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Numerical Solutions of the Convolution‐Hypernetted Chain Integral Equation for the Pair Correlation Function of a Fluid. II. The Hard‐Sphere Potential
1.We also refer to this as the approximation
2.M. Klein and M. S. Green, J. Chem. Phys. 39, 1367 (1963);
2.see also, M. Klein, thesis, University of Maryland, 1962, and references in each.
3.J. G. Kirkwood, E. K. Maun, and B. J. Alder, J. Chem. Phys. 18, 1040 (1950).
4.M. N. Rosenbluth and A. W. Rosenbluth, J. Chem. Phys. 22, 881 (1954);
4.W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27, 1207 (1957)., J. Chem. Phys.
5.T. Wainwright and B. J. Alder, Nuovo Cimento, Suppl. 9, 116 (1958);
5.B. J. Alder and T. Wainwright, J. Chem. Phys. 27, 1209 (1957);
5.B. J. Alder and T. Wainwright, 31, 459 (1959); , J. Chem. Phys.
5.T. Wainwright and B. J. Alder, AEC Report UCRL 5251 (1958);
5.B. J. Alder and T. Wainwright, Proc. Intern. Symp. Transport Processes Statist. Mech. (Brussels) 1956, New York (1958).
6.G. S. Rushbrooke and H. J. Scoins, Proc. Roy. Soc. (London) A216, 203 (1953).
7.G. S. Rushbrooke and P. Hutchinson, Physica 27, 647 (1961).
8.S. Katsura, Phys. Rev. 115, 1417 (1959) and references therein.
9.Our values, having been read from a graph, are probably not the precise values of Alder et al. The magnitudes of the errors introduced by this, however, should be small compared to the amount by which our results deviate from the Alder values.
10.G. S. Rushbrooke and H. I. Scoins, Phil. Mag. 42, 582 (1951);
10.B. R. A. Nijboer and L. Van Hove, Phys. Rev. 85, 777 (1952);
10.R. W. Hart, R. Wallis, and L. Pode, J. Chem. Phys. 19, 139 (1951).
11.The equations of state of Fig. 3 were calculated using the virial coefficients of Table II. The quantity from which is calculated is given in Table V.
12.Some authors13,14 have suggested the use of the five known virial coefficients in an exponential series. The corresponding exponential would then be used at higher densities. Their suggestion is given a valid basis by Eq. (11) whose form holds for as well. Their use of the first terms of the partially known exponential at high densities is, however, only an approximation.
13.H. W. Woolley, Fourth Hypervelocity Impact Sym. Eglin Air Force Base, Florida (1960) ASTIA AD‐244‐476.
14.Y. Rocard, Rev. Sci. 90, 387 (1952).
15.Equivalent assumptions are made in the Monte Carlo calculation.
16.An extrapolation based on higher differences indicates that a singularity might not occur at any density. These higher differences, however, are of questionable validity.
17.Another way of seeing this at intermediate densities (i.e., Region II) is to note that, while the mean of and for is a slight overestimation of the correct fourth virial coefficient, the mean of and for this approximation represents an underestimation of the correct fifth virial coefficient. These tend to compensate each other.
18.One can compute from the pair correlation function by Fourier transformation. We have attempted to do this for the Born‐Green equation using the pair correlation functions which were published by Kirkwood, Maun, and Alder for several values of their parameter λ. The values of so obtained increased with λ for λ small, went through a maximum, and decreased. This is considerably different from the behavior of as obtained from our solutions. We have considered this to be wrong and have attributed the error to an insufficient number of significant figures in the published quantities. As a result we do not have a second Born‐Green equation of state with which to make comparisons.
19.These were calculated from expressions published by Nijboer and Van Hove.10
20.J. A. Prins, and H. Petersen, Physica 3, 147 (1936).
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