+ $7’):
7 > 0,
Fig. 4. Plot of experimental values of broadening Parameter r*/2q versus inverse pressure squared_ The solid points correspond to medium probe power (170 JAWand the open point thi lower probe power (70 pW). The full line is a weighted least-squares straight-line fit to the medium power data, which was used to predict the corresponding broken line for the low power data.
used and have been combined into 11 groups with the same pressure and power. A quite pronounced residual pressure dependence of y*/p is apparent and needs to be explained. . At high pressures (short relaxation times --SO ns) we might expect that the ftite sampling window (-20 ns) cause distortions of the observed decay profile. F was investigated by using a revised fitting function -.
236
D_A. Andrews / Resonance in microwavespecmm of formaldehyde
for a sample containing --SO?&of contaminant to be -4% lower than that for pure formaldehyde, a correction could be applied to the data, which produced changes in y*lp of at most 1%. The effects of Doppler and wall broadening were estimated to be equivalent to an extra gas pressure of 0.6 and 0.2 mTorr, and a,& contribute less than 1% even at the lowest pressrrre(60 mTorr). The next order correction term to the decay profile due to finite probe power has already been given for the case of equal 7’s by eq. (9) By computer fitting simulated decay curves with this distortion term incIuded, it was shown that the effective 7. was shifted to
Table 1 Results of fitting to different functions (see eqs. (11) and
(13) for defiitiom of F(r) and 7)
ml 1 14-r 1 +r+g 1 +&r
:
A&age
f=iznP
residual (5%)
(MHz/Torr)
1.8
a.7 17.3 23.0
0-a CL5 0.8
17.3
gives a fmal -r-j&p
= 22.96 ” 0.3 MHz
TOIT-‘.
In view of the disagreement with previous theoretical treatments the data has also been fitted to a series of alternative proposed expressions, viz.
or (13)
y(r) = Al + A3 exp(-r)F(r); =A1
i-As;
T>O ?-GO,
(14)
where the constant a is approhately 2.60. Since in for the cases: a waveguide cell the electric field strength varies across the ceil dimensions, an average over the nonF(T) = 1, (a) uniform field distribution of the pump and probe F(7) = 1 f r, (b) should be performed. However this will only change the absolute value of the cone&ion and not its form: (c) F(s)= 1 t7+$P, which is proportional to the power and inversely (d) F(T)= 1 +A~T_ proportional to the square of the pressure. We note that incomplete suppression of the pump The results are presented in table 1. It% apparent that. field during the off cycle can also mod+ the effecthe present proposed formula (c) provides a superior tive r* by increasing the apparent YE. This gives rise fit to the-data, with residuals approaching the random to a correction term sinmar to that in eq_ (13) and noise Ievel in the.data (0.4%)). The other formulae give with the same l/p* dependence, and may be allowed larger residuals and interestingly a much lower apparfor in the same way as for the probe power. ent T*. In his study Glotieux f 141 observed the nonThe data shown in fig. 4 haa therefore been fitted exponential nat& o_fthe decay but.due to limited signal-to-noise found himself unable~to,d+inguish to a straight line of the above form (13). The fitted between these different ti&rtg functio&: solid line ia for the higher probe power (170 MW) data, but by using the fitted parameters from -this . !C.. _-.., I .::_;._ -. .: ‘-. data set we can obtain a predicted line for the more _..~$ c;ncl&o& L :.. -._.A-1 _ ~1 ::_: : imprecise lower probe case (broken l&j which is in. -:. : ._. z.. .-.: :..: ~ reasonable agreement with the data_ _: .I-.-.. :A fo&_l-& do;Gyek&=& &tr;meiei -& -: By extrapolating to zero_lIpa p’e obtainthe value : been-~oris~~tea_~~-~~_~~ && &&&&+. .2296 Z@z ?orr-! for the effec_tive.broad&rig :. @ ~~ i-_& r&l&_o;i.o;f z. ..-&.&e &b&-m& __. parametery’i~tip. Grn+timating the~uncertainty in:-. -: this resu& a-l%~c&brat+~uncer-t&r~ haa been- :: _
D.A. Andrcws / Rcsonanck in miaowavc
Though comparison Mth self-relaxation r&es .obtained from other me+&mentsis made difficult by the lack of any l/T1 data for’either the ri or 7~ transition, a number of workers however, have measured l/T* (7~) with the results shown in table 2. All are slightly less than the ~*/27rp value obtained here and infer by -eq_(7) a value for &y:rw+-&/Zap of 23 5 2 1 .O MIIz Torr;’ . Although this rate is marginally greater than 7@n~, the large values of T,/T,that characterise rotational relaxation in ammonia [ 1$1 are not present. A possible explanation of this different behaviour lies in the reduced influence of rotational resonance in formahlehyde [9,11,27]. In the case of NH, a very large number of inversion transitions overlap in frequency to within a few MHz, with the result that intermolecular exchange of rotational energy becomes very efficient. In formaldehyde the corresponding K-doubling transitions differ by many GHz, and so resonant exchange of energy is not the completely dominant process observed in NH, _Calculations by Prakash and Boggs 1281 Indicate that .direct transitions between levels which enhance the ratio TJT, , proba&z=d2,,> bly account for -20% of all relaxation process& contributing to 7~. This may be contrasted withthe case of HCN where direct I-doubling transitionS contribute less-than 5% to the total relaxation rate [ 131. Thus formaldehyde would appear to be au interesting intermediate &&e between the strongly dipolar rotational resonance in NHs and the weakly dipolar relaxation found in linear molecules such as OCSand HCN.
