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NEUTRON SCATTERING STUDIES

OF DENSITY FLUCTUATIONS

IN LIQUID ARGON AND NEON

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. J.M. DIRKEN, IN HET OPENBAAR TE VERDEDIGEN TEN OVERSTAAN VAN HET COLLEGE VAN DEKANEN OP 5 SEPTEMBER 1985 TE 16.00 UUR

DOOR

ADRIANUS ANTONIUS VAN WELL

geboren te 's-Hertogenbosch, natuurkundig ingenieur

TR diss

1447

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Dit proefschrift is goedgekeurd door de p r o m o t o r e n : PROF.DR.IR. L.A. DE GRAAF

PROF.DR. J.J. VAN LOEF

"de weg der wetenschap is kommervol" prof. prlwytzkofsky

aan mijn ouders voor sanneke en daan

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5

-CONTENTS PREFACE

References

CHAPTER 1

DENSITY FLUCTUATIONS IN LIQUID ARGON: I . Coherent dynamic structure factor along t h e

1 2 0 - K Isotherm obtained by neutron scattering

Abstract I . INTRODUCTION 11 . DEFINITIONS I I I . EXPERIMENTAL A . Spectrometer B . Samples

1 . Argon and container 2 . Vanadium C. Measurements I V . DATA REDUCTION

A . General

B . Correction for background C . Normalization

D . Correction for multiple scattering and d u t y - c y c l e overlap E . Correction for resolution

F . Conversion to S ( k , u ) at rectangular ( k , G j ) - g n d G . Assessment of data quality

V. RESULTS

A . Dynamic structure factor B . Frequency moments

C. Longitudinal current correlation function VI . SUMMARY AND CONCLUSIONS

Appendix A Appendix B References

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CHAPTER 2 67 DENSITY FLUCTUATIONS IN LIQUID NEON

STUDIED BY NEUTRON SCATTERING

Abstract 6 8 I . INTRODUCTION 69 I I . EXPERIMENT AND DATA REDUCTION 7 3

A . Measurements 7 3 B . Corrections 74 C . Quality checks 7 8 I I I . RESULTS AND DISCUSSION 80

A . Static structure factor 80 B . The second frequency moment a n d quantum effects 8 2

C . Dynamic structure factor, short-wavelength

heat and sound modes 86 D . Generalized hydrodynamics description 9 3

E . Discussion 96 I V . CONCLUSIONS 100

Appendix. The t h r e e - p o l e approximation of S ( k , w ) 101

References 104

CHAPTER 3 107 DENSITY FLUCTUATIONS IN LIQUID ARGON:

1 1 . Coherent dynamic structure factor at large wave numbers

Abstract 1 0 8 I . INTRODUCTION 109 I I . EXPERIMENT 111

A . Measurements 111 B . Data reduction 112 III . DYNAMIC STRUCTURE FACTOR 117

IV. COMPARISON WITH THEORY 125 A . Classical system 125 B . Quantum system 130 V. SUMMARY AND CONCLUSIONS 132

Appendix A. Details of some corrections 134 Appendix B. Large-k behavior of S ( k , w )

for a classical system 136

References 139

SAMENVATTING (Summary In Dutch) 141

CURRICULUM VITAE 144

7

-PREFACE

This thesis consists of three publications describing inelastic neutron scattering experiments on liquid argon and n e o n . From neutron scattering data information can be obtained about the microscopic dynamical behaviour of a liquid. Here, the word microscopic is meant for processes that take place on a time scale comparable with the time between two collisions experienced by one particle (in our case an argon or a neon atom) , which is of the order of 1 ps = 10 s, and on a length scale comparable with the diameter of a p a r ­ ticle, which is of the order of 1 nm - 10 m. It is impossible a n d , moreover, useless to follow in detail the motions and positions of each of the 10 particles present in 1 cm of liquid. The appropriate "tools" used to describe the microscopic dynamical behaviour of a fluid are time-dependent correlation functions. The correlation function of special interest in this thesis is the d e n s i t y - d e n s i t y correlation function G ( r , t ) , also called the Van Hove correlation function , which is directly connected with the cross section for inelastic neutron and light scattering, and therefore experimen­ tally accessible.

For a classical system in thermodynamic equilibrium at temperature T, c o n ­ sisting of N particles in a volume V, G ( r , t ) is defined by

G ( r , t ) = - < n t O . 0 ) n ( r , t ) > , n where N n ( r , t ) = Z 6 [ r - R . ( t ) ] j=i J

is the microscopic number d e n s i t y , R. {t) the position of particle j at time t, the brackets denote an ensemble average, and n = < n ( r , t ) > = N/V is the mean number d e n s i t y . G ( r , t ) is proportional to the probability of finding a particle at position r at time t, given there was a particle in the origin at

time t - 0 . An alternative way of interpreting this correlation function is the following. The microscopic density fluctuations around the mean number density n , which are present as a result of the thermal motion of the p a r t i ­

cles, may be unraveled into their spectral components by making a Fourier analysis. To this end the Fourier transform of G (r, t) , the intermediate

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8

-F ( k , t ) » [ e 'k"rG ( r , t ) dr , V

which describes the decay in a time t of a plane-wave like disturbance, of the microscopic number density , with wavelength X = 2 n / k .

The double differential cross section for neutron scattering is related to the dynamic structure factor S ( k , w ) , the frequency spectrum of F ( k , t ) ,

S ( k , w ) = [ e ~I U tF < k , t ) d t .

For an isotropic system, which will be considered h e r e , F ( k , t ) and S(k,C*>) depend only on the magnitude k of k and nol on its direction.

T h u s , in an inelastic neutron scattering experiment density fluctuations are probed in the double Fourier s p a c e , as is the case in inelastic light scattering. The main difference between the two techniques is that the k values probed by neutrons are of the order of 1 - 100 nm"1, representing wavelengths of 0 . 1 < X < 10 nm, whereas in light scattering k and X are three

orders of magnitude smaller and larger, respectively. Whereas in the case of light scattering density fluctuations in a liquid are studied with wavelengths that cover a few thousand particles (the hydrodynamic regime), in neutron scattering the probed wavelength covers only a few particles, or even less than one particle (the kinetic regime) . Besides light and neutron scattering there is a third, pseudo-experimental, technique to study the microscopic dynamical behaviour of a fluid, v i z . , computer molecular dynamics simulations. In this technique, made possible by the advent of large and fast computers, the track of each particle in a system of a few hundred is calcu­ lated . The resulting correlation functions are usually presented in time space rather than in frequency space .

Although in the past decades many elaborate studies have been devoted to ( 2 )

the microscopic dynamical behaviour of liquids , the dynamics of even sim­ ple liquids, such as liquefied noble gases, are still not fully understood. The reason for this is twofold. (i) Due to the relatively low neutron inten­ sity avialable at the present neutron sources, it is very cumbersome and time consuming to obtain accurate neutron scattering d a t a , which form the basis of information about the microscopic dynamical behaviour. Moreover, in order to extract this information it is of vital importance that the necessary

correc 9 correc

-tions to the experimental data are performed adequately, implying that the used correction procedures should be of very high quality. For this reason a substantial part of this thesis is dedicated to the careful reduction and analysis of the experimental d a t a . (ii) Of the three best known states of aggregation of a many body system, the description of the liquid state is inherently the most complicated. This is due to the facts that in the liquid, in contrast to the g a s , the particles are close together and almost continu­ ously interacting with each other, a n d , in contrast to the solid, the p a r t i ­ cles do not exhibit a long-range order.

As to the experimental point, we will follow the methods described in more detail by Verkerk . In Chapter 1 account is given of a neutron scattering experiment on liquid argon of high statistical accuracy, performed at the High Flux Reactor of the Institut Laue-Langevin in Grenoble, and of the c a r e ­ ful corrections that were applied. The final data and their estimated u n c e r ­ tainties prove to be reliable from three independent consistency c h e c k s . The

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physical interpretation of these d a t a , partly discussed elsewhere , will be extended in Chapter 2 .

