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Existence of spin symmetry species in reorienting methyl groups at high temperatures
Volume
CHEMICAL
80, number 2
REPLY
PHYSICS
i June 1981
LETTERS
TO COMMENT BY S. CLOUCH
EXISTENCE
OF SPIN SYMMETRY
SPECIES
IN REORIENTING
METHYL
GROUPS
AT HIGH TEMPERATURES S. EMID Laboratoriwn Received
voor Technwche Natwrkttnde,
Technische Hogeschooi
Delft, 2600 GA Delfr, The Netherlands
19 March 1981; in final form 2 April I981
Reorientations of methyl groups in solids are symmetry restricted. Consequently long-lived spin symmetry species and reorientations coexist even at high temperatures. This is a refmement of the usual reorientation model. The possibility of creating an annular momentum imbalance between symmetry species is shown. The possibfity of observing exponential relaxation is dis&ssed.
First we confess that we appreciate Clough’s
com-
ment [I]. It offers us the opportunity to go into more details of the physicaI implications of our previous work [2]. In reply to CIough’s comment and to resolve the problems he mentioned we show, using the dynamics of methyl groups in solids, that relatively long-lived high-temperature spin symmetry species (HTSSS) (Clough calls this RTSSS) coexist with high-temperature rapid random reorientations (HTRRR) (Clough calls this the hopping model). After making transparent what the physical meaning is of the mathematical formalism, we point out that HTSSS is compatible with the traditional HTRRR, the former being in fact a refinement of the latter. We also discuss that the many reports of exponential relaxations referred to by Clough do fit into the HTSSS model. FinaIIy we recall that the HTRRR and HTSSS models predict identical coupled relaxation. To study the dynamics of methyl groups in solids we start with the harnihonian ff=i!fs
+ffR
frr,
f&R
+&
++p
2
(11
cf. ref. [2] (the terrnHsL in ref. [2] is not relevant here and is therefore omitted). Hs, HR and HL are the unperturbed spin, free rotor and lattice (phonon) hamihonians, respectively_HSR is the magnetic spin~~~*~~tion, HR~ is the static hindering potenRp is the non-magnetic rotor-phonon inter0 009--2614/S l/0000-0000/$02.50
0 North-Holland
action. We proceed along well-defined steps, omitting details not relevant to the discussion. The free rotor eigenfunctions Gti (4, t) and eigenvalues en of the nth rotor level are the solutions of the Schrddinger equation
i.e.
En = fiGr2/21 >
n = 0, -t 1, rt 2, . . ..
(3)
where 7in is the value of the angular momentum L (L = -ifi a/d@) and I is the moment of inertia of the methyl group. We note that the plus and minus vaiues of n physically mean rotations of the methyl groups in opposite directions, say clockwise for n > 0 and anticfockwise for n < 0. Next we classify I&~ in the A, Ea and Eb symmetry representations of the triangular rotor. We thus attach a symmetry label X on the eigenfunctions and denote them by I&, . Bloch’s theorem requires that &.&+2_rr/3)=&-&),
X3 = 1
(4)
(the periodicify is imposed by the CT3symmetry), where h = 1, expf2zif3) and exp(-2ti/3) for X =: A,
Publishing Company
393
Volume
CHEMICAL
80, number 2
PHYSICS
and so causes symmetry-restricted
Ea and Eb, respectrvely. From eqs. (3) and (4) one easily finds
S = A
for ?t = 0. f 3. + 6, ___,
(5a)
x = E”
for iz = I___.-5,
-2,
I, 4, -._] .
(5b)
,y = Eb
for 12= {.... -4,
-I,?.
5, . ..} .
