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Is ultrasound a suitable probe to determine the structure of the vortex lattice in 3He-A?
PHYSICAi
Physica B 178 (1992) 262-265 North-Holland
Is ultrasound a suitable probe to determine the structure of the vortex lattice in 3He-A? J.P. P e k o l a a, G . M . K i r a a, A . J . M a n n i n e n a a n d B . N . Kiviladze b aLow Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland blnstitute of Physics of the Georgian Academy of Sciences, 380077 Tbilisi, Georgia
We consider the propagation of ultrasound in 3He-A transverse to the axes of vortices in a regular array. Owing to the sound propagation anisotropy, sharp changes in attenuation should be experimentally observed in the directions of symmetry. The lattice can be oriented by an applied magnetic field, tilted with respect to the rotation axis.
Several experimental methods have been employed to investigate the structure of individual vortices in 3He-A [1]. Measurements by cw N M R have indicated the existence of nonsingular, axially nonsymmetric vortices in magnetic fields in excess of the dipole field H D = 3 mT. Transmission of negative ions revealed continuous vortices as well, but these measurements alsc ~ave the first indication of singular vortices ir He-A. Most recently, ultrasonic transmission experiments, with pulses propagating along the rotation axis, have demonstrated two types of nonsingular vortices, with a transition close to H D, separating topologically different individual vortices [2]. In this paper we discuss the possibility of using zero sound, propagating transversally to the rotation axis, for observing the two-dimensional lattice found by individual vortices. The basic idea is to control, by a suitably oriented magnetic field H, (a) the orbital /-texture of 3He-A in each unit cell and (b) the vortex lattice vector a, with respect to the sound propagation axis q. Let us first consider the orientation of the texture in each unit cell far from the vortex axis. The/-vector couples to the magnetic field via the spin-orbit interaction: if the component of H transverse to 12 exceeds H D, the dipolar energy 0921-4526/92/$05.00
fD turns l into the plane transverse to H [3]. Rotation by an angular velocity 12 forces 1 to lie in the plane perpendicular to 12 because of the coupling between superflow and the orbital /texture [4]; this energy, however, is rather weak, only about 10 2 times fD, but, according to earlier ultrasonic experiments in low magnetic fields [5], it exceeds the ever present orienting effects due to, e.g., heat flows and container walls. Therefore, in a tilted magnetic field and under rotation, we are left with only one preferred orientation of I (far from the vortex axis), namely the direction perpendicular to both 12 and H. The orientation of the vortex lattice in 3He-A has been studied theoretically by Ohmi [6]. The preference for a triangular lattice over a square array was pointed out already by Tkachenko [7]. Ohmi concludes that the orientation energy for the vortex lattice is due to asymptotic deviations of the orbital texture from a uniform direction. In the case of the commonly accepted SeppfilaVolovik (SV) nonsingular vortices [8], the orientation of the lattice in the minimum energy state is always such that a is within 3.2 ° from the asymptotic direction of the /-vector. However, the energy holding the lattice in the preferred direction is on the order of 10 -6 times smaller than fD. Because the orientation energy is so
© 1992- Elsevier Science Publishers B.V. All rights reserved
J.P. Pekola et al. / Ultrasound and the structure of the vortex lattice in 'He-A weak, it remains an experimental task to find whether a two-dimensional periodic vortex array, with its orientation determined by an external magnetic field, will be realized in 3He-A. In the following analysis we assume that this is the case, and that a is aligned with the asymptotic direction of l. In superfluid 3He-A, attenuation a and velocity c of ultrasound are anisotropic and depend on the angle /3 between the /-vector and the sound wave vector q, according to the following formulae: OL= all cos4~ qL 2% sin2/3 c0s2/3 + a s sin4/3 , c = c o - Acll c0s4/3 - 2Ac c sin2/3
(1)
Core region of
Vortex unit cell
Ullrasound rays propagating through the lattice Fig. 1. Ultrasound propagates perpendicular to the vortex axes.
COS2/3
- zXc± sin4/3 .
