Temperature dependence of the atomic self-motion in liquid argon

Physica 50 ( 1970) 51 l-523

TEMPERATURE

o North-Holland Publishing Co.

DEPENDENCE

SELF-MOTION P. ZANDVELD,

OF THE

IN LIQUID

C. D. ANDRIESSE,

ATOMIC

ARGON

J. D. BREGMAN,

A. HASMAN

and J. J. VAN LOEF Interwniversitair Reactor Instituut, Delft, Nederland Received 2 July 1970

synopsis Quasi-elastic neutron-scattering experiments are performed on liquid Ar in orthobaric states at 85.7, 103.5 and 115.9 K, from which the mean square atomic displacement is derived for times between 0 and 2.25 ps. The self-diffusion coefficients found are in good agreement with those obtained by tracer-diffusion measurements. The data are interpreted in terms of a modified Langevin-diffusion model and the stochastic model for the velocity autocorrelation function. It is found that by applying the stochastic model good agreement is found with computer results of Rahman. The “caging effect” in the velocity autocorrelation function gradually disappears within the temperature range investigated. It is concluded that the diffusive modes are damped by a mechanism which depends on the temperature in a way different from that in a liquid metal. 1. I&rod&ion. In recent years computer studies have given clear and detailed information on the atomic self-motion in simple liquids such as Arip s), which is usually expressed in terms of the velocity autocorrelation

function. On the other hand inelastic incoherent neutron-scattering experiments also can give valuable information on the atomic self-motion, though it is very difficult to extract information about this correlation function from the experiments in a direct way. Recently the well-known results of Dasannacharya and Raos) on inelastic neutron scattering by liquid Ar near the triple point were used by Casanova and Levia) in conjunction with the interatomic potential as derived from the isotope separation factors) in order to determine the time-evolution of the velocity autocorrelation function and the self-diffusion coefficient. The latter turned out to be in good agreement with the results of tracer-diffusion experiments69 7). An important parameter in studying atomic self-motion is the temperature. Andrus and coworkers*) performed quasi-elastic neutron-scattering experiments on liquid Ar in a temperature range from 87 to 124 K. In contrast to Dasannacharya and Rao they concluded that continuous diffusion cannot explain their results. Therefore it was considered useful to investigate the 511

512

ZANDVELD,

ANDRIESSE,

BREGMAN,

HASMAN

AND VAN LOEF

temperature dependence of incoherent neutron scattering by liquid Ar more closely by considering neutron momentum transfers which are well away from the first neutron-diffraction peak in Ar. Our experimental data, which are given in terms of width functions

in the next section,

are interpreted

in

section three. The conclusions are summarized in the final section, whereas in an appendix the way the data are handled is treated in more detail. 2. Experimental data. Using the Delft rotating-crystal spectrometers) we have studied the inelastic scattering of 4.1 A neutrons by liquid Ar for momentum transfers Q between 0.8 and 1.5 A-1. The Ar was held in a 10.6 cm diameter cylindrical container of aluminium, so that about 10yO of the incident neutrons are scattered and about 30% are absorbed by the sample. Experimental points are chosen along the vapour pressure cwve at temperatures of 85.7, 103.5 and 115.9 K respectively. Furthermore we measured the scattering by solid Ar at 80.5 K.

Fig. 1. Averaged intensities of a part of the time-of-flight spectrum from liquid Ar at 85.7 K (circles) and from solid Ar at 80.5 K (triangles) measured with 4.1 A neutrons under a scattering angle of 58“; the intensities are normalized to the top of the elastic peak.

The neutron spectra from Ar are characterized by an elastic peak, which in case of the liquid is broadened by diffusive modes. In addition an inelastic peak is observed in the solid, which is caused by acoustic phonons. In fig. 1 this peak is compared with the corresponding part of the spectrum measured from liquid Ar at 85.7 K. From the absence of any inelastic contribution to

SELF-MOTION

the latter

it is concluded

should be strongly

IN LIQUID

ARGON

that possible phonon-like

excitations

513

in the liquid

dampedt.

