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The Logarithmic Anomaly in the Pressure

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The Logarithmic Anomaly in the Pressure The Logarithmic Anomaly in the Pressure Coefficient temperatureswere referred to theupper lambda of Helium Close to the Lambda Line" point in orderto correct for small changes in the thermometer or the measuring circuit. Points on the lambda curve were found by warm- ing the sample slowly at constant volume. Because H. A. Kierstead of the large change in thermal conductivity at the lambda transition, the heating curve showed a sharp Argonne National Laboratory, Argonne, 111. break,which could be located to within amicro- degree.The sample was held at this temperature until equilibriumwas established before reading Introduction themanometer. Then the density was changed slightly by raisingthe oilin the left limb of the Severalinvestigators have reported logarithmic manometer,and the new lambda pointwas found. anomalies in the heat capacity [I], thermal expan- The break in the heating curve was especially sharp sion[2-6], and pressure coefficient [7-81of liquid with solid present in the cell, because the heat of helium in theneighborhood of the lambda transi- meltingincreased the apparent heat capacity. tion.Unfortunately, all of theseexperiments ex- Thus the upper lambda point could be determined cept those of Lounasmaa 171 and of Lounasmaa and directly, rather than as the intersection of two inde- Inconnection with theseexperiments we have Kaunisto [8] were done on the vapor pressure curve, pendently measured curves. measured the change of pressure and density with where the anomalous behavior is confined to a very The pressure coefficient p,.=(dP/aT),.was deter- temperaturealong the lambda line. These experi- narrowtemperature interval. The experiments of minedby keeping the oil level -in the leftlimb of ments, which are reported in detail elsewhere [lo]. Lounasmaaand Kaunisto [8] atvarious pressures the manometer constant and measuring the height can be summarized by the equations alongthe lambda curve show that the anomalous of oil in the right limb at successively lower tempera- behavior is spreadout over a larger temperature tures.After the lambda point was passed, as intervalathigher pressures. Consequently, we judged by thethermal response, the temperature have measured the pressure coefficient, &=(aP/aT),., was raised slowly in order to determine the lambda of liquid helium along the isochore which crosses point exactly. thelambda line at the melting curve. This inter- section is called the upper lambda point (TA,,PA,). Results Experimental Procedure Small changes in the sample density could be made The pressure coefficient PI.= (dP/dT),. was meas- by raising the oil in the left limb of the manometer, ured along an isochore which crossed the lambda The apparatus usedin these experiments is essen- forcinggas to condense into G. A change of 0.01 line very close to Ti,. Results are shown in figure 2, tially thesame as that used by Lounasmaa [7]. mm in the oil level caused a change of 2 x 10-9 plottedagainst log,o(T-Tk). The equation of where TA is the lambda temperature for the pressure Therefore, it will be described only briefly here. It g/cm3 in thedensity. theline is P and density p, and TA'is the temperature of the is shown schematically in figure 1. Temperatures were measured with a germanium upperlambda point, 1.7633 OK. Heliumgas was purified in atrap (not shown) resistance thermometer in apotentiometer circuit p,=9.39+7.6910glo(T-T~) atm/OK, (1) immersed in liquid helium, and was condensed into .usingdoublea potentiometer. At theupper Discussion the piezometer G through the low-temperature valve lambda pointthe thermometer resistance was where Ti is the lambda transition temperature for A, which was kept closed during measurements. G 1483 R, andits sensitivity was 636 microdegrees theisochore. This graph contains points taken on Thelogarithmic dependence of PI. on T-TA wasisolated from the liquid helium bath by the perohm. With 50 pA of measuringcurrent, the two differentcooling runs. The lambda point for is similar to thatfound by Lounasmaa [7] and by vacuumcase B. Itstemperature was controlled resistancecould be measured witha precision of one run was 15 pdeg below Ti', and for the other it Lounasmaaand Kaunisto [8] atlower densities, by the heater F and by pumping on liquid helium 0.002 R. Themeasuring current heated the ther- was 5 pdeg above. These small differences are not and is consistentwith their observation that the in E. mometer above the cell temperature; but thiseffect regardedas significant, but the observed lambda two constants in eq (1) both increase with increas- Changes in thepressure on the sample were was reproducible, and the same measuring current pointwas used in eachrun in plotting T-TA. ingdensity. The measurements of Lounasmaa measured by the oil manometer K and read with a wasused in calibratingthe