Neutron scattering study of the dynamics in dense argon gas

NEUTRON SCATTERING STUDY OF THE DYNAMICS IN DENSE ARGON GAS

NEUTRON SCATTERING STUDY OF THE

DYNAMICS IN DENSE ARGON GAS

proefschrift

ter verkrijging van de graad van

doctor in de technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus

dr.ir. C.J.D.M. Verhagen, hoogleraar

in de afdeling der Technische Natuurkunde,

voor een commissie uit de senaat te verdedigen

op woensdag, 17 september 1969 te 14 uur

door

CORNELIS DIRK ANDRIESSE

natuurkundig ingenieur

geboren te Leeuwarden

dit proefschrift is goedgekeurd door de promotor PROF.DR. J.J. VAN LOEF.

Aan mijn ouders Aan Engeltje

View of the pressure control and gas handling system described in chapter 3.5: the characters correspond to those in fig. 11.

Neutron scattering study of the dynamics in dense argon gas page 18 19 81 92 93 100 105 106 line* 13 fb 6 ft 4 fb 18 ft 3 ft 10 ft 13 ft 12 fb E R E A T A instead of (see also 2. 3) ( 4 ) , d ' dotted line phenomena by self correlation k g T / ( n p ) strain c h a r a c t e r i s t i c effects read (see also 2.4) C ( 4 ) _ dashed line phenomena observed by self correlation function k g T / p

small number, strain c h a r a c t e r i s t i c delay effects

C O N T E N T S

Page

Chapter 1 Introduction 11

Chapter 2 Analysis of atomic dynamics in fluids 15

2 . 1 Introduction 15 2.2 Correlation functions and some of their general

p r o p e r t i e s 15 2 . 3 Motions of individual atoms 19

2.4 Collective motions of atoms 24 2.5 Response model for collective motions 28

Chapter 3 Experimental method 35

3.1 Introduction 35 3.2 Neutron spectroscopic method 36

. 1 General r e m a r k s on the measurement of S(Q, w )

with neutrons 36 . 2 Rotating crystal s p e c t r o m e t e r 38

. 3 Correction procedures for instrumental effects 41 3.3 T e m p e r a t u r e and p r e s s u r e stability of the gaé sample 42

3.4 T e m p e r a t u r e control system 43

. 1 Cryostat 43 . 2 T e m p e r a t u r e measurement and control 46

3.5 P r e s s u r e control and g a s handling system 49

. 1 Design considerations 49

.2 Apparatus 52 . 3 Handling of the gas 57

page

Chapter 4 Experimental r e s u l t s 59

4 . 1 Introduction 59 4 . 2 Physical data characterizing the experimental runs 60

4 . 3 Time-of-flight spectra 61 4.4 The scattering law 68

4. 5 Accuracy 76 Chapter 5 Discussion 79 5.1 Introduction 79 5.2 Equilibrium p r o p e r t i e s 80 5.3 T r a n s p o r t properties 84 . 1 General features 84 .2 Interference effects 86 . 3 Width functions 90 5.4 Density inhomogeneities 93 5.5 Concluding r e m a r k s 95 Appendix 98 References 101 Sjmibols 104 Summary 106 Samenvatting 108 Acknowledgment

_no

r l . O

-0.5

Coherent intermediate scattering function for dense argon gas at 145.5 K and 3 6 . 1 atm (see chapter 4 . 4 ) .

C H A P T E R 1

I N T R O D U C T I O N

The study of t r a n s p o r t phenomena in dense g a s e s deals with ties a s diffusion, viscosity and heat conduction. These macroscopic proper-ties reflect the way in which the atoms move and transfer momentum and energy to each other. In kinetic theory the specific features of atomic mo-tions a r e derived from the form of the atomic interaction potential, using simple Newtonian mechanics to describe the motions in the ensemble of Interacting a t o m s . The problem to be solved in the kinetic theory of dense g a s e s and liquids is difficult since the number of atoms to deal with is hdge whereas no simplifying assumptions can be made on the local geometry a s is possible in the case of c r y s t a l s . In order to simplify the problem schematic interaction potentials can be considered. It may be mentioned that the Van der Waals theory of dense g a s e s actually amounts to assuming a very long range attractive part of the potential (the "molecular field") and a very steep ("hard c o r e " ) repulsive part. However, such assumptions a r e too crude to account for the dynamics in a gas at the onset of condensation, especially not far from the c r i t i c a l point.

Under these c i r c u m s t a n c e s it would be very useful to have direct ex-perimental knowledge of atomic dynamics, in order to compare it both with m e a s u r e m e n t s of the macroscopic transport coefficients and with predictions by kinetic theory. Atomic dynamics can be investigated using two different methods. The first consists of the solution of the hierarchy of equations of motion for a large set of atoms by a computer using a suitably chosen in-teraction potential. Such computer experiments have proven to be a power-11

ful tool for understanding the atomic dynamics in systems having a well defined interaction potential like argon. The second method involves m e a s u r -ing the inelastic scatter-ing of some type of radiation. The intensity of the scattered radiation is proportional to the F o u r i e r transform of the c o r r e lation of spacedependent densities at different t i m e s . F o r fluids the c h a r a c -t e r i s -t i c c o r r e l a -t i o n s ex-tend on a spiace and -time scale of a few A and ps respectively. Therefore it is most useful to apply the radiation of slow neutrons which have a wavelength comparable to interatomic spacings and an energy comparable to the kinetic energy of the a t o m s . The last proper-ty implies that inelastic effects may be detected relatively easy so that in-formation on atomic dynamics can be derived from the spectrum of scattered neutrons.

Some c h a r a c t e r i s t i c features of the method of inelastic scattering of slow neutrons may be summarized h e r e . Neutrons a r e scattered by the atomic nucleus with a probability which is determined by the type and the spin state of this nucleus. When the sample contains various isotopes or when the nucleus of a mono-isotopic substance has a spin, the resulting scattering is diffuse since the distribution of isotopes or spin orientations in the sample will generally be at random. This type of scattering is called incoherent. Only when the scattering c e n t e r s a r e identical, i . e . when the atomic nuclei a r e s i m i l a r and without spin, interference between the scat-tered waves i s possible. In that c a s e the scattering i s called coherent. It can be shown that incoherent scattering accounts for the self motion of the atoms and coherent scattering for collective motions. The inelastic s c a t t e r -ing by liquids may be divided into s m a l l - e n e r g y - t r a n s f e r (quasi-elastic) scattering and intrinsically inelastic scattering. The first i s due to low en-ergy diffusive motions of the atoms and the latter to phonon-like collective motions. In the case of molecular fluids the rotational and internal vibra-tional modes of motion give r i s e to inelastic scattering a l s o .

The objective of this investigation is to provide experimental data on the atomic dynamics in a simple dense gas close to condensation using the technique of slow neutron scattering. Such data do not yet exist since the neutron scattering by g a s samples with their relatively low density is r a t h e r small, unless they contain nuclei with a large scattering c r o s s - s e c t i o n such a s hydrogen. However, hydrogen condensates at very low t e m p e r a t u r e s and 12

the handling of it r e q u i r e s a lot of experimental c a r e . F u r t h e r m o r e , hydro-gen is diatomic and it should be treated quantummechanically which will make the interpretation of data more difficult. In hydrogeneous compounds, such a s methane, the internal modes of motions conceal the translational motions of the molecules which a r e of p r i m a r y interest. Some y e a r s ago it was found that one particular isotope of argon ( Ar, natural abundance 0.337 moI%) has a very large scattering c r o s s - s e c t i o n . Using a s a sample argon, enriched in this isotope, it should be possible to acquire the desired neutron scattering data for a gas without internal d e g r e e s of freedom.

The choice of argon is made for different r e a s o n s . F i r s t of all argon h a s been studied extensively so that a l a r g e compilation of data from theory and experiment exists already. Secondly there a r e no extreme cryogenic r e -quirements nor serious problems with r e s p e c t to the n e c e s s a r y p r e s s u r e

Of.

c e l l s . F u r t h e r m o r e , the use of a sample completely enriched in A r (the nucleus has no spin) should provide a completely coherent scattering pattern, reflecting the ensemble average of correlations between position (and motion)

dftitity

of all the atoms at different t i m e s . This will be especially valuable for the understanding of the momentum and energy transport properties, which a r e , in principle, related to the viscosity and heat conduction of the g a s .

