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Pair potential for argon from neutron diffraction at low density
Physica 57 (1972) 191-204 o North-Holland Publishing Co.
PAIR FROM NEUTRON
POTENTIAL
FOR ARGON
DIFFRACTION
AT LOW DENSITY
C. D. ANDRIESSE* Interuniversitair Reactor Instituut, Delft, Nederland E. LEGRAND Studiecentrum VOOY Kernenergie, S.C.K./C.E.N., Received
Mel, BelgiB
11 June 1971
Synopsis A diffraction experiment is described in which 1.065 A neutrons are incident on gaseous ssAr at a density of 2.52 x 1021 atoms/cm3 and temperature of 141.3 K. Information accurate to 1% is obtained on the structure factor S(Q) in the wavenumber range 0.5 A-1 < Q < 6.3 A-1. Using both HNC and the Percus-Yevick theory the pair potential q(r) for Ar is derived from a set of S(Q) data. This set of data is extended to Q = 0 and Q = 13 A-i in such a way that, within the core of the potential, the pa.ir correlation function is zero and the correct hydrodynamic limit for S(Q) is obtained. The locations of V(Y) = 0 and dq(r)/dr = 0 are found to be (3.28 f 0.01) A and (3.81 f 0.01) A, respectively, whereas the depth e/k = (124 & 4) K. The pair potential, which is tabulated, is compared with the one discussed by Dymond and Alder and some analytic potentials. In the appendices a new correction to the static approximation and a relation for the damping of oscillations in S(Q) at large Q values are calculated.
1. Ilztroduction. A direct way to determine the interaction potential of two atoms is to measure the diffraction of X rays or neutrons by a system of these atoms at a density which is sufficiently low that screening of the pair interaction by other atoms can be neglected. For a dilute gas the diffraction pattern is the Fourier transform of the atomic distribution function in its asymptotic (p + 0) form p exp[ -&(r)], where p is the average number density, q(r) the pair potential and /? = (kT)-1, K being Boltzmann’s constant and T the temperature. Since in diffraction experiments a finite density is always needed, one has to apply results from density expansions in structural theory. It should be verified then that the result for v(r) is only little affected by the approximations of, e.g., hypernetted chain (HNC)
* Present address : Ruimte-Onderzoek, Groningen .
WSN-gebouw
191
Paddepoel, Rijksuniversiteit,
192 or the Percus-Yevick are small.
C. D. ANDRIESSE AND E. LEGRAND theory, and that corrections
for non-additivity
indeed
In the case of argon the knowledge of v(r) so far is based almost exclusively on the application of kinetic theory to macroscopic properties. By carefully considering all known properties of Ar Dymond and Alderi) have constructed a numerical potential function, which substantially deviates from analytic potentials like those of Lennard-Jones or Kihara. Though atomicbeam scattering techniques in principle can give information on V(Y) without relying on kinetic theory, the study of Ar-Ar scattering23 a) has not progressed sufficiently that scattering data could be inverted directly4). The only existing direct information on v(r) of Ar, derived from X-ray scattering on the (dense) gass), appears to be in conflict with other evidence637). Furthermore systematic differences are observed between these X-ray results and recent neutron-scattering data on a part of the structure factor of gaseous Ar in a corresponding states). Neutron-scattering experiments on gaseous Ar therefore seem to be useful. In this paper we present results of diffraction measurements on isotopically pure s6Ar at about l/10 of the density in the solid state. The problems we met in determining p(r) with reasonable precision were considerable. Because of the weakness of the scattering and the lack of a pronounced structure in the gas, the measurements took a long time, during which many parameters of the experimental setup had to be kept constant. Furthermore an accurate knowledge was required of the various correction factors. We therefore elaborate on the description of the experimental procedure and the applied corrections, given in sections 2 and 3, respectively. The results for the pair potential are discussed in section 4. In the appendices a new correction to the static approximation and a relation for the damping of oscillations in the diffraction pattern at large scattering angles are calculated. 2. Experimental procedure. The density chosen for our experiment is ten times lower than in the solid. This is a compromise between the requirements that p should be as low as possible and that a lo/o-accurate diffraction experiment contains useful information. For a given density of the gas the approximate form of the distribution function p exp[-_Bq(Y)] is most pronounced at the lowest possible temperature, i.e. at the onset of condensation. The condensation temperature at & of the density in solid Ar is about 140 K and the corresponding vapour pressure about 30 atm. The latter temperature and pressure therefore were relevant for our experiment . In order to cool the gas, the Ar-sample holder was placed in a liquidnitrogen cryostat with 1 mm thick aluminium windows for the neutron beam. It has been kept at the desired temperature by heating two zener
PAIR
i
/
POTENTIAL
//
/
/
193
FOR ARGON
0
,
2
Jcm
Fig. 1. Horizontal cross section of the sample holder with the cadmium masks and its position with respect to the neutron beam at 47” scattering angle; A and B are trajectories of twice-scattered neutrons which can and cannot be detected, respectively.
diodes on the small tube connecting holder and liquid-nitrogen reservoir, using a proportional control system with a Rosemount EC 1050 platinum resistance as sensor. The latter also served to measure the sample temperature and it has been calibrated against the vapour-pressure curve of Ar using a calibrated Wallace and Tiernan FA 233 bourdon gauge as pressure indicator. In order to obtain the desired pressure, we applied a technique similar to that described earlier using a simplified version of the gashandling systems). The main differences were that the oil-pressure part was replaced by a system in which a pressure could be built up from a gas cylinder, and that instead of the oil-pressure balance the bourdon gauge was used, mounted in the isotope-gas part of the pressure system. Furthermore a more compact version of the diaphragm systemlo) has been built, which now could be operated without thermostat. With this equipment the temperature and pressure of the sample were stable to within about 3 x 1O-a, as a result of which the density remained constant within 1 x 10-s. The gas sample consisted of 99.56 mol y0 ssAr, which has the large scattering cross section us of (73.7 * 0.4) brr) and, for 1 A neutrons, an absorption cross section oa of (6.5 f 1.O) brs). The sample was kept at 141.3 K under 3 1.07 atm, from which follows a density of 2.52 x 1021 atoms/ems. The holder of the gas (see fig. 1) was an 8.5 mm thick plane slab of aluminium, in which an array of cylindrical holes has been drilled of 7.5 mm diameter at separations of 1.0 mm; from this follows an effective thickness t of 5.25 mm. The transmission of the sample T = exp[-$(a, + a,)] therefore was 0.90. Against both sides of the holder 0.5 mm thick cadmium masks were employed, accurately situated in front of the three central holes of the
194
C. D. ANDRIESSE
holder. volume The on the
AND
E. LEGRAND
These masks determined the geometry for the effective scattering of the sample (cf. section 3). neutron-diffraction measurements on the gas have been performed powder diffractometer at the BR21a), using 1.065 A neutrons from
the (111) planes of a copper monochromator in reflection. The sample, mounted on the goniometer table, was rotated at one half the speed of the BFa detector. At the sample position the beam of (2.5 & 0.2) x 106 neutrons /second was 42 mm high and 3% mm wide with a horizontal divergence of 0.5”. Luminosity differences in the beam have been determined by neutron photography, since they played a role in calculating the effective scattering volume of the sample. Two different fission chambers with separated power supplies and amplifiers were used to monitor the beam in order to detect eventual long-term variations in the electronics. Though good counting statistics were desirable, we found it practical to limit the counting time at each detector position to about 30 min. On the other hand we measured at a large number of different detector positions between 5” and 65” scattering angle 20 and repeated each run several times. The number of counts collected in this way was near 12,000 for each detector position. A similar procedure has been followed to measure the scattering from the empty sample holder, which, between the aluminium diffraction peaks, was as low as 600. Actually we alternately made measurements with the sample holder filled and with the sample holder empty. The corresponding measurements have been compared with each other and only those numbers have been added, which within the statistical spread appeared to be identical. Finally we added the intensities counted at two subsequent detector positions. Using this procedure it proved to be possible to collect lo/o-accurate scattering data during an extended period of time (a few months). 3. Evalzcatiolz of the exfieriwaental data. We have taken into account corrections which could amount to about IO-2 or more, viz. for background scattering, effective scattering volume, multiple scattering and for the static approximation. No correction was needed for the finite linewidth of the diffractometer (about 0.7” at 20” scattering angle), since calculations with Eckart’s formula14) showed that they are less than 10-a. A corrected angle-dependent intensity S(0) has been obtained from
44 - w. + mwi v(ed -.- p. + 773(8,)-j v(e)
s(e) = Ace,)
~(1
-
o1)-
f(e).
