We have carried out a high-resolution x-ray study of the mass-density fluctuations and a light-scattering study of the bend-mode-director fluctuations associated with the nematic-smectic-$A$ phase transition in 4-cyano-4′-octylbiphenyl (8CB). This phase transition is found to be nearly second order with a first-order temperature jump of less than 10 mK. The peak scattering intensity, $\sigma \left({\overrightarrow{\mathrm{q}}}_0\right)$, with ${\overrightarrow{\mathrm{q}}}_0=(0, 0, \frac{2\pi }{d})$, where $d$ is the layer spacing, and the longitudinal correlation length, ${\xi }_{\parallel }$, exhibit single power-law divergences for $5\times {10}^{-5\mathrm{}}<|1-\frac{T}{T_c}|<\mathrm{}2\mathrm{}\times {10}^{-2\mathrm{}}$ with critical exponents of $\gamma =1.26\mathrm{}\pm 0.06$ and ${\nu }_{\parallel }=0.67\mathrm{}\pm 0.02$, respectively. These exponents agree within the errors with the predicted $d=3\mathrm{}$, $n=2\mathrm{}$ values appropriate to de Gennes' superconductor analogue model. The ratio of the longitudinal to transverse correlation lengths, $\frac{{\xi }_{\parallel }}{{\xi }_{\perp }}$, increases gradually with decreasing reduced temperature, changing by a factor of 2.6 over the above temperature range. This corresponds to an effective exponent difference $\sqrt{\parallel }-\surd \mathit{†}=0.16\mathrm{}$. The bend mode elastic constant $K_3$, which is predicted to have a divergent contribution ${\left(\frac{k_{B}T{q}_o^2}{24\pi }\right)}^2{\xi }_{\parallel }$, exhibits single power law behavior for ${10}^{-4\mathrm{}}\le 1-\frac{T}{T_c}<{10}^{-2\mathrm{}}$, with ${\nu }_{\parallel }=0.62\mathrm{}\pm 0.03$ in reasonable agreement with the x-ray results. The predicted amplitude is, however, wrong by a factor of four. Measurements of $B$, the layer compressibility coefficient in the smectic-$A$ phase, yield $\frac{B}{K_1}\sim {(T_c-T)}^{\phi }$ with $\phi =0.26\mathrm{}\pm 0.06$, in agreement with previous experiments in N-$p$-cyanobenzylidene-$p$-octyloxyanilene (CBOOA) and octyloxycyanobenzylidene (80CB), but in marked disagreement with the superconducting analogue model which predicts $\phi ={\nu }_{\parallel }^1=0.67\mathrm{}$.

• Received 12 June 1978

DOI:https://doi.org/10.1103/PhysRevB.19.1657