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Sound absorption of superfluid3He-B under strong magnetic field
PHYSICA
Physica B 194-196 (1994) 819-820 North-Holland
Sound absorption of Superfluid
aHe-B under strong Magnetic Field
M.Ashida ~ and K.Nagai ~ "Department of Physics, Yamaguchi University, Yamaguchi 753, Japan ~Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-hiroshima 724, Japan Theory of the sound absorption of superfluid aIIe-B under arbitrary magnetic field is presented in the collisionless regime. It is shown that the imaginary J = 1, Jz = 0 mode together with pair breaking excitations yield an interesting absorption spectrum depending upon magnetic field. We suppose that the external magnetic field is applied along z-axis and the distorted order parameter of the B phase is given by [1, 2]
(A~:, Ay, A~,) = (A1/~=, A,lSrj, A2/~.,).
(1)
where 15 is the unit vector along p. In this case the excitation energy of the Bogoliubov quasiparticle is given by
Zo : ~(~ + IAI2 + o3~/4
-
a~LE z
,
In Figs. 1 and 2, we illustrate the frequency dependence of ( when T/Tc = 0 . 9 0 , C O L / A B W = 0.05 and F~ = -0.70 [4]. Here, ABW is the energy gap of the BW state at T = OK without magnetic field. We first calculate A1, A2 and ~L numerically by minimizing the free energy to obtain A1/ABW = 0.531, A~/ABw = 0.507 and the effective field (OL/ABW = 0.146. Next, using these parameters, we calculate ( numerically.
o.1
(2)
I
I
I
~
I [ 1 1 1 1
I
J where ~r = 4-1 , lAB is the energy gap defined by 2
IAI
^2
- A,(p: +
+
2 ^2
,
/
0.0
(3)
and
(1.3
E, = X/(~ + IAz I=
(4)
The effective Larmor frequency O~L is determined by a self-consistent equation ~0L : COL -- 4 f ~ S z , where WL is the Larmor frequency 7 H and f~ is the antisymmetric part of the s-wave Fermi liquid interaction and S, is the total spin polarization. In this report, we discuss the sound propagation parallel to the external magnetic field. As long as the particle-hole symmetry is assumed, the dispersion relation at arbitrary magnetic field is given by a known form [3] c~ - c ~
c~
A
--
1+~F~4 1 s l+f~ 5 1+ ~f~( 9
(5) '
where c is a (complex) sound velocity and cl is the first sound velocity. But the expression of ( at the presence of the magnetic field is too complicated to write down explicitly here. We discuss on the w dependence of ( in this report. When F~ is not so large, Re( is proportional to the sound velocity shift and Im( is proportional to the sound attenuation.
-o.1
-0.2
f
[
0.0
B I
I
[
[
0.5
1
I
tl
I
1.0
i
b
I
I
.5
co/ABW Fig. 1 Real part of ( When the sound propagates parallel to the magnetic field, only the collective modes with J~ = 0 couple to the zero sound. In the presence of the magnetic field, the total angular momentum J is no longer a good quantum number. But for the sake of convenience we label the modes by their J value at the absence of magnetic field. The vertical line at w / A B w = 0.809 in these figures represents the delta-function peak from the collective mode J = 2, J~ = 0 which correspond to the squashing mode at w = X/i~/5A in the BW state. The peak ( and anti-peak ) near W/ABW = 1 in Fig. 2 comes from the collective mode J = 1,J, = 0. This spectrum is very similar to the experimental results by Ling
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0990-X
820
et al [5] and Saunders et al. [6] Note that their experiments were done in the geometry q A_ H and they identified this structure with the collective modes J = 1, J~ = :kl following Schopohl and Tewordt [7, 8], whereas our result is for the q II H geometry. However, recently McKenzie and Sauls [9] discussed that the collective modes J = 1, J~ = +1 do not couple to zero sound. Our result, therefore, suggests that the anomalous absorption observed by the experiments is also due to t h e J = l J~ = 0 m o d e . 0.05
i
,
J
,
i
I
;
I
I
[
I
0.04
,u,3"
0.03 0.02
-
0.01
I--@~ 0.0
O. O0
E
E
]
0.5
]
]
uj
.0
hand, the velocity shift shows the anomaly at C, D and F rather than at the pair breaking edges A and B. We also calculate ~ at WL/ABW = 0.08, 0.10, 0.12 and 0.15 and find drastic change in the absorption spectrum under the strong magnetic field. The anomaly at the point C is clearly seen when WL/ABW = 0.08, 0.10 and 0.12. Also the point D shows a weak anomaly. The measurement of the points C, D and E enables to determine experimentally the values of A1, A2 and the effective field &L. When WL/ABW 2~ 0.08, the absorption peak of the squashing mode is in the continuum of the pair breaking absorption and has a finite width. Increasing the magnetic field, the peak position of the squashing mode moves to lower frequency side lowering its height and broadening its width. At WL/ABW = 0.15, the squashing mode has no longer a sharp peak. Detailed account of the present study shall be reported elsewhere.