@cl
It is important to notice that clockwise as well as dntrclockwise rotating methyl groups are present withm each symmetry species_ This observation is crucial III resolving the problems raised by Clough [I] as we show below, but before doing so we first consider the other terms in eq. (1). The effect of the hindering porential ERR, considered as a perturbation, is to adnliv different values of ?? w&in the same symmetry species [3], since fi,, is of A symmetry in the rotor coordinate, i.e. _ [I,,(@ f 2x/3) = NRL(@) _ (6) Therefore the chrssitication in A, EL and Eb symmetry species is not changed by fi,,. The effect of the spur hamiltonian Hs and the Fermi statistics is taken into account in the usual way ds i0110ws. Smce rotation through 2rr/3 implies simultaneous exchanges of two pairs of protons on the methyl triangle, the Pauli princrple requires that spin and rotor product eigenstates should be of A symmetry, i e of the form
For the identiticatron of each of these products it is therefore sufficient to specify e.g. the spin symmetry. One can thus speak about A, Ea and Eb spin symmetry species. Now we introduce the lattice H,_, assumed to remain in thermal equilibrium at the lattrce temperature T, which is arbitrary and can be as high as room temperature, and the rotor-phonon interaction HRp mducing rapid transitions The crucial point is this: Since HRp is non-magnetic, it cannot change the spin symmetry. As a consequence of eq. (7) it causes only transitions between rotor states $& and $&, having the same symmetry. This property of HRp is well known. To put the same thing in another way [X], NRp is of A symmetry with respect to the rotor: l
ff~p(@ + 2x/3) =&p(@) 394
LETTERS
(8)
?
June 1981
transitions:
(9) Now suppose II > 0, then @It represents methyl groups in clockwise rotation. HRP excites those methyl groups to the rz’th rotor level of the same symmetry_ When 12’ < 0 thrs means that the methyl groups reorient into an anticlockwise rotation. Considering the instants of excitations to be random and defining the average relaxation time of the rotor states to be re we get the random reorientation model, which is symmetry restricted in the sense of relation (9) We thus arrive at the main conclusion: Since thermally activated rotor transitions are symmetry restricted, the resulting symmetry-restricted rapid random reorientations (SRRRR) occurring on a short time scale ~~ do not effect the lifetime T, of the (high-temperature) spin symmetry specres (HTSSS). Therefore HTSSS and SRRRR coexist; in fact SRRRR is accommodated within HTSSS. Tt is determined by the spin-rotor interaction HSR (this being modulated by the random reorientations), Tt is called a spinlattice relaxation time or spin symmetry conversion time, since relaxation converts A into E31b and vice versa, or E” into Eb and vice versa, spin symmetry species. As re can be of the order of lo-lo s, one can indeed have T, 25 lOlo 7, or even longer, since in practice Tr > 1 s. Also the SRRRR is indeed a refinement of the usual random reorientation (HTRRR) model [ 11. We can display the relationships between the different models as: HTRRR = SRRRR E HTSSS .
cw
We have presented a rather detailed analysis since the above finding is apparently non-trivial, as we should learn from Clough’s comment [l] _ Now we deal with another interesting aspect, namely rotational polarization defined as the population difference between Ea and Eb spin symmetry species [4,5] _ Since it has been established that long-lived spin symmetry species exist, rotational polarization persists on the long time scale r,. The existence of rotational polarization of symmetry species has been established experimentally and unambiguously [6,7], so there is no doubt about this. However, the intriguing question remains whether it is also possible to create an angular momentum imbalance between methyl groups belong-
Volume 80, number 2
ing to the Ea and Eb symmetry species, which we have called rotational polarization in rotor space [Z]. We now discuss “he feasibility of the latter type of rotational polarization. We consider the ensemble of methyl groups to be in complete equilibrium. Complete equilibrium means not only equilibrium within the symmetry species, but also equilibrium between the symmetry species. Then the Boltzmann distribution of the 12th rotor level of X rotor symmetry
1 June 1981
CHEMICAL PHYSICS LETTERS
subensemble has a definite angular momentum. Of course, Its value depends on the temperature, and because of the alternating values of n, cf. eqs. (5b) and (SC), it decreases with increasing temperature_ (We note that afthough M.)$ --j 0 as T increases, the lifetimes of the spin symmetry species are still of order TI -1
In equilibrium, from eqs. (13) and (14)
is given by
F(L)&=O.
where NX is the fraction of methyl groups of X symmetry and the subscript eq denotes its equliibriurn value_ We can thus divide the ensemble of methyl groups into three subensembles corresponding to the A, E” and Eb symmetry species, The expectation value of the angular momentum Ufl in each subensemble can be computed. In equilibrium
(12) Since there are as many clockwise as antlclockwise rotating methyl groups for any H E A, cf. eq. (5a) and
ref. [3], UP eq = u
03)
and we call the A subensemble stationary.
