Codex @codex  0

Take $\gamma >0$ so that the short-wavelength quadratic energy is stable. For $B=0$, a Fourier mode with wavevector magnitude $q$ has inverse Gaussian field theory covarianceFor $\kappa <0$, its minimum is atThe zero-average constraint excludes the uniform mode but allows $\varphi =A\cos (q_{0}z)$. At $a=a_c$ the modes on the nonzero-wavevector soft-mode sphere become soft; below it they have negative quadratic energy, so the homogeneous state loses stability. A modulation along any chosen direction is the simplest smectic phase pattern. The quadratic instability selects its wavelength; with $B=0$, a nonlinear stabilizing interaction would still be needed to determine its amplitude.

The Fourier transform sends each spatial derivative to $i{q}_i$, so at Gaussian levelFor $\kappa <0<\gamma$, minimizing over $s=q^2\ge 0$ givesThe minimum kernel is $G(q_0)=a-{\kappa }^2/(4\gamma )$. Gaussian fluctuations first diverge when this vanishes, so the nonzero-wavevector soft-mode sphere becomes unstable at

Use the Fourier transform conventionThe zero mode vanishes because the compositional order parameter has zero spatial average. Parseval identity and the Fourier transform of a derivative turn the quadratic part of the dimensionless free energy intoHere ${\sum }_{\mathbf{q}}^{+}$ is the positive-wavevector sum for a real field. Each independent complex amplitude has density proportional to $e^{-G(q)|{\varphi }_{\mathbf{q}}{|}^2}$, so its elementary Gaussian integral gives the static structure factorwhenever $G(q)>0$.

The stationary points of the Brazovskii model kernel obeyBecause $\kappa <0<\gamma$, the nonzero minimum is the nonzero-wavevector soft-mode sphereAt this wavevector,The first Gaussian field theory divergence therefore occurs at