Ab Initio Values of the Thermophysical Properties of Helium as Standards.Recent quantum mechanical calculations of the interaction energy of pairs of helium atoms are accurate and some include reliable estimates of their uncertainty. We combined these ab initio ab initio prep. lawyer Latin for "from the start," as "it was legal ab initio." results with earlier published results to obtain a helium-helium interatomic potential that includes relativistic retardation effects over all ranges of interaction. From this potential, we calculated the thermophysical properties of helium, i.e., the second virial coefficients, the dilute-gas viscosities, and the dilutegas thermal conductivities of [He.sup.3], [He.sup.4], and their equimolar mixture from 1 K to [10.sup.4] K. We also calculated the diffusion and thermal diffusion coefficients of mixtures of [He.sup.3] and [He.sup.4]. For the pure fluids, the uncertainties of the calculated values are dominated by the uncertainties of the potential; for the mixtures, the uncertainties of the transport properties also include contributions from approximations in the transport theory. In all cases, the uncertainties are smalle r than the corresponding experimental uncertainties; therefore, we recommend the ab initio results be used as standards for calibrating instruments relying on these thermophysical properties. We present the calculated thermophysical properties in easy-to-use tabular form. Key words: diffusion coefficient; helium; intermolecular potential; second virial; thermal conductivity; thermal diffusion factor; thermophysical standards; transport properties; viscosity. Accepted: July 20, 2000 Available online: http://www.nist.gov/jres 1. Introduction Today, the most accurate values of the thermophysical properties of helium at low densities can be obtained from two, very lengthy, calculations. The first calculation uses quantum mechanics and the fundamental constants to obtain, ab initio, a potential energy [varphi](r) for the helium-helium ([He.sub.2]) interaction at discrete values of the interatomic separation r and also limiting forms of [varphi] (r) at large r (see Fig. 1). The second calculation uses standard formulae from quantum-statistical mechanics and the kinetic theory of gases to obtain the thermophysical properties of low-density helium from [varphi](r). Here, we report the results of the second calculation spanning the temperature range 1 K to 1 [10.sup.4] K for the second virial coefficient B (T), the viscosity [eta](T), the thermal conductivity [lambda](T), the mass diffusion coefficient D(T), and the thermal diffusion factor [[alpha].sub.T](T) for [He.sup.3], [He.sup.4], and their equimolar mixture. Our results, together with estimates of their uncertain ties, are presented in easy-to-use tabular form in Appendix A. For the pure fluids, the statistical-mechanics calculations make negligible contributions to the uncertainties of the tabulated properties; therefore, we estimated the uncertainties of the results by varying [varphi](r) within its uncertainty and examining the consequences. For the equimolar mixture, the results from different orders of approximation in the transport theory are compared to estimate their contribution to the uncertainties. The present results can be applied to many problems in metrology; here we mention a few. Low-density helium is used in primary, constant-volume, gas thermometry [1]; primary, dielectric-constant gas thermometry [2]; and in interpolating gas thermometry (required by ITS-90 in the temperature range 3 K to 24.6 K) [3]. These applications require the extrapolation of measurements to zero pressure. if the present values of B(T) are used for such extrapolations, the results may be more accurate and the probability of detecting systematic errors in the measurements will be increased. Low-density helium can be used to calibrate acoustic resonators for acoustic thermometry and for measuring the speed of sound in diverse gases. Spherical acoustic resonators [4] may be calibrated using the present values of [lambda](T), B (T), and temperature derivatives dB/dT and [d.sup.2]B/[dT.sup.2]. The same properties together with [eta](T) may be used to calibrate cylindrical acoustic resonators [5]. Other instruments that might be calibrated with the help of the present results include the vibrating wire viscometer [6], the Greenspan acoustic viscometer [7], and the Burnett apparatus [8] for making very accurate measurements of the equation of state of moderately dense fluids. The present work contrasts with a long tradition of using semi-empirical models for [varphi](r) to correlate the thermophysical property data for helium and the other monatomic gases [9, 10, 11]. These semi-empirical models combined limited ab initio results with critically evaluated and judiciously selected experimental data to determine the function [varphi](r) that correlates as much data as possible. In this work, we did not consider experimental results until all of the calculations were completed as in [12, 13]. The ab initio results were then compared to the sets of data that others had selected as inputs to semi-empirical models. In every case that we examined, the ab initio values of the thermophysical properties agreed with the data within plausible estimates of their combined uncertainties. This manuscript is organized as follows: Sec. 2 reviews the ab initio results for [varphi](r) and our analytic representation of them. Section 3 outlines the steps in calculating the thermophysical properties of helium from [varphi](r). Each step includes a description of the precautions that were taken to insure that imperfections of the numerical methods did not adversely affect the results. Section 4 estimates the uncertainty of the ab initio helium pair potential and how it propagates into the uncertainties of the calculated properties. Section 5 describes the tabulated results and methods for their use. Section 6 compares the calculated properties with selected measurements. Section 7 summarizes the present results and the prospects for future refinements. 2. Ab Initio Values for the [He.sub.2] Potential Energy Functions [varphi](r) Table 1 lists recent ab initio values of [varphi](r) at selected values of r (3.0 bohr Niels Henrik David 1885-1962. Danish physicist. He won a 1922 Nobel Prize for his investigation of atomic structure and radiations. His son Aage Niels Bohr (born 1922), also a physicist, shared a 1975 Nobel Prize for discovering the asymmetry of atomic nuclei. 2.1 Long-Ranges: r > 8 bohr The asymptotic long-range attractive behavior of our preferred potential [[varphi].sub.00](r) is represented by the two-body dispersion coefficients [C.sub.n](n = 6, 8, ...) in the multipole expansion. These coefficients have been calculated, ab initio, by two independent groups [22, 23] using a sum-over-states formalism with explicitly electron-correlated wave functions to describe the states. The independent calculations [22, 23] differed by less than 1 in the fourth digit. This small difference makes a negligible contribution to the uncertainties of the thermophysical properties calculated from [[varphi].sub.00](r). 2.2 Short_Ranges: r less than 3 bohr Ceperley and Partridge [15] obtained values of (varphi)(r) at small r using a quantum Monte Carlo (QMC) method. The QMC method is exact insofar as it requires no mathematical or physical approximations beyond those in the Schrodinger equation and the method yields estimates of the uncertainties of [varphi](r). Komasa [24] used a variational method to obtain rigorous upper bounds to [varphi](r) in the range 0.01 bohr [less than or equal to] r [less than or equal to] 15 bohr. At some values of r, the variational values of [varphi](r) are less than the QMC values; however the differences between the values are usually within twice the QMC uncertainties. Thus, we used the variational values to determine [[varphi].sub.00](r) and we have evidence that the QMC uncertainties are reasonable. At smaller values of r the variational and QMC results are inconsistent. For example, at r = 1 bohr (not plotted), Komasa reports [varphi](1 bohr) = [286.44 [+ or -] 0.03] X [10.sup.3] K, and Ceperley and Partridge report [varphi][l bohr] = [291.9 [+ or -] 0.6] X [10.sup.3] K. We are unable to resolve this inconsistency; however, the inconsistency does not affect the thermophysical properties in the temperature range 1 K to [10.sup.4] K. Komasa provides two values for the well depth at 5.6 bohr, [epsilon]/[k.sub.B] = - 10.947 K using a 1200-term basis set and [epsilon]/[k.sub.B] = - 10.978 K using a 2048 term basis set. The second value is 0.3 % lower. Komasa's calculations at other values of r used the 1200-term basis set. We speculate that comparable reductions in [varphi](r) would occur if Komasa's variational calculation were repeated with the larger basis set at all values of r. 2.3 Intermediate Ranges: 3 > r > 8 bohr At intermediate ranges, we considered the seven relevant publications cited in Table 1. Anderson et al. [16] report exact QMC results that have relatively large uncertainties. Klopper and Noga [17] used an explicitly correlated coupled cluster [CCSD(T)] method that resulted in the limiting value for the well depth of [epsilon]/ [k.sub.B] = - 10.68 K at 5.6 bohr. Then, they estimated the effects of quadruple substitutions to be -0.32 K at 5.6 bohr (and -1.9 K at 4.0 bohr) by comparing their results to the full configuration interaction (FCI) calculation of van Mourik and van Lenthe [25]. This extrapolation to a complete basis set resulted in [epsilon]/ [k.sub.B] = -(11.0 [+ or -] 0.03) K, which agrees with the QMC results of Anderson [16]. Korona et al. [18] used symmetry-adapted perturbation theory (SAPT) to calculate values for [varphi](r) with uncertainties that they estimated to be the larger of 0.3 % or 0.03 K in the range 3 bohr [less than or equal to] r [less than or equal to] 7 bohr. The SAPT well-depth is [epsilon]/[k.sub.B] = -(11.06 [+ or - ] 0.03) K, the lowest of all ab initio results; however, it also agrees with the QMC result [16] within the latter's uncertainty. While this project was in progress, two groups extended the CCSD CCSD - Canadian Council on Social Development CCSD - Canadian Cultural Society of the Deaf CCSD - Central Columbia School District (Bloomsburg, PA) CCSD - Charleston County School District (South Carolina) CCSD - Clark County School District CCSD - Clarkstown Central School District (New City, New York) CCSD - Cobb County School District (Georgia) CCSD - Collins Cobuild Student's Dictionary CCSD - Command Communications Service Designator(T) calculations of Klopper and Noga [17]. These groups (de Bovenkamp and Duijneveldt [20]; and van Mourik and Dunning [21]) used different techniques to extrapolate the results of Klopper and Noga [17] to an infinite basis set. Gdanitz [19] also published calculations labeled r12-MRACPF in which he extrapolated his results to an infinite basis set by yet another method. These three recent publications and the variational results of Komasa [24] indicate that the SAPT [18] results in the region around r = 4.0 bohr are too attractive by approximately 0.05 K (Fig. 1, lower panel). Nevertheless, we used the SAPT intermediate-range results in determining the potential [[varphi].sub.00](r) and we used the differences between the SAPT and the other results to determine alternative potentials that were used to estimate the uncertainties of the thermophysical properties. Our decisions are based on three observations. First, we recalled that Komasa's [24] variational result at 5.6 bohr decreased 0.3 % upon increasing the basis set from 1200 terms to 2048 terms. If Komasa's result at 4.0 bohr (292.784 K) were decreased by 0.3 %, it would be 291.906 K, in agreement with the SAPT value of 291.64 K. Gdanitz suggested that the decrease at 4.0 bohr might be less than 0.3 % because the variation method is more accurate at smaller separations [26]. Second, we noted that the two extensions of Klopper and Noga' work [17] are not independent. The two decompose [varphi](r) in several components, the largest of which were calculated best by Klopper and Noga. Thus, the uncertainties of these results may be dominated by those of Klopper and Noga. (van Mourik and Dunning [21] state, "It is likely that the corrected curve is the most accurate available to date for [He.sub.2] interactions". In effect, they asserted that Klopper and Noga's interaction energies are more accurate than their own complete basis set extrapolated energies.) Third, Bukowski et al. [27] argue that thei r own Gaussian-type geminals (GTG) computation bounds the larger components of Klopper and Noga's CCSD(T) computations and they suggest that Klopper and Noga's results may be too high by approximately 0.3 K at 4 bohr and by approximately 0.04 K at 5.6 bohr. If Bukowski et al.'s suggestion is correct and if one decreases the CCSD(T) values of [varphi](r) accordingly, then they all would agree with the SAPT results. Ultimately, additional calculations will resolve these issues. 