We note however that the Ea and Eb subensembles have nonzero resulting angular momenta. From eqs. (5b), (5c), (11) and (12) it can be shown that
,L)~=-(L)&b>o*
04)
Therefore we call the Ea subensemble that of clockwise and the Eb subensemble that of anticlockwise rotating methyl groups in the sense of eq. (14). The interpretation of the A, Eb and Ea spin symmetry species as representing stationary, clockwise and anticlockwise rotating methyl groups in refs. [4,7] should be understood in this sense. We must admit, however, that this was not clear in our previous work, and Clough’s comment has prompted us to write the present anaiysis. Although the individual methyl groups within the Ea or Eb symmetry execute random reorientation on a time scale TV, the equilibrium distribution of each
(15)
However, as predicted in refs. [4,5] and measured in refs. [6,7], it is possible to get a relaxation-induced non~qu~b~um E” - Eb population difference, Consequently an imbalance of angular momentum can be created by relaxation, such that
~LF>4LFb)
(L>“=o.
06)
The lifetime of such an induced rotational polarization in rotor space will be the same as that of the induced rotational polarization of the spin symmetry species itself, namely of order T1, since the two rotatronal polarizations are mutually coupled by eq. (7)_ An imbalance of angular momentum can also occur in equilibrium. It is known that a magnetic moment is associated with the angular momentum of rotating methyl groups [8]. It then follows that there is a resulting magnetic moment associated with the resulting angular momentum of the E3. and Eb subensembles, respectively. The magnetic field will shift the effective energy levels 1.51 of the Ea and Eb subensembles in opposite directions, leaving the effective ener,v level of the A subensemble unaltered. This Zeeman effect makes the Ea population differ from the Eb population. With the help of eq. (12) one easily sees that there will be an Imbalance of angular momentum. The experiment described in ref. [2] was meant to measure this effect. Some evidence of the expected effect was observed, but more carefully designed experiments are necessary and are being prepared, After this discussion, showing the coexistence of high-temperature spin symmetry species with rapid random reorientation, also the existence of rotational polarization in spin as welI as in rotor space, we now discuss the problem of exponential relaxation [9-l I] (referred to by Clough as refs. [20-221). We note that the solution of this kind of problem has already been given in ref. [IZ] (~o~espon~ng to ref. 151 of Clough’s 395
Volume 80, number 2
CHEhlICAL
PHYSICS LETTERS
paper). Exponential Zeeman relaxation follows from the HTSSS model when: (1) The Zeeman relaxation is observed after saturation; this was done in most of the cases in refs. [9,1 I] _ (2) The number of protons per molecule relaxed by one methyl group becomes large (> 10) as in all cases in ref. [lo] and some in ref. [9] _ (3) Intermolecular relaxation is not negIigibIe, as III rcfs. 19-l l] where there are many non-methyl protons (except perhaps for glycine [9]. but see below). (4) wore $ 1 where o. is the Larmor frequency_ Sum:: more than one of the above factors apply to refs. [9-l 11, the observed exponential relaxation can be explained within the HTSSS model or equivalently the symmetry restricted spin-diffusion (SRSD) model [4,5] However. as shown in ref. [ 121, the relaxation rare is not equal to the expression grven by the compiete spin drffusion model (i.e. the usual spin temperature theory). Smce the latter model has been applied in refs. [9- 1 I 1, strictly speaking the Interpretation of the expenments is not quite correct. It may also lead to pu/c.hng results, like the absence of intermolecl;lar rcl.rx.rtron in 2-methyl pyridine [ 11,131, or to results wtuch seem to be in conflict with the existing Iirerature. like exponential relaxation [9] in contrast to the earlier found non-exponential relaxation [ 141 for glycine. We recall that a partly deuterated single crystal was used in ref. [14] with the crystal so oriented that the methyl groups were magnetically equivalent so that the non-exponentiality could not be due to powder averaging of anisotropic relaxation [ 1] _ Moreover, upon request of the