Here A(8) is the intensity measured with the sample holder filled; Ba and B(8) are, respectively, the angle-independent and angle-dependent components of the scattering measured with the sample holder empty; T is the transmission of the Ar sample; v(0) the effective scattering volume; 28, a
PAIR
certain scattering
POTENTIAL
FOR ARGON
195
angle; M the ratio of multiple to primary scattering
01that fraction of M which is not counted in the detector;
and
f(0) the correction
factor to the static approximation. The structure factor S(Q) is obtained from S(0) by changing 0 into the wave number Q = 471.sin(B)/&, & being the neutron wavelength used. The value of Bo has been determined from the intensity obtained at large 8 between the diffraction peaks of the aluminium of the sample holder; the remainder of the background scattering, B(O), is due to these diffraction peaks and the pronounced air scattering near the through-going neutron beam. The effective scattering volume v(0) of the sample has been determined from the geometry of the sample holder with its cadmium masks (see fig. l), taking into account the inhomogeneities of the beam intensity. If the cadmium plates would have no thickness and the beam would be perfectly homogeneous, v(0) should be angle-independent. In the actual situation there has been a gradual decrease starting at 28 = 5” to 0.86 of the initial value at 28 = 65”. The multiple scattering has been determined with the aid of a table given by Brockhouse et al. 15) with results for plane slabs. Detailed calculations by Zandveldle) have proved that even for a dense fluid the isotropic approximation underlying the results of Brockhouse et al. is very good. If no cadmium masks were employed we could use the tabulated value of M, which in our case is 0.170. However, there is a part 01of M which is scattered into the direction of the detector from between the two cadmium plates, which is not counted (see fig. 1). The value of CYhas been estimated from cy = exp[--p$w(o, + o&)], where ze,is the width of the opening in the masks, and it is found to be 0.78. In order to subtract M( 1 - a) we had to normalize the background- and volume-corrected intensity to its asymptotic value. This means that we had to make a choice for 28,. Since, however, f(0) still has to be taken into account, there is some uncertainty on the height of the asymptote. We chose 28, = 500, where f(f3) is still below 10-s and the structure factor should be close to crossing the asymptote. The correction to the static approximation f(0) is calculated using (A.8) derived in appendix I. We here remark that our result for f(f3)differs somewhat from the results obtained by Placzek17) and Ascarelli and Caglioti l*) . In order to verify whether systematic errors were introduced in handling the experimental data as described above, we have analysed S(Q) in the following way 19). First a smooth curve was fitted through the experimental points. This curve has been extended somewhat arbitrarily to Q = 0 and to Q = 13 A-l so that the thermodynamic condition for S(0) is satisfied, i.e. S(0) = 3.12, and that for Q > 6 A-1 S(Q) = 1. Then the curve was trans-
196
C. D. ANDRIESSE
AND
E. LEGRAND
formed into the atomic distribution function g(r) according Prins relation 00 1 dQ sin(Q4 QWQ) - 11. g(r) = 1 + -2x2pr
to the Zernike-
(24
s 0
The result for g(r) has been corrected to give the value zero from Y = 0 up to about r = 3.0 A, a value slightly below the core diameter of Ar. This corrected g(r) function was transformed back to S(Q) using co
dr sin(Qr) r[g(r) -
S(Q)= 1 + 5
w
11,
s
0
and it has been compared to the experimental data. After a correction of this result, in order to improve the fit of the experimental data and to retain the correct value of S(O)?, the g(r) function was calculated again, etc. As
Fig. 2. Corrected experimental 2.52 x
1021 atoms/cm3
data on the structure factor S(Q) of gaseous ssAr at
and 141.3 K (dots), compared with the curve used to compute the pair potential (see text).
t It was found that the best interpolation between Q = 0 and Q = 0.35 A-1 closely follows the lorentzian
S(Q) = S(O)/(l
+ LsQs)
with
L = 5.0A; this L value is not se) from X-ray small-
much different from the values found by Thomas and Schmidt angle scattering of Ar near the critical point.
PAIR
this procedure
converged
possible systematic
POTENTIAL
FOR
197
ARGON
rapidly to a physically
meaningful
result for g(y),
errors in our result should be small.
In fig. 2 the result for S(Q) obtained after five iterations is compared with the corrected experimental data. Using this curve we again computed g(r) and also the direct correlation function c(r) given by m
1
C(Y) = ___ dQ%Q~lQ[l 2xspr J’
(3)
- &]a
0
From g(r) and C(I) the potential
was derived
TABLE Numerical
data
by applying
the results of
I
of the pair potential
q(r)/k
our neutron-diffraction qlk
for Ar derived
from
experiment qlk
y/k (K)
(i)
(L
(K)
(i)
3.10
195
3.66
-117.5
4.22
-91
3.12
160
3.68
-
119.3
4.24
-89
3.14
130
3.70
-
120.7
4.26
-86
3.16
105
3.72
-
121.9
4.28
-84
3.18
82
3.74
- 122.8
4.30
-81
3.20 3.22
61
3.76
-
4.32
-79
43 27
3.78
-
123.8
4.34
-77
3.24
3.80
-
124.0
4.36
-75
3.26
13
3.82
-
123.9
4.38
-73
123.5
(K)
0
3.84
- 123.6
4.40
-70
3.30
-
12
3.86
-123.1
4.42
-68
3.32
-
23
3.88
-
122.4
4.44
-66
3.34
-
33
3.90
-
121.5
4.46
-64
3.36
-
43
3.92
-
120.5
4.48
-62
3.38
-
51
3.94
-119.3
4.50
-60
3.40
-
59
3.96
-118
4.60
-51
3.42
-
67
3.98
-
4.70
-43
3.44
-
74
4.00
-115
4.80
-36
3.46
-
80 85
4.02 4.04
-113
4.90
-111
5.00
-30 -24
90 95 -100 -104
4.06 4.08 4.10
-109 -107
5.10 5.20
4.12
-105 -103
5.30 5.40
4.14
-101
5.50
3.60 3.62
-107 -110 -113
4.16 4.18
-
99
6.00
3.64
-
4.20
-
96 94
6.50 7.00
3.28
3.48 3.50 3.52 3.54 3.56 3.58
115.5
116.5
-19 - 14.5 -11 -9 -6 -4 -2
7.5
198
C. D. ANDRIESSE
HNC and the Percus-Yevick &(r)
= g(r) -
c(r) -
AND
theorysi),
E. LEGRAND
respectively
1 + W(r)l,
(4)
Bd4 = ln[l - WlsP)!. The potentials
(5)
obtained using (4) and (5) appeared to be identical within
0.2 K. In table I the average of the values for p(r)/K is listed for the range 3.1 A to 7.0 A. In the region of the potential well (3.35 a to 4.8 A) the absolute accuracy of our data is estimated to be & 4 K. Outside this region the inaccuracy is somewhat larger. This estimate is based on the variations in v(r) as obtained in the Fourier transformation procedure. 4. Discussion. We first go into the two questions whether, for the relevant density and temperature, the structural theories are reliable and whether corrections for non-additivity can be neglected. From the striking similarity of the results obtained with HNC and the Percus--Yevick theory it can be concluded that in the density expansion of c(r) those graphs, which both theories have in common, are the important ones. Verletis) has tested the theory by computing S(Q) for fluids with Lennard-Jones interaction using both molecular dynamics and the Percus-Yevick theory. He has found perfect agreement between “experiment” and theory, at least for critical densities and lower. This gives confidence in the validity of the theory for the low-density state considered in our experiment. Non-additivity effects have been estimated by Copeland and Kestner 7),
-180+~---
Fig. 3. Comparison (DA)
numerical
5
6
of our result for the pair potential
potential
and the Lennard-Jones (KH)
analytic
(LJ),
potentials.
