Resonance Damping in Ferromagnets and Ferroelectrics arXiv:1001.2845v1 [cond-mat.other] 16 Jan 2010 The phenomenological equations of motion for the relaxation of ordered phases of magnetized and polarized crystal phases can be developed in close analogy with one another. For the case of magnetized systems, the driving magnetic field intensity toward relaxation was developed by Gilbert. For the case of polarized systems, the driving electric field intensity toward relaxation was developed by Khalatnikov. The transport times for relaxation into thermal equilibrium can be attributed to viscous sound wave damping via magnetostriction for the magnetic case and electrostriction for the polarization case. PACS numbers: 76.50.+g, 75.30.Sg I. INTRODUCTION It has long been of interest to understand the close analogies between ordered electric polarized systems, e.g. ferroelectricity , and ordered magnetic systems, e.g. fer- romagnetism . At the microscopic level, the source of such ordering must depend on the nature of the electronic en- ergy spectra. The relaxation mechanism into thermal equilibrium state must be described by local electric field fluctuations for the electric polarization case and by mag- netic intensity fluctuations for the magnetization case; Specifically, the field fluctuations for each case tric relaxation of an ordered polarization is the Landau- Khalatinikov-Tani equation $^{15–17}$. This equation can be simply modeled 18–21 with effective electrical circuits $^{22–25}$. Information memory applications 26–29 of such polarized system are of considerable recent interest $^{30–32}$. The unification of the magnetic Gilbert-Landau- Lifshitz equations and the electric Landau-Khalatnikov- Tani equations via the relaxation time tensor depends on the notion of a nonequilibrium driving field . For the magnetic case, the driving magnetic intensity H$_{d}$ deter- mines the relaxation of the magnetization via the torque equation A. Widom Physics Department, Northeastern University, Boston, MA USA S. Sivasubramanian NSF Nanoscale Science & Engineering Center for High-rate Nanomanufacturing, Northeastern University, Boston MA USA C. Vittoria and S. Yoon Department of Electrical and Computer Engineering, Northeastern University, Boston, MA USA Y.N. Srivastava Physics Department and & INFN, University of Perugia, Perugia IT determine the relaxation time tensor for both cases via the fluctuation-dissipation formula 1–4 We have unified the theories of relaxation in ordered po- larized systems and ordered magnetized systems via the Kubo transport time tensor in Eqs.(1) and (2). The transport describing the relaxation of or- dered magnetization is the Landau-Lifshitz-Gilbert equation $^{5–7}$. This equation has been of considerable recent interest 8–10 in describing ordered magnetic reso- nance phenomena $^{11–14}$. The equation describing the elec- wherein γ is the gyromagnetic ratio. For the electric case, the driving electric field E$_{d}$ determines the relaxation of the polarization via the equation of motion for an ion of charge ze The unification of both forms of relaxation lies in the close analogy between the magnetic driving intensity H$_{d}$ and the electric driving field E$_{d}$ . In Sec.II the thermodynamics of ordered magnetized and polarized systems is reviewed. The notions of mag- netostriction and electrostriction are given a precise ther- modynamic definition. In Sec.III, the phenomenology of the relaxation equations are presented. The magnetic driving intensity H$_{d}$ and the electric driving field E$_{d}$ are defined in terms of the relaxation time tensor Eq.(2). In Sec.IV, we introduce the crystal viscosity tensor. From a Kubo formula viewpoint, the stress fluctuation correlax- G pol ij ( r , r $^{′}$, t ) = 1 ¯ h ∫ β 0 〈 Δ E$_{j}$ ( r $^{′}$, − iλ )Δ E$_{i}$ ( r , t ) 〉 dλ, G mag ij ( r , r $^{′}$, t ) = 1 ¯ h β 0 〈 Δ H$_{j}$ ( r $^{′}$, − iλ )Δ H$_{i}$ ( r , t ) 〉 dλ, wherein β = ¯ h k$_{B}$T , (1) τ$_{ij}$ = ∫ ∞ 0 lim V →∞ [ 1 V ∫ V ∫ V G$_{ij}$ ( r , r $^{′}$, t ) d $^{3}$r d $^{3}$r ′ ] dt. (2) ˙ M = γ M × H$_{d}$ , (3) m ¨ r = ze E$_{d}$ . (4)