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Spin-lattice relaxation in partially converted CH4 above 0.3 K
El Alami Ariba, H. Glättli
To cite this version:
El Alami Ariba, H. Glättli. Spin-lattice relaxation in partially converted CH4 above 0.3 K. Journal de Physique, 1983, 44 (12), pp.1351-1354. �10.1051/jphys:0198300440120135100�. �jpa-00209722�
Spin-lattice relaxation in partially converted CH4 above 0.3 K
El Alami Ariba (*) and H. Glättli
Service de Physique du Solide et de Résonance Magnétique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France
(Reçu le 10 mai 1983, accepté le 8 août 1983)
Résumé. 2014 Le temps de relaxation spin-réseau des protons dans CH4 a été mesuré dans la gamme de concentration des espèces F de 3 % à 37 % et à des températures allant de 0,3 K à 4,2 K. Un modèle simple de relaxation est pro-
posé qui tient compte du voisinage isomérique d’une molécule relaxante.
Abstract. 2014 The proton spin-lattice relaxation time has been measured in the range of F-species concentrations between 3 % and 37 % and at temperatures from 0.3 K to 4.2 K. A simple relaxation model is proposed which takes into account the spin-isomer environment of a relaxing molecule.
Classification Physics Abstracts
76.60E
The solid methanes, although composed of one of
the simplest molecules, exhibit a surprisingly rich
amount of interesting features in their orientational
behaviour, dependent on isotopic modification, tem- perature, pressure and spin-species composition.
Despite some recent conflicting evidence [1] it seems by now well established, that CH4 at low pressure ( 200 bar) and below 20 K is in a partially ordered phase, called antiferrorotational, jS-phase or phase II [2, 3] with eight sublattices. On two of these sublattices,
the molecules rotate almost freely while on the remaining six, they are restricted to tunnelling motion
between the equilibrium orientations imposed by the
electrostatic field. This field is different in orientation,
but equal in magnitude on these six sublattices. One
can speak, therefore, of two sites, containing respec-
tively free rotors and oriented molecules and the
phase II is then composed of 1/4 of (almost) free rotat- ing and of 3/4 of (almost) oriented molecules.
In CH4, the imperfect orientational localization on
both sites leads, through the Pauli principle, to three
different spin species, with different total spin I per molecule : A or meta (I = 2), F or ortho (I = 1), and
E or para (1= 0).
At helium temperatures, the free rotors are mostly
in the ground state (J = 0, I = 2) since the first excited rotational state (J = 1, I = 1) is at 12.7 K and tran-
sitions J = 0 H J = 1, are fast [4-6]. The populations
of higher levels can be neglected and we speak of
J = 0 and J = 1 molecules on the free rotor sites.
The ground torsional state of the oriented molecules is split into three manifolds, labelled A(1 = 2), F(I = 1) and E(I = 0) with degeneracies 5, 9 and 2 respectively and splittings A AF =-- 2 K, A FE -- 1 K.
Transitions between these levels need again spin
conversion. We speak of A-, F- or E-type molecules.
Spin conversion on the oriented sites is very slow and it is impractical to study pure CH4 at helium tempera-
tures with equilibrium spin species populations.
Paramagnetic impurities can speed up the conver- sion considerably. Oxygen impurities are easily incor- porated into CH4. However, they may be inhomoge- neously distributed inside the solid, they put large
strains on the lattice and they lead to a conversion speed, which is only weakly dependent on magnetic
field. For these reasons, we used the more difficult
procedure of y-irradiation at 4.2 K which creates CH3
and H radicals.
In this way, one obtains a strongly field-dependent,
or resonant, conversion. This makes it possible to change the conversion speed by simply changing the magnetic field and even to tune-in selectively the
F -+ A or the E -+ F conversion. We can, therefore, study CH4 with spin species populations both well
known and far from equilibrium with the lattice. Such
a study, i.e. spin-lattice relaxation time (T1 ) measure-
ments, has already been done for highly converted CH4 where, to a first approximation, only isolated
F-molecules exist in a pure A environment [7].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198300440120135100
1352
In the present work, we report results on T 1 in the
intermediate range, where the F-species concentration is not small.
1. Relaxation model.
The magnetic Hamiltonian responsible for spin-
lattice relaxation is the intramolecular dipolar inter-
action Jeàntra. Due to the symmetry of the A state,
Jeàntra has only vanishing matrix elements between the different ml states within the A manifold. Therefore,
A molecules do not relax. This is not true for the F states. We call the spin-lattice relaxation time of an
F-molecule T 1. In a mixture of A- and F-species, a unique relaxation time T 1 is observed. This indicates that’the intermolecular dipolar interaction is efficient to establish an internal equilibrium between the Zeeman energies of F- and A-species. If PA and PF are
the populations of A and F molecules (PA includes
also the free rotor 1 = 2 molecules)
where the right hand side is simply the ratio of the Zee-
man specific heat of the F-species to that of all protons.
