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H− ionic radius in alkali halide crystals from the local mode frequency of U-centres
Volume 12, nmnlmx. 4
PHYSICS LETTERS
quency waves, and (b) the spin relaxation in Case (B) is oflectively governed by tlm orbital relaxation tLme. We now turn to the diecussimz of the effective magnetic field accompanied by ~ e sound wave. The interaction of the sotmd wave w i ~ the e l e c tronic spLns o r i g i n a t e s f r o m (1) the e r d i n a r y e l e c i r o n - l a t t i c e interaction v i a the ~pin-orbtt coupling (the Elliot mechanism), an~ (2) the d i r e c t modulaUon of the s p i n - o r b i t COt~Hng by t]~ ~),t~d wave (tke O v e r h a u s e r mechanism). As Yafet ~-I has shown, the main t e r m s ~f the two m e c h a n i s m s cancel e a c h other if the m a t e r i a l s have a c e n t r e of inversion. The interaction is t h e r e f o r e p r o p o r t.tonal to Cq2Ag, C being the deformation potential and Ag the g shift. We confine o u r s e l v e s to the discussion on s e m i m e t a ~ like bismuth which has a v e r y l a r g e g shift. According to Yafet 2), the m a t r i x e l e m e n t for spin r e v e r s a l is roughly given
by ]Mk~q, ~ ; k, t[ ~ Cq2Ag a ~R,
~
c~_~=~
.
We should r e m a r k that the ln~gltudlnal and t r a n s v e r s e waves g i v e ~ tse to the fields of the s a m e o ; ' d e r of magnitude. The e f f e c t i v e f i e l d (9) is s , . m l l e r than ~_h~ of G e r a s i m e n k o by a factor ~ k i / q , ~f being the wave v e c t o r at the F e r m i surface. If we take for bismuth • = I0 W / c m 2, O ~
c / ec,
I0 e ,
A g = 200, w = 109 s e c - 1 , N = 2.3 x 10 l'l E / = 0.017 eV, WoT = 10, we have at r e s o n a n c e Hx ~ 200 g a u s s ,
a ~ 10 cm -1 .
(10)
In any cane we could expect the ultrm$onlc spin r e s o n a n c e by using sufficiently high freqttvncy wave,, at low t e m p e r a t u r e s . The author should e x p r e s s his ~ to P r o f e s s o r M o r r e ! H. Cohen, wl~ s u g g e s t e d the probl e m while the author wa~ at ~ e Institute for the Study of Metals of the University of Chicago.
(8)
where c is a length of the o r d e r of the lattice constant, ~ the lattice displacement. The effective nzagnetic field H can be defined by equating the matrix element of the perturbation, /~@" / / e x p (iq. r) ((~: the Pauli spin m-o.trix), b e tween plane waves to the e x p r e s s i o n (8). We ~hus obtain
Hx
15 O c t c ~ r 19~4
(9)
##s 2
R ef~Te;~c~ S I) V . I . G e r a a i ~ , Soviet Phym.-JETP 13(1961)410. 2) Y.Yatet, Solid state physics, erie. F.Se~tz stud D. Turnlmll, Vol. 14 (1963) p. I. 3) H.C.Torrey, Phys.Rev. 104 (1956) 563; I.Kaplan, Phys. R~v. 115 (19E9) 575. 4) I.M. LifshHz, M.Ya.Azbel' and . l . C ~ r a s ~ o , J. Phys. Chem. ~ l t d a 1 (19~.~) 164. 5) M.H.Cohe~, M . J . ~ m ~ d W.A.Harrb~m~. Phys.Rev. 11V (1960) 937.
