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| -rw-r--r-- | main.tex | 21 |
1 files changed, 14 insertions, 7 deletions
@@ -271,8 +271,8 @@ be demonstrative of how the stiffness is calculated and prove useful in expressing the functional form of the stiffness. Then we will compute the strain stiffness using some tricks from functional calculus. -The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ to a field -linearly coupled to it is given by +The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ to a +thermodynamically conjugate field is given by \begin{equation} \begin{aligned} &\chi^\recip(x,x') @@ -305,11 +305,18 @@ Near the unordered--modulated transition this yields \label{eq:susceptibility} \end{equation} with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and -$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and $\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths. We must emphasize that -this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ -cannot couple linearly to a uniform magnetic field. The object defined in -\eqref{eq:sus_def} is most readily interpreted as proportional to the two-point -connected correlation function +$\xi_\parallel=(|\Delta\tilde +r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is +the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and +$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths. +Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde +r|)^{-1}$ does not diverge at the unordered--modulated transition. Though it +anticipates a transition with Curie--Weiss-like divergence at $\Delta\tilde +r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$. We must +emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ +or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field. The object +defined in \eqref{eq:sus_def} is most readily interpreted as proportional to +the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$. The strain stiffness is given in a similar way to the inverse susceptibility: we |