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Relaxation phenomena associated with microwave pumping of rotational transitions
Volume 34, number
1
1 July 1975
CHEMICAL PHYSICS iETTERS
RELAXATION PHENOMENA ASSOCIATED WlTH MICROWAVE PUMPING OF ROTAT:ONAL. TIUNSITIONS C. BOTTCHER
Received 28 March 1975
It is shown that the Bloch equations describing a hvo-lcvcl sysrem in a radiation field can be extended to take proper account of the degenerate magnetic sublevels. By solving the equation for a J = 0 to / = 1 transition in a linear rotor WC show that the saturated linewidth depends on the realignment rime T4 as well as on the usual Tl and T,. Further WC show that the ratio T:/Tz which enters the ururated linewidth is not exactly unity, and that itsdeviation from unity is related to the rate of collisional transitions betvreen the two pumped levels. Thus it nxy be possiklc to measure collision rates of astrophysical interest which could not be obtained in any other way. Some approximate formulae are derived assuming that only long-range interactions
arc present.
Interest in collisionally induced transitions between rotational levels in molecules with prominent microwave spectra has been considerably stimulated
by the discovery of complex molecules in interstellar space [1 j . In cold dense clouds the most important rates are those for collisions between H2 and mincer constituents with observable microwave lines, notably HZCO and H3COH. Since the theoretical prediction of rotational cross sections in complex systems is in its infAncy [2] it is natural to consider ways of mensuring these cross sections in the laboratory. Crossed molecular beam techniques might seem ideal, but they are very expensive and cannot at present distinguish energy losses less than 0.01 eV, ruling out direct measurements on most rotational processes. Thus spectroscopic techniques seem initially more promising (this is not to say that beam studies on systems like H, f H,CO would not be extremely valuable). A number of experiments on astrophysics! systems have already beer. reported [3]. Finally, recent ad-
vances in microwave spectroscopic
techniques
[4]
make it possible to extract much -more information than is contained in classical line broadening measurements [5] In this letter we shall derive the correct phenomenologiczl equations describing pumping on the J = 0
to J = 1 transition
of a linear rotor and show liow the relaxation times involved are related to collision parameters. In future communications the theory outlined here will be used to interpret experiments on astrophysical systems*. A fuller account of the theory will be Fublished elsewhere [6]. The response of an N-state quanta1 system to coherent radiation is most readily studied by setting up Bloch equations_ The statistically averaged density matrix for one system [7] is expanded in a set of N2 basic operators pa, = cb;Oi
such that bi = t6,,,v where ( ) denotes a quanta1 average and “a? a statistical average. Then the Bloc11 equations describe the time evolution of bi in the applied radiation field with the addition of phenom-
enological terms to account for non-radiative relaxation
processes_
Suppose
&= E exp [i(ky - ot)]
+ Measurements
on Hz-H2CO
the electric
field is
+ C.C.
and HZ-HJCOH
(I)
are current-
ly kinp carried oul in this department by Dr. J.G. Baker and Mr. C. Feuikde. 143
CHEMICAL PHYSICS LETTERS
~o!ume 34, number 1
and the dipole moment of the molecules is ~1. The hamiltoniti describing the interaction between rddiation and molecules is u=
~-‘/2P&,
(2
1 July 1975
T1 and Tq. We sl=lI F.OW show that transitions to the other sublevels of J = 1, viz., M = + 1 cannot be neglected 2nd that two further relaxation times must be introduced. The density matrix for the four level system IJM) = IOO), IlO), II l), I1 -1) can be expanded in a basis of 16 operators. These are chosen to be: the unit matrix, the population difference L
-L
where P is the dipole moment (polarisation) operator, whose matrix elements are give:? below. For a system with only two levels IJM> = IOO?, (10) the density matrix can be expanded in the Pacli matrices (1, al , u2, u3). In terms of physical quantities u1 = P,, u2 = Q, 2nd u3 = Iv where Q is the velocity dipole moment and N is the population difference. If we transform all operators to 2 rotating frame of reference,
N=
d=
AZ = 3-“‘(311-J2),
esp(-~iwCa3)SZexp(~iwla3),
and discard the rapidly oscillating tonian (2) becomes simply, c =F.P,
F=
(3) terms,
A
(4)
To simplify notition we shall from now on take all operators in the rotating frame anti drop the cravat; the same symbol will be used for an operator and its expectation value. A dot denotes time differentiation, wx the transition frequency and w. = o - wX the distance from the line centrlz. To write down the Bloch equations we introduce two relaxation rates, 71 = 1 /Z-l .associated withNand 72 = 112-z associated with Q, P. In thermal equilibrium Q, P vanish while N is non zero, say M Then with the electric field a!ong the z-axis fv
Q,=-PNPz =w&
-r&
[+?;+4F2(Y;/Yl)l
+iA
(5)
-I,
1).