.5
/AB w Fig. 2 Imaginary part of Pair breaking excitations also give rise to structures in the sound absorption spectrum. The pair breaking occurs at w = 2E+, 2E_, E_ + E+ and E_ - E+. Because of the anisotropy of the energy gap, for small w the attenuation occurs in some part of the Fermi surface and for w larger than some critical frequency the attenuation occurs on entire region of the Fermi surface. Therefore the sound absorption has several small anomalies like the cusp found in 3IIe-A. [10] We plot these points by small circles in Fig. 1 and Fig. 2. The point A,B,... has the frequency WA, wB, " ", respectively. The small hill from w = 0 to wa = goc in Fig. 2 is due to the mode w = E_ E+. The attenuation edge at co = cob = 2A~, C~L is the starting point of the mode co = 2E+. And other characteristic frequencies are defined by wc = 2A2, WD = 2A1, WE : ,v/4A~ + uS~ and coF = 2 a l When the applied magnetic field is small ( about w L / A B w % 0.05 ), the anomalies in the attenuation at points C, D and F are too small to observe. On the other
References
[1] N.Schopohl, J. Low Temp. Phys. 49, 347 (1982). [2] M.Ashida and K.Nagai, Prog. Theor. Phys. 74, 949 (1985). [3] P. WSlfle, Phys. Rev. B14, 89 (1976). [4] D.S.Greywal], Phys. Rev. 1327, 2747 (1983). [5] R.Ling, J.Saunders and E.R.Dobbs, Phys. Rev. Left. 59,461 (1987). [6] J.Saunders, R.Ling, W.Wojtanowski and E.R.Dobbs,J. Low Temp. Phys. 79, 75 (1990). [7] N.Schopohl and L.Tewordt, J. Low Temp. Phys. 45, 67 (1981). [8] N.Schopohl and L.Tewordt, J. Low Temp. Phys. 57, 601 (1984). [9] R.H.McKenzie and J.A.Sauls, preprint. [10] M.Ashida and K.Nagai, Prog. Theor. Phys. 70, 1672 (1983).
Physica B 194-196 (1994) 819-820 North-Holland
Sound absorption of Superfluid
aHe-B under strong Magnetic Field
M.Ashida ~ and K.Nagai ~ "Department of Physics, Yamaguchi University, Yamaguchi 753, Japan ~Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-hiroshima 724, Japan Theory of the sound absorption of superfluid aIIe-B under arbitrary magnetic field is presented in the collisionless regime. It is shown that the imaginary J = 1, Jz = 0 mode together with pair breaking excitations yield an interesting absorption spectrum depending upon magnetic field. We suppose that the external magnetic field is applied along z-axis and the distorted order parameter of the B phase is given by [1, 2]
(A~:, Ay, A~,) = (A1/~=, A,lSrj, A2/~.,).
(1)
where 15 is the unit vector along p. In this case the excitation energy of the Bogoliubov quasiparticle is given by
Zo : ~(~ + IAI2 + o3~/4
-
a~LE z
,
In Figs. 1 and 2, we illustrate the frequency dependence of ( when T/Tc = 0 . 9 0 , C O L / A B W = 0.05 and F~ = -0.70 [4]. Here, ABW is the energy gap of the BW state at T = OK without magnetic field. We first calculate A1, A2 and ~L numerically by minimizing the free energy to obtain A1/ABW = 0.531, A~/ABw = 0.507 and the effective field (OL/ABW = 0.146. Next, using these parameters, we calculate ( numerically.
o.1
(2)
I
I
I
~
I [ 1 1 1 1
I
J where ~r = 4-1 , lAB is the energy gap defined by 2
IAI
^2
- A,(p: +
+
2 ^2
,
/
0.0
(3)
and
(1.3
E, = X/(~ + IAz I=
(4)
The effective Larmor frequency O~L is determined by a self-consistent equation ~0L : COL -- 4 f ~ S z , where WL is the Larmor frequency 7 H and f~ is the antisymmetric part of the s-wave Fermi liquid interaction and S, is the total spin polarization. In this report, we discuss the sound propagation parallel to the external magnetic field. As long as the particle-hole symmetry is assumed, the dispersion relation at arbitrary magnetic field is given by a known form [3] c~ - c ~
c~
A
--
1+~F~4 1 s l+f~ 5 1+ ~f~( 9
(5) '
where c is a (complex) sound velocity and cl is the first sound velocity. But the expression of ( at the presence of the magnetic field is too complicated to write down explicitly here. We discuss on the w dependence of ( in this report. When F~ is not so large, Re( is proportional to the sound velocity shift and Im( is proportional to the sound attenuation.