As to the t h e o r y , we will concentrate on two a s p e c t s . First, we will interpret the decay of density fluctuations in terms of the decay of the most important eigenmodes in the liquid in the spirit of the theoretical work of

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De Schepper and Cohen . They calculated the eigenmodes of a h a r d - s p h e r e system, based on a revised Enskog t h e o r y . The three most important eigenmodes for the density fluctuations c a n , in the hydrodynamic regime, be identified with one heat and two sound modes. It appeared that for liquid argon this three-modes description is valid far into the kinetic regime, but the b e h a ­ viour of these "extended" heat and sound modes differ considerably from the

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hydrodynamic behaviour . In Chapter 2 we recur to this interpretation extensively and make use of it to analyze the experimental results of liquid n e o n . The description in terms of eigenmodes proves to be very successful and the parameters characterizing these modes can be determined rather a c c u r a t e ­ l y . The description of the dynamical behaviour in terms of wavelength-dependent transport coefficients, however, appears to be less satisfactory. Having results both of liquid neon and argon at corresponding thermodynamic states makes it possible, for the first time, to compare the microscopic dynamical behaviour of two liquids In detail. Except for some

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1 0

-subtle d i f f e r e n c e s , the overall agreement between liquid argon and neon is good. With regard to the second theoretical a s p e c t , if wavelengths of density fluctuations are considered which are small compared to the interparticle d i s t a n c e s , the dynamical behaviour is dominated by the free streaming of the particles. The accuracy of the experimental large-k results for liquid argon, described in Chapter 3 , makes it possible to confront the experimental data with theories, that describe the transition to the free streaming behaviour. This is done in three w a y s . The results are compared with a theory for a classical system consisting of particles interacting via a smooth i n t e r p a r t i ­ cle potential, the same for a quantum system, and with a theory for a classi­ cal h a r d - s p h e r e system.

In conclusion, we like to remark t h a t , on one h a n d , the neutron scattering data of modest statistical a c c u r a c y , as obtained from the 2-MW Hoger Onderwijs Reactor at the Interuniversitair Reactor Instituut in Delft, have shown to reveal remarkably valuable information on the microscopic dynamical behaviour of liquids. On the other h a n d , the more accurate r e s u l t s , originat­ ing from more intense neutron sources, make it possible to confirm a n d s u b ­ stantiate these first r e s u l t s .

References

( 1 ) L . v a n H o v e , P h y s . R e v 9 3 , 2 4 9 ( 1 9 5 4 ) .

( 2 ) s e e , e . g . , J . P . Boon a n d S . Y i p , "Molecular Hydrodynamics" (McGraw-Hill, New York, I 9 6 0 ) ; J . P . H a n s e n and I.McDonald, "Theory of Simple Liquids" (Academic P r e s s , London, 1 9 7 6 ) .

( 3 ) P . V e r k e r k , P h . D. T h e s i s , University of Technology, Delft ( 1 9 8 5 ) .

( 4 ) I . M . d e S c h e p p e r , P . V e r k e r k , A . A . v a n Well, and L . A . d e Graaf, P h y 5 . R e v . Lett. 5 0 , 9 7 4 ( 1 9 8 3 ) ; I . M . d e S c h e p p e r , E . G . D . C o h e n , and M . J . Z u i l h o f , P h y s . L e t t . A 1 0 1 , 3 9 9 0 9 8 4 ) ; M . J . Z u i l h o f , E . G . D . C o h e n , and I . M . d e S c h e p p e r , P h y s . L e t t . A 1 0 3 , 1 2 0 ( 1 9 8 4 ) ; I . M . d e Schepper, P . V e r k e r k , A . A . v a n Well, and L . A . d e Graaf, P h y s . L e t t . A 1 0 4 , 2 9 ( 1 9 8 4 ) . ( 5 ) I . M . d e Schepper and E . G . D . C o h e n , J . S t a t . Phys . 2 7 , 2 2 3 ( 1 9 8 2 ) ;

E . G . D . C o h e n , I . M . d e . S c h e p p e r , and M . J . Z u i l h o f , Physica B+C 1 2 7 , 2 8 2 ( 1 9 8 4 ) .

CHAPTER 1

DENSITY FLUCTUATIONS IN LIQUID ARGON: I . Coherent dynamic structure factor along the 1 2 0 - K isotherm

obtained b y neutron scattering

A . A . van Well, P . Verkerk, and L . A . de Graaf Interuniversitair Reactor Instituut, 2 6 2 9 JB Delft, The Netherlands

J . - B . Suck

Institut Laue-Langevin 156X Centre de Tri, 3 8 0 4 2 Grenoble Cédex, France

J . R . D . Copley

Institut Laue-Langevin 156X Centre de Tri, 3 8 0 4 2 Grenoble Cédex, France and Department of Physics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

P h y s . R e v . A 3 1 , 3391 ( 1 9 8 5 )

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12

-Abstract

Coherent dynamic structure factors S ( k , to) obtained by means of thermal neutron inelastic scattering are presented. The experiments were performed on

liquid Ar at four densities along the 120-K isotherm covering a range of

wave numbers k from 4 . 2 to 3 9 . 0 nm . The neutron time—of—flight spectra are corrected for all known experimental effects with an improved data-reduction system. Special attention is paid to corrections for multiple scattering, d u t y - c y c l e overlap, and instrumental resolution. The importance of various correction steps is shown. The reliability of the corrected data is assessed by means of two independent consistency c h e c k s , v i z . , the detailed-balance condition and the first frequency moment of S ( k , t o ) . The 5 { k , w ) data are presented both as a function of k at fixed u and as a function of to at fixed k . The peak height and full width at half maximum of S ( k , to) at fixed k are shown for all densities together with the srnall-k (hydrodynamic) and large-k (free gas) asymptotes. The frequency moments of S ( k , t o ) , evaluated up to the fourth moment, are consistent with results from computer simulations and from theoretical calculations. The longitudinal current correlation function C (k.to) , derived from the experimental S ( k , t o ) , is examined and both its peak position (yielding the dispersion curve for longitudinal current fluctua­ tions) and its peak height (a measure of the life time of the fluctuations) are discussed.

13

-I . -INTRODUCT-ION

Thermal fluctuations in a fluid in equilibrium can be studied over dif­ ferent frequency and wavelength scales using a variety of techniques. In the

regime of interest in statistical physics wavelengths are of the order of

interatomic distances I and frequencies are of the order of the inverse of the time between collisions T. Recent theoretical work in this field ( s e e , e . g . , Refs. 1 and 2 ) has largely focused on the dynamic structure factor S( k . to) (the double Fourier transform of the d e n s i t y - d e n s i t y correlation f u n c ­ tion) , since it is a fundamental function which can be obtained experimental­ ly . For dense gases and liquids, where t - 0 . 1 nm and T - 1 p s , the only experimental tool to obtain S( k, to) is inelastic thermal neutron scattering ( I N S ) , since thermal neutron wavelengths are of order £ and energies are of order fi/f. Refs . 3 - 7 review INS experiments on simple ( i . e . , monatomic) fluids. Since 1980 the following INS studies of classical simple fluids have been reported. Söderström et a l . measured liquid lead at 6 2 3 and 1 1 7 3 K, covering wavenumber ranges 4 < k < 6 8 nm and 15 < k < 76 nm , r e s p e c t i v e

-( 9 )

l y . Dahlborg and Olsson studied liquid bismuth at 5 8 0 K in the k range from 5 to 16 nm . Groome et a l . determined S ( k , t o ) for xenon gas at five thermodynamic states along the 3 0 3 - K isotherm, while Egelstaff et a l . examined krypton at 12 dense gas s t a t e s , ranging from the critical to 0 . 8 times the triple point d e n s i t y , along the 2 9 7 - K isotherm.

Due to the higher neutron fluxes that have become available at new research reactors, and to recent improvements to data-correction p r o c e d u r e s , the quality of INS data has steadily improved starting with the well-known work of Skbld et a l . on liquid argon near its triple point. These improvements have stimulated theoretical physicists to develop new theories. Also, the advent of large fast computers has made accurate molecular-dynamics simulations possible. These three approaches and in particular their mutual interactions have contributed to our present knowledge and understanding of the fluid s t a t e ,

In order to test current theories, and more specifically to test theories of the temperature and density d e p e n d e n c e of S ( k , t o ) , precise experimental data on fluids in a wide range of the phase diagram are indispensable. Using the 2MW reactor at Delft, Verkerk demonstrated the feasability of s t u d y

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-J P K

ing the density dependence of S ( k , t o ) in liquid Ar along the 120-K isotherm. These results motivated us to repeat part of the measurements at the High Flux Reactor of the Institut Laue-Langevin in Grenoble in order to obtain higher statistical accuracy, simultaneously extending the covered ( k , u ) region to higher k values. In this paper we report the results of the latter measurements and postpone the physical interpretation and comparison with theory to a forthcoming p a p e r .

After giving the relevant formulae in Section II, the experimental details are discussed in Section III. In Section IV great emphasis is given to the data-reduction procedure, which is crucial in connection with high-accuracy INS experiments. In Section V the final results are presented, and Section VI contains a summary and some concluding remarks.

