A comparison of the experimental radial distribution function at the highest density, with one calculated by molecular dynamics, indicated substantial agreement. 2 with the 90% confidence interval scaled to the total gas scattering function (•, preliminary sample intensity estimate; O, data; d, rejected data; === ~ confidence interval; - -. Extrapolation). The effect of temperature and bulk density on the width of the main intensity peak in the argon intensity function, j(s) at j(s) = 1. Complete SY!llbols, this work; Open symbols, Mikolaj10.
Effect of temperature and bulk density on the height of the main peak in argon. Argon radial atomic density function for steps 1, 2 and 3 (Parabola at small r is 4TTr2p). Net Argon Radial Distribution Function for No. This work; • , Molecular Dynamics of Verletl7, same density but higher temperature).
Direct correlation function of argon operation No. The direct correlation function of argon no. Potential, Watt Percussion-Yevick Approximation19) Argon Direct Correlation Function. The depth of the argon potential well as predicted by the Percus-Yevick equation (Open symbols, this work; •, Mikolaj and Pings13).
INTRODUCTION
However, apart from this complication, direct comparisons of radial distribution functions cannot be made with full rigor due to an uncertainty estimate. This improvement was analyzed by repeating measurements of one of the states measured by Honeywell and transformed by Mikolaj. A comparison of the two independently measured intensity patterns shows a moderate discrepancy; a discrepancy which, incidentally, was predicted by Levesque and Verlet.15 The details of the intensity discrepancy and the changes it has affected in the radial distribution function will be discussed during the development of the thesis.
Since one of the main uses of the experimental structure functions is to test models or theories of the liquid state, comparisons will be made with radial. Furthermore, a few radial distributions calculated by Fehder21 using two-dimensional molecular dynamics will be presented to support the claim of the existence of the subsidiary peak. The validity of the Percus-Yevick assumption will be subjected to analysis with the data contained herein.
The uncertainty, 6Is(s), in Is(s) is derived (See Appendix 3) from the assumption37 that x-ray scattering is governed by Poisson statistics, for which is the square of the standard deviation. Before presenting the net radial distribution functions and the direct correlation functions obtained for the six conditions investigated, the details of the experimental method and the data reduction scheme will be summarized.
EXPERIMENTAL
An aluminized Mylar heat shield, F, reduces radiant heat transfer to the exposed portion of the cell. This thennometer was in the non-irradiated part of the cell and immersed in the argon. The intensity of the scattered radiation was detected with a sodium iodide (thallium-activated) scintillation counter.
Control of the main heater was switched from the thermocouple to the platinum resistance sensor. Is(e) is the total scatter of the sample, i.e. the sum of the coherent, which contains the structure information, and the incoherent scatter. With the exception of the uncertainty band for the structure functions, the technique is Mikolaj's.
The wavelength behavior of the mass absorption coefficient is taken from the International Crystallographic Tables,40. This is why the incident beam was kept at the top of the argon.
RESULTS A. General
The hysteresis behavior of the Bourdon coil was carefully avoided by never pressurizing the gauge above the gauge pressure. To subtract the cell intensity from the cell and sample intensity, the latter must be scaled by Q, the ratio of the cell plus sample reference angular intensity to the blank cell reference. A typical plot of the normalized data has been presented in Note that over 40 of the 237 angle measurements have been.
The first peak and the right shoulder of the second peak of the argon scattering curve are clearly visible at six degrees and to the right of 13°. The values of the dispersion-corrected atomic scattering factor and the incoherent scattering factor used to derive i(s) from the experimental data are listed in Table III. Before the data could be subtracted, values of absorption coefficients, ASSC, ACSC and ACC for incoherent and coherent scattering were calculated for each angle where the scattering data were taken.
Representative values of the absorption coefficients for each run, estimated to be accurate to three significant figures, are listed in Table V. The location of the first minimum is between s of 0.55 and 0.69 for the first five runs, but shifts significantly toward the origin of s of about 0.14 for the highest density. The effect of the dual-filter monochromatization was to reduce the size of the first peak, thereby reducing the peak width.
Without the generations of uncertainty, it would be. be extremely difficult to prudently report the structure. the basic features, the three main peaks, are distinct. Before presenting the net radial distribution functions calculated from the divergence-corrected intensity functions, it is important to note that the effects of divergence correction have a small effect oh h(r). In the lower half of Figure 23, the correction effect is seen to be so small that the two h(r) curves are almost identical.