: :. Table 2
spcctntm
of formaldehyde
231
Acknowledgement
‘& research was supported by the Science Research Council under contract GR/A/g9641_ I would like to thank Dr. J.G. Baker for many helpful SU~~&~OIIS during the progress of this work and for commenting on the manuscript.
Appendix
The appliC&On of tensor operators to the problem of rotational energy relaxation has been treated by a number of authors [23,25,26]. Using the notation of Bottcher and Feuillade, the L-order multipole moments for the population difference and polarisation are represented by tensor operators Ns and & where for a J 2 -+ 2 transition L runs from 0 to 4. For linearly polarised radiation it is sufficient to consider only the M = 0 component of the tensor, so the suffDcwill be suppressed, assumed zero_ For the pump transitionl3 + 3 the population is represented by a set of operators ti where L runs from 0 to 6_ An important property of the tensor operators is that cross-relaxation can only occur between tensors of the SClltte order, thus simplifying the description of population transfer in double resonance. A-set of modified Bloch equations replacing eqs. (1) may be written down:
NL’ with IL ~ L’I = 1, thus only add order & and even order NL’ are included. A completely independent set of equations for even QL and odd NL’ may be similarly written down, but since the observabie absorption signd measured by this experiment is proportional to Q’ , only the first set of equations is required. If the usual assumptions of low probe power (no =turation) and small values of cross-relssation ylv~ty~ < yN[, are made then the equations further simpIify to -Q’
=K,@
-Jjo = y&V
- P)
x [l +y=T++yo2r2],
(A.4)
where y* =
7Q1 +4 iP”(7,q0
fP2(YW2
+7fi
+ 7,.@ -
4Ql)}/@
27~,)
+P’).
(A.5)
and
+ ylvOrvO(rP -- iP),
= y,vv’N i TN’ &,ZW’ ,
-Ii”
= y,&P
-w
=y#“.
(A-6)
- IP), (A.?)
These equations mzy bc solxd transform technique to obtain e-‘Q”
_F?.
References
by the usual Laplace [ l] T. Amw~o and R.H. Schwendcman, J. Chem. Phys. 65 (1976) 5133. [Z.] R.A. Creswell, S.R. Brown and R.H. Schwendeman, J. Chem. Phys. 64 (1976) 1620. [3] J.C. hIcGurk, T.G. Schmalz and W.H. Fly~arc. Advan. Chcm. Phys. 25 (1974) 1. [4] S.L. Coy, J. Chem. Phys. 63 (1975) 5145. [S] R-II. Schwcndeman, Arm. Rev. Phys. Chcm. 29 (1978) 537. [6] C. l3ottcher. Chem. Phys. Letters 34 (1975) 143. [7] D.A. Andrew& J.G. B&X, c. Bottcher and C. Feuillode, Chem. Phyr 46 (1080) 351. [8] T. 9x1, Phyn Rev. A18 (1975) 2570. [9] W.E. Hokc, D.R. Bauer and W.H. Flygxe, J. Chem. Phys. 67 (1977) 3454. [IO] J. EUis and J.G. Baker, in preparation. [ 1 l] T. Okx, Molcc. Phys. 9 (?973) 127. [ 121 U.L Utind and W.H. Flyprc. J. Chem. Phys. 45 (1966) 2421. [13] HA. Rnbitz and R.G. Cordon. J. Chem. Phys. 53 (1970) :.!15. 1141 P. Glori+x, 2. Naturforsch. 34a (1979) 609. [IS] S.R. Brown, J. Chem. Fhyr 60 (1974) 172’3.