H- IONIC RADIUS IN ALKALI HALIDE CRYSTALS FROM THE LOCAL MODE FREQUENCY OF U-CENTRES R. FIESCHI, G. F. NARDELLI and N. TERZI lstit~to di Fisica dell'Untversltl# dt Mllcmo, Milcmo, ltaly Received I1 September 1964
In a recent paper Zavl, I) showed that the discrepanc~ between the experimental data on the U-centre induced infrared absorption in alkali halide crystals and the theoretical (rigid ions) local m o d e frequencies of these centres m a y disappear ff these theoretical values are reduced by the ratio of trazusverse opfJe m o d e frequency to the longitudinal optic one at zero wave vector in the perfect lattice. According to us however ana, ss defect give,-
290
rise to a x elevant coupling between the transverw~ and the longitudinal modes (only at zero w~ve v e c tor does the coupling vanish) so that it is no longer possible to consider separately the contributlc~ of the transverse or longitudinal m o d e s in the local modes. The above d i s c r e p a n c y , indeed, m a y be explained in t e r m s of a change in the f o r c e - c o n stant at the defect s i t e and this change allows, in turn~ an e s t i m a t e of the ionic r a d i u s for tho H" in alkali halide c r y s t a l s .
Volume 12, m,mher 4
PHY,~(,'S L E T T E R S
The sec,,1~r equaUxm for the local mode f r e qtumcy 2) ~o2 read~ as follows:
d~ {I+ (L-~)-~ ~(~)} =0, where L is the dyruun/cal m = ~ t x of the perfect crystal and the change of m a s s and the c ~ n ~ e of i o r c e c o m ~ n t at the tmpm-tty site a r e t ~ , m into
account in the perturbaC~ ~(~2). T.~e ~ ictus Born model 3) and the shell model 4) with polartsable negative tons were cone/tiered, aud for both models the fractional changee of force constant ~o, or X, that must be included In A(~0~ in o r d e r to fit the experimental Incal mode frequencies 5), were evaluated for Uceniree in NaCI, KCI and KI at room temperature. The re su l t s are shown in table 1. Table 1 C=ymUd A ~ B oveHa~ ~'X°to~e ,~ matrioe~ n.n. force ocmstm~ (r ) (shell model) 8.21 9.~3 10.20
NRC~
KC1 KI
-0.62 -0.83 -0.68
-0.43 -0.52 -0.52
The overlap n.n. ; force ommtan~ are tu units of e2/~, v volame of t~e umlt cell, e electron charge. A ~ B are ~efh~l In the text,
In the rigid imm model Xo (in units of A+ 2B, where
=
-
(2r o is the side of the cubic unit cell) arises e~~ r e l y f r o m the change of the overlap potential o~ the other hand, in the shell model an imp~rt~mt centrlb~ton to X can ~ also f r o m the cl~m6es of electzo~c p o t t Y and shell cl~rge, so on~ can wrlte x = x + ~pol, where ~o Ires the same meaning ms in tlm rigid ions model. The last s t a ~ m e ~ t will be proven in a forthcoming paper 6). ~o o r x can be related to the H" ionic radius through the knowledge of a W~Litsble H" - alkali tons overlap pot~uUal, ff the n.n. elast/c r e ~ = ~ t/on is k~own as a funcUon of the lmpm-/ty ionic r~dtus itself. R is g ~ a l l y mummed that th~ W"
au~au b o , ~ , in ~
halide crystms, display
a remarkable ion/c character, so that the HuggtnsMayer form
ma~ be ueed to represent th~ overlap potenttal
15October1964
between the H" and the net~N)uring alkali ions. The value 0.339 ~ f o r the constant p 7) and the Furo/and Tosi 7) values for the host cr y st al radii were employed. A relationship between the n e ~ e s t - n e t g h b o u r displacement and the ionic radius of impurity ions was obtained for the NaC1 matrix by plotting the n.n. ~ displacements calculated by Fuk~! 