In o r d e r to interpret the measured spectra it is n e c e s s a r y to analyse the atomic dynamics in the dense gas in t e r m s of the correlations between the atomic positions (densities) or motions (currents). Since kinetic theory does not yet provide us with a solution of the many body problem the c o r r e -lations a r e to be discussed in t e r m s of simplifying models for the atomic motion. In chapter 2 a survey is given of various models described in the l i t e r a t u r e . Since these models do not adequately account for coherent s c a t t e r -ing from low density fluids a response model is introduced. The experi-mental method is described in chapter 3 with a sp)ecial emphasize on the handling of a small quantity of valuable A r g a s . The r e s u l t s of the scattering experiments a r e presented in chapter 4. Finally in chapter 5 a d i s -cussion is given of the t r a n s p o r t p r o p e r t i e s measured, whereas some at-tention is payed also to the equilibrium projjerties and fluctuations in the g a s .

C H A P T E R 2

ANALYSIS OF ATOMIC DYNAMICS IN FLUIDS

2 . 1 INTRODUCTION

The analysis of atomic dynamics of m a t t e r is relatively easy in the two limiting c a s e s of the ideal gas and the ideal c r y s t a l . In the first c a s e all the motions in the system a r e essentially individual and there a r e no collective interactions whereas in the second case all the motions of the system a r e essentially collective. The presence in a system of both col-lective and individual motions, a s found in fluids, complicates its dynamical p r o p e r t i e s .

T h e r e i s , however, a general method for describing the dynamics of m a t t e r , which can be used not only for the two limiting c a s e s but also for intermediate situations. This method is based on space-time correlation functions. In 2.2 we shall give a general account of this method and apply it to various models to describe both individual (2.3) and collective motions of atoms ( 2 . 4 ) . A m o r e detailed discussion of collective motions is given in 2 . 5 .

2.2 CORRELATION FUNCTIONS AND SOME OF THEIR GENERAL PROPERTIES

The dynamic structure S(Q,w) of a system is the s p a c e - t i m e - F o u r i e r transform of the generalized atomic distribution function G(£, t), defined a s

G ( £ . t ) = i Z < 5 ( r + r (0) - r (t)| > (2.1)

i, j •' •'

for classical systems (Van Hove, 1954) (N = number of atoms, r = position vectors, t = time, < . . . > denotes a thermal average) The sum in (2.1) can be evaluated for i = j , giving the self part G ( r , t ) , whereas the remaining t e r m s (i ^ j) give the distinct part G . ( £ , t ) . In the static approximation (t = 0)

G ( r , 0 ) = G g ( r , 0 ) + G ^ ( r , 0 ) = 6 (r) + p g ( r ) , (2.2)

where g ( r ) is the pair correlation function and p the average number densi-ty. A similar relation can be given for the space-transform of G(£, t), F ( Q , t) called the intermediate scattering function, and the t i m e - t r a n s f o r m of F ( Q , t), the dynamic s t r u c t u r e S(Q,w) (see fig. 1 with the definitions). The self part of these functions is often r e f e r r e d to as the incoherent part, whereas the complete functions a r e denoted with coherent. The dynamic structure S(Q,co) is also called the scattering law, since it gives the r e l a -tive intensities of the radiation scattered by the system with momentum transfer hQ and energy transfer hco. The differential scattering c r o s s s e c -tion for slow neutrons is related to the scattering law in the following way (see fig. 2 with the definitions)

2

d ^ = l F | - [ < b ' > S (Q.CO) + « b > 2 - < b 2 » S^ (Q.o.) ] , (2.3)

where b is the scattering length and k and k . a r e the wave vectors of scattered and impinging neutrons. F o r a system with identical s c a t t e r e r s

2 2

( < b > - < b >) = 0, which means that there is no incoherence.

In our study the time behaviour of the scattering functions is of p r i m a --13

ry interest. F o r very small times (t < 10 s) all systems behave s i m i l a r -ly; their dynamics r e s e m b l e the dynamics of an ideal g a s . Therefore gener-al relations exist for the relaxation at smgener-all times, which can be presented using a time expansion of the intermediate scattering function

F .(Q,t) = 2 ^ ^^ = 2 - V r ! , (2.4) s,d^-i ' ^ n! . n n! * s , d ^

3t ^^Q n 16

G(r t) real space real time

• •

•

•

•

•

• •

G(r,t) = ( - L j JdQe*0-r F(0,t)

Fig. 1. Scheme of the scattering functions.

diDe-'<^t S(Q,ü)

kj wavevector irKoming neutrons, k wovevector scottercd neutrons Q = k - k i ,

Q-Vk^+kf -2kkiCosiji »iu>-h^kf-k')/(2mn), h . h / ( 2 n ) , h Planck's constant, mp neutron mass.

for small energy transfer: kj-k a mnui/(hk|). u > 0 in the case thot energy isgoined by thescatterer.

The functions k^ ' are the n moments of the scattering law , 3 " F ( Q , t ) a n oo j , ^ «(") = = ] = J d w e '*^S(Q,co) ] = '' t ^ o 3^ - t ^ O = ( - i ) " / d w w ° S ( Q , w ) (2.5) - o o

and the expressions derived a r e known a s the moment relations. F o r c l a s s i -cal systems the zeroth and second moments a r e (De Gennes, 1959)

ï , ( ° ) = l , (2.6)

kj°^ = S(Q) - 1, (2.7)

( 2 . 8 )

(2.9) respectively.

Here S(Q) is the structure factor (see a l s o 2.3), k_ is Boltzmann's constant, T the t e m p e r a t u r e and M the scattering m a s s . F o r classical systems the odd elements of J vanish a s the scattering law is s y m m e t r i c . The first two of these relations apply to the static approximation, the last two a r e relevant in the ideal gas c a s e . The first relation t ^ ' = 1 means that the

° s

incoherent scattering is not Q dependent (diffuse scattering). F u r t h e r m o r e (2) ~

Ï ,^ ' = 0 e x p r e s s e s the fact that in an ideal gas there is no correlation between the velocities of different atoms. The higher moment relations depend on particular interactions of the system. Recently the following a p -proximate relations for the fourth moments a r e derived for a fluid (Scho-field, 1966) ,A\ / ' Q ^ k „ T \ ^ k„T ^ \ M / ^ P ^ Q Z t ( 4 ) f Q y ^ B' + (4/3)G' ^2 m - V M y

pk_T ^ ^ ' ' ^

i.

Here D i s the self diffusion coefficient (see also 2.3); B' and G' a r e formal expressions for some space integrals over the pair correlation function and derivatives of the atomic interaction potential, which a r e identical to the instantaneous bulk and shear moduli respectively. F o r reasons of simplicity in (2.11) the fourth moment for coherent scattering is given r a t h e r than

(4) _ (4) (4) d ~ « " «s

Using the former expressions we shall discuss a number of models for the fluid, on the base of which approximate relations for the scattering law of a dense gas can be calculated. The justification of the use of models r a t h e r than rigorous r e s u l t s is that there is still no complete theory based on first principles that gives a satisfactory account of atomic dynamics in fluids.

F u r t h e r m o r e , we t r e a t argon a s a c l a s s i c a l system. Quantum effects a r e important whenever the de Broglie wavelength is large compared to interatomic distances and secondly whenever the time h/(k„T) is small com-pared to the time scale of the atomic motions.

Egelstaff (1967) has discussed the conditions under which the c l a s s i -cal limit of the scattering law gives a reasonable description. F o r argon at the relevant t e m p e r a t u r e s the first effect is negligible. The second effect, however, may be significant at l a r g e energy t r a n s f e r s which a r e to be ex-pected in neutron scattering by a g a s . We take it into account by introducing the detailed balance condition into the c l a s s i c a l scattering law (see 3 . 2 . 1 ) .

2 . 3 MOTIONS OF INDIVIDUAL ATOMS

Let us s t a r t with the almost trivial case of atomic motions in the ideal g a s . The probability n(v) for a velocity in the interval (v^, v + dv) is given by the Maxwell-Boltzmann distribution. Now, the probability G (£, t) for the displacement of an atom from the origin to a volume d£ around £ in a time t is just equal to the probability n ( v ) d ( v ) : G (r, t) dr = n ( r / t )

- 3 ~ ~ ~ ~ d ( r / t ) = n ( r / t ) t d r . Therefore, inserting for n ( r , t) the Maxwell-Boltzmann distribution, one obtains

On F o u r i e r transforming we find

f ^\^

2)

F ^ ( Q , t ) = e x p \ - ^ t ' j , (2.13)

Ss(Q-) = ( 2 1 ; ^ ) ' ^ ^ " ' (• V ^ ) • ^ ' ' " ^

All functions a r e Gaussian. Since there a r e no correlations between different atoms in the ideal gas, the distinct p a r t s of the scattering functions a r e zero (^ ,^ = 0). The ideal gas is a limiting case for the next models.