2.4 Algebraic Representations of ab initio Values of [varphi] (r) We calculated the thermophysical properties of helium six times, each using a different function to represent ab initio values of [varphi](r). We fitted two of these six functions, [[varphi].sub.00] and [[varphi].sub.B] to our own selections among the published ab initio values. The third function, [[varphi].sub.SAPT], had already been fitted by others to ab initio results and used to calculate thermophysical properties. [18] We fitted the fourth, [[varphi].sub.A], to the same ab initio results used to obtain [[varphi].sub.SAPT]; however, we added one additional fitting parameter. Thus, differences between the thermophysical properties computed from [[varphi].sub.SAPT] and [[varphi].sub.A] provide one indication of the sensitivity of the properties to the algebraic representation of the ab initio "data". The last two functions are denoted [[[varphi].sup.-].sub.A] and [[[varphi].sup.+].sub.A] To obtain [[[varphi].sup.-].sub.A], we decreased the ab initio short-range results [15] by their claimed uncertaintie s and decreased the intermediate-range SAPT results by 0.1 % and re-fitted them. Then, we increased the ab initio results by their claimed uncertainties and the SAPT results by 0.1% and fitted them to obtain [[[varphi].sup.+].sub.A] The differences between the thermophysical properties calculated using [[varphi].sub.A], [[[varphi].sup.-].sub.A], [[[varphi].sup.+].sub.A] and [[varphi].sub.SAPT] are analogous to the uncertainties of measured values of thermophysical properties conducted in a single laboratory and analyzed using different methods. In the present case, the differences between the thermophysical properties calculated from [[varphi].sub.A], [[[varphi].sup.-].sub.A], [[[varphi].sup.+].sub.A], and [[varphi].sub.SAPT] are much smaller than the differences between those calculated from [[varphi].sub.00], [[varphi].sub.A] and [[varphi].sub.B]. 2.4.1 [[varphi].sub.00] We used [[varphi].sub.00] to calculate the thermophysical properties tabulated in Appendix A. In our judgement, [[varphi].sub.00] is the best representation of the ab initio results available at the time of this writing. The subscript "00" identifies [[varphi].sub.00] by the year in which we began using it. The ab initio results fitted by [[varphi].sub.00](r) come from three sources: (1) at small r(1 [less than] r [less than] 2.5 bohr), the results of the variational calculation from Komasa [24], (2) at intermediate r (3 bohr [less than] r [less than] 7 bohr), the SAPT results from Korona et al. [18], (3) at large r, the asymptotic constants from the "exact" dispersion coefficients of Bishop and Pipin [22] and the higher order dispersion coefficients determined from the approximate relations presented by Thakkar [29]. The algebraic representation of [[varphi].sub.00](r) is a modification of the form given by Tang and Toennies [9]. The representation is the sum of repulsive ([[varphi].sub.rep]) and attractive ([[varphi].sub.alt]) terms: [[varphi].sub.00](r) = {[[varphi].sub.rep](r) + [[varphi].sub.att](r), 0.3 [less than or equal to] r/bohr [less than] [infinity] [[varphi].sub.rep](0.3 bohr) + [[varphi].sub.alt](0.3 bohr), 0 [less than or equal to] r/bohr [less than]0.3 [[varphi].sub.rep](r) = A exp([a.sub.1] r + [a.sub.2][r.sup.2] + [a.sub.-1][r.sup.-1] + [a.sub.-2][r.sup.-2], (1) [[varphi].sub.alt](r) = -[[[sigma].sup.8].sub.n=3] [f.sub.2n](r)[C.sub.2n]/[r.sup.2n][1 - ([[[sigma].sup.2n].sub.k=0] [([delta]r).sup.k]/k!)exp(-[delta]r)]. Equation (1) includes the factor[f.sub.2n](r) that accounts for the relativistic retardation of the dipole-dipole (n = 3) term applied over all r. This factor changes the behavior of the dipole-dipole term from [r.sup.-6] to [r.sup.-7] at very large r, and it was taken from Jamieson et al. [30]. When the expressions for the retardation of the higher dispersion terms [C.sub.8] and [C.sub.10] given by Chen and Chung [23] were applied to [[varphi].sub.00], the well depth changed by only 0.0014 K out of 11 K. the resulting changes in the calculated thermophysical properties were much smaller than their uncertainties; thus, we used the approximation [f.sub.2n](r)[equivalent] 1 for n [greater than] 3. (Note: retardation is included when calculating the thermophiysical properties; however, by convention, it is not included when comparing Eq. (1) to the ab initio results.) We also considered the adiabatic correction of the helium dimer dimer /di·mer/ (di´mer) 1. a compound formed by combination of two identical molecules. 2. a capsomer having two structural subunits. di·mer (d given by Komasa et al. [28]. The effects of this correction were also much smaller than those from the uncertainties in [varphi](r); thus, we omitted this correction. The definition of [[varphi].sub.00](r) in Eq. (1) is broken into two ranges. If this were not done, [[varphi].sub.00](r) would have a suprious maximum at very small values of r. As indicated in Eq. (1), the break-point was set at 0.3 bohr. The dispersion coefficients ([C.sub.6], [C.sub.8],... [C.sub.16]) in Eq. (1) and Table 1 were held fixed [22, 29]. The values of the remaining parameters in Table 1 ([a.sub.-2], [a.sub.-1], [a.sub.1], [a.sub.2], and [delta]) were determined by fitting [[varphi].sub.00](r) to the ab initio results. When fitting [[varphi].sub.00] the ab inition results were weighted in proportion to the reciprocal of the uncertainty squared, where the uncertainties were taken (when available) from the publications that presented the results. [15, 18, 20, 21, 24]. 2.4.2 [[varphi].sub.SAPT] Korona et al. fitted their SAPT results and the QMC values of Ceperley and Partridge [15] to the algebraic expression One or more characters or symbols associated with algebra; for example, A+B=C or A/B. of Tang and Toennies [9] while holdign constant the asymptotic dispersion coefficients of Bishop and Pipin [22]. They included higher order dispersion coefficient determined with combining rules of Thakkar [29] and retardation effects of the [C.sub.6] dispersion coefficient as given by Jamieson et al. [30]. Janzen and Aziz [11] calculated the thermophysical properties of helium using [[varphi].sub.SAPT] and they 'judged it to be the most accurate characterization of the helium interaction yet proposed." We believe that [[varphi].sub.00] is more accurate than [[varphi].sub.SAPT] because it uses the recent, accurate variational results of Komasa [24] instead of the earlier short range QMC values of Ceperley and Partridge [15]. 2.4.3 [[varphi].sub.A], [[[varphi].sup.-].sub.A], and [[[varphi].sup.+].sub.A] In an attempt to ascertain how uncertainties in the interaction energies propagate into the thermophysical properties we constructed alternative potentials which differed in the choice of ab initio results, and in the form of the algebraic expression. The first alternative, denoted [[varphi].sub.A], was obtained by fitting the exact same ab initio results from [18, 22, 24, 29] as [[varphi].sub.SAPT]. The algebraic expression of Tang and Toennies [9] was modified by adding a [a.sub.3][r.sup.3] to the exponent of the repulsive term, such that [[varphi].sub.rep] = A exp [a.sub.1]r + [a.sub.2][r.sup.2] + [a.sub.3][r.sup.3]). The additional [a.sub.3][r.sup.3] term enables [[varphi].sub.A] to fit the SAPT ab initio results within 0.1 % in two regions r = 3 bohr and at r [greater than] 6 bohr where [[varphi].sub.SAPT] [18] deviates from the ab initio results slightly greater than 0.1 %. To obtain obtain [[[varphi].sup.-].sub.A], we decreased the ab initio short-range [15] and long-range [22] results by their claimed uncertainties and decreased the intermediate-range SAPT results by 0.1 % and the long-range dispersion coefficients by 0.08 %. Equation (1) was then re-fitted to obtain [[[varphi].sup.-].sub.A]. We then increased the ab initio results by their claimed uncertainties and the intermediate-range SAPT results by 0.1 % and again fitted them to obtain [[[varphi].sup.+].sub.A]. 2.4.4 [[varphi].sub.B] The potential [[varphi].sub.B], uses the CCSD(T) results of van Mourik and Dunning [21] and of van de Bovenkamp and van Duijneveldt [30] instead of the SAPT results of Korona et al. [18] in the intermediate range of 3 bohr [less than] r [less than] 7 bohr. To fit these values the algebraic expression of Tang and Toennies [9] was modified again by adding a [a.sub.-1][r.sup.-1] and [a.sub.-2][r.sup.-2] to the exponent of the repulsive term, such that [[varphi].sub.rep] = A exp ([a.sub.1]r + [a.sub.2][r.sup.2] + [a.sub.-1][r.sup.-1] + [a.sub.-1][r.sup.-2]). 2.5 Comparison of [[varphi].sub.00], [[varphi].sub.SAPT], [[varphi].sub.A], and [[varphi].sub.B] Table 2 and the lower panel of Fig. 1 display the changes in [varphi](r) resulting from alternate choices among the ab initio results. The differences between the thermophysical properties calculated using [[varphi].sub.00], [[varphi].sub.SAPT], [[varphi].sub.A], and [[varphi].sub.B] are analogous to the differences between measurements of thermophysical properties conducted in different laboratories using different methods and they are used to estimate the uncertainties of the results for pure [He.sup.3] and pure [He.sup.4]. Table 3 lists some characteristic properties of the potentials that we have used. They include the well depth [epsilon]/[k.sub.B], the locations of the zero ([sigma]) and of the minimum ([r.sub.m]) of the potential, and the energy of the bound state ([E.sub.b]) of a pair of [He.sup.4] atoms. Following Janzen and Aziz [31], we estimated the number of Efimov states [N.sub.E] from the scattering length and the effective range with the result [N.sub.E] = 0.77 [+ or -] 0.01 for [[varphi].sub.00]. Because [N.sub.E] [less than] 1 for all potentials in Table 2, Efimov states are unlikely to exist. A discussion of these properties of the interatomic potential for helium can be found in Ref. [31]. 3. Numerical Calculations and Their Uncertainties Here, we outline the steps required to calculate the thermophysical properties of helium from the inter-atomic potential. We also describe the precautions that were taken to insure that the uncertainties in the results from approximations in statistical mechanics and in the numerical methods were both smaller than the uncertainties results from different choices for [varphi](r). The initial steps of calculating the thermophysical properties that depend upon pairs of helium atoms are all the same. (1) The Schrodinger equation for the scattering of a helium atom at the energy E in the potential [varphi](r) is separated in spherical coordinates, (2) the radial part of the wave function is expanded in partial waves [[psi].sub.l](r) of angular momentum l, (3) several nodes of the scattered wave are located far from the scattering atom, and (4) the phase shifts [[delta].sub.l] of the scattered wave are determined and (5) summed with appropriate statistics to obtain cross sections. The summations account for large symmetry effects at low temperatures [32]. Thus, separate summations are required for [He.sup.3] and [He.sup.4] and their mixtures when calculating the second virial coefficient and the transport properties. The final step (6) is an integration over energy that is appropriate to the thermophysical property under consideration. 3.1 Integration of the Radial Schrodinger Equation The Schrodinger equation is separated in spherical coordinates and decomposed into angular momentum states to obtain ([d.sup.2]/d[r.sup.2] + [k.sup.2] - l(l + 1)/[r.sup.2] - 2[micro]/h [varphi](r)) [[psi].sub.l](r) = 0 (2) where h is Planck's constant [14] divided by 2[pi], [micro] is the reduced mass [micro] = ([m.sub.1] + [m.sub.2])/[m.sub.1][m.sub.2], k = [(2[micro]E).sup.1/2]/h is the wave number, and E is the energy of the incoming wave. Equation (2) is integrated to obtain the perturbed wave function [[psi].sub.l](k, r). The location [r.sub.n] of the nth zero (or node) of the wave function [[psi].sub.l](k, r) was found using a five point Aiken interpolation formula with values of [[psi].sub.l](k, r) near the n th node. The integration was performed using Numerov's method [33] as implemented in [34] and [35]. At each energy, [r.sub.n] was recalculated using successively smaller step sizes. The calculation was terminated when halving the step size changed [r.sub.n] less than [10.sup.-9] X [r.sub.n]. We verified that the tolerance [10.sup.-9] X [r.sub.n] was sufficiently small that further reductions of the step-size did not change the thermophysical properties beyond the tolerances given in Table 6. The final sizes of the integration steps are listed in Table 4. 3.2 Calculation of Phase Shifts, [[delta].sub.l](k, n) The relative phase shifts, [[delta].sub.l](k, n) of the outgoing partial wave were evaluated from the relation [[delta].sub.l](k, n) = arctan [j.sub.l](k, [r.sub.n])/[n.sub.l](k, [r.sub.n]) (3) where [j.sub.l](k, [r.sub.n]) and [n.sub.l](k, [r.sub.n]) are the Bessel and Neuman functions for angular momentum quantum number l and wave number k. In practice, the phase shifts were evaluated at groups of three consecutive nodes. If the phase shift did not change by more than [10.sup.-8] X [[delta].sub.l](k, n) between the first and last of the three nodes, it was assumed that n ( and [r.sub.n]) were sufficiently large that additional effects of the potential were negligible, and the calculation was terminated. Otherwise, the calculation was continued to larger values of r, and the test was repeated. We verified that the tolerance [10.sup.-8] X [[delta].sub.l](k, n) is consistent with the uncertainties of the thermophysical properties listed in Table 6. 