The 1962 He³ Scale of Temperatures II. Derivation

Abstract

An Experimental Thermodynamic Equation (ETE) temperature scale valid from 0.2 to 2.0 °K has been calculated for He³. The scale is based on new comparisons, (P₃, P₄),of He³ and He⁴ vapor pressures above 0.9 °K; on the 1958 He⁴ temperature scale; and on the best available data for several thermodynamic properties of He³ from 0.2 to 2.0 °K.

The T₆₂ Full-Range Working Equation (FWE) scale,

$$\ln P_3=-2.49174/T+4.80386-0.286001T+0.198608T^2-0.0502237T^3+0.00505486T^4+2.24846\text{\hspace{0.17em}ln\hspace{0.17em}}T$$

fits the ETE scale and the (P₃, T₅₈) data and is therefore valid for use from 0.2 to the critical point, 3.324 °K. The maximum deviation from the ETE scale is 0.4 mdeg and the standard deviation from the input data is 0.25 mdeg. The fit to the seven recalculated isotherms of Keller in the range of the 1962 He³ scale can be determined by converting Keller’s P₄’s to equivalent P₃’s, using direct P₄ to P₃ interpolation equations. The fit of the 1962 He³ scale is as good as the fit of the 1958 He⁴ scale to the same isotherms, the average displacements of the two scales both being 1.5 mdeg below the isotherms. The average standard deviations for (T₆₂−T_iso) and for (T₅₈−T_iso) are 1.2 and 1.0 mdeg, respectively, for these seven isotherms

1. Introduction

At the time He³ was first liquified [39]² a comparison of its vapor pressures, P₃, with those of He⁴, P₄, was given, along with a careful determination of its critical pressure, P_c, which was found to be 875 mm Hg. Subsequently, Abraham, Osborne, and Weinstock (AOW) presented [5] more accurate (P₃, P₄) comparisons, additional critical point data in agreement with ref. [1], and an empirical temperature scale, T_K, based on a modification [5] of the 1948 He⁴ scale [41]. They found it possible to fit all their data, ranging from 0.011 (P_c (1.02 °K) up to P_c, to an equation for log (P/T^5/2)having only three fitted powers of T.

In 1953, Chen and London [42] criticised the form of the T_K equation and attempted to fit the same (P₃,P₄) data of AOW to an equation having a proper theoretical form for extrapolation to 0 °K. Although they fitted coefficients of five powers of T, and omitted data at pressures above 0.4 P_c, the fit of their equation was not satisfactory above 0.1 P_c.

In 1957 Sydoriak and Roberts [9] extended the measured range of pressures and temperatures down to 0.000074 P_c and 0.45 °K using two different paramagnetic salts calibrated against He³ above 1° for the temperature measurement. At that time two newer He⁴ temperature scales were in use, so two He³ scales were calculated: the T_E He³ scale, based on the 1955lE He⁴ scale [43], and the T_L He³ scale, based on the 1955L He⁴ scale [25].

Only by abandoning attempts to retain an analytical expression of proper theoretical form for extrapolation to 0 °K did Sydoriak and Roberts find it possible to get a good fit to their data in combination with those of AOW. To cover the four orders of magnitude range in P₃ only four fitted powers of T were needed.

Following the adoption of the 1958 He⁴ scale [2] the present authors made a proposal [1] to the Seventh International Conference of Low Temperature Physics held at the University of Toronto in 1960 for a new He³ vapor pressure scale to be based on the 1958 He⁴ scale and on various thermodynamic properties of He³. The proposed procedure was similar to that used for the existing T_E and T_L scales except that newly available specific heat data [3, 4] could now be included instead of using a calculated “spin entropy” [44] term. In addition, a different magnetic temperature conversion was being studied for the paramagnetic salt data [9] intended to be used to extend the scale below 1 °K.

The proposal was favorably received by members of the conference, with some reservations as to the feasibility of including vapor pressure data obtained with an iron alum thermometer.

We have subsequently abandoned incorporation of any paramagnetic salt data into the scale derivation except for measurements of specific heat using a cerium magnesium nitrate thermometer [3].

