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Compressibility and correlation length of helium-4 near the critical point
Volume 41A, number 4
PHYSICS LETTERS
9 October 1972 +
of the electrons in the F 8 minimum of m* = 0.0255 m 0 [1], we have calculated the electron temperature as a function of electric field following a procedure given in ref. [5]. Fig. 2 shows the dependence of the electron temperature on the electric field strength. An analysis of these data in terms of relevant scattering mechanisms is in progress and will be published elsewhere.
16
TL= Z,.2 K
o
o
1c y, v
t
,A
20
AO
60 80 E (mV/cm)
100
120
140
We are indebted to Prof. K. Seeger for suggesting to investigate hot electron effects in this material. Prof. Seeger would like to express his gratitude to Prof. Busch and Dr. Yuan of the ETH, Zfirich for supplying the sample material.
Fig. 2. Dependence of the electron temperature Te on the applied electric field E.
References erate this increase in electron temperature alters the electron energy only by a very small amount and therefore the mobility, which depends on the mean electron energy, doe not change. Because of the fact that the hole mobility is more than an order of magnitude smaller than the electron mobility at this temperature and concentration [2], the influence of the holes on the SdH experiment can be neglected. Using a value of the effective mass
352
[ 1] B.L. Booth and A.W. Ewald, Phys. Rev. 168 (1968) 796. [2] C.F. Lavine and A.W. Ewald, J. Phys. Chem. Solids 32 (1971) 1121. [3] S.H. Groves and W. Paul, Phys. Rev. Lett. 11 (1963) 194. [4] R.A. Isaacson and F. Bridges, Solid State Commun. 4 (1966) 635. [5 ] H. Kahlert and G. Bauer, phys. stat. sol. (b) 46 (1971 ) 535. [6] G. Bauer and H. Kahlert, Phys. Rev. B5 (1972) 556.
Volume 41A, number 4
PHYSICS LETTERS
9 October 1972
COMPRESSIBILITY AND CORRELATION LENGTH OF HELIUM-4 NEAR THE CRITICAL POINT A. TOMINAGA and Y. NARAHARA Department of Applied Physics, Tokyo University of Education, Tokyo, Japan Received 26 June 1972 From light scattering experiments we have obtained the temperature dependence of the correlation length and the compressibility of 4He in the critical region. We have observed the temperature dependence of the critical opalescence in helium-4. Our experimental setup consists of a helium-neon laser, an attenuator and associated focusing optics, a sample cell in the cryostat and a cooled photomultiplier in a photon counting mode operation. We monitored and manually adjusted the laser power (about 3 mW) to better than 1%. The sample cell was made of OFHC copper with four plane parallel windows coated with antireflection dielectric. The temperature of the cell was regulated to better than 20 #K by an electronic temperature controller [ 1 ]. The pressure of the sample was measured with a pressure gauge with sensitivity 25/a Hg. We determined the critical temperature by observing maximum scattered intensities. Thus determined Tc a n d P c were 5189.808 -+ 0.03 mK and 1706.008 + 0.025 mmHg in T58 respectively. According to the Einstein and Ornstein-Zernike theories', the scattered intensities are proportional to
(ap/ala)T/[1 + ( r ~ ) 21 l - n / 2 , where p and/a are density and chemical potential of the sample, K is the scattering vector, ~ is the longrange correlation length, and 7/is a measure of the departure from Ornstein-/Zernike behavior. For T < T c, the incoming beam was focussed just below the phase boundary and we measured the scattered light intensities at a right angle along the coexistence curve in the temperature range 2X 10 -5 < 1 - TIT c < 4X 10 -2. Our results are shown in fig. 1. Assuming the following power law dependeiace ( a p / a / a ) T " ( I - T / T O - " ~ and ~ = ~0(1 ~T/Tc) - d we obtained 7 = 1.180 ± 0.01, u ' = 0 . 5 9 ~ g ~4 and ~0 = 1.59_+0~95 .~. As the attenuation of the incoming
o t--
o ° 0
104
103 i 10 2
z 10
10-5
10-4
10-3
( Tc-T } / T o
10.'-2
Fig. 1. The intensity of scattered light at a right angle versus reduced temperature along liquid side of the coexistence curve. The solid line corresponds to the zero angle limit and ~' = 1.180. The deviation from the solid line at very near the critical temperature Tc is explained by the Ornstein-Zernike theory. beam [2] is negligibly small (about 1% for our temperature closest to Tc), we assumed that the deviations from the straight line for 1 - TIT c < 4X 10 -4 in fig. 1 are entirely due to the temperature dependence of Gee fig. 2). As 77 is expected to be very small by Fisher [10] and our present data are not sufficiently accurate to detect it directly, we assumed rl is equal to
zero.