Now let us consider a fluid in which structure exists and interactions take place. F o r c l a s s i c a l systems the self part of the intermediate s c a t t e r -ing function can generally be written

F g ( Q . t ) = i j 2 < e x p ( i Q . ( r . ( t ) - r . ( 0 ) ) j > . (2.15)

In the approximation that Q. ( £ . ( t ) - £ . ( 0 ) ) has a Gaussian probability distribution, the thermal average can be taken

F g ( Q , t ) = e x p [ - ^ < Q . ( r . ( t ) - r . ( 0 ) ) > 1 = exp ( - Q ^ r ( t ) j . (2.16)

This is called the Gaussian approximation and it is believed to be reasonably good for fluids (Sjölander, 1965). The width function F (t) for the ideal gas can be found by comparing (2.13) and (2.16)

I^(^)ideal gas = W ^ ' ^'-''^ which is drawn in fig. 3 (curve A). It can be shown that for a c l a s s i c a l

system

1 t

r ( t ) = - j ƒ dt' (t - f ) < v(0). v ( t ' ) > , (2.18) o

where < v ( 0 ) . v ( t ) > is the velocity selfcorrelation function. F o r the ideal g a s this function is constant since there a r e no interactions. Hence F (t) is quad-ratic in t. However, for the diffusive motions in fluids the velocity self-correlation function will decay after some time and will tend to z e r o . This

r(t)

Fig. 3. Width functions for an ideal gas (A), a dense fluid (B) and a fluid described by Langevin diffusion (C); 7 is a friction frequen-cy given by (2.28).

means that F (t) becomes linear in t for long times:

r ( t ) = constant + D t, (2.19)

where

D = 3 ƒ d t < v ( 0 ) . v ( t ) > o

(2.20)

is the self diffusion coefficient. In the case that one considers long t i m e s only the constant in (2.19) will be much s m a l l e r than Dt. These long times a r e relevant for thermal neutron scattering by dense fluids so that the ap-proximation

•^(^^dense fluid = ^ ' (2.21)

is often useful.

Substitution of (2.21) i n t o ( 2 . 1 6 ) g i v e s for G „ ( r , t )

G ^ ( r , t ) = ( 4 , r D t ) ' " ' " exp ( - -^-^), (2.22)

which is just a solution of F i c k ' s diffusion equation

DV^Gg (r, t) = - | ^ Gg (r, t ) . (2. 23)

F o r this reason (2.21) is known a s the simple diffusion model for F (t) (see fig. 3, curve B). The scattering law has the shape

S s ( Q ' " ) = ^ ^ ^ ^ 2 ' (2.24) which is Lorentzian in contrast with (2.14). Expression (2.21) is not c o r r e c t

to describe the short time behaviour since in the simple diffusion model the second moment relation (2.8) is violated

£^(2) ^±- exp (- Q^Dt) 1 =

^ bC t^O

As in our experiment we will investigate the atomic motions in a dense gas r (t) is somewhere in between quadratic in t and linear in t. In this r e -gion deviations from the ideal gas will be slight only. Following Egelstaff and Schofield (1962) we introduce an interpolation function

= D(\A77-2

r ( t ) . , = D (V t + r - r ) , (2.25) ^ ' m t e r p . ^ o o " ^ '

which is quadratic in t for small times (t « r ) and linear in t for long t i m e s (t » r ). The p a r a m e t e r T is a m e a s u r e for the time in which the atomic motions reach the asymptotic behaviour, indicated in (2.19). This particular form is chosen in order to take advantage of an integral r e p r e -sentation of the modified Bessel function K,, with which the scattering law can be evaluated 2 2 exp(DQ r^) DQ r „ ^ ^ ^ ^ ^ _ _ _ ^ ^ S ^-i S„(Q,w) = ~ / ? — 5 = 7 K, ( r „ V w + (DQ^)^). (2.26) / . , + ( D 0 ' ^ ) ^ 2 This expression reduces to the Lorentzian form (2.24) for DQ -^ 0

2

and to the Gaussian form (2.14) for DQ -*°°- One can apply this

lation form using tables for K, (x). Then there is the question of the physi-cal meaning of the time r . It is c l e a r that T m e a s u r e s some friction during the diffusive p r o c e s s , causing a certain delay after which the dif-fusive displacements of the atoms take place. This has been considered in the Langevin model for atomic motions (Ramakrishnan, 1959), which we will discuss now.

According to this model atomic motions in fluids a r e described by d^r dr

M * - ^ - + 7 M * - ; T - = f (t) , ( 2 . 2 7 )

dt'' °^ ~

where _f(t) is a rapidly fluctuating driving force, M is an effective atomic m a s s and 7 i s a friction frequency, given by the Einstein-Stokes relation

k T

y = ~ - • (2.28)

M D

The introduction of an effective m a s s is based on the notion that in a dense fluid the motion of an atom is impeded by its neighbours and that virtually a number of a t o m s must be moved for the diffusive motion of one atom. The ratio M^/M is expected to be density dependent; for the case of a g a s near condensation it should be very near to unity. The width function, calculated for the Langevin model is (Schofield, 1960)

r ( t ) T . „ _ H n = D t t + i ( e x p ( - 7 t ) - l l ] , (2.29) 'Langevin ' 7

which i s a l s o drawn in fig. 3 (curve C ) . Unfortunately no analytic form for S (Q, co) can be obtained from this width function. The asymptote for long times is D ( t - l / - y ) , giving a z e r o width for t = I / 7 . This time may be identi-fied with T in (2.25) and it could be interpreted a s the time during which atoms make unperturbed motions, preceeded and terminated by collisions (free diffusion).

T h e r e i s still a l a r g e number of other models on the self motion in fluids, but they apply m o r e specifically to triple point liquids, where dif-fusion takes place in jumps r a t h e r than in quasi-continuous flow. We need not to d i s c u s s these models h e r e .

•••) note added in proof: S.J. Cocking pointed out to us that In this case the scattering law can be written as an infinite sum of Lorentzian functions, see Singwl and Sjölander, Phys. Rev. 119(1960)863.

2.4 COLLECTIVE MOTIONS OF ATOMS

The coherent scattering law S(Q,w) accounts for the dynamics of the whole ensemble of scattering nuclei in the fluid, and therefore it contains interference effects. A rigorous calculation of these effects needs the so-lution of a many particle problem. Though recently much p r o g r e s s has been made in approximative methods to solve such problems (Nelkin and Ranga-nathan, 1967; K e r r , 1968; Singwi et a l . , 1968), the r e s u l t s obtained a r e

still not entirely satisfactory, even for small Qvalues which is often r e -ferred to a s the hydrodynamic region of S(Q,co).

A well-known feature, caused by interference effects in atomic mo-tions, is the propagation of sound waves in liquids. Ordered density fluctu-ations cause the Brillouin scattering of light and inelastic scattering of neutrons at low Q (see fig. 4). T h e r e is a remote resemblance with collec-tive motions in crystal lattices, and concepts a s quasi-lattice, quasi-phonons and quasi-zones a r e applied to triple points liquids. Such concepts a r e used in describing the collective modes of motion in liquids (Singwi, 1965). We will not describe this model h e r e , since the density of the fluid to be stud-ied is too low to make a comparison with crystal-lattice dynamics meaning-ful.

S(Q,u))

Fig. 4. Illustration of propagating modes; the velocity of propagalion is ( t g a ) " ^ .

An interesting quantity for discussing the collective modes in fluids 2

is CO S(Q,co). In the hydrodynamic limit it gives the Brillouin peaks, where-a s in generwhere-al it is the F o u r i e r trwhere-ansform of the correlwhere-ation of c u r r e n t s in the system (Rahman, 1968). A plot of the position of its maximum v e r s u s Q will give a dispersion curve of the c u r r e n t density fluctuations, which r e -sembles a phonon dispersion curve for low Q. It should be pointed out that such a plot is very sensitive to the shape of S(Q,w).

A useful starting point for the calculation of interference effects is the convolution approximation for G , (r, t) proposed by Vineyard (1958),

G ^ ( r , t ) = p ƒ d r ' G g ( r - r ' , t ) g ( r ' ) , vol.

where g(£) is the pair distribution function (see fig. 5).