-
= Q’(O)
Q’ (1) = Q’ (m) + (jJ” + 0’) c --7=’
+?.TIz!jil +yg,Q’,
-2
Q](r)
to zero, the following result is obtained
(1
_
e-7Q’r)
Q
(A-3) By making the assumption I(y,+ L _yd~)(+~~ L.. ~Q;)[_~y~~yIV~‘and-putting in ihe startin~conditions using ek. (A.$with timk derktives put equal
[ 161 H. hGder, W. Schrepp and H:.Dreizlcr, 2. Naturforsch. 311(1976> 1419. I Ii] P_ Glorieux and B. &cke, Chem. Phyf 4 (1974) 120. [ 18) D.A_~And&vr. in preparation. : [ 19) 3. Blundell; Dipl. Adnn. Stud. in Physks, University of hfmch~ter (1979)_ . ” [ 201P. Glorieur; Ph. D. The%, Universiti des Sciences et Tczhniques de Lillc,.France (1976). [21] H,Jetter. E.F: P-n. CL. No& J-C. &hk ad. W_Hi.Fly+, J. Chem. Phys_ 59 (197!) 1796. ‘. ‘.. j : 122]C Feuillade, Chini. Phyr Letters 41 (1976) 529.
--
,__’
[23]
C- Bot!chcr 127.
2nd C. Feuillade.
Chem.
Phys. 54 (1981)
1241 1I.C. Torrcy, Phys. Rev. 76 (1949) 1059. [?S] W. Liu and R.A. Marcus, J. Chem. Phys. 63 (1975) 272. 1261 D.A. Coombe and R.F. Snider, J. Chem. Phyr 67 (1977) 2659.
[27] P.W. Anderson, Phys Rev. 75 (1949) 647. 1281 V. Pnkxh nnd J.E. 30~s. J. Chem. Phys. 57 (1472)
2599. [29] D.V. Rogers and J.A. Roberts. [30]
(1973) 200. G. Deslmon,
H. Dreizlcr
J. hlol. Spectry.
and Il. hlfidcr.
46
to be published.
..
Ndrth-Ho&and PublishingCompany
TIME-RESOLVED
FOUR-LEVEL
OF FORMALDEtHYDE -.
DOUBiE
RESONANCE IN THE MICROWAVE SPECTRUM
_
David A. &DREW Department of Physics, Schuster Lobmat&,
Received29 Juiy 1980
The Univerdy. Manchester Ml3 9PL:UK
-
The technique o; tram&t double resonance has been applied to the (2 11 G 2&-(311~ 313)~ system in the microwave s$ectmm of formaldehyde. An effective broadkng parameter 7*/27rp = 22.96 f 0.3 MHz Ton--’ was measured and theory is presented to reldte this to other measured multipole relaxation rates.
1_ Introduction
:. Over the past twenty years, much effort has gone into measuring relaxation times for rotational transitions in polar molecules; These measurements fall into two main cl-s: linf%hape analysis of static spectral profies [1,2] and time-resolved measurements of transient phenomena [3-5]_ Analysis of these experiments has usually been confined to a determination of the conventional TI and T2 population and polarisation relaxation times corresponding to a two-level system; however recent theoretical [6-S] and experimental Work [9,10] has indicated that, in a multi-level system in which collisional transfer of population is important, the different types of experiment can give quite different soits of information about relaxation. proce&s. A promistii t&hnique in this respkt is &a! of *e-resolved dotible resonance (TDR).. Steady+ate do_uble tesonance,pticu@rly in the hands of.Oka and co-workers [ll] has&&n very &cessful in &l&dating
midable technical problems associated with the rapid switching of very Iarge voltages applied to the s+nple. Instead the microwave radiation is itqlf modulated by means of a PIN diode.before it enters the sample cell. The systems most commonly studied by double resonance are three-level systems, in which pump and probe transitions share a common energy level, and the pump field directly affects the probe populatio.ns tid polarisations through this common level. Analysis of the transient response for systems of this sort is complicated by this direct coupling and will form the subject of a following paper [X3]. This present paper is confined to the study of four-level double resonances in which the pump can only affect the probe transition by collisional transfer of population. Because the effects of these crossrelaxation processes are smaller than the direct relaxi-. tion,.typic&y by an order of magnitude, the signals observed aF less.intense than those produced by Stark-switching by this factor. However, the Q b&ch
232
D.A. Andrews/
Resonancein
microwavespectrum
offormaldehyde
1 :
-.