3.3 Calculation of the Second Virial Coefficient, B(T) The second virial coefficient was obtained by adding two or three terms; the first term is a thermal average [B.sub.th](T), the second term is that of an ideal gas [B.sub.ideal](T), and the third term is the bound state term [B.sub.bound](T), which applies to [He.sup.4], but not to [He.sup.3] because a bound state exists only for [He.sup.4]. 3.3.1 The Thermal Average Term [B.sub.th](T) The thermal average term [B.sub.th](T) is [B.sub.th] = [[[integral].sup.[infinity]].sub.0] k exp( - [k.sup.2]/[k.sub.B]T) [[[sigma].sup.[infinity]].sub.l=0] (2l + 1) [[delta].sub.l] (k, n [similar and equal to] [infinity]) dk (4) where [k.sub.B] is the Boltzmann constant, and [[delta].sub.l](k, n [similar and equal to] [infinity]) is the phase shift at large enough separation that the potential to longer perturbs the outgoing wave function [32, 36]. Equation (4) contains both a sum and an integral with the limits 0 and [infinity]. Truncating the sum and the integral at a finite upper bound is a potential source of error. At each value of k, the sum was computed until the addition of six phase shifts did not change the sum by more than [10.sup.-8] of its value. At the lowest energies, this condition was met after adding seven phase shifts; at the highest energies, hundreds of phase shifts were added. At this step of the calculation, symmetry effects are incorporated. The unweighted sum [Eq. (4)] is carried out over all values of l only when calculating the interaction virial coefficient of mixtures of [He.sup.3] and [He.sup.4], because these atoms are distinguishable and follow Boltzmann statistics. For pure [He.sup.3] and [He.sup.4], weighted sums are performed over the even and odd values of l using the formulas [[sigma].sub.BE] = [s + 1/2s + 1] [[sigma].sub.even] + [s/2s + 1] [[sigma].sub.odd] (5) [[sigma].sub.FD] = [s + 1/2s + 1] [[sigma].sub.odd] + [s/2s + 1] [[sigma].sub.even] where s is the spin quantum number (0 for [He.sup.4]; 1/2 for [He.sup.3]), BE stands for Bose-Einstein statistics for bosons boson: see elementary particles; Bose-Einstein statistics. ([He.sup.4]), and FD stands for Fermi-Dirac statistics for ferminos ([He.sup.3]). Details on this calculation can be found in Ref. [32]. The integral in Eq. (4) was evaluated using a standard integration routine, DQAGI [37]. This routine is designed for semi-infinite or infinite intervals and automatically uses nonlinear transformation and extrapolation to achieve user-specified absolute and relative tolerances for a user-specified function. The relative error was set to [10.sup.-8]. If the integrator could not achieve this accuracy, an error message would have been reported the problem. 3.3.2 The Ideal-Gas and Bound State Terms [B.sub.ideal](T) and [B.sub.bound](T) The ideal-gas contribution [B.sub.ideal](T) is negative for BE and positive for FD, and zero for Boltzmann statistics as given by [B.sub.ideal] = [+ or -][N.sub.A] [2.sup.-5/2] [[lambda].sup.3] (6) where [N.sub.A] is the Avagodro constant and [lambda] [equivlent] [h/([micro][k.sub.B]T)].sup.1/2] is the "thermal wavelength." The ideal-gas term is important only at low temperatures; it is 1/10 of B(T) at 5 K and 1/100 of B(T) at 75 K. The ideal-gas term is a function of fundamental physical constants and the resulting standard uncertainty is on the order of [10.sup.-6]. For [He.sup.4], the bound state term [B.sub.bound](T) is [B.sub.bound] = - [N.sub.A] [2.sup.-3/2] [[lambda].sup.3]([e.sup.[E.sub.b]/[k.sub.B]T] - 1) (7) where [E.sub.b] is the energy of the bound state. The bound state term is 1/1000 of B(T) at 3 K and 1/100 of B(T) at 0.4 K. [E.sub.b] was determined from integrating the Schrodinger equation; thus, it depended upon the integration step size. Decreasing step sizes were used until consecutive values of [E.sub.b] differed by less than [10.sup.-6] X [E.sub.b]. This numerical uncertainty is much smaller than the 18% difference between [E.sub.b] determined from [[[varphi].sup.-].sub.A] and that determined from [[[varphi].sup.+].sub.A] (Table 3). The sum of the numerical uncertainties in the calculation of B(T) is at most [10.sup.-5] X B(T). This is insignificant compared with the uncertainty of B(T) which arises from the uncertainty of the potential [varphi](r). For example, the uncertainty of B(T) resulting from the uncertainty of [varphi](r) is 0.0022 X B(T) at 300 K; the relative uncertainties at other temperatures are listed in Table 6. 3.4 Calculation of the Transport Properties In order to calculate the transport properties, we used the numerical methods outlined above to obtain the phase shifts as functions of the wave number and angular momentum quantum number. Then we computed the sums over the phase shift that determine the quantum cross sections, [Q.sup.(1)], [Q.sup.(2)], [Q.sup.(3)] ...[Q.sup.(n)], etc. [38]. The cross sections were integrated with respect to energy to obtain the temperature-dependent collision integrals. Finally, the transport properties were calculated using the appropriate combinations of the collision integrals. 3.4.1 Calculation of the Quantum Cross Sections [Q.sup.(n)] The quantum cross sections are functions involving the sums of the phase shifts that depend upon the symmetry of the interacting atoms. The sums over the even and the odd values of l are needed separately: [[Q.sup.(1)].sub.odd] = 4[pi]/k [[[sigma].sup.[infinity]].sub.l=1,3,5...] (2l + 1) [sin.sup.2] [[delta].sub.l] [[Q.sup.(1)].sub.even] = 4[pi]/k [[[sigma].sup.[infinity]].sub.l=0,2,4...] (2l + 1) [sin.sup.2] [[delta].sub.l] (8) and then weighted sums are computed. To evaluate [Q.sup.(1)] for Bose-Einstein (BE) or Fermi-Dirac (FD) statistics the sums are weighted with the spin-dependent quotients, as shown in Eq. (5). As for the case of the second virial coefficient, the sums in Eq. (8) extend to l = [infinity]. The sum was continued until the addition of six more phase shifts changed the cross section by less than [10.sup.-8] of its value. Cross sections with moments up to n = 6 are required to calculate the collision or omega integrals used in the higher order approximations for the transport properties. The equations for these calculations are given by Ref. [38]. 3.4.2 Calculation of the Collision Integrals [[omega].sup.(n,s)] The reduced collision integrals were evaluated from the equation [[omega].sup.(n,s)*] ([T.sup.*]) = [{(s + 1)! [T.sup.*(s+2)]}.sup.-1] X [[[integral].sup.[infinity]].sub.0] [Q.sup.(n)*]([E.sup.*])[e.sup.-[E.sup.*]/[T.sup.*]] [E.sup.*(s+1)] d[E.sup.*] (9) where the superscript * indicates that both the energy and the temperature were scaled by the well-depth of [[varphi].sub.00] and [Q.sup.(n)*] was scaled by the value [Q.sup.(n)] for a rigid sphere of radius [r.sub.m], the location of the minimum of [[varphi].sub.00] (Table 3; See Ref. [32]). In order to evaluate of Eq. (9), the quantum cross sections [Q.sup.(n)*] must be calculated at each energy E used for the quadrature. We calculated a table of [Q.sup.(n)*] as a function of [E.sup.*] and used a 5 point Aiken interpolation to determine values of [Q.sup.(n)*] between tabulated values. The intervals in the table were determined such that the interpolated values had a uncertainty of less that [10.sup.-6] X [Q.sup.(n)*]. Equation (9) was integrated using the automated quadrature routine DQAGI [37], discussed in Sec. 3.1.3, with the tolerance set to [10.sup.-8]. The numerical methods used to calculate the collision integrals yielded results with a relative uncertainty of less than [10.sup.-5]. 3.4.3 Calculation of the Transport Properties From the Collision Integrals The transport properties of dilute gases are calculated using combinations of the collision integrals in approximations of increasing complexity and accuracy. The viscosity and thermal conductivity of pure [He.sup.3] and [He.sup.4] were calculated to the 5th order approximation [39]. The equimolar mixture thermal conductivity [40] and thermal diffusion factors [41] were calculate to the 3rd order, and the diffusion coefficient and mixture viscosity were calculated to 2nd order. Figure 2 shows the effects of truncating the order of the calculation. The changes in [eta] and [lambda] for [He.sup.4] and equimolar mixtures of [He.sup.4] and [He.sup.3] are compared at four temperatures upon increasing order of the approximation. The calculations converge very well; 2nd to 3rd order results in less than a 0.1 % change, 3rd to 4th order results in less than a 0.01 % change, and 4th to the 5th order less than 0.001 %. The behavior of the other transport properties ([eta] and [lambda] for pure [He.sup.3], [D.sub.12], and [[alpha].sub.T]) is similar to that shown in Fig. 2. Figure 2 shows that the change in [eta] and [lambda] of the equimolar mixture from 1st to 2nd order, is very close to that of pure [He.sup.4]. These results show that only calculations of the 2nd order contribute any significant uncertainty to the calculated properties. From these observations, we conclude that the relative uncertainty of [eta] and [D.sub.12] for the equimolar mixture ranges from 0.01 % to 0.04 % in the temperature range 10 K [less than or equal to] T [less than or equal to] [10.sup.4] K. Figure 2, together with the equivalent figure for pure [He.sup.3], suggest that, at T [greater than] 100 K, the accuracy of the calculated [eta] and [D.sub.12] for the equimolar mixture might be improved if one extrapolated from 2nd order to 5th order by following the curves for pure [He.sup.4] and [He.sup.3]. The rapid reduction of the uncertainty of the calculated viscosity with increasing order of approximation is not sensitive to [varphi](r); Viehland et al. [39] obtained similar results for the viscosity of rigid atoms that interact via (12-6) Lennard-Jones potentials. 3.5 Interpolation as a Function of Temperature The tables in Appendix A list values of the second virial coefficient, the transport properties, and their first derivatives as functions of temperature. The temperature intervals were chosen so that the errors from linear interpolation would be smaller than the uncertainties propagated from the uncertainties of the interatomic potential. Table 5 lists bounds of the interpolation errors, and the unweighted average over the entire temperature range. Below 10 K, the interpolation errors increase because the temperature derivatives of the properties increase. 3.6 Classical Calculation We made an important check of the entire calculation of each thermophysical property. To do so, we performed the relatively simple classical calculation [32] which is valid at high temperatures where the ratio of the de Broglie Victor Maurice, comte de Broglie, 1647–1727, was marshal of France and fought in the wars of King Louis XIV. His son François Marie, duc de Broglie, 1671–1745, marshal of France, fought at Malplaquet (1709), in the War of the Polish Succession, and in the War of the Austrian Succession. wavelength h[(2[pi]mkT).sup.1/2] to atomic diameter [sigma] is much less than 1. Figures 3 and 4 show that the classical calculations of the viscosity and of the second virial coefficient asymptotically approach the quantum results. 4. Uncertainties of the Thermophysical Properties From the Uncertainty of the Potential We now evaluate how the uncertainty of the ab initic values of [varphi](r) propagates into the uncertainty of calculated thermophysical properties. To do so, we calculated the properties with each of the potentials discussed in Sec. 2 and we plotted the results as deviations from the results obtained for [[varphi].sub.00](r). Figure 3 shows these deviations for the viscosity of [He.sup.4]. In Fig. 3, the width of the shaded band surrounding the curve [[varphi].sub.A] spans the range of results obtained with [[[varphi].sup.-].sub.A] to those obtained with [[[varphi].sup.+].sub.A]. Similar bands could have been placed about the results from those obtained with [[varphi].sub.00], [[varphi].sub.B], and [[varphi].sub.SAPT]; they were omitted for clarity. We took the differences in the alternative potentials as an accurate estimate of the uncertainty in [[varphi].sub.00]. By comparing the properties calculated from each alternative potential, we estimated the actual uncertainty propagated into each reported thermophysical property. Figure 3 shows that as the temperature is increased from 1 K to 10 K, the relative uncertainty of the viscosity [u.sub.r]([eta]) of [He.sup.4] decreases from 0.4 % to 0.1 %. In this temperature range, the discrepancies among the potentials are comparable to the uncertainty of each potential, as indicated by the width of the shaded band. In the range 10 K [less than] T [less than] 1000 K, the difference between the results obtained using [[varphi].sub.00] and the results obtained with [[varphi].sub.B] and [[varphi].sub.SAPT] lead us to conclude that [u.sub.r]([eta]) is approximately 0.08 %. If the discrepancies between the ab initio results around 4.0 bohr could be resolved, then [u.sub.r]([eta]) would be reduced by nearly a factor of three in this tem perature range. In the range 1000 K [less than] T [less than] [10.sup.4] K, we also conclude [u.sub.r]([eta]) [approximate] 0.08 %. In this temperature range, the results from [[varphi].sub.00] and [[varphi].sub.B] are more reliable than the results from [[varphi].sub.A] and [[varphi].sub.SAPT] having been fit to the short-range variational calculations of Komasa [5] as discussed in Sec. 2, above. The relative uncertainty of the second virial coefficient [u.sub.r](B) of [He.sup.4] can be judged from Fig. 4. In the range 1 K [less than] T [less than] 10 K, [u.sub.r](B) [approximate] 1 %. At T [approximate] 23.4 K, B(T) passes through zero. There, [u.sub.r](B) diverges; however, the uncertainty of B, u(B) [approximate] 0.3 [cm.sup.3.][mol.sup.