Our value, 7' = 1.180 is consistent with 1.1 +- 0.1 353
Volume 41 A, number 4
PHYSICS LETTERS
those of other substances [ 2 - 4 ] within experimental uncertainty. As c~' is not smaller than 0.12, the inequality
1000
5OO
\
2 - a ' 1> 3v'
o~ 20O
lOO
,
,
,
,, ,,I 5 10 ( !- T/~c.~lO 5
i 20
Fig. 2. Temperature dependence o1"the correlation length along the liquid side of coexistence curve. The data points were obtained assuming the Ornstein-Zernike theory and .-/= 1.180. The solid line shows ~ = 1.59(1-T/Tc)-0.59 A. by Roach [6] on 4He and 1.17 +- 0.03 by Wallace and Meyer [8] on 3He but it is far from Garfunkel's [9] result 1.4 if')' = 7'. This disagreement may be due to the large background in their experiment. By the way, our preliminary experiment showed 3' = 1.21 -+ 0.05 along the critical isochore. Note that with regard to the critical experiment of compressibility the critical index 7' of helium is rather small, compared with 1.21 [4], 1.23 [2] and 1.219 [3] o f X e , SF 6 and CO 2 respectively. Among critical indices, there are inequalities a'+2/3+7'~>2
and
7'~>~(6-1).
Using 0.352 + 0.003 by Roach and Douglass [5] 1or and our value for 7', gives a ' ~> 0.12 and 6 ~< 4.35 which are consistent with c( by Modlover's specific heat measurements [7] and 6 by Kiang's calculations [11 ] on the liquid-droplet model. For the correlation length, our results agree with
354
9 October 1972
requires v' ~< 0.63, which may improve the range of uncertainties to u' = 0.59 + 0.04 and ~0 = 1.59 -+ 0.95 A. To our knowledge, this is the first report of correlation length on helium. The authors wish to acknowledge the assistance of Mr. A. Nakazawa and Mr. K. Matsumoto in both conducting the experiments and constructing the apparatus.
References ]1 ] A. Tominaga, to be published in Cryogenics 12 (Oct. 1972). [2] V.G. Puglieli and N.C. Ford, Jr., Phys. Rev. Lett. 25 (1970) 143. [3] J.H. Lunacek and D.S. Cannel, Phys. Rev. Lett. 27 (1971) 841. [4] I.W. Smith, M. Giglio and G.B. Benedek, Phys. Rev. Lett. 27 (1971) 1556. [5] Pat R. Roach and D.tt. Douglass, Phys. Rev. Lett. 17 (1966) 1083. [6} Pat R. Roach, Phys. Rev. 170 (1968) 213. [7] M.R. Moldover, Phys. Rev. 182 (1969) 342. [8] B. Wallace, Jr., and H. Meyer, Phys. Rev. A2 (1970) 1563. I91 M.P. Garfunkel, W.I. Goldburg and C.M. Huang, Proc. of 12th Intern. Conf. on Low Temp. Phys., Kyoto (1970) p. 59. [10] M.E. Fisher, J. Math. Phys. 5 (1964) 944. 111] C.S. Kiang, Phys. Rev. Lett. 24 (1970)47.
