Full-text resources of PSJD and other databases are now available in the new Library of Science.
Visit https://bibliotekanauki.pl


Preferences help
enabled [disable] Abstract
Number of results
2016 | 129 | 6 | RK.129.6.1-1-RK.129.6.1-3

Article title

Rado-Weertman Boundary Equation Revisited in Terms of the Free-Energy Density of a Thin Film



Title variants

Languages of publication



Historically, the first boundary conditions to be formulated and used in the theory of ferromagnetic thin films, the Rado-Weertman (RW) conditions, have a general advantage of being a simple differential equation, 2A_{ex} ∂m/∂n - K_{surf}m = 0. A key role in this equation is played by the phenomenological quantity K_{surf} known as the surface anisotropy energy density; A_{ex} denotes the exchange stiffness constant, and m is the amplitude of the transverse component of dynamic magnetization. In the present paper we use a microscopic theory to demonstrate that the surface anisotropy energy density of a thin film is directly related with its free-energy density, a fact not observed in the literature to date. Using two local free-energy densities F^{surf} and F^{bulk}, defined separately on the surface and in the bulk, respectively, we prove that K_{surf} = d(F^{surf} - F^{bulk}), where d is the lattice constant. The above equation allows to determine the explicit configuration dependence of the surface anisotropy constant K_{surf} on the direction cosines of the magnetization vector for any system with a known formula for the free energy. On the basis of this general formula the physical boundary conditions to be fulfilled for a fundamental uniform mode and surface modes to occur in a thin film are formulated as simple relations between the surface and bulk free-energy densities that apply under conditions of occurrence of specific modes.



  • Surface Physics Division, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland


  • [1] G.T. Rado, J.R. Weertman, J. Phys. Chem. Solids 11, 315 (1959), doi: 10.1016/0022-3697(59)90233-1
  • [2] C. Vittoria, Microwave Properties of Magnetic Films, World Sci., Singapore 1993, doi: 10.1142/9789814354387
  • [3] H. Puszkarski, Acta Phys. Pol. A 38, 217, 899 (1970)
  • [4] H. Puszkarski, Progr. Surf. Sci. 9, 191 (1979), doi: 10.1016/0079-6816(79)90013-3
  • [5] H. Puszkarski, P. Tomczak, Sci. Rep. 4, 6135 (2014), doi: 10.1038/srep06135
  • [6] H.T. Diep, Theory of Magnetism - Application to Surface Physics, World Sci., Singapore 2014, doi: 10.1142/8994
  • [7] L. Wojtczak, Magnetic Thin Films, Wydawnictwo Uniwersytetu Łódzkiego, Łódź 2009 (in Polish)

Document Type

Publication order reference

YADDA identifier

JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.