(band) Kihara
with the Dymond-Alder (K) and Klein-Hanley
PAIR
who found that triple-dipole bution.
According
POTENTIAL
interactions
to their calculations
199
FOR ARGON
give the most substantial based on the Kingston
contri-
composite
potentialss), which has a shape not very different from our result, the decrease of the minimum of the potential due to these interactions is, at 2.52 x 1021 atoms/cm3 and 141.3 K, equivalent with 3.3 K. Though this is just within the experimental inaccuracy of our result, it still is not a negligible effect, which should be traced in a further investigation of S(Q) over a range of densities in the gas phase. For such experiments a precision of at least a few times 10-s is a prerequisite. In fig. 3 our result is compared with the Dymond-Alder numerical potentiall) and with the following analytic potentials: Lennard- Jones (12, 6) V(Y)/& = 4[(o/V”
-
(o/VI,
(6)
where E/K = 119.8 K and 0 = 3.405 A as determined by Levelt23) virial-coefficient data mainly at supercritical temperatures, Kihara (12, 6)
from
(7) with E/k = 163.7 K, c = 3.150 A and y = 0.164 as determined by Weir et al.24) from second virial-coefficient data at subcritical temperatures only, and: Klein-Hanley (m, 6, 8) 25) m - +z
- 8)
(r/lm) ’
1
(8)
with E/k = 153 K, 0 = 3.292 A (the relation between CTand the position of the potential minimum I m is discussed below), m = 11 and y = 3.0 as determined from both virial and viscosity data over the wide temperature range from about 100 K to about 2000 K. From fig. 3 it is evident that our result resembles most the Dymond-Alder potential. Though a discrepancy is observed in the depth E, the general shape is similar, whereas the values of CJand rm essentially are the same. It is of interest to note that according to Tanaka and Yoshinos6), who studied the ultraviolet absorption spectrum of the Ars molecule, F/k should be about 132 K, which is in between our value and that of Dymond and Alder. One notices that the value lrn for the Lennard-Jones potential is quite correct, but that c is too high whereas the outer wall of the well is not sufficiently steep. Our result clearly rules out the approximation of the Kihara potential, which is much too deep and has too low values for c and rm. There also is a discrepancy with the Klein-Hanley potential. We tried to adjust the parameter y in this potential so that our value for the ratio G/lm = (0.862 & 0.004) is obtained, and found that, for m = 11, y be close to
C. D. ANDRIESSE
200
AND
E. LEGRAND
more it appeared that for m values between 9 and 18 no positive y is compatible with our result. It therefore seems questionable whether a better representation of the actual potential can be obtained only by adding an (r&)8 term. For the limited range of 3.68 A to 3.94 A our result can be approximated bY @)/E
+
1 = ‘ic (7 -
G?p,
(9)
with c/k = (124 & 4) K, lm = (3.81 & 0.01) A and C = (4.5 5 1) A-2. The constant C is close to the value 4.93 A-2 for &-rdsv(r)/drs of the LennardJones potential at I = lm. However, the latter potential is less symmetric with respect to the position of the minimum. We note that the lattice constant a = (5.31109 f 0.00008) A for the f.c.c. structure of solid Ar at 4.25 Ka7) gives a nearest-neighbour distance which is (0.05 i 0.01) A below our value for rm. This possibly is an effect of the zero-point energy in the crystal lattice, as has been discussed by Horton and Leechss). With regard to the long-range part of the potential we note that for Y > 5 A our data can be described within the estimated error by v(r) = C(@/Y~ with the theoretical predictionsa) of the van der Waals constant C(6) of (5.95 f 0.54) x 10-7s Jm 6. It should be stressed that our data on the repulsive part of the potential, notably for v(r) > 0, cannot be very accurate, since it has been impossible for us to trace the details of the weak oscillatory behaviour of S(Q) for Q > 4 A-r. In appendix II we suggest that the measurement of phase and damping of the oscillations at large Q values in the liquid structure factor can be used to find the derivative of the core potential. Here an expression is derived, which relates the hardness of the core to the damping of these oscillations relative to the hard-sphere case. However, accurate information on S(Q) for liquid Ar at various temperatures is lacking. We are indebted to E. J. Jansen, who assisted in Acknowledgement. the measurements and performed the various Fourier transformations. It is a pleasure to thank J. Baudeweyns, H. P. E. Perre, M. Van Roy and R. H. Vis for their technical assistance and F. Lafhe for the drawings. The comment by Professor J. J. van Loef on the manuscript greatly helped to improve the presentation. One of us (C.D.A.) wants to express his gratitude to M. N&e de MCvergnies and J. Nihoul for his pleasant stay at the S.C.K./ C.E.N. in Mol.
PAIR
POTENTIAL
APPENDIX
201
FOR ARGON
I
In calculating a correction to the static approximation we have tried to unify the treatments of Placzek17) and of Ascarelli and Cagliotiis). The former actually consists in an expansion of &/)3, & being the wavelength of neutrons incident on the sample and 3Lthe wavelength after scattering by the sample, in terms of the frequency cr)defined in (A. 1). Then j dc&/lS(Q, o) is evaluated in terms of the frequency moments of the scattering law S(Q, LC)), some of which are theoretically known. The latter treatment takes into account that in a diffraction experiment the above integration is performed at constant scattering angle 213 rather than at constant Q. In order to account for this, a model is assumed for S(Q, o), in which Q and cr) are expressed in 28 and 1, the former remaining constant and the latter being the integration variable. However, the model assumed for S(Q, w) violates the detailed-balance condition, which in the case of a gas may be serious. In our calculation we prefer to evaluate J dw&/iZS(Q, W) rather than to integrate over 1, assuming for S(Q, o) the form given by Ascarelli and Caglioti modified with the neutron recoil term and expanding &/l and in powers of W. We shall consider terms up to quadratic in cc). The relation between w, ilg and 1 is
(A-1) where ti is Planck’s
constant
divided by 2x and m the neutron mass. Hence
(A4 with oi = ~+%/(wz$)
= 1.09 x 1014 s-1 for the diffractometer
used.