-1]. In the range 100 K [less than] T [less than] [10.sup.4] K, [u.sub.r](B) of [He.sup.4] gradually declines from 0.4% to 0.1 %. The uncertainties for each property are summarized in Table 6 from comparisons similar to those provided in Figs. 3 and 4 and described in the preceding paragraphs. These uncertainties are much lower than those from measurements; thus, the corresponding values of the properties listed in the Appendices can be used as standards. 5. Results The results of the present calculations for [He.sup.4], [He.sup.3], and their equimolar mixture are listed in Tables A1, A2, and A3 in Appendix A. These tables contain the second virial coefficient B for the pure species and the interaction second virial [B.sub.12] where [B.sub.mix] = [[x.sup.2].sub.1][B.sub.11] + 2[x.sub.1][x.sub.2][B.sub.12] + [[x.sup.2].sub.2][B.sub.22]. The zero-density viscosity, thermal conductivity and their equimolar mixture. The diffusion coefficient at 101.325 kPa (one atmosphere), and the thermal diffusion factor. Derivatives with respect to temperature are provided to facilitate interpolation and for use in calculating acoustic virial coefficients. The tables for pure [He.sup.4] and [He.sup.3] contain the self-diffusion coefficient calculated without symmetry effects (Boltzmann statistics), and the thermal diffusion factor of mixture of 99.999% [He.sup.4] or [He.sup.3] respectively. The tables span the temperature interval 1 K [less than or equal to] T [less than or equal to] [10.sup.4] K. The highest temperature is well below the first excited state of helium (2 X [10.sup.5] K) and well below 2.91 X [10.sup.5] K, the highest value of the ab initio results used to determine [[varphi].sub.A]. In order to calculate the thermophysical properties between the tabulated temperatures, we recommend interpolation using the cubic polynomial f(T) such that f(T) = a (T - [T.sub.1])+b(T - [T.sub.2]) + {c(T - [T.sub.1])+ d(T - [T.sub.2])}(T - [T.sub.1])(T - [T.sub.2]) a = f([T.sub.2])/[delta]T b = f([T.sub.1])/[delta]T c = {f'[([T.sub.2])/([delta]T).sup.2]} - {[(a + b)/([delta]T).sup.2]} d = {f'[([T.sub.1])/([delta]T).sup.2]} - {[(a + b)/([delta]T).sup.2]}, (10) where f' = df/dT and [delta]T = [T.sub.2] - [T.sub.1]. The calculated values listed in the Tables are accurate to the uncertainties discussed in Sec. 4. Equation (10) contributes an additional uncertainty from the interpolation discussed as in Sec. 3. 6. Comparison With Measurements In this section we compare the values of the thermophysical properties calculated [[varphi].sub.00] with the best experimental values. In nearly every case, the experimental values agree with the calculated values within their combined uncertainties, and the calculated properties have the smaller uncertainties. 6.1 Second Virial Coefficient Figure 5 displays the deviations of various experimental values of B (T) for [He.sup.4] from [B.sub.00](T) calculated using [[varphi].sub.00]. The dashed curves in Fig. 5 represent the values of [B.sub.A](T), calculated using [[varphi].sub.A], and the dash-dot-dot curves the values of [B.sub.B](T), calculated from [[varphi].sub.B]. Also shown in Fig. 5 and summarized in Table 7, are measured values of B (T) along with their reported uncertainties. In nearly every case, [B.sub.00](T) agrees with the experimental values within the uncertainties of the experimental values. The maximum uncertainties of [B.sub.00](T) are estimated by comparing the variances with [B.sub.A](T) and [B.sub.B](T). These uncertainties are much smaller than the experimental uncertainties (see the dash-dot-dot curve in Fig. 5.). At very low temperatures B (T) is sensitive to the shape of the potential well. Figure 5 shows that the lower well depth of [[varphi].sub.00] predicted by Korona et al. [18] reproduces the low temperature measurements better than the shallower well depth predicted by Van de Bovenkamp and van Duijneveldt [20] and by Van Mourik and Dunning [21]. To further strengthen this argument, it is known that the low temperature second virial measurements have not been corrected for contributions from the third virial coefficient C(T). For [He.sup.4] [42], the size of this "third virial correction" can be seen in the top panel of Figure 5. In that panel, the solid circles show the values B(T) before they were corrected in Ref. [43], and the open circles show the values after the correction. The correction for C(T) lowers the second virial values bringing them further in line with [B.sub.00](T) and away from [B.sub.B](T) indicating a preference for the lower well depth. Table 7 provides two numerical measures of the differences between experimental values of B (T) and those calculated using [[varphi].sub.00]. One measure is the mean of the absolute values of the differences [B.sub.exp] - [B.sub.00] and the second is the range of these differences. The final colunm of Table 7 lists the range of the uncertainties reported by the experimenters. In nearly all cases the experimental uncertainties exceed the differences [B.sub.exp] - [B.sub.00]. Figure 6 compares [B.sub.exp](T) of [He.sup.3], deduced from the measurements of Matacotta et al. [47], with [B.sub.00](T). There is an obvious trend in the deviations which is larger than the experimental uncertainties below 5 K. Probably, the trend would be removed if [B.sub.exp](T) was corrected for the for effects of the third virial coefficient of [He.sup.3] [48] as discussed above for [He.sup.4] [42]. 6.2 Viscosity Figures 7 and 8 and Table 8 compare the zero-density viscosity [[eta].sub.00], calculated using [[varphi].sub.00], with measured values from many sources. The experimental results are typically reported at 101.325 kPa where the density dependence is negligible in comparison with experimental uncertainties. Figure 7 shows the viscosity of [He.sup.3] and [He.sup.4] at low temperatures where the large quantum effects lead to important differences between the isotopes. The [[eta].sub.00](T) values are in good agreement with the measurements of Becker et al. [49]. Figure 8 displays the fractional deviations of various values of [[eta].sub.exp], of [He.sup.4] from [[eta].sub.00]. In nearly every case they are smaller than the uncertainties provided by the experimenters. In Fig. 8, the barely visible dashed curve represents [[eta].sub.A] calculated using [[varphi].sub.A], and the dash-dot-dot curve [[eta].sub.B], calculated from [[varphi].sub.B]. The differences between these curves are a measure of the ab initio uncertainties which are much smaller than the reported experimental uncertainties. 6.3 Thermal Conductivity Figure 9 and Table 9 compare the values of the zero-density thermal conductivity of [He.sup.4] calculated using [[varphi].sub.00] with measured values from several sources. The experimental thermal conductivities are typically reported at 101.325 kPa, however the density dependence is negligible compared to the experimental uncertainties. As it was the case for B and [eta], most of the values of [[lambda].sub.exp] differ from [[lambda].sub.00] by an amount comparable to the uncertainty of the measurements. The differences between the values [[lambda].sub.00], [[lambda].sub.A], and [[lambda].sub.B] calculated using [[varphi].sub.00], [[varphi].sub.A] and [[varphi].sub.B] are much smaller than the uncertainties of the measurements. Table 9 lists the root mean square of the relative differences [delta][lambda]/[[lambda].sub.00] [equivalent] ([[lambda].sub.exp] -- [[lambda].sub.00])/[[lambda].sub.00] and the range of these relative differences. The final column of Table 9 lists the range of the uncertainties reported by the exp erimenters. 6.4 Diffusion Coefficient [D.sub.12](T) Figure 10 and Table 10 compare the values of the mutual diffusion coefficient for an equimolar mixture of [He.sup.3] and [He.sup.4] at one atmosphere (101325 Pa). Figure 10 shows the deviations of [D.sub.12,exp] taken from three sources, from [D.sub.12,00], where [D.sub.12,00] was calculated using [[varphi].sub.00]. Because the diffusion coefficient is difficult to measure, the uncertainties of the experimental values are comparatively large; therefore, the relative deviations of the values calculated using [[varphi].sub.A] and [[varphi].sub.B] are not visible in Fig. 10. The difference in [D.sub.12] on going from the first to the second order approximation is practically the same as seen for the viscosity in Fig. 2. Table 10 lists the root-mean-square of the relative differences [delta]D/[D.sub.00] [equivalent] ([D.sub.exp] - [D.sub.00])/[D.sub.00] and the range of these relative differences as well as the range of the uncertainties reported by the experimenters. 6.4 Thermal Diffusion Factor [[alpha].sub.T] The thermal diffusion factor [[alpha].sub.T] is a complicated function of temperature and concentration and only a few, relatively inaccurate measurements are available. Figure 11 compares [[alpha].sub.T,exp] for an equimolar mixture of [He.sup.3] and [He.sup.4] to the [[alpha].sub.T,00] values calculated from [[varphi].sub.00]. The values of [[alpha].sub.T,A] and [[alpha].sub.T,B] calculated from [[varphi].sub.A] and [[varphi].sub.B] are also shown, only differing at low temperatures. In the first-order approximate calculation of the transport properties, [[alpha].sub.T] is identically zero; thus, we compared the second-order transport-theory results to the third-order results to estimate the uncertainties of the ab initio results from truncating the transport theory. Going from the second to third order increased [[alpha].sub.T] by 0.56 % at 10 K and by 0.36 % at 10,000 K. The thermal diffusion factor is very difficult to measure the typical relative uncertainties are 4 % to 8 %. Owing to the experimental difficulties, the ca lculated values would be more accurate than any experimentally determined value. 7. Conclusion We have reviewed the recent ab initio calculations of [varphi](r) for helium. We represented one of the most accurate ab initio values of [varphi](r) by the algebraic expression [[varphi].sub.00](r) and we estimated its uncertainty by comparing the various ab initio calculations. For the thermophysical properties, the most significant uncertainties occur near 4.0 bohr. Using [[varphi].sub.00](r), we calculated B,[eta], [lambda], [D.sub.12], and [[alpha].sub.T]. The numerical methods used in these calculations contributed negligible uncertainty to the results. In all cases, the uncertainties of the calculated thermo-physical properties propagated from the uncertainties in [[varphi].sub.00](r) were much less than the uncertainties of published measurements. Therefore, the calculated values should be used as standard reference values. The large number of recent ab initio calculations of [varphi](r) demonstrate that this is an active field of research. In the near future, ab initio calculations will surely reduce the uncertainty of [varphi](r) near 4.0 bohr, further reducing the uncertainties in the calculated properties. Improved ab initio calculations of the molar polarizability of helium and of the dielectric virial coefficients are also under way. These may well lead to an ab initio standard of pressure based on measurements of the dielectric constant of helium near 273.16 K [70]. 8. Appendix A. Calculations The results of the present calculations for [He.sup.4], [He.sup.3], and their equimolar mixture are listed in Tables A1, A2, and A3, respectively. These tables contain the second virial coefficient for the pure species, the interaction second virial coefficient [B.sub.12], the zero-density viscosity and thermal conductivity, the diffusion coefficient, and the thermal diffusion coefficient. Derivatives with respect to temperature are provided to facilitate interpolation and for use in calculating acoustic viral coefficients. 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Moldover, Can a pressure standard be based on capacitance measurements?, J. Res. Natl. Inst. Stand. Technol. 103 (2), 167-175 (1998); K. Szalewicz, private communication. (71.) J. J. Hurly, W. L. Taylor, and F. R. Meeks, Thermal-Diffusion Factors at Low-Temperatures for Gas-Phase Mixtures of Isotopic Helium, J. Chem. Phys. 96 (5), 3775-3781 (1992). (72.) W. L. Taylor, Thermal Diffusion factor for the [He.sup.3]- [He.sup.4] system in the quantum region, J. Chem. Phys. 58 (3), 834-840 (1972). (73.) W. L. Taylor and S. Weissman, Thermal Diffusion Factors for the [He.sup.3]- [He.sup.4] system, J. Chem. Phys. 55 (8), 4000-4004 (1971). (74.) B. B. McInteer, L. T. Aldrich, and A. O. Nier, The Thermal Diffusion Constant of helium and the Separation of [He.sup.3] by Thermal Diffusion, Phys. Rev. 72 (6), 510-511 (1947). (75.) W. W. Watson, A. J. Howard, N. E. Miller, and R. M. Shiffrin, Isotopic Thermal Diffusion Factors for Helium and neon at Low Temperatures, Z. Naturforsch. Teal A 18A, 242-245 (1963).