8) against the corresponding ionic radius of the substituted halogen ions ~J; i~ the KCI m ~ t ~ x this relationship was deduced f~om the former on the basis o[ the discrete elastic theory, by comparing Fukai's values of n.n. displacement around a subetltuted bromine in NRC1 mad KCI crystals. W e were not able to find the analogous relationship for iodide crystal matrices, since no data about the elastic relaxation around impurities ar e available for such crystals. In the framework of the rigid ions model the H- Ionic radius in NaCI and KCI crystal matrices, deduced from the values of Xo, is found to be close to 1 ~ and the n.n. displacement is found to be inwards, amounting to -0.05 A and -0.035 A in in NaCI and KC1 respectively. This H- crystal radius strongly disagrees with Pauling~s p r ~ i / c t/on 10) which sets the H- crystal radius between those of Br" ~_ndI - , On the contrary, a result consiEtent with the Panl/n~ s predtctlon is found when the present values o! X ar e discussed on the basis of the shell model. In ~ case~ by t~.k!n~ the value ~, = 1.9,~3 11) for the H- electronic polarisabil/ty in crystais, assumln~ that a single electron contributes to its shell charge, and using the theoretical results contained in the quoted forthcoming p _aper, .~poJ is found to bave nearly the same value as z o f table 1, so that a vasinhing change of the overlap f o r c e s (i. e. ~) must be assumed in NaCI and KCI cr y st al m at r i ces. The value 1.78 ~ for the I f ionic radtus ts found consistently in both crystals. The same value is found also in KI crystal m at r i ces, ff an inward n. ~ displacement of -5°/o m assumed. Note tlmtin the above d e t c r m , n ~ o n of the H" ionic rmtins the polarissbUtty c~ the positive ions has been partially t s k m i~zo account by ad~n~ suitable correct/ram to the values for a and for the shell charge of H ' . ~ey~
es
1) G.8.Zavt, 8ovlet Phys.-45olld State 5 (1963) 792. Z) LM. Llfohflz, 8uppl.Huovo C/mento 4 (1956) 716;
E.W. Mort/roll mKl R.B.PoUs, Phys.Rev. I00 0955) 52S. ~ 8 094O) 51S. 291
Volume 12, number 4
PHYSICS
4) A.D.B.Woods, '7. Cochran and B. N. Brockh~mo, Phys. Rev. 119 (1960) 980. 5) See e . g . G . S c h r a e f e r , ,;.Phy~,.Chem.8olida 12 (1960) 233. 6) R. Fteschi, G.F.Nardelli and N. Terzl, to be published. 7) F.G.Fumi and M.P.To~f, J . C ~ m . 8 o l i d s 25 (196~ 31;
SPIN-LATTICE
RELAXATION
LETTERS
1.50~tfl~er 1964
v~e also M.P.T~mi an~ F.G. Furni, J.Pbye. Cbern. Solids 25 (1964) 45. 8) ¥. Fttkal, Prec. Int. Conf. tm C~stzd I==ttice Defects, Tokio, 1962. 9) Landolt-Bornstein Tables - eel.I, part 4 (Springer, Berlin, 1955). 10) R . f . C a l d e r , W. Cochran, D.Grifflth~ and R.D. Lowde, J. Phys. Cbem.5olida 23 (19C23 621
IN S O L I D
ETIIYI, ENE
M, J. R. H O C H * and F. A. R U S H W O R T H Department of Natu~'al Philosophy. The Univer.~ity, St.And~ews, Scott.. Recelv.~ ~.2 September 1964
Proton Magnetic Resonance studies of the a b s o r p t i o n spectrum and spin-Lattice relaxation time (T1) have be~n carried out at 22.6 M c / s on polycrystalline ethylene in the temperature interval 20°K to the m. pt. 104°K using the ste,~,~ystate spectrometer and cryostat described previously 1). F r o m the second m o m e n t calculatlone, unir,g the accurately known molecular and crystal structure parameters, it appears likely that the ethylene molecules tmdergo 180 ° flip reorientations abou~ their figure (C = C) axes. Below 50°K the molecular reorientation is too slow (< 5 × I04 c/~J to narrow the spectrum although the effects of rapid torsional oscillations still give a marked temperature dependence. In the second m o m e n t cldcuLations the effect of torsional oscillations was allowed for in the zero point approximation. Between 50°K and the m. pt. the reorientation rate is sufficient to narrow the spectrum. The T 1 is shown in fig. 1, where an unexpected anomaly occurred below 50OK. A correlationtime analysis of the T I data was carried out assuming a Debye spectral density function 1
~-
r'I - XlL7
~c
4 vc
G
+ C~o2Te2 + 1 + 4-a,~o--2,rc~_ ,l1
(1)
,~,~re v. is the correlation time and C I a con~ n t ob~ined from the m i n i m u m value of 7'I. A ; ,or of log ~0oTc against I/T for values of T > 50OK gives a value ~'f 2.0 ± 0.3 k cal/mole for the upCommonwealth Scholar. Now at Physics Department, University of Natal, Pietermaritzbttrt;, &,Jth-Afrlca. ~'~2
s°o-T--
-r L-f
......... :...... T-~ .