II. DEFINITIONS

The dynamic structure factor S ( k , u ) represents the frequency spectrum of density fluctuations with wavelength X = 2 n / k , and is the Fourier transform of the intermediate scattering function F ( k , t ) :

i (^ - i u t S ( k , w ) = — dt e F ( k , t ) ,

2TT J

- , , ,v 1 2 - i k - r . ( O ) ik-r.,(t) F ( k , t ) = — h < e j e i > .

The brackets denote an equilibrium ensemble average at temperature T and number density n = N / V , with M the number of particles and V the volume of the system; r . f t) is the Heisenberg position operator of particle j at time t . The quantity measured in an INS experiment, using the lime-of-flight (TOF) technique, is the double differential cross section d o/dfidX,, which i s , in the first Born approximation, related to S ( k , u ) by

d2o nfto, X

= B - J L S ( kfw ) . ( 2 )

dfidX m x4

f Af

with dQ the solid angle into which the neutron Is scattered, X and X the

15

-wavelengths of the incident and scattered neutrons (Xf is proportional to the TOF of the neutron from the sample to the detector) , ft Planck's constant divided by 2TT , m the mass of the neutron, a, the bound atom cross section; k and u> are the momentum and energy transfers in units of ft:

k = k - kf l u = (E - E , ) / f i . ( 3 )

o f o f

and k. = lk.1 = 2TT/X. . i i l

S ( k , u ) satisfies the detailed balance condition

SflU

S(k,G>) - e S(k,-G)) , ( 4 )

with 3 = l / knT , kn being Boltzmann's constant. t) ü

Since liquid argon can in first approximation be considered a classical system, we will present most of our results in the form of the symmetrized dynamic structure factor

S ( k , u ) = exp T - Yz&ftu + - k - 1 S ( k , w ) , ( 5 )

with M the mass of one particle of the system. Eq. ( 5 ) gives a quasi-classical

( 1 4 )

approximation of S ( k , o j ) that is exact for an ideal g a s .

We will also consider the longitudinal current correlation function C _ ( k , t ) , defined by

Cf(k,t) = - i I - F(k,t) , (6)

1 k2 d t2 and its frequency spectrum

1 f05 - iw t i i2

C . ( k , u ) = — dt e C0( k , t ) = Ü - S ( k , u ) . ( 7 )

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16 -2, e r Nk 1 N C. . . ( k . t ) = —j E < (v.(O)-k) (v.,(t)-k) x j . j ' = l J J exp [ - i k - ( r . ( 0 ) - r . , U ) ) ]> „ J J c£ ( 8 )

where v.(t) and r.(t) are the velocity and position of particle j at time t,

C (k,ui) = — S ( k , u ) . ( 9 ) k2

III. EXPERIMENTAL

A. Spectrometer

Time of flight spectra were obtained with the IN4 spectrometer at the Institut Laue-Langevin which is at present the most suitable instrument for accurate S C k, oj) measurements on fluids because of the combination of a high neutron flux on the sample, a long flight path, and a large detector area. It was operated v/ith two phased rotating pyrolytic-graphite crystals as mono-chromator. The incident wavelength, determined in a separate run with two low efficiency detectors 4.873 m apart, was 0.2544(2) nm [corresponding to a neu­ tron energy of 12.635(20) meV] . 234 detector tubes of 2 . 5 cm diameter, 30 cm active length, filled with 0 . 4 MPa (4 bar) 3He, were arranged in 57 groups

by combining 1,2,3 or 6 detectors. The k resolution [full width at half maxi­ mum, FWHM] for elastic scattering, which arises from the angular resolution due to the finite size of the detectors and of the sample, is given as a function of k in Fig. 1 ( a ) . The detector arrangement was chosen to obtain the best angular resolution for k values around the peak of S(k) (at k : 20 nm-1)

where changes in S(k,cj) as a function of k are expected to be most pro­ nounced. Note that the finite TOF resolution discussed below gives an extra contribution to the k resolution (of the order of a few tenths of 1 nm-1) .

The detectors, fixed at a distance of 4.003 m from the sample, covered an

17 -TE, . 0

<

0.5 1.5 in CL "a i.o

<

0.5 0 1 (a)

v—

1 1 (b) i i I

-...

-2 0 30 4 0 k/nm

F i g . 1 . (a) The calculated experimental wave-vector resolution Ak (FWHM) (b) the measured frequency resolution Au (FWHM), both for elastically scattered neutrons.

angular range from 9 to 106 degrees, and were placed in such a way as to avoid Bragg scattering from the aluminium container. Spectra were recorded in 512 time channels 8 us wide. The region covered in the (k.u)-piane is indi­ cated in Fig.2; each line represents one of the 57 scattering angles. The TOF system was triggered by a pulse from the monochromator twice per revolution. The time between two pulses, defining the period of one duty cycle, was 4170 us. Two boron chambers in the monochromatic beam were used as monitors. The intensity of monitor 1 , placed in front of the sample, is proportional to the incoming neutron flux, was used for normalization. Monitor 2 was placed behind the sample.

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The TOF resolution (for u=0) of the spectrometer was determined from the elastic scattering of a vanadium sample (see Sees. III. B. 2 and IV.C). The relative TOF resolution measured at the detectors (FWHM), At/t, varied from 2 . 7 % to 3 . 8 %. This resulted in an absolute frequency resolution. Au, as shown in Fig . 1 ( b ) . 0.2 X/nm 0.3 0,4 CH NR 400

Fig. 2 . The kinematic region for the experiment. TOF channel numbers, scattered neutron wavelengths, and frequency transfers, are indicated. Crosses represent the aluminium Bragg peaks. The dashed and dashed-dotted lines are the sound "dispersion" curves, (j =c k, for

meas-r s s

Urements a and d respectively.

19

-B. Samples 1 . Argon and container

3 6Ar is a purely coherent scatterer with a bound atom cross section ob =

77.85(40) b< 1 5 ) and an absorption cross section a^ = 7( 1 ) b for 12 .6 meV neu­ t r o n s '1 6' . The sample used was composed of 99.6 mol% MAr and impurities of

,0Ar H , N , and O . The container was made of a capillary of 5052 aluminium

with inner diameter 0.74 mm and wail thickness 0.28 mm. The 5 m long capil­ lary was bent back and forth into 30 parallel tubes, with their center lines

1 .3 mm apart, mounted in an aluminium frame covered with cadmium. A descrip­ tion of a similar container is given in Ref.17. The calculated transmissions of the samples are listed in Table I. The plane through the container tubes made an angle of 143° with respect to the incoming beam (see Fig. 3) . In this way all 30 tubes were exposed to the incoming beam and could be seen by all detectors. The container, mounted in a liquid-^ cryostat with 1 mm thick Al windows, was surrounded by a 0 . 2 mm thick Al heat radiation shield. This shield was kept at about the same temperature as the sample, reducing the temperature gradient across the sample to less than 5 ItlK. The sample tempera­ ture was measured with a calibrated platinum resistor placed within the alu­ minium frame of the container. The pressure of the sample was measured with a pressure transducer, connected to the filling line of the container.

Cd - covered Al- frame

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20

2. Vanadium

In order to calibrate the spectrometer, i . e . , to determine the relative efficiencies of the detector groups and to measure the TOF resolution, we determined the elastic part of the scattering from a vanadium plate at room temperature. The elastic fraction is described by the Debije-Waller factor exp(2ak ) where a = 3 3 . 5 x 10 nm at room temperature . The bound-atom cross sections for incoherent and coherent scattering are o. - 4.97{5) b, a

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- 0 . 0 2 9 ( 2 ) b , respectively, and the absorption cross section a = 7 . 1 9 ( 4 ) b for 12.6-meV neutrons . The number density at room temperature is n = 7 0 . 5 nm . The 2.1-mm-thick vanadium plate was covered with a cadmium mask of 41 x 90 mm , the same dimensions as that part of the argon sample irradiated by the neutrons and seen by the detectors, and was also placed at an angle of 143° with respect to the incoming beam. In order to obtain reli­ able data for absolute normalization and for the TOF resolution, it is impor­ tant to use the same geometry for vanadium and for the argon sample: increas­ ing the height of the mask from 41 mm to 61 mm the TOF resolution increased 5% for the smaller scattering angles to 10% for the larger angles. The vana­ dium sample was placed in a thin-walled AI box filled with argon gas to avoid the more intense air scattering.