Selected ab initio values of [varphi](r).
(1 bohr = 0.052 917 721 nm)
[varphi](3.0 bohr)/K [varphi](4.0 bohr)/K
Ceperley and Partridge [15] 3800 [+ or -] 100
Anderson et al. [16] 3812 [+ or -] 96.0
Klopper and Noga [17] 294.5
292.6
Korona et al. [18] 3759.959 [+ or -] 11.3 291.64 [+ or -] 0.9
Komasa [24] 3768.469 292.784
Gdanitz [19] 3768.813 293.025
3768.0 [+ or -] 0.8 292.7 [+ or -] 0.4
van de Bovenkamp and 293.48
Duijneveldt [20] 292.72 [+ or -] 0.02
van Mourik and Dunning 293.498
[21] 292.578
[varphi](5.6 bohr)/K
Ceperley and Partridge [15]
Anderson et al. [16] -11.01 [+ or -] 0.10
Klopper and Noga [17] -10.68
-11.00
Korona et al. [18] -11.06 [+ or -] 0.03
Komasa [24] -10.947
-10.978
Gdanitz [19] -10.947
-11.05 [+ or -] 0.10
van de Bovenkamp and -10.95
Duijneveldt [20] -10.99 [+ or -] 0.02
van Mourik and Dunning -11.00 [+ or -] 0.03
[21] -10.99
Remarks
Ceperley and Partridge [15] "exact" QMC
Anderson et al. [16] "exact" QMC
Klopper and Noga [17]
corrected to FCI
Korona et al. [18] SAPT
Komasa [24] (1200 term)
(2048 term)
upper bound
Gdanitz [19]
extrapolated to [infinity] basis set
van de Bovenkamp and
Duijneveldt [20] corrected to FCI
van Mourik and Dunning
[21] corrected to FCI
Parameters for Eq. (1) in atomic units (1 bohr = 1
Bo = 0.052 917 721 nm, 1 hartree = 1
Ha = 3.157 746 5 X [10.sup.5] K)
Property (unit) [[varphi].sub.00]
[10.sup.-6] A (K) 2.83379199
[a.sub.1] ([Bo.sup.-1]) -1.986231822
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) -5.034284240
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 0.0
[a.sub.-1] (Bo) -0.3514929118
[a.sub.-2] ([Bo.sup.2]) 0.1101468439
[delta] ([Bo.sup.-1]) 2.00788607
[C.sub.6] (Ha*[Bo.sup.-6]) 1.46097780
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.4117855
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83691250
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.265
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.644
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.275
Property (unit) [[varphi].sub.A]
[10.sup.-6] A (K) 2.02311
[a.sub.1] ([Bo.sup.-1]) -1.84827
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) -7.55879
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 1.82924
[a.sub.-1] (Bo) 0.0
[a.sub.-2] ([Bo.sup.2]) 0.0
[delta] ([Bo.sup.-1]) 2.03451
[C.sub.6] (Ha*[Bo.sup.-6]) 1.46098
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.41179
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83691
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.265
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.644
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.275
Property (unit) [[[varphi].sup.+].sub.A]
[10.sup.-6] A (K) 2.03130
[a.sub.1] ([Bo.sup.-1]) -1.85059
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) -7.50314
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 1.71078
[a.sub.-1] (Bo) 0.0
[a.sub.-2] ([Bo.sup.2]) 0.0
[delta] ([Bo.sup.-1]) 2.02137
[C.sub.6] (Ha*[Bo.sup.-6]) 1.45981
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.41066
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83544
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.262
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.638
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.273
Property (unit) [[[varphi].sup.-].sub.A]
[10.sup.-6] A (K) 2.01529
[a.sub.1] ([Bo.sup.-1]) -1.84616
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) -7.60470
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 1.93491
[a.sub.-1] (Bo) 0.0
[a.sub.-2] ([Bo.sup.2]) 0.0
[delta] ([Bo.sup.-1]) 2.04780
[C.sub.6] (Ha*[Bo.sup.-6]) 1.46215
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.41291
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83838
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.268
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.650
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.277
Property (unit) [[varphi].sub.B]
[10.sup.-6] A (K) 3.12631
[a.sub.1] ([Bo.sup.-1]) -2.01639
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) -4.67475
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 0.0
[a.sub.-1] (Bo) -0.47972
[a.sub.-2] ([Bo.sup.2]) 0.16755
[delta] ([Bo.sup.-1]) 2.01997
[C.sub.6] (Ha*[Bo.sup.-6]) 1.46098
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.41179
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83691
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.265
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.644
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.275
Property (unit) [[varphi].sub.SAPT]
[10.sup.-6] A (K) 2.07436426
[a.sub.1] ([Bo.sup.-1]) -1.88648251
[10.sup.2] [a.sub.2] ([Bo.sup.-2]) 6.20013490
[10.sup.3] [a.sub.3] ([Bo.sup.-3]) 0.0
[a.sub.-1] (Bo) 0.0
[a.sub.-2] ([Bo.sup.2]) 0.0
[delta] ([Bo.sup.-1]) 1.94861295
[C.sub.6] (Ha*[Bo.sup.-6]) 1.46097780
[10.sup.-1] [C.sub.8] (Ha*[Bo.sup.-8]) 1.4117855
[10.sup.-2] [C.sub.10] (Ha*[Bo.sup.-10]) 1.83691250
[10.sup.-3] [a] [C.sub.12] (Ha*[Bo.sup.-12]) 3.265
[10.sup.-4] [a] [C.sub.14] (Ha*[Bo.sup.-14]) 7.644
[10.sup.-6] [a] [C.sub.16] (Ha*[Bo.sup.-16]) 2.275
(a.)Calculated using combining rules of Thakkar [29]
Properties of the fitted helium potentials.
(1 A = [10.sup.-10] m)
Property (unit) [[varphi].sub.00] [[varphi].sub.A]
[epsilon]/[k.sub.B] (K) 11.054 11.063
[r.sub.m] (bohr) 5.6039 5.6034
[r.sub.m] (A) 2.9654 2.9652
[sigma] (bohr) 4.9873 4.9870
scattering length (A) 83.68 82.00
effective range (A) 7.24 7.24
bound state/[k.sub.B] (mK) 1.90 1.98
Property (unit) [[varphi].sub.B] [[[varphi].sup.-].sub.A]
[epsilon]/[k.sub.B] (K) 10.974 11.074
[r.sub.m] (bohr) 5.6097 5.6034
[r.sub.m] (A) 2.9685 2.9625
[sigma] (bohr) 4.9922 4.9868
scattering length (A) 96.91 85.30
effective range (A) 7.30 7.26
bound state/[k.sub.B] (mK) 1.39 1.83
Property (unit) [[[varphi].sup.+].sub.A]
[epsilon]/[k.sub.B] (K) 11.052
[r.sub.m] (bohr) 5.6034
[r.sub.m] (A) 2.9652
[sigma] (bohr) 4.9873
scattering length (A) 78.90
effective range (A) 7.22
bound state/[k.sub.B] (mK) 2.51
Integration step sizes used in a given energy
range to locate the nth zero of [[psi].sub.l](k,r)
Integration step size Applicable range
([cm.sup.-1]) ([cm.sup.-1])
0.0001 0.0 [less than or equal to]
k [less than or equal to] 0.01
0.001 0.01 [less than or equal to]
k [less than or equal to] 0.4
0.01 0.4 [less than or equal to]
k [less than or equal to] 10.0
0.05 10.0 [less than or equal to]
k [less than or equal to] 100.0
0.1 100.0 [less than or equal to] k
Relative uncertainties from interpolating
between tabulated temperatures
Max
(1 K to 10 K)
[delta]B/B X [10.sup.6] 187
[delta][eta]/[eta] X [10.sup.6] 107
[delta][lambda]/[lambda] X [10.sup.6] 85.5
[delta][D.sub.12]/[D.sub.12] X [10.sup.6] 1.95
[delta][[alpha].sub.T]/[[alpha].sub.T]
X [10.sup.6] 288
Max
(10 K to [10.sup.4] K)
[delta]B/B X [10.sup.6] 95.3
[delta][eta]/[eta] X [10.sup.6] 3.23
[delta][lambda]/[lambda] X [10.sup.6] 3.24
[delta][D.sub.12]/[D.sub.12] X [10.sup.6] 1.93
[delta][[alpha].sub.T]/[[alpha].sub.T]
X [10.sup.6] 6.55
Average
(1 K to [10.sup.4] K)
[delta]B/B X [10.sup.6] 18.0
[delta][eta]/[eta] X [10.sup.6] 3.01
[delta][lambda]/[lambda] X [10.sup.6] 3.05
[delta][D.sub.12]/[D.sub.12] X [10.sup.6] 0.38
[delta][[alpha].sub.T]/[[alpha].sub.T]
X [10.sup.6] 6.43
Relative uncertainty of thermophysical
properties of pure [He.sup.4] and [He.sup.3]