The model assumed for the scattering law is
M being the atomic mass, which has a correct zeroth, first and, except for the quantum correction, second moment. In this form we have to expand S*(Q)/Q in powers of o. The o dependence of Q and S(Q) in the exponent can be neglected if only terms up to quadratic in o are considered. From
202
C. D. ANDRIESSE
AND
E. LEGRAND
we derive, using (A.l)
l-f:-&
Q=Qt
(A-5)
W
where Qz = 4x sin(8)/&, i.e. the “static” value of the wavenumber. After expanding S(Q) around S(Qg) we find
S8(Q) Q
S'(Qd
-Qi +
c (> z
2 (&
1+ t
-
i&l
$21)
+
i&z:
+
i32)
1
(A4
.
Here zi = QiS’(Qi)/S(Qz) and zz = QiS”(QJ/S(Qi), whereas S’(Qt) and S”(Q6) are the first and second derivatives of S(Qz). Inserting (A.6) into (A.3), (A.3) and (A.2) into j dw&/ilS(Q, W) and then performing the integration leads to the result
(3 -
$1
+
$2; +
4z2).
(A.7
The first term S(Qi) is the classic result of the static approximation, the other two terms give, in powers of Qf, corrections to this approximation. It may be noted that in our treatment the contribution involving the second derivative of S(Qi) and the Q! coefficient, viz. 3
?VQi
64
/3M@~u;
Q:s"(Qi)
s(Qi) '
which can be important in the analysis of subsidiary maxima of S(Q), has the same coefficient as in the treatment by Ascarelli and Caglioti. In our experiment on gaseous Ar the curvature of S(Q) is only slight. For this reason we can neglect terms with derivatives of S(Qe). Furthermore it is found that, for the Qf of interest, the first correction term in (A.7) is much larger than the second. For our purpose it seemed to be sufficient therefore to subtract from the measured intensity the fraction
(see section 3). This fraction is very small and it becomes is as high as 7.8 k-1.
1OV2only when Qi
PAIR POTENTIAL FOR ARGON
APPENDIX
203
II
We here present a simple calculation, which suggests that accurate measurements of phase and damping of the oscillations in the structure factor of fluids at high density can give direct information on the derivative of the core potential. It is known that liquid structures can be described remarkably well by hard spheresso), but there are some minor discrepancies which have been ascribed to the finite hardness of the core31). Consider the potential v(r) = vo[l -
r/~ol,
(A.9
for I between 0 and ~0, which is zero for r > ~0. The value of ~0 is assumed to be much larger than the thermal energy B-1. The slope -IJ&O will be considered as the effective hardness of the core and it will depend on j3. Since we are going to consider large Q values only, one can apply the dilutegas approximation for the distribution function g(r), i.e., g(r) = exp[-/$(r)]. relation If we insert (A.9) and then calculate S(Q) with the Zernike-Prins (2b), we obtain for ~F,O> 1,
(A. 10) The crossing points of the asymptote, are approximately given by
which measure
tg(Qro) = QroP- (Q~o)~/(Bvo)I-~>
the phase of S(Q), (A.ll)
which may be solved by a graphical method. For ~V)O-+ 00 one obtains the well-known solutions for the hard-sphere case somewhat below Qro = cross (1z + 3) x, n = 2, 3, . . . For decreasing &JO the solutions eventually the above values for QYO and approach Qro = (n + 1) x, which implies a shift of the oscillations in S(Q) to higher Q. The damping of the oscillations, d, relative to the hard-sphere case is found to be approximately
d = 11+ (Q~o)~/(Bvo)~I-~>
(A. 12)
if one assumes that the maxima and minima of S(Q) are located close to QYO= nx. This clearly shows the increase of damping by a decrease of &JO. We remark that both this phase shift and enhanced damping have been foundsi) and that notably (A. 12) can be used with some success to determine the derivative of the core potential at different heights (/?-1 - E), E being the potential depth.
204
PAIR POTENTIAL
FOR ARGON
REFERENCES 1) Dymond, J. H. and Alder, B. J., J. them. Phys. 51 (1969) 309. 2) Baratz, B. and Andress, R. P., J. them. Phys. 52 (1970) 6145. 3) Cavallini, M., Gallinaro, G., Meneghetti, L., Stoles, G. and Valbusa, U., Chem. Phys. Letters 7 (1970) 303. 4) Buck, U., J. them. Phys. 54 (1971) 1923. 5) Mikolaj, P. G. and Pings, C. J., J. them. Phys. 46 ( 1967) 1401. 6) Levesque, D. and Verlet, L., Phys. Rev. Letters 20 (1968) 905. 7) Copeland, D. A. and Kestner, N. R., J. them. Phys. 49 (1968) 5314. 8) Hasman, A. and Zandveld, P., Phys. Letters 34A (1971) 112. 9) Andriesse, C. D., Physica 48 (1970) 61. 10) Andriesse, C. D., Rev. sci. Instrum. 39 (1968) 400. 11) Krohn, V. E. and Ringo, G. R., Phys. Rev. 148 (1966) 1303. 12) McMurtrie, G. E. and Crawford, D. P., Phys. Rev. 77 (1950) 840. 13) Legrand, E., Thesis Catholic University of Leuven, 1966. 14) Eckart, C., Phys. Rev. 51 (1937) 735. 15) Brockhouse, B. N., Corliss, L. M. and Hastings, J. M., Phys. Rev. 98 (1955) 1721. 16) Zandveld, P., Internal report Interuniversiteir Reactor Instituut, Delft, 1969. 17) Placzek, G., Phys. Rev. 86 (1952) 377. 18) Ascarelli, P. and Caglioti, G., Nuovo Cimento 43B (1963) 375. 19) Verlet, L., Phys. Rev. 165 (1968) 201. 20) Thomas, J. E. and Schmidt, P. W., J. them. Phys. 39 (1963) 2506. 21) Rice, S. A. and Gray, P., The statistical mechanics of simple liquids, Interscience Publ. (New York, 1965). 22) Kingston, A. E., J. them. Phys. 42 (1965) 719. 23) Levelt, J. M. H., Physica 26 (1960) 361. 24) Weir, R. D., Wynn Jones, I., Rowlinson, J. S. and Saville, G., Trans. Faraday Sot. 63 (1967) 1320. 25) Klein, M. and Hanley, H. J. M., J. them. Phys. 53 (1970) 4722. 26) Tanaka, Y. and Yoshino, K., J. them. Phys. 53 (1970) 2012. 27) Peterson, 0. G., Batchelder, D. N. and Simmons, R. O., Phys. Rev. 158 (1966) 703. 28) Horton, G. K. and Leech, J. W., Proc. Phys. SOC. 82 ( 1963) 8 16. 29) Gordon, R. G., J. them. Phys. 48 (1968) 3929. 30) Ashcroft, N. W., Physica 35 (1967) 148. 31) Page, D. I., Egelstaff, P. A., Enderby, J. E. and Wingfield, B. R., Phys. Letters 29A (1969) 296.