An alternative procedure for establishment of the low-temperature end of the new He³ scale has been thoroughly discussed [10] in a report to the Fourth Symposium on Temperature, its Measurement and Control in Science and Industry, Columbus, Ohio, March 1961. In this method the thermodynamic consistency of the (P,T) data can be examined point by point. The method showed [45] that the AOW data could not be combined with the 1958 He⁴ scale to yield a thermodynamically consistent scale in the range from 1 to 2 °K. A detailed discussion of the inconsistency, which is equivalent to several milli-degrees, is given in a companion paper [7] to this one, hereafter referred to as Part III. Because of this inconsistency new (P₃,P₄)comparisons were undertaken [45] in an improved apparatus designed to mininize errors due to He⁴ film reflux. The results are reported in detail in another companion paper [46] to this one, hereafter referred to as Part I.

Since the measurements reported in Part I provide an explanation for the thermodynamic inconsistency, below T_λ, of a scale based on the AOW data, we have used only (P₃,P₄)data given in Part I in deriving the 1962 He³ scale, reserving the AOW data above T_λ for the purpose of checking the final scale.

The second and third virial coefficients of He³ are needed to establish the low-temperature end of the present scale. For this purpose Keller’s isotherm data [12] has been reanalyzed, using the method of multiple variable least squares [47, 48]. For a further discussion of the method and results we refer to Part III, in which the scale derived below is examined for consistency with isotherm, paramagnetic salt, and latent heat data.

2. The High-Temperature Working Equation Scale

The three steps required to arrive at a full range equation are those discussed in detail for a Method I derivation in ref. [10].

The first step is to derive a working equation scale by which one can interpolate between the(P₃,P₄)data points of Part I converted to a(P₃,T₅₈) table of data. The primary use of this scale is for making small corrections to experimental quantities of second order importance which enter into the thermodynamic treatment of the (P₃,T₅₈) data. It will suffice to state that we used a working equation scale which fits the (P₃,T₅₈) data with a variance of 0.25 mdeg and a maximum deviation of 0.6 mdeg. (In other He³ scale derivations a high-temperature working scale was considerably more important, since it was also used in the determination of paramagnetic salt calibration equations [9] and to assign temperatures to He³ latent heat data [10].)

3. The Experimental Thermodynamic Scale

The second step is to derive an analytical expression for the thermodynamic vapor pressure equation (see eqs (2) and (4) of ref. [10]) which can be written as follows, putting on the left those terms which can be evaluated from existing thermodynamic data:

$$\ln P_3+f(T)+f\left(C_{\text{sat}},T\right)-ϵ+f\left(V_L,P_3,T\right)=-\frac{a}{RT}-\frac{b}{R}.$$ (1)

The values of P₃ and T₅₈ used are given in table 1, which is a portion of the data of Part I, excluding the lowest three data points, at P₄<40_μ, because the calculated He⁴ film reflux pressure drop was excessive.

Table 1. He³ vapor pressure data^a used in deriving the 1962He³ scale.

P₃ is the measured He³ vapor pressure in mm Hg at 0 °C and standard .gravity, δP₃ the estimated maximum error in P₃, and δT₆₂ the temperature error equivalent to δP₃. Temperature T₅₈ is obtained from the measured He⁴ vapor pressure. P₄, and the 1958 He⁴ temperature scale. δT₅₈ is the estimated maximum error in T₅₈ equivalent to the maximum estimated error in our measurement of P₄, excluding possible errors in the 1958 He⁴ scale itself. The last column showing the deviations of the 1962 He³ scale from the input data, exhibits a random scatter above 2 °K. At lower temperatures there appears to be some small regularity in the misfit.

In fitting the coefficients of the equation for the 1962 He³ scale, the weight given to each data point was a function of both δP₃ and δT₅₈.^bAs expected, for almost all of the data points (T₆₂−T₅₈)<(δT₆₂+δT₅₈), since the right side of the inequality is the estimated maximum error. On average (δ T₆₂+δT₅₈)was 1.7 times (T₆₂−T₅₈)

The last entry in the table is the measured value of the critical point.^a In addition to the pressure measurement error listed, there is an uncertainty of ±1.5 mm Hg in the location of the critical point.

