Free energy of dipolar hard spheres: The virial expansion under the presence of an external magnetic field

A method for calculation of the free energy of dipolar hard spheres under the presence of an applied magnetic field is presented. The method is based on the virial expansion in terms of density as well as the dipolar coupling constant λ, and it uses diagram technique. The formulas and the diagrams, needed to calculate the second B2 and third B3 virial coefficients, are derived up to the order of ∼λ3, and compared to the zero-field case. The formula for B2 is the same as in the zero-field case; the formula for B3, however, is different in an applied field, and a derivation is presented. This is a surprising result which is not emphasized in standard texts, but which has been noticed before in the virial expansion for flexible molecules (Caracciolo et al., 2006; Caracciolo et al., 2008). To verify the correctness of the obtained formulas, B2 and B3 were calculated within the accuracy of λ2, which were applied to initial magnetic susceptibility. The obtained expression fully coincides with the well-known theories (Morozov and Lebedev, 1990; Huke and Lücke, 2000; Ivanov and Kuznetsova, 2001), which used different methods to calculate the initial magnetic susceptibility.