-o.1
-0.2
f
[
0.0
B I
I
[
[
0.5
1
I
tl
I
1.0
i
b
I
I
.5
co/ABW Fig. 1 Real part of ( When the sound propagates parallel to the magnetic field, only the collective modes with J~ = 0 couple to the zero sound. In the presence of the magnetic field, the total angular momentum J is no longer a good quantum number. But for the sake of convenience we label the modes by their J value at the absence of magnetic field. The vertical line at w / A B w = 0.809 in these figures represents the delta-function peak from the collective mode J = 2, J~ = 0 which correspond to the squashing mode at w = X/i~/5A in the BW state. The peak ( and anti-peak ) near W/ABW = 1 in Fig. 2 comes from the collective mode J = 1,J, = 0. This spectrum is very similar to the experimental results by Ling
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E0990-X
820
et al [5] and Saunders et al. [6] Note that their experiments were done in the geometry q A_ H and they identified this structure with the collective modes J = 1, J~ = :kl following Schopohl and Tewordt [7, 8], whereas our result is for the q II H geometry. However, recently McKenzie and Sauls [9] discussed that the collective modes J = 1, J~ = +1 do not couple to zero sound. Our result, therefore, suggests that the anomalous absorption observed by the experiments is also due to t h e J = l J~ = 0 m o d e . 0.05
i
,
J
,
i
I
;
I
I
[
I
0.04
,u,3"
0.03 0.02
-
0.01
I--@~ 0.0
O. O0
E
E
]
0.5
]
]
uj
.0
hand, the velocity shift shows the anomaly at C, D and F rather than at the pair breaking edges A and B. We also calculate ~ at WL/ABW = 0.08, 0.10, 0.12 and 0.15 and find drastic change in the absorption spectrum under the strong magnetic field. The anomaly at the point C is clearly seen when WL/ABW = 0.08, 0.10 and 0.12. Also the point D shows a weak anomaly. The measurement of the points C, D and E enables to determine experimentally the values of A1, A2 and the effective field &L. When WL/ABW 2~ 0.08, the absorption peak of the squashing mode is in the continuum of the pair breaking absorption and has a finite width. Increasing the magnetic field, the peak position of the squashing mode moves to lower frequency side lowering its height and broadening its width. At WL/ABW = 0.15, the squashing mode has no longer a sharp peak. Detailed account of the present study shall be reported elsewhere.
.5
/AB w Fig. 2 Imaginary part of Pair breaking excitations also give rise to structures in the sound absorption spectrum. The pair breaking occurs at w = 2E+, 2E_, E_ + E+ and E_ - E+. Because of the anisotropy of the energy gap, for small w the attenuation occurs in some part of the Fermi surface and for w larger than some critical frequency the attenuation occurs on entire region of the Fermi surface. Therefore the sound absorption has several small anomalies like the cusp found in 3IIe-A. [10] We plot these points by small circles in Fig. 1 and Fig. 2. The point A,B,... has the frequency WA, wB, " ", respectively. The small hill from w = 0 to wa = goc in Fig. 2 is due to the mode w = E_ E+. The attenuation edge at co = cob = 2A~, C~L is the starting point of the mode co = 2E+. And other characteristic frequencies are defined by wc = 2A2, WD = 2A1, WE : ,v/4A~ + uS~ and coF = 2 a l When the applied magnetic field is small ( about w L / A B w % 0.05 ), the anomalies in the attenuation at points C, D and F are too small to observe. On the other
References
[1] N.Schopohl, J. Low Temp. Phys. 49, 347 (1982). [2] M.Ashida and K.Nagai, Prog. Theor. Phys. 74, 949 (1985). [3] P. WSlfle, Phys. Rev. B14, 89 (1976). [4] D.S.Greywal], Phys. Rev. 1327, 2747 (1983). [5] R.Ling, J.Saunders and E.R.Dobbs, Phys. Rev. Left. 59,461 (1987). [6] J.Saunders, R.Ling, W.Wojtanowski and E.R.Dobbs,J. Low Temp. Phys. 79, 75 (1990). [7] N.Schopohl and L.Tewordt, J. Low Temp. Phys. 45, 67 (1981). [8] N.Schopohl and L.Tewordt, J. Low Temp. Phys. 57, 601 (1984). [9] R.H.McKenzie and J.A.Sauls, preprint. [10] M.Ashida and K.Nagai, Prog. Theor. Phys. 70, 1672 (1983).