‘. I:..:.-
RQ621 klystrons giving into the cell 100 mW a&d 0.2 mW for the pump and probe respectively. The punm was modulated using a Microwave Associates PIN ’ diode switch with a U ns switching time and >30 dB isolation. A Hewlett-Packard ‘%faffleGron” low-pass tiher and isolator were used to prevent the pump radiation falling on the detector. The isolator in addition to providing a good match for the probe radiation will absorb 90% of incident pump power with * PP
Fig. 1. Schematicd&am of apparatus. Key: A ampiitier. AT attenuator, BA boxcar avenger, CK clock generator,CM capacitancemanometergauge,CP cross@d coupIer,D micrcwoe detector, F low-passfilter,G samplega;trinput. I X-band isolator,Kl Oki30 V12 klystron,K2 EM R9621 kfysnon, S PIh diodeswitch.SC waveguide Stark cell. fl microprocessor-baszdcontrol unit.
2. Experiment A schematic d&ram of the apparatus is given in fig. 1_The system is based around a conventional Hewlett-Packard X-band Stark cell with, as signal sources, the phase-stabilized OKI 3OVi2 and Ehfl
only a few percent reflected, thus improving the impedance match for pump, as well as supplementing the rejection of *-hefilter. The transient signal from the
J-band detector (el PV) was amplified by 90 dB using wideband amplifiers (risetime ~3 ns overall) and fed to a boxcar signal averager. The minimum gatewidth available on the boxcar was 10 ns and the effective resolution of the camplete system was measured to be 20 ns_ A 1 MB crystal clock divided down to give a 125kHz squarewavewasusedtodrivethe PINdiode modulator and to give a reference signal to trigger the boxcar averager 1 MSbefore the pump was switched Off. The data acquisition system [I91 controlled by a hlC68CKlmicroprocessor, provided a stepped ramp voltage to sweep the time delay of’the boxcar; the
F
DA; Andrews / Resonnnce in microwave spectrum of formnldehyde analogue output from the avetiger was synchronously converted to an 8_bit digital form and accumulated in memory. The results could be dispIayed on an oscilloscope or punched out onto tape for further analysis. Typically a double sweep over 64 channels would take 2 miu witha box& time constant of 300 ms. A typical signal response as the pump power Is switched offisshowninfig.2. The pump and probe frequencies were kept exactlyonresonanceforthe3,,+3,, and2,,+2,2 resonances at 28975 and 14.458 GHz respectively, by phaselocking each to an harmonic of an UHF frequency synthesizer. The pressure in the cell was mot&red by a.capacitan? manometer gauge. Data was taken over a series of pressures from 60 to 160 mTorr and at low and moderate probe powers (70 and 170 ccw)_ To obtain an acc\lrate t&e-scale calibration for the boxcar, the time delay produced by the boxcar was compared with that’produced by an oscilloscope with a delayed timebase; this was in turn caIiirated Using a 100 MHz reference sinewave to generate markers every 10 ns.
3_ Theory The theory of transient double resonance phen&nena has been given by several authors [13-16,20-221, so only a simplified treatment adequate for the pres-
233
ent experiment will be given here,Consid& the fourlevel system .&own in figi 3, assuming for the moment that the levels are non-degenerate. The difference in fractional populations for the pump and probe transitions will be represented by W andM respectively, which in the absence of applied fields will relax back to their thermal equilibrium vahxes w and B with characteristic rates 7~ and ye_ A change in W however will also produce a change in N by cross-relaxation, and this is represented by the ~NW term, coupling the population differences. Thus for the situation of pump field off and finite probe field e(t) = .$ cos wr exactly on resonance w = wg we may write down the simplified Bloch equations [3,7,23] I -8,’ = -2FQ
+ ~&IV - rs, + ~lVw(w - @),
(la)
~=~N+Y~Q, -??=y&v-
(lb) m)+y~&v-B),
(14
where Q is related to the inquadrature component of the polarisation yDiinduced by the probe field E(r) by
where No is the total number density of absorbing molecules and &,b the transition dipole moment. Q may also be related to the observed absorption Q per unit length of the probe radiation by (r = --wNo!&&&
a&
F = .&abl~
-
A couple of comments may be made about these equations. Firstly, the pump polarisation does not appear explicitly in any of these equations and its decay therefore does not contribute.to the observed signal (unlike the three-level case). Secondly, we have as&med that the population sums (n, + nb) and . (n, + na) are *e-independent. This approximafion is particularly good f&r molecules with energy leveLs formi$g closely spaced doublets, such as formaldeHyde, but more generally it has been shown [7], *ht. the coupling of population sum to popiilation difference only contributes to second drder in the. observabl&. Similarly populati&&ansf$r o&o‘f th& syste&.