PHYSICS LETTERS
9 October 1972 +
of the electrons in the F 8 minimum of m* = 0.0255 m 0 [1], we have calculated the electron temperature as a function of electric field following a procedure given in ref. [5]. Fig. 2 shows the dependence of the electron temperature on the electric field strength. An analysis of these data in terms of relevant scattering mechanisms is in progress and will be published elsewhere.
16
TL= Z,.2 K
o
o
1c y, v
t
,A
20
AO
60 80 E (mV/cm)
100
120
140
We are indebted to Prof. K. Seeger for suggesting to investigate hot electron effects in this material. Prof. Seeger would like to express his gratitude to Prof. Busch and Dr. Yuan of the ETH, Zfirich for supplying the sample material.
Fig. 2. Dependence of the electron temperature Te on the applied electric field E.
References erate this increase in electron temperature alters the electron energy only by a very small amount and therefore the mobility, which depends on the mean electron energy, doe not change. Because of the fact that the hole mobility is more than an order of magnitude smaller than the electron mobility at this temperature and concentration [2], the influence of the holes on the SdH experiment can be neglected. Using a value of the effective mass
352
[ 1] B.L. Booth and A.W. Ewald, Phys. Rev. 168 (1968) 796. [2] C.F. Lavine and A.W. Ewald, J. Phys. Chem. Solids 32 (1971) 1121. [3] S.H. Groves and W. Paul, Phys. Rev. Lett. 11 (1963) 194. [4] R.A. Isaacson and F. Bridges, Solid State Commun. 4 (1966) 635. [5 ] H. Kahlert and G. Bauer, phys. stat. sol. (b) 46 (1971 ) 535. [6] G. Bauer and H. Kahlert, Phys. Rev. B5 (1972) 556.
Volume 41A, number 4
PHYSICS LETTERS
9 October 1972
COMPRESSIBILITY AND CORRELATION LENGTH OF HELIUM-4 NEAR THE CRITICAL POINT A. TOMINAGA and Y. NARAHARA Department of Applied Physics, Tokyo University of Education, Tokyo, Japan Received 26 June 1972 From light scattering experiments we have obtained the temperature dependence of the correlation length and the compressibility of 4He in the critical region. We have observed the temperature dependence of the critical opalescence in helium-4. Our experimental setup consists of a helium-neon laser, an attenuator and associated focusing optics, a sample cell in the cryostat and a cooled photomultiplier in a photon counting mode operation. We monitored and manually adjusted the laser power (about 3 mW) to better than 1%. The sample cell was made of OFHC copper with four plane parallel windows coated with antireflection dielectric. The temperature of the cell was regulated to better than 20 #K by an electronic temperature controller [ 1 ]. The pressure of the sample was measured with a pressure gauge with sensitivity 25/a Hg. We determined the critical temperature by observing maximum scattered intensities. Thus determined Tc a n d P c were 5189.808 -+ 0.03 mK and 1706.008 + 0.025 mmHg in T58 respectively. According to the Einstein and Ornstein-Zernike theories', the scattered intensities are proportional to
(ap/ala)T/[1 + ( r ~ ) 21 l - n / 2 , where p and/a are density and chemical potential of the sample, K is the scattering vector, ~ is the longrange correlation length, and 7/is a measure of the departure from Ornstein-/Zernike behavior. For T < T c, the incoming beam was focussed just below the phase boundary and we measured the scattered light intensities at a right angle along the coexistence curve in the temperature range 2X 10 -5 < 1 - TIT c < 4X 10 -2. Our results are shown in fig. 1. Assuming the following power law dependeiace ( a p / a / a ) T " ( I - T / T O - " ~ and ~ = ~0(1 ~T/Tc) - d we obtained 7 = 1.180 ± 0.01, u ' = 0 . 5 9 ~ g ~4 and ~0 = 1.59_+0~95 .~. As the attenuation of the incoming
o t--
o ° 0
104
103 i 10 2
z 10
10-5
10-4
10-3
( Tc-T } / T o
10.'-2
Fig. 1. The intensity of scattered light at a right angle versus reduced temperature along liquid side of the coexistence curve. The solid line corresponds to the zero angle limit and ~' = 1.180. The deviation from the solid line at very near the critical temperature Tc is explained by the Ornstein-Zernike theory. beam [2] is negligibly small (about 1% for our temperature closest to Tc), we assumed that the deviations from the straight line for 1 - TIT c < 4X 10 -4 in fig. 1 are entirely due to the temperature dependence of Gee fig. 2). As 77 is expected to be very small by Fisher [10] and our present data are not sufficiently accurate to detect it directly, we assumed rl is equal to
zero.
