Russell J. Donnelly
541-346-4226 (Tel)
541-346-5861 (Fax)

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The Observed Properties of Liquid Helium
at the Saturated Vapor Pressure

Chapter 1. Density, Thermal Expansion, and Dielectric Constant

The density of liquid 4He at saturated vapor pressure (SVP) has been measured by a number of authors (see the chronological bibliography at the end of this chapter). We have chosen as our source the recent work of Niemela and Donnelly 40, who measured the dielectric constant of liquid helium and resolved many of the discrepancies which existed in the older literature.

Dielectric constants were converted to density using the Clausius-Mossotti relation

where M = 4.0026 g/mol is the molecular weight of helium and is the molar polarizability. They adopted the value M =0.123296 cm3/mol deduced by Harris-Lowe and Smee30 from their measurements of dielectric constant and Kerr and Taylor's18 measurement of the density.

Although Niemela and Donnelly did not make any measurements below the density minimum, they provided a continuous representation of the density and expansion coefficient to zero temperature using theoretical methods described in detail in their paper obtaining values of the density which are described by the following equation up to 1.344K

where . The coefficients of Eq.(2) are given in Table 1.1. The authors fit the reduced density

to a function of the form:

where t=T-T. Here T =2.1768k , and P=0.1461087 g/cm3 . Because they wished to tabulate the expansion coefficient over the entire range of temperatures, it was desirable that not only the density, but also its first derivative should smoothly join to the calculated values given by Eq.(1.2). This was best achieved by using Eq.(1.2) up to 1.344K and using Eq.(1.4) for data in the ranges 1.334k T Tand T< T < 4.9k. The resulting set of equations and their derivatives provide a continuous representation of the density and expansion coefficient from near 0 K to 4.9 K. The coefficients for Eq.(1.4) are given in Table 1.1. The mean fractional deviation of the density data from the fit is 0.5 x10-6. The density is tabulated in g/cm3. Multiply the entries by 1000 to convert to kg/m3.

Table 1.1. Coefficients for Eqs.(2), (4), (5) and (6).

 i ai X 103 bi X 103 mi X 103 si X 103 1.344K-T T-4.9 K 1.344K-T T-4.9 K 0-1.344 K 0-1.344 k 1 -7.57537 -7.94605 3.79937 -30.3511 -1.26935 -0.117818 2 6.87483 5.07051 1.86557 -10.2326 7.12413 1.64045 3 4.88345 -3.00636 -16.7461 -6.18750 4 0 0.240720 8.75342 13.4293 5 0 -2.45749 -11.3971 6 0 1.53454 2.94176 7 0 -0.308182

In Figure 1.1 we plot Eqs. 1.2 and 1.4 for the density. The following expressions give the deduced thermal expansion coefficient: For 0.15 T 1.344 :

For 1.33 T< T and T < T < 4.9

The coefficients are listed in Table 1.1. The method of obtaining Eq. (1.5) is explained by Niemela and Donnelly. Notice that the average fractional deviation of Eq.(1.5) from the theoretical values on which it is based is 1x10-6, so that its valid temperature range begins 150 mK above absolute zero. Eq.(1.6) was derived from Eq.(1.4). Also be aware that this data was not taken in great temperature detail near the lambda point.

Figure 1.2 shows the expansion coefficient over the entire range of temperatures.

The dielectric constant can be obtained as follows:

This roundabout method of getting the dielectric constant arises because Niemela and Donnelly fitted the density (their main goal) after converting individual dielectric constant measurements to density by means of Eq.(1.1).

Fig. 1.1. The recommended values of the density of liquid 4He as a function of temperature at the saturated vapor pressure.

Figure 1.2. The recommended values of the thermal expansion coefficient of liquid 4He at the saturated vapor pressure as a function of temperature.

Figure 1.3. Detail of the recommended values for the thermal expansion coefficient of liquid 4He near the lambda transition.

Figure 1.4. The recommended values of the thermal expansion coefficient of liquid 4He at low temperatures.

Table 1.2. Recommended values of the dielectric constant, density and thermal expansion coefficient of liquid 4He at saturated vapor pressure.