XY
above, the orientation tensor Awhose cartesian
A, -CiA,
=J2
= + i(J+J, ++I?), (8)
2’
J and A are already familiar in optical pumping [9] and are associated with the relaxation rates y3 = 1/7’s, generality we consider el74 = I /Tq . For completely liptically polarised radiation, F = F’ i iF” = (cX ei+, 0, F,).
(9)
There are now 15 Bloc11 equations which can be written concisely in tensor notation: we use [&I to deriote the symmetric trace!ess dyadic product of two vectors, iab] =ub +6a
Then
8=
-2(a*b)l.
[putringg
P =&iv
q&-r+&
= (8/3)‘/2]
- F”.P)
+7#?-,v),
i-F’ X J - F”.A
-gF’N+F”XJ+F’-A-
t w,,Q - y,P, w,P-
y2Q,
(10)
~=F’xP+F”xQ-~,J,
It can be shown [G] that the absorption coefficient is proportionai to (-Q,); so from the steady-state solution of (5) rhe line profile is (-Q,P
xx
lir =g(F’*Q
=WQ, _ iy#Lv),
l,l,
The operators P, Q defined vector J and the aligment components are
the hamil-
3-1$&.-
6-L/2diag(-3,
A.= [F’p] (6)
in agreeinent wirh refs. [4?8] . The frequency and dccay time of Transient signals are ~:asi.ly worked from (5). To avoid complicaied formulae we only quote the-result for w. = 0 and F > 17, - ~~1 the frequency and damping constant are
-
[F’Q] - YEA.
If one again takes the radiation to be linearly polarked (F, = 0) eqs. (lO).reduce to (set g’ = 2-lj2 g) r;i =gFQs
l
-Y,@--N),
k, = -g’FQ,-*/4AI, & = -gFlV + g’FAz - o. Pz - r2Qt, i; = ~oQ,-Y;pz, where N, Q, and Pz ire now coupled
(11) to A,.
The ab-
Volume
34, number
CHEMICAL
1
to (-Qz) sorption coefficient is still proportional we fiid in place. of (6) that the line proftie is (42,)
-
k-$ +Y: +4&,(2/r,
l/r,>1 -I -
f
PHYSICS
and
(12)
LETZ’ERS
path S-matrices for colJision induced transitions within I, II are S(l), S(H) the relaxation rate associated withA
is
If w,-, = 0 and F is much larger than the differences
c
between the relaxation rates we fmd that the frequency of transients is the same as in (7) but that the damping is given by r = i(3y2
+2y,
fy4)_
r;
not on the true T,
= _:(T, + 25).
The transient time T;”
depends
damping
involves
a different
A1,111 A
where the average is over all classical paths and a max-
(13)
Thus the static linewidth but on an effective time
1 JuJ;l 1975
effective
wellian distribution of u, 11is the number density of perturbers and u the centre-of-mass velocity of a perturber-absorber system. If A.is one component of an irreducible tensor (14) can be greatly simplified, as in the work of O.mont [ 1!] ‘. It is instructive to appiy (17) to the two level situation described by (5). If (...I is used to abbreviate bJ(-..)I av we find that
= 3( I /T4 + 2/T1 )-’ .