..... i . k = -=_
'--
' i',
~[I
0-1 2'0
I i
i
.
.
......,
.
.
.
.
.
.
i
I .... ! \
i \i
'
~!/
TI]
/,0 60 80 ~1{30 ' TEHPERATURE (°K }
Fig. 1. The variation with temporature of the spinlattice relaxation time in solid ethylene. p a r e n t a c t i v a t i o n e n e r g y E a g o v e i n i n g the r e orientation prucess. It is n e c e s s a r y to c o n s i d e r the p( s s i b i l i t y of b a r r i e r p e n e t r a t i o n by tunnelling of the m o l e c u l e s about t h e i r f i g u r e axes. T u n n e l l i n g of CH 3 g r o u p s w a s c o n s i d e r e d by P o w l e s and Gut~wsky 2) and by SteJskal and Gutowsky 3), while a g e n e r a i i s e d theoreftc~.~ t r e a t m e n t ol tun ,elll~.. wa~ given by D a s 4). ~f it is a s s u m e d that the t m r r i e r h i n d e r i n g r e o r t e n t ; l t i o n of the m o l e c u l e s In the l a t t i c e is of the form V = -~Vo(l + cos 2~, where ~ols the angle of rotatbm, and furthermore assun-lng a~ an apprc0clma'.ion that this barrier is time -ud temperature independent, then substitution int ) t~e Schr0-
PHYSICS LETTERS
quency waves, and (b) the spin relaxation in Case (B) is oflectively governed by tlm orbital relaxation tLme. We now turn to the diecussimz of the effective magnetic field accompanied by ~ e sound wave. The interaction of the sotmd wave w i ~ the e l e c tronic spLns o r i g i n a t e s f r o m (1) the e r d i n a r y e l e c i r o n - l a t t i c e interaction v i a the ~pin-orbtt coupling (the Elliot mechanism), an~ (2) the d i r e c t modulaUon of the s p i n - o r b i t COt~Hng by t]~ ~),t~d wave (tke O v e r h a u s e r mechanism). As Yafet ~-I has shown, the main t e r m s ~f the two m e c h a n i s m s cancel e a c h other if the m a t e r i a l s have a c e n t r e of inversion. The interaction is t h e r e f o r e p r o p o r t.tonal to Cq2Ag, C being the deformation potential and Ag the g shift. We confine o u r s e l v e s to the discussion on s e m i m e t a ~ like bismuth which has a v e r y l a r g e g shift. According to Yafet 2), the m a t r i x e l e m e n t for spin r e v e r s a l is roughly given
by ]Mk~q, ~ ; k, t[ ~ Cq2Ag a ~R,
~
c~_~=~
.
We should r e m a r k that the ln~gltudlnal and t r a n s v e r s e waves g i v e ~ tse to the fields of the s a m e o ; ' d e r of magnitude. The e f f e c t i v e f i e l d (9) is s , . m l l e r than ~_h~ of G e r a s i m e n k o by a factor ~ k i / q , ~f being the wave v e c t o r at the F e r m i surface. If we take for bismuth • = I0 W / c m 2, O ~
c / ec,
I0 e ,
A g = 200, w = 109 s e c - 1 , N = 2.3 x 10 l'l E / = 0.017 eV, WoT = 10, we have at r e s o n a n c e Hx ~ 200 g a u s s ,
a ~ 10 cm -1 .