DA. Androvs
234
/ Remncttce
in microwave spectrum of fomialdehyde and the startingvalues Q(O) and N(O) have been
7~w(W - @) in eq. (la)_ This approximation essentially neglects the effect of the probe in transferring population to the pumped transition, thus changing its reaction on the probeThe solution of this set of equations is most elegantly performed using Laplace transform techniques [24], which when applied to eqs. (I) give
exponent&& and damped sinusoids driven by the probe field (probe nutations), and agrees with the form given by Glorieux $201 if his approximation 7N = 7rv = yQ = ,y is made:
0 = p& - N(0) + ‘yiy(R - R/p)
Q(t) = QW - QX e-”
*r&9
@a)
- W/P) - 2@!,
0=&j-
Q(O)+7&
o=pW--
W(O)‘7”@_
eliminated by solutions of-the steady-state equations. The form of eq. (4)is seen to be a combination of
CZ!Ffi, Rip),
(2b) (2c)
where the transformed variables fi, Q and i? are functions of the dummy variable p only. Solving for D and rii in terms of R and substituting into eq. (2a) gives
This equation may be converted by an inverse Laplace transform to give Q(f), which has two different forms depending on whether 4F* is greater or less than &rN - yQi* _ Taking the case of large probe power (4F= >$7~-7Q~2)theSOhtiOUhasthefO~Otig
Eq. (4) is a complicated expression to use for the analysis of the experimental data, especially.since an average has to be taken over the non-uniform power distribution in the cell_ At high pressures however, the transient response is dominated by relaxation processes and it is worthwhile to obtain a simpler expression to fust order in the probe power. TO this end consider the case of small probe power (4F2 < 7’) by returning to eq. (3) and putting 4F2 = 0 in the denominators This is essentially equivalent to deleting the fit term on the rhs of eq. (la), which would normally produce saturation effects for the probe transition, and are negligible at low probe payer. inverting the Laplace transform of the thus modified eq- (3) and =ummg h’ - YQ 1, fTN - TQi * rQ > such that we can neglect second order terms Irw - -ye I* compared to 7’ , the exponentials may be expanded to give Q(t)= Q(m) - QA[l +7*t
(6)
+:r”t’]exp(-7*t),
fern
where 7* = tk
YQ *%7,
= -w,YNi+’
+ TN - 2rQ), [w(o)
-
(7)
WhQrN-
It may be noted_that the pumping conditions only appear in eq. (6) through the coefficient QA, so the forrnof equation is true even for non-uniform power distriiution across a waveguide. This present form differs somewhat unexpectedly from that obtained .,.. by Brown [15]., namely
&j-= Q(->7 QA(1 .* 70
ai&-&
:--.
1 (8)
D-4. &drews/
Resonance in mictottave spectrum offormaldehyde
235
direct transfer of population from pumped to probed levels’is absent. The correctness of eq. (6j may be confirmed by taking the expression (5) and making ari expansion in powers of F*/7*, obtaining Q(r) = Q(i) - QA emn [l f 7t f 4 y* tz
which agrees with eq. (6> in the limit F*/yZ + 0, if the y’s are equal. Throughout this section the assumption has been made that the levels are non-degenerate even though eachJ state in fact has (25 + 1) fold degeneracy. It has been shown however that this degeneracy may be elegantly taken into account by use of a tensor pperator formalism [23,25,26]. In this approach the populations and polarisations are represented by a series of spherical tensors 7$) whose symmetry properties greatly simplify the analysis of complex multi-level systems. As shown in the appendix the analysis of this section may be readily carried through for the multidegenerate case to yield an equation of exactly the same form aseq. (7), but with a _ different interpretation of 7*, namely 7* = 7& + 4-{$(7fl+ + fi’(7,z
7fl-
27Qx)
+ 7$ - 27Q1))@’ -Pfi*>-’ ,
(10)
where 7p, 7rrp, etc. are the relaxation parameters for the individual spherical tensors p, No, etc., and the coefficients fi are given by eq. (A-6) in the appendix. The effective relaxation parameter is seen to be a weighted means of-the relaxation,parameters associated with the different multipole tensors contributing to ihe signal.