Equation 1 has already been proposed in the early pioneering work on spin-lattice relaxation in solid
CH4 [8]. It implies that only transitions within the F- manifold are important for relaxation. This assump- tion seems reasonable for not too small F-species
concentrations and for vanishing J = 1 free rotors.
Within the framework of the model outlined above,
the information on the dynamics in CH4 is contained in T’. For a given Larmor frequency, T F may depend explicitly on temperature and on PF, as shown before.
To separate the two parameters, we used the same method as in the earlier work at lower PF : at the field
for A--+ F resonant conversion (12 kG) and at a given temperature, the equilibrium population is
obtained in typically one day. Figure 1 shows a plot
of the evolution of T1 during conversion (at T =
0.62 K) both on and off resonance. The ratio of the two conversion speeds (50) is close to an earlier value measured at T = 1.1 K [9]. The field is then switched off resonance (9.4 kG) and T1 is measured as a func- tion of temperature in a time short compared to
Fig. 1. - Evolution of T1 1 at 0.62 K on and off resonant
conversion.
off-resonance conversion. Standard pulse techniques
at 40 MHz have been used and in all cases, perfectly exponential relaxation has been observed over at least one decade. The results are shown in figure 2.
The low temperature limit is imposed by our 3He
cryostat. At the high temperatures, there are two ways to violate the approximation of constant population
within one set of measurements : conversion of the oriented molecules may come in, if the change in temperature is not fast enough. Besides, the free
rotor J = 1 molecules may start to influence Ti,
since conversion is fast [4, 6]. It is evident that the most converted samples are particularly sensitive
to error. The dashed line in figure 2 gives a limit
of confidence for the constant population. It is based
on the following assumptions : (i) conversion is assumed to be instantaneous on the free rotor sites, and the intrinsic relaxation of a J = 1 molecule is
comparable to that of an oriented molecule. This constitutes then a parallel relaxation path. (ii) The change in temperature is always fast compared to the
off-resonant conversion of the oriented molecules and PF can be regarded as constant up to at least 4 K.
The dotted line gives then the limit, where J = 1
molecules contribute 5 % to the relaxation rate, i.e.
an amount comparable to the experimental accuracy.
From the equilibrium temperature of each set of
measurements (intersection with the equilibrium curve), one can infer the population through the cor- responding Boltzmann factors. The calculation is
complicated by the fact that the tunnelling splittings
are population dependent. We used the form AAF - dAF(1 - aPF) with a = 0.31 found from a fit to the inelastic neutron scattering results of Heidemann
Fig. 2. - T1 1 as a function of temperature for different F-
species concentrations, together with the thermal equili-
brium line.
et al. [10] above 1 K and dAF = 1.89 K is the pure A limit measured recently [11]. The equilibrium PF populations obtained in this way are shown as para- meters in figure 2.
We can now obtain the dependence of T 1 on PF
for a given temperature. As can be seen from the nearly horizontal lines in figure 2, T1 (PF) is the same
at all temperatures below 1 K but changes markedly
above 1 K and at low concentrations. This is shown in figure 3 where T 1 (P F) is plotted at low tempera-
ture and at the confidence limit, i.e. 2.5 K. The PF dependence is seen to be less pronounced at high temperature. This is true at low PF. At high PF, we
find the same values at all temperatures. We have included a value taken from reference 12 measured at 4.2 K. Since we do not know the precise conditions
of this experiment, the error bar for PF is given by the
two extreme possible cases : no conversion of oriented molecules (infinite tunnelling temperature) or com- plete conversion to equilibrium at 4.2 K.
No new data or assumptions have been used in
figure 3. It is just a different way of presenting the original data of figure 2. We can, however, try to go
one step further : using (1) we can plot separately T 1
as a function of PF. The result is shown in figure 4.
Fig. 3. - ( T 1 ) -1 as a function of P, at 0.4 K and 2.5 K.
Fig. 4. - (T F)- ’ I as a function of PF at 0.4 K and 2.5 K.
The lines are fits to equation 5.
The determination of PFa through its exponential dependence on temperature and tunnel splitting is the
main error source in our measurements, as shown in
figure 3. The same error propagates through the popu-
lation factor to T F. This is indicated in figure 4 by the oblique error bars. They mean that the true value is always close to the error line. Some correlation between different measurements, due to systematic
errors, exists certainly although it is difficult to eva-
luate. The true dependence T F(p I F) is then most likely
a smooth curve similar to one of those depicted in figure 4.
The decrease in T Fat low PF, clearly seen for the low temperature points is very peculiar. It seems difficult
to explain in terms of a one-particle model. The
answer may be in a modification of the correlation function for reorientation as a function of spin species
concentrations. Since a detailed quantum mechanical
treatment of the dynamics of methane is still lacking
at present, we resort to a crude model which is able to
reproduce the two remarkable features of figure 4,
i.e. the decrease of T Fboth I with decreasing PF at low temperatures and with decreasing temperature at low PF.
As a straightforward generalization of a one-par- ticle model, we assume that T F of a particular F-
molecule is dependent on its spin-species environment.