By measuring T2, TT/T2 and Ty* one can in principle extract Tl and T4. It does not seem possible to obtain T3 using only linearly polarised radiation; in contrast to the optical
case, one cannot
observe
the polar-
isation of spontaneously emitted radiation. I-Iowever, it may be feasible to insert a probe field E, and a detector sensitive to radiation polarised in this way. One can solve (1 I) if the proble field is weak (&E,) and after a lengthy analysis it is found that the static linewidth for the absorption of the component of the field parallel to the x-axis is the same as (12) wirh an additional term in the denominator,
F2(T4 - T,) sin@(w, T2 cos Q - sin 0). In this way one can readily measure (T, - T3). A more elaborate calculation without the assumption that E,
where ~(1 j, ~(2) are the total inelastic
cross section out of the upper, lower states of the pumped transiiion and o12 is the cross section for inelastic transitions
between
the upper and lower states themselves.
We have neglected an interference term which ought to be small. The significant consequence of (1.5) is that Yl - 72 =
2(q&
so by measuring both Tl and T, one can isolate u12. Thus rhe possibility is opened ip of studying an entirely new range of cross sections. A generalisation of (16) holds when transitions to degenerate sublevels are allowed for f6] . We will not write out here the rather complicated analysis for the four level system, but simply quote the resu!ts obtained when long-range forces are dominant. Let the leading long-range term be a 21--29pole interaction and let the multipole moments of the absorber and perturber be Q* and Q,, Then
T3 = T4 = 2T, = ; T, =
(u,,+ ,
(17)
where
a0
=-2(L-CL+11 1) -irbk
,W
L = f +g and C& is a constant * This reference considers onIy result is derived in ref. [6] .
(e.g., C, 1 = S/9, C,,
th; exe
1 = II; a more
=
general
I45
Volume 34, number 1
CHEMICAL
5/8), For dipole-dipole interactions Tz is indepenu, i.e., of both temperature and reduced .mass. dent For self-interactiag CCS molecules (Q = 0.7 128 debye = Ci.281 eat,) we find
of
T2 = 12&mtorr, which is less than ha!f the va..ue reported in ref. [4] _ The value of T2 would be increased by including short-mnge forces. The ratios of relaxation times given in (17) would be the same in an!’ strong-coup!ing model. The experimental values of TT* and T, are consistent with predicted ratio (TT*/T2) = 1 SO but not conclusively so. Clearly an exciting subjec; is unfolding for both experimentalists and theoreticians. The immediate priorities are (a) to measure (,r;/r. j for one system and test our prediction of a value different from unity and (b) to calculate relaxation times in a framework more accurate than the simple long-range approximations quoted above. ‘The latter program would involve some model of the potential surface at small and moderate separations and a fully quantal description of the scattering process; it would then be possible to compare calculated and measured relaxation times in a meaningful way. If the systems chosen for detailed study are Hz-H,CC and Hz-H3 COH one
PHYSICS
LETTERS
1 July
lras the added stimulus
of astrophysical
1975
applications.
References [i 1 D. ter Haar and XI-A. Pelling, Rept. Progr. Phys. 37 (1974) 481. (2j S. Green and P. Thaddeus, Astrophys. J. (1975), to be oublished. [3] D.V. Rodgers and J.A. Roberts, I. hlol. Specttry. 46 (1973) 200: P.B. Foreman, Phys. Letters
K.-R. Chien and S.G. Kukolich, 29 (1974) 29.5;
Chem.
R.M. Lees and KS. Haque, Can. J. Phys. 52 (1974) 2250. (11 J.C. XlcGurk, R.T. Hoffman and W.H. Flygare, J. Chem. Phys. 60 (1974) 2922; J.C. XicGuk, H. Mader, R.T. Hoffman, T.G. Schmalz [51 [61 [71 [RI
and W-H. Flygare, J. Chem. Phys. 61 (1974) 3759. G. Bimbaum, Advan. Chcm. Fhys. 12 (1967) 487. C. Bottchei, Proc. Roy. SOC. A (1975), to be published.
U. Fsno, Rev. Mod. Phys. 29 (1957) 74. hlicrowave specC.H. Townes and A.L. Schawlow, troscopy (McGraw-Hill, New York, 1955) p. 371. [91 H.S.W. hlassey and E.H.S. Burhop, Electronic and ionic impact phenomena, Vol. 3 (Oxford Univ. Press, London, 1973). [lOI hi. Baranger, in: Atomic and maleculz processes, ed. D.R. Bates (Academic Press, New York, 1362) p. 493. [Ill A. Omont, J. Phys. (Paris) 26 (1965) 26.
