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\sect{Ising model of adsorption sites (Exam2003 Q3)}
A large number ${M>>1}$ of adsorption sites are ordered along the
length of a ring.
Between every two adsorption sites a spin ${\sigma_{i}=\pm 1}$ is
located. The ring is soaked in a gas temperature ${T}$ and it's
chemical potential. At most, one particle gas can be adsorbed to a
given site ${n_{i}=0,1}$.
The adsorption energy is ${\varepsilon >0}$ if the two adjacent
spins are in the same direction.
The adsorption energy is ${-\varepsilon}$ if the adjacent spins are
in the opposite direction.
Assume positive adsorption energy that gives "priority" to the
promagnetical order.
Write the expression for the energy ${E\left[\sigma _{i},
n_{i}\right]}$ of a given configuration.
Write the transfer matrix ${T}$ that is shown in the calculation of
the grand canonical distribution function ${Z\left(\beta,
\mu\right)}$ of the system.
(Guideline: carry out the sum over the occupation options. Define
${T}$ for the remaining sum over she spins).
Find the self values of the transfer matrix.
Write expressions for the basic function ${F\left(T,\mu\right)}$ and
for the adsorbed particles ${N=\Sigma\langle n_{i}\rangle}$.
Write an expression for the correlation length ${\xi}$ that
characterize the arrangement of the spins in the system.