The measured neutron spectra are considered to be a convolution of the scattering law with the resolution function of the spectrometer, which is determined by the shape of the elastic peak for solid Ar. We have taken the resolution function into account by applying the Fourier-transformation technique which is resumed in the appendix 12). As a result the experimental data are given directly in terms of the intermediate scattering function. We obtain a function

B(Q,t) = fFs (Q>t) + (1 - f) F(Q>t)> where the second

term can be related

(1)

to the first term using the substi-

tution 13)

F(Q> f) = S(Q) Fs (Q/,/s (Q), t)-

(2)

Here f is the incoherent fraction of the total scattering, F,(Q, t) and F(Q, t) are the self-part and the total (coherent) intermediate scattering function, respectively, and S(Q) is the structure factor. The approximation made by (2) will not introduce significant errors as long as the structure factor is small. reasonably well by For Q below 1.5 A-l, F,(Q, t) can be approximated

WQ, 4 = exd-Q2rW

[I+ ib2(4~Q2rP)121~

(3)

where as(t) accounts for most of the non-gaussian behaviour of F,(Q, t) (its value has been numerically determined by Rahmani)) and y(t) is called the width function. It is well known that y(t) represents the mean square atomic displacement, divided by 6, which for times longer than 0.5 ps4) can be described by its asymptotic form

y(t) =

c + DC

(4)

where C is a constant and D the self-diffusion coefficient. Using f = 0.343) and the relevant values for S(Q) obtained by X-ray scatteringrJ), we have numerically transformed our data on B(Q, t) into y(t) for three different Q values. The results averaged over the three Q values are fitted by the method of least squares to the asymptotic form (4) for times longer than 0.5 ps (see figs. 2-4). Values of C and D are given in table I.

* The existence of inelastic scattering by liquid Ar is somewhat controversial; and coworkerslo)

Chen

observed an inelastic peak for Ar at 85 K, which was not found by

Dasannacharya and Rao3) ; Skijld and Larssonll) existence of this peak at 94 K.

could not directly

confirm

the

514

ZANDVELD,

ANDRIESSE,

BREGMAN, HASMAN AND VAN LOEF TABLE I

Experimental

values for C and D at three different temperatures

A

D (lo-gmss-1)

&

85.7 f 0.6 103.5 f 0.8 115.9 f 0.6

-0.04 -0.05 -0.10

f 0.02 f 0.03 + 0.02

0.6a2

1.6 f 0.2 2.8 & 0.2 5.0 & 0.1

r

65.7 K

r

-0.2 Fig. 2. Experimental data on y(l) at 85.7 K; left: data at Q = 0.8, 1.1 and 1.5 A-1 indicated by circles, squares and triangles respectively; right: averaged data with least-squares fit to (4).

r

l.ORC

103.5 K

.

0.4 -

v

.

I

;V

I

1

2

I

A

3’PS ” -0.2

Fig. 3. Experimental

“y ,’

A

1

I

I

-t I

2

I

3

data on y(t) at 103.5 K, see fig. 2.

I

4PS

SELF-MOTION

IN LIQUID

1.4A2-

ARGON

515

r

.

ii

1.2 -

Y(')i.O -

.

. . .

i0.8

.,: 7 .

Ck6 -

I

.i’ :

I

I

I

I

1

2

3

I 4 PS

Bt

-0.2L

Fig. 4. Experimental data on y(t) at 115.9 K, see fig. 2.

3. Irtterpretation. Our data on the self-diffusion coefficient appear to be in satisfactory agreement with those obtained by tracer-diffusion techniques 697) (see fig. 5). H owever, it follows from fig. 5 that our data probably do not obey an exponential law in the temperature range investigated. It should be noticed that the results of recent neutron-scattering experiments by Andrus and coworkerss) are in disagreement with ours. The main difference with our experiment is that these authors measured in a region of Q-values where the coherent scattering dominates, so that their result for the selfmotion is very sensitive to shortcomings of the model used to account for interference effects. The parameter C represents the mean-square displacement of the vibrating atoms, diminished by the mean square of the length of a diffusive step. All our values for C appear to be negative. The result of Rahman’s computer study for liquid Ar at 94 K is positive (0.033 As) i), whereas the neutron data by Dasannacharya and Rao for liquid Ar at 84.5 K a) result in a C value which is close to zero (0.014 As) 4). In view of the relatively large errors found in C we intend to perform neutron-scattering experiments on a mixture of s6Ar and aOAr, for which the incoherently scattered fraction is increased from 0.34 to the largest possible value 0.70.

516

ZANDVELD.