Our value, 7' = 1.180 is consistent with 1.1 +- 0.1 353
Volume 41 A, number 4
PHYSICS LETTERS
those of other substances [ 2 - 4 ] within experimental uncertainty. As c~' is not smaller than 0.12, the inequality
1000
5OO
\
2 - a ' 1> 3v'
o~ 20O
lOO
,
,
,
,, ,,I 5 10 ( !- T/~c.~lO 5
i 20
Fig. 2. Temperature dependence o1"the correlation length along the liquid side of coexistence curve. The data points were obtained assuming the Ornstein-Zernike theory and .-/= 1.180. The solid line shows ~ = 1.59(1-T/Tc)-0.59 A. by Roach [6] on 4He and 1.17 +- 0.03 by Wallace and Meyer [8] on 3He but it is far from Garfunkel's [9] result 1.4 if')' = 7'. This disagreement may be due to the large background in their experiment. By the way, our preliminary experiment showed 3' = 1.21 -+ 0.05 along the critical isochore. Note that with regard to the critical experiment of compressibility the critical index 7' of helium is rather small, compared with 1.21 [4], 1.23 [2] and 1.219 [3] o f X e , SF 6 and CO 2 respectively. Among critical indices, there are inequalities a'+2/3+7'~>2
and
7'~>~(6-1).
Using 0.352 + 0.003 by Roach and Douglass [5] 1or and our value for 7', gives a ' ~> 0.12 and 6 ~< 4.35 which are consistent with c( by Modlover's specific heat measurements [7] and 6 by Kiang's calculations [11 ] on the liquid-droplet model. For the correlation length, our results agree with
354
9 October 1972
requires v' ~< 0.63, which may improve the range of uncertainties to u' = 0.59 + 0.04 and ~0 = 1.59 -+ 0.95 A. To our knowledge, this is the first report of correlation length on helium. The authors wish to acknowledge the assistance of Mr. A. Nakazawa and Mr. K. Matsumoto in both conducting the experiments and constructing the apparatus.
References ]1 ] A. Tominaga, to be published in Cryogenics 12 (Oct. 1972). [2] V.G. Puglieli and N.C. Ford, Jr., Phys. Rev. Lett. 25 (1970) 143. [3] J.H. Lunacek and D.S. Cannel, Phys. Rev. Lett. 27 (1971) 841. [4] I.W. Smith, M. Giglio and G.B. Benedek, Phys. Rev. Lett. 27 (1971) 1556. [5] Pat R. Roach and D.tt. Douglass, Phys. Rev. Lett. 17 (1966) 1083. [6} Pat R. Roach, Phys. Rev. 170 (1968) 213. [7] M.R. Moldover, Phys. Rev. 182 (1969) 342. [8] B. Wallace, Jr., and H. Meyer, Phys. Rev. A2 (1970) 1563. I91 M.P. Garfunkel, W.I. Goldburg and C.M. Huang, Proc. of 12th Intern. Conf. on Low Temp. Phys., Kyoto (1970) p. 59. [10] M.E. Fisher, J. Math. Phys. 5 (1964) 944. 111] C.S. Kiang, Phys. Rev. Lett. 24 (1970)47.