P3	± δP3	± δT62	T58	± δT58	T62−T58
mm Hg	mm Hg	mdeg	deg	mdeg	mdeg
5.254	0.003	0.11	0.89848	0.78	–0.2
6.671	.003	.09	.94311	.56	–.2
8.692	.004	.10	.99612	.39	.3
10.176	.004	.09	1.03014	.48	.3
11.041	.004	.08	1.04850	.41	.2
13.487	.005	.09	1.09572	.42	.0
20. 001	.008	.11	1.19772	.27	.0
28.395	.011	.12	1.30024	.09	.1
38.512	.014	.13	1.39994	.68	.0
50.234	.018	.13	1.49553	.58	.1
66. 069	.027	.17	1.60410	.53	–.3
81. 978	.060	.32	1.69703	.48	–.6
82.515	.028	.15	1.69980	.48	–.5
102.75	. 060	.28	1.80113	.18	–.0
103.65	.060	.28	1.80548	.18	–.2
103.81	.060	.28	1.80627	.05	–.2
124.23	.070	.29	1.89616	.06	–.5
125.86	.060	.25	1.90300	.05	–.6
151.88	.070	.26	2.00293	.11	–.1
153.49	.035	.13	2.00845	.11	.2
181.58	.048	.16	2.10432	.11	.2
196.61	.045	.14	2.15174	.10	.2
203.29	.055	.17	2.17207	.10	.1
205.83	.055	.16	2.17945	.10	.4
214.47	.080	.23	2.20521	.09	.0
227.21	.059	.17	2.24144	.09	.1
248.77	.080	.21	2.30010	.08	–.1
288.86	.090	.22	2.40069	.07	–.1
333.14	.090	.20	2.50157	.06	.0
381.81	.110	.22	2.60250	.06	.3
433.39	.100	.18	2.70121	.05	.1
491.85	.110	.19	2.80389	.05	.0
555.00	.110	.17	2.90588	.04	.0
617.09	.130	.19	2.99913	.04	–.3
657.11	.109	.16	3.05549	.08	–.1
657.78	.110	.16	3.05632	.07	.0
658.95	.110	.16	3.05792	.07	.0
721.41	.114	.15	3.14129	.07	.1
729.70	.114	.15	3.15203	.07	.1
738.63	.116	.15	3.16356	.07	.0
804.98	.120	.14	3.24556	.06	–.1
806.35	.120	.14	3.24724	.06	–.1
853.73	.140	.16	3.30238	.06	.3
873.00	.300	.34	3.3240	.47	.6

^aRef. [46] (Paper I of this series).

^bRef. [7] (Paper III of this series).

The second term is

$$f(T)=-i-(5/2)\ln T$$ (2)

where i is the chemical constant, i = 5.31733.

For the remaining terms on the left we write the thermodynamic function and its empirical equivalent as follows. The calculable part of the specific heat term is

$$f\left(C_{\text{sat}},T\right)=\frac{1}{RT}{\int }_{T_m}^{T}dT\text{'}{\int }_{T_m}^{T\text{'}}\left(C_{\text{sat}}/T\prime \prime \right)dT\text{\hspace{0.17em}}\prime \prime ,$$ (3a)

where C_sat is the specific heat of saturated liquid He³. Smoothed values of the C_sat data of Brewer, Sreedhar, Kramers, and Daunt [3] and data points of Wein-stock, Abraham, and Osborne [4] are shown in table 2, and are used to obtain an empirical equation for C_sat,

$$C_{\text{ sat, }x}/R=0.25154+0.47485T-0.54064T^2+0.406356T^3-0.082729T^4$$ (3b)

for 0.2<T<2°. In this and the following equations and tables the subscript, x, is used to designate either an empirical function of T fitted to a thermodynamic quantity or an experimental interpolation equation.

Table 2. Values at selected temperatures, T, of the specific heat function^a, f(C_sat,T) appearing in the thermodynamic equation for In P₃.

The function is evaluated by means of an explicit equation^b, C_sat,_x, fitted to smoothed C_sat data of Brewer, Sreedhar, Kramers, and Daunt c below 1 °K and plotted data points of Weinstock, Abraham, and Osborne^d above 1 °K. The misfit of the equation, ΔC=C_sat,_x–C_sat,_obs, is seen to have a random scatter.