(2)
H e r e c o is the velocity of zero sound in normal Fermi liquid, and the values of the parameters a i and Ac i depend on temperature, pressure, ultrasound frequency, etc. In calculating the amplitude of the sound wave, after it has passed through the vortex lattice perpendicular to the rotation axis, we use geometrical acoustics, i.e. diffraction of sound from vortices is not taken into account. We also assume that each sound ray propagates straightforwardly through a hexagonal array of vortices. The basic equation for sound attenuation is
dA = -a(r)A
263
ds .
(3)
passes, and the integrals are calculated over the path of the ray in each cell. We further assume that all unit cells are identical. The attenuation integral in each cell, for a given direction of sound propagation, then depends only on the normal distance b of the sound ray from the vortex axis. Using the W i g n e r - S e i t z approximation with the cell radius R = V ~ / m 3 ~ , eq. (4) can be written in the form k/R2-(j Ab)2
ln(~-----~) = - l i m ~, Nj ab~o ] Ab
f
ith unit cell
(4)
In the final expression we sum over those unit cells of the vortex lattice through which the ray
a(r) d s A b
-~/R2-(j Ab)2
R
=- f g(bo, b) f -R
H e r e d A is the change in the amplitude A of the sound when it has propagated an infinitesimal distance ds in a medium with attenuation coefficient ~(r). After the sound ray, with original amplitude A0, has propagated through a lattice of vortices in the perpendicular direction with respect to their axes (see fig. 1), its amplitude is such that
f
_X/R2_b2
a(r) dbds. (5)
H e r e Nj is the n u m b e r of unit cells through which the ray passes in such a way that b is in the range ( j - 1 ) A b < b < j Ab, and the sum is calculated over different values of j. In the last expression, we have changed the sum over j into an integral over b, and we have defined the distribution g(b o, b) as N / A b - + g(bo, b) when Ab ~ 0 and j Ab ~ b. The variable b 0 indicates that the distribution depends on the distance b 0 of the ray from the axis of the first vortex through which it passes. The distribution also depends on the direction of sound propagation in the lattice. In most cases, g(b o, b) can be ap-
264
J.P. Pekola et al. / Ultrasound and the structure of the vortex lattice in 3He-A
proximated by a constant. However, this is not the case for some special propagation directions through a vortex array, e.g., if the ray is parallel with one of the nearest-neighbor base vectors of the hexagonal lattice. Let us next calculate the attenuation integral of eq. (5) for g(bo, b) =- L/'rrR 2 = constant; here L is the total path length of the sound, and R is again the Wigner-Seitz radius of a unit cell. The amplitude of all rays, after passing through the lattice, is then the same, i.e. we can consider A 0 and A as the total amplitude of the sound wave before and after it has passed through the vortices, respectively. We assume that the lattice consists of nonaxisymmetric continuous Sepp/il/iVolovik vortices of type v [8]. For the /-texture in each unit cell, we use the expression [9] i(r, q~)= - 2 cos ~'(r) + ()) sin q~ ÷ i c o s q~) sin ~(r),
~rrr/r c, ~r(r) = lTr ,
(6)
r<~Q rc
H e r e r and q~ are the polar coordinates, .f determines the asymptotic direction of t h e / - v e c t o r far from the axis of the vortex, and r e = 1 0 50 ~m is the radius of the vortex core. In the case of sound propagating in the xy-plane and making an angle T with the x-axis, the attenuation integral (5) can be calculated easily in polar coordinates: /A\
ln~o)
= L[-a±-
(D21 cos2T ÷ D22 sin2T)
X (t~c -- G ± ) -- (D41
COS4")/÷
As eq. (7) shows, the amplitude of sound, after it has passed through a regular lattice of SVvortices, depends on the angle y between the sound wave vector q and the asymptotic direction of t h e / - v e c t o r far from the vortex axis, but not on the structure of the vortex lattice. This is the case when g(bo, b) is constant. However, as we saw earlier, the form of g(b o, b) becomes anomalous when the sound propagates along one of the symmetry directions of the lattice, and g(b o, b) cannot be approximated by a constant. In fig. 2 we show the results of our computer simulations of sound propagation in a hexagonal lattice of SV-vortices. We have divided the incoming wave front into several rays, followed each ray separately through the lattice by assuming that it propagates straight in its original direction, and calculated the change in the amplitude and phase of each ray after it has gone through the whole sample of 3 H e - A . To determine the average amplitude of the sound wave, A, the amplitudes of the rays have first been summed, taking their phase differences into
I
"" A ~5 V
(7)
where
'
O41 = 1 - g D43 = 9
(-~) 2 .