ANDRIESSE,

BREGMAN,

HASMAN

AND VAN LOEF

Fig. 5. Experimental data on the self-diffusion coefficient; comparison of data by Cini-Castagnoli and Riccie) (triangles) and data by Naghizadeh and Rice’) (squares) with those deduced from figs. 2-4 (circles with error bar). The full line corresponds to diffusion according to an exponential law with an activation energy of 3.4 kJ/mol.

It is useful to interpret C and D in terms of the velocity function $(t), which is related to y(t) by y(t) = (/Wf)-1 /dt.(l -

autocorrelation

1’) $(t’).

(5)

Here p = (kT) -1, k is Boltzmann’s constant, T the temperature and A4 the atomic mass. It follows from the fourth-moment theorem of the scattering law15) that ds#(O)/dts = -/(3M),

(6)

in which U is the interatomic potential and the average is taken over the classical configuration space. The constant C can be related to (see below), whereas

According Schofieldr6),

to the Langevin-diffusion model modified by Egelstaff the velocity autocorrelation function is given by

2/G(t)= @M*D)s

[ts + (/%b?*D)s]-*.

and

(7)

SELF-MOTION

IN LIQUID

ARGON

517

Here the effective mass M* may be different from the atomic mass. The ratio M*/M indicates the number of atoms involved in the diffusive motion of one atom, (M*/M - 1) being called the dynamic coordination By comparing (7) and (6) it is found that M*/M

= 3&?D)[M(

whereas substitution

PU>]-4,

number-IT).

(8)

of (7) into (5) gives

C = -_BM*D2.

(9)

The effective mass at each temperature is calculated from our experimental data using (9) and compared with that obtained from (8) with the values of according to Boato and coworkerss) (see fig. 6). The agreement is quite satisfactory indicating that the model by Egelstaff and Schofield properly accounts for the short-time behaviour of #(t).

1

O*U

120K

Fig. 6. Data on M*/iW, deduced from our experiment (circles with large error bar), compared with those deduced from (8) with (PU) according to Boato and coworkerss) (triangles with small error bar).

Since the area of 4(t) has to be proportional to D and not, as in the case of (7), to DM*/M, this means that for long times the actual.$(t) must deviate from (7). This deviation may be caused by vibrational modes which are not taken into account in the above model. According to Rahman and coworkers a complete physical description of the self-motion can only be obtained using at least four independent parametersrs). In their stochastic model for yG(t)they introduce damping parameters for the vibrational and diffusive modes, viz. r and 6, a factor which weighs the relative importance of both modes and a parameter characterizing the frequency spectrum of the vibrations.

518

ZANDVELD,

According

ANDRIESSE,

HASMAN

AND

VAN

LOEF

to this modelis)

z/~(t)= (O’/CJJD)~eXp(-@) x

BREGMAN,

{cos(rwt)

-

r

+

3/&@Tdm

~0~eXp(--l’wlt)

sin(wit)).

(10)

Here wi = o( 1 + P)-*, Wn is a Debye frequency relevant for the spectrum of vibrations in the liquid and (W’/WD)~the fraction of modes participating in diffusion. Substitution of (10) into (5) gives D =

(m’/mD)3/(pMc)

(11)

and c = 3/(@&2,j(

1 - d/m,,)- D/C.

(12)

The four-parameter expression (10) can be reduced to one containing three parameters in case the damping of vibrational modes is strong. From the relative width of the inelastic peak for solid (polycrystalline) Ar, shown in fig. 1, which is due to a superposition of phonons in the various crystal directions, we estimate that the damping parameter r is about 0.2. From the shape of the neutron spectrum measured for the liquid as compared to that observed for the solid it is concluded that in liquid Ar the value of I’ is such that ~1 N LO/~,which gives 4(t) == (IB’/OD)~eXp(-[t)

+

3/mLWrdm u2 exp(-&)(l

-

cot),

(lOa)

when terms of the order r-2 are neglected*. In order to be able to express the velocity autocorrelation function in terms of the experimental parameters, we have to estimate Con so that we can derive (O’/Wn)3 and 5 from (11) and (12). Assuming a linear dependence between On and the thermal conductivity (compare e.g. Zimanis)), values for Wn are calculated from the ratio of the thermal conductivity of liquid20) to that of solid Ar, for which the Debye frequency has been recently determinedsi). We find cI)n = 5.5, 4.6 and 3.9 ps-l for T = 85.7, 103.5 and 115.9 K respectively. The values for (O’/CBD)~and 5, which are compatible with these frequencies, are given in fig. 7. It is found that the fraction of modes participating in diffusion rapidly increases with temperature, as it should. The damping of the diffusive modes appears to increase slightly with the temperature, which is somewhat unexpected. According to Sjblander”2) which is the effective damping para5 should be compared with (@'W*D)-l, meter in the modified Langevin-diffusion model; we find that the product t It should be noticed that according to Rahman and coworkersls) the saturation value of r is about I’ = 2; for this value (1 Oa) is a good approximation of (10).