T	Csat,x	ΔCsat	fx(Csat,T)
deg	cal mole–1 deg–1	mcal mole–1 deg–1
0.2	0.652b	8c	0.000
.3	.706	−8	.0243
.4	.752	−5	.0665
.5	.793	0	.1109
.6	.831	8	.1539
.7	.870	3	. 1946
.8	.911		.2329
.9	.957		.2691
1. 0	1.009		.3034
1.1	1.069	−4, −9d,e	.3362
1.2	1.136	+1, −9, +4	.3675
1.3	1.211	+6,+4	.3977
1.4	1.294	+5,–1	.4269
1.5	1.385	−6	.4553
1.6	1.482	–13,+14	.4831
1.7	1.585	+2	.5104
1.8	1. 692	+16, −4	.5372
1.9	1. 800	−20, +2	.5637
2.0	1.908	+5	.5899

^aEq (3a)

^bEq (3b)

^cRef. [3]

^dRef. [4]. Both Tand C_sat were adjusted to the T₅₈ scale; e.g., C_sat was multiplied by the value of d T₅₈/d Tk at each data point.

^eWhere multiple entries appear they refer to data points nearest to the indicated temperatures.

The lower limit of the fit, 0.2 °K, was arbitrarily selected within the range of measured values of C_sat. By this choice the lower limit of reliability of the 1962 He³ scale is chosen to be 0.2 °K. By inserting eq (3b) , the exact theoretical expression (3a) is converted to the experimental interpolation equation for T_m=1.0° and 0.2<T<2°:

$$f_x\left(C_{\text{sat}},T\right)=\frac{0.39332}{T}-0.57013+0.237426T-0.090344T^2+0.033863T^3-0.0041364T^4+0.25154\ln T.$$ (3c)

For convenience we have taken T_m to be 1.0 °K, although any other temperature in the range of eq (3b) could as well have been selected. The effect of the specific heat term on the ETE scale below 1 °K is easier to identify with this choice of T_m.

The vapor volume term is

$$ϵ=\ln \left(P_3{V}_G/RT\right)-\frac{2B}{V_G}-\frac{3C}{2V_G^2}.$$ (4a)

For the vapor volume, V_G, we used the inverse volume expansion form of the equation of state

$$P{V}_G=RT\left(1+\frac{B}{V_G}+\frac{C}{V_G^2}\right).$$ (4b)

For the second and third virial coefficients of He³ we used the equations found in Part III in the multiple parameter least squares analysis of Keller’s isotherm data [12],

$$B=4.942-\frac{270.986}{T}{\text{cm}}^3/\text{mole}$$ (4c)

and

$$C=2866/\sqrt{T}{\text{cm}}^6/{\text{mole}}^2.$$ (4d)

Since a high-speed calculator was available, it was possible to use the implicit form (4a), with P₃ being taken from the (P₃,T₅₈) data of table 1. Table 3 shows values of the term and of its component parts. The liquid volume term is

$$f\left(V_L,P_3,T\right)=-\frac{1}{RT}{\int }_0^{P_3}{V}_{L}d{P}_3$$ (5a)

or

$$f_x\left(V_L,P_3,T\right)=-0.005554T^3-0.000163T^4,T<2°\text{K.}$$ (5b)

The coefficients of (5b) were evaluated by fitting the next to last column of table 4, which is calculated from the smoothed [49] V_L data of Taylor and Kerr [50] and of Sherman and Edeskuty [51] and from the working equation (for ΔP₃).

Table 3. The vapor volume term^a, ϵ, in the equation for In P₃, as evaluated by an iterative procedure, using a highspeed digital computer.

For their general usefulness the table also shows solutions of the cubic equation^b for the vapor volume V_g, and values of the second^c and third^d vir-coefficients, B and C, based on reanalysis^e of the isotherms of Keller^f.