(10)
In any cane we could expect the ultrm$onlc spin r e s o n a n c e by using sufficiently high freqttvncy wave,, at low t e m p e r a t u r e s . The author should e x p r e s s his ~ to P r o f e s s o r M o r r e ! H. Cohen, wl~ s u g g e s t e d the probl e m while the author wa~ at ~ e Institute for the Study of Metals of the University of Chicago.
(8)
where c is a length of the o r d e r of the lattice constant, ~ the lattice displacement. The effective nzagnetic field H can be defined by equating the matrix element of the perturbation, /~@" / / e x p (iq. r) ((~: the Pauli spin m-o.trix), b e tween plane waves to the e x p r e s s i o n (8). We ~hus obtain
Hx
15 O c t c ~ r 19~4
(9)
##s 2
R ef~Te;~c~ S I) V . I . G e r a a i ~ , Soviet Phym.-JETP 13(1961)410. 2) Y.Yatet, Solid state physics, erie. F.Se~tz stud D. Turnlmll, Vol. 14 (1963) p. I. 3) H.C.Torrey, Phys.Rev. 104 (1956) 563; I.Kaplan, Phys. R~v. 115 (19E9) 575. 4) I.M. LifshHz, M.Ya.Azbel' and . l . C ~ r a s ~ o , J. Phys. Chem. ~ l t d a 1 (19~.~) 164. 5) M.H.Cohe~, M . J . ~ m ~ d W.A.Harrb~m~. Phys.Rev. 11V (1960) 937.
H- IONIC RADIUS IN ALKALI HALIDE CRYSTALS FROM THE LOCAL MODE FREQUENCY OF U-CENTRES R. FIESCHI, G. F. NARDELLI and N. TERZI lstit~to di Fisica dell'Untversltl# dt Mllcmo, Milcmo, ltaly Received I1 September 1964
In a recent paper Zavl, I) showed that the discrepanc~ between the experimental data on the U-centre induced infrared absorption in alkali halide crystals and the theoretical (rigid ions) local m o d e frequencies of these centres m a y disappear ff these theoretical values are reduced by the ratio of trazusverse opfJe m o d e frequency to the longitudinal optic one at zero wave vector in the perfect lattice. According to us however ana, ss defect give,-
290
rise to a x elevant coupling between the transverw~ and the longitudinal modes (only at zero w~ve v e c tor does the coupling vanish) so that it is no longer possible to consider separately the contributlc~ of the transverse or longitudinal m o d e s in the local modes. The above d i s c r e p a n c y , indeed, m a y be explained in t e r m s of a change in the f o r c e - c o n stant at the defect s i t e and this change allows, in turn~ an e s t i m a t e of the ionic r a d i u s for tho H" in alkali halide c r y s t a l s .
Volume 12, m,mher 4
PHY,~(,'S L E T T E R S
The sec,,1~r equaUxm for the local mode f r e qtumcy 2) ~o2 read~ as follows:
d~ {I+ (L-~)-~ ~(~)} =0, where L is the dyruun/cal m = ~ t x of the perfect crystal and the change of m a s s and the c ~ n ~ e of i o r c e c o m ~ n t at the tmpm-tty site a r e t ~ , m into
account in the perturbaC~ ~(~2). T.~e ~ ictus Born model 3) and the shell model 4) with polartsable negative tons were cone/tiered, aud for both models the fractional changee of force constant ~o, or X, that must be included In A(~0~ in o r d e r to fit the experimental Incal mode frequencies 5), were evaluated for Uceniree in NaCI, KCI and KI at room temperature. The re su l t s are shown in table 1. Table 1 C=ymUd A ~ B oveHa~ ~'X°to~e ,~ matrioe~ n.n. force ocmstm~ (r ) (shell model) 8.21 9.~3 10.20
NRC~
KC1 KI
-0.62 -0.83 -0.68
-0.43 -0.52 -0.52
The overlap n.n. ; force ommtan~ are tu units of e2/~, v volame of t~e umlt cell, e electron charge. A ~ B are ~efh~l In the text,
In the rigid imm model Xo (in units of A+ 2B, where