present author, measurements have been done by Brom’s group 1151 on glycine powder [9] at room temperature and on 2- as well as 3-methylpyridine [ 1 l] at 100 K, at a resonance frequency of 10 MHz and using the 90”-f-90” pulse program. It is interesting to note that in all three cases non-exponential relaxation was measured. For glycine the short time constant is =l 1 ms and the long time one 42 ms, the latter being consistent with ref. [9] ; for 3-methylpyridine the short time constant is =0.6 s and the long time one 41 s, the latter being consistent with ref. Ill]. Therefore, the conclusion must be that only the longest of the two time constants has been measured in ref. [9] as well as in ref. [l l] , as is to be expected in a recovery from saturation measurement. A full account of Brom’s measurements wrll be published elsewhere _ Remark. There are several ways to decide whether 396
1 June 1981
the non-exponentiallty is due to the coupled relaxatrons of the HTSSS model or to uncoupled anisotropic or inhomogeneous (by spin temperature gradient [ 11) relaxation. The first type is dependent on preparation, so the result of the 90”-t-90” pulse program is different from that of the sat--t--90° pulse program, see e.g. ref. [ 121. The second type is independent of preparation and gives the same result for those two programs. Another conclusive test is to use the pulse program developed recently [7] which serves as a convenient method to measure the relaxation-induced rotational polarization. The latter effect is absent for the uncoupled relaxations_ A last remark mentioned by Clough is that the HTRRR (or hoppmg [ 11) model itself can lead to non-exponential relaxation [16,17] (refs. [23,24] of Clough’s paper). This remark is indeed correct with regard to the treatment by Hilt and Hubbard [ 161. As for the analysis by Mehring and Raber [ 171, however, we should note that they used the HTSSS model, as is evident from p. 1117 of ref. [17]. In the context of Clough’s comment the above would be a paradox. In our view, however, it does not matter at all which model is used since we have shown, cf. relation (lo), that the HTRRR and HTSSS models are equivalent. They lead to exactly the same coupled relaxation equations as already noted in the very first paper on SRSD [ 181, It should be mentioned that this is also the only model which, among other things, provided the obvious explanation [4,5] for the Haupt effect [19] at low temperatures, and at the same time could predict its high-temperature analogue [5] _ The latter has now been confirmed experimentally [20]. Concluding we can say that we have resolved all the problems raised by Clough [ 11, using conventional methods. This little prece of work J. Schmidt, on the occasion testimony that his constant towards understanding was
is dedicated to Professor of his 60th birthday, as a stimulance for the quest not idle.
References [ll S. Ciough, Chem. Phys. Letters 80 (1981) 389. [2] S. Emid, Chem. Phys. Letters 72 (1980) 189. [3] A.E. Zweers and H-B. Brom, Physica 85B (1977) 223.
Volume 80, number 2
CHEMICAL PHYSICS LE-JTERS
[4] S. Emid, R A. Wind and S. Clough, Chem. Phys. Letters 33 (1974) 269. [S ] S. Emid and R.A Wind, Chem. Phys. Letters 33 (1975) 269. [6] R.A. Wind, S. Emid and J.F.JM_ Pourquie, Phys. Letters 53A (1975) 310. [7] S. Emid, Chem. Phys. Letters 77 (1981) 323. [ 81 W. Gordy and R.L. Cook, Microwave molecular spectroscopy (Interscience, New York, 1970) p. 375. [9] E-R. Andrew, W.S. Hinshaw, M.G. Hutchins and R.O.I. Sjiiblom, Mol. Phys. 31 (1976) 1479. [ 10 ] E R. Andrew, D.J. Bryant and E M. Cashell, Chem Phys. Letters 69 (1980) 551. [ 11 J W. Muher-Warmuth, R. Schuler, M. Prager and A. Kohmar, J. Chem. Phys. 69 (1978) 2382.
1 June 1981
[12] R-A_ Wind, S. Emid, J.F.J.M. Pourquie and J. Smidt, J. Chem. Phys. 67 (1977) 2436. 1131 W. Miiller-Warmuth, private communication (1976). 1141 M-F. Baud, Bull. Am. Phys. Sot. 14 (1969) 600. [ 151 H-B. Brom, University of Leyden, private communication. [ 161 R L. Hilt and P.S. Hubbard, Phys. Rev. 134A (1964) 392. [ 171 M. Mehring and R.H. Raber, J. Chem. Phys. 59 (1973) 1116. [18] S. Emid and R-A. Wind, Chem. Phys. Letters 27 (19743 312. [19] J. Haupt, Phys. Letters 38A (1972) 389. [20] S. Emtd and J. Smidt, Chem. Phys. Letters 77 (198:) 318.
