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3He spin diffusion coefficient and temperature variation of roton gap energy in dilute 3He-4He mixtures
Volume 47A, number 5
PHYSICS LETTERS
3He SPIN DIFFUSION ROTON
COEFFICIENT
GAP ENERGY
22 April 1974
AND TEMPERATURE
IN DILUTE 3He-4He
VARIATION
OF
MIXTURES
K. FUKUDA, Y. HIRAYOSHI and A. HIRAI
Department of Physics, Faculty of Science, Kyoto University, Kyoto, Japan Received 11 March 1974 The spin diffusion coefficient (Ds) of 3He in dilute 3He-4He mixtures has been measured and analysed, following Khalatnikov theory and taking the temperature variation of the roton gap energy (h) into account. The initial purpose of this study was to measure D s precisely around the X point, in connection with the reported anomaly of NMR signals [1,4] D s was measured by using a standard Carr-Purcell method at 10 MHz [5, 6]. As an example of the experimental results, fig. 1 shows the temperature dependence o f D s at SVP for a 2.5 3He solution. This result is consistent with the previously reported ones [ 7 - 1 1 ] . The ?t point of this solution is about 2.14 K. A r o u n d this temperature, we could not find any anomaly in D s so far, though the signal intensity (amplitude o f the free induction decay in our case) showed some anomaly, which was used to determine the )t point indeed. Details on this problem will be discussed separately.
*K: •
"
T;
8 ,q
** 7
cILl CD
g 6 "5
0¢
1,2 1.4 1.6 1.8 2.0 2.2 "K Fig. 2. The temperature dependence of the roton gap energy, • denote the roton gap energy analysed from the data of fig. 1, x denote the results of the neutron diffraction by Henshaw and Woods.
In our 3He concentration range and in our temperature range of fig. 1, D s is determined mainly by the collision of the 3He quasiparticles with rotons. Then, following Khalatnikov and Zharkov [12], with a roton gas model,
o o o o
c
o
D s = A exp (,5/kT),
o
(1)
o
(33 0
(J
g
o
~3
%0
c o.
t
Tx I
1.6
I)8
2.10
212 *K
Fig. 1. The measured spin diffusion coefficient D s as a function of temperature for 3He-aHe solution of 2.5 3He concentration. Th is 2.14°K.
where A, which contains the scattering cross section between the 3He quasil~articles and rotons, is considered temperature independent within our experimental accuracy. If we assume A to be temperature independent, we obtain the value o f A/k of 13.5 K with A o f 1.3 × 10 - 7 cm2/sec from the data o f fig. 1. This value of &/k is consistent with the reported values [7, 8] in the same 3He concentration and temperature range as ours, but this value o f z~/k is too large compared with the reasonable roton gap energy of 8.6 K at 0 K. Now we have attempted to analyse the data o f 377
Volume 47A, number 5
PHYSICS LETTERS
fig. 1, taking the temperature dependence of A into account. At 1.64 K, D s = 5.0 X 10 - 4 cm2/sec (our experimental result of fig. 1) and 2x/k is assumed to be 8.4 K. Then, we have A = 3.0 X 10 . 6 cm2/sec. With this value of A, the experimental temperature dependence o l D s in fig. 1 gives the temperature dependence of A. Fig. 2 shows the result of this analysis, where the temperature variation of the roton gap energy for pure 4He obtained by the neutron diffraction experiment by Henshaw and Woods [13] is shown for comparison. We note also that the value of A of 3.0 X 10 - 6 cm2/sec is more consistent than 1.3 X 10 7 cm2/sec, comparing with the available experimental data [12, 14, 15]. Recently, the roton gap energy of 3He 4He mixtures at 0 K was determined to be independent on the 3He concentration by two roton Raman scattering experiment [ 16, 17]. With this information in mind, it seems our present analysis is more reasonable than the analysis with the temperature independent roton gap energy, and we may think the temperature variation of the roton gap energy is nearly the same in dilute 3 H e - 4 H e mixture as in pure 4He, just as the value of the roton gap energy at 0 K itself.