propagated from the differences between
potentials
2000 [delta]B/B X [10.sup.4] [delta][eta]/[eta] [delta][lambda]/
X [10.sup.4] [lambda] X [10.sup.4]
2 80 40 40
5 89 17 17
10 125 6.3 6.5
20 559 5.4 5.2
50 91 8 8
100 43 7.7 7.7
200 29 6.7 6.8
300 22 6.1 6.2
400 19 5.7 5.8
500 17 5.5 5.5
1000 14 4.8 4.8
2000 11 4.6 4.6
2000 [delta][D.sub.12]/ [delta][[alpha].sub.T]/
[D.sub.12] X [10.sup.4] [[alpha].sub.T] X [10.sup.4]
2 56 301
5 15 84
10 5.8 32
20 4.5 10
50 6.6 7.5
100 6.4 4.4
200 5.8 4.5
300 5.4 4.1
400 5.1 7.1
500 4.9 9.8
1000 4.6 19.9
2000 5.6 33
Deviations of [B.sub.exp] from [B.sub.00]
Calculated using [[varphi].sub.00]
Authors [reference] Temp. range [less than]\[B.sub.exp]
- [B.sub.00l]\[greater than]
(K) ([cm.sup.3]*[mol.sup.-1])
Berry [42] 2.60 to 27.10 0.41
Gugan and Michel [43] 4.22 to 27.10 0.13
[corrected for C(T)]
Gugan and Michel [43] 4.23 to 27.17 0.15
Kemp et al. [44] 27.10 to 172.01 0.06
Gammon [45] 98.15 to 1474.85 0.05
Kell et al. [46] 273.15 to 623.15 0.03
Waxman and Davis [8] 298.15 0.08
Matacotta et al. [47] 1.47 to 20.30 0.42
Authors [reference] Range ([B.sub.exp] - [B.sub.00])
([cm.sup.3]*[mol.sup.-1])
Berry [42] 0.14 to 0.83
Gugan and Michel [43] 0.04 to 0.20
[corrected for C(T)]
Gugan and Michel [43] 0.03 to 0.33
Kemp et al. [44] -0.05 to 0.11
Gammon [45] -0.07 to 0.07
Kell et al. [46] -0.05 to 0.05
Waxman and Davis [8] 0.08
Matacotta et al. [47] -0.14 to 0.96
Authors [reference] Reported
uncertainties
([cm.sup.3]*[mol.sup.-1])
Berry [42] 0.20 to 1.00
Gugan and Michel [43] 0.20 to 0.70
[corrected for C(T)]
Gugan and Michel [43] 0.01 to 0.07
Kemp et al. [44] 0.13 to 0.16
Gammon [45] 0.05 to 0.06
Kell et al. [46] 0.01 to 0.15
Waxman and Davis [8] 0.01
Matacotta et al. [47] 0.20 to 1.00
Relative deviations of [[eta].sub.exp]
from [[eta].sub.00] calculated using [[varphi].sub.00]
Temperature
range (K)
Wakeham et al. [50] 298 to 793
Maitland and Smith [51] 80 to 2000
Vogel [52] 294.5 to 647.9
Kestin et al. [53] 298 to 973
Clark and Smith [54] 77.5 to 373
Dawe and Smith [55] 293 to 1600
Coremans et al. [56] 20.4 to 77.8
Kestin and Wakeham [57] 298 to 473
Johnston and Grilly [58] 79 to 296
Kalelkar and Kestin [59] 298 to 1121
Becker et al. [49] [He.sup.4] 1.3 to 4.2
Becker et al. [49] [He.sup.3] 1.3 to 4.2
Kestin et al. [60] 298 to 778
Guevara et al. [61] 1100 to 2150
100X
[([delta][eta]/[[eta].sub.00]).sub.rms]
Wakeham et al. [50] 0.22
Maitland and Smith [51] -0.51
Vogel [52] 0.06
Kestin et al. [53] 0.20
Clark and Smith [54] 0.58
Dawe and Smith [55] -1.16
Coremans et al. [56] 4.03
Kestin and Wakeham [57] 0.16
Johnston and Grilly [58] -0.96
Kalelkar and Kestin [59] -0.31
Becker et al. [49] [He.sup.4] 5.15
Becker et al. [49] [He.sup.3] 2.73
Kestin et al. [60] 0.35
Guevara et al. [61] -0.90
Range of
100 X [delta][eta]/[[eta].sub.00]
Wakeham et al. [50] 0.12 to 0.32
Maitland and Smith [51] -2.27 to -0.51
Vogel [52] 0.02 to 0.12
Kestin et al. [53] 0.08 to 0.30
Clark and Smith [54] 0.18 to 1.21
Dawe and Smith [55] -2.43 to 0.45
Coremans et al. [56] 1.77 to 6.33
Kestin and Wakeham [57] 0.10 to 0.22
Johnston and Grilly [58] -1.60 to -0.18
Kalelkar and Kestin [59] -2.16 to 0.46
Becker et al. [49] [He.sup.4] -1.65 to 9.57
Becker et al. [49] [He.sup.3] -0.13 to 4.37
Kestin et al. [60] 0.08 to 0.57
Guevara et al. [61] -3.93 to 0.31
Reported
uncertainties (%)
Wakeham et al. [50] 0.2 to 0.5
Maitland and Smith [51] 1.5
Vogel [52] 0.3
Kestin et al. [53] 0.1 to 0.3
Clark and Smith [54] 0.5
Dawe and Smith [55] 1.0
Coremans et al. [56] 3.0
Kestin and Wakeham [57] 0.3
Johnston and Grilly [58] 3.0
Kalelkar and Kestin [59] 0.5
Becker et al. [49] [He.sup.4] 5.0
Becker et al. [49] [He.sup.3] 5.0
Kestin et al. [60] 0.1 to 0.3
Guevara et al. [61] 0.65
Relative deviations of [[lambda].sub.exp]
from [[lambda].sub.00]
Temperature
range (K)
Wakeham et al. [50] 298.15 to 973.15
Haarman [62] 328.15 to 468.15
Jody et al. [63] 400 to 2500
Assael et al. [64] 308.15
Acton and Kellner [65] 3.3 to 20.0
Kestin et al. [66] 300.65
100 X
[([delta][lambda]/[[lambda].sub.00]).sub.rms]
Wakeham et al. [50] 0.23
Haarman [62] 0.43
Jody et al. [63] 2.34
Assael et al. [64] 0.20
Acton and Kellner [65] 0.66
Kestin et al. [66] 0.13
Range of
100 X [delta][lambda]/[[lambda].sub.00]
Wakeham et al. [50] 0.07 to 0.36
Haarman [62] -0.54 to -0.32
Jody et al. [63] -4.67 to -0.41
Assael et al. [64] -0.20
Acton and Kellner [65] -0.34 to 1.30
Kestin et al. [66] -0.13
Reported
uncertainties (%)
Wakeham et al. [50] 0.2 to 0.5
Haarman [62] 0.3
Jody et al. [63] 2.0 to 4.7
Assael et al. [64] 0.2
Acton and Kellner [65] 1.0
Kestin et al. [66] 0.3
Deviations of the [D.sub.12,exp] from
[D.sub.12,00] calculated using [[varphi].sub.00]
Temperature 100 X
range (K) [([delta]D/[D.sub.00]).sub.rms]
Liner and Weissman [67] 303 to 806 1.613
Bendt [68] 14.4 to 296.0 3.876
DuBro and Weissman [69] 76.5 to 888.3 4.142
Range of Reported
100 X [delta]D/[D.sub.00] uncertainties (%)
Liner and Weissman [67] -4.4 to 0.55 1.3 to 4.7
Bendt [68] -6.9 to 5.6 2.0 to 4.0
DuBro and Weissman [69] -6.7 to -1.1 5.0
Thermophysical properties of [He.sup.4] as a
function of temperature, where (-2) is X
[10.sup.-2]
(T) B dB/dT
(K) ([cm.sup.3]*[mol.sup.-1]) ([cm.sup.3]*[mol.sup.-1]*[K.sup.-1])
1.0 -474.449 664.861
1.2 -369.743 411.102
1.4 -302.255 276.706
1.6 -255.395 198.357
1.8 -220.996 149.180
2.0 -194.640 116.448
2.25 -169.200 88.972
2.5 -149.421 70.388
2.75 -133.560 57.209
3.0 -120.530 47.499
3.5 -100.334 34.377
4.0 -85.360 26.103
4.5 -73.788 20.526
5 -64.566 16.578
6 -50.774 11.477
7 -40.944 8.419
8 -33.581 6.440
9 -27.859 5.084
10 -23.285 4.114
11 -19.547 3.397
12 -16.435 2.850
14 -11.556 2.087
16 -7.910 1.591
18 -5.088 1.251
20 -2.842 1.007
22 -1.017 8.27(-1)
23 -0.227 7.54(-1)
24 0.494 6.90(-1)
25 1.155 6.33(-1)
(T) [d.sup.2]B/d[T.sup.2] [eta] d[eta]/dT
(K) ([cm.sup.3]*[mol.sup.-1]*[K.sup.-2] ([micro]Pa*s) ([micro]Pa*[K.sup.-1])
1.0 -1771.941 3.279(-1) 5.296(-2)
1.2 -891.823 3.395(-1) 6.980(-2)
1.4 -500.007 3.573(-1) 1.096(-1)
1.6 -304.265 3.834(-1) 1.504(-1)
1.8 -197.490 4.171(-1) 1.860(-1)
2.0 -135.022 4.573(-1) 2.149(-1)
2.25 -89.028 5.146(-1) 2.419(-1)
2.5 -61.860 5.775(-1) 2.600(-1)
2.75 -44.818 6.440(-1) 2.705(-1)
3.0 -33.586 7.123(-1) 2.749(-1)
3.5 -20.391 8.492(-1) 2.708(-1)
4.0 -13.372 9.815(-1) 2.573(-1)
4.5 -9.272 1.106 2.407(-1)
5 -6.706 1.222 2.244(-1)
6 -3.849 1.433 1.974(-1)
7 -2.415 1.620 1.783(-1)
8 -1.615 1.791 1.649(-1)
9 -1.134 1.951 1.551(-1)
10 -8.269(-1) 2.102 1.474(-1)
11 -6.215(-1) 2.246 1.412(-1)
12 -4.790(-1) 2.385 1.359(-1)
14 -3.021(-1) 2.648 1.274(-1)
16 -2.026(-1) 2.895 1.206(-1)
18 -1.425(-1) 3.131 1.151(-1)
20 -1.039(-1) 3.356 1.104(-1)
22 -7.81(-2) 3.573 1.064(-1)
23 -6.84(-2) 3.678 1.046(-1)
24 -6.02(-2) 3.782 1.029(-1)
25 -5.32(-2) 3.884 1.013(-1)
(T) [lambda] d[lambda]/dT
(K) (mW*[m.sup.-1]*[K.sup.-1]) (mW*[m.sup.-1]*[K.sup.-2])
1.0 2.624 4.512(-1)
1.2 2.713 5.030(-1)
1.4 2.839 7.759(-1)
1.6 3.026 1.097
1.8 3.276 1.395
2.0 3.581 1.642
2.25 4.022 1.872
2.5 4.510 2.023
2.75 5.028 2.108
3.0 5.560 2.143
3.5 6.628 2.110
4.0 7.658 2.005
4.5 8.630 1.878
5 9.537 1.754
6 11.180 1.547
7 12.650 1.400
8 14.000 1.295
9 15.250 1.218
10 16.440 1.157
11 17.570 1.108
12 18.660 1.066
14 20.720 9.990(-1)
16 22.660 9.458(-1)
18 24.510 9.021(-1)
20 26.270 8.654(-1)
22 27.970 8.338(-1)
23 28.800 8.196(-1)