=
-
(2r o is the side of the cubic unit cell) arises e~~ r e l y f r o m the change of the overlap potential o~ the other hand, in the shell model an imp~rt~mt centrlb~ton to X can ~ also f r o m the cl~m6es of electzo~c p o t t Y and shell cl~rge, so on~ can wrlte x = x + ~pol, where ~o Ires the same meaning ms in tlm rigid ions model. The last s t a ~ m e ~ t will be proven in a forthcoming paper 6). ~o o r x can be related to the H" ionic radius through the knowledge of a W~Litsble H" - alkali tons overlap pot~uUal, ff the n.n. elast/c r e ~ = ~ t/on is k~own as a funcUon of the lmpm-/ty ionic r~dtus itself. R is g ~ a l l y mummed that th~ W"
au~au b o , ~ , in ~
halide crystms, display
a remarkable ion/c character, so that the HuggtnsMayer form
ma~ be ueed to represent th~ overlap potenttal
15October1964
between the H" and the net~N)uring alkali ions. The value 0.339 ~ f o r the constant p 7) and the Furo/and Tosi 7) values for the host cr y st al radii were employed. A relationship between the n e ~ e s t - n e t g h b o u r displacement and the ionic radius of impurity ions was obtained for the NaC1 matrix by plotting the n.n. ~ displacements calculated by Fuk~! 8) against the corresponding ionic radius of the substituted halogen ions ~J; i~ the KCI m ~ t ~ x this relationship was deduced f~om the former on the basis o[ the discrete elastic theory, by comparing Fukai's values of n.n. displacement around a subetltuted bromine in NRC1 mad KCI crystals. W e were not able to find the analogous relationship for iodide crystal matrices, since no data about the elastic relaxation around impurities ar e available for such crystals. In the framework of the rigid ions model the H- Ionic radius in NaCI and KCI crystal matrices, deduced from the values of Xo, is found to be close to 1 ~ and the n.n. displacement is found to be inwards, amounting to -0.05 A and -0.035 A in in NaCI and KC1 respectively. This H- crystal radius strongly disagrees with Pauling~s p r ~ i / c t/on 10) which sets the H- crystal radius between those of Br" ~_ndI - , On the contrary, a result consiEtent with the Panl/n~ s predtctlon is found when the present values o! X ar e discussed on the basis of the shell model. In ~ case~ by t~.k!n~ the value ~, = 1.9,~3 11) for the H- electronic polarisabil/ty in crystais, assumln~ that a single electron contributes to its shell charge, and using the theoretical results contained in the quoted forthcoming p _aper, .~poJ is found to bave nearly the same value as z o f table 1, so that a vasinhing change of the overlap f o r c e s (i. e. ~) must be assumed in NaCI and KCI cr y st al m at r i ces. The value 1.78 ~ for the I f ionic radtus ts found consistently in both crystals. The same value is found also in KI crystal m at r i ces, ff an inward n. ~ displacement of -5°/o m assumed. Note tlmtin the above d e t c r m , n ~ o n of the H" ionic rmtins the polarissbUtty c~ the positive ions has been partially t s k m i~zo account by ad~n~ suitable correct/ram to the values for a and for the shell charge of H ' . ~ey~
es
1) G.8.Zavt, 8ovlet Phys.-45olld State 5 (1963) 792. Z) LM. Llfohflz, 8uppl.Huovo C/mento 4 (1956) 716;
E.W. Mort/roll mKl R.B.PoUs, Phys.Rev. I00 0955) 52S. ~ 8 094O) 51S. 291
Volume 12, number 4
PHYSICS