4. D%ti kalysis and discussion
..
-’
The data from each individual run was leastsquares fit to the 4 parameter formula: v(t) =Ai +A3 exp(-r)(l
+ r+ $7’):
7 > 0,
Fig. 4. Plot of experimental values of broadening Parameter r*/2q versus inverse pressure squared_ The solid points correspond to medium probe power (170 JAWand the open point thi lower probe power (70 pW). The full line is a weighted least-squares straight-line fit to the medium power data, which was used to predict the corresponding broken line for the low power data.
used and have been combined into 11 groups with the same pressure and power. A quite pronounced residual pressure dependence of y*/p is apparent and needs to be explained. . At high pressures (short relaxation times --SO ns) we might expect that the ftite sampling window (-20 ns) cause distortions of the observed decay profile. F was investigated by using a revised fitting function -.
236
D_A. Andrews / Resonance in microwavespecmm of formaldehyde
for a sample containing --SO?&of contaminant to be -4% lower than that for pure formaldehyde, a correction could be applied to the data, which produced changes in y*lp of at most 1%. The effects of Doppler and wall broadening were estimated to be equivalent to an extra gas pressure of 0.6 and 0.2 mTorr, and a,& contribute less than 1% even at the lowest pressrrre(60 mTorr). The next order correction term to the decay profile due to finite probe power has already been given for the case of equal 7’s by eq. (9) By computer fitting simulated decay curves with this distortion term incIuded, it was shown that the effective 7. was shifted to
Table 1 Results of fitting to different functions (see eqs. (11) and
(13) for defiitiom of F(r) and 7)
ml 1 14-r 1 +r+g 1 +&r
:
A&age
f=iznP
residual (5%)
(MHz/Torr)
1.8
a.7 17.3 23.0
0-a CL5 0.8
17.3
gives a fmal -r-j&p
= 22.96 ” 0.3 MHz
TOIT-‘.
In view of the disagreement with previous theoretical treatments the data has also been fitted to a series of alternative proposed expressions, viz.
or (13)
y(r) = Al + A3 exp(-r)F(r); =A1
i-As;
T>O ?-GO,
(14)
where the constant a is approhately 2.60. Since in for the cases: a waveguide cell the electric field strength varies across the ceil dimensions, an average over the nonF(T) = 1, (a) uniform field distribution of the pump and probe F(7) = 1 f r, (b) should be performed. However this will only change the absolute value of the cone&ion and not its form: (c) F(s)= 1 t7+$P, which is proportional to the power and inversely (d) F(T)= 1 +A~T_ proportional to the square of the pressure. We note that incomplete suppression of the pump The results are presented in table 1. It% apparent that. field during the off cycle can also mod+ the effecthe present proposed formula (c) provides a superior tive r* by increasing the apparent YE. This gives rise fit to the-data, with residuals approaching the random to a correction term sinmar to that in eq_ (13) and noise Ievel in the.data (0.4%)). The other formulae give with the same l/p* dependence, and may be allowed larger residuals and interestingly a much lower apparfor in the same way as for the probe power. ent T*. In his study Glotieux f 141 observed the nonThe data shown in fig. 4 haa therefore been fitted exponential nat& o_fthe decay but.due to limited signal-to-noise found himself unable~to,d+inguish to a straight line of the above form (13). The fitted between these different ti&rtg functio&: solid line ia for the higher probe power (170 MW) data, but by using the fitted parameters from -this . !C.. _-.., I .::_;._ -. .: ‘-. data set we can obtain a predicted line for the more _..~$ c;ncl&o& L :.. -._.A-1 _ ~1 ::_: : imprecise lower probe case (broken l&j which is in. -:. : ._. z.. .-.: :..: ~ reasonable agreement with the data_ _: .I-.-.. :A fo&_l-& do;Gyek&=& &tr;meiei -& -: By extrapolating to zero_lIpa p’e obtainthe value : been-~oris~~tea_~~-~~_~~ && &&&&+. .2296 Z@z ?orr-! for the effec_tive.broad&rig :. @ ~~ i-_& r&l&_o;i.o;f z. ..-&.&e &b&-m& __. parametery’i~tip. Grn+timating the~uncertainty in:-. -: this resu& a-l%~c&brat+~uncer-t&r~ haa been- :: _