397
CHEMICAL
80, number 2
REPLY
PHYSICS
i June 1981
LETTERS
TO COMMENT BY S. CLOUCH
EXISTENCE
OF SPIN SYMMETRY
SPECIES
IN REORIENTING
METHYL
GROUPS
AT HIGH TEMPERATURES S. EMID Laboratoriwn Received
voor Technwche Natwrkttnde,
Technische Hogeschooi
Delft, 2600 GA Delfr, The Netherlands
19 March 1981; in final form 2 April I981
Reorientations of methyl groups in solids are symmetry restricted. Consequently long-lived spin symmetry species and reorientations coexist even at high temperatures. This is a refmement of the usual reorientation model. The possibility of creating an annular momentum imbalance between symmetry species is shown. The possibfity of observing exponential relaxation is dis&ssed.
First we confess that we appreciate Clough’s
com-
ment [I]. It offers us the opportunity to go into more details of the physicaI implications of our previous work [2]. In reply to CIough’s comment and to resolve the problems he mentioned we show, using the dynamics of methyl groups in solids, that relatively long-lived high-temperature spin symmetry species (HTSSS) (Clough calls this RTSSS) coexist with high-temperature rapid random reorientations (HTRRR) (Clough calls this the hopping model). After making transparent what the physical meaning is of the mathematical formalism, we point out that HTSSS is compatible with the traditional HTRRR, the former being in fact a refinement of the latter. We also discuss that the many reports of exponential relaxations referred to by Clough do fit into the HTSSS model. FinaIIy we recall that the HTRRR and HTSSS models predict identical coupled relaxation. To study the dynamics of methyl groups in solids we start with the harnihonian ff=i!fs
+ffR
frr,
f&R
+&
++p
2
(11
cf. ref. [2] (the terrnHsL in ref. [2] is not relevant here and is therefore omitted). Hs, HR and HL are the unperturbed spin, free rotor and lattice (phonon) hamihonians, respectively_HSR is the magnetic spin~~~*~~tion, HR~ is the static hindering potenRp is the non-magnetic rotor-phonon inter0 009--2614/S l/0000-0000/$02.50
0 North-Holland
action. We proceed along well-defined steps, omitting details not relevant to the discussion. The free rotor eigenfunctions Gti (4, t) and eigenvalues en of the nth rotor level are the solutions of the Schrddinger equation
i.e.
En = fiGr2/21 >
n = 0, -t 1, rt 2, . . ..
(3)
where 7in is the value of the angular momentum L (L = -ifi a/d@) and I is the moment of inertia of the methyl group. We note that the plus and minus vaiues of n physically mean rotations of the methyl groups in opposite directions, say clockwise for n > 0 and anticfockwise for n < 0. Next we classify I&~ in the A, Ea and Eb symmetry representations of the triangular rotor. We thus attach a symmetry label X on the eigenfunctions and denote them by I&, . Bloch’s theorem requires that &.&+2_rr/3)=&-&),
X3 = 1
(4)
(the periodicify is imposed by the CT3symmetry), where h = 1, expf2zif3) and exp(-2ti/3) for X =: A,
Publishing Company
393
Volume
CHEMICAL
80, number 2
PHYSICS
and so causes symmetry-restricted
Ea and Eb, respectrvely. From eqs. (3) and (4) one easily finds
S = A
for ?t = 0. f 3. + 6, ___,
(5a)
x = E”
for iz = I___.-5,
-2,
I, 4, -._] .
(5b)
,y = Eb
for 12= {.... -4,
-I,?.
5, . ..} .
