Magnetostriction and thermal expansion of HoAl$_3$(BO$_3$)$_4$

The magnetostriction and thermal expansion of rare-earth aluminoborate HoAl$_3$(BO$_3$)$_4$ have been studied theoretically. The calculated field and temperature dependences of the multipole moments of the Ho$^{3+}$ ion in HoAl$_3$(BO$_3$)$_4$ made it possible to describe the known experimental data and to predict possible anomalies of thermal expansion. It has been shown that, for the direction of the field $\mathbf{B}\parallel\mathbf{c}$, the nonmonotonic character of magnetostriction along the axis a is determined by the multipole moments, the main of which is $\beta_J\langle O^0_4\rangle$. For $\mathbf{B}\parallel\mathbf{a}$ and $\mathbf{B}\parallel\mathbf{b}$, the maximum moments are $\beta_J\langle O^2_4\rangle$ and $\alpha_J\langle O_2^2\rangle$; their variation with the field and temperature explain well the form of magnetostriction. It has been established that the greater value of magnetostriction $\Delta a/a$ for $\mathbf{B}\parallel\mathbf{b}$ than for $\mathbf{B}\parallel\mathbf{a}$ and the greater value of magnetostriction for the field in the basal plane than for $\mathbf{B}\parallel\mathbf{c}$ are caused by greater variations in the field of actual multipole moments.