T	VG	B	C	ϵ
deg	cm3/mole	cm3/mole	(cm3/mole)2
0.2	1.03×109	–1350	6409	0.0000
.3	9.97×106	–898.3	5233	.0001
.4	8.87×105	–672. 5	4532	.0008
.5	1.95×105	–537. 0	4053	. 0027
.6	6.83×104	–446. 7	3700	.0065
.7	3.123×104	–382.2	3426	.0122
.8	1.692×104	–333.8	3204	.0195
.9	1.028×104	–296. 2	3021	.0284
1.0	6780	–266. 0	2866	.0384
1.1	4747	–241. 4	2732	.0495
1.2	3478	–220. 9	2616	.0613
1.3	2640	–203. 5	2513	.0738
1.4	2061	–188.6	2422	.0868
1.5	1646	–175.7	2340	. 1003
1.6	1339	–164.4	2266	. 1141
1.7	1106	–154.5	2198	.1282
1.8	925.6	–145.6	2136	. 1427
1.9	782.8	–137.7	2079	.1574
2.0	667.9	–130.6	2027	.1723

^aEq (4a)

^bEq (4b)

^cEq (4c)

^dEq (4d)

^eRef. [7]

^fRef. [12]

Table 4. The liquid volume term, f (V_L, P₃, T), in the equation for In P₃ is calculated by numerical integration at selected temperatures, T, using smoothed values ^a, V_L, of liquid volume data of Kerr and Taylor^b and Sherman and Edeskuty. ^c.

For P₃ the high-temperature working equation is used. Column 3 shows values^d calculated from a power series in Tⁿ fitted to column 2. The last two columns show the mismatch and its equivalent in millidegrees.

T	f(VL,P3,T)	fx(VL,P3,T)	fx−f	ΔT
deg				mdeg
0.2	0.0000	0.0000	0.0000	0.0
.4	.0000	.0004	.0004	.0
.6	.0005	.0012	.0007	.1
.8	.0021	.0029	.0008	.1
1.0	.0052	.0057	.0005	.1
1.2	.0099	.0099	–.0000	–.0
1.4	.0163	.0158	–.0005	–.2
1.6	.0244	.0238	–.0006	–.2
1.8	.0342	.0341	–.0001	–.0
2.0	.0459	.0470	.0011	.6

^aRef. (49)

^bRef. (50)

^cRef. (51)

^dEq. (5b)

On the right side of eq (1) we have those terms of the thermodynamic equation which cannot be adequately calculated from existing data on He³:

$$a=L_{\text{o}}-{\int }_0^{T_m}{C}_{\text{sat}}dT$$ (6)

and

$$b=S_L\left(T_m\right)$$ (7)

where L_o is the value of the latent heat of vaporization at absolute zero, and S_L(T_m) is the liquid entropy at T_m.

Using the ordinary method of least squares analysis and weighting each data point equally in this step of the derivation of the 1962 He³ scale, we find

$$a/R=2.09842\pm 0.00070\text{ for }{T}_m=1.0°$$ (8a)

and

$$b/R=1.08360\pm 0.00046\text{ for }{T}_m=1.0°.$$ (8b)

By combining all the above functions we obtain an experimental thermodynamic equation scale (ETE) which is valid from 0.2 to 2.0 °K.

Because of the complexity of the ETE equation, and the fact that it is implicit in the pressure, iterative solutions were obtained with the aid of an electronic digital computer. A table in steps of 1 mdeg was prepared for comparison with the work-ins; equation scale.

4. The 1962 He³ Full-Range Working Equation Scale

To obtain an expression valid over the full range from 0.2 °K to the critical point we now fit selected portions of eq (1) to a power series in Tⁿ, using as input pressures all of the (P₃,T₅₈) data of table 1. Using the method of multiple variable least squares analysis [47, 48] we fit

$$\ln P_3+f(T)+f_x\left(C_{\text{sat}},T\right)+\frac{a}{RT}+\frac{b}{R}=\sum _{n=1}^4{d}_n{T}^n.$$ (9a)

Note that in this fitting the vapor and liquid volume terms are expected to be fitted by the power series. To be acceptable it will therefore be necessary to demonstrate not only that the scale fits the input data but also that the scale of (9a) agrees with the ETE scale below the range of the input data, i.e., below P₃ = 5.254 mm.