thermometer. It was Since this isochore meets the melting curve at the at 13 atm pressure (T~=2.023 OK) are of particular cathetometer to 0.01 mm,which corresponded to calibrated against the vapor pressure of helium in E lambda point, the measurements in figure 2 refer to interestsince he found that p,. fittedan equation abouta millionth of anatmosphere. The absolute on the T58 scale [9], using a tube separate from the 1 HeI. One pointwas measured in supercooled of the same form as (1) to within 2 x 1O-5 OK of the pressure was read to 0.01 atm on the Bourdon gage pumpingtube. The change in thethermometer He II, where wasfound to be -27.6 at T- TA lambda point for both He I and He II, the constants P, which was calibrated against a dead weight tester. betar resistance at the upper lambda point after warming =- 1.08 X lO-s. This point is not plotted in figure beingdifferent for the two cases. In thesame toroom, temperature and recooling corresponded paperLounasmaa presents measurements of the *Based on work performed under the auspices of the U.S. Atomic Energy '2, but it is approximately the value to be expected Commission, to atemperature change of lessthan lO-4 OK. All 'for T- TA = 1.08 X lO-s. compressibility KT= p"(dp/dP),r along an isotherm 92 93 passingthrough the same lambda point. He References 151 E. Maxwell and C. E. Chase, Physica 24, 5139 (19%). 161 E. C. Kerranti R. D. Taylor, Ann. Phys. (N.Y.) 26. 902 finds no evidencefor anything but a slow linear [I] W. hl. Fairbank, M. I. Buckingham, and C. F. Kellers, 96.5). Proc. 5th Int. Conf. Low Temp. Phys., Madison, Wis- 171 0.V. Lounasmaa, Phys. Rev. 130, 847 (1963). variation of KT with P- PA to within 1O-3 atm from consin (1957), p. 50: C. F. Kellers, Thesis, Duke Uni- 181 0.\’. Lounasmaa and L. Kaunisto, Ann. Acad. Sci. Fenn. the lambda line (corresponding to 2 x 10-5 OK for versity (1960). AVI. No. 59 ( 1960): Bull. Am. Phys. Soc. 5, 290 I 1960). T-Ti) and a smalldiscontinuity at the lambda [2] K. K. Atkins and M. H. Edwards, Phys. Rev. 97, 1429 (91 H. van Dijk. M. Durieux, J. R. Clement,and J. K. Logan, ( 1955). National Bureau of Standards Monograph 10, 1960. point. [3] M. H. Edwards, Can. J. Phys. 36, 884 (1958). 1101 H. A. Kierstead, 138, A1504 (1965), 141 E, Maxwell, C. E. Chase, and W. E. Millet, Proc. 5th lnt. (111 M. J. Buckingham and W. M. Fairbank, Progress in Low Sincean isochore cannot cross the lambda line Conf. Low Temp. Phys., Madison,Wisconsin (19571, TemperaturePhysics C. I. Gorter, ed.) Vol. 111, ch. III with a slope greater in magnitude than that of the p. 53. (North Holland Publishing Company, Amsterdam, 1961). lambda line, (dPld7‘)~is a lower bound for PI.(both are negative); so eq (1) must break down for some smallvalue of T-TA. Onthe other hand, there is no suchrestriction on KT. Infact, Buckingham The Nature of the Cooperative Transition and Fairbank [ll] have shown that if Cp becomes infinite at the lambda line, ~7.mustalso. The exper- iments seem to suggest that Pr becomes infinite and M. J. Buckingham KT does not. This situation can be examined more closely with the help of the equation University of Western Australia, Nedlands, Western Australia 1. Introduction action solution, each cell is properly described as a piece of‘ one or other of the phases, but in the ab- A thermodynamicsingularity arises in statis- sence of the interaction these are mixed at random. We plan to measure KT near the upper lambdapoint tical mechanics only in the limit N+ m, where N Thequalitative effect of the interactionsbetween in the near future. is the number of atoms in thesystem. By regard- cells is thus the segregation or phase separation of thecells. The entropyassociated with thissepa- To investigatethe nature of thelambda trans- ingan infinite system as made up of an infinite number of large but finite parts, called cells, with ration is constant(depending on composition) in formation, it is very important to study how KT goes the two phase region and is, of course, zero in the to infinity and how approaches (~P/~T)A(if M atoms each, the singularity is seen to arise from PI. one phase region-except for a very small tempera- indeed they do) at temperatures so close to TAthat theinteractions between the cells-which for M large correspond to surface interactions, since the tureinterval in the immediate neighborhood of cq (11 breaks down his is probablyonly pos- This equation shows that KT will not vary much the transition temperature. siblenear the upper lambda point. In Lounas- interatomicforces are short-ranged. For the same until PI. becomesnearly equal to (dP/dT)A. If P,. At first sight the very small magnitude of the mix- maa’s experiment at 13 atm the logarithmic relation reasoninteractions between other than nearest becomes equal to (dP/dT)h, as must happen if Cp ing entropy-associated with hutone degree of predicts that, for T- TA= lO-6, PI.will be less than neighborcells are zero.
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