Table I. Experimental conditions of the measurements

measurement a b c d e f g description 3 SAr, 120K, 2 36Ar, 120K, 11 3 6Ar, 1 2 0 K . 2 7 3 6Ar, 1 2 0 K . 4 0 empty container vanadium plate V-background MPa 5 MPa MPa MPa , 120K 300K T r ' > 0 . 9 1 6 0 . 9 1 2 C . 9 0 S 0 . 9 0 6 0 . 9 8 9 0 . 7 4 2 1 . 0 0 0

s

3

>

0 . 0 7 2 0 . 0 7 5 0 . 0 7 9 0 . 0 8 1 0 . 1 0 5 .. 3) time ( h ) 2 6 . 8 2 4 . 0 31 . 3 2 5 . 4 2 4 . 7 9 . 8 3 . 5 number of runs 6 6 8 7 7 4 1 f<> 4 . 7 0 2 0 0 0 0 3 0 0 0 4 0

i) Tr : total transmission (calculated)

2) 5 : fraction scattered by the sample (calculated) 3) time : total measuring time

4) f : percentage of rejected data on the basis of statistical checks ( Sec. III. C)

21

-C. Measurements

Experiments on Ar were performed at four densities in the liquid range along the 120-K isotherm, with successive density increments of approximately 5% (denoted as measurements a - d, see Table I) . A measurement with empty container (measurement e) determined the background scattering. The normali­ zation and the resolution measurement were combined using a vanadium plate as scattering sample (measurement f) . An "empty spectrometer" measurement (meas­ urement g) served to determine the background in the vanadium measurement.

The experimental conditions of of the argon samples are listed in Table II. AT . and Ap . reflect variations in the measured T and p durind the

rel 'rel p 5

experiment. The number density n, An ,, and An . are determined from T and

r rel abs

p and their uncertainties (from Ref .21 ) .

Each measurement was divided into several runs. The total measuring time and number of runs are listed in Table I (30 h measuring time of the argon sample results in an intensity, integrated over the 57 different spectra, of 3 x 10 counts) . The statistical consistency of these runs was checked. during the course of the experiment as described in Ref. 22, enabling us to detect malfunctioning of some detectors ( e . g . , increased electronic noise) at an early stage. The runs were summed and stored on magnetic tape together with their variances.

Table II. Thermodynamic conditions of argon

measureme a b c d nt T ( K ) 1 1 9 1 1 9 1 1 9 1 2 0 . 9 8 . 9 6 . 9 6 . 0 3

ATrei'

(K) 0 . 0 4 0 . 0 4 0 . 0 7 0 . 0 3 P (MPa) 2 . 0 1 1 1 . 4 9 2 6 . 9 8 3 9 . 4 4 APrel' (MPa) 0 . 0 4 0 . 0 3 0 . 0 6 0 . 0 6 n3 > (nrrr3) 1 7 . 6 0 1 8 . 5 1 1 9 . 5 1 2 0 . 1 1 A nr e l 3 ) ( n m -3) 0 . 0 0 7 0 . 0 0 5 0 . 0 0 6 0 . 0 0 3 A na b s 3 ) ( n m- 3) 0 . 0 7 0 . 0 6 0 . 0 4 0 . 0 4 1) absolute uncertainty AT , - 0 . 5 K 2) absolute uncertainty Ap . = 0 . 1 MPa

3) from Ref.21 (n at the critical and triple points are 8 . 0 7 7 and 2 1 . 3 4 3 nm , respectively)

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22

-IV. DATA REDUCTION

A . General

In order to extract S ( k , u) data from the experimental TOF spectra, the latter must be corrected for background ( i . e . , time-independent background and container s c a t t e r i n g ) , multiple scattering, d u t y - c y c l e overlap, detector efficiency, self-shielding, and experimental TOF resolution. The spectra should also be normalized absolutely, and it is convenient to convert the derived cross sections d a/dSldX to S ( k , u ) on a rectangular ( k , u > ) - g r i d . The large amount of experimental data with high statistical accuracy motivated us to devise a new system of computer programs to perform the corrections and conversions described a b o v e . A detailed description is given in Ref. 2 3 .

Fig. 4 shows a flow diagram of the corrections and conversions which were applied. The correction for multiple scattering and d u t y - c y c l e overlap was done in an iterative fashion.

We emphasize that all corrections are described in detail in Ref. 23 and in the next sections we shall only discuss those correction steps that reveal relevant information about the present experiment, illustrating them with results from measurement b .

As a first step all measurements were normalized to the same unit of incoming neutron flux, corresponding to a measuring time of about 8 h .

B . Correction for background

A first correction for the time-independent background, due to electronic noise and fast n e u t r o n s , was applied by subtracting a constant level from each spectrum. This level was determined from the 2 5 consecutive channels

with lowest intensity and was on average (in the order the measurements were

made) 5 . 7 , 5 . 5 , 2 . 1 , 2 . 0 , 0 . 7 , 1 . 2 , and 0 . 4 counts per detector tube per

TOF channel per 8 hours for measurements d, c, a, b , e, f, and g, respective­

l y . After the first two measurements we improved the shielding around the detectors, yielding a lower background intensity for the remaining measure­ ments . If there is negligible " d u t y - c y c l e overlap" (meaning that a neutron loses sufficient energy to be detected one or more d u t y - c y c l e s l a t e r ) , this constant level will indeed be a good estimate of the time-independent b a c k ­ ground. If, however, d u t y - c y c l e overlap is not negligible the subtracted

23 model "N --. S(k,oj) J ^ i f experimental \ \\xx\. spectray1 \ correction for background « normalisation -d2a-/dIidX

w

correction for multiple scattering and duty - cycle overlap

w

correction for detector efficiency \\ correction for resolution

w

correction for self shielding « conversion d2a-/dftd\~S(k,üj) \\ interpolation to rectangular(k,üj)-grid ( " I

I "'

'

M J

F f g . 4 . Flow diagram of the data reduction. The correction for multiple

scattering and d u t y - c y c l e overlap is applied iteratively.

level will contain a contribution from the overlap, and the time-independent background will be «determined during the correction for multiple scattering

( S e c . I V . D ) .

In Fig. 5 spectra measured at five representative scattering angles are shown for measurement b together with the corresponding empty-container

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24

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-In Fig. 6 vanadium spectra at the same detector angles and the corresponding background are displayed.

The argon (vanadium) spectra were next corrected for container (spectrome­ ter) scattering, taking into account the (calculated) attenuation of this

, (23) scattering by the sample

C. Normalization

The argon spectra were normalized absolutely using the elastic part of the vanadium spectra. The vanadium TOF spectra were corrected for the wavelength dependence of the detector efficiency and then for inelastic scattering, approximated by the double differential cross section for incoherent one-phonon scattering*2 4'. The normalized vibrational density of states Z(uK

needed in the latter correction, was determined as follows. For w > 7 . 5 ps Z(U) was determined from the TOF spectra at the 10 largest scattering angles of the vanadium measurement and for o < 7 . 5 ps"1 Z(u) was approximated by the Debije model. Demanding Z(u) to be continuous at (o = 7 . 5 ps yields &D = 380

K. This value agrees very well with values in the literature [Ref.18, GD = 360(30) Ki Ref.25, 0D = 390 K] .

The argon TOF spectra S,, were normalized to d2o/dfidX using the relation­

ship

(10)

Fig. 5 . Time-of-flight spectra at five representative scattering angles f from measurement b with corresponding empty-container intensity, indi­ cated by a continuous line, both after correction for time-independent background.

F i g . 6 . Vanadium time-of-flight spectra (error bars) and vanadium back­ ground (continuous line), at the same scattering angles as in Fig.5^ both after correction for time-independent background. At tp = 9 6 . 7 5 the background spectrum and part of the vanadium spectrum is enlarged, showing the inelastic part of the vanadium scattering.

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26

-where F. = n A o S ( k . ) / ( n A 4TIA\ ) , i and i are indices for the scattering

i v v v v i a a J

anöle and TOF channel respectively, n (n ) is the number density of vanadium

t ' v a

(argon) , A (A ) is the illuminated volume of vanadium (aréon) seen by the s v a » i detector, o is the incoherent bound atom cross section of vanadium, AX is

v

the TOF channel width in wavelength units, and S (k.) is the elastic strtic-V I

ture factor of vanadium, including the Debije-Waller factor, multiple scattering, and self shielding, calculated according to Copley et a l . [S (k.) ranged from 0 . 7 4 to 0 . 8 5 ] . The elastic parts of the vanadium TOF

V ' e£ r. &%

spectra, V,'., were divided by XT V 7 . , for later use in the correction for TOF IJ J U

resolution.

We estimate the absolute normalization of the final Ar results to be a c c u ­ rate within a few percent. largely due to inaccuracies in A and A. . These stem from the 3-4% tolerance on the inner diameter of the container tubes and from imperfect placements of the cadmium shielding on the aluminium frame of the sample container and of the cadmium window on the vanadium.