The solution of the analysis, combining coefficients of identical powers of T, is the full range working equation scale, FWE,

$$\ln P_3=-2.49174/T+4.80386-0.286001T+0.198608T^2-0.0502237T^3+0.00505486T^4+2.24846\ln T0.2<T<3.324°\text{K}.$$ (9b)

The upper limit is the critical point temperature consistent with the redetermination of the critical pressure found in Part I to be at 873.0 mm Hg.

As shown in figure 1, a comparison of the two tables we generate from the ETE scale and the FWE scale shows excellent agreement: nowhere below 2 °K do the scales differ by more than 0.4 mdeg. Equation (9b) is therefore in effect an experimental thermodynamic scale from 0.2 to 2.0 °K and an empirical scale above 2 °K.

Figure 1. The solid line is the deviation of the 1962 He³ scale (i.e., the Full-range Working Equation, eq 9b) from the Experimental Thermodynamic Equation (ETE) scale in the range of validity of the ETE scale.

Plotted points are deviations of the 1962 He³ scale from the (P₃,P₄) input data: T₆₂(P₃)— T₅₈(P₄). The + represents data at the critical point.

A comparison of the fit of the 1962 He³ scale to the input (P₃,T₅₈) data is given in table 1 and figure 1. The standard deviation of the data from the scale is 0.25 mdeg.

The He⁴ lambda point occurs at P_4,λ=37.80 mm Hg (corrected to 0 °C and standard gravity). On the 1958 He⁴ scale this corresponds to 2.1720 °K. In Part I a direct interpolation procedure is described by which the value of P₃ which corresponds to P_4,λis 203.25 mm Hg. Hence T_62,λ=2.1721 °K, for P₃=203.25 in good agreement with the value of this fixed point on the 1958 He⁴ scale of temperatures.

The most fundamental test which can be given to the 1962 He³ scale is its fit to He³ and He⁴ isotherm data, of which those of Keller [11, 12] are the most complete and accurate. Although Keller used a He⁴ thermometer in most of his isotherms, his observed P₄’s can be related to P₃’s by the direct interpolation equations described in Part I. In Part III Keller’s isotherms have been reanalyzed by the method of multiple variable least squares. Results of the comparison with the 1962 He³ and with the 1958 He⁴ scales are shown in figure 2. We note that the weighted average of T_{N, 62} is 1.52 mdeg below T_iso, whereas for the same isotherms T_{N, 58} averages 1.50 mdeg below T_iso. Since for these isotherms T_n,62—T_iso and T_N,62—T_iso have average standard deviations of 1.2 mdeg and 1.0 mdeg, respectively, we conclude that not much would be gained by basing a He³ scale more directly on the isotherm data. In other words, the 1958 He⁴ scale, which is itself fundamentally based on these isotherm data, has evidently been an adequate interpolation parameter for fitting the He³ data to these isotherms.

Figure 2. Deviations of temperature scales from Keller’s (ref. [11, 12]) isotherm temperatures.

T_iso, as reanalyzed in Part III (ref. [7]). ● (T₆₂—T_iso) for He⁴ isotherms; ■ (T₆₂—T_iso) for He³ isotherms; ◯, ☐ (T₅₈—T_iso) for He⁴ and He³ isotherms, respectively. To get T₆₂, Keller’s He⁴ vapor pressure thermometer readings are converted to equivalent P₃’s by means of direct P₄−to−P₃ interpolation equations derived in Part I (ref. [46]). Lengths of bars for the 1958 He⁴ scale deviations are equal to the standard deviation for T_iso as calculated in the analysis of the isotherm data. For 1962 He³ scale deviations we add on the standard deviation of the conversion from P₄ to equivalent P₃. The T₆₂ and T₅₈ bars terminate in solid and open triangles respectively.

Table 5 shows vapor pressures calculated from eq (9b) in steps of 10 mdeg. In table 6 we show dP/dT and T₆₂—T_x in 0.1 deg steps, where T₆₂ corresponds to eq (9b) and T_x to the various He³ scales in use in the past. We also have included on the far right the two He⁴ scale differences (T_L55—T₅₈) and (T_55E—T₅₈), as given in the NBS Monograph 10 “The 1958 He⁴ Scale of Temperatures” [2]. In a few recent publications an attempt has been made to “correct” the T_L and T_E He³ scales by adding these He⁴ scale “correction” terms. To convert these temperatures to the 1962 He³ scale it is necessary to apply the sum of columns 3 and 6 (or 4 and 7) to the “corrected T_L” or “corrected T_E” scales.

