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The velocity of second sound near the lambda transition in superfluid4He
PHYSICA
Physica B 194-196 (1994) 733-734 North-Holland
T h e V e l o c i t y of S e c o n d S o u n d N e a r t h e L a m b d a
T r a n s i t i o n in S u p e r f l u i d 4He*
M. J. Adriaans, D. R. Swanson, and J. A. Lipa Department of Physics, Stanford University, Stanford, CA 94305-4060, USA We report new high precision data on the velocity of second sound close to the lambda transition of 4He at the saturated vapor pressure. The second sound wave is detected by a powdered salt pill of copper ammonium bromide in direct contact with the liquid helium. With a thermometer noise of < 10-1°K/v/-Hz we are able to make measurements in the linear regime to within 20 nanokelvin of the lambda point. We have obtained reasonable agreement with renormalization group theory predictions after allowance for the effect of gravity. The results are also compared with previous experiments. 1. I N T R O D U C T I O N Experiments performed close to the lambda point of superfluid 4He are a powerful test of the renormalization group (RG) theory, which makes predictions about critical behavior close to the second order phase transition, but complications arise as the transition temperature is approached. For example, signal strengths need to be kept low to avoid nonlinearities in the response. In this paper we report new measurements of the second sound velocity near the lambda transition in a cylindrical resonator. The second sound velocity U is U S = TS2ps/Cppn, where T is the temperature, S is the entropy, Pn and p, are the normal fluid and superfluid densities, and Cp is the specific heat at constant pressure. The superfluid density in helium displays critical behavior which is described by p, = [k0(1 + k1~)]c¢(1 + Dp~a), where ~ and A are critical exponents whose values are predicted by RG theory, and are determined experimentally by fits to data. In particular RG theory predicts = 0.672+0.002 [1]. The coefficients, k0, kl, and Dp are also determined by fitting to experimental data. In our analysis the reduced temperature is defined by 1 - T/Tx, with Tx measured at the bottom of the cell. Gravity induces an inhomogeneity in the superfluid causing the lambda transition temperature to depend on the height of the f u i d sample. The correction for this gravity effect is ATx/Ah = 1.273#K/era. Previous high resolution second sound experiments [2,3] have taken * Work supported by NASA contract numbers JPL-957448 and NGT-50856.
data down to ~ --~ 10 -7, with errors becoming significant near the transition temperature. 2. E X P E R I M E N T
The second sound resonator has been described in detail in [4]. Briefly, the cell consists of a 1.3 cm high cylindrical cavity formed from epoxy with a heater at the b o t t o m and a paramagnetic salt pill detector at the top. To improve the thermal stability and general performance of the previous experiment performed by Marek, Lipa, and Phillips (MLP) with the same resonator [2], the cell is thermally coupled to the innermost stage of a 4-stage thermal isolation system inside a vacuum can. In addition, a low temperature valve and reservoir are attached to the cell, and a thermal shield completely encloses the final stage, allowing far better thermal isolation of the experiment. Second sound was generated by applying an AC signal to the heater and measuring the response from the SQUID output at twice the drive frequency with a spectrum analyzer. AC powers as low as 2 n W / c m 2 were used near the lambda point. The data covers the approximate range of reduced temperature from 10 -8 < ¢ < 10 -2. Data were also taken near 1.65 K, the maximum in the second sound velocity, where its value is known to high accuracy [5]. By measuring the resonant frequency of the cell at this maximum its height was accurately determined. Determination of the lambda transition temperature in any high resolution experiment is crucial. In this experiment Tx is found by slowly heating the cell at a rate of approximately
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
SSDI 0921-4526(93)E0947-F
734
23
0.8
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I
I
I
I
I
0.6 .