D . Correction for multiple s c a t t e r i n g a n d d u t y - c y c l e overlap

In order to calculate the contribution of multiple scattering and d u t y - c y c l e overlap to the experimental differentia! cross sections, the experiment was simulated on the computer using the Monte Carlo program MSCAT

( 2 7 )

described by Copley . The sample scattering was described by a model S ( k , u O and for the container both coherent and incoherent elastic scattering were simulated. Since the real S ( k , U)) is not known until after the complete data reduction, the correction procedure was applied iteratively as indicated in Fig. 4 .

In each simulation run the histories of 4 0 0 0 neutrons were followed and the scattering intensities at 16 scattering angles ranging from 9° to 106° were calculated for 4 3 TOF channels with flight times ranging from 1.1 to 8 . 0 ms (the flight time for elastic scattering was 2 . 5 7 tns) . Seven of the c h a n ­ nels were chosen at flight times long enough to fall into the next d u t y - c y c l e , in order to determine the d u t y - c y c l e - o v e r l a p contribution.

( 2 8 )

Using a revised version of MSCAT it was possible to distinguish the following types of scattering: single scattering by the argon sample (denoted by s) and by the container (c) , multiple scattering ( i . e . , double, triple, e t c . , scattering) in the sample only (ss) and in the container only (cc) , and

multiple scattering with at least one scattering event in the sample and one in the container, the last collision before detection being in the sample

(cs) or in the container (sc) .

We have assumed that the experimental TOF spectra were corrected for c-scattering and for cc-sca tiering in the correction for background described a b o v e . They must stil! be corrected for s s - , s c - , and c s - s c a t t e r i n g . Even

( 2 9 )

using the improved method of calculation' "" , the simulation by MSCAT of the detector response to coherent elastic scattering is inefficient and with 4 0 0 0 neutrons the statistical accuracy of sc was very poor. Therefore the assump­ tion sc^cs was made and 2cs (rather than cs+sc) was used for the purpose or correction. In a separate run with MSCAT in which an incoherently scattering container was simulated sc equaled cs within 10%. From this test and the fact that cs + sc constitute only a minor fraction of the total scattering (see Fig. 7) cs + sc - 2cs seems to be an acceptable approximation.

The simulated intensities were converted to normalized differential cross sections d a/dQd\ and multiplied by the proper detector efficiency E ( X ) . Examples of simulated spectra at two scattering angles (measurement b , last iteration step) are shown in Fig. 7 . Apparently the multiple scattering is fairly isotropic, but because S ( k ) is small at the smaller scattering angles its contribution is relatively more important in this r a n g e . The simulated cross sections were interpolated from the scattering angles and TOF channels used in the MSCAT simulation to the experimental o n e s . At this stage the "total" simulated intensity, t . . , can be compared with the normalized e x p e r ­ imental differential cross sections (d ö / d f ï d X ) . . . Here

ij

.sim , sim sim - sim, , , , ,

t . . - e. ( s . . + s s . . + 2 c s . . ) 4 e . o.. , ( 1 1 ) ij J U iJ ij J-™ ij with s i m sim „ sim o.. - s. . + s s . . + 2c5. . ij i , j+m ï , j+m i , j+m

where i and j are the experimental detector and channel number respectively, m - T / A t , At being the experimental TOF channel width and T the period of one d u t y - c y c l e ; e . o.. is the d u t y - c y c l e overlap contribution. In Fig. 8 the nor­ malized experimental differential cross sections of measurement b are shown together with ts l of the last iteration s t e p , the contribution of the multi­ ple scattering and overlap, t . . - e . s . . , being represented by the dashed

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28

line. The experimental and simulated spectra agree very well, meaning t h a t , since the input model for MSCAT is assumed to be correct in this c a s e , both the simulation and the interpolation procedure function

satisfactorily-In situations where the intensity due to d u t y - c y c l e overlap is not negli­ gible a part of it was inevitably subtracted during the correction for b a c k ­ ground (Sec .IV.B) • Using the simulation d a t a , the time-independent background was redetermined, as described in Ref ■ 2 3 , such that subtraction of the multi­ ple scattering and d u t y - c y c l e overlap would lead to a zero mean of the first 50 TOF channels. These channels correspond to energy transfers Fin with Id) I > 60 ps . At these energies the contribution of single scattering is assumed to be negligible. For examples see R e f . 2 3 .

<3 b io-*= 9°

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Fig. 7 . Simulated spectra (by means of MSCAT) at two scattering angles f

for measurement b (last iteration step) . The arrow indicates the incom­ ing wavelength and the shaded vertical bars the experimental dead time of the TOF analyzer. Open circles. s-scattering; triangles, s s - s c a t t e r i n g ; squares, cs-scattering; solid circles, all overlap scattering (s + ss + 2 c s ) ; dashed line, all multiple scattering plus d u t y - c y c l e overlap. All intensities are multiplied w' t h the proper detector efficiency. [ 1 attometer (am) = 10 m; 1 am/sr = l b / A s r ] .

F i g . a . Normalized double diffe­

rential cross-sections of measu­ rement b (error bars) at the same scattering angles as in Fig. 5 , simulated spectra ( b y means of MSCAT, last iteration s t e p ) , i n ­ cluding single scattering, multi­ ple scattering and d u t y - c y c l e overlap (continuous line) , and simulated multiple scattering a n d d u t y - c y c l e overlap ( d a s h e d l i n e ) . The arrow indicates the position with zero energy transfer.

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Since the correction for multiple scattering and d u t y - c y c l e overlap is sensitive to the choice of the model S ( k , u ) used in MSCAT, it was performed iteratively as indicated in Fig. 4 . Convergence was reached within three to four iteration s t e p s . The model used in the first step had no adjustable parameters and for the correction the "subtraction method" was used

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30

[subtracting f x (if!" - E j s * ™ ) , where f is a factor, close to 1, d e t e r ­ mined by comparing the averaged norm of the experimental data and of the simulated data ] . For the subsequent steps a model with adjustable parame­ ters, determined from the S ( k , u) -results of the former s t e p , was taken and the correction was performed with the "factor method" (multiplying the e x p e r ­ imental spectra by t, sf^/tf™) . Details of the different models are given in Appendix ft. In order to provide some insight into the influence of the model and of the correction method (subtraction or factor) the results of the first and the last iteration step for measurement b are discussed in Appendix A .

E . Correction for resolution

After the correction for detector efficiency (see Fig. 4) the argon spectra were corrected for TOF resolution by the method described in R e f . 3 0 . The resolution function was assumed to be independent of TOF channel and was set equal to the normalized elastic part of the vanadium spectrum. In this method the number of data points after the correction , with spacing AX. (i denotes the scattering a n g l e ) , is mainly determined by the shape of the resolution function and the statistical uncertainties in the measured spectrum. As a result the number of data points after the correction is less than the number of points before. The idea behind the method is that no significant informa­ tion about the corrected spectrum can be obtained with a resolution better than AX.. In Fig. 9 TOF spectra of measurement b are given before and after the resolution correction. Note that AX. is different for different s c a t t e r ­ ing angles. For the calculation of the standard deviations of the spectra after the resolution correction, the statistical errors in the resolution function were neglected.

F . Conversion to S ( k . u ) at rectangular ( k , u ) - g r i d

The fully corrected double differential cross section d V d M X after c o r ­ rection for self-shielding (see Fig. 4) was converted to the symmetric S(k,u>) using E q s . ( 2 ) and ( 5 ) . At this stage S ( k , u ) is known at discrete scattering angles <p. (see Fig. 2) at equidistant X values with increment AX. ( S e c . I V . E ) . In order to compare the results with theory and to calculate derived proper­ ties such as frequency moments and FWHM of S ( k , u ) , both at fixed k, the

31

-obtained results were interpolated to a rectangular ( k , OJ ) - g r i d . The i n t e r p o ­ lation , for which v/e used cubic spline routines, was performed in two s t e p s .

Fig. 9 . Double differential cross

section before (error bars) and after (closed circles with error bars) correction for t i -me-of-flight resolution of m e a s u ­ rement b at the same scattering angles as in Fig . 5 . 2 1 4 2 40 20 4 2 8 11

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32

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i '

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ö 3 ,

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Fig. 1 0 . Symmetrized dynamic structure factor, S ( k , u ) , from measurement b

at 12 discrete co-values. + , energy-gain d a t a ; x , energy-loss d a t a . The abscissae of the data points are determined by the experimental scattering angles.

33

-First, for each scattering angle (p., indicated by a solid line in F i g . 2 , S ( k , 0)) was interpolated from equidistant X values to discrete equidistant w values in the range - 4 0 < u . < 13 ps . In F i g . 1 0 S ( k , u ) is displayed for 12 values of l u l as a function of k (for measurement b) . For each to value the results at 57 scattering angles are shown.