D22 = 2 \ R /
D42 = g
I
0
i
i
1
i
i
0
20
40
60
80
100
T (deg)
l(rc) 2
2
I
2
-20
\R/
I
D42 sinaT cos2T
+ D43 sin4T)(% - 2a~ + % ) ] ,
D21 = 2 _ ( r c ~
I
'
Fig. 2. Numerically simulated average attenuation ( a ) of ultrasound propagating into the direction 3' (see the inset). The simulation conditions are the following: T/Tc=0.8 , P = 26 bar, f = 35.15 MHz. Attenuation and velocity coefficients are [10] % = 2 . 0 5 c m 1 a c = 6 . 7 1 c m 1, a i =0.52cm-1, c o = 3 9 0 m / s , Ac11=0.0042c o, Ac e=0.0081c o and Ac 1 = 0.0054c o. The lattice is hexagonal and its dimensions are r c = 3 x l 0 Sm, R = 6 . 3 x l 0 Sm.
J.P. Pekola et al. / Ultrasound and the structure o f the vortex lattice in 3He-A
account, and the sum has been divided by the n u m b e r of rays. The average attenuation ( a ) presented in fig. 2 is calculated from A using the formula
In
(8)
where A 0 is the transmitted amplitude and L the total path length of the sound. The results of the simulations agree with eq. (7), except for the anomalous attenuation maxima at y = 0 °, 30 °, 60 ° , and 90 ° , which are the directions of high s y m m e t r y in the hexagonal vortex lattice. In these directions the average attenuation can also be calculated analytically using geometrical acoustics. The results of analytical and numerical calculations coincide, except that according to the analytical result, the attenuation peak at y = 0 ° is even higher than in fig. 2; this difference is due to the Wigner-Seitz approximation used in our numerical simulations. T o conclude, we have suggested a possibility for observing directly the symmetry of the vortex lattice in 3He-A. Our method is based on measuring the attenuation of ultrasound propagating transversaUy with respect to the vortex lattice in an inclined magnetic field. Our analysis is based on geometrical acoustics, which is not strictly valid when the wavelength A = c~ f ~ (400 m / s ) / ( 3 5 MHz) = 11 ~xm is of the order of the vortex cores. Nevertheless, and even though the velocity of sound is inhomogeneous, we can m a k e the following conclusions. Assuming that the orientation of the lattice can be controlled by an applied magnetic field H, tilted
265
with respect to 82, one should see anomalous m a x i m a or minima in the ultrasound attenuation as a function of the angle between q and the c o m p o n e n t of H in the plane perpendicular to ~ . These anomalies should occur at angles of high symmetry. Even if the vortex lattice has some local nonidealities, the anomalies will not vanish, as was confirmed by computer simulations using disturbances which were small compared with the size of the unit cell.
Acknowledgement We thank G. Kharadze, O.V. L o u n a s m a a and G. Volovik for useful discussions.
References [I] P. Hakonen, O.V. Lounasmaa and J. Simola, Physica B 160 (1989) 1. [2] J.P. Pekola, K. Torizuka, A.J. Manninen, J.M. Kyynfirfiinen and G.E. Volovik, Phys. Rev. Lett. 65 (1990) 3293. [3] M.M. Salomaa and G.E. Volovik, Rev. Mod. Phys. 59 (1987) 533. [4] G.E. Volovik and P.J. Hakonen, J. Low Temp. Phys. 42 (1981) 503. [5] J.M. Kyyn/iriiinen, J.P. Pekola, K. Torizuka, A.J. Manninen and A.V. Babkin, J. Low Temp. Phys. 82 (1991) 325. [6] T. Ohmi, J. Low Temp. Phys. 56 (1984) 183. [7] V.R. Tkachenko, Soy. Phys. JETP 22 (1966) 1282. [8] H.K. Seppiil/i and G.E. Volovik, J. Low Temp. Phys. 51 (1983) 279. ]9] A.L. Fetter, J. Low Temp. Phys. 67 (1987) 145. [10] W. Wojtanowski, private communication (1990).