[Jt=t]

Duo]

Fig. 5.

[ i ]

0

On the convolution approximation- At time t = 0 there is an atom i in the origin and an atom j in a unit volume around r ' with probability g(£')- The proba-bility that atom j has wandered to a unit volume around r_ in t i m e t is Gg(£-£', t) g ( £ ' ) . The probability for finding another atom than i in the latter volume at time t is given by the space integral (2.30).

(2.30)

On F o u r i e r transforming we find

F ^ ( Q , t ) = [ S ( Q ) - 1 ] F ^ ( Q , t ) , (2.31)

where the structure factor S(Q) is given by

The coherent scattering law then becomes

S(Q,co) = S(Q) Sg(Q,co). (2.33) This approximation thus directly r e l a t e s the coherent scattering law

to the incoherent one discussed above. F o r t = 0 it is c o r r e c t , moreover it should be c o r r e c t for very long distances and times also. However, it then fails to predict the Brillouin peaks, assuming a reasonable model for G ( r , t ) in this limit a s given by (2.22). F o r intermediate times it fails be-cause according to the approximation t h e r e is a finite chance that two atoms a r e at the same moment at the same place. F u r t h e r m o r e it violates certain moment relations. In spite of its shortcomings, it has the m e r i t of being conceptually easy, which makes it attractive to add further concepts for im-provement. We briefly indicate how corrections to the convolution approxi-mation can be obtained. The distinct part of the intermediate scattering function, F , (Q, t), can be written a s (compare (2.15))

F . ( Q , t ) = -i 2 < e x p ( - i Q . r (0)) exp ( i Q . r ( t ) ) > . (2.34)

N .^. - - J - - 1

Consequently the approximation (2.31) is equivalent to separating the thermal average into a static and a dynamic part respectively a s follows

< e x p ( - i Q . £ . ( 0 ) ) e x p ( i Q . £ j ( 0 ) ) > < e x p ( - i Q . £ . ( 0 ) e x p ( i Q . r . ( t ) ) > . (2.35)

The first t e r m leads to the time-indep)endent factor [ S ( Q ) - 1 ] , wherea s the second lewhereads to the time depiendent fwhereactor F (Q, t). F r o m (2. 34) c o r -rection t e r m s can be calculated for (2.31) for small times (Nijboer and Rahman, 1966). One has an interpolation function for intermediate times if the corrections a r e written in such a functional form that they converge for l a r g e t i m e s , giving Üierefore a long time behaviour just a s predicted by the convolution approximation (which may be inadequate!) (Glass and Rice, 1968). However, such interpolation functions always will be somewhat a r b i t r a r y since they a r e not based on physical arguments for the behaviour at inter-mediate t i m e s .

Let us t r a c e an atom in the liquid which diffuses away from its equi-librium position, e . g . the origin. In equiequi-librium we know the probability g(£) of finding other atoms at distance r from the origin. If there is instan-26

taneous relaxation in the liquid then the structure of the liquid about the dif-fusing atom will maintain its equilibrium configuration throughout the motion. This is in fact stated by (2.31), which can be rewritten a s

t

F ^ ( Q , t ) = [ S ( Q ) - 1] ƒ d f F ( Q , t - t ' ) 6 ( t ' ) . (2.36) u — — Q

ta-in words: accordta-ing to the convolution approximation the relaxation of the structure is a s fast a s the incoherent one, which accounts for the dif-fusive motion of one atom. However, the forces in a liquid a r e not suf-ficiently strong to r e q u i r e instantaneous relaxation. The forces do, however, drive the atoms in the liquid towards the equilibrium configuration. The lat-ter relaxation will be delayed with respject to the incoherent one. Rahman (1964) h a s studied the time delay in a computer calculation, simulating the atomic motions in liquid argon at 94 K, by comparing G j ( r , t) and G (£, t). In order to improve the convolution approximation he proposed an empirical relation for the time t ' that should be inserted in G ( r , t ) instead of t

s —

t' = t - T [ 1 - exp ( - t / T ) - ( t / r ) ^ e x p ( - t ^ / T ^ ) ] . (2.37) -12

The time constant T has the value 1.0 x 10 s for the system studied by Rahman. The functional form chosen for t' is such that the relation for the second moment of S(Q,w) according to (2.9) is satisfied. F u r t h e r m o r e , the relation for the fourth moment of S(Q,co) given by (2.11) can be s a t i s -fied by a suitable dependence of r on Q (Desai and Yip, 1968). Another way to improve the convolution approximation is given by Sköld (1967). Instead of Q he sufasitutes

Q' = Q / Vs(Q) (2.38)

in S (Q,cj) with the r e s u l t that the relation for the second moment of S(Q, w) is satisfied. We s t r e s s the point that this applies to the coherent scattering law r a t h e r than to the distinct part only. It may be verified that this a p -proximation also leads to a delayed decay of F , (Q, t). However, both the transformed time t ' and the transformed wave vector Q' a r e somewhat a r t i -ficial expressions introduced in order to account for delay in the actual relaxation of the fluid s t r u c t u r e . Therefore we shall try to describe the relaxation with a physical model for the delay.

2.5 RESPONSE MODEL FOR COLLECTIVE MOTIONS

Let us first note that 6(t) in (2.36) can be replaced by a function of time that is different from z e r o a l s o for t ^ 0. In this way one may account for delay effects with r e s p e c t to the instantaneous relaxation of the fluid. In a preliminary note it is proposed to replace 6(t) by a decreasing function of time with a finite value for t = 0 (Andriesse, 1968b). However, such a substitution does not give the c o r r e c t short time limit for F , (Q, t) a s it yields F . ( Q , 0) = 0 instead of F , ( Q , 0) = S ( Q ) - 1 . In order to describe the collective motions in a fluid on the b a s i s of the motion of one atom, a function is needed that first p r e s c r i b e s instantaneous relaxation and then, with this relaxation a s a reference, a function that d e s c r i b e s the Q depend-ent delay. We note these functions by 6 (t) and f(Q, t).

Physically f(Q, t) may be considered a s a response function of the fluid. This is made plausible by the following reasoning. Let us select one atom, that we call blue, and travel with it when it diffuses away. Surrounding atoms, which we call red, a r e accelerated and move away from their in-itial equilibrium positions a s a consequence of the motion of the blue atom. The particular motions of the red atoms may be considered a s due to an external perturbation whereas the blue atom may be considered to remain at r e s t . F r o m this point of view it looks a s if the red atoms, due to an external force, first move away from their positions around the blue atom and then, due to interatomic forces, a r e moving back to their initial posi-tions. The mathematical description of these motions is given by a response function.

As a r e s u l t of these arguments (2.36) should be modified into t F ^ ( Q , t ) = [ S ( Q ) - 1 ] ƒ d f F g ( Q . t - f ) [ 6 ( f ) + f ( Q , t ' ) ] = o t = [ S ( Q ) - 1 ] [ F g ( Q . t ) + ƒ d f F g ( Q , t - t ' ) f ( Q , t ' ) 1 . (2,39) o

The coherent scattering law can now be written a s

S(Q,Cü) = S(Q)Sg(Q,co) [ 1 + c(Q)f(Q,co) 1 , (2.40 28

where c(Q), the space-transform of the direct correlation function, appears a s an abbreviation for 3 ( Q ) = l - s J ö ) (2.41) and f(Q,w) is given by f(Q-w) = ^ /"dt e x p ( i w t ) f ( Q , t ) . (2.42) - oo

It might be s t r e s s e d that this approach gives the specific features of the fluid response directly in t e r m s of the scattering c r o s s - s e c t i o n (2.3). F u r t h e r m o r e it simply shows the modification introduced in the convolution approximation, which is retained for c(Q)f(Q,cj) -* 0. Since for a dilute fluid c(Q) is n e a r to z e r o the deviations a r e small, i . e . we can in general expand around special solutions obtained with the convolution approximation.