4) A.D.B.Woods, '7. Cochran and B. N. Brockh~mo, Phys. Rev. 119 (1960) 980. 5) See e . g . G . S c h r a e f e r , ,;.Phy~,.Chem.8olida 12 (1960) 233. 6) R. Fteschi, G.F.Nardelli and N. Terzl, to be published. 7) F.G.Fumi and M.P.To~f, J . C ~ m . 8 o l i d s 25 (196~ 31;
SPIN-LATTICE
RELAXATION
LETTERS
1.50~tfl~er 1964
v~e also M.P.T~mi an~ F.G. Furni, J.Pbye. Cbern. Solids 25 (1964) 45. 8) ¥. Fttkal, Prec. Int. Conf. tm C~stzd I==ttice Defects, Tokio, 1962. 9) Landolt-Bornstein Tables - eel.I, part 4 (Springer, Berlin, 1955). 10) R . f . C a l d e r , W. Cochran, D.Grifflth~ and R.D. Lowde, J. Phys. Cbem.5olida 23 (19C23 621
IN S O L I D
ETIIYI, ENE
M, J. R. H O C H * and F. A. R U S H W O R T H Department of Natu~'al Philosophy. The Univer.~ity, St.And~ews, Scott.. Recelv.~ ~.2 September 1964
Proton Magnetic Resonance studies of the a b s o r p t i o n spectrum and spin-Lattice relaxation time (T1) have be~n carried out at 22.6 M c / s on polycrystalline ethylene in the temperature interval 20°K to the m. pt. 104°K using the ste,~,~ystate spectrometer and cryostat described previously 1). F r o m the second m o m e n t calculatlone, unir,g the accurately known molecular and crystal structure parameters, it appears likely that the ethylene molecules tmdergo 180 ° flip reorientations abou~ their figure (C = C) axes. Below 50°K the molecular reorientation is too slow (< 5 × I04 c/~J to narrow the spectrum although the effects of rapid torsional oscillations still give a marked temperature dependence. In the second m o m e n t cldcuLations the effect of torsional oscillations was allowed for in the zero point approximation. Between 50°K and the m. pt. the reorientation rate is sufficient to narrow the spectrum. The T 1 is shown in fig. 1, where an unexpected anomaly occurred below 50OK. A correlationtime analysis of the T I data was carried out assuming a Debye spectral density function 1
~-
r'I - XlL7
~c
4 vc
G
+ C~o2Te2 + 1 + 4-a,~o--2,rc~_ ,l1
(1)
,~,~re v. is the correlation time and C I a con~ n t ob~ined from the m i n i m u m value of 7'I. A ; ,or of log ~0oTc against I/T for values of T > 50OK gives a value ~'f 2.0 ± 0.3 k cal/mole for the upCommonwealth Scholar. Now at Physics Department, University of Natal, Pietermaritzbttrt;, &,Jth-Afrlca. ~'~2
s°o-T--
-r L-f
......... :...... T-~ .
..... i . k = -=_
'--
' i',
~[I
0-1 2'0
I i
i
.
.
......,
.
.
.
.
.
.
i
I .... ! \
i \i
'
~!/
TI]
/,0 60 80 ~1{30 ' TEHPERATURE (°K }
Fig. 1. The variation with temporature of the spinlattice relaxation time in solid ethylene. p a r e n t a c t i v a t i o n e n e r g y E a g o v e i n i n g the r e orientation prucess. It is n e c e s s a r y to c o n s i d e r the p( s s i b i l i t y of b a r r i e r p e n e t r a t i o n by tunnelling of the m o l e c u l e s about t h e i r f i g u r e axes. T u n n e l l i n g of CH 3 g r o u p s w a s c o n s i d e r e d by P o w l e s and Gut~wsky 2) and by SteJskal and Gutowsky 3), while a g e n e r a i i s e d theoreftc~.~ t r e a t m e n t ol tun ,elll~.. wa~ given by D a s 4). ~f it is a s s u m e d that the t m r r i e r h i n d e r i n g r e o r t e n t ; l t i o n of the m o l e c u l e s In the l a t t i c e is of the form V = -~Vo(l + cos 2~, where ~ols the angle of rotatbm, and furthermore assun-lng a~ an apprc0clma'.ion that this barrier is time -ud temperature independent, then substitution int ) t~e Schr0-