@cl
It is important to notice that clockwise as well as dntrclockwise rotating methyl groups are present withm each symmetry species_ This observation is crucial III resolving the problems raised by Clough [I] as we show below, but before doing so we first consider the other terms in eq. (1). The effect of the hindering porential ERR, considered as a perturbation, is to adnliv different values of ?? w&in the same symmetry species [3], since fi,, is of A symmetry in the rotor coordinate, i.e. _ [I,,(@ f 2x/3) = NRL(@) _ (6) Therefore the chrssitication in A, EL and Eb symmetry species is not changed by fi,,. The effect of the spur hamiltonian Hs and the Fermi statistics is taken into account in the usual way ds i0110ws. Smce rotation through 2rr/3 implies simultaneous exchanges of two pairs of protons on the methyl triangle, the Pauli princrple requires that spin and rotor product eigenstates should be of A symmetry, i e of the form
For the identiticatron of each of these products it is therefore sufficient to specify e.g. the spin symmetry. One can thus speak about A, Ea and Eb spin symmetry species. Now we introduce the lattice H,_, assumed to remain in thermal equilibrium at the lattrce temperature T, which is arbitrary and can be as high as room temperature, and the rotor-phonon interaction HRp mducing rapid transitions The crucial point is this: Since HRp is non-magnetic, it cannot change the spin symmetry. As a consequence of eq. (7) it causes only transitions between rotor states $& and $&, having the same symmetry. This property of HRp is well known. To put the same thing in another way [X], NRp is of A symmetry with respect to the rotor: l
ff~p(@ + 2x/3) =&p(@) 394
LETTERS
(8)
?
June 1981
transitions:
(9) Now suppose II > 0, then @It represents methyl groups in clockwise rotation. HRP excites those methyl groups to the rz’th rotor level of the same symmetry_ When 12’ < 0 thrs means that the methyl groups reorient into an anticlockwise rotation. Considering the instants of excitations to be random and defining the average relaxation time of the rotor states to be re we get the random reorientation model, which is symmetry restricted in the sense of relation (9) We thus arrive at the main conclusion: Since thermally activated rotor transitions are symmetry restricted, the resulting symmetry-restricted rapid random reorientations (SRRRR) occurring on a short time scale ~~ do not effect the lifetime T, of the (high-temperature) spin symmetry specres (HTSSS). Therefore HTSSS and SRRRR coexist; in fact SRRRR is accommodated within HTSSS. Tt is determined by the spin-rotor interaction HSR (this being modulated by the random reorientations), Tt is called a spinlattice relaxation time or spin symmetry conversion time, since relaxation converts A into E31b and vice versa, or E” into Eb and vice versa, spin symmetry species. As re can be of the order of lo-lo s, one can indeed have T, 25 lOlo 7, or even longer, since in practice Tr > 1 s. Also the SRRRR is indeed a refinement of the usual random reorientation (HTRRR) model [ 11. We can display the relationships between the different models as: HTRRR = SRRRR E HTSSS .
cw
We have presented a rather detailed analysis since the above finding is apparently non-trivial, as we should learn from Clough’s comment [l] _ Now we deal with another interesting aspect, namely rotational polarization defined as the population difference between Ea and Eb spin symmetry species [4,5] _ Since it has been established that long-lived spin symmetry species exist, rotational polarization persists on the long time scale r,. The existence of rotational polarization of symmetry species has been established experimentally and unambiguously [6,7], so there is no doubt about this. However, the intriguing question remains whether it is also possible to create an angular momentum imbalance between methyl groups belong-
Volume 80, number 2
ing to the Ea and Eb symmetry species, which we have called rotational polarization in rotor space [Z]. We now discuss “he feasibility of the latter type of rotational polarization. We consider the ensemble of methyl groups to be in complete equilibrium. Complete equilibrium means not only equilibrium within the symmetry species, but also equilibrium between the symmetry species. Then the Boltzmann distribution of the 12th rotor level of X rotor symmetry
1 June 1981
CHEMICAL PHYSICS LETTERS
subensemble has a definite angular momentum. Of course, Its value depends on the temperature, and because of the alternating values of n, cf. eqs. (5b) and (SC), it decreases with increasing temperature_ (We note that afthough M.)$ --j 0 as T increases, the lifetimes of the spin symmetry species are still of order TI -1
In equilibrium, from eqs. (13) and (14)
is given by
F(L)&=O.
where NX is the fraction of methyl groups of X symmetry and the subscript eq denotes its equliibriurn value_ We can thus divide the ensemble of methyl groups into three subensembles corresponding to the A, E” and Eb symmetry species, The expectation value of the angular momentum Ufl in each subensemble can be computed. In equilibrium
(12) Since there are as many clockwise as antlclockwise rotating methyl groups for any H E A, cf. eq. (5a) and
ref. [3], UP eq = u
03)
and we call the A subensemble stationary.