It is interesting to note that expression (2.39) is s i m i l a r to the " m a s -ter equation", which d e s c r i b e s the time behaviour of the velocity distribu-tion funcdistribu-tion (Prigogine and Résibois, 1961). The latter equadistribu-tion is derived from first principles and it gives the decay in two t e r m s also. The first t e r m d e s c r i b e s the contribution, at time t, of the initial excitations which interact with each other in order to give a state without correlation. The second term e x p r e s s e s the fact that the interactions in general a r e non instantaneous events, thus correlating the distribution function at a given time t to the same functions at previous t i m e s . Therefore this t e r m is written in the form of a time convolution, in which the kernel is compara-ble to the space-transform of f(Q,t)

1 3

-f ( £ . t ) = ( i ^ ) ; d Q e x p ( i Q . £ ) -f ( Q , t ) . (2.43)

- oo

In the appendix f(Q, t) is shown to be related also to the generalized density-density response function (Pines, 1966).

The problem now is to find an expression for f(Q, t) that suitably a c -counts for the delayed relaxation of F . ( Q , t). We can introduce a function that satisfies some moment relations. On differential ting (2.39) it is found that

3<^"^f(Q.t)

7^-^— ] = O for n = O, 1, 2 (2.44)

^^ t^O

expressing the fact that f(Q, t) should be odd in order to get S(Q,cj) even. F u r t h e r m o r e it is found from (2.8 through 11) that

(2.45)

3f(Q.t) J

3t

3^f(Q,t)

Bt^

. (2) ^ \ ^

s M '

(^.J-

Q^k„TY S(Q) [2 + k„T/(MD^Q^)] - K

Let us consider a simple form for f(Q,t) that for t^.0 satisfies f(Q. 0) = 0 besides the relations (2.45) and (2.46). F u r t h e r m o r e it should converge for large t, i . e .

f(Q,t) = t / a e x p ( - 7 ^ t ) . (2.48) This equation contains no p a r a m e t e r s additional to those that can be

calculated using the above conditions. F r o m (2.45) it can be found that

a = M / ( Q \ g T ) , (2.49) whereas (2.46) and (2.49) yield

7 2 _ 1 Q\T S ( Q ) ( 2 + kgT/(MD^Q^)} - K

o " ' 3 M S(Q) - 1 • (2.50)

The constant K is a m e a s u r e for the rigidity of the fluid and it can be calculated from the viscous constants B' and G'. F o r a triple point liquid

we find from data, given by Egelstaff (1967), K •>• 100. F o r a fluid at low densities and high t e m p e r a t u r e s we obtain from data, given by Schofield (1968), K = 2 6 / 9 . In both c a s e s K exceeds the t e r m S(Q) I 2 + k^T/(MD'^Q^)], ex-cept for very small Q values which we will not consider h e r e (Q < 0 . 1 A ). It is reasonable to a s s u m e that this is the case also for fluids at intermedi-ate densities and t e m p e r a t u r e s . We therefore write instead of (2.50) the approximate relation

1 Q ' I ^ B T _K

M S ( Q ) - 1 (2.51)

There a r e two r e m a r k s that should be made h e r e . F i r s t it should be remembered that for Q->. oo, S(Q) ->• 1, which implies that f(Q,t) converges for large Q. This is a n e c e s s a r y condition for the existence of the F o u r i e r transform given in (2.43). Secondly there a r e no r e a l solutions for y at Q values corresponding to S ( Q ) < 1 . This means that f(Q, t) oscillates for S ( Q ) < 1 whereas for S ( Q ) > 1 the function just has one maximum and then it d e c r e a s e s monotonically. If we consider (2.50) instead of the approximation (2.51) it is evident that 7 will not be purely imaginary for S ( Q ) < 1 nor purely real for S ( Q ) > 1 but in general complex. This implies that the oscil-lations of f(Q, t) for S ( Q ) < 1 a r e damped, in particular for S ( Q ) ' ' l or c ( Q ) * 0 . The behaviour of f(Q, t) in the two different regions is drawn in fig. 6.

Fig. 6. Illustration of the behaviour of the response function f(Q, t) in the regimes S ( Q ) < 1 and S ( Q ) > 1 .

The physical meaning of both r e g i m e s is the following.

Atoms, regularly spaced at distances corresponding to Q values in the peak of S(Q), quickly respond to the displacement of one of them. This response is strongly damped a s a consequence of the relatively large forces between these a t o m s . However, atoms out of this regular a r r a y , at distances c o r r e -sponding to the Q values for which S ( Q ) < 1 , only slowly respond to such a displacement. In this case the restoring forces a r e relatively small. This leads to a weakly damped response, characterized by an oscillatory atomic motion. Such oscillatory motions cause a peak in the frequency spectrum, i . e . the motions give r i s e to propagating modes comparable to sound waves. The t i m e - F o u r i e r transform of (2.48) is calculated using (2.42), which gives

, 7 2 . C 0 2

f ( Q > ^ ) = ^ ° 2 ^ ^ 2 2 (2.52) ^ o '

T h e r e a r e poles for the w values given by 2

^ 2 = - 7 2 _ X B K— , , i-o\ o 3 M l-S(Q) ' '' ^^^' which i s the dispersion relation for these modes. F o r small Q and dense fluids the s t r u c t u r e factor S(Q) can be considered constant, so that the speed of propagation is given by

^-^-(j^rmY- (2-54)

Inserting the values S(Q)=« 0 . 1 , K = 100 and k^T/M « 1.7 x 10^ m^s"2, which a r e typical for the triple point liquid of argon, we find from (2.54) V =ï 800 m s . This value is in agreement with the measured velocity of ultrasound in liquid argon of 820 m s (Sköld and Larsson, 1967). Since

-1 e e

for increasing Q the factor {l-S(Q)} "^ Q (with > 0) the dispersion of the propagating modes has an upward curvature, which is in agreement with the theory of Brillouin scattering (Mclntyre and Sengers, 1968). T h e r e is no point of inflection in (2.53) beyond which the curvature is negative. Such a behaviour would be expected if the disf)ersion r e s e m b l e s the phonon disper-sion in a crystal lattice. However, a s pointed out by Rahman (1968), the modes cease to propagate when Q i n c r e a s e s because of the viscous damping. This is illustrated in fig. 4. The use of (2.50) instead of the approximation 32

(2.51) would account for this effect. The damping is large for c(Q) ~ 0. Therefore one coiüd say that at Q values for which c'(Q) ~ 0 the dispersion given by (2.53) is spurious. This i s the case far outside the hydrodynamic region for dense fluids and at any Q value for dilute fluids.

We now calculate the dispersion of c u r r e n t density fluctuations in the fluid (see 2.4). The l a t t e r propagate outside the hydrodynamic region also whereas the dispersion relation is readily measured in neutron scattering experiments. At the same time the width A to of the quasi-elastic p)eak is calculated.

We r e s t r i c t ourselves to dilute fluids for which c'(Q) i s about z e r o . Consequently the deviations from the convolution approximation will be small, so that we expand around the solutions for

2 S ( Q , A w ) = S ( Q , 0 ) , (2.55)

3S(Q,w) S(Q,w)

= = - 2 = , (2.56)

aco w

given by the convolution approximation. We select for S (Q,w) both (2.14) and (2.24) since the m o r e r e a l i s t i c interpolation (2.26) is too complicated to use for expansion p r o c e d u r e s .

The solutions of (2.55) for the case of a gas (using (2.14), index g) and of a liquid (using (2.24), index 1) a r e

1/2

AcoS = Q [ 2 In 2 k g T / M ] (1 + O^) , (2.57)

A w ^ = DQ^ (1 + n b , (2.58) B 1

where n ° ' a r e small c o r r e c t i o n t e r m s . Similarly the solutions of (2.56)

' ^ ^ m a x = Q l 2 k B T / M l ' ' d + « / > . (2-59)

w^ - ^ ° ° . (2.60) max ^ '

The latter r e s u l t is a consequence of the fact that (2.24) violates the 2 2

second moment relation (2.8). In the case that 7 » w for the relevant frequencies, the following approximate relations a r e obtained

n g = - ^ ' S 3 ( Q ) s 2 ( Q ) , (2.61)

n l = - ^ c ^ Q ) s 2 ( Q ) ^ ^ ^ ^ , (2.62)

n S = . J 7 ê ^ Q ) s 2 ( Q ) . (2.63)

It should be noted that n ^ = JÏ ®. Therefore the ratio Aco ^ / w ^ is

m max independent of Q. This suggests that a plot of experimental data for

Au;/co v e r s u s Q should be useful. The value of the ratio is Vln 2 = 0.831 ' max -i ,

or l e s s , since for a dense fluid co ^oo . The peak width A co shows a max '^

broadening effect for c^(Q)<0 and a narrowing effect for c^(Q)>0. The n a r -rowing effect of Aco has been predicted by De Gennes (1959) and it is found indeed in the coherent scattering from liquids. Analogous to the narrowing effect of Aco we find for c^(Q)>0 a local minimum in the dispersion relation (2.59) for c u r r e n t density fluctuations. This is in agreement with recent investigations using computer calculations (Rahman, 1968) and neutron s c a t t e r -ing r e s u l t s (S-ingwi et a l . , 1968).