21
,.. 0.4 O ..~
•
.~ 0.2
,~ 19
L 17 L b
~ T
0 -0.2 -0.4
15
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-0.6 i
-600
r%
-400
0 T - Tx (nK) -200
I
I
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I
I
200
-0.8
i
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Figure 1. Lambda point determination. 20 nK/sec with a low DC voltage applied to the heater. When the temperature reaches the local lambda point at the b o t t o m of the cell, normal fluid enters the cell. This restricts the heat flow through the cell and the detector records a sudden drop in the heating rate (Figure 1). Tx is chosen as the last point just before the drop. The lambda point is scanned at least 10 times before and after each data set to verify that the T;~ reading is consistent. The deviation of these readings from their average value is approximately 4- 30 nK. 3. R E S U L T S The data was analyzed using the approach developed by Swanson, et al. [6]. Two-fluid hydrodynamics with corrections for gravityi expanded to first order in Vps, is solved in the resonator using the shooting method. Using this model, we obtained a preliminary best fit value for ( = 0.67016 -4- 0.00008. Preliminary values for the other parameters are k0 = 2.368 4-0.002, Dp = 0.539 4- 0.016, and a T~ shift of-32.7 nK. The remaining parameters were fixed at A = 0.5 and kl - -1.74. Goldner and Ahlers [3] (GA) reported the results from their second sound experiment with ( = 0.6705 4- 0.0006. The model used by GA neglected the effects of gravity because their cell was short (1.2mm) and T~ was measured at the middle of the cell. The reanalyzed data from MLP give a best fit value for = 0.6708 4- 0.0004 [6]. Figure 2 shows a deviation plot between the most recent data and the
i 10-s
i 10 -6
t
J 10 .4
I lff 2
Reduced Temperature e Figure 2. Deviation plot of the second sound frequency data from a least squares fit. preliminary best fit function. The "glitch" in the data at ¢ -~ 10 -5 appears to be due to the fact that the cell is filled from the bottom. The fill line extends 9 cm below the b o t t o m of the cell and it is the cell's main thermal link to the reservoir. When normal fluid first enters the b o t t o m of the fill line at AT~ = (9cm)*(1.273pK/cm) -- 10-bK, the cell becomes isolated due to thermal gradients that can now exist in the fill line. Further analysis is being done to explore these small effects. Our experiment has shown that it is possible to determine ( to about a part in 104. It is now a theoretical challenge to derive a prediction to similar accuracy. REFERENCES
[1] D. Z. Albert, Phys. Rev. B 25, 4810 (1982), and J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980). [2] D. Marek, J. A. Lipa, and D. Phillips, Phys. Rev. B 38, 4465 (1988). [3] L. S. Goldner and G. Ahlers, Phys. Rev. B 45, 13129 (1992). L . S . Goldner, Ph.D. Thesisl University of California at Santa Barbara (1991). [4] T. C. P. Chui and D. Marek, J. Low Temp. Phys. 73, 161 (1988). [5] R. T. Wang, W. T. Wagner, and R. J. Donnelly, J. Low Temp. Phys. 68, 409 (1987). [6] D. R. Swanson, T. C. P. Chui, and J. A. Lipa, Phys. Rev. B 46, 9043 (1992).
Physica B 194-196 (1994) 733-734 North-Holland
T h e V e l o c i t y of S e c o n d S o u n d N e a r t h e L a m b d a
T r a n s i t i o n in S u p e r f l u i d 4He*
M. J. Adriaans, D. R. Swanson, and J. A. Lipa Department of Physics, Stanford University, Stanford, CA 94305-4060, USA We report new high precision data on the velocity of second sound close to the lambda transition of 4He at the saturated vapor pressure. The second sound wave is detected by a powdered salt pill of copper ammonium bromide in direct contact with the liquid helium. With a thermometer noise of < 10-1°K/v/-Hz we are able to make measurements in the linear regime to within 20 nanokelvin of the lambda point. We have obtained reasonable agreement with renormalization group theory predictions after allowance for the effect of gravity. The results are also compared with previous experiments. 1. I N T R O D U C T I O N Experiments performed close to the lambda point of superfluid 4He are a powerful test of the renormalization group (RG) theory, which makes predictions about critical behavior close to the second order phase transition, but complications arise as the transition temperature is approached. For example, signal strengths need to be kept low to avoid nonlinearities in the response. In this paper we report new measurements of the second sound velocity near the lambda transition in a cylindrical resonator. The second sound velocity U is U S = TS2ps/Cppn, where T is the temperature, S is the entropy, Pn and p, are the normal fluid and superfluid densities, and Cp is the specific heat at constant pressure. The superfluid density in helium displays critical behavior which is described by p, = [k0(1 + k1~)]c¢(1 + Dp~a), where ~ and A are critical exponents whose values are predicted by RG theory, and are determined experimentally by fits to data. In particular RG theory predicts = 0.672+0.002 [1]. The coefficients, k0, kl, and Dp are also determined by fitting to experimental data. In our analysis the reduced temperature is defined by 1 - T/Tx, with Tx measured at the bottom of the cell. Gravity induces an inhomogeneity in the superfluid causing the lambda transition temperature to depend on the height of the f u i d sample. The correction for this gravity effect is ATx/Ah = 1.273#K/era. Previous high resolution second sound experiments [2,3] have taken * Work supported by NASA contract numbers JPL-957448 and NGT-50856.
