High-precision magnetization, $M(T,H),$ data have been taken along the c axis (easy direction of magnetization) of a high-purity Gd single crystal in the critical region near the ferromagnetic-paramagnetic phase transition. Elaborate data analyses demonstrate that the single power laws, by themselves, do not adequately describe the observed field dependence of M at the Curie point $T_C,$ ${M(T}_C,H),$ and the temperature variations of spontaneous magnetization, $M(T,0\mathrm{}),$ and initial susceptibility, $\chi \mathrm{}\left(T\right),$ in the asymptotic critical region $|\epsilon |=|(T-T_C{)/T}_C|<~2\times {10}^{-3},$ but do so only when the multiplicative logarithmic corrections (LC), predicted by the renormalization group (RG) calculations for dipolar Ising (spin dimensionality $n=1)$ spin systems at the upper marginal space dimension $d^{*}=3,$ are taken into account. Such data analyses also permit the first accurate determination of LC exponents ${(x}^{\prime },x),$ the asymptotic critical exponents $\beta ,$ $\gamma ,$ and $\delta ,$ and critical amplitudes $\mathrm{B̂},$ $\hat{\Gamma },$ and $\mathrm{D̂}$ for $M(T,0\mathrm{}),$ $\chi \mathrm{}\left(T\right),$ and ${M(T}_C,H).\mathrm{}$ The exponents $x^{\prime },$ x, $\beta ,$ $\gamma ,$ and $\delta ,$ as well as the universal amplitude ratio $R_{\chi }=\mathrm{D̂}{B}^{\delta -1}\hat{\Gamma }$ possess the same (within the uncertainty limits) values as those yielded by the RG calculations for a $d=3\mathrm{}$ uniaxial dipolar ferromagnet. Moreover, the presently determined values of $\beta ,$ $\gamma ,$ and $\delta ,$ together with the reported value of the specific heat critical exponent $\alpha ,$ obey the scaling relations $\beta +\gamma =\beta \delta$ and $\alpha +2\mathrm{}\beta +\gamma =2\mathrm{}$ accurately. By establishing that gadolinium belongs to the $d=3,$ $n=1\mathrm{}$ dipolar static universality class, the present results resolve the long-standing controversy surrounding the nature of the asymptotic critical behavior of Gd.

• Received 6 April 1999

DOI:https://doi.org/10.1103/PhysRevB.60.12166