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The non-saturating anisotropy of induced torque in potassium
Volume 39A, number 4
PHYSICS LETTERS
22 May 1972
THE NON-SATURATING ANISOTROPY OF INDUCED TORQUE IN POTASSIUM J.S. LASS Fysisch Laboratorium, Katholieke Universiteit, Ni/megen, TheNetherlands
Received 24 March 1972
—
The anisotropy of induced torque in potassium observed by Schaefer and Marcus can be explained as a result 10% departure of sample shape from sphericity.
Recent measurements by Schaefer and Marcus [1] of induced torque on small single crystals of potassium showed remarkable twofold anisotropy of the torque for all magnetic fiulds. At higher fields the torque developed an increasingly larger component with fourfold symmetry. These anisotropies have lead Overhauser [2] to conclude that the Fermi surface of potassium is neither simply connected nor of cubic symmetry. We wish to show that a departure with twofold symmetry of the sample shape from a sphere is sufficient to produce the observed anomalies, without assuming any anisotropy of the resistivity or of the Fermi surface. The magnitude and crystalographic orientation dependence of the measured torque suggest that the distorted shape developed during solidification of the sample from a molten droplet. Calculations of the torque induced in an ellipsoidshaped sample haye been made earlier [3] , but a general expression has not been given sofar. It can be easily derived by solving Maxwell’s equation V x ~ ‘j= —B/c with a boundary condition at the surface of the ellipsoid jn = 0. Proceeding in analogy with Visscher and Falicov [4] one can show that the torque N= —(det~)[Bx 2irR5 (eyeB)] (I) 1 Sc2 where —
~[Tr(~~)~’
~+]..1
~ is the resistivity tensor, = StS, and S is a linear transformation of coordinates such that r~ = ~, where re lies on the surface of the ellipsoid and r’ 5 on the sur-
ofa
face of a sphere with a radius R. The symmetric part of tensor ~ leads to the torque component Nn in the direction of the axis of rotation. Using eq. (1) we have considered a flattened ellipsoid of rotation with its short semiaxis (bR, b ~ 1) tilted through an angle ~pfrom the axis of rotation fi in a plane which is rotated about fi by an angle 0 measured from the plane perpendicular to the magnetic field (B .L fi). As a measure of the departure2)sin2p. from rotational We symmetry we have denoted i~= (1 — btensor with nonfurther assumed a magnetoresistivity zero elements ~ = = =p 0q and = —‘~~= p0w~r(where q is a dimensionless coefficient, c~is the cyclotron frequency and r is the relaxation time). The explicit formula for N~is lengthy, but one Important feature appears clearly. While at low fields (~~r 20,2q) at oscilatory terms are present proportional to ~cos fairly high fields (s~~r>> 2q) non-saturating terms proportional to n2sin20cos20(w~r/2q)2dominate the result. The physical picture behind these non-saturating effects of sample shape is analogous to that of open orbits in transverse magnetoresistance. With the exception of highly symmetrical configurations, the boundary conditions for the current require large electric fields to be set up, causing extra dissipation to rise as B2. To make a comparison of this model with the experimental results of Schaefer and Marcus [1] we used the exact formula withb 0.90,~~,randq1 + 0.025 (~~r). The latter assumption of linear magnetoresistance is consistent with a large number of independent studies of potassium. The results are shown in fig. 1. Comparison with figs. 1 and 5 of ref. [I] shows excellent 343
PHYSICS LETTERS
Volume 39A, number 4
22 May 1972
somewhere on the opposite side. This would be caused by the solid-liquid interface sweeping through the centre of the sample later than through most of the surface
which is more effectively cooled, so that quite a narrow conical region of molten material at the warmest side of the sample would colapse inward to make up for the volume contraction at the centre. The position of this “dimple” could very well be controlled by the fastest
crystal growing speed of the various crystalographic planes. We conclude that the experimental results of Schaefer
60
500
e
90’
Fig. 1. The induced torque in arbitrary units for a sphere (b 1) and a flattened ellipsoid of rotation (b < 1) as a function of magnetic field and orientation. Linear magneto-resistance has been assumed, and angle ~ chosen equal to ir. The torque is an even function of 0 about the points 0 = 0” and 0 = 90°. For K with a resistance ratio of 2850, ~~CT = 20 at B = 1 tesla. proportional to (1
+
0.025 ~
and Marcus [1] are not in disagreement with the measurements [5] on large (22 mm diameter, r~ 0) spheres which showed no torque anisotropy. The magnitude of the linear magnetoresistance appears to be very similar in both cases.
The author wishes to acknowledge the hospitality of the group of Prof. P. Wyder at the University of Nijmegen.
References agreement, achieved essentially with only one adjustable parameter, i.e. b = 0.90, since q is determined exactly by the slopes of the curves for 0 = 0°and 90°.
It therefore appears that a large number of the samples used in ref. [1] had a caliper along one of the (110) axis smaller by about 10%. One possible explanation would be that as the droplet of molten potassium from which the sample originated solidified gradually from one point on the surface, a “dimple” developea
344
[1] J.A. Schaefer and J.A. Marcus, Phys. Rev. Letters 27 (1971) 935.
[2] A.W. Overhauser, Phys. Rev. Letters 27 (1971) 938 [3] J.S. Lass and A.B. Pippard, I. Phys. E: J. Sci. Instrum., 3(1970)136. [4] P.B. Visscher and L.M. Falicov, Phys. Rev. B2 (1970) 1518. [5] J.S. Lass, J. Phys. C 3 (1970) 1926.