PAIR FROM NEUTRON
POTENTIAL
FOR ARGON
DIFFRACTION
AT LOW DENSITY
C. D. ANDRIESSE* Interuniversitair Reactor Instituut, Delft, Nederland E. LEGRAND Studiecentrum VOOY Kernenergie, S.C.K./C.E.N., Received
Mel, BelgiB
11 June 1971
Synopsis A diffraction experiment is described in which 1.065 A neutrons are incident on gaseous ssAr at a density of 2.52 x 1021 atoms/cm3 and temperature of 141.3 K. Information accurate to 1% is obtained on the structure factor S(Q) in the wavenumber range 0.5 A-1 < Q < 6.3 A-1. Using both HNC and the Percus-Yevick theory the pair potential q(r) for Ar is derived from a set of S(Q) data. This set of data is extended to Q = 0 and Q = 13 A-i in such a way that, within the core of the potential, the pa.ir correlation function is zero and the correct hydrodynamic limit for S(Q) is obtained. The locations of V(Y) = 0 and dq(r)/dr = 0 are found to be (3.28 f 0.01) A and (3.81 f 0.01) A, respectively, whereas the depth e/k = (124 & 4) K. The pair potential, which is tabulated, is compared with the one discussed by Dymond and Alder and some analytic potentials. In the appendices a new correction to the static approximation and a relation for the damping of oscillations in S(Q) at large Q values are calculated.
1. Ilztroduction. A direct way to determine the interaction potential of two atoms is to measure the diffraction of X rays or neutrons by a system of these atoms at a density which is sufficiently low that screening of the pair interaction by other atoms can be neglected. For a dilute gas the diffraction pattern is the Fourier transform of the atomic distribution function in its asymptotic (p + 0) form p exp[ -&(r)], where p is the average number density, q(r) the pair potential and /? = (kT)-1, K being Boltzmann’s constant and T the temperature. Since in diffraction experiments a finite density is always needed, one has to apply results from density expansions in structural theory. It should be verified then that the result for v(r) is only little affected by the approximations of, e.g., hypernetted chain (HNC)
* Present address : Ruimte-Onderzoek, Groningen .
WSN-gebouw
191
Paddepoel, Rijksuniversiteit,
192 or the Percus-Yevick are small.
C. D. ANDRIESSE AND E. LEGRAND theory, and that corrections
for non-additivity
indeed
In the case of argon the knowledge of v(r) so far is based almost exclusively on the application of kinetic theory to macroscopic properties. By carefully considering all known properties of Ar Dymond and Alderi) have constructed a numerical potential function, which substantially deviates from analytic potentials like those of Lennard-Jones or Kihara. Though atomicbeam scattering techniques in principle can give information on V(Y) without relying on kinetic theory, the study of Ar-Ar scattering23 a) has not progressed sufficiently that scattering data could be inverted directly4). The only existing direct information on v(r) of Ar, derived from X-ray scattering on the (dense) gass), appears to be in conflict with other evidence637). Furthermore systematic differences are observed between these X-ray results and recent neutron-scattering data on a part of the structure factor of gaseous Ar in a corresponding states). Neutron-scattering experiments on gaseous Ar therefore seem to be useful. In this paper we present results of diffraction measurements on isotopically pure s6Ar at about l/10 of the density in the solid state. The problems we met in determining p(r) with reasonable precision were considerable. Because of the weakness of the scattering and the lack of a pronounced structure in the gas, the measurements took a long time, during which many parameters of the experimental setup had to be kept constant. Furthermore an accurate knowledge was required of the various correction factors. We therefore elaborate on the description of the experimental procedure and the applied corrections, given in sections 2 and 3, respectively. The results for the pair potential are discussed in section 4. In the appendices a new correction to the static approximation and a relation for the damping of oscillations in the diffraction pattern at large scattering angles are calculated. 2. Experimental procedure. The density chosen for our experiment is ten times lower than in the solid. This is a compromise between the requirements that p should be as low as possible and that a lo/o-accurate diffraction experiment contains useful information. For a given density of the gas the approximate form of the distribution function p exp[-_Bq(Y)] is most pronounced at the lowest possible temperature, i.e. at the onset of condensation. The condensation temperature at & of the density in solid Ar is about 140 K and the corresponding vapour pressure about 30 atm. The latter temperature and pressure therefore were relevant for our experiment . In order to cool the gas, the Ar-sample holder was placed in a liquidnitrogen cryostat with 1 mm thick aluminium windows for the neutron beam. It has been kept at the desired temperature by heating two zener
PAIR
i
/
POTENTIAL
//
/
/
193
FOR ARGON
0
,
2
Jcm
Fig. 1. Horizontal cross section of the sample holder with the cadmium masks and its position with respect to the neutron beam at 47” scattering angle; A and B are trajectories of twice-scattered neutrons which can and cannot be detected, respectively.
diodes on the small tube connecting holder and liquid-nitrogen reservoir, using a proportional control system with a Rosemount EC 1050 platinum resistance as sensor. The latter also served to measure the sample temperature and it has been calibrated against the vapour-pressure curve of Ar using a calibrated Wallace and Tiernan FA 233 bourdon gauge as pressure indicator. In order to obtain the desired pressure, we applied a technique similar to that described earlier using a simplified version of the gashandling systems). The main differences were that the oil-pressure part was replaced by a system in which a pressure could be built up from a gas cylinder, and that instead of the oil-pressure balance the bourdon gauge was used, mounted in the isotope-gas part of the pressure system. Furthermore a more compact version of the diaphragm systemlo) has been built, which now could be operated without thermostat. With this equipment the temperature and pressure of the sample were stable to within about 3 x 1O-a, as a result of which the density remained constant within 1 x 10-s. The gas sample consisted of 99.56 mol y0 ssAr, which has the large scattering cross section us of (73.7 * 0.4) brr) and, for 1 A neutrons, an absorption cross section oa of (6.5 f 1.O) brs). The sample was kept at 141.3 K under 3 1.07 atm, from which follows a density of 2.52 x 1021 atoms/ems. The holder of the gas (see fig. 1) was an 8.5 mm thick plane slab of aluminium, in which an array of cylindrical holes has been drilled of 7.5 mm diameter at separations of 1.0 mm; from this follows an effective thickness t of 5.25 mm. The transmission of the sample T = exp[-$(a, + a,)] therefore was 0.90. Against both sides of the holder 0.5 mm thick cadmium masks were employed, accurately situated in front of the three central holes of the
194
C. D. ANDRIESSE
holder. volume The on the
AND
E. LEGRAND
These masks determined the geometry for the effective scattering of the sample (cf. section 3). neutron-diffraction measurements on the gas have been performed powder diffractometer at the BR21a), using 1.065 A neutrons from
the (111) planes of a copper monochromator in reflection. The sample, mounted on the goniometer table, was rotated at one half the speed of the BFa detector. At the sample position the beam of (2.5 & 0.2) x 106 neutrons /second was 42 mm high and 3% mm wide with a horizontal divergence of 0.5”. Luminosity differences in the beam have been determined by neutron photography, since they played a role in calculating the effective scattering volume of the sample. Two different fission chambers with separated power supplies and amplifiers were used to monitor the beam in order to detect eventual long-term variations in the electronics. Though good counting statistics were desirable, we found it practical to limit the counting time at each detector position to about 30 min. On the other hand we measured at a large number of different detector positions between 5” and 65” scattering angle 20 and repeated each run several times. The number of counts collected in this way was near 12,000 for each detector position. A similar procedure has been followed to measure the scattering from the empty sample holder, which, between the aluminium diffraction peaks, was as low as 600. Actually we alternately made measurements with the sample holder filled and with the sample holder empty. The corresponding measurements have been compared with each other and only those numbers have been added, which within the statistical spread appeared to be identical. Finally we added the intensities counted at two subsequent detector positions. Using this procedure it proved to be possible to collect lo/o-accurate scattering data during an extended period of time (a few months). 3. Evalzcatiolz of the exfieriwaental data. We have taken into account corrections which could amount to about IO-2 or more, viz. for background scattering, effective scattering volume, multiple scattering and for the static approximation. No correction was needed for the finite linewidth of the diffractometer (about 0.7” at 20” scattering angle), since calculations with Eckart’s formula14) showed that they are less than 10-a. A corrected angle-dependent intensity S(0) has been obtained from
44 - w. + mwi v(ed -.- p. + 773(8,)-j v(e)
s(e) = Ace,)
~(1
-
o1)-
f(e).