D.A. Andrcws / Rcsonanck in miaowavc
Though comparison Mth self-relaxation r&es .obtained from other me+&mentsis made difficult by the lack of any l/T1 data for’either the ri or 7~ transition, a number of workers however, have measured l/T* (7~) with the results shown in table 2. All are slightly less than the ~*/27rp value obtained here and infer by -eq_(7) a value for &y:rw+-&/Zap of 23 5 2 1 .O MIIz Torr;’ . Although this rate is marginally greater than 7@n~, the large values of T,/T,that characterise rotational relaxation in ammonia [ 1$1 are not present. A possible explanation of this different behaviour lies in the reduced influence of rotational resonance in formahlehyde [9,11,27]. In the case of NH, a very large number of inversion transitions overlap in frequency to within a few MHz, with the result that intermolecular exchange of rotational energy becomes very efficient. In formaldehyde the corresponding K-doubling transitions differ by many GHz, and so resonant exchange of energy is not the completely dominant process observed in NH, _Calculations by Prakash and Boggs 1281 Indicate that .direct transitions between levels which enhance the ratio TJT, , proba&z=d2,,> bly account for -20% of all relaxation process& contributing to 7~. This may be contrasted withthe case of HCN where direct I-doubling transitionS contribute less-than 5% to the total relaxation rate [ 131. Thus formaldehyde would appear to be au interesting intermediate &&e between the strongly dipolar rotational resonance in NHs and the weakly dipolar relaxation found in linear molecules such as OCSand HCN.
: :. Table 2
spcctntm
of formaldehyde
231
Acknowledgement
‘& research was supported by the Science Research Council under contract GR/A/g9641_ I would like to thank Dr. J.G. Baker for many helpful SU~~&~OIIS during the progress of this work and for commenting on the manuscript.
Appendix
The appliC&On of tensor operators to the problem of rotational energy relaxation has been treated by a number of authors [23,25,26]. Using the notation of Bottcher and Feuillade, the L-order multipole moments for the population difference and polarisation are represented by tensor operators Ns and & where for a J 2 -+ 2 transition L runs from 0 to 4. For linearly polarised radiation it is sufficient to consider only the M = 0 component of the tensor, so the suffDcwill be suppressed, assumed zero_ For the pump transitionl3 + 3 the population is represented by a set of operators ti where L runs from 0 to 6_ An important property of the tensor operators is that cross-relaxation can only occur between tensors of the SClltte order, thus simplifying the description of population transfer in double resonance. A-set of modified Bloch equations replacing eqs. (1) may be written down:
NL’ with IL ~ L’I = 1, thus only add order & and even order NL’ are included. A completely independent set of equations for even QL and odd NL’ may be similarly written down, but since the observabie absorption signd measured by this experiment is proportional to Q’ , only the first set of equations is required. If the usual assumptions of low probe power (no =turation) and small values of cross-relssation ylv~ty~ < yN[, are made then the equations further simpIify to -Q’
=K,@
-Jjo = y&V
- P)
x [l +y=T++yo2r2],
(A.4)
where y* =
7Q1 +4 iP”(7,q0
fP2(YW2
+7fi
+ 7,.@ -
4Ql)}/@
27~,)
+P’).
(A.5)
and
+ ylvOrvO(rP -- iP),
= y,vv’N i TN’ &,ZW’ ,
-Ii”
= y,&P
-w
=y#“.
(A-6)
- IP), (A.?)
These equations mzy bc solxd transform technique to obtain e-‘Q”
_F?.
References
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-
= Q’(O)
Q’ (1) = Q’ (m) + (jJ” + 0’) c --7=’
+?.TIz!jil +yg,Q’,
-2
Q](r)
to zero, the following result is obtained
(1
_
e-7Q’r)
Q
(A-3) By making the assumption I(y,+ L _yd~)(+~~ L.. ~Q;)[_~y~~yIV~‘and-putting in ihe startin~conditions using ek. (A.$with timk derktives put equal
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,__’
[23]
C- Bot!chcr 127.
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H. Dreizlcr
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and Il. hlfidcr.
46
to be published.