Table 5. He³ vapor pressures on the 1962 He³ scale at 0 °C and standard gravity, 980.665 cm/sec².

The units of pressure are microns (10^–3 mm) of mercury below 1 °K and millimeters of mercury at higher temperatures

T	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0. 20	0.012	0. 024	0.046	0.084	0.144	0.239	0. 382	0. 592	0. 891	1.308
0.30	1.877	2. 636	3. 633	4. 921	6. 561	8. 619	11.173	14. 304	18.105	22. 673
0. 40	28.11	34. 54	42.08	50. 86	61.01	72.68	86. 02	101.17	118. 31	137. 61
0.50	159.2	183.3	210.1	239.8	272.5	308.5	347.9	391.1	438.0	489.1
0.60	544.4	604.3	668.9	738.4	813.0	893.0	978. 7	1070.1	1167. 6	1271.4
0.70	1381	1498	1622	1753	1892	2038	2192	2355	2525	2704
0.80	2892	3089	3295	3511	3736	3971	4216	4472	4739	5016
0. 90	5304	5603	5914	6237	6572	6918	7277	7649	8034	8431
1.00	8.842	9. 267	9.704	10.156	10. 622	11.102	11. 597	12.106	12. 631	13.170
1.10	13. 725	14. 295	14. 881	15. 484	16.102	16. 737	17. 388	18.056	18. 741	19.443
1.20	20.163	20. 900	21. 655	22. 428	23.220	24.029	24. 857	25. 704	26. 571	27. 456
1.30	28.360	29. 285	30. 229	31.193	32.177	33.181	34. 206	35.252	36. 319	37. 407
1.40	38. 516	39. 646	40.799	41.973	43.169	44. 388	45. 629	46. 893	48.179	49. 489
1.50	50.822	52.178	53. 558	54. 961	56.389	57.840	59. 316	60.817	62. 342	63.892
1.60	65.467	67.068	68. 694	70.345	72.022	73. 726	75. 455	77. 211	78. 993	80.802
1.70	82. 638	84. 501	86.391	88. 309	90.254	92. 228	94. 229	96. 258	98.315	100.402
1.80	102. 516	104. 660	106. 833	109.035	111.266	113. 527	115. 818	118.138	120. 489	122.870
1. 90	125.282	127. 724	130.197	132. 701	135.236	137. 803	140. 401	143.031	145. 692	148.386
2.00	151.112	153. 870	156. 661	159. 485	162.342	165. 232	168.155	171.112	174.102	177.126
2.10	180.184	183. 276	186. 403	189.564	192. 760	195. 990	199. 256	202. 557	205. 894	209.266
2. 20	212. 673	216.117	219.597	223.113	226.665	230. 255	233. 881	237. 544	241. 244	244.982
2. 30	248. 757	252. 570	256. 420	260. 309	264. 236	268. 202	272. 206	276. 249	280. 331	284. 452
2. 40	288. 613	292. 813	297.053	301.333	305. 653	310.013	314. 414	318.855	323.337	327. 861
2.50	332.425	337.031	341. 679	346. 368	351.100	355. 874	360. 690	365. 549	370. 450	375.395
2. 60	380.383	385. 414	390. 489	395. 608	400. 771	405. 978	411. 230	416. 526	421. 868	427. 254
2.70	432.686	438.164	443. 687	449. 256	454. 872	460. 534	466. 242	471. 998	477. 801	483. 651
2. 80	489. 549	495. 495	501.488	507. 531	513. 622	519. 762	525. 951	532.189	538. 477	544.815
2. 90	551. 203	557. 642	564.131	570.672	577.264	583. 907	590. 602	597.349	604.149	611.002
3.00	617.907	624. 866	631. 879	638. 945	646.066	653. 241	660. 472	667. 757	675.098	682. 496
3.10	689. 949	697. 459	705.026	712. 650	720.332	728.072	735. 871	743.728	751. 644	759. 620
3.20 3. 30	767.656 851.406	775. 753 860.130	783. 910 868. 918	792. 128 877. 773	800.408	808. 750	817.155	825. 622	834.153	842. 747

Table 6. The temperature derivative dP₃/dT₆₂ and deviations of various He³ temperature scales from the present 1962 He³ scale.

