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Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in d Dimensions

1973; American Physical Society; Volume: 8; Issue: 7 Linguagem: Inglês

10.1103/physrevb.8.3363

0556-2805

Amnon Aharony,

Magnetic properties of thin films

The exact renormalization-group approach is used to study the critical behavior for $T>{T}_{c}$, $H=0$ of a uniaxial ferromagnetic (or ferroelectric) system in $d$ dimensions, with exchange and dipolar interactions between the (single-component) spins. Normal Ising-like behavior is retained for $t=\frac{T}{{T}_{c}}\ensuremath{-}1\ensuremath{\gg}\stackrel{^}{g}=\frac{{(g{\ensuremath{\mu}}_{B})}^{2}}{J{a}^{d}}$, where $J$ is the exchange parameter, $g{\ensuremath{\mu}}_{B}$ is the magnetic moment per spin, and $a$ is the lattice spacing. Crossover to a characteristic dipolar behavior occurs when ${t}^{\ensuremath{\varphi}}\ensuremath{\approx}\stackrel{^}{g}$, where $\ensuremath{\varphi}=1+\frac{\ensuremath{\epsilon}}{6}$ (to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$). For $t\ensuremath{\ll}\stackrel{^}{g}$, the leading temperature singularity in the Fourier transform of the spin-spin correlation function $\ensuremath{\Gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ becomes ${\ensuremath{\xi}}^{2}\ifmmode\times\else\texttimes\fi{}{[1+{(\ensuremath{\xi}q)}^{2}\ensuremath{-}{h}_{0}{(\ensuremath{\xi}{q}^{z})}^{2}+{g}_{0}{(\frac{{q}^{z}}{q})}^{2}]}^{\ensuremath{-}1}$, where ${h}_{0}$ and ${g}_{0}$ are of order $\stackrel{^}{g}$ and $\ensuremath{\xi}(t)$ varies as ${t}^{\ensuremath{-}\frac{1}{2}}$ for $d>3$, as ${t}^{\ensuremath{-}\frac{1}{2}}{|\mathrm{ln}t|}^{\frac{1}{6}}$ for $d=3$, and as ${t}^{\ensuremath{-}\ensuremath{\nu}}$ with $\frac{1}{2\ensuremath{\nu}}=1\ensuremath{-}\frac{(3\ensuremath{-}d)}{6}+O({(3\ensuremath{-}d)}^{2})$ for $d<3$. The susceptibility displays the expected demagnetization effects, namely, $({\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{-}{g}_{0}\mathcal{D})\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}2}$. The experimental situation is mentioned briefly.