data down to ~ --~ 10 -7, with errors becoming significant near the transition temperature. 2. E X P E R I M E N T
The second sound resonator has been described in detail in [4]. Briefly, the cell consists of a 1.3 cm high cylindrical cavity formed from epoxy with a heater at the b o t t o m and a paramagnetic salt pill detector at the top. To improve the thermal stability and general performance of the previous experiment performed by Marek, Lipa, and Phillips (MLP) with the same resonator [2], the cell is thermally coupled to the innermost stage of a 4-stage thermal isolation system inside a vacuum can. In addition, a low temperature valve and reservoir are attached to the cell, and a thermal shield completely encloses the final stage, allowing far better thermal isolation of the experiment. Second sound was generated by applying an AC signal to the heater and measuring the response from the SQUID output at twice the drive frequency with a spectrum analyzer. AC powers as low as 2 n W / c m 2 were used near the lambda point. The data covers the approximate range of reduced temperature from 10 -8 < ¢ < 10 -2. Data were also taken near 1.65 K, the maximum in the second sound velocity, where its value is known to high accuracy [5]. By measuring the resonant frequency of the cell at this maximum its height was accurately determined. Determination of the lambda transition temperature in any high resolution experiment is crucial. In this experiment Tx is found by slowly heating the cell at a rate of approximately
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved
SSDI 0921-4526(93)E0947-F
734
23
0.8
, I , l , , I
I
I
I
I
I
0.6 .
21
,.. 0.4 O ..~
•
.~ 0.2
,~ 19
L 17 L b
~ T
0 -0.2 -0.4
15
L
-0.6 i
-600
r%
-400
0 T - Tx (nK) -200
I
I
[
I
I
200
-0.8
i
400
Figure 1. Lambda point determination. 20 nK/sec with a low DC voltage applied to the heater. When the temperature reaches the local lambda point at the b o t t o m of the cell, normal fluid enters the cell. This restricts the heat flow through the cell and the detector records a sudden drop in the heating rate (Figure 1). Tx is chosen as the last point just before the drop. The lambda point is scanned at least 10 times before and after each data set to verify that the T;~ reading is consistent. The deviation of these readings from their average value is approximately 4- 30 nK. 3. R E S U L T S The data was analyzed using the approach developed by Swanson, et al. [6]. Two-fluid hydrodynamics with corrections for gravityi expanded to first order in Vps, is solved in the resonator using the shooting method. Using this model, we obtained a preliminary best fit value for ( = 0.67016 -4- 0.00008. Preliminary values for the other parameters are k0 = 2.368 4-0.002, Dp = 0.539 4- 0.016, and a T~ shift of-32.7 nK. The remaining parameters were fixed at A = 0.5 and kl - -1.74. Goldner and Ahlers [3] (GA) reported the results from their second sound experiment with ( = 0.6705 4- 0.0006. The model used by GA neglected the effects of gravity because their cell was short (1.2mm) and T~ was measured at the middle of the cell. The reanalyzed data from MLP give a best fit value for = 0.6708 4- 0.0004 [6]. Figure 2 shows a deviation plot between the most recent data and the
i 10-s
i 10 -6
t
J 10 .4
I lff 2
Reduced Temperature e Figure 2. Deviation plot of the second sound frequency data from a least squares fit. preliminary best fit function. The "glitch" in the data at ¢ -~ 10 -5 appears to be due to the fact that the cell is filled from the bottom. The fill line extends 9 cm below the b o t t o m of the cell and it is the cell's main thermal link to the reservoir. When normal fluid first enters the b o t t o m of the fill line at AT~ = (9cm)*(1.273pK/cm) -- 10-bK, the cell becomes isolated due to thermal gradients that can now exist in the fill line. Further analysis is being done to explore these small effects. Our experiment has shown that it is possible to determine ( to about a part in 104. It is now a theoretical challenge to derive a prediction to similar accuracy. REFERENCES
[1] D. Z. Albert, Phys. Rev. B 25, 4810 (1982), and J. C. Le Guillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980). [2] D. Marek, J. A. Lipa, and D. Phillips, Phys. Rev. B 38, 4465 (1988). [3] L. S. Goldner and G. Ahlers, Phys. Rev. B 45, 13129 (1992). L . S . Goldner, Ph.D. Thesisl University of California at Santa Barbara (1991). [4] T. C. P. Chui and D. Marek, J. Low Temp. Phys. 73, 161 (1988). [5] R. T. Wang, W. T. Wagner, and R. J. Donnelly, J. Low Temp. Phys. 68, 409 (1987). [6] D. R. Swanson, T. C. P. Chui, and J. A. Lipa, Phys. Rev. B 46, 9043 (1992).