24 29.610 8.063(-1)
25 30.410 7.938(-1)
(T) D(101.3 kPa) [[alpha].sub.T]
(K) ([10.sup.-4]*[m.sup.2]*[s.sup.-1])
1.0 7.154(-5) 4.147(-2)
1.2 9.622(-5) 5.098(-2)
1.4 l.240(-4) 5.716(-2)
1.6 1.560(-4) 6.162(-2)
1.8 1.927(-4) 6.501(-2)
2.0 2.345(-4) 6.764(-2)
2.25 2.943(-4) 7.011(-2)
2.5 3.624(-4) 7.190(-2)
2.75 4.386(-4) 7.318(-2)
3.0 5.228(-4) 7.409(-2)
3.5 7.138(-4) 7.525(-2)
4.0 9.328(-4) 7.594(-2)
4.5 1.178(-3) 7.646(-2)
5 1.446(-3) 7.693(-2)
6 2.050(-3) 7.784(-2)
7 2.735(-3) 7.871(-2)
8 3.495(-3) 7.949(-2)
9 4.326(-3) 8.014(-2)
10 5.226(-3) 8.068(-2)
11 6.190(-3) 8.110(-2)
12 7.217(-3) 8.143(-2)
14 9.451(-3) 8.186(-2)
16 1.191(-2) 8.208(-2)
18 1.459(-2) 8.216(-2)
20 1.749(-2) 8.214(-2)
22 2.058(-2) 8.206(-2)
23 2.220(-2) 8.200(-2)
24 2.387(-2) 8.193(-2)
25 2.558(-2) 8.185(-2)
26 1.762 5.83(-1) -4.73(-2) 3.985 9.980(-2) 31.200 7.821(-1)
28 2.840 4.98(-1) -3.78(-2) 4.181 9.706(-2) 32.740 7.606(-1)
30 3.766 4.30(-1) -3.07(-2) 4.373 9.460(-2) 34.240 7.412(-1)
35 5.587 3.08(-1) -1.93(-2) 4.833 8.941(-2) 37.840 7.004(-1)
40 6.917 2.29(-1) -1.28(-2) 5.269 8.523(-2) 41.260 6.675(-1)
45 7.921 1.76(-1) -8.94(-3) 5.686 8.176(-2) 44.530 6.402(-1)
50 8.698 1.38(-1) -6.46(-3) 6.087 7.882(-2) 47.670 6.171(-1)
60 9.806 8.86(-2) -3.66(-3) 6.851 7.406(-2) 53.650 5.798(-1)
70 10.537 5.98(-2) -2.25(-3) 7.572 7.035(-2) 59.290 5.507(-1)
80 11.038 4.16(-2) -1.47(-3) 8.260 6.734(-2) 64.680 5.270(-1)
90 11.389 2.95(-2) -9.98(-4) 8.920 6.483(-2) 69.846 5.073(-1)
100 11.640 2.11(-2) -7.03(-4) 9.558 6.270(-2) 74.833 4.906(-1)
120 11.947 1.07(-2) -3.78(-4) 10.775 5.923(-2) 84.360 4.633(-1)
140 12.098 4.92(-3) -2.18(-4) 11.932 5.650(-2) 93.405 4.419(-1)
160 12.160 1.50(-3) -1.32(-4) 13.039 5.428(-2) 102.063 4.245(-1)
180 12.167 -6.18(-4) -8.32(-5) 14.105 5.242(-2) 110.403 4.099(-1)
200 12.140 -1.96(-3) -5.34(-5) 15.137 5.083(-2) 118.474 3.975(-1)
225 12.077 -2.99(-3) -3.10(-5) 16.386 4.914(-2) 128.241 3.842(-1)
250 11.994 -3.59(-3) -1.78(-5) 17.596 4.769(-2) 137.701 3.729(-1)
275 11.900 -3.93(-3) -9.74(-6) 18.772 4.644(-2) 146.897 3.630(-1)
300 11.799 -4.10(-3) -4.69(-6) 19.919 4.534(-2) 155.863 3.544(-1)
325 11.695 -4.18(-3) -1.46(-6) 21.040 4.436(-2) 164.625 3.467(-1)
350 11.591 -4.19(-3) 6.15(-7) 22.138 4.348(-2) 173.205 3.398(-1)
375 11.486 -4.15(-3) 1.95(-6) 23.215 4.268(-2) 181.621 3.336(-1)
400 11.383 -4.09(-3) 2.81(-6) 24.272 4.196(-2) 189.889 3.279(-1)
450 11.183 -3.93(-3) 3.66(-6) 26.338 4.069(-2) 206.028 3.179(-1)
500 10.991 -3.74(-3) 3.88(-6) 28.344 3.960(-2) 221.707 3.094(-1)
600 10.637 -3.36(-3) 3.66(-6) 32.211 3.783(-2) 251.926 2.955(-1)
700 10.319 -3.01(-3) 3.19(-6) 35.922 3.643(-2) 280.913 2.846(-1)
800 10.033 -2.72(-3) 2.72(-6) 39.506 3.529(-2) 308.912 2.757(-1)
900 9.774 -2.47(-3) 2.32(-6) 42.986 3.434(-2) 336.095 2.682(-1)
1000 9.538 -2.25(-3) 1.99(-6) 46.378 3.353(-2) 362.590 2.618(-1)
1200 9.124 -1.91(-3) 1.48(-6) 52.946 3.221(-2) 413.878 2.515(-1)
1400 8.770 -1.65(-3) 1.14(-6) 59.280 3.117(-2) 463.334 2.434(-1)
1600 8.462 -1.44(-3) 9.01(-7) 65.427 3.033(-2) 511.329 2.368(-1)
1800 8.190 -1.28(-3) 7.27(-7) 71.421 2.963(-2) 558.122 2.313(-1)
2000 7.947 -1.15(-3) 5.97(-7) 77.286 2.904(-2) 603.905 2.267(-1)
2500 7.437 -9.08(-4) 3.90(-7) 91.499 2.904(-2) 714.835 2.267(-1)
3000 7.026 -7.45(-4) 2.73(-7) 105.216 2.904(-2) 821.876 2.267(-1)
3500 6.684 -6.28(-4) 2.00(-7) 118.560 2.904(-2) 925.992 2.267(-1)
4000 6.393 -5.41(-4) 1.53(-7) 131.612 2.904(-2) 1027.819 2.267(-1)
4500 6.141 -4.73(-4) 1.20(-7) 144.429 2.904(-2) 1127.805 2.267(-1)
5000 5.918 -4.19(-4) 9.68(-8) 157.054 2.904(-2) 1226.279 2.267(-1)
6000 5.542 -3.39(-4) 6.62(-8) 181.847 2.904(-2) 1419.639 2.267(-1)
7000 5.232 -2.83(-4) 4.79(-8) 206.173 2.904(-2) 1609.339 2.267(-1)
8000 4.972 -2.41(-4) 3.61(-8) 230.154 2.904(-2) 1796.322 2,267(-1)
9000 4.747 -2.09(-4) 2.81(-8) 253.873 2.904(-2) 1981.249 2.267(-1)
10000 4.551 -1.84(-4) 2.24(-8) 277.393 2.904(-2) 2164.607 2.267(-1)
26 2.734(-2) 8.177(-2)
28 3.101(-2) 8.159(-2)
30 3.485(-2) 8.140(-2)
35 4.522(-2) 8.087(-2)
40 5.664(-2) 8.034(-2)
45 6.907(-2) 7.980(-2)
50 8.246(-2) 7.928(-2)
60 1.120(-1) 7.831(-2)
70 1.452(-l) 7.741(-2)
80 1.818(-1) 7.659(-2)
90 2.216(-1) 7.584(-2)
100 2.646(-1) 7.514(-2)
120 3.597(-1) 7.390(-2)
140 4.666(-1) 7.281(-2)
160 5.847(-1) 7.184(-2)
180 7.137(-1) 7.097(-2)
200 8.532(-1) 7.017(-2)
225 1.042 6.927(-2)
250 1.246 6.845(-2)
275 1.466 6.769(-2)
300 1.700 6.699(-2)
325 1.949 6.634(-2)
350 2.212 6.573(-2)
375 2.489 6.516(-2)
400 2.780 6.462(-2)
450 3.403 6.361(-2)
500 4.078 6.270(-2)
600 5.584 6.108(-2)
700 7.290 5.968(-2)
800 9.189 5.843(-2)
900 11.278 5.731(-2)
1000 13.551 5.628(-2)
1200 18.639 5.445(-2)
1400 24.430 5.286(-2)
1600 30.909 5.144(-2)
1800 38.060 5.015(-2)
2000 45.872 4.897(-2)
2500 68.240 4.637(-2)
3000 94.577 4.415(-2)
3500 124.803 4.219(-2)
4000 158.862 4.043(-2)
4500 196.713 3.882(-2)
5000 238.324 3.734(-2)
6000 332.735 3.466(-2)
7000 441.969 3.229(-2)
8000 565.960 3.014(-2)
9000 704.683 2.816(-2)
10000 858.143 2.633(-2)
Thermophysical properties of [He.sup.3] as a
function of temperature, where (-2) is X [10.sup.-2]
T B dB/dT
(K) ([cm.sup.3]*[mol.sup.-1]) ([cm.sup.3]*[mol.sup.-1]*[K.sup.-1])
1.0 -237.503 174.583
1.2 -206.501 137.501
1.4 -181.829 110.586
1.6 -161.811 90.544
1.8 -145.295 75.286
2.0 -131.470 63.444
2.25 -117.097 52.073
2.5 -105.208 43.422
2.75 -95.226 36.710
3.0 -86.736 31.410
3.5 -73.089 23.709
4.0 -62.616 18.504
4.5 -54.332 14.836
5 -47.616 12.159
6 -37.392 8.599
7 -29.972 6.405
8 -24.336 4.958
9 -19.909 3.953
10 -16.338 3.226
11 -13.397 2.682
12 -10.933 2.264
14 -7.038 1.675
16 -4.101 1.287
18 -1.811 1.018
20 0.022 8.242(-1)
22 1.519 6.795(-1)
23 2.168 6.206(-1)
24 2.762 5.688(-1)
25 3.308 5.229(-1)
26 3.810 4.822(-1)
28 4.703 4.132(-1)
30 5.471 3.573(-1)
35 6.987 2.568(-1)
40 8.097 1.915(-1)
45 8.936 1.467(-1)
50 9.586 1.148(-1)
60 10.509 7.365(-2)
70 11.114 4.929(-2)
80 11.524 3.384(-2)
90 11.808 2.355(-2)
100 12.005 1.642(-2)
120 12.237 7.620(-3)
140 12.336 2.752(-3)
160 12.360 -1.030(-4)
180 12.339 -1.842(-3)
200 12.291 -2.924(-3)
225 12.206 -3.728(-3)
250 12.107 -4.168(-3)
275 12.000 -4.392(-3)
300 11.889 -4.485(-3)
325 11.776 -4.495(-3)
350 11.664 -4.455(-3)
375 11.554 -4.383(-3)
400 11.445 -4.291(-3)
450 11.236 -4.080(-3)
500 11.038 -3.857(-3)
600 10.673 -3.434(-3)
700 10.349 -3.068(-3)
T [d.sup.2]B/d[T.sup.2] [eta] d[eta]/dT
(K) ([cm.sup.3]*[mol.sup.-1]*[K.sup.2]) ([micro]Pa*s) ([micro]Pa*[K.sup.-1])
1.0 -218.592 5.561(-1) 5.280(-1)
1.2 -156.563 6.608(-1) 5.126(-1)
1.4 -115.265 7.593(-1) 4.688(-1)
1.6 -86.881 8.472(-1) 4.086(-1)
1.8 -66.837 9.224(-1) 3.439(-1)
2.0 -52.348 9.850(-1) 2.830(-1)
2.25 -39.409 1.047 2.187(-1)
2.5 -30.305 1.096 1.701(-1)
2.75 -23.736 1.134 1.360(-1)
3.0 -18.897 1.165 1.136(-1)
3.5 -12.431 1.215 9.298(-2)
4.0 -8.637 1.261 9.003(-2)
4.5 -6.209 1.306 9.430(-2)
5 -4.605 1.355 1.006(-1)
6 -2.732 1.461 1.114(-1)
7 -1.752 1.576 1.172(-1)
8 -1.191 1.695 1.191(-1)
9 -8.465(-1) 1.814 1.185(-1)
10 -6.238(-1) 1.931 1.166(-1)
11 -4.732(-1) 2.047 1.141(-1)
12 -3.676(-1) 2.159 1.115(-1)
14 -2.350(-1) 2.377 1.062(-1)
16 -1.593(-1) 2.584 1.014(-1)
18 -1.130(-1) 2.783 9.726(-2)
20 -8.309(-2) 2.974 9.365(-2)
22 -6.286(-2) 3.158 9.049(-2)
23 -5.517(-2) 3.248 8.906(-2)
24 -4.870(-2) 3.336 8.771(-2)
25 -4.318(-2) 3.423 8.643(-2)
26 -3.847(-2) 3.509 8.523(-2)
28 -3.091(-2) 3.677 8.301(-2)