378
22 April 1974
References [1] V.N. Grigor'ev et al., Zh. Eksp. i Teor. Fiz. 56 (1969) 21 [Soviet Phys. JETP 29 (1969) 11]. [2] B.T. Beal, J. Hatton and R.B. Harrison, Phys. Lett. 21 (1966) 147. [3] E.P. Horvitz, Phys. Rev. A1 (1970) 1708. [4} S. Saito, Phys. Lett. 43A (1973) 241. [5] H.Y. Carr and E.M. Purcell, Phys. Rev. 94 (1954) 630. [6] S. Meiboom and D. Gill, Rev. Sci. Instrum. 29 (1958) 688. [7] R.L. Garwin and H.A. Reich, Phys. Rev. 115 (1959) 1478. [81 D.C. Chang and H.E. Rorschach, J. Low Temp. Phys. 10 (1973) 245. [9] J.E. Opfer, K. Luszczynski and R.E. Norberg, Phys. Rev. 172 (1968) 192. [10] D.K. Beigelsen and K. Lusczynski, Phys. Rev. A1 (1971) 1060. [ 11 ] R.B. Harrison and J. Hatton, J. Low Temp. Phys. 6 (1972) 43. [12] I.M. Khalatnikon and V.M. Zharkov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1108 [Soviet Phys. JETP 5 (1957) 905]. [131 D.G. Henshaw and A.D.B. Woods, Phys. Rev. 121 (1961) 1266. [14] T.P. Ptukha, Zh. Eksp. i Teor. Fiz. 40 (1961) 1583 [Soviet Phys. JETP 13 (1961) 1112]. [15] G.A. Herzlinger and J.G. Kind, Phys. Lett. 40A (1972) 65. [16] R.L. Woerner, D.A. Rockwell and T.J. Greytak, Phys. Rev. Lett. 30 (1973) 1114. [17] C.M. Surko and R.E. Slusher, Phys. Rev. Lett. 30 (1973) 1111.
PHYSICS LETTERS
3He SPIN DIFFUSION ROTON
COEFFICIENT
GAP ENERGY
22 April 1974
AND TEMPERATURE
IN DILUTE 3He-4He
VARIATION
OF
MIXTURES
K. FUKUDA, Y. HIRAYOSHI and A. HIRAI
Department of Physics, Faculty of Science, Kyoto University, Kyoto, Japan Received 11 March 1974 The spin diffusion coefficient (Ds) of 3He in dilute 3He-4He mixtures has been measured and analysed, following Khalatnikov theory and taking the temperature variation of the roton gap energy (h) into account. The initial purpose of this study was to measure D s precisely around the X point, in connection with the reported anomaly of NMR signals [1,4] D s was measured by using a standard Carr-Purcell method at 10 MHz [5, 6]. As an example of the experimental results, fig. 1 shows the temperature dependence o f D s at SVP for a 2.5 3He solution. This result is consistent with the previously reported ones [ 7 - 1 1 ] . The ?t point of this solution is about 2.14 K. A r o u n d this temperature, we could not find any anomaly in D s so far, though the signal intensity (amplitude o f the free induction decay in our case) showed some anomaly, which was used to determine the )t point indeed. Details on this problem will be discussed separately.
*K: •
"
T;
8 ,q
** 7
cILl CD
g 6 "5
0¢
1,2 1.4 1.6 1.8 2.0 2.2 "K Fig. 2. The temperature dependence of the roton gap energy, • denote the roton gap energy analysed from the data of fig. 1, x denote the results of the neutron diffraction by Henshaw and Woods.
