Visualization of Time-Reversal Symmetry in Magnetic Point Groups
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I.
INTRODUCTION
TIME reversal symmetry refers to the replacement, in an equation, of the time variable t by t. If the timereversed equation is identical to the original equation, then the equation is said to have time-reversal symmetry. Consider as an example the (classical) equation of motion of a charged particle with charge q and mass m in a stationary electric field E(r): d2 r m 2 ¼ qEðrÞ dt
qij ðBÞ ¼ qji ðBÞ ½1
where r is the position variable of the particle. If we replace t by t, then this equation remains unchanged, so that motion in a stationary electric field has timereversal symmetry. This symmetry lies at the basis of Onsager’s principle of microscopic reversibility, which states that, if the velocities of particles in a physical system that is slightly perturbed from equilibrium are all simultaneously reversed (i.e., t fi t), then all particles will retrace their former trajectories in the opposite sense.[1] It can be shown that, as a direct consequence of this principle, the principal transport properties of a material must be described by symmetric second rank tensors, Kij (thermal conductivity), rij (electrical conductivity), and Dij (diffusivity). The situation becomes more complex when magnetic fields are involved. Consider the equation of motion for the same charged particle in a stationary magnetic field B(r): m
d2 r dr ¼ q BðrÞ dt2 dt
½2
where the force on the right-hand side is the velocitydependent Lorentz force. If we replace t by t, then the right-hand side appears to change sign, so that the MARC DE GRAEF, Professor, is with the Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. Contact e-mail: [email protected] Manuscript submitted December 1, 2009. Article published online February 26, 2010 METALLURGICAL AND MATERIALS TRANSACTIONS A
equation does not have time-reversal symmetry. To restore the validity of the principle of microscopic reversibility, one must also change the sign of the magnetic field, i.e., replace B by B. This has profound consequences on galvanomagnetic and thermomagnetic material properties. For instance, in the presence of a magnetic field, a material’s resistivity tensor, qij, must satisfy the following symmetry constraint: ½3
This relation states that the effect of a magnetic field on the element qxy of the resistivity tensor is the opposite of its effect on the element qyx; in other words, the resistivity tensor is no longer a symmetric tensor. A similar relation holds for the thermal conductivity tensor, Kij, and the diffusivity tensor, Dij. For magnetic properties, the Peltier and Seebeck tensors are related to each other by Kelvin’s relation: pij(B) = T Rji(B).[2] Material tensors for equilibrium properties must have at least the symmetry of the underlying crystal structure; this is commonly known as Neumann’s principle.[3] The standard crystallographic symmetries are described by the 32 point groups, i.e., those combinations of rotations, mirrors, and the inversion, that are compatible with the 14 Bravais lat
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