Probability distribution function of dipolar field in two-dimensional spin ensemble

Abstract

We theoretically determine the probability distribution function of the net field of the random planar structure of dipoles which represent polarized particles. At small surface concentrations c of the point dipoles this distribution is expressed in terms of special functions. At the surface concentrations of the dipoles as high as 0.6 the dipolar field obey the Gaussian law. To obtain the distribution function within transitional region c < 0.6, we propose the method based on the cumulant expansion. We calculate the parameters of the distributions for some specific configurations of the dipoles. The distribution functions of the ordered ensembles of the dipoles at the low and moderate surface concentrations have asymmetric shape with respect to distribution medians. The distribution functions allow to calculate various physical parameters of two-dimensional interacting nanoparticle ensembles.The final publication is available at www.epj.org,

Effect of dipolar interactions on optical nonlinearity of two-dimensional nanocomposites

“…As shown in the Ref. [15] for such a system, W (E) becomes the Gaussian distribution at high nanoparticle surface concentrations c = Ns/S > 0.6, where N is the number of the particles having been excited, S is the area of the sample, s is the area of the particle. Hence, we cannot apply directly the approach of Ref.…”

Impact of interparticle dipole–dipole interactions on optical nonlinearity of nanocomposites

“…( 5) is done under the assumption that the number of the dipoles with one direction is balanced by dipoles with opposite alignment owing to the random nature of the excitement. Thus, this expansion has only even-numbered terms [14], the odd-numbered λ n are zero. Then, following Ref.…”

“…This approach is employed for theoretical treatment of the dipole-dipole interactions in the resonance methods [13], dilute magnetic systems [17] and ferroelectric dipole glasses [18]. In a fashion similar to the two-dimensional model [14], we express C(ρ) in powers of ρ (the negative cumulant expansion). Thus, the distribution function W (E) after the inverse Fourier transform can be formulated in integral form,…”

“…[12], previously obtained distribution functions of the net dipolar field projection E [13] was utilized. For volume concentrations (volume fractions) of the excited nanoparticles c > 0.1 the Gaussian distribution was used as distribution functions of the random field E. In this work, we employ for bulk samples the negative cumulant expansion proposed for the two-dimensional distribution of dipoles [14,15]. This will allow us to obtain the third-order selfinduced optical susceptibility for volume fractions c 0.1 which frequently occur in experiments.…”

“…In this work, we employ for bulk samples the negative cumulant expansion proposed for the two-dimensional distribution of dipoles [14,15]. This will allow us to obtain the third-order selfinduced optical susceptibility for volume fractions c 0.1 which frequently occur in experiments.…”