SELF-MOTION

IN LIQUID

519

ARGON

2 ps-’

I

I

I

O60

1

100

I

12OK

LT Fig. 7. Parameters of the stochastic

-0.21 0

0.4

0.0

modeli*) see text.

1.2

pertinent to different temperatures,

1.6

2.0

2.4 ps

-t

Fig. 8. Velocity autocorrelation function of liquid Ar according to the stochastic model with parameters for 85.7 K (full line) 103.5 K (dashed line) and 115.9 K [dotted line), compared to the one calculated by Rahman for 94 K (circles).

520

ZANDVELD,

ANDRIESSE,

&VM*D N 0.4 is independent

BREGMAN,

HASMAN

of the temperature

AND

within

VAN

LOEF

the experimental

error. The temperature dependence of [ therefore is compatible with that of M*/M which is found in using the modified Langevin-diffusion model. Since the decay of M*/M with increasing temperature is much more pronounced than in liquid Nasa), it is concluded that the damping mechanism of diffusive modes should be different in the two cases. In fig. 8 the velocity autocorrelation function (1 Oa) with the parameters shown in fig. 7 is compared with $(t) calculated for liquid Ar at 94 K by Rahmanl). In view of the remarkable resemblance we conclude that the stochastic model gives a satisfactory description of #(t). The “caging effect”, which manifests itself in the negative part of #(t) for 85.7 and 103.5 K, appears to have the correct magnitude. Furthermore it is shown that for increasing temperature this effect gradually disappears and that at 115.9 K, where M* is found to be close to M, a more Langevin-type of diffusive motion is appropriate. In the orthobaric states investigated the densities differ; the disappearance of the “caging effect” is both due to an increase of the kinetic energy and to a decrease of the density. It should be noticed that the correlation function described by the stochastic model initially decays somewhat too fast. It does not seem likely though, that this seriously affects the qualitative conclusions on the temperature dependence of the parameters considered. From neutron-scattering measurements on liquid Ar 4. Conclusions. experimental parameters have been derived which, together with estimates of the Debye frequency, determine the temperature dependence of the velocity autocorrelation function given by the stochastic model. The time integral over this function, which is proportional to the self-diffusion coefficient, is found to increase with the temperature in essential agreement with the results of tracer-diffusion measurements. At the same time the “caging effect” in this function gradually disappears. Furthermore it is concluded that the diffusive modes are damped by a mechanism which depends on the temperature

in a way different

from that in a liquid metal.

APPENDIX

The time-of-flight resolution of a rotating-crystal spectrometer is the result of two contributions, one being due to the velocity spread of the neutron pulse incident on the sample and the other due to the spread in flight path from the reflection point in the rotating crystal to the detection point (see fig. 9). The neutron flux scattered by the sample into the direction #J is

@(+, A’) = @,N-&

s

dil” $

“,“; exp(--bw)

S(Q, UJ) r@“),

(A.1)

SELF-MOTION

IN LIQUID

ARGON

521

DETECTOR

Fig. 9. Scheme of the neutron flight from the crystal to a detector in the rotatingcrystal spectrometer (two characteristic paths are shown).

where ~Z---

1 a

1” -

1’

10

a” + 1’ 10

1; (fljlN)2)

(A.2)

(44.3) In the above formulae r,(A) is a normalized energy-resolution function, N the number of scatterers, g the cross section, S(Q, UJ)the classical scattering law, 10 the average wavelength and $0 the total intensity of the incident neutron pulse, m the neutron mass and ti Planck’s constant divided by 2x. One should notice that the peak width of the function r&“), which is constant on energy scale, rapidly increases on the corresponding wavelength scale; for small 1’ the energy resolution practically becomes a delta function of the wavelength. Due to differences in flight path the intensity #(c#, A) measured at the position of the detectors is a convolution of @(#, A’) and a normalized flight path resolution function rf($, A) $444 Defining

= P’@(9,

A’) rr($+,il -

the Fourier transform

A’).