In our 3He concentration range and in our temperature range of fig. 1, D s is determined mainly by the collision of the 3He quasiparticles with rotons. Then, following Khalatnikov and Zharkov [12], with a roton gas model,
o o o o
c
o
D s = A exp (,5/kT),
o
(1)
o
(33 0
(J
g
o
~3
%0
c o.
t
Tx I
1.6
I)8
2.10
212 *K
Fig. 1. The measured spin diffusion coefficient D s as a function of temperature for 3He-aHe solution of 2.5 3He concentration. Th is 2.14°K.
where A, which contains the scattering cross section between the 3He quasil~articles and rotons, is considered temperature independent within our experimental accuracy. If we assume A to be temperature independent, we obtain the value o f A/k of 13.5 K with A o f 1.3 × 10 - 7 cm2/sec from the data o f fig. 1. This value of &/k is consistent with the reported values [7, 8] in the same 3He concentration and temperature range as ours, but this value o f z~/k is too large compared with the reasonable roton gap energy of 8.6 K at 0 K. Now we have attempted to analyse the data o f 377
Volume 47A, number 5
PHYSICS LETTERS
fig. 1, taking the temperature dependence of A into account. At 1.64 K, D s = 5.0 X 10 - 4 cm2/sec (our experimental result of fig. 1) and 2x/k is assumed to be 8.4 K. Then, we have A = 3.0 X 10 . 6 cm2/sec. With this value of A, the experimental temperature dependence o l D s in fig. 1 gives the temperature dependence of A. Fig. 2 shows the result of this analysis, where the temperature variation of the roton gap energy for pure 4He obtained by the neutron diffraction experiment by Henshaw and Woods [13] is shown for comparison. We note also that the value of A of 3.0 X 10 - 6 cm2/sec is more consistent than 1.3 X 10 7 cm2/sec, comparing with the available experimental data [12, 14, 15]. Recently, the roton gap energy of 3He 4He mixtures at 0 K was determined to be independent on the 3He concentration by two roton Raman scattering experiment [ 16, 17]. With this information in mind, it seems our present analysis is more reasonable than the analysis with the temperature independent roton gap energy, and we may think the temperature variation of the roton gap energy is nearly the same in dilute 3 H e - 4 H e mixture as in pure 4He, just as the value of the roton gap energy at 0 K itself.
378
22 April 1974
References [1] V.N. Grigor'ev et al., Zh. Eksp. i Teor. Fiz. 56 (1969) 21 [Soviet Phys. JETP 29 (1969) 11]. [2] B.T. Beal, J. Hatton and R.B. Harrison, Phys. Lett. 21 (1966) 147. [3] E.P. Horvitz, Phys. Rev. A1 (1970) 1708. [4} S. Saito, Phys. Lett. 43A (1973) 241. [5] H.Y. Carr and E.M. Purcell, Phys. Rev. 94 (1954) 630. [6] S. Meiboom and D. Gill, Rev. Sci. Instrum. 29 (1958) 688. [7] R.L. Garwin and H.A. Reich, Phys. Rev. 115 (1959) 1478. [81 D.C. Chang and H.E. Rorschach, J. Low Temp. Phys. 10 (1973) 245. [9] J.E. Opfer, K. Luszczynski and R.E. Norberg, Phys. Rev. 172 (1968) 192. [10] D.K. Beigelsen and K. Lusczynski, Phys. Rev. A1 (1971) 1060. [ 11 ] R.B. Harrison and J. Hatton, J. Low Temp. Phys. 6 (1972) 43. [12] I.M. Khalatnikon and V.M. Zharkov, Zh. Eksp. i Teor. Fiz. 32 (1957) 1108 [Soviet Phys. JETP 5 (1957) 905]. [131 D.G. Henshaw and A.D.B. Woods, Phys. Rev. 121 (1961) 1266. [14] T.P. Ptukha, Zh. Eksp. i Teor. Fiz. 40 (1961) 1583 [Soviet Phys. JETP 13 (1961) 1112]. [15] G.A. Herzlinger and J.G. Kind, Phys. Lett. 40A (1972) 65. [16] R.L. Woerner, D.A. Rockwell and T.J. Greytak, Phys. Rev. Lett. 30 (1973) 1114. [17] C.M. Surko and R.E. Slusher, Phys. Rev. Lett. 30 (1973) 1111.