C H A P T E R 3

EXPERIMENTAL METHOD

3.1 INTRODUCTION

Studies of the atomic dynamics of m a t t e r using the method of inelastic scattering of neutrons r e q u i r e a monochromatic neutron beam and a method for the analysis of the neutrons scattered by the sample. The determination of the scattering law S(Q,co) actually amounts to the measurement of the energy and momentum t r a n s f e r s of neutrons to the sample. In general only small energy t r a n s f e r s a r e to be expected in the scattering of neutrons by fluids. F o r that reason slow or long-wavelength (4 A) neutrons a r e used, the energy of which is lower than the t h e r m a l energy k^T of the atomic motions in the fluid. Some general considerations concerning the m e a s u r e -ment of S(Q,co) and a description of the spectrometer a r e given in 3 . 2 .

The main objective of this investigation is the determination of S(Q,co) for gaseous argon close to condensation. In this case the density i s low. Moreover, for normal argon the coherent and incoherent scattering c r o s s -sections a r e small and of the same o r d e r of magnitude. This means that the scattered intensity will be very weak and for a considerable part diffuse. In the analysis of experimental data it is important to separate c o r -rectly the contributions due to coherent and incoherent scattering. This can be done, in principle, by the m e a s u r e m e n t of neutron scattering from argon in various isotopic compositions. Since the stable isotopes of argon have no nuclear spin, a sample consisting of only one isotope should s c a t t e r slow neutrons coherently. In this case the coherent contribution, which is of

p r i m a r y i n t e r e s t for the study of collective motions in the gas, is isolated in the most d i r e c t way. Therefore, and in view of the low density of the gas sample, argon highly enriched in Ar was chosen for our studies a s the scattering c r o s s - s e c t i o n of this nucleus is very large: 72 + 5 barn (Henshaw, 1957; Chrien et a l . 1962).

The required t e m p e r a t u r e and p r e s s u r e stability of the gas sample is discussed in 3 . 3 . A description of the cryostat and the measurement and control of the t e m p e r a t u r e a r e given in 3 . 4 . As a sample of the A r gas is very expensive the available amount for our experiments was r e s t r i c t e d to a few g r a m only. The handling of this limited and valuable quantity of the gas required some special techniques which a r e discussed in 3 . 5 .

3.2 NEUTRON SPECTROSCOPIC METHOD

3 . 2 . 1 G e n e r a l r e m a r k s o n t h e m e a s u r e m e n t of S ( Q , c o ) w i t h n e u t r o n s

The m e a s u r e m e n t of S(Q,co) involves a double scattering technique, for which in general bright neutron sources and luminous s p e c t r o m e t e r s a r e n e c e s s a r y . F i r s t one should select a monochromatic and collimated neutron beam, the flux of which wiU be a factor 10 or m o r e lower than the flux n e a r the core of the nuclear r e a c t o r . Secondly a selection has to be made for neutrons scattered by the sample with well-defined energy and momen-tum t r a n s f e r s . F u r t h e r m o r e the scattered fraction of the incoming radiation should be made small in order to avoid appreciable multiple scattering. These factors together attenuate the intensity of the neutron radiation by a

13

factor like 10 . The thermal flux of the H . O . R . at Delft a t the entrance 12 -2 -1

of the beam tube is about 3 x 10 cm s at 2 MW. As the neutron wave-length used (4 A) is selected from a low intensity part of the thermal neutron spectrum, low counting r a t e s a r e to be expected so that the back-ground radiation is relatively important. Moreover, long counting times (some ciays) a r e n e c e s s a r y to collect data with sufficient statistical accuracy.

In determining S(Q,co) one can proceed along two different lines, a p -plying either the t h r e e - a x i s method (Iyengar, 1967) or the chopped-beam method with time-of-flight analysis (Brugger, 1967). Though for accurate 36

m e a s u r e m e n t s of S(Q,co) a t h r e e - a x i s spiectrometer is to be preferred, a timeofflight s p e c t r o m e t e r is in general much more luminous. Our s c a t t e r -ing experiments a r e performed with a rotat-ing c r y s t a l s p e c t r o m e t e r in which the crystal monochromatizes and chops the neutron beam at the same time. With this s p e c t r o m e t e r we m e a s u r e the time-of-flight of neutrons scattered at various angles (j> . Since the time-of-flight is proportional to the neutron wavelength X, neutron spectra a r e obtained. One can easily derive that the energy and momentum t r a n s f e r s hco and hQ depend on <p and X in the follow-ing way (see fig. 2 with the definitions)

2TT h ,_1 1 ' " n X^ X2 ^ 2 , 2 Q = 47r 2 2 X . + X - 2X. Xcos 0 ,„ „, ^ 1 (3.2)

where X. i s the wavelength of incoming neutrons and m the neutron m a s s . Elimination of X from the above equations gives solutions for Q(co) with X. and 4> a s p a r a m e t e r s . Examples of these solutions a r e presented in fig. 7, where X. and <t> correspond to p a r a m e t e r s of the rotating crystal s p e c t r o -m e t e r used. F r o -m this figure it follows that in a ti-me-of-flight spectru-m the constant <t> method c o v e r s a large range of Q values when the scattering is not e l a s t i c . 2.8

Q[&-n

u.[x10"s'J

Fig. 7. Frequency-wavevector values covered by the rotating crystal spectrometer used.

In o r d e r to select constant Q values from the intensity pattern 1(0, X), maps have to be drawn of I(<J>,X = constant) for a number of suitably s e -lected values of X . In these maps the values I(Q, X = constant) can be found by interpolation. The resulting I(Q,X) i s actually equivalent to the

differ-2

ential scattering c r o s s - s e c t i o n (d a/dS2dX) which is related to the scattering law by

d2a 27rNh ^ i , hco h2Q2 , ^ , 2 . ^ „ , ^ , ,„ „,

d J m = - 1 ^ 7 T ^ ^ P l - 2 i r T - 8 N C T 1 < ^ > S ( Q ' " ) (3-3) n X B B

in the case of coherent scattering. Except for the exponential factor, this follows from ( 2 . 3 ) and (dco/dX) calculated from (3.1). The exponential fac-tor, which takes into account the detailed balance condition (Schofield, 1960),

-4

c o r r e c t s for recoil effects in the system. The factor X gives a very high contribution for small X . This means that in spite of poor counting s t a t i s -tics the transformation of I(Q, X) to S(Q,to), involving multiplication with

4

X , r e s u l t s in a well-defined wing co < 0 a t the neutron energy gain side of the quasi-elastic peak of S(Q,co). On the other hand the wing co > 0 at the neutron energy loss side will be badly defined. Due to this effect the sym-metry of S(Q,co), i . e . the validity of the first moment relation, cannot in general be checked for broad peaks.

3 . 2 . 2 R o t a t i n g c r y s t a l s p e c t r o m e t e r

In our experiments the rotating crystal spectrometer i s used, which is described in detail in the thesis of De Graaf (1968). A diagram of the s p e c t r o m e t e r i s given in fig. 8.

Neutrons a r e produced in the r e a c t o r core by fission and thermalized in light w a t e r . The quasi-Maxwellian shape of the neutron spectrum i s sketched in the insert of fig. 8. A 3 cm thick beryllium reflector of fast neutrons i s placed at the entrance of the radiation tube in front of the r e actor core in order to soften the neutron spectrum in the beam. The q u a r t z -filter further r e m o v e s fast neutrons from the beam and it reduces the gamma-ray background. The polycrystalline berylliumfilter, 40 cm in length, cooled down to 77 K, s c a t t e r s all neutrons from the beam for which X < 3.95 A, corresponding to twice the l a r g e s t spacing of lattice planes in Be. The cold neutrons transmitted through the filters a r e collimated before they impinge 38

on the moncx;hromator. This is a cylindrical (diameter 4 cm, length 5 cm) Pb single crystal rotating with a speed of about 14000 rpm around a h o r i -zontal axis, that coincides with the [211] c r y s t a l direction. Neutrons of 4.06 A wavelength a r e scattered from the (111) planes at a Bragg-angle

O

2Ö = 89 two times per revolution.