(A.4

of $(4, A) by

(A.5)

522

ZANDVELD,

ANDRIESSE,

BREGMAN,

HASMAN

AND

VAN

LOEF

we find

b4.6) with

(A.7)

R(#, 2) = h(t) Rr(+>t), Re(t) and Rf($, t) being defined in a similar way as Y($, t), and B(c$, t) =

rdw exp[it{ 1 -ca

x exp(-_bo)(

1-

(1 + a~)-*}]

&zco)-1 S(Q, w).

(A-8)

In (A.8) B(+, t) can be replaced by B(Q, t) if the momentum changes, inherent to a constant 4 experiment, can be neglected. On expanding (1 + a~)-*, (1 - Qaw)-1 and e-bm in powers of w we obtain

W2 + . . .

S(Q,4.

If the width of S(Q, cc)) is small compared to the characteristic with order of magnitude lb - &l-r, one has B(Q, t) =

rdw exp(itQa w) S(Q, w), -00

(A.9 frequency

(A. 10)

which directly gives the intermediate scattering function on a time scale reduced by &a. The experimental information is given by 1z = 256 points, equidistant in 1s). Since n is a power of 2 we can use a fast Fourier-transform program 24) to calculate B(Q, t) on a computer. The computation time for a fast Fourier transformation is proportional to n x slog(n), compared to n x +z for a conventional transformation. In our case this gives a reduction factor of computation time of 32, which enables us to obtain B(Q, t) from the timeof-flight spectra within a few minutes on a IBM 360/65 computer. It follows from expansion (A.9) that for long times B(Q, t) will have an imaginary component, though S(Q, w) is real and symmetric in w. From this a criterion can be derived for the validity of the approximation underlying (A. 10). A full account of this method to analyse neutron time-of-flight spectra will be published elsewheress) .

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523

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

Rahman, A., Phys. Rev. 136 (1964) A405. Verlet, L., Phys. Rev. 159 (1967) 98. Dasannacharya, B. A. and Rao, K. R., Phys. Rev. 137 (1965) A417. Casanova, G. and Levi, A., ch. 8 in: Physics of simple liquids, eds. H. N. V. Temperley and coworkers, North-Holland Publ. Comp. (Amsterdam, 1968). Boato, G., Casanova, G. and Levi, A., J. them. Phys. 37 (1962) 201. Cini-Castagnoli, G. and Ricci, F. P., Nuovo Cimento 15 (1965) 795. Naghizadeh, J. and Rice, S. A., J. them. Phys. 36 (1962) 2710. Andrus, W. S., Muether, H. R. and Palevsky, H., Phys. Letters 26.4 (1968) 152. de Graaf, L. A., Physica 40 (1969) 497. Chen, S. H., Eder, 0. J., Egelstaff, P. A., Haywood, B. C. G. and Webb, F. J., Phys. Letters 19 (1965) 269. Skijld, K. and Larsson, K. E., Phys. Rev. 161 (1967) 102. Bregman, J. D., internal report I.R.I., Delft, 1970. Skold, K., Phys. Rev. Letters 19 (1967) 1023. Eisenstein, A. and Gingrich, N. S., Phys. Rev. 62 (1942) 261. de Gennes, P. G., Physica 25 (1959) 825. Egelstaff, P. A. and Schofield, P. J., Nuclear Sci. Engng. 12 (1962) 260. Bonamy, L. and Galatry, L., Physica 46 (1970) 25 1. Rahman, A., Singwi, K. S. and Sjolander, A., Phys. Rev. 126 (1962) 997. Ziman, J. M., Electrons and phonons, Clarendon Press (Oxford, 1960) 291. Ikenberry, L. D. and Rice, S. A., J. them. Phys. 39 (1963) 1561. Finegold, L. and Phillips, N. E., Phys. Rev. 177 (1969) 1383. Sjblander, A., ch. 7 in: Thermal neutron scattering, ed. P. A. Egelstaff, AC. Press (London, 1965). Cocking, S. J., J. Phys. C 2 (1969) 2047. Brenner, N. M., M.I.T. report AD 657019, Lexington (Mass.) 1967. Bregman, J. D. and de Mul, F. F. M., to be published.