Second, S ( k , u ) and the estimated standard deviations were interpolated to equidistant k values k, = k + ( i - l ) A k , i = l , . . . , 6 4 . where k = 4 . 2 nrrT a n d

^ I O o

Ak - 0 . 6 nm , at each value of Id). I . For this interpolation three cases were

J F

distinguished:

i ) W,=0 - A cubic spline interpolation through the experimental data was u s e d .

i i ) 0 < I W.I < 13 ps . A spline interpolation was performed through the energy -loss data ( u > 0 ) , resulting in S ( k . , u . ) , as well as through the

' „ _ l J

energy-gain data (to < 0) , yielding S ( k . , u . ) , with estimated standard deviations o . . and a,.. For each k. the final result, S(k,,u>.) , was

' J ' J V . . ( i . taken to be the weighted mean of S and S with weights w . . and w . .

S i j i j given by

to*)'

+ A', r ,

(i2)

IJ IJ where

A.. =: ft | ST(k.,co,)-S ( k . , u . ) I .

ij ' i J i J '

Systematic errors and a possible underestimation of the propagation of the statistical error, resulting in a discrepancy between energy-loss and energy-gain d a t a , were incorporated into the final estimate of the standard deviation a..

IJ

a2. = ( w+. + w T . ) "1 . ( 1 3 ) IJ IJ ' J

i i i ) IU, I > 13 ps . Only energy-gain data are available, a cubic-spline data smoother is used for the interpolation.

Interpolated values of S(k,U>) at fixed k are shown in Fig. 11 for five k values for measurements b and d .

(19)

(o) (b)

O 5 10 0 10 20

Fig . 11 . (a) Symmetrized dynamic structure factor S (k , u ) and ( b ) longitu­

dinal current correlation function C . ( k , w ) at five values of wavenumber k , for measurements b (error bars) and d (continuous l i n e ) . The v e r t i ­ cal dashed and full lines in ( b ) indicate, the peak positions u> of C ( k , u ) of measurement b and d, respectively, determined as explained in the t e x t . Note the difference in the td scales of ( a ) and ( b ) .

35

G. Assessment of data quality

Two relationships can be used to measure the quality of the experimental S ( k , u ) d a t a .

The first relationship is the detailed balance condition C E q . ( 4 ) J which implies that S ( k , u ) is symmetric in to. In the present case the energy-loss and energy-gain data can be compared for I Id I < 13 ps . For this purpose a quality factor Q.. is defined:

Q2. = ( r+. )2 + ( r T . )2 , ( 1 4 )

IJ IJ 1J with residuals

rT. = J S " ( k , , u . ) - S ( k . , w . ) \ to.. .

If S ~ ( k , u ) ) were normally distributed around S ( k , u ) Q.. would follow a X -distribution with one degree of freedom. Then the expectation value of Q..

would be equal to 1 and the variance 2 . In Table III, for the four measure­

ments Q(u>) is given, i . e . , the square root of GT-, averaged over all k values and different finite cointervals u . < w < W. , containing N„ pairs of r e s i

-Ji h s R F

d u a l s ,

Q ( u ) - [ N "1 Z E Q2. 1 % . ( 1 5 )

L R j = ji i U J

Q ( CJ ) may be considered a coarse measure of the ratio of systematic error to the estimated statistical error. The results in Table III show that the s y s ­ tematic errors do not grossly exceed the statistical errors and that the energy-gain and energy-loss data agree rather well (see also Fig. 10) .

The second relationship is the exact expression for the first moment <o> of S ( k , w ) :

<Gi> . = w0 = fik2/2M , ( 1 6 )

exact R

with fio the recoil e n e r g y . In Fig. 12 the ratio of the experimental and

theoretical first frequency moments < ( J > / DR is shown. For k < 10 nm_ 1 the fraction of <u> determined by the model used for large w is considerable

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36

Table III. The quality factor Q(td) (defined in text)

for measurements a - d u-interval (ps_ 1> 0 - 2 2 - 4 4 - 6 6 - 6 8 - 1 0 1 0 - 1 2 1 2 - 1 3 NR 571 5 4 4 5 1 3 4 7 7 4 3 5 3 8 4 138 a 1 . 5 5 2 . 2 0 2 . 1 9 1 . 9 2 1 . 9 2 1 . 7 0 1 . 8 5 Q ( w ) b 1 . 4 7 2 . 0 0 1 . 7 7 1 . 8 6 1 . 8 4 1 . 7 9 1 . 8 4 c 1 . 5 3 1 . 7 9 1 . 7 1 1 . 9 7 2 . 4 9 1 . 8 1 1 . 6 4 d 2 . 0 1 1 . 9 2 1 . 9 1 1 . 7 0 2 . 8 5 1 . 6 7 1 . 8 9 3 0 6 2 1 . 9 3 1 . 7 9

( F i g . 2 and S e c . V . B ) , and therefore <u> is less reliable in this k range. The experimental and exact <0J> are in excellent agreement for measurements a, b , and d, and k > 10 nm . For measurement c , however, <w> is 15-20% too high in the range 10 < k < 2 5 nm , for reasons unknown to u s .

Note that the detailed balance condition gives information about the q u a l ­ ity of the data at small and intermediate frequencies, whereas the first moment is more sensitive to higher frequencies.

V. RESULTS

A . Dynamic structure factor

We first discuss the asymptotic behavior of the dynamic structure factor S ( k , u ) at small and large wave number k . In order to define the terms "small" and "large" we consider a h a r d - s p h e r e system, where the two essential units of length are the mean free path 9. and the diameter d . Although argon has a continuous interaction potential a "corresponding" hard-sphere system can be defined by choosing an appropriate hard-sphe

sities I is of the order of 0 . 1 d (see Table IV)

defined by choosing an appropriate h a r d s p h e r e diameter d . At liquid d e n

-At large k, i . e . , k » I \ S ( k , w ) tends to the free gas limit

L ( k , u ) = — — exp f - V: i / / ( v k ) F Vlri v , k I °

( 1 7 )

with v2 = Cj3M)"' for a classical system. First order quantum effects enter E q . ( 1 7 ) only as a correction to v2 of order K. For argon at the present c o n ­ ditions this correction is less than 2 % and is neglected in the remainder of this p a p e r . 1.2 h o**>

(c

1.0 0.8 1.0

as

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/ H + * K

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5

10 20 30 4 0

k/nnr'

F i g . 1 2 . Ratio of the experimental first frequency moment of S ( k , u ) , < u > , and t h e exact value wR ~ ftk /2M a s function of wave number k . Measurements a - d are depicted with solid circles, squares, triangles,

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Table IV. Thermodynamic and transport properties

All quantities are calculated for Ar at 1 2 0 . 0 K assuming that c , X , a , D , and r , and n "* M . mass M = 5 9 . 7 3 x 10 kg

thermal velocity v - (|3M) = 1 6 6 . 6 ms~ hard sphere (HS) diameter d = 0 . 3 4 3 nm

L e n n a r d - J o n e s (LJ) parameters a = 0 . 3 3 6 nm; kR~~ z = 1 2 3 . 2 K

31 32

thermodynamic condition HS: reduced density

mean free path LJ: reduced density

reduced temperature s p e c , heat at const .pressure s p e c . h e a t at const.volume

adiabatic sound velocity structure factor at k = 0 nd=

Ha

c F c V ■jfa C ( 1 0 "

do"

c / c P (ms 2 3J K - ' ) V

)

S ( 0 ) 0 . 7 1 0 0 . 0 3 3 0 . 6 6 8 0 . 9 7 4 8 . 7 1 3 . 0 0 ( 9 ) 2 . 9 0 6 3 2 ( 6 ) 0 . 2 0 2 0 . 7 4 7 0 . 0 2 9 0 . 7 0 2 0 . 9 7 4 7 . 6 4 3 . 0 9 ( 9 ) 2 . 4 8 7 2 4 ( 8 ) 0 . 1 3 1 0 . 7 8 7 0 . 0 2 5 0 . 7 4 0 0 . 9 7 4 6 . 9 7 3 . 2 1 ( 9 ) 2 . 1 7 8 2 4 ( 8 ) 0 . 0 8 9 d 0 . 8 1 2 0 . 0 2 3 0 . 7 6 3 0 . 9 7 4 6 . 7 0 3 . 2 8 ( 9 ) 2 . 0 4 8 8 6 ( 9 ) 0 . 0 7 2 21 2 1 21 2 1 shear viscosity thermal conductivity thermal diffusivity sound damping factor self-diffusion constant

n(10

kgm s )

X( l O ^ W n T ' r C1) a=X(nc ) " ' ( 1 0 " "8m2r '

r ( i o « r ' )

D(IO'VV')

