Temperature dependence of the rotational life times in solid H2

Solid State Communications, Vol. 18, pp. 685-687, 1976.

Pergamon Press.

Printed in Great Britain

TEMPERATURE DEPENDENCE OF THE ROTATIONAL LIFE TIMES IN SOLID H2* R.F. Buzerak, M. Chart and H. Meyer Department of Physics, Duke University, Durham, NC 27706, U.S.A.

(Received 22 August 1975 by R.H. Silsbee) We have studied the temperature dependence of the nuclear transverse and longitudinal relaxation times in solid H2 for ortho concentrations 10 -3 < X < 10 -2 between 0.4K and the triple point, 13.9 K. We find a striking temperature dependence in 7"2 over the whole temperature range. This new effect is ascribed to a coupling between molecular rotation and lattice vibrations, which is brought into evidence by the narrow width of the spectral density of the rotational fluctuations at low X. SOLID ORTHO and para H2 mixtures present a number of analogies to a dilute antiferromagnet. 1 It is well known that in the solid state, the ortho molecules have a rotational angular momentum J = 1, an electric quadrupole moment eQ and the nuclear spins are parallel with a total spin I = 1. They correspond to a magnetic ion. Para-molecules are spherically symmetric with zero angular momentum and spin, and give no NMR signal. Neighboring J = 1 molecules are coupled via electric quadrupole-quadrupole interaction, which is the dominant anisotropic interaction under conditions of saturated vapor pressure. The characteristic parameter of this interaction is then Fo = 6e2Q2/25RS where R is the distance between nearest neighbors and with an effective value F[ks = 0.82 K. x Nuclear relaxation times offer a means to study the spectral densities of the rotational fluctuations as a function of the dilution of the J = 1 species, and recently there has been progress in the calculation of the shape of.these spectral densities over a large range of the molefraction X of the J = 1 molecules. 2 - 5 In this paper we describe the results for X < 10 -2 of a systematic study of the nuclear longitudinal and transverse relaxation times over the range 1 0 - 3 < X < 2 x 10 -1. These measurements, at the frequencies 6o/21r of 5.3 and 29 MHz, were carried out over the temperature range 0.4 ~< T <~ 13.7 K, the upper temperature being close to the triple point. Our data are in fact an extension of the work by Hardy and Gaines 6 who carried out such measurements at 4.2 K. Our most striking result is a strong temperature dependence of T2 which is unrelated to the classical diffusional linenarrowing. This new effect is especially noticeable for X < 0.01 where the linewidth is determined primarily by * Research supported by a grant from the Army Research Office Durham and from the Office of Naval Research.

the life time of the rotational states and which should be independent of temperature over our experimental range. We believe that this temperature dependence of T2 is caused by a broadening of the rotational levels by interaction with lattice vibrations as T is increased. The effect becomes more visible with decreasing X as the rotational life times become longer (i.e., the width of the spectral density becomes narrower). To our knowledge, such a rotation-vibration interaction has not been reported before in other solids. The observable nuclear relaxation rates T~ I and T~ ~ can be taken in a first approximation as the sum of the inter molecular and intra-molecular nuclear dipolar contributions. In the temperature region below about 9 K, where classical diffusion motion is not observed, we can speak of "rigid lattice" (R.L.) relaxation rates and (T~nter)R1.L" decreases uniformly with decreasing X. On the other hand (T~ntra) -1 is determined by the effect of rotational fluctuations that modulate the intramolecular nuclear dipolar field, and this phenomenon gives a similar result as exchange-narrowing in paramagnets 7 or motional narrowing. Theory 2 - 5 shows that approximately (T~ntra)-1 ~ X -s/3 and a simple calculation based on previous experimental evidence s indicates that (T~ntr~) ~ (T~nter)R L near X = 10 -2. In our concentration range the calculated (T~nter) -~ is always much smaller than (T~ntr~)-1 even in the diffusion region. Hence, for the remainder of this paper we write T~ = T]ntra. For X <~8 x 10 -3, the NMR line width is determined mostly by the width of the rotational fluctuation spectral density, 4 hr~ ~, which is of the order of 20FX s/3. It is clear that in this concentration range, r~ ~becomes comparable with col and we expect a frequency dependence for both T~ntra and T~nt~a which is predicted by theory and which has already been observed 6 by Hardy and Gaines for T]. The relevant equations are 4

685

686

Vol. 18, No. 6

ROTATIONAL LIFE TIMES IN SOLID H2 1.0

'

'

'

'

I

'

'

'

r

I

'

i

0.~

,

T 2 (E I. D end spin echo) 2 9 MHz

'

' '1 .... I

"G

Q5

,/

o.~

E to OJ

. 3 ' 0 0 0 . 0 ~

T~ Cmsec)

.~._,,.,0.05 j

"E

~-~ o.m

//

/

I

I

I

I

[

I

I

I

I

I

o x • • o

o

,

,

0.22 0.31 0.46 0.60 0.77

,

I ....

I

I

lO

2o

,

,

I ,

T(K)

I

Fig. 1. Transverse relaxation time/'2 for very low ortho molefraction X as a function of temperatures. The fre. quency was 29 MHz.

Fig. 2. The temperature dependence T~tra(T) -T~tra(0.05 K) for various concentrations X, as obtained from Fig. 1. The dashed line has a slope of 1.6.

LOngitudinol

q=-I

o

./ x

I0 T (K)

1 2rt [~c2 *~ Tptra - ~ 2

'''

o

5

5

,

-/

'T'

0

/

/x

F-

0.05

•,~

•

I

o.1

x//

x •

q2J(q')(--q °~)

Reloxotion

2 9 MHz 5,3 MHz

Time

TI 0.5~

X : 0 6"/*

O.