PHYSICS LETTERS
22 May 1972
THE NON-SATURATING ANISOTROPY OF INDUCED TORQUE IN POTASSIUM J.S. LASS Fysisch Laboratorium, Katholieke Universiteit, Ni/megen, TheNetherlands
Received 24 March 1972
—
The anisotropy of induced torque in potassium observed by Schaefer and Marcus can be explained as a result 10% departure of sample shape from sphericity.
Recent measurements by Schaefer and Marcus [1] of induced torque on small single crystals of potassium showed remarkable twofold anisotropy of the torque for all magnetic fiulds. At higher fields the torque developed an increasingly larger component with fourfold symmetry. These anisotropies have lead Overhauser [2] to conclude that the Fermi surface of potassium is neither simply connected nor of cubic symmetry. We wish to show that a departure with twofold symmetry of the sample shape from a sphere is sufficient to produce the observed anomalies, without assuming any anisotropy of the resistivity or of the Fermi surface. The magnitude and crystalographic orientation dependence of the measured torque suggest that the distorted shape developed during solidification of the sample from a molten droplet. Calculations of the torque induced in an ellipsoidshaped sample haye been made earlier [3] , but a general expression has not been given sofar. It can be easily derived by solving Maxwell’s equation V x ~ ‘j= —B/c with a boundary condition at the surface of the ellipsoid jn = 0. Proceeding in analogy with Visscher and Falicov [4] one can show that the torque N= —(det~)[Bx 2irR5 (eyeB)] (I) 1 Sc2 where —
~[Tr(~~)~’
~+]..1
~ is the resistivity tensor, = StS, and S is a linear transformation of coordinates such that r~ = ~, where re lies on the surface of the ellipsoid and r’ 5 on the sur-
ofa
face of a sphere with a radius R. The symmetric part of tensor ~ leads to the torque component Nn in the direction of the axis of rotation. Using eq. (1) we have considered a flattened ellipsoid of rotation with its short semiaxis (bR, b ~ 1) tilted through an angle ~pfrom the axis of rotation fi in a plane which is rotated about fi by an angle 0 measured from the plane perpendicular to the magnetic field (B .L fi). As a measure of the departure2)sin2p. from rotational We symmetry we have denoted i~= (1 — btensor with nonfurther assumed a magnetoresistivity zero elements ~ = = =p 0q and = —‘~~= p0w~r(where q is a dimensionless coefficient, c~is the cyclotron frequency and r is the relaxation time). The explicit formula for N~is lengthy, but one Important feature appears clearly. While at low fields (~~r 20,2q) at oscilatory terms are present proportional to ~cos fairly high fields (s~~r>> 2q) non-saturating terms proportional to n2sin20cos20(w~r/2q)2dominate the result. The physical picture behind these non-saturating effects of sample shape is analogous to that of open orbits in transverse magnetoresistance. With the exception of highly symmetrical configurations, the boundary conditions for the current require large electric fields to be set up, causing extra dissipation to rise as B2. To make a comparison of this model with the experimental results of Schaefer and Marcus [1] we used the exact formula withb 0.90,~~,randq1 + 0.025 (~~r). The latter assumption of linear magnetoresistance is consistent with a large number of independent studies of potassium. The results are shown in fig. 1. Comparison with figs. 1 and 5 of ref. [I] shows excellent 343
PHYSICS LETTERS
Volume 39A, number 4
22 May 1972
somewhere on the opposite side. This would be caused by the solid-liquid interface sweeping through the centre of the sample later than through most of the surface
which is more effectively cooled, so that quite a narrow conical region of molten material at the warmest side of the sample would colapse inward to make up for the volume contraction at the centre. The position of this “dimple” could very well be controlled by the fastest
crystal growing speed of the various crystalographic planes. We conclude that the experimental results of Schaefer
60
500
e
90’
Fig. 1. The induced torque in arbitrary units for a sphere (b 1) and a flattened ellipsoid of rotation (b < 1) as a function of magnetic field and orientation. Linear magneto-resistance has been assumed, and angle ~ chosen equal to ir. The torque is an even function of 0 about the points 0 = 0” and 0 = 90°. For K with a resistance ratio of 2850, ~~CT = 20 at B = 1 tesla. proportional to (1
+
0.025 ~
and Marcus [1] are not in disagreement with the measurements [5] on large (22 mm diameter, r~ 0) spheres which showed no torque anisotropy. The magnitude of the linear magnetoresistance appears to be very similar in both cases.
The author wishes to acknowledge the hospitality of the group of Prof. P. Wyder at the University of Nijmegen.
References agreement, achieved essentially with only one adjustable parameter, i.e. b = 0.90, since q is determined exactly by the slopes of the curves for 0 = 0°and 90°.
It therefore appears that a large number of the samples used in ref. [1] had a caliper along one of the (110) axis smaller by about 10%. One possible explanation would be that as the droplet of molten potassium from which the sample originated solidified gradually from one point on the surface, a “dimple” developea
344
[1] J.A. Schaefer and J.A. Marcus, Phys. Rev. Letters 27 (1971) 935.
[2] A.W. Overhauser, Phys. Rev. Letters 27 (1971) 938 [3] J.S. Lass and A.B. Pippard, I. Phys. E: J. Sci. Instrum., 3(1970)136. [4] P.B. Visscher and L.M. Falicov, Phys. Rev. B2 (1970) 1518. [5] J.S. Lass, J. Phys. C 3 (1970) 1926.