1 0 . 6 ( 3 ) 0 . 8 6 ( 3 ) 5 . 6 ( 2 ) 2 0 ( 2 ) 0 . 6 8 ( 1 ) 1 2 . 7 ( 5 ) 0 . 9 7 ( 3 ) 6 . 9 ( 2 ) 2 0 ( 2 ) 0 . 5 8 ( 1 ) 1 5 . 4 ( 5 ) 1 7 . 4 ( 5 ) 3 4 1 . 1 1 ( 3 ) 1 . 2 0 ( 4 ) 34 8 . 2 ( 2 ) 8 . 9 ( 3 ) 2 0 ( 2 ) 2 1 ( 2 ) 3 5 0 . 4 9 ( 1 ) 0 . 4 3 ( 1 ) 36

1) Enskog mean free path = tllV2 nd g ( d ) l with SH S( d ) the pair correlation function at contact ( gl l c( d ) from K e f . 3 3 )

2) S ( 0 ) = kBT ( 3 n / aP)T = kBT - ( / ( M cs :) = 1\/c^ 3) T = & [ ( e + 4 l l / 3 ) ( n M ) ~ ' + ( f - l >al with c the bulk viscosity

i) from CMD simulation for a LJ system at conditions k„T7e---0.97 and no = 0 . 6 9 2 and 0 . 7 6 2 , using the interpolation formula D ~ a + on

(22)

At small k , i . e . , k « d , fluctuations with wavelength X » d are probed and the liquid may be considered as a continuum. The dynamic behaviour is governed by the linearized NavierStokes equations resulting in the h y d r o d y

-A0 S, ( k , w ) = — h -, ^s r z c,+ ( to+tos ) t a nip u + z0 n l- (to+Uc z r - ( u - us) tanct ( 1 8 )

with A = ("Y-l ) S ( 0 ) /-Y , z = akS,

o o A - S ( 0 ) / ( 2 f ) , z = Tk2,

s s w = cck , tanrp= C f ^ - D a + r ] k / c .

The symbols for thermodynamic and transport properties used in the above equations are explained in Table IV and their numerical values for 3bAr at the experimental conditions are also given. The present neutron scattering results are in the regime d < k < i" , intermediate between hydrodynamics and free streaming.

Three-dimensional representations of the fully corrected S ( k , to) for meas­ urements a and d are shown in Fig. 1 3 . Considerable changes are apparent as a result of increasing the density n by 14 %. Representative numerical results for 5 ( k , t o ) together with an estimate of its standard deviation CEq. ( 13) ] are tabulated in Tables V - VIII. C In these tables only a selection of the avail­ able data is listed. S ( k , to) data at 64 k values for the four thermodynamic conditions are obtainable from the authors ] . In addition four derived properties of S ( k , t o ) , v i z . , 5 ( k ) , the FWHM at fixed k of S ( k , t o ) , and the peak position and peak height of C J k . t o ) , are given. These quantities will be d i s ­ cussed in more detail below.

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(23)

42

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(24)

Sik) F W H M ( p s- 1) « „ ( p s_ l) C , l * , < J „ ) H O "8 m ' s o, (ps"1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5 0 9.5 4.2 0.093(31 1.58(21)

')

0.0330(20) 0.0311(13) 0.0264(11) 0 0208(91 0.0162(8) 0.0133(7) 0.0116(7) 0.0106(6) 0.0097(5) 0.0088(4) 0.0082(5) 0.0066(6) T A U L E V I ] 8.4 0.115(1) 3.08(17) 8.55(28) 0.175(8) 0.0209(6) 0.0203(4) 0.0190(4) 0.0171(4) 0.0152(4) 0.0135(4) 0.0122(3) 0.0111(2) 0.0102(3) 0.0093(2) 0.0084(2) 00066(2) 0.0056(1) 0.0048(1) 0.004HI) 0.003611) 0.0032(1) 0.0030(1) 0.0027(1) 0.0025(1) 0.0023(1) 0.0021(1) 00018(1) 0.0017(1) 0.00)6(1) 0.0014(0)

Symmeln/.ed dynamic struct 12.6 0 213(1) 4.61U3) 8.59(18) 0.130(4) 0.0305(5) 0.0302(3) 0.0294(3) 00282(3) 0.0267(3) 00250(3) 0.0233(2) 0.0216(2) 0.0200(2) 0.0185(2) 0,0171(2) 0.0141(2) 0.0118(2) 00100(2) 0.0086(1) 0,0075(11 0.006611) 0.0057(1) 0.00510) 0.0045(1) 0.0040(1) 0 0036(1) 0.0032(1) 0.0028(1) 0.0026(1) 00023(1) 162 0.671(4) 3.74(71 4 89(16) 0.144(3) 0.1322(18) 0.1305(14) 0.1260(16) 0.1192(16) 0.1110(13) 0.1022(10) 00933(9) 0.0846(9) 0.0763(8) 0.0685(7) 0.0613(6) 0.0466(5) 0.0360(5) 0.0282(5) 0.0225(4) 0 0185(4) 0.0151(3) 0 0120(2) 0.0099(3) 00079(2) 0.0064(2) 0.0056(3) 0.0044(1) 0.0037(1) 0.0032(1) 0.0026(1)