Fig. 8. Schematic diagram of the rotating crystal spectrometer. A - reactor core; B - light water moderator; C - radiation tube; D - beryllium reflector; E - quartz filter; F - cold berylliiun filter; G - collimator; H - rotating lead crystal; J - collimator; K - sample; L - one of the seven detectors; M - monitor; N - multichannel memory; S - signals; 1 - flight path. Insert: neutron spectra.

The pulsed, monochromatic neutrons having a velocity of 1000 m s p a s s through a second collimator before reaching the sample. Using a flight path of 1.22 m the scattered neutrons from one pulse a r r i v e at different

times at each of the seven d e t e c t o r s . These detectors a r e LiF-ZnS scintillators, placed at various scattering angles <t> between 19.5 and 92.0 . They subtend a solid angle i2 = 0.008 steradian and their counting efficiency is about 30% for the relevant neutron e n e r g i e s . Much attention has been given to reduce the neutron background by using borated paraffin, B,C and cadmium a s shielding m a t e r i a l s . However, a s the flight path is not evacu-ated, there i s still a considerable background due to scattering by a i r just behind the sample, which on the average is comparable to the intensities scattered by the sample itself. The intensity in the presence of the samples has been 30 to 55 counts/min per detector and a typical background intensity (both time dependent and time independent) 20 counts/min per detector. The signal of each detector is fed into a group of 256 channels of a multichannel memory and a separate group of 256 channels is fed by signals from both BFo monitors. During our experiments the channel length has been 8 ^ s . An optical system at the rotating c r y s t a l , giving a signal twice per r e v o -lution, s t a r t s the clock of the time-of-flight unit. In order to reduce the dead time of the s p e c t r o m e t e r a certain delay is given to this signal, during which neutrons from one pulse may fly from the crystal to somewhere be-tween the sample and the d e t e c t o r s . Readout of the memory is possible by parallel printer and punched tape.

The (elastic) spectral line width of the spectrometer At is about 65 MS and it is mainly determined by the width of the neutron pulse produced by the rotating c r y s t a l . The wavelength distribution AX/X of the neutron r a d i -ation selected by this extended c r y s t a l causes a spread in time-of-flight of about 50 JUS. F u r t h e r m o r e the time of rotation of the c r y s t a l during which neutrons a r e reflected into the direction of the second collimator (burst time) depends both on the divergence of the beam in this collimator and on the diameter and the angular velocity of the c r y s t a l . In principle the burst width can be varied by changing the collimator. However, a s the total width At is approximately determined by the square root of the sum of the squares of different contributions to the time spread, the decrease in burst width will not affect At very much w e r e a s it leads to a rapid d e c r e a s e of counting r a t e s . Therefore the burst width chosen is 50 a s which is equal to the time-of-flight spread due to the distribution AX/X. In this design a reason-ably narrow line width is combined with a good luminosity, which is im-portant considering the available neutron flux.

3 . 2 . 3 C o r r e c t i o n p r o c e d u r e s f o r i n s t r u m e n t a l e f f e c t s

The spectra measured with the rotating crystal s p e c t r o m e t e r a r e folded with the instrumental line function. Though in our measurements the peaks of S(Q,co) were r a t h e r broad, the effect of the non-zero line width cannot be neglected. F u r t h e r m o r e , in order to c a r r y out the interpolation mentioned in 3 . 2 . 1 it is necessary that the intensities obtained with the different de-t e c de-t o r s a r e normalized. Therefore we have measured de-the scade-tde-tering by a 1.0 mm thick polythene foil located at the sample position. The scattering by this hydrogeneous compound is 98% incoherent whereas the Debye-Waller factor determining the intensity of the elastic peak is within 5% independent of the scattering angle. As a r e s u l t this experiment gives the desired nor-malization within a few procent. F r o m the integrated intensities scattered at different 0 , we determine correction factors to the argon spectra. M o r e -over the same m e a s u r e m e n t yields the line function of the s p e c t r o m e t e r since the scattering from the foil is essentially e l a s t i c .

The unfolding of the spectra and the line function offers a particular problem. Therefore in general S(Q,co), calculated on the base of some models, is folded with this function and then compared with the measured s p e c t r a . However, there is an elegant method to eliminate the line function a s is demonstrated by Dasannacharya and Rao (1965). Their method consists in dividing the F o u r i e r time t r a n s f o r m s of the measured scattering law and the instrumental line function. In order to facilitate a direct comparison with models on the atomic motion, we have performed the transformation of the scattering law S ( Q , C J ) to F ( Q , t). In our data for F (Q, t) the line function is divided out. Data of the width A of S(Q,co) a r e obtained using the

approxi-2 approxi-2 approxi-2

mate relation A = A - A , where A and A a r e the widths of the m r m r

measured peak and the instrumental line.

The efficiency of the neutron detectors depends on the neutron energy (wavelength). Therefore the measured time-of-flight spectra a r e somewhat deformed. A correction for this effect h a s been made using the wavelength dependent efficiency of the different detectors (De Graaf, 1968).

Other c o r r e c t i o n s , not related to instrumental effects, a r e discussed in 4 . 3 .

3.3 TEMPERATURE AND PRESSURE STABILITY OF THE GAS SAMPLE

The triple point values of temperature and p r e s s u r e for argon a r e 83.8 K and 0.68 atm and the corresponding c r i t i c a l values a r e 150.7 K and 48.0 atm respectively. These data indicate the whole region which i s , in principle, of interest for m e a s u r e m e n t s on argon near the coexistence region, though we shall only select points in the dense region near the critical point. Therefore a cryostat is needed to cool the gas and a system in which it can be compressed. In this chapter we consider the stability r e q u i r e m e n t s for the t e m p e r a t u r e and p r e s s u r e of the sample. To do this one should like to know a p r i o r i how sensitive the neutron scattering is influenced by the t e m p e r a t u r e and p r e s s u r e of the g a s . It is generally known that some t r a n s -port properties diverge at the phase transition and that they can be studied only by requiring very good stabilization of temperature and p r e s s u r e . These properties, however, apply to macroscopic s y s t e m s . T h e r e is no reason to expect that on a microscopic scale the transport properties will show any anomalous behaviour. We will discuss this point further in 5 . 4 . This means that the neutron scattering probably will not be very sensitive to variations of temperature and p r e s s u r e . Therefore we expect that a t e m p e r a t u r e v a r i -ation A T = 0.1 K and a p r e s s u r e vari-ation AP = 0 . 1 atm a r e stiU accepta-ble.

These stability r e q u i r e m e n t s , however, a r e r a t h e r severe a s the counting times in the neutron scattering experiment a r e long. Therefore the long-term variation or drift of the control systems should be avoided. Drift is caused by slow changes of the p a r a m e t e r s in these s y s t e m s and it can be prevented or reduced by selecting suitable m a t e r i a l s and by special design. The temperature control of the sample, therefore, is to be based on a sta-ble temperature sensor (platinum wire) and the further components of the control system, including a bridge and signal amplifiers, should be stable a s well. F o r the same reason the p r e s s u r e control system is to be based on a standard like the p r e s s u r e balance.

3.4 TEMPERATURE CONTROL SYSTEM

3 . 4 . 1 C r y o s t a t

In order to obtain the t e m p e r a t u r e s needed a cryostat is built using liquid nitrogen (normal boiling point 77. 3 K) a s coolant (fig. 9). This coolant is stored in a cylindrical copper tank, 13 l i t e r in volume, with spiralized filling and a i r admittance tubes from stainless steel. The tank is evacuated by an Edwards E 0 2 air-cooled oil diffusion pump to a p r e s s u r e of about

-4

10 T o r r measured with an ionization gauge. As the axis of the cryostat is horizontal the tank is supported by nylon s p a c e r s which r e s t in the alumini-um outer jacket. The l a r g e dimensions of the tank (diameter 25 cm, length 33 cm) a r e chosen in o r d e r to reduce the number of times at which refilling is n e c e s s a r y so that the temperature setting of the gas sample may be d i s turbed. Evaporation of 13 liter of liquid nitrogen needs 2 x 10 joule w h e r e -a s the he-at le-ak from the surroundings due to r-adi-ation -and conduction is about 20 watt. This means that liquid nitrogen stays in the tank for 30 hour Nevertheless liquid nitrogen was supplemented every 6 hour , in order to a s c e r t a i n a good t e m p e r a t u r e stability of its copper walls. In fact during r e -filling we measured a variation of the sample temperature of 0.03 K or l e s s .