1
1 July 1975
CHEMICAL PHYSICS iETTERS
RELAXATION PHENOMENA ASSOCIATED WlTH MICROWAVE PUMPING OF ROTAT:ONAL. TIUNSITIONS C. BOTTCHER
Received 28 March 1975
It is shown that the Bloch equations describing a hvo-lcvcl sysrem in a radiation field can be extended to take proper account of the degenerate magnetic sublevels. By solving the equation for a J = 0 to / = 1 transition in a linear rotor WC show that the saturated linewidth depends on the realignment rime T4 as well as on the usual Tl and T,. Further WC show that the ratio T:/Tz which enters the ururated linewidth is not exactly unity, and that itsdeviation from unity is related to the rate of collisional transitions betvreen the two pumped levels. Thus it nxy be possiklc to measure collision rates of astrophysical interest which could not be obtained in any other way. Some approximate formulae are derived assuming that only long-range interactions
arc present.
Interest in collisionally induced transitions between rotational levels in molecules with prominent microwave spectra has been considerably stimulated
by the discovery of complex molecules in interstellar space [1 j . In cold dense clouds the most important rates are those for collisions between H2 and mincer constituents with observable microwave lines, notably HZCO and H3COH. Since the theoretical prediction of rotational cross sections in complex systems is in its infAncy [2] it is natural to consider ways of mensuring these cross sections in the laboratory. Crossed molecular beam techniques might seem ideal, but they are very expensive and cannot at present distinguish energy losses less than 0.01 eV, ruling out direct measurements on most rotational processes. Thus spectroscopic techniques seem initially more promising (this is not to say that beam studies on systems like H, f H,CO would not be extremely valuable). A number of experiments on astrophysics! systems have already beer. reported [3]. Finally, recent ad-
vances in microwave spectroscopic
techniques
[4]
make it possible to extract much -more information than is contained in classical line broadening measurements [5] In this letter we shall derive the correct phenomenologiczl equations describing pumping on the J = 0
to J = 1 transition
of a linear rotor and show liow the relaxation times involved are related to collision parameters. In future communications the theory outlined here will be used to interpret experiments on astrophysical systems*. A fuller account of the theory will be Fublished elsewhere [6]. The response of an N-state quanta1 system to coherent radiation is most readily studied by setting up Bloch equations_ The statistically averaged density matrix for one system [7] is expanded in a set of N2 basic operators pa, = cb;Oi
such that bi = t6,,,v where ( ) denotes a quanta1 average and “a? a statistical average. Then the Bloc11 equations describe the time evolution of bi in the applied radiation field with the addition of phenom-
enological terms to account for non-radiative relaxation
processes_
Suppose
&= E exp [i(ky - ot)]
+ Measurements
on Hz-H2CO
the electric
field is
+ C.C.
and HZ-HJCOH
(I)
are current-
ly kinp carried oul in this department by Dr. J.G. Baker and Mr. C. Feuikde. 143
CHEMICAL PHYSICS LETTERS
~o!ume 34, number 1
and the dipole moment of the molecules is ~1. The hamiltoniti describing the interaction between rddiation and molecules is u=
~-‘/2P&,
(2
1 July 1975
T1 and Tq. We sl=lI F.OW show that transitions to the other sublevels of J = 1, viz., M = + 1 cannot be neglected 2nd that two further relaxation times must be introduced. The density matrix for the four level system IJM) = IOO), IlO), II l), I1 -1) can be expanded in a basis of 16 operators. These are chosen to be: the unit matrix, the population difference L
-L
where P is the dipole moment (polarisation) operator, whose matrix elements are give:? below. For a system with only two levels IJM> = IOO?, (10) the density matrix can be expanded in the Pacli matrices (1, al , u2, u3). In terms of physical quantities u1 = P,, u2 = Q, 2nd u3 = Iv where Q is the velocity dipole moment and N is the population difference. If we transform all operators to 2 rotating frame of reference,
N=
d=
AZ = 3-“‘(311-J2),
esp(-~iwCa3)SZexp(~iwla3),
and discard the rapidly oscillating tonian (2) becomes simply, c =F.P,
F=
(3) terms,
A
(4)
To simplify notition we shall from now on take all operators in the rotating frame anti drop the cravat; the same symbol will be used for an operator and its expectation value. A dot denotes time differentiation, wx the transition frequency and w. = o - wX the distance from the line centrlz. To write down the Bloch equations we introduce two relaxation rates, 71 = 1 /Z-l .associated withNand 72 = 112-z associated with Q, P. In thermal equilibrium Q, P vanish while N is non zero, say M Then with the electric field a!ong the z-axis fv
Q,=-PNPz =w&
-r&
[+?;+4F2(Y;/Yl)l
+iA
(5)
-I,
1).