Columns 3,4, and 5 give differences of the T_E, T_L and T_K He³ scales respectively from the 1962 He⁴ scale in the form (T₆₂–T_x) expressed in millidegrees. In Columns 6 and 7 are reproduced the deviations of the 1955 He⁴ scales from the accepted 1958 He⁴ scale as these deviations have been utilized by some investigators in an attempt to “update” the T_E and T_L He³ scales.

T	dP3/dT62	T62−TLa	T62−TEa	T62−Tkb	(TL55−T58)c	(T55E−T58)c
°K	mm Hg/°K	mdeg	mdeg	mdeg	mdeg	mdeg
0.2	0.001
.3	.066	−5.0	−6.3
.4	.592	−5.8	−7.4
.5	2.283	−6.2	−8.1
.6	5.756	−6.2	−8.3
.7	11.362	−5.8	−8.2		–1.1	1.0
.8	19.234	−5.2	−7.8		–1.2	1.1
.9	29.379	−4.4	−7.3		–1.3	1.2
1.0	41.745	−3.6	−6.7	−4.0	–1.5	1.3
1.1	56.26	−2.9	−6.1	−3.7	–1.6	1.4
1.2	72.84	−2.1	−5.4	−3.2	–1.7	1.5
1.3	91.44	–1.5	−4.9	−2.5	–1.8	1.5
1.4	111.99	−0.9	−4.4	–1.6	–1.9	1.6
1.5	134.45	–.4	−3.9	−0.7	−2.0	1.6
1.6	158.78	.0	−3.4	0.2	−2.1	1.5
1.7	184.94	.3	−3.1	1.1	−2.1	1.4
1.8	212.92	.5	−2.8	2.0	−2.2	1.1
1.9	242.68	.7	−2.4	2.7	−2.2	0.8
2.0	274.22	.9	−2.0	3.4	−2.2	0.6
2.1	307.51	1.1	–1.6	3.9	−2.2	0.6
2.2	342.57	1.4	–1.1	4.3	−2.2	0.8
2.3	379.40	1.6	−0.6	4.5	−2.1	0.5
2.4	418.03	1.9	0.0	4.5	−2.0	0.0
2.5	458.53	2.2	0.6	4.4	–1.9	−0.4
2.6	500.96	2.5	1.3	4.1	–1.8	−0.7
2.7	545.46	2.7	1.9	3.5	–1 5	−0.9
2.8	592.18	2.9	2.4	2.7	–1.3	–1.1
2.9	641.34	2.8	2.7	1.5	−0.9	–1.1
3.0	693.22	2.5	2.9	0.0	−0.6	–1.1
3.1	748.16	1.9	2.6	−2.0	−0.2	–1.1
3.2	806.61	0.6	1.7	−4.7	0.2	−0.9
3.3	869.11	–1.3	0.1	−8.0	0.7	−0.7

^aRef. [9]

^bRefs. [5, 41]

^cRef. [2]

Detailed (P,T) and (T,P) tables have been published [8] by the Los Alamos Scientific Laboratory and in Part IV of this series.

In order to make the advantages of He³ as a vapor pressure thermometer more widely accessible, specially purified He³ is being made available for purchase³ for thermometry through the United States Atomic Energy Commission isotopes program.

If it were possible the authors would acknowledge in detail the aid given by numerous cryogenists in the course of the development and evaluation of the 1962 He³ scale of temperatures and the discussions concerning the scale by the Advisory Committee on Thermometry of the International Committee on Weights and Measures. We are especially indebted to F. G. Brickwedde, B. M. Abraham, H. van Dijk, M. Durieux, R. I. Joseph, W. E. Keller, D. W. Osborne, and J. H. van Vleck. Since our correspondence ran to several dozen letters it would be impractical to specify all of the participants and to elaborate on their various contributions.