Here A(8) is the intensity measured with the sample holder filled; Ba and B(8) are, respectively, the angle-independent and angle-dependent components of the scattering measured with the sample holder empty; T is the transmission of the Ar sample; v(0) the effective scattering volume; 28, a
PAIR
certain scattering
POTENTIAL
FOR ARGON
195
angle; M the ratio of multiple to primary scattering
01that fraction of M which is not counted in the detector;
and
f(0) the correction
factor to the static approximation. The structure factor S(Q) is obtained from S(0) by changing 0 into the wave number Q = 471.sin(B)/&, & being the neutron wavelength used. The value of Bo has been determined from the intensity obtained at large 8 between the diffraction peaks of the aluminium of the sample holder; the remainder of the background scattering, B(O), is due to these diffraction peaks and the pronounced air scattering near the through-going neutron beam. The effective scattering volume v(0) of the sample has been determined from the geometry of the sample holder with its cadmium masks (see fig. l), taking into account the inhomogeneities of the beam intensity. If the cadmium plates would have no thickness and the beam would be perfectly homogeneous, v(0) should be angle-independent. In the actual situation there has been a gradual decrease starting at 28 = 5” to 0.86 of the initial value at 28 = 65”. The multiple scattering has been determined with the aid of a table given by Brockhouse et al. 15) with results for plane slabs. Detailed calculations by Zandveldle) have proved that even for a dense fluid the isotropic approximation underlying the results of Brockhouse et al. is very good. If no cadmium masks were employed we could use the tabulated value of M, which in our case is 0.170. However, there is a part 01of M which is scattered into the direction of the detector from between the two cadmium plates, which is not counted (see fig. 1). The value of CYhas been estimated from cy = exp[--p$w(o, + o&)], where ze,is the width of the opening in the masks, and it is found to be 0.78. In order to subtract M( 1 - a) we had to normalize the background- and volume-corrected intensity to its asymptotic value. This means that we had to make a choice for 28,. Since, however, f(0) still has to be taken into account, there is some uncertainty on the height of the asymptote. We chose 28, = 500, where f(f3) is still below 10-s and the structure factor should be close to crossing the asymptote. The correction to the static approximation f(0) is calculated using (A.8) derived in appendix I. We here remark that our result for f(f3)differs somewhat from the results obtained by Placzek17) and Ascarelli and Caglioti l*) . In order to verify whether systematic errors were introduced in handling the experimental data as described above, we have analysed S(Q) in the following way 19). First a smooth curve was fitted through the experimental points. This curve has been extended somewhat arbitrarily to Q = 0 and to Q = 13 A-l so that the thermodynamic condition for S(0) is satisfied, i.e. S(0) = 3.12, and that for Q > 6 A-1 S(Q) = 1. Then the curve was trans-
196
C. D. ANDRIESSE
AND
E. LEGRAND
formed into the atomic distribution function g(r) according Prins relation 00 1 dQ sin(Q4 QWQ) - 11. g(r) = 1 + -2x2pr
to the Zernike-
(24
s 0
The result for g(r) has been corrected to give the value zero from Y = 0 up to about r = 3.0 A, a value slightly below the core diameter of Ar. This corrected g(r) function was transformed back to S(Q) using co
dr sin(Qr) r[g(r) -
S(Q)= 1 + 5
w
11,
s
0
and it has been compared to the experimental data. After a correction of this result, in order to improve the fit of the experimental data and to retain the correct value of S(O)?, the g(r) function was calculated again, etc. As
Fig. 2. Corrected experimental 2.52 x
1021 atoms/cm3
data on the structure factor S(Q) of gaseous ssAr at
and 141.3 K (dots), compared with the curve used to compute the pair potential (see text).
t It was found that the best interpolation between Q = 0 and Q = 0.35 A-1 closely follows the lorentzian
S(Q) = S(O)/(l
+ LsQs)
with
L = 5.0A; this L value is not se) from X-ray small-
much different from the values found by Thomas and Schmidt angle scattering of Ar near the critical point.
PAIR
this procedure
converged
possible systematic
POTENTIAL
FOR
197
ARGON
rapidly to a physically
meaningful
result for g(y),
errors in our result should be small.
In fig. 2 the result for S(Q) obtained after five iterations is compared with the corrected experimental data. Using this curve we again computed g(r) and also the direct correlation function c(r) given by m
1
C(Y) = ___ dQ%Q~lQ[l 2xspr J’
(3)
- &]a
0
From g(r) and C(I) the potential
was derived
TABLE Numerical
data
by applying
the results of
I
of the pair potential
q(r)/k
our neutron-diffraction qlk
for Ar derived
from
experiment qlk
y/k (K)
(i)
(L
(K)
(i)
3.10
195
3.66
-117.5
4.22
-91
3.12
160
3.68
-
119.3
4.24
-89
3.14
130
3.70
-
120.7
4.26
-86
3.16
105
3.72
-
121.9
4.28
-84
3.18
82
3.74
- 122.8
4.30
-81
3.20 3.22
61
3.76
-
4.32
-79
43 27
3.78
-
123.8
4.34
-77
3.24
3.80
-
124.0
4.36
-75
3.26
13
3.82
-
123.9
4.38
-73
123.5
(K)
0
3.84
- 123.6
4.40
-70
3.30
-
12
3.86
-123.1
4.42
-68
3.32
-
23
3.88
-
122.4
4.44
-66
3.34
-
33
3.90
-
121.5
4.46
-64
3.36
-
43
3.92
-
120.5
4.48
-62
3.38
-
51
3.94
-119.3
4.50
-60
3.40
-
59
3.96
-118
4.60
-51
3.42
-
67
3.98
-
4.70
-43
3.44
-
74
4.00
-115
4.80
-36
3.46
-
80 85
4.02 4.04
-113
4.90
-111
5.00
-30 -24
90 95 -100 -104
4.06 4.08 4.10
-109 -107
5.10 5.20
4.12
-105 -103
5.30 5.40
4.14
-101
5.50
3.60 3.62
-107 -110 -113
4.16 4.18
-
99
6.00
3.64
-
4.20
-
96 94
6.50 7.00
3.28
3.48 3.50 3.52 3.54 3.56 3.58
115.5
116.5
-19 - 14.5 -11 -9 -6 -4 -2
7.5
198
C. D. ANDRIESSE
HNC and the Percus-Yevick &(r)
= g(r) -
c(r) -
AND
theorysi),
E. LEGRAND
respectively
1 + W(r)l,
(4)
Bd4 = ln[l - WlsP)!. The potentials
(5)
obtained using (4) and (5) appeared to be identical within
0.2 K. In table I the average of the values for p(r)/K is listed for the range 3.1 A to 7.0 A. In the region of the potential well (3.35 a to 4.8 A) the absolute accuracy of our data is estimated to be & 4 K. Outside this region the inaccuracy is somewhat larger. This estimate is based on the variations in v(r) as obtained in the Fourier transformation procedure. 4. Discussion. We first go into the two questions whether, for the relevant density and temperature, the structural theories are reliable and whether corrections for non-additivity can be neglected. From the striking similarity of the results obtained with HNC and the Percus--Yevick theory it can be concluded that in the density expansion of c(r) those graphs, which both theories have in common, are the important ones. Verletis) has tested the theory by computing S(Q) for fluids with Lennard-Jones interaction using both molecular dynamics and the Percus-Yevick theory. He has found perfect agreement between “experiment” and theory, at least for critical densities and lower. This gives confidence in the validity of the theory for the low-density state considered in our experiment. Non-additivity effects have been estimated by Copeland and Kestner 7),
-180+~---
Fig. 3. Comparison (DA)
numerical
5
6
of our result for the pair potential
potential
and the Lennard-Jones (KH)
analytic
(LJ),
potentials.