The argon sample holders a r e made of a massive slab-type block of aluminium in which several parallel cylindrical holes a r e bored a s shown in fig. 9. Aluminium is chosen for its high transparancy for neutrons ( m a c r o -scopic total scattering c r o s s - s e c t i o n 0.098 cm ) and its good conduction of heat (heat conduction coefficient 2.2 watt cm K ). Different sample holders a r e needed in order to reduce the effect of multiple scattering. The number of atoms in the sample must be chosen such that only 10% of the incident neutrons a r e scattered. Therefore the sample thickness should be about 0 . 1 / o p , where a is the scattering c r o s s section and P the number density. As p v a r i e s rapidly in the g a s p^ase along the coexistence curve, the thickness of the samples should range from 2 mm near the c r i t i c a l p)oint to a few cm at t e m p e r a t u r e s of 30 K below T . Two sample holders a r e used having 62.5 mm long holes, 8.4 mm and 4.7 mm in diameter respectively, separated from each other by at l e a s t 1 mm thick walls. The large

dimen-2

sions a r e chosen in o r d e r to cover the 4 x 5 cm neutron beam in the r o -tating crystal s p e c t r o m e t e r completely. The effective argon thickness of the

-|20 cm

15

10

5

O

Fig. 9. Cryostat. A - sample chamber; B - aluminium/iron connection; C - capil-lary tube; D - axis incoming neutron beam; E - heat radiation shield; F - cadmium layer; G - window; H - heat conductor; J - liquid nitrogen tank; K - spiralized filling tubes; L - connection to vacuum pump; M - outer j a c k e t .

c h a m b e r s was 5.19 mm and 3.15 mm respectively and the effective alumini-um thickness was in both c a s e s about 3.5 m m .

The normal on the slab-plane is inclined to about 30° with r e s p e c t to the axis of the incoming neutron beam so that the length of the neutron flight paths in the sample do not differ more than about 30% for the various scattering angles. A cylindrical heat shield of aluminium, 10 mm thick, envelopes the sample chamber and it h a s been kept a t approximately the same temperature a s the l a t t e r . The inner sides of this shield a r e cadmium plated in o r d e r to absorb scattered neutrons. In this shield 0.1 mm thick aluminium windows a r e fitted a t the position of the neutron beam.

Both sample chamber and shield a r e connected to the liquid nitrogen tank by supports acting a s heat conductors. The argon sample holder is fixed in a small m a s s i v e cylindrical copper body by s c r e w s , using indium foils and some g r e a s e in order to a s s u r e a good heat contact. This block, on which two heating elements a r e mounted (zener diodes, maximum power dissipation 5 watt each), is soldered to a copper tube a s a heat conductor (outer diameter 6 mm, inner d i a m e t e r 4 mm, length 45 mm). The heat flow through the latter from the sample chamber to the liquid nitrogen tank is such that, without heating the diodes, a temperature of 84 K is reached in the sample chamber. By dissipating about 10 watt the t e m p e r a t u r e can be r a i s e d to the c r i t i c a l t e m p e r a t u r e of argon. The heat shield is supported by cylindrical aluminium r o d s 10 mm in diameter, holding the zener diodes a s heating elements and connected to the liquid nitrogen tank by s c r e w s . The rods have a good conductance, n e c e s s a r y to obtain t e m p e r a t u r e s n e a r to 84 K in the shield. During our m e a s u r e m e n t s at about 145 K the power dissipated in these zener diodes consequently has been relatively high (15 watt).

The connection of the stainless steel filling capillary (outer diameter 3 mm, inner diameter 0. 9 m m ) to the aluminium argon chamber offered some difficulty due to differences in the thermal expansion coefficients of both m e t a l s . The use of a steel capillary r a t h e r than an aluminium filling tube with its relatively l a r g e volume is prescribed by the limited amount of Of.

A r isotope gas available. After some failures of mechanical connections using soft metal rings from lead or indium as s e a l s , we obtained a reliably

tight transition made from 10 mm thick plates of 99. 5% pure aluminium and ordinary iron, attached to each other by "explosion" soldering (Nobel Dynamit A . G . , Troisdorf n e a r Cologne, Germany). A cylindrical piece is taken from these plates, through which a 1 mm diameter axial hole is bored. The alu-minium part then is welded to the bcxly of the argon chamber and the capil-lary is soldered into the iron part.

3 . 4 . 2 T e m p e r a t u r e m e a s u r e m e n t a n d c o n t r o l

Platinum w i r e s a r e used a s temperature s e n s o r s . Miniature platinum r e s i s t a n c e elements (Rosemount REC 1050) and a calibrated platinum r e -sistance t h e r m o m e t e r (RosemouHï E 109-100) a r e placed in 3.3 mm diame-ter cylindrical holes which a r e bored in the massive p a r t s of sample cham-ber and heat shield. The elements a r e g r e a s e d and fit in the holes accu-rately, in order to make sure that chamber and sensor have the same t e m p e r a t u r e . The miniature elements REC 1050 a r e used for temperature control. At the same time they give a rough indication of the t e m p e r a t u r e .

-3 Their reproducibility of the resistance value R at the ice point is 2 x 10 The precision element E 109-100 h a s an accuracy of 2 x 10 and it is cali-brated at the ice point and steam point within 0.02 K (calibration current 1 mA).

The relation between the r e s i s t a n c e and the temperature of a platinum wire is given by

R,jjt/R^ = 1 + AT^ + BT^^ + C (T^ - 100) T*^ , (3.4)

where T* = T - 273.15 (T temperature in K) and A, B and C a r e constants which depend, however, on the density of impurities and dislocations of the

-4

w i r e . As long a s we r e q u i r e an accuracy of 2 x 10 only the dependence of the constant A on these properties of the wire needs to be considered. The other two then can be treated as constants which a r e known from the calibration of other platinum w i r e s . The two values of the r e s i s t a n c e R_x, measured a t the ice point and steam point, a r e thus sufficient for a com-plete calibration of this t h e r m o m e t e r . With the calibration constants

R = 99.958 ohm,

°

-3

A = 3.9826 X 10 , B = -0.587 X 1 0 ' ^ , C = -4.32 X 10"^^,

an extensive table is made relating the r e s i s t a n c e value with the t e m p e r a t u r e s . Using the platinum t h e r m o m e t e r we have measured the absolute t e m p e r a t u r e of the sample chamber within 0.05 K a c c u r a t e , since the bridge, with which we determined the r e s i s t a n c e values (see below), is m o r e accurate than the t h e r m o m e t e r .

In order to determine t e m p e r a t u r e gradients we have used REC 1050 elements and a l s o copper-constantan thermceouples. These m e a s u r e m e n t s a r e accurate to within 0.2 K. We have found a small temperature difference in the sample chamber when the heat shield was kept at the same t e m p e r a -t u r e a s -the sample chamber. This is due -to large -t e m p e r a -t u r e gradien-ts along the thin windows of the shield. In practice we lowered the t e m p e r a t u r e of the massive walls of this shield somewhat so that the effective t e m -p e r a t u r e is com-parable to the one of the sam-ple chamber. (The effective t e m p e r a t u r e is defined a s the t e m p e r a t u r e at which the net heat flow from shield to sample chamber is z e r o . ) In that case temperature gradients in the sample chamber a r e s m a l l e r than 0.2 K and very probably within 0 . 1 K.

The platinum r e s i s t a n c e s have four leads (0.2 mm diameter copper wire) so that c u r r e n t and potential m e a s u r e m e n t s can be taken in order to eliminate the wire r e s i s t a n c e s . The c u r r e n t used for m e a s u r e m e n t s with the precision r e s i s t a n c e is about 2 mA, giving approximately the same selfheat-ing a s at the calibration ( 0 . 1 mUIiwatt). Use h a s been made of a Leeds & Northrup K5 compensator with an overall precision of 5 x 10 , the line-arity being 1 x 10 . The Bleeker standard r e s i s t a n c e , with which the plati-num r e s i s t a n c e s a r e compared, was 50J2 to within 2 x 10 . These accu-r a c i e s all exceed the accuaccu-racy of the best theaccu-rmometeaccu-r used.

A temperature control is n e c e s s a r y , since the temperature will change 0 . 1 K or m o r e when the boiling p r e s s u r e of the liquid nitrogen v a r i e s m o r e

-3

than 10 atm and when the t e m p e r a t u r e outside the cryostat changes more than one degree. We have used a proportional control system. The r e