Physica B 178 (1992) 262-265 North-Holland
Is ultrasound a suitable probe to determine the structure of the vortex lattice in 3He-A? J.P. P e k o l a a, G . M . K i r a a, A . J . M a n n i n e n a a n d B . N . Kiviladze b aLow Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland blnstitute of Physics of the Georgian Academy of Sciences, 380077 Tbilisi, Georgia
We consider the propagation of ultrasound in 3He-A transverse to the axes of vortices in a regular array. Owing to the sound propagation anisotropy, sharp changes in attenuation should be experimentally observed in the directions of symmetry. The lattice can be oriented by an applied magnetic field, tilted with respect to the rotation axis.
Several experimental methods have been employed to investigate the structure of individual vortices in 3He-A [1]. Measurements by cw N M R have indicated the existence of nonsingular, axially nonsymmetric vortices in magnetic fields in excess of the dipole field H D = 3 mT. Transmission of negative ions revealed continuous vortices as well, but these measurements alsc ~ave the first indication of singular vortices ir He-A. Most recently, ultrasonic transmission experiments, with pulses propagating along the rotation axis, have demonstrated two types of nonsingular vortices, with a transition close to H D, separating topologically different individual vortices [2]. In this paper we discuss the possibility of using zero sound, propagating transversally to the rotation axis, for observing the two-dimensional lattice found by individual vortices. The basic idea is to control, by a suitably oriented magnetic field H, (a) the orbital /-texture of 3He-A in each unit cell and (b) the vortex lattice vector a, with respect to the sound propagation axis q. Let us first consider the orientation of the texture in each unit cell far from the vortex axis. The/-vector couples to the magnetic field via the spin-orbit interaction: if the component of H transverse to 12 exceeds H D, the dipolar energy 0921-4526/92/$05.00
fD turns l into the plane transverse to H [3]. Rotation by an angular velocity 12 forces 1 to lie in the plane perpendicular to 12 because of the coupling between superflow and the orbital /texture [4]; this energy, however, is rather weak, only about 10 2 times fD, but, according to earlier ultrasonic experiments in low magnetic fields [5], it exceeds the ever present orienting effects due to, e.g., heat flows and container walls. Therefore, in a tilted magnetic field and under rotation, we are left with only one preferred orientation of I (far from the vortex axis), namely the direction perpendicular to both 12 and H. The orientation of the vortex lattice in 3He-A has been studied theoretically by Ohmi [6]. The preference for a triangular lattice over a square array was pointed out already by Tkachenko [7]. Ohmi concludes that the orientation energy for the vortex lattice is due to asymptotic deviations of the orbital texture from a uniform direction. In the case of the commonly accepted SeppfilaVolovik (SV) nonsingular vortices [8], the orientation of the lattice in the minimum energy state is always such that a is within 3.2 ° from the asymptotic direction of the /-vector. However, the energy holding the lattice in the preferred direction is on the order of 10 -6 times smaller than fD. Because the orientation energy is so
© 1992- Elsevier Science Publishers B.V. All rights reserved
J.P. Pekola et al. / Ultrasound and the structure of the vortex lattice in 'He-A weak, it remains an experimental task to find whether a two-dimensional periodic vortex array, with its orientation determined by an external magnetic field, will be realized in 3He-A. In the following analysis we assume that this is the case, and that a is aligned with the asymptotic direction of l. In superfluid 3He-A, attenuation a and velocity c of ultrasound are anisotropic and depend on the angle /3 between the /-vector and the sound wave vector q, according to the following formulae: OL= all cos4~ qL 2% sin2/3 c0s2/3 + a s sin4/3 , c = c o - Acll c0s4/3 - 2Ac c sin2/3
(1)
Core region of
Vortex unit cell
Ullrasound rays propagating through the lattice Fig. 1. Ultrasound propagates perpendicular to the vortex axes.
COS2/3
- zXc± sin4/3 .