(band) Kihara
with the Dymond-Alder (K) and Klein-Hanley
PAIR
who found that triple-dipole bution.
According
POTENTIAL
interactions
to their calculations
199
FOR ARGON
give the most substantial based on the Kingston
contri-
composite
potentialss), which has a shape not very different from our result, the decrease of the minimum of the potential due to these interactions is, at 2.52 x 1021 atoms/cm3 and 141.3 K, equivalent with 3.3 K. Though this is just within the experimental inaccuracy of our result, it still is not a negligible effect, which should be traced in a further investigation of S(Q) over a range of densities in the gas phase. For such experiments a precision of at least a few times 10-s is a prerequisite. In fig. 3 our result is compared with the Dymond-Alder numerical potentiall) and with the following analytic potentials: Lennard- Jones (12, 6) V(Y)/& = 4[(o/V”
-
(o/VI,
(6)
where E/K = 119.8 K and 0 = 3.405 A as determined by Levelt23) virial-coefficient data mainly at supercritical temperatures, Kihara (12, 6)
from
(7) with E/k = 163.7 K, c = 3.150 A and y = 0.164 as determined by Weir et al.24) from second virial-coefficient data at subcritical temperatures only, and: Klein-Hanley (m, 6, 8) 25) m - +z
- 8)
(r/lm) ’
1
(8)
with E/k = 153 K, 0 = 3.292 A (the relation between CTand the position of the potential minimum I m is discussed below), m = 11 and y = 3.0 as determined from both virial and viscosity data over the wide temperature range from about 100 K to about 2000 K. From fig. 3 it is evident that our result resembles most the Dymond-Alder potential. Though a discrepancy is observed in the depth E, the general shape is similar, whereas the values of CJand rm essentially are the same. It is of interest to note that according to Tanaka and Yoshinos6), who studied the ultraviolet absorption spectrum of the Ars molecule, F/k should be about 132 K, which is in between our value and that of Dymond and Alder. One notices that the value lrn for the Lennard-Jones potential is quite correct, but that c is too high whereas the outer wall of the well is not sufficiently steep. Our result clearly rules out the approximation of the Kihara potential, which is much too deep and has too low values for c and rm. There also is a discrepancy with the Klein-Hanley potential. We tried to adjust the parameter y in this potential so that our value for the ratio G/lm = (0.862 & 0.004) is obtained, and found that, for m = 11, y be close to
C. D. ANDRIESSE
200
AND
E. LEGRAND
more it appeared that for m values between 9 and 18 no positive y is compatible with our result. It therefore seems questionable whether a better representation of the actual potential can be obtained only by adding an (r&)8 term. For the limited range of 3.68 A to 3.94 A our result can be approximated bY @)/E
+
1 = ‘ic (7 -
G?p,
(9)
with c/k = (124 & 4) K, lm = (3.81 & 0.01) A and C = (4.5 5 1) A-2. The constant C is close to the value 4.93 A-2 for &-rdsv(r)/drs of the LennardJones potential at I = lm. However, the latter potential is less symmetric with respect to the position of the minimum. We note that the lattice constant a = (5.31109 f 0.00008) A for the f.c.c. structure of solid Ar at 4.25 Ka7) gives a nearest-neighbour distance which is (0.05 i 0.01) A below our value for rm. This possibly is an effect of the zero-point energy in the crystal lattice, as has been discussed by Horton and Leechss). With regard to the long-range part of the potential we note that for Y > 5 A our data can be described within the estimated error by v(r) = C(@/Y~ with the theoretical predictionsa) of the van der Waals constant C(6) of (5.95 f 0.54) x 10-7s Jm 6. It should be stressed that our data on the repulsive part of the potential, notably for v(r) > 0, cannot be very accurate, since it has been impossible for us to trace the details of the weak oscillatory behaviour of S(Q) for Q > 4 A-r. In appendix II we suggest that the measurement of phase and damping of the oscillations at large Q values in the liquid structure factor can be used to find the derivative of the core potential. Here an expression is derived, which relates the hardness of the core to the damping of these oscillations relative to the hard-sphere case. However, accurate information on S(Q) for liquid Ar at various temperatures is lacking. We are indebted to E. J. Jansen, who assisted in Acknowledgement. the measurements and performed the various Fourier transformations. It is a pleasure to thank J. Baudeweyns, H. P. E. Perre, M. Van Roy and R. H. Vis for their technical assistance and F. Lafhe for the drawings. The comment by Professor J. J. van Loef on the manuscript greatly helped to improve the presentation. One of us (C.D.A.) wants to express his gratitude to M. N&e de MCvergnies and J. Nihoul for his pleasant stay at the S.C.K./ C.E.N. in Mol.
PAIR
POTENTIAL
APPENDIX
201
FOR ARGON
I
In calculating a correction to the static approximation we have tried to unify the treatments of Placzek17) and of Ascarelli and Cagliotiis). The former actually consists in an expansion of &/)3, & being the wavelength of neutrons incident on the sample and 3Lthe wavelength after scattering by the sample, in terms of the frequency cr)defined in (A. 1). Then j dc&/lS(Q, o) is evaluated in terms of the frequency moments of the scattering law S(Q, LC)), some of which are theoretically known. The latter treatment takes into account that in a diffraction experiment the above integration is performed at constant scattering angle 213 rather than at constant Q. In order to account for this, a model is assumed for S(Q, o), in which Q and cr) are expressed in 28 and 1, the former remaining constant and the latter being the integration variable. However, the model assumed for S(Q, w) violates the detailed-balance condition, which in the case of a gas may be serious. In our calculation we prefer to evaluate J dw&/iZS(Q, W) rather than to integrate over 1, assuming for S(Q, o) the form given by Ascarelli and Caglioti modified with the neutron recoil term and expanding &/l and in powers of W. We shall consider terms up to quadratic in cc). The relation between w, ilg and 1 is
(A-1) where ti is Planck’s
constant
divided by 2x and m the neutron mass. Hence
(A4 with oi = ~+%/(wz$)
= 1.09 x 1014 s-1 for the diffractometer
used.