30 -2.520(-2) 3.841 8.101(-2)
35 -1.593(-2) 4.235 7.674(-2)
40 -1.067(-2) 4.610 7.328(-2)
45 -7.476(-3) 4.969 7.038(-2)
50 -5.423(-3) 5.314 6.792(-2)
60 -3.091(-3) 5.973 6.392(-2)
70 -1.905(-3) 6.596 6.078(-2)
80 -1.243(-3) 7.190 5.823(-2)
90 -8.470(-4) 7.762 5.609(-2)
100 -5.961(-4) 8.313 5.427(-2)
120 -3.186(-4) 9.367 5.130(-2)
140 -1.826(-4) 10.369 4.896(-2)
160 -1.096(-4) 11.328 4.705(-2)
180 -6.771(-5) 12.253 4.545(-2)
200 -4.241(-5) 13.148 4.408(-2)
225 -2.354(-5) 14.231 4.262(-2)
250 -1.253(-5) 15.280 4.137(-2)
275 -5.884(-6) 16.301 4.029(-2)
300 -1.787(-6) 17.296 3.933(-2)
325 7.679(-7) 18.268 3.848(-2)
350 2.364(-6) 19.220 3.772(-2)
375 3.350(-6) 20.155 3.703(-2)
400 3.939(-6) 21.073 3.641(-2)
450 4.413(-6) 22.865 3.531(-2)
500 4.428(-6) 24.606 3.437(-2)
600 3.961(-6) 27.962 3.283(-2)
700 3.370(-6) 31.183 3.162(-2)
T [lambda] d[lambda]/dT
(K) (mW*[m.sup.-1]*[K.sup.-1]) (mW*[m.sup.-1]*[K.sup.-2])
1.0 5.756 5.459
1.2 6.839 5.317
1.4 7.864 4.893
1.6 8.786 4.309
1.8 9.585 3.677
2.0 10.259 3.073
2.25 10.942 2.421
2.5 11.481 1.909
2.75 11.909 1.535
3.0 12.258 1.275
3.5 12.816 1.004
4.0 13.294 9.302(-1)
4.5 13.762 9.487(-1)
5 14.249 1.002
6 15.310 1.116
7 16.466 1.187
8 17.671 1.217
9 18.890 1.218
10 20.102 1.204
11 21.295 1.181
12 22.463 1.155
14 24.720 1.102
16 26.876 1.054
18 28.939 1.011
20 30.923 9.735(-1)
22 32.837 9.407(-1)
23 33.770 9.258(-1)
24 34.688 9.117(-1)
25 35.593 8.985(-1)
26 36.486 8.859(-1)
28 38.234 8.628(-1)
30 39.939 8.420(-1)
35 44.033 7.976(-1)
40 47.928 7.615(-1)
45 51.658 7.313(-1)
50 55.249 7.057(-1)
60 62.088 6.640(-1)
70 68.559 6.313(-1)
80 74.735 6.047(-1)
90 80.667 5.825(-1)
100 86.394 5.635(-1)
120 97.341 5.326(-1)
140 107.740 5.082(-1)
160 117.698 4.883(-1)
180 127.293 4.717(-1)
200 136.581 4.574(-1)
225 147.821 4.422(-1)
250 158.711 4.293(-1)
275 169.298 4.180(-1)
300 179.621 4.081(-1)
325 189.710 3.992(-1)
350 199.590 3.913(-1)
375 209.283 3.842(-1)
400 218.804 3.777(-1)
450 237.392 3.662(-1)
500 255.451 3.564(-1)
600 290.258 3.404(-1)
700 323.649 3.278(-1)
T D(101.3 kPa) [[alpha].sub.r]
(K) ([10.sup.-4]*[m.sup.2]*[s.sup.-1])
1.0 1.910(-4) 6.797(-2)
1.2 2.706(-4) 8.532(-2)
1.4 3.629(-4) 9.794(-2)
1.6 4.654(-4) 1.077(-1)
1.8 5.756(-4) 1.150(-1)
2.0 6.913(-4) 1.200(-1)
2.25 8.412(-4) 1.235(-1)
2.5 9.952(-4) 1.247(-1)
2.75 1.152(-3) 1.242(-1)
3.0 1.312(-3) 1.224(-1)
3.5 1.639(-3) 1.171(-1)
4.0 1.980(-3) 1.112(-1)
4.5 2.338(-3) 1.057(-1)
5 2.715(-3) 1.012(-1)
6 3.531(-3) 9.498(-2)
7 4.431(-3) 9.149(-2)
8 5.413(-3) 8.964(-2)
9 6.475(-3) 8.868(-2)
10 7.612(-3) 8.819(-2)
11 8.822(-3) 8.795(-2)
12 1.010(-2) 8.782(-2)
14 1.287(-2) 8.769(-2)
16 1.589(-2) 8.757(-2)
18 1.916(-2) 8.743(-2)
20 2.266(-2) 8.724(-2)
22 2.639(-2) 8.702(-2)
23 2.834(-2) 8.690(-2)
24 3.035(-2) 8.678(-2)
25 3.240(-2) 8.665(-2)
26 3.451(-2) 8.652(-2)
28 3.889(-2) 8.626(-2)
30 4.348(-2) 8.599(-2)
35 5.580(-2) 8.531(-2)
40 6.933(-2) 8.464(-2)
45 8.400(-2) 8.400(-2)
50 9.978(-2) 8.339(-2)
60 1.345(-1) 8.226(-2)
70 1.734(-1) 8.125(-2)
80 2.161(-1) 8.033(-2)
90 2.627(-1) 7.949(-2)
100 3.128(-1) 7.872(-2)
120 4.236(-1) 7.735(-2)
140 5.479(-1) 7.616(-2)
160 6.852(-1) 7.510(-2)
180 8.349(-1) 7.415(-2)
200 9.968(-1) 7.329(-2)
225 1.216 7.231(-2)
250 1.452 7.142(-2)
275 1.707 7.060(-2)
300 1.978 6.985(-2)
325 2.266 6.915(-2)
350 2.571 6.849(-2)
375 2.891 6.787(-2)
400 3.228 6.729(-2)
450 3.948 6.621(-2)
500 4.729 6.524(-2)
600 6.469 6.351(-2)
700 8.439 6.201(-2)
800 10.058 -2.758(-3) 2.842(-6) 34.294 3.063(-2) 355.903 3.176(-1)
900 9.796 -2.496(-3) 2.400(-6) 37.314 2.981(-2) 387.217 3.090(-1)
1000 9.557 -2.275(-3) 2.042(-6) 40.259 2.910(-2) 417.739 3.016(-1)
1200 9.139 -l.923(-3) 1.515(-6) 45.959 2.796(-2) 476.824 2.897(-1)
1400 8.782 -1.658(-3) 1.160(-6) 51.457 2.706(-2) 533.800 2.804(-1)
1600 8.472 -1.452(-3) 9.133(-7) 56.793 2.633(-2) 589.092 2.728(-1)
1800 8.199 -1.288(-3) 7.355(-7) 61.996 2.572(-2) 643.000 2.665(-1)
2000 7.955 -1.155(-3) 6.031(-7) 67.087 2.521(-2) 695.745 2.611(-1)
2500 7.443 -9.112(-4) 3.926(-7) 79.424 2.521(-2) 823.544 2.611(-1)
3000 7.031 -7.471(-4) 2.741(-7) 91.331 2.521(-2) 946.864 2.611(-1)
3500 6.688 -6.296(-4) 2.013(-7) 102.913 2.521(-2) 1066.813 2.611(-1)
4000 6.396 -5.417(-4) 1.535(-7) 114.243 2.521(-2) 1184.125 2.611(-1)
4500 6.143 -4.737(-4) 1.206(-7) 125.369 2.521(-2) 1299.316 2.611(-1)
5000 5.920 -4.196(-4) 9.704(-8) 136.328 2.521(-2) 1412.766 2.611(-1)
6000 5.543 -3.393(-4) 6.638(-8) 157.849 2.521(-2) 1635.532 2.611(-1)
7000 5.234 -2.829(-4) 4.799(-8) 178.965 2.521(-2) 1854.081 2.611(-1)
8000 4.973 -2.412(-4) 3.616(-8) 199.781 2.521(-2) 2069.499 2.611(-1)
9000 4.748 -2.093(-4) 2.813(-8) 220.370 2.521(-2) 2282.550 2.611(-1)
10000 4.552 -1.842(-4) 2.243(-8) 240.786 2.521(-2) 2493.792 2.611(-1)
800 10.633 6.068(-2)
900 13.044 5.948(-2)
1000 15.669 5.839(-2)
1200 21.541 5.645(-2)
1400 28.225 5.476(-2)
1600 35.699 5.325(-2)
1800 43.949 5.189(-2)
2000 52.961 5.064(-2)
2500 78.760 4.790(-2)
3000 109.131 4.556(-2)
3500 143.985 4.349(-2)
4000 183.256 4.164(-2)
4500 226.896 3.996(-2)
5000 274.868 3.841(-2)
6000 383.707 3.561(-2)
7000 509.625 3.314(-2)
8000 652.548 3.090(-2)
9000 812.447 2.885(-2)
10000 989.327 2.695(-2)
Thermophysical properties of an
equimolar binary mixture of [He.sup.3]
-- [He.sup.4] as a function of
temperature, where (-2) is X [10.sup.-2]
T B dB/dT
(K) ([cm.sup.3]*[mol.sup.-1]) ([cm.sup.3]*[mol.sup.-1]*[K.sup.-1])
1.0 -338.460 362.783
1.2 -278.477 248.027
1.4 -236.166 180.498
1.6 -204.675 137.383
1.8 -180.297 108.166
2.0 -160.849 87.440
2.25 -141.430 69.017
2.5 -125.902 55.906
2.75 -113.192 46.238
3.0 -102.590 38.900
3.5 -85.895 28.672
4.0 -73.329 22.034
4.5 -63.518 17.477
5 -55.638 14.211
6 -43.756 9.939
7 -35.212 7.347
8 -28.767 5.654
9 -23.731 4.486
10 -19.687 3.646
11 -16.369 3.021
12 -13.597 2.543
14 -9.233 1.872
16 -5.957 1.433
18 -3.411 1.130
20 -1.380 9.125(-1)
22 0.276 7.508(-1)
23 0.993 6.851(-1)
24 1.649 6.273(-1)
25 2.250 5.763(-1)
T [d.sup.2]B/d[T.sup.2] [eta] d[eta]/dT
(K) ([cm.sup.3]*[mol.sup.-1]*[K.sup.-2]) ([micro]Pa*s) ([micro]Pa*[K.sup.-1]
1.0 -761.653 4.147(-1) 2.503(-1)
1.2 -428.348 4.629(-1) 2.363(-1)
1.4 -264.495 5.102(-1) 2.378(-1)
1.6 -174.857 5.582(-1) 2.416(-1)
1.8 -121.639 6.067(-1) 2.437(-1)
2.0 -88.058 6.555(-1) 2.433(-1)
2.25 -61.471 7.159(-1) 2.397(-1)
2.5 -44.632 7.751(-1) 2.338(-1)
2.75 -33.437 8.327(-1) 2.267(-1)
3.0 -25.720 8.884(-1) 2.191(-1)
3.5 -16.178 9.943(-1) 2.046(-1)
4.0 -10.848 1.093 1.921(-1)
4.5 -7.632 1.187 1.820(-1)
5 -5.579 1.276 1.737(-1)
6 -3.248 1.443 1.612(-I)
7 -2.058 1.599 1.519(-1)
8 -1.387 1.747 1.446(-1)
9 -9.800(-1) 1.889 1.384(-1)
10 -7.184(-1) 2.024 1.331(-1)
11 -5.425(-1) 2.155 1.284(-1)
12 -4.199(-1) 2.281 1.242(-1)
14 -2.667(-1) 2.522 1.171(-1)
16 -1.799(-1) 2.751 1.113(-1)
18 -1.270(-1) 2.968 1.064(-1)
20 -9.303(-2) 3.177 1.022(-1)
22 -7.015(-2) 3.377 9.863(-2)
23 -6.148(-2) 3.475 9.701(-2)
24 -5.418(-2) 3.571 9.549(-2)
25 -4.799(-2) 3.666 9.406(-2)
T [lambda] d[lambda]/dT
(K) (mW*[m.sup.-1]*[K.sup.-1]) (mW*[m.sup.-1]*[K.sup.-2])
1.0 3.802 2.449
1.2 4.260 2.185
1.4 4.690 2.129
1.6 5.117 2.149
1.8 5.550 2.175
2.0 5.986 2.183
2.25 6.529 2.160
2.5 7.063 2.108
2.75 7.582 2.039
3.0 8.082 1.961
3.5 9.024 1.810
4.0 9.896 1.685
4.5 10.714 1.589
5 11.490 1.517
6 12.952 1.415
7 14.328 1.341
8 15.639 1.282
9 16.894 1.230
10 18.101 1.185
11 19.265 |