We note however that the Ea and Eb subensembles have nonzero resulting angular momenta. From eqs. (5b), (5c), (11) and (12) it can be shown that
,L)~=-(L)&b>o*
04)
Therefore we call the Ea subensemble that of clockwise and the Eb subensemble that of anticlockwise rotating methyl groups in the sense of eq. (14). The interpretation of the A, Eb and Ea spin symmetry species as representing stationary, clockwise and anticlockwise rotating methyl groups in refs. [4,7] should be understood in this sense. We must admit, however, that this was not clear in our previous work, and Clough’s comment has prompted us to write the present anaiysis. Although the individual methyl groups within the Ea or Eb symmetry execute random reorientation on a time scale TV, the equilibrium distribution of each
(15)
However, as predicted in refs. [4,5] and measured in refs. [6,7], it is possible to get a relaxation-induced non~qu~b~um E” - Eb population difference, Consequently an imbalance of angular momentum can be created by relaxation, such that
~LF>4LFb)
(L>“=o.
06)
The lifetime of such an induced rotational polarization in rotor space will be the same as that of the induced rotational polarization of the spin symmetry species itself, namely of order T1, since the two rotatronal polarizations are mutually coupled by eq. (7)_ An imbalance of angular momentum can also occur in equilibrium. It is known that a magnetic moment is associated with the angular momentum of rotating methyl groups [8]. It then follows that there is a resulting magnetic moment associated with the resulting angular momentum of the E3. and Eb subensembles, respectively. The magnetic field will shift the effective energy levels 1.51 of the Ea and Eb subensembles in opposite directions, leaving the effective ener,v level of the A subensemble unaltered. This Zeeman effect makes the Ea population differ from the Eb population. With the help of eq. (12) one easily sees that there will be an Imbalance of angular momentum. The experiment described in ref. [2] was meant to measure this effect. Some evidence of the expected effect was observed, but more carefully designed experiments are necessary and are being prepared, After this discussion, showing the coexistence of high-temperature spin symmetry species with rapid random reorientation, also the existence of rotational polarization in spin as welI as in rotor space, we now discuss the problem of exponential relaxation [9-l I] (referred to by Clough as refs. [20-221). We note that the solution of this kind of problem has already been given in ref. [IZ] (~o~espon~ng to ref. 151 of Clough’s 395
Volume 80, number 2
CHEhlICAL
PHYSICS LETTERS
paper). Exponential Zeeman relaxation follows from the HTSSS model when: (1) The Zeeman relaxation is observed after saturation; this was done in most of the cases in refs. [9,1 I] _ (2) The number of protons per molecule relaxed by one methyl group becomes large (> 10) as in all cases in ref. [lo] and some in ref. [9] _ (3) Intermolecular relaxation is not negIigibIe, as III rcfs. 19-l l] where there are many non-methyl protons (except perhaps for glycine [9]. but see below). (4) wore $ 1 where o. is the Larmor frequency_ Sum:: more than one of the above factors apply to refs. [9-l 11, the observed exponential relaxation can be explained within the HTSSS model or equivalently the symmetry restricted spin-diffusion (SRSD) model [4,5] However. as shown in ref. [ 121, the relaxation rare is not equal to the expression grven by the compiete spin drffusion model (i.e. the usual spin temperature theory). Smce the latter model has been applied in refs. [9- 1 I 1, strictly speaking the Interpretation of the expenments is not quite correct. It may also lead to pu/c.hng results, like the absence of intermolecl;lar rcl.rx.rtron in 2-methyl pyridine [ 11,131, or to results wtuch seem to be in conflict with the existing Iirerature. like exponential relaxation [9] in contrast to the earlier found non-exponential relaxation [ 141 for glycine. We recall that a partly deuterated single crystal was used in ref. [14] with the crystal so oriented that the methyl groups were magnetically equivalent so that the non-exponentiality could not be due to powder averaging of anisotropic relaxation [ 1] _ Moreover, upon request of the present author, measurements have been done by Brom’s group 1151 on