In the lattice, there are $N$ nearly independent ions, each ion has a spin of $1/2$ and a magnetic moment of $\mu_{0}$. The system is in a uniform external field $B$ and at temperature $T$. Calculate:
(1) the partition function;
(2) the entropy $\sigma$;
(3) the average energy $E$;
(4) the fluctuation of the magnetic moment $\Delta M=\sqrt{( M-\overline{M} )^{2}}$.

Options:
A. $$
( \mathrm{e}^{h} + \mathrm{e}^{-h} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{c o s h}(h) + \mathrm{s i n h}(h) \big) - \theta \big]
$$
$$
- N \mu_{0} B \mathrm{c o s h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{\mu_{2}} {\operatorname{c o s h} \left( {\frac{\mu_{2} B} {k T}} \right)}
$$
B. $$
( \mathrm{e}^{c} + \mathrm{e}^{-c} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{e}^{c} + \mathrm{e}^{-c} \big) - \gamma \mathrm{s i n h} \gamma \big]
$$
$$
- N \mu_{0} B \mathrm{c o s h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{\mu_{1}} {\operatorname{s i n h} \left( {\frac{\mu_{1} B} {k T}} \right)}
$$
C. $$
( \mathrm{e}^{e} + \mathrm{e}^{-e} )^{N}
$$
$$
N k \bigg[ \mathrm{l n} \bigg( \mathrm{e}^{e} + \mathrm{e}^{-e} \bigg) - \chi \mathrm{t a n h} \chi \bigg]
$$
$$
- N \mu_{1} B \mathrm{s i n h} \Bigl( \frac{\mu_{1} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{2 \mu_{0}} {\operatorname{c o s h} \left( \frac{2 \mu_{0} B} {k T} \right)}
$$
D. $$
( \mathrm{e}^{g} + \mathrm{e}^{-g} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{s i n h}(g) + \mathrm{c o s h}(g) \big) - \zeta \big]
$$
$$
- N \mu_{0} B \mathrm{s i n h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{\mu_{2}} {\operatorname{s i n h} \left( {\frac{\mu_{2} B} {k T}} \right)}
$$
E. $$
( \mathrm{e}^{f} + \mathrm{e}^{-f} )^{N}
$$
$$
N k \bigg[ \mathrm{l n} \bigg( \mathrm{e}^{f} + \mathrm{e}^{-f} \bigg) - \eta \mathrm{s i n h} \eta \bigg]
$$
$$
N \mu_{0} B \mathrm{c o s h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{2 \mu_{1}} {\operatorname{t a n h} \left( \frac{2 \mu_{1} B} {k T} \right)}
$$
F. $$
( \mathrm{e}^{j} + \mathrm{e}^{-j} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{c o s h}(j) + \mathrm{e}^{j} \big) - \lambda \big]
$$
$$
N \mu_{1} B \mathrm{t a n h} \Bigl( \frac{\mu_{1} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{3 \mu_{1}} {\operatorname{c o s h} \left( \frac{3 \mu_{1} B} {k T} \right)}
$$
G. $$
( \mathrm{e}^{i} + \mathrm{e}^{-i} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{s i n h}(i) + \mathrm{e}^{-i} \big) - \kappa \big]
$$
$$
- N \mu_{1} B \Bigl( \frac{\mu_{1} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{3 \mu_{0}} {\operatorname{t a n h} \left( \frac{3 \mu_{0} B} {k T} \right)}
$$
H. $$
( \mathrm{e}^{b} + \mathrm{e}^{-b} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{e}^{b} + \mathrm{e}^{-b} \big) - \beta \mathrm{t a n h} \beta \big]
$$
$$
- N \mu_{0} B \mathrm{s i n h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{\mu_{0}} {\operatorname{t a n h} \left( {\frac{\mu_{0} B} {k T}} \right)}
$$
I. $$
( \mathrm{e}^{\alpha}+\mathrm{e}^{-\alpha} )^{N}
$$
$$
N k [ \mathrm{l n} ( \mathrm{e}^{\alpha}+\mathrm{e}^{-\alpha} )-\alpha\mathrm{t a n h} \alpha]
$$
$$
- \ N \mu_{0} B \mathrm{t a n h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
{\sqrt{N}} {\frac{\mu_{0}} {\operatorname{c o s h} \left( {\frac{\mu_{0} B} {k T}} \right)}}
$$
J. $$
( \mathrm{e}^{d} + \mathrm{e}^{-d} )^{N}
$$
$$
N k \big[ \mathrm{l n} \big( \mathrm{e}^{d} + \mathrm{e}^{-d} \big) - \delta \mathrm{c o s h} \delta \big]
$$
$$
N \mu_{0} B \mathrm{t a n h} \Bigl( \frac{\mu_{0} B} {k T} \Bigr)
$$
$$
\sqrt{N} \frac{\mu_{0}} {\operatorname{s i n h} \left( {\frac{\mu_{0} B} {k T}} \right)}
$$
Answer: I