XY
above, the orientation tensor Awhose cartesian
A, -CiA,
=J2
= + i(J+J, ++I?), (8)
2’
J and A are already familiar in optical pumping [9] and are associated with the relaxation rates y3 = 1/7’s, generality we consider el74 = I /Tq . For completely liptically polarised radiation, F = F’ i iF” = (cX ei+, 0, F,).
(9)
There are now 15 Bloc11 equations which can be written concisely in tensor notation: we use [&I to deriote the symmetric trace!ess dyadic product of two vectors, iab] =ub +6a
Then
8=
-2(a*b)l.
[putringg
P =&iv
q&-r+&
= (8/3)‘/2]
- F”.P)
+7#?-,v),
i-F’ X J - F”.A
-gF’N+F”XJ+F’-A-
t w,,Q - y,P, w,P-
y2Q,
(10)
~=F’xP+F”xQ-~,J,
It can be shown [G] that the absorption coefficient is proportionai to (-Q,); so from the steady-state solution of (5) rhe line profile is (-Q,P
xx
lir =g(F’*Q
=WQ, _ iy#Lv),
l,l,
The operators P, Q defined vector J and the aligment components are
the hamil-
3-1$&.-
6-L/2diag(-3,
A.= [F’p] (6)
in agreeinent wirh refs. [4?8] . The frequency and dccay time of Transient signals are ~:asi.ly worked from (5). To avoid complicaied formulae we only quote the-result for w. = 0 and F > 17, - ~~1 the frequency and damping constant are
-
[F’Q] - YEA.
If one again takes the radiation to be linearly polarked (F, = 0) eqs. (lO).reduce to (set g’ = 2-lj2 g) r;i =gFQs
l
-Y,@--N),
k, = -g’FQ,-*/4AI, & = -gFlV + g’FAz - o. Pz - r2Qt, i; = ~oQ,-Y;pz, where N, Q, and Pz ire now coupled
(11) to A,.
The ab-
Volume
34, number
CHEMICAL
1
to (-Qz) sorption coefficient is still proportional we fiid in place. of (6) that the line proftie is (42,)
-
k-$ +Y: +4&,(2/r,
l/r,>1 -I -
f
PHYSICS
and
(12)
LETZ’ERS
path S-matrices for colJision induced transitions within I, II are S(l), S(H) the relaxation rate associated withA
is
If w,-, = 0 and F is much larger than the differences
c
between the relaxation rates we fmd that the frequency of transients is the same as in (7) but that the damping is given by r = i(3y2
+2y,
fy4)_
r;
not on the true T,
= _:(T, + 25).
The transient time T;”
depends
damping
involves
a different
A1,111 A
where the average is over all classical paths and a max-
(13)
Thus the static linewidth but on an effective time
1 JuJ;l 1975
effective
wellian distribution of u, 11is the number density of perturbers and u the centre-of-mass velocity of a perturber-absorber system. If A.is one component of an irreducible tensor (14) can be greatly simplified, as in the work of O.mont [ 1!] ‘. It is instructive to appiy (17) to the two level situation described by (5). If (...I is used to abbreviate bJ(-..)I av we find that
= 3( I /T4 + 2/T1 )-’ .