(2)
H e r e c o is the velocity of zero sound in normal Fermi liquid, and the values of the parameters a i and Ac i depend on temperature, pressure, ultrasound frequency, etc. In calculating the amplitude of the sound wave, after it has passed through the vortex lattice perpendicular to the rotation axis, we use geometrical acoustics, i.e. diffraction of sound from vortices is not taken into account. We also assume that each sound ray propagates straightforwardly through a hexagonal array of vortices. The basic equation for sound attenuation is
dA = -a(r)A
263
ds .
(3)
passes, and the integrals are calculated over the path of the ray in each cell. We further assume that all unit cells are identical. The attenuation integral in each cell, for a given direction of sound propagation, then depends only on the normal distance b of the sound ray from the vortex axis. Using the W i g n e r - S e i t z approximation with the cell radius R = V ~ / m 3 ~ , eq. (4) can be written in the form k/R2-(j Ab)2
ln(~-----~) = - l i m ~, Nj ab~o ] Ab
f
ith unit cell
(4)
In the final expression we sum over those unit cells of the vortex lattice through which the ray
a(r) d s A b
-~/R2-(j Ab)2
R
=- f g(bo, b) f -R
H e r e d A is the change in the amplitude A of the sound when it has propagated an infinitesimal distance ds in a medium with attenuation coefficient ~(r). After the sound ray, with original amplitude A0, has propagated through a lattice of vortices in the perpendicular direction with respect to their axes (see fig. 1), its amplitude is such that
f
_X/R2_b2
a(r) dbds. (5)
H e r e Nj is the n u m b e r of unit cells through which the ray passes in such a way that b is in the range ( j - 1 ) A b < b < j Ab, and the sum is calculated over different values of j. In the last expression, we have changed the sum over j into an integral over b, and we have defined the distribution g(b o, b) as N / A b - + g(bo, b) when Ab ~ 0 and j Ab ~ b. The variable b 0 indicates that the distribution depends on the distance b 0 of the ray from the axis of the first vortex through which it passes. The distribution also depends on the direction of sound propagation in the lattice. In most cases, g(b o, b) can be ap-
264
J.P. Pekola et al. / Ultrasound and the structure of the vortex lattice in 3He-A
proximated by a constant. However, this is not the case for some special propagation directions through a vortex array, e.g., if the ray is parallel with one of the nearest-neighbor base vectors of the hexagonal lattice. Let us next calculate the attenuation integral of eq. (5) for g(bo, b) =- L/'rrR 2 = constant; here L is the total path length of the sound, and R is again the Wigner-Seitz radius of a unit cell. The amplitude of all rays, after passing through the lattice, is then the same, i.e. we can consider A 0 and A as the total amplitude of the sound wave before and after it has passed through the vortices, respectively. We assume that the lattice consists of nonaxisymmetric continuous Sepp/il/iVolovik vortices of type v [8]. For the /-texture in each unit cell, we use the expression [9] i(r, q~)= - 2 cos ~'(r) + ()) sin q~ ÷ i c o s q~) sin ~(r),
~rrr/r c, ~r(r) = lTr ,
(6)
r<~Q rc
H e r e r and q~ are the polar coordinates, .f determines the asymptotic direction of t h e / - v e c t o r far from the axis of the vortex, and r e = 1 0 50 ~m is the radius of the vortex core. In the case of sound propagating in the xy-plane and making an angle T with the x-axis, the attenuation integral (5) can be calculated easily in polar coordinates: /A\
ln~o)
= L[-a±-
(D21 cos2T ÷ D22 sin2T)
X (t~c -- G ± ) -- (D41
COS4")/÷
As eq. (7) shows, the amplitude of sound, after it has passed through a regular lattice of SVvortices, depends on the angle y between the sound wave vector q and the asymptotic direction of t h e / - v e c t o r far from the vortex axis, but not on the structure of the vortex lattice. This is the case when g(bo, b) is constant. However, as we saw earlier, the form of g(b o, b) becomes anomalous when the sound propagates along one of the symmetry directions of the lattice, and g(b o, b) cannot be approximated by a constant. In fig. 2 we show the results of our computer simulations of sound propagation in a hexagonal lattice of SV-vortices. We have divided the incoming wave front into several rays, followed each ray separately through the lattice by assuming that it propagates straight in its original direction, and calculated the change in the amplitude and phase of each ray after it has gone through the whole sample of 3 H e - A . To determine the average amplitude of the sound wave, A, the amplitudes of the rays have first been summed, taking their phase differences into
I
"" A ~5 V
(7)
where
'
O41 = 1 - g D43 = 9
(-~) 2 .