The model assumed for the scattering law is
M being the atomic mass, which has a correct zeroth, first and, except for the quantum correction, second moment. In this form we have to expand S*(Q)/Q in powers of o. The o dependence of Q and S(Q) in the exponent can be neglected if only terms up to quadratic in o are considered. From
202
C. D. ANDRIESSE
AND
E. LEGRAND
we derive, using (A.l)
l-f:-&
Q=Qt
(A-5)
W
where Qz = 4x sin(8)/&, i.e. the “static” value of the wavenumber. After expanding S(Q) around S(Qg) we find
S8(Q) Q
S'(Qd
-Qi +
c (> z
2 (&
1+ t
-
i&l
$21)
+
i&z:
+
i32)
1
(A4
.
Here zi = QiS’(Qi)/S(Qz) and zz = QiS”(QJ/S(Qi), whereas S’(Qt) and S”(Q6) are the first and second derivatives of S(Qz). Inserting (A.6) into (A.3), (A.3) and (A.2) into j dw&/ilS(Q, W) and then performing the integration leads to the result
(3 -
$1
+
$2; +
4z2).
(A.7
The first term S(Qi) is the classic result of the static approximation, the other two terms give, in powers of Qf, corrections to this approximation. It may be noted that in our treatment the contribution involving the second derivative of S(Qi) and the Q! coefficient, viz. 3
?VQi
64
/3M@~u;
Q:s"(Qi)
s(Qi) '
which can be important in the analysis of subsidiary maxima of S(Q), has the same coefficient as in the treatment by Ascarelli and Caglioti. In our experiment on gaseous Ar the curvature of S(Q) is only slight. For this reason we can neglect terms with derivatives of S(Qe). Furthermore it is found that, for the Qf of interest, the first correction term in (A.7) is much larger than the second. For our purpose it seemed to be sufficient therefore to subtract from the measured intensity the fraction
(see section 3). This fraction is very small and it becomes is as high as 7.8 k-1.
1OV2only when Qi
PAIR POTENTIAL FOR ARGON
APPENDIX
203
II
We here present a simple calculation, which suggests that accurate measurements of phase and damping of the oscillations in the structure factor of fluids at high density can give direct information on the derivative of the core potential. It is known that liquid structures can be described remarkably well by hard spheresso), but there are some minor discrepancies which have been ascribed to the finite hardness of the core31). Consider the potential v(r) = vo[l -
r/~ol,
(A.9
for I between 0 and ~0, which is zero for r > ~0. The value of ~0 is assumed to be much larger than the thermal energy B-1. The slope -IJ&O will be considered as the effective hardness of the core and it will depend on j3. Since we are going to consider large Q values only, one can apply the dilutegas approximation for the distribution function g(r), i.e., g(r) = exp[-/$(r)]. relation If we insert (A.9) and then calculate S(Q) with the Zernike-Prins (2b), we obtain for ~F,O> 1,
(A. 10) The crossing points of the asymptote, are approximately given by
which measure
tg(Qro) = QroP- (Q~o)~/(Bvo)I-~>
the phase of S(Q), (A.ll)
which may be solved by a graphical method. For ~V)O-+ 00 one obtains the well-known solutions for the hard-sphere case somewhat below Qro = cross (1z + 3) x, n = 2, 3, . . . For decreasing &JO the solutions eventually the above values for QYO and approach Qro = (n + 1) x, which implies a shift of the oscillations in S(Q) to higher Q. The damping of the oscillations, d, relative to the hard-sphere case is found to be approximately
d = 11+ (Q~o)~/(Bvo)~I-~>
(A. 12)
if one assumes that the maxima and minima of S(Q) are located close to QYO= nx. This clearly shows the increase of damping by a decrease of &JO. We remark that both this phase shift and enhanced damping have been foundsi) and that notably (A. 12) can be used with some success to determine the derivative of the core potential at different heights (/?-1 - E), E being the potential depth.
204
PAIR POTENTIAL
FOR ARGON
REFERENCES 1) Dymond, J. H. and Alder, B. J., J. them. Phys. 51 (1969) 309. 2) Baratz, B. and Andress, R. P., J. them. Phys. 52 (1970) 6145. 3) Cavallini, M., Gallinaro, G., Meneghetti, L., Stoles, G. and Valbusa, U., Chem. Phys. Letters 7 (1970) 303. 4) Buck, U., J. them. Phys. 54 (1971) 1923. 5) Mikolaj, P. G. and Pings, C. J., J. them. Phys. 46 ( 1967) 1401. 6) Levesque, D. and Verlet, L., Phys. Rev. Letters 20 (1968) 905. 7) Copeland, D. A. and Kestner, N. R., J. them. Phys. 49 (1968) 5314. 8) Hasman, A. and Zandveld, P., Phys. Letters 34A (1971) 112. 9) Andriesse, C. D., Physica 48 (1970) 61. 10) Andriesse, C. D., Rev. sci. Instrum. 39 (1968) 400. 11) Krohn, V. E. and Ringo, G. R., Phys. Rev. 148 (1966) 1303. 12) McMurtrie, G. E. and Crawford, D. P., Phys. Rev. 77 (1950) 840. 13) Legrand, E., Thesis Catholic University of Leuven, 1966. 14) Eckart, C., Phys. Rev. 51 (1937) 735. 15) Brockhouse, B. N., Corliss, L. M. and Hastings, J. M., Phys. Rev. 98 (1955) 1721. 16) Zandveld, P., Internal report Interuniversiteir Reactor Instituut, Delft, 1969. 17) Placzek, G., Phys. Rev. 86 (1952) 377. 18) Ascarelli, P. and Caglioti, G., Nuovo Cimento 43B (1963) 375. 19) Verlet, L., Phys. Rev. 165 (1968) 201. 20) Thomas, J. E. and Schmidt, P. W., J. them. Phys. 39 (1963) 2506. 21) Rice, S. A. and Gray, P., The statistical mechanics of simple liquids, Interscience Publ. (New York, 1965). 22) Kingston, A. E., J. them. Phys. 42 (1965) 719. 23) Levelt, J. M. H., Physica 26 (1960) 361. 24) Weir, R. D., Wynn Jones, I., Rowlinson, J. S. and Saville, G., Trans. Faraday Sot. 63 (1967) 1320. 25) Klein, M. and Hanley, H. J. M., J. them. Phys. 53 (1970) 4722. 26) Tanaka, Y. and Yoshino, K., J. them. Phys. 53 (1970) 2012. 27) Peterson, 0. G., Batchelder, D. N. and Simmons, R. O., Phys. Rev. 158 (1966) 703. 28) Horton, G. K. and Leech, J. W., Proc. Phys. SOC. 82 ( 1963) 8 16. 29) Gordon, R. G., J. them. Phys. 48 (1968) 3929. 30) Ashcroft, N. W., Physica 35 (1967) 148. 31) Page, D. I., Egelstaff, P. A., Enderby, J. E. and Wingfield, B. R., Phys. Letters 29A (1969) 296.