glycine powder [9] at room temperature and on 2- as well as 3-methylpyridine [ 1 l] at 100 K, at a resonance frequency of 10 MHz and using the 90”-f-90” pulse program. It is interesting to note that in all three cases non-exponential relaxation was measured. For glycine the short time constant is =l 1 ms and the long time one 42 ms, the latter being consistent with ref. [9] ; for 3-methylpyridine the short time constant is =0.6 s and the long time one 41 s, the latter being consistent with ref. Ill]. Therefore, the conclusion must be that only the longest of the two time constants has been measured in ref. [9] as well as in ref. [l l] , as is to be expected in a recovery from saturation measurement. A full account of Brom’s measurements wrll be published elsewhere _ Remark. There are several ways to decide whether 396
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the non-exponentiallty is due to the coupled relaxatrons of the HTSSS model or to uncoupled anisotropic or inhomogeneous (by spin temperature gradient [ 11) relaxation. The first type is dependent on preparation, so the result of the 90”-t-90” pulse program is different from that of the sat--t--90° pulse program, see e.g. ref. [ 121. The second type is independent of preparation and gives the same result for those two programs. Another conclusive test is to use the pulse program developed recently [7] which serves as a convenient method to measure the relaxation-induced rotational polarization. The latter effect is absent for the uncoupled relaxations_ A last remark mentioned by Clough is that the HTRRR (or hoppmg [ 11) model itself can lead to non-exponential relaxation [16,17] (refs. [23,24] of Clough’s paper). This remark is indeed correct with regard to the treatment by Hilt and Hubbard [ 161. As for the analysis by Mehring and Raber [ 171, however, we should note that they used the HTSSS model, as is evident from p. 1117 of ref. [17]. In the context of Clough’s comment the above would be a paradox. In our view, however, it does not matter at all which model is used since we have shown, cf. relation (lo), that the HTRRR and HTSSS models are equivalent. They lead to exactly the same coupled relaxation equations as already noted in the very first paper on SRSD [ 181, It should be mentioned that this is also the only model which, among other things, provided the obvious explanation [4,5] for the Haupt effect [19] at low temperatures, and at the same time could predict its high-temperature analogue [5] _ The latter has now been confirmed experimentally [20]. Concluding we can say that we have resolved all the problems raised by Clough [ 11, using conventional methods. This little prece of work J. Schmidt, on the occasion testimony that his constant towards understanding was
is dedicated to Professor of his 60th birthday, as a stimulance for the quest not idle.
References [ll S. Ciough, Chem. Phys. Letters 80 (1981) 389. [2] S. Emid, Chem. Phys. Letters 72 (1980) 189. [3] A.E. Zweers and H-B. Brom, Physica 85B (1977) 223.
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[4] S. Emid, R A. Wind and S. Clough, Chem. Phys. Letters 33 (1974) 269. [S ] S. Emid and R.A Wind, Chem. Phys. Letters 33 (1975) 269. [6] R.A. Wind, S. Emid and J.F.JM_ Pourquie, Phys. Letters 53A (1975) 310. [7] S. Emid, Chem. Phys. Letters 77 (1981) 323. [ 81 W. Gordy and R.L. Cook, Microwave molecular spectroscopy (Interscience, New York, 1970) p. 375. [9] E-R. Andrew, W.S. Hinshaw, M.G. Hutchins and R.O.I. Sjiiblom, Mol. Phys. 31 (1976) 1479. [ 10 ] E R. Andrew, D.J. Bryant and E M. Cashell, Chem Phys. Letters 69 (1980) 551. [ 11 J W. Muher-Warmuth, R. Schuler, M. Prager and A. Kohmar, J. Chem. Phys. 69 (1978) 2382.
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[12] R-A_ Wind, S. Emid, J.F.J.M. Pourquie and J. Smidt, J. Chem. Phys. 67 (1977) 2436. 1131 W. Miiller-Warmuth, private communication (1976). 1141 M-F. Baud, Bull. Am. Phys. Sot. 14 (1969) 600. [ 151 H-B. Brom, University of Leyden, private communication. [ 161 R L. Hilt and P.S. Hubbard, Phys. Rev. 134A (1964) 392. [ 171 M. Mehring and R.H. Raber, J. Chem. Phys. 59 (1973) 1116. [18] S. Emid and R-A. Wind, Chem. Phys. Letters 27 (19743 312. [19] J. Haupt, Phys. Letters 38A (1972) 389. [20] S. Emtd and J. Smidt, Chem. Phys. Letters 77 (198:) 318.
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