D22 = 2 \ R /
D42 = g
I
0
i
i
1
i
i
0
20
40
60
80
100
T (deg)
l(rc) 2
2
I
2
-20
\R/
I
D42 sinaT cos2T
+ D43 sin4T)(% - 2a~ + % ) ] ,
D21 = 2 _ ( r c ~
I
'
Fig. 2. Numerically simulated average attenuation ( a ) of ultrasound propagating into the direction 3' (see the inset). The simulation conditions are the following: T/Tc=0.8 , P = 26 bar, f = 35.15 MHz. Attenuation and velocity coefficients are [10] % = 2 . 0 5 c m 1 a c = 6 . 7 1 c m 1, a i =0.52cm-1, c o = 3 9 0 m / s , Ac11=0.0042c o, Ac e=0.0081c o and Ac 1 = 0.0054c o. The lattice is hexagonal and its dimensions are r c = 3 x l 0 Sm, R = 6 . 3 x l 0 Sm.
J.P. Pekola et al. / Ultrasound and the structure o f the vortex lattice in 3He-A
account, and the sum has been divided by the n u m b e r of rays. The average attenuation ( a ) presented in fig. 2 is calculated from A using the formula
In
(8)
where A 0 is the transmitted amplitude and L the total path length of the sound. The results of the simulations agree with eq. (7), except for the anomalous attenuation maxima at y = 0 °, 30 °, 60 ° , and 90 ° , which are the directions of high s y m m e t r y in the hexagonal vortex lattice. In these directions the average attenuation can also be calculated analytically using geometrical acoustics. The results of analytical and numerical calculations coincide, except that according to the analytical result, the attenuation peak at y = 0 ° is even higher than in fig. 2; this difference is due to the Wigner-Seitz approximation used in our numerical simulations. T o conclude, we have suggested a possibility for observing directly the symmetry of the vortex lattice in 3He-A. Our method is based on measuring the attenuation of ultrasound propagating transversaUy with respect to the vortex lattice in an inclined magnetic field. Our analysis is based on geometrical acoustics, which is not strictly valid when the wavelength A = c~ f ~ (400 m / s ) / ( 3 5 MHz) = 11 ~xm is of the order of the vortex cores. Nevertheless, and even though the velocity of sound is inhomogeneous, we can m a k e the following conclusions. Assuming that the orientation of the lattice can be controlled by an applied magnetic field H, tilted
265
with respect to 82, one should see anomalous m a x i m a or minima in the ultrasound attenuation as a function of the angle between q and the c o m p o n e n t of H in the plane perpendicular to ~ . These anomalies should occur at angles of high symmetry. Even if the vortex lattice has some local nonidealities, the anomalies will not vanish, as was confirmed by computer simulations using disturbances which were small compared with the size of the unit cell.
Acknowledgement We thank G. Kharadze, O.V. L o u n a s m a a and G. Volovik for useful discussions.
References [I] P. Hakonen, O.V. Lounasmaa and J. Simola, Physica B 160 (1989) 1. [2] J.P. Pekola, K. Torizuka, A.J. Manninen, J.M. Kyynfirfiinen and G.E. Volovik, Phys. Rev. Lett. 65 (1990) 3293. [3] M.M. Salomaa and G.E. Volovik, Rev. Mod. Phys. 59 (1987) 533. [4] G.E. Volovik and P.J. Hakonen, J. Low Temp. Phys. 42 (1981) 503. [5] J.M. Kyyn/iriiinen, J.P. Pekola, K. Torizuka, A.J. Manninen and A.V. Babkin, J. Low Temp. Phys. 82 (1991) 325. [6] T. Ohmi, J. Low Temp. Phys. 56 (1984) 183. [7] V.R. Tkachenko, Soy. Phys. JETP 22 (1966) 1282. [8] H.K. Seppiil/i and G.E. Volovik, J. Low Temp. Phys. 51 (1983) 279. ]9] A.L. Fetter, J. Low Temp. Phys. 67 (1987) 145. [10] W. Wojtanowski, private communication (1990).