We neglect the E-type and the disordered (all are
J = 0) molecules. We assume that T F is only depen-
dent on the number i of nearest neighbour F-molecules not on their particular spatial arrangement. One can then write
where Pi is the probability for an F-molecule to be surrounded by iF-type neighbours and cxi is its relaxa- tion speed. For a random distribution
In order to obtain a minimum in (T1 )-1 we are
led to assume
with ai an increasing function of i. The physical pic-
ture corresponding to the above assumptions shows
two distinct, independent relaxation mechanisms.
One involving a single, isolated F-molecule (cxo) and
another involving mutual reorientations of nearest
neighbour pairs. Equation 4 implies that the coupling
of neighbouring F-molecules due to the electrostatic interactions is sufficiently strong to make transitions within the F-level of a single molecule negligible as
soon as there is at least one F-type neighbour. The simplest assumption for the ai is ai = ia which means
1354
that the speed of the pair process is proportional to
the number of nearest neighbour F-molecules. (4)
becomes then
The solid line in figure 4 is a plot of (5) with a = 6.5 s-1
and ao = 10 s-1. It fits reasonably well the data at 2.5 K. The dashed and dash-dotted lines are obtained with the same a (6.5 s-’) and two different values of ao
(18 and 23) to give some idea of the possible spread
in the fit due to the inaccuracy of the measurements at low temperatures. It is not unreasonable that the pair
process is found to be temperature independent. On
the other hand, the relaxation of the isolated molecules ao is slightly more efficient at 0.4 K than at 2.5 K.
This can be understood by assuming a correlation time T, for this process which, as is usual, decreases
with increasing temperature. In this case, the decrease of TY with decreasing temperature means that we are in the region of rapid motion, or, in the framework of
an exponential correlation function rote 0.62. Since
our measuring frequency was ro/2 1t = 40 MHz this
means that T, 2.5 x 10-9 s. Such an upper limit for the correlation time is not in contradiction with recent work on the frequency dependence of T1 [12-14].
The E-species have been partly ignored in the above treatment. This means that we have taken into account their contribution (or rather their absence of contri-
bution) to the Zeeman specific heat (Eq. 1) and we
have assumed Boltzmann equilibrium amongst the A-,
F- and E-species for each given F concentration. Any
other assumption would only slightly change the
values of PF and T F. 1 For the dynamics, however (Eq. 2) we have assumed that E-species are equivalent
to A, the important property being non-F.
2. Conclusions.
The T 1 measurements reported here for the interme- diate concentration range of F-species 3 %§ to 37 %
cover the region where the F-molecules become pre-
dominantly isolated i.e. not surrounded by other F- species. The aim of these measurements is to extract the intrinsic relaxation speed of the F-species. At low concentration, the fast varying specific heat term
becomes a severe limitation to the precision with
which both PF and T F I can be obtained. We have
proposed a probably over-simplified model which
distinguishes basically isolated F-molecules from those surrounded by other nearest-neighbour F- species. This model is able to reproduce the maximum in T F at an intermediate concentration. A detailed check of the model for PF below a few percent would be worthwhile but is made difficult experimentally
due to the need to convert CH4 to a state of well
known PF. Only a microscopic treatment of the mole- cular reorientations taking into account the spin species environment could put the above model on a
firmer theoretical basis. Such a theory has not been attempted up to now.
References
[1] PROKHVATILOV, A. I., ISAKINA, A. P. and KRUPSKII, I. N., Solid State Commun. 42 (1982) 59.
[2] JAMES, H. M. and KEENAN, T. A., J. Chem. Phys. 31 (1959) 12.
[3] YAMAMOTO, T., KATAOKA, Y. and OKADA, K., J.
Chem. Phys. 66 (1977) 2701.
[4] CODE, R. F. and HIGINBOTHAM, J., Can. J. Phys. 54 (1976) 1248.
[5] NIJMAN, A. J. and BERLINSKY, A. J., Can. J. Phys. 58 (1980) 1049.
[6] BUCHMAN, S., CANDELA, D., VETTERLING, W. T. and POUND, R. V., Phys. Rev. B 26 (1982) 1459.
[7] BOUCHET, B. and GLÄTTLI, H., J. Physique Lett. 42 (1981) 103.
[8] DE WIT, G. A. and BLOOM, M., Can. J. Phys. 43 (1965)
986.
[9] GLATTLI, H., SENTZ, A. and EISENKREMER, M., Phys.
Rev. Lett. 28 (1972) 871.
[10] HEIDEMANN, A., PRESS, W., LUSHINGTON, K. J. and MORRISON, J. A., J. Chem. Phys. 75 (1981) 4003.
[11] PRESS, W., private communication.
[12] AGOSTINELLI, E., and CALVANI, P., Phys. Lett. A 95 (1983)118.
[13] CALVANI, P. and DE LUCA, F., J. Chem. Phys. 73 (1980)
167.
[14] BOUCHET, B., to be published.
