Temperature dependence of single-particle kinetic energy in fluid parahydrogen

ELSEVIER

Physica B 234-236 (1997) 334-336

Temperature dependence of single-particle kinetic energy in fluid parahydrogen C. A n d r e a n i a, D . C o l o g n e s i a, A. F i l a b o z z i a, M . N a r d o n e b aDipartimento di Fisica e lstituto Nazionale di Fisica della Materia, Universit~ Torvergata, Via Ricerca Scientifica 1, 00133 Roma, Italy bDipartimento di Fisica e Istituto Nazionale di Fisica della Materia, Universitfi Roma III, Via Vasca Navale 86, 00154 Roma, Italy

Abstract

We present experimental measurements of the single molecule dynamics in fluid parahydrogen. The experiment consists of an inelastic neutron scattering determination of the translational kinetic energy, (Ek), as a function of temperature at a density close to the triple point. Experimental values of (Ek) are found to behave differently from a harmonic solid-like calculation. Keywords: Hydrogen; Momentum distribution; Quantum effects

In the present work we report the measurements of the mean translational kinetic energy, (Ek), in fluid parahydrogen as a function of temperature at constant density, performed using inelastic neutron scattering (INS). INS is now a well-established technique which allows to derive, within the framework of the impulse approximation (IA), the momentum distribution In(p)] in many monatomic systems [1-3]. In molecular systems, when intramolecular degrees of freedom can be decoupled from the intermolecular ones, INS can still be used to measure (Ek), I-4-6]. In an isotropic system, initially in its ground state (e.g. parahydrogen in the J = 0 rotational level), the scattering cross section can be written as d2a "~ ki d~do~ ] = ~ S(q, o9)

*Corresponding author.

=~f~b2f.(q)Sn(P)

× 6 hco - E,

h2q2 2M

hZq "P'~d ~/- J ~

(1)

The summation is meant over all the n intramolecular excited states; M is the molecular mass; kl and kf are the final and initial momenta of the neutron; f , and E, are, respectively, the structure factor and the excitation energy for the nth level. Following Eq. (1), the measured scattering spectra, at fixed q, will consist of a set of peaks centered at h2q 2 Ec,(q) = E, + 2---M(2) and broadened by the translational motion of the molecule. The measurements have been performed using the chopper spectrometer MARI at the ISIS pulsed

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C. Andreani et al. / Physica B 234-236 (1997) 334-336 n e u t r o n source, with a n e u t r o n incident energy of Eo = 500 meV, such that a m o m e n t u m transfer in the range between 0.5 ~,- ~ < q < 20 A - ~ and an energy transfer in the range between 0 < h~o < 480 meV were achieved. The n o r m a l h y d r o g e n conversion to p a r a h y d r o g e n was obtained at T = 17 K with the aid of paramagnetic catalyst inserted in the scattering cell, that was formed by two coaxial aluminium tubes with a 0.5 m m gap filled with the sample. Experimental S(q, co) (see Fig. 1) has been fitted in order to estimate the parameters in Eq. (1), where the s u m m a t i o n ran only over the first three rotational transitions (0 ~ 1, 0 ~ 3, 0 ~ 5) allowed by the h y d r o g e n incoherent cross section. The molecular n(p) has been assumed to have a Gaussian shape [7], with a variance a~ = 2 ( E k ) M / 3 , and the fit parameters were, besides the ap, the structure factors f,(q) and the peak positions E¢,(q). Small corrections, coming from fast-neutron b a c k g r o u n d and from multiple scattering, were also taken into account. The instrumental resolution was assumed to be a c o m b i n a t i o n of Gaussians coming from the moderator, the choppers, etc. The final fit results are shown in Fig. 2, where (Ek > values for the two densities of the measurement (p = 22.41 molecules n m -3, p = 10.45 molecules nm -3) are plotted as a function of T. The same figure also reports the classical prediction (dashed line): (Ek > = 3 kBT, and the result of a harm o n i c solid-like calculation (full line), obtained using a spectral density of the velocity autocorrelation function 9(0)) w o r k e d out from the incoherent structure factor S~(q, e)) by means of [-8]:

g(o~) ~o~l-s~(q'~)] =

q2

q-o

~FS¢(q,~)l

~ o , - - ~ - .

k S(q)q ] ~ o

,

(3)

where Ss(q, co) has been evaluated using the Vineyard a p p r o x i m a t i o n [9] from a previous measurement of coherent S¢(q, o~) in liquid parah y d r o g e n at T = 14.7 K and p = 22.83 molecules n m -3 1-10]. In a h a r m o n i c solid-like calculation [7], is given by =~

h~ocoth

~

g(~o)d~.

(4)

335 I

I

80

"~.

60

i

v

"

~ ' 40 k

2O

0

400

200

c0(rneV) Fig. 1. Inelastic structure factor measured by MAR1 spectrometer on liquid parahydrogen at T = 17 K, p = 22.41 molecules nm-3, with q = 11.0 A 1. Two rotational peaks are evident: 0 ~ 3 (the highest) and 0 ~ 5 (on the right side).

I

100

/ 50 v / /

/

/ 0

t

i

/

/

/

/

/

/

I

/

/

/

i i

J

I

i

~

20

,

I 40

,

t

t 60

T (K) Fig. 2. Translational kinetic energy as a function of temperature for parahydrogen. Squares with error bars are experimental results for p = 22.41 molecules nm 3, the triangle corresponds to p = 10.45 molecules nm -3 while the circle with error bar is from [5]. Dashed line is the classical behavior while full line is the harmonic model from [10] data, corresponding to T = 14.7K.

We note that experimental values are larger than the classical prediction, deviations being more p r o n o u n c e d for the higher density. H o w e v e r the h a r m o n i c behavior does not reproduce the experimental results accurately, showing a less steep temperature slope. It seems that this h a r m o n i c

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C. .4ndreani et al. / Physica B 234-236 (1997) 334-336

model is inadequate to describe the temperature dependence of our data, while this is not the case of normal fluid 4He [11]. However, one has to bear in mind that some variations in the spectral density 9(co) with temperature are likely to occur, causing changes of the average frequency (co) that describes the oscillating motion of a molecule in the potential well caused by its nearest neighbors: as temperature rises, molecules are involved in harder collisions and (co) increases. References [1] R.O. Hilleke et al., Phys. Rev. Lett. 52 (1984) 847. [2] M. Snow, Y. Yang and P.E. Sokol, Europhys. Lett. 19 (1992) 403.

I-3] H.K. Anderson et al., Physica B 181 (1992) 865. [4] J.A. Young, and J.U. Koppel, Phys. Rev. A 135 (1964) 603. [5] W. Langel et al., Phys. Rev. B 38 (1988) 11275. [6] K.W. Herwig et al., Phys. Rev. B 41 (1990) 96. [7] V.F. Sears, Can. J. Phys. 63 (1985) 68. [8] P.A. Egelstaff, An Introduction to Liquid State (Clarendon Press Oxford, 1992). [9] G.H. Vineyard, Phys. Rev. 110 (1958) 999. [10] K. Carneiro, M. Nielsen and J.P. McTague, Phys. Rev. Lett. 30 (1973) 481. 1-11] C. Andreani, A. Filabozzi, M. Nardone, F.P. Ricci and J. Mayers, Phys. Rev. B 50 (1994) 12 744.