The height of the first maximum varies from 1.09 to 1.44 over the density range, with Run # 4 being lower than the trend. The first coordination number, an estimate of the average number of nearest neighbors, was calculated by determining the area under the function 4nr~;[r g(r)] sym. The subscript sim means that the function r g(r) is made symmetric about the vertical line by producing the maximum of the first peak by the mirror image of the leading edge as the second half of the peak.
DISCUSSION OF RESULTS
The structure factor at low and intense s is lower, the right half of the second peak is higher, and the third peak is unchanged. If there was a fourth peak approaching 7.2, its size relative to the height of the third peak at s of 5.4 would be predicted to be. It was not possible to identify a peak of this magnitude in any of the six in- .
The position in s-s:pace of the maximum in i(s) was predicted from Percus-Yevick theory and the Lennard-Jones potential and for hard spheres by Verlet,17 using. Within experimental precision, the positions of the maximum in the six intensity samples produce an r0 that fits this model. In addition, the position of the second peak maximum occurs at a smaller value of r.
A comparison of the net radial distribution function for Run #6 with one calculated by Verlet using molecular dynamics is depicted in Figure 35. The height of the first peak in these conditions does not increase uniformly as the density increases and the temperature decreases. This is consistent with the fact that the height of the first peak of h(r) for Run 4 is lower than the trend indicated for the other five densities.
The use of the radial distribution function, g(r) to calculate a coordination number for argon has been discussed by Mikolaj et. The magnitude of the first coordination number depends on which method was used to calculate it. One approach to understanding the properties of fluids is based on attempts to calculate the radial distribution function, g(r).
Secondary features are observed on the right shoulder of the main peak in the direct correlation function. The first maximum in i(s) is lower than the density trend, and the width of the third peak is larger than others at lower and. Mikolaj, "An X-Ray Diffraction Study of the Structure of Fluid Argon", dissertation, California Institute of Technology, Pasadena, California (1965).
Levelt, "Measurements of the compressibility of argon in the gaseous and liquid phases", PhD thesis, Amsterdam. Effect of temperature and bulk density on the width of the main intensity peak as a function of argon intensity, j(s) at j(s) c 1 (solid symbols, this work; open symbols, MikolajlO).
SUMMARY OF THE SMOOTH INI'ENSITY FUNCI'IONS
BEFORE DIVERGENCE CORRECTION
COHERENT SCATTERING FROM A LIQUID COMPOSED OF O:NE TYPE OF SPHERICALLY SYMMETRIC ATOMS. (This derivation follows that of
The time-averaged atomic scattering factor can be calculated from H.artree-Foch wave equations4 for the electron density in the atom. The atomic positions of all the atoms in the liquid at time t can be described by a delta function. This equation can now be formally integrated over the whole space since the kernel of the integral vanishes for reasonably small r.
For a spherically symmetric distribution, p (-;) is therefore only a function of the magnitude of 1 -r 1 or r and . e c ) co 2TTTT. 0 (s) is measured from smin to sma.x. smin is determined by the fact that forward scattering cannot be measured in high beam, whereas sma.x is bounded above by 4n/.
APPENDIX 2
The a and ~ heading means that the associated intensity, whether for the cell or the cell and sample data, is the experimental scattering intensity measured at 0 for either an a or ~ filter on the incident X-ray beam. ACSC (e) measures the volume-averaged reduction in cell scattering caused by absorption in both the sample and the cell. ACC (0) measures the volume-averaged reduction in cell spreading caused by absorption in the cell.
ACCcoh (e) is the ACC coefficient estimated for the coherent wavelength over the entire distance traveled, L. Ye inc (e) is the fraction of Ic that dissipates with the incoherent wavelength shift. ASSC (e), ASSC (e) are the estimated ASSC coefficients for coherent and incoherent diffracted wavelengths N is the number of atoms of the irradiated sample.
I s is the intrinsic scattering power of the sample atoms y8coh is the fraction of coherent radiation in Is. 8 is the fraction of incoherent radiation in Is In this experiment, the voltage windows in the pulse. The fraction , y s coh of the coherent distribution from the sample ' depends on the structure p (r) and can be calculated from I (e).