arc factor from n

k ( 192 2.037(8) 2 15(2) 2.99(42) 0 163(2) Sik, 0.6973(52) 0.6732(36) 0.6124(33] 0.5324(291 0 4493(27) 0.3741(251 0.3072(201 0.2498(17) 0.2039(14) 0.1691(23) 0,1425(34) 0.095CK17) 0.0668(9) 0.0492(12) 0.0339(7) 0.0266(8) 0.0201(7) 00151(9) 0.0120(4) 00094(4) 0.0073(2) 0.0057(2) 0.0052(3] 00041(2] 0.003211; 0 0026( 11 « s u r c m c n l c r r T1) 22.2 1.284(8) 4 11(8) 4 56(25] 0.151(2) j ) (ps) 0.2379(28) 0.2359(20) 0.2302(19) 0.2209(17) 0.2085(17) 0.1939(22) 0.1785(27) 0.1634(29) 0.1489(27) 0.1352(24) 0.1223(21) 00950(16) 0.0741(12) 0.0580(6) 0.0464(4) 0.0369(4) 0.0287(4) 0.0229(3) 0.0187(3) 0.0150(3) 0.0119(2) 00097(2) 0.0083(2) 0.00680) 0.0055(2) 0.0043(1) i, = 19 51 n m " 25.8 0.70214) 7 67(6) 7.77(17) 0 1220) 0.0740(3) 0.0735(4) 0.0720(7) 0.0699(10) O.0675OI) 0.0650(11) 0.0628(9) 0.0609(6) 0.0592(4) 0.0574(6) 0.0555(9) 0.0501(9) 0.0444(4) 0 0399(4) 0.0357(3) 0 0313(3) 0.0273(3) 0.0239(3) 0.0211(2) 0.0186(2) 0.0)64(2) 0.0145(2) 0.0127(1) 0.0110(1) 0 0096(1) 00084(2) 30 0 0.727(3) 7 56(9) 8.99(20) 0 103(3) 0.0722(7) 0.0718(5) 0.0708(5) 0.0692(6) 0.0671(7) 0 0649(9) 0.062500) 0.O6O3O1) 0.0581(10) 0.0561(9) 0.0540(7) 0.0488(5) 0.0432(6) 0.0382(7) 0.0341(4) 0.0305(3) 0.0273(2) 00242(2) 0.0216(2) 0.0195(2) 0.0176(3) 00160(3) 0.0144(4) 0.0130(3) 0.0115(4) 0.0103(2) 34.2 1.048(4) 742(7) 8.1708) 0.097(3) 0.1087(6) 0.1083(4) 0.1072(4) 0,1055(5) 0.1032(5) 0.1005(6) 0.0975(7) 0.0942(8) 0.090719) 0.0871(9) 0.0834(9) 0.0742(8) 0.0651(8) 0.0571(8) 0.0503(7) 0.0443(7) 0.0388(6) 0.0338(6) 0.0295(4) 0.0260(3) 0 0229(3) 0.0203(3) 0 0178(4) 0.0155(5) 0.0137(5) 0.0125(3) 39.0 1.14003) 10.44(21) 8.64(20) 0.101(5) 0.0913(5) 0.0912(4) 0.0909(6) 0.0905(8) 0.0913(23) 0.0908(24) 0.0899(22) 0.0886(20) 0.0870(17) 0.0849(15) 0.0825(19) 0.0760(20) 0.069503) 0.063806) 0.058504) 0.0534(17) 0 0480(16) 0.0427(14) 0.037903) 0.033803) 0.0302111) 00270(9) 0.024OO0) 0.021300) 0.0188(8) 0.0165(4) T A B L E V I I . iConU, 30.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 0.0012(0) 0.0009(0) 0.0005(0) 0.0020(1) 0.0016(0) 0.0012(0) 0.0008(0) 0.0005(0) 0.0004(0) 0 0 0 0 3 ( 0 ) 0.0002(0) OOOOUO) 0.0001(0) O.OOOCXO) 0 0 0 2 2 ( 1 ) 0.0015(1) 0 0 0 1 1 ( 0 ) O.OOOS(O) 0.0006(0) 0.0005(0) 0 0 0 0 4 ( 0 ) 0.0003(0) 0.0<302(0) 0 0002(0) 0.0001(0) 0.0024(2) 0.0016(1) 0.0012(1) 0.0008(1) 0.0006(1) 0.0005(1) 0.00041.1) 0 . 0 0 0 3 ; i) 0 . 0 0 0 3 : i) 0.000211) 0.0O021I) 0.0041(3) 0.0025(2) 0 . 0 0 1 9 0 ) 0.0013(1) 0.0009(1) 0.0007(1) 0 0005(0) 0.0004(0) 0.0003(0) 0.0002(0) 0.0001(0) 0 0070(1) 0.0052(2) 0.0038(1) 0.0028(1) 0 0 0 2 0 0 ) 0.0014(1) 000)0(1) 0.00070) 0.00050) 0.0004(1) 0.0003(1) 0 0092(1) 0.007OO) 0 0 0 5 5 ( 1 ) 0 . 0 0 4 3 0 ) 0 0 0 3 2 ( 1 ) 0 . 0 0 2 4 0 ) 0.0018(1) 0 . 0 0 1 4 0 ) 0.0010O) 0.0008(1) 0.0006(0) 0.0108(2) 0.0082(2) 0,0064(1) 0.0050(3) 0.0039(3) 0 0030(3) 0.0023(2) 0.0018(2) 0.0013(2) 0.0010(2) 0.0003(2) 0.0145(2) 0.0114(2) 0 . 0 0 8 8 0 ) 0.0063(3) 0.0053(3) 0.0041(3) 0.0032(3) 0.0024(3) 0.0019(3) 0.0014(3) 0,0011(3) T A B L E V I I I . S y m m e t r i z e d d y n a m i c s t r u c t u r e factor from m e a s u r e k ( n n r ' l 19 2 22.2 S{k) F W H M ( p s "1 *« ( p s "1) Sitk,am) (tO' 0.08713) 1.95(28) 0.027007) 0.025901) 0.023200) 0.0196(9) 0.0160(8) 0.0131(8) 0.0111(6) 0.1040) 3.4106) 8.27(22) 0.149(5} 0.0178(4) 00175(3) 0.0168(3) 0.0157(3) 0.0144(3) 0.0130(3) 0.0116(3) 0.192(2) 4.81(18) 8.67(23) 0 124(3) 0.0264(6) 00262(4) 0.0257(4) 0.0248(4) 0.0237(3) 0.0224(3) 0.021013) 0.619(6) 3.71(10) 5.26(21) 0.140(4) 0.1202(23) 0.118609) 0.1145(24) 0.1083(26) 0.1007(24) 0.0924(21) 0.084006) 2.08001) 1.96(3) 3.05(67) 0 157(2) S(A,a 0.7694(63) 0.7350(49) 0.6533(52) 0,5544(49) O.460H45) 0.3768(37) 0 3027(28) 1,30301) 3,73(9) 4,4806) 0.143(2) ) (ps) 0.2601(33) 0.257CK23) 0 2483(23) 0.2354(23) 0.2193(26) 0 2014(31) 0.1834(34) 0.684(6) 7.5802) 7.99(23) 0.118(2) 0 0704(4) 0.0701(7) 0.0693(12) 00681(17) 0.0666(20) 0.0648(22) 00629(22) 0.706(2) 7 45(6) 9.46(23) 0.099(3) 0.0699(6) 0.0696(4) 0.0688(4) 0.0675(4) 0.0659(4) 0.0640(5) 0.0619(5) 1.047(5) 7.03(5) 8.12(26) 0.094(2) 0.1126(7) 0.1122(5) 0.1109(5) 0.1088(5) 0.1061(5) 0.1029(6) 0.0993(6) 1.15306) 9.99(21) 8.61(23) 0,099(2) 0.0942(7) 0.09«(6) 00941(7) O.094O19) 0.0953(36) 0.0952(35) 0.0947(31)

(25)

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ü * ] 9 » i q « q i ' ! O i f l C i « q « i O ï i p o p p p p p p q < 47

k/nnrr

k/nm-F f g . 1 3 . Three-dimensional plot of the symmetrized dynamic structure fac­ tor S ( k , u ) from measurements a (left) and d (right) .

The FWHM of S ( k , u ) at fixed k is determined, and an estimate of its stan­ dard deviation is made, as described in Appendix B. In Fig. 14(a) S ( k , 0 ) =

FWHM/FWFMf are shown. Here S ( k , 0 ) / [ S ( k ) Sf( k , 0 ) ] and in Fig.14(b) FWHM

~ i V?

S , ( k , 0 ) = (VTn v k ) and FWHM, = 2(2«n2) v k are the free-gas values and

f o i o 0

S(k) is the experimentally determined zeroth frequency moment <u > (Sec.V.B). This particular way of displaying the peak value and width of S ( k , u ) is cho­ sen in order to give a maximum of information in the plots. At large k both displayed quantities will approach unity. The hydrodynamic asymptotes S, ( k , 0 ) = ("(-1 )S(0) /(trjak2) and FWHM, = 2ak2 are indicated for conditions a and d. As

(12)

found before , FWHM exhibits an oscillatory behavior around the diffusion expression 2Dk2 [indicated in F i g . 1 4 ( b ) ] , which is the value at small k for

the self part of the dynamic structure factor defined by setting j equal to j ' in E q . ( l ) . Since experimental values of the self-diffusion coefficient of argon at the examined thermodynamic states are not available, values of D

(26)

^ 9 H

obtained with computer molecular dynamics (CMD) simulations were u s e d . Although the S ( k » w ) results of these simulations agree very well with the present argon data ' , the adaptation of D values from CMD to argon should be considered with the necessary caution.

The well-known De Gennes narrowing in FWHM is visible ai k :: 20 nm ; it shifts to larger k with increasing d e n s i t y . At k < 12 nm FWHM is roughly linear in k C see Fig . 1 4 ( c ) ] , b u t , although the error bars are relatively large in this k range, a significant d e n s i t y - d e p e n d e n t deviation from this linear behavior [ F i g . 1 4 ( b ) ] , in accordance with the change in the h y d r o d y n a m -ic limit, is a p p a r e n t . A more or less linear behavior as a function of k of FWHM in the same k range can also be observed in the gas state of

( 1 1 1 (^Pi)

krypton . De Schepper et a l . discuss the FWHM as a function of k and its d e p e n d e n c e on the interaction potential in more detail. They show t h a t , particularly in the small k - r e g i o n , FWHM is very sensitive to the choice of interaction potential.

B . Frequency moments

The n frequency moments of S ( k , ( d ) and S(k,(*>) are defined by + a>

<un> = [ Dn 5 ( k , u ) dw

and - $w ( 1 9 )

< un> - [ wn S ( k , u ) d u .

Here < u > Is t h e moment including quantum effects and <OJ > its classical approximation. The odd moments <w > are equal to z e r o . The moments have been calculated from the experimental S ( k , w) by numerical integration. E s t i ­ mates of their standard deviations were obtained from the standard deviations of the experimental S (k ,[•)) d a t a , taking into account, in an approximate way

(see Ref. 2 3 ) , correlations due to interpolation from TOF-scale to w - s c a l e . At discrete k values experimental data are available for w < W. ( k ) , W. ( k ) being determined by the kinematic region covered by the experiment ( F i g . 2 ) .

Fig . 1 4 . ( a ) Peak values of the symmetrized dynamic structure factor

divided by the experimental static structure factor and the free gas limit, S * ( k , 0 ) = S ( k , 0 ) / [ S ( k ) Sf( k , 0 ) ] . Key as In Fig. 12 . The h y d r o d y

-- 49

(a)

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k/nm

namic asymptote is indicated for measurements a ( d a s h e d line) and d ( d a s h e d - d o t t e d l i n e ) . ( b ) Full width at half maximum of 5 ( k , u ) divided by the free gas v a l u e , FWHM* = FWHM/FWHM,. The hydrodynamic asymptote is indicated b y the two steeper straight lines and the simple diffusion by the other two lines for measurements a ( d a s h e d lines) and d ( d a s h e d - d o t t e d lines) . The dotted curve represents the experimental time-of-flight resolution.

straight l i n e .

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