By measuring T2, TT/T2 and Ty* one can in principle extract Tl and T4. It does not seem possible to obtain T3 using only linearly polarised radiation; in contrast to the optical
case, one cannot
observe
the polar-
isation of spontaneously emitted radiation. I-Iowever, it may be feasible to insert a probe field E, and a detector sensitive to radiation polarised in this way. One can solve (1 I) if the proble field is weak (&E,) and after a lengthy analysis it is found that the static linewidth for the absorption of the component of the field parallel to the x-axis is the same as (12) wirh an additional term in the denominator,
F2(T4 - T,) sin@(w, T2 cos Q - sin 0). In this way one can readily measure (T, - T3). A more elaborate calculation without the assumption that E,
where ~(1 j, ~(2) are the total inelastic
cross section out of the upper, lower states of the pumped transiiion and o12 is the cross section for inelastic transitions
between
the upper and lower states themselves.
We have neglected an interference term which ought to be small. The significant consequence of (1.5) is that Yl - 72 =
2(q&
so by measuring both Tl and T, one can isolate u12. Thus rhe possibility is opened ip of studying an entirely new range of cross sections. A generalisation of (16) holds when transitions to degenerate sublevels are allowed for f6] . We will not write out here the rather complicated analysis for the four level system, but simply quote the resu!ts obtained when long-range forces are dominant. Let the leading long-range term be a 21--29pole interaction and let the multipole moments of the absorber and perturber be Q* and Q,, Then
T3 = T4 = 2T, = ; T, =
(u,,+ ,
(17)
where
a0
=-2(L-CL+11 1) -irbk
,W
L = f +g and C& is a constant * This reference considers onIy result is derived in ref. [6] .
(e.g., C, 1 = S/9, C,,
th; exe
1 = II; a more
=
general
I45
Volume 34, number 1
CHEMICAL
5/8), For dipole-dipole interactions Tz is indepenu, i.e., of both temperature and reduced .mass. dent For self-interactiag CCS molecules (Q = 0.7 128 debye = Ci.281 eat,) we find
of
T2 = 12&mtorr, which is less than ha!f the va..ue reported in ref. [4] _ The value of T2 would be increased by including short-mnge forces. The ratios of relaxation times given in (17) would be the same in an!’ strong-coup!ing model. The experimental values of TT* and T, are consistent with predicted ratio (TT*/T2) = 1 SO but not conclusively so. Clearly an exciting subjec; is unfolding for both experimentalists and theoreticians. The immediate priorities are (a) to measure (,r;/r. j for one system and test our prediction of a value different from unity and (b) to calculate relaxation times in a framework more accurate than the simple long-range approximations quoted above. ‘The latter program would involve some model of the potential surface at small and moderate separations and a fully quantal description of the scattering process; it would then be possible to compare calculated and measured relaxation times in a meaningful way. If the systems chosen for detailed study are Hz-H,CC and Hz-H3 COH one
PHYSICS
LETTERS
1 July
lras the added stimulus
of astrophysical
1975
applications.
References [i 1 D. ter Haar and XI-A. Pelling, Rept. Progr. Phys. 37 (1974) 481. (2j S. Green and P. Thaddeus, Astrophys. J. (1975), to be oublished. [3] D.V. Rodgers and J.A. Roberts, I. hlol. Specttry. 46 (1973) 200: P.B. Foreman, Phys. Letters
K.-R. Chien and S.G. Kukolich, 29 (1974) 29.5;
Chem.
R.M. Lees and KS. Haque, Can. J. Phys. 52 (1974) 2250. (11 J.C. XlcGurk, R.T. Hoffman and W.H. Flygare, J. Chem. Phys. 60 (1974) 2922; J.C. XicGuk, H. Mader, R.T. Hoffman, T.G. Schmalz [51 [61 [71 [RI
and W-H. Flygare, J. Chem. Phys. 61 (1974) 3759. G. Bimbaum, Advan. Chcm. Fhys. 12 (1967) 487. C. Bottchei, Proc. Roy. SOC. A (1975), to be published.
U. Fsno, Rev. Mod. Phys. 29 (1957) 74. hlicrowave specC.H. Townes and A.L. Schawlow, troscopy (McGraw-Hill, New York, 1955) p. 371. [91 H.S.W. hlassey and E.H.S. Burhop, Electronic and ionic impact phenomena, Vol. 3 (Oxford Univ. Press, London, 1973). [lOI hi. Baranger, in: Atomic and maleculz processes, ed. D.R. Bates (Academic Press, New York, 1362) p. 493. [Ill A. Omont, J. Phys. (Paris) 26 (1965) 26.