 T90(K) p(g/cm3) 103(k-1) T90(K) p(g/cm3) 103(k-1) 0.00 1.057255 1.451397E-1 0.000 2.50 1.057135 1.448402E-1 39.4 0.10 1.057255 1.451397E-1 0.001 2.55 1.057017 1.445467E-1 41.8 0.15 1.057255 .451396E-1 10.004 2.60 1.056892 1.442368E-1 44.1 0.20 1.057255 1.451395E-1 0.011 2.65 1.056761 1.439114E-1 46.3 0.25 1.057255 1.451395E-1 0.018 2.70 1.056625 1.435712E-1 48.4 0.30 1.057255 1.451393E-1 0.028 2.75 1.056482 1.432164E-1 50.6 0.35 1.057255 1.451391E-1 0.042 2.80 1.056334 1.428472E-1 52.7 0.40 1.057255 1.451388E-1 0.058 2.85 1.056180 1.424638E-1 54.8 0.45 1.057254 1.451384E-1 0.080 2.90 1.056020 1.420661E-1 57.0 0.50 1.057254 1.451377E-1 0.107 2.95 1.055854 1.416538E-1 59.2 0.55 1.057254 1.451368E-1 0.139 3.00 1.055683 1.412269E-1 61.5 0.60 1.057253 1.451356E-1 0.175 3.05 1.055505 1.407850E-1 63.9 0.65 1.057253 1.451342E-1 0.214 3.10 1.055322 1.403279E-1 66.3 0.70 1.057252 1.451324E-1 0.254 3.15 1.055132 1.398551E-1 68.7 0.75 1.057251 1.451304E-1 0.292 3.20 1.054936 1.393663E-1 71.3 0.80 1.057250 1.451281E-1 0.325 3.25 1.054733 1.388611E-1 74.0 0.85 1.057249 1.451257E-1 0.348 3.30 1.054523 1.383390E-1 76.7 0.90 1.057248 1.451232E-1 0.357 3.35 1.054307 1.377997E-1 79.5 0.95 1.057247 1.451207E-1 0.345 3.40 1.054084 1.372427E-1 82.5 1.00 1.057246 1.451183E-1 0.309 3.45 1.053853 1.366675E-1 85.5 1.05 1.057246 1.451163E-1 0.242 3.50 1.053615 1.360736E-1 88.7 1.10 1.057245 1.451150E-1 0.138 3.55 1.053369 1.354605E-1 92.0 1.15 1.057245 1.451144E-1 -0.008 3.60 1.053115 1.348278E-1 95.3 1.20 1.057245 1.451151E-1 -0.200 3.65 1.052853 1.341748E-1 98.9 1.25 1.057246 1.451173E-1 -0.442 3.70 1.052583 1.335009E-1 103 1.30 1.057248 1.451215E-1 -0.737 3.75 1.052305 1.328054E-1 106 1.35 1.057250 1.451281E-1 -1.08 3.80 1.052017 1.320877E-1 110 1.40 1.057254 1.451373E-1 -1.45 3.85 1.051720 1.313467E-1 115 1.45 1.057259 1.451493E-1 -1.87 3.90 1.051414 1.305817E-1 119 1.50 1.057265 1.451646E-1 -2.36 3.95 1.051097 1.297914E-1 124 1.55 1.057273 1.451837E-1 -2.91 4.00 1.050770 1.289745E-1 129 1.60 1.057282 1.452071E-1 -3.53 4.05 1.050432 1.281296E-1 134 1.65 1.057293 1.452352E-1 -4.23 4.10 1.050082 1.272549E-1 140 1.70 1.057307 1.452686E-1 -4.99 4.15 1.049719 1.263483E-1 146 1.75 1.057323 1.453079E-1 -5.84 4.20 1.049343 1.254075E-1 153 1.80 1.057341 1.453538E-1 -6.79 4.25 1.048952 1.244297E-1 160 1.85 1.057362 1.454070E-1 -7.86 4.30 1.048545 1.234117E-1 168 1.90 1.057387 1.454684E-1 -9.07 4.35 1.048121 1.223498E-1 177 1.95 1.057416 1.455394E-1 -10.5 4.40 1.047677 1.212398E-1 187 2.00 1.057449 1.456217E-1 -12.2 4.45 1.047213 1.200768E-1 198 2.05 1.057488 1.457181E-1 -14.4 4.50 1.046725 1.188552E-1 211 2.10 1.057534 1.458340E-1 -17.7 4.55 1.046211 1.175686E-1 225 2.15 1.057594 1.459840E-1 -24.7 4.60 1.045669 1.162098E-1 241 2.20 1.057643 1.461049E-1 9.64 4.65 1.045095 1.147706E-1 258 2.25 1.057596 1.459877E-1 20.7 4.70 1.044485 1.132419E-1 279 2.30 1.057526 1.458148E-1 26.4 4.75 1.043836 1.116131E-1 301 2.35 1.057443 1.456071E-1 30.5 4.80 1.043143 1.098727E-1 328 2.40 1.057349 1.453727E-1 33.8 4.85 1.042400 1.080076E-1 358 2.45 1.057246 1.451162E-1 36.7 4.90 1.041602 1.060033E-1 392

1.1 Chronological Bibliography for Density and Expansion Coefficient