X = 0.32%

0

X =O.2 2"/*

TI

(rnsec'l

iI

(la) q=-2 (25

1

2rt[~ 2 .t

x : 0,77%

o

q=-I T (K)

+

~d2q~,1.{:>( -=_

q~

t

2T~t,= 1

(Ib)

where c and d are respectively the spin rotation and the spin-spin dipolar coupling constants and the Jtqi)(co) are the spectral densities of the fluctuations of certain angular momentum operators. It is now important to note that for X < 10 -2 we have ~ / k n ~ 0.4 K, our lowest temperature in the experiment. Hence, the spectral densities should be described by the high temperature approximation, which shows them to be temperature in-

dependent. 2,4 The J~O(co) have roughly a Lorentzian shape 1'4'9 and for cor~ ,~ 1, T~atra "~ T~ntra while for ~ r ~ l >> 1, T~atra(tJ) >~ T~at~a(6o = 0) and T~ntza(co) 3T~ntra(~ = 0). This last feature is the analogy of the famous "10/3" effect in paramagnetic systems with strong exchange 7 and we have indeed observed a frequency dependence of T2 in our experiments. However, the frequency variation of T~ntra is comparatively small to that of T~ntra and to a first approximation we put for our experimental conditions: T~intra- [2_~ )]-1 t~) ~ tic: + ~-ab]"~(0 •

[n

(2)

;

,,

;

;

,;

,i

T(K)

Fig. 3. Longitudinal relaxation time T~ for very low X as a function of T and at frequencies of 29 and 5.3 MHz. Using the theory of reference 4 and the numerical values for c and d, this gives an order-of-magnitude relation T~ntra(~o) ~ 1 x 10-12r~1 (see).

(3)

In Fig. 1 we show our T2 results at co/2rr = 29 MHz for several molefractions X. For X ~< 4.6 x 10 -3, these were obtained from the initial slope of a semilogarithmic plot of the free indiction decay signal vs time. Above this concentration,/'2 was determined from standard spin-echo techniques. Under conditions where both methods were used, they were found to agree to within 10%. We note the strong temperature dependence of T2 and also that of a plot of/'2 extrapolated to T ~ 0.4 K vs X gives nearly the expected X s/3 dependence. Hence it appears that the data below about 1 K are representative of rotational fluctuations alone. To get an order-of-magnitude information on the rotational life-time, we make the crude assumption that the general shape of the spectral density function does not depend on the width. Equation (3) now suggests that the spectral density width is r -~ = ~-~ + 5r -~ and accordingly we plot in Fig. 2 the temperature variation

Vol. 18, No. 6

ROTATIONAL LIFE TIMES IN SOLID H 2

of T ~ ira after correcting the original data for intermolecular dipolar broadening. For several molefractions the data had to be extrapolated to 0.4 K which introduced systematic uncertainties. We suggest that this broadening 5r -~ might possibly be caused by p h o n o n rotation interaction. The temperature dependence is somewhat stronger than one might expect for rotational fluctuations due to a one-phonon process. It is too weak to be caused by a lattice or impurity mode. 1° Using equation (3), an order-of-magnitude estimate gives ~r -l ~ 6 x 106 Tl"6sec-t.

(4)

Because of these arguments we believe that the temperature variations of the free induction decays reported earliern for a sample o f X = 3 x 10 -a between T = 0.3 and 4.2 K cannot be simply interpreted in terms of a crystalline field splitting of the rotational levels Jz = +- 1, 0 for isolated J = 1 molecules. By contrast to the/'2 data, the temperature variation of T~ at 29 MHz and at 5.3 MHz is less dramatic

687

but still significant and this is shown in Fig. 3. When the temperature increases, Tl passes through a broad maximum near 7 K and then decreases. These data are also expected to reflect the influence of rotation-phonon interaction on the rotational life times, but they cannot by analyzed as simply as the 7'2 results. Our results of T1 and T2 as a function of X both for 29 and 5.3 MHz are in good agreement with those by Hardy and Gaines 7 at 4.2K. A complete report of these results including those in the classical diffusion regime for X ~< 0.2 will be prepared for publication in the near future.

Acknowledgements - The authors gratefully acknowledge a number of stimulating discussions on the phononrotation interactions and helpful correspondence with Professor A.B. Harris. They further appreciate helpful communications with Professor T. Nakamura and Professor W.N. Hardy.

REFERENCES 1.

HARRIS A.B.,J. Appl. Phys. 42, 1574 (1971).

2.

HARRIS A.B.,Phys. Rev. B2, 3495 (1970).

3.

HAMA J.,INUZUKAT.&NAKAMURAT.,Progr. Theor. Phys. 48, 1769 (1972).

4.

FUJIO M., HA/VIAJ. & NAKAMURA T., Solid State Commun. B, 1091, (1973) and to be published.

5.

EBNER C.& MYLES C.,Phys. Rev. BI1,2339 (1975); MYLES C.& EBNER C.,Phys. Rev. B (in print).

6.

HARDY W.N. & GAINES J.R., (as quoted in reference 4).

7.

KUBO R. & TOMITA K., Prog. Theor. Phys. 9,888 (1954).

8.

PEDRONIP.,CHANM.,SCHWEIZERR.&MEYERH.,J. LowTemp. Phys. 19,537(1975).

9.

We neglect here the effect of crystalline field splitting of the rotational states [(CONSTABLE J. & GAINES J.R., Solid State Commun. 9, 155 (1971)] on the spectral density shape.

10.

We acknowledge a helpfull conversation and correspondence on this point with Professor R. Orbach.

11.

HARDY W.N. & GAINES J.R., Phys. Rev. Lett. 19, 1417 (1967).