A paramagnetic solid is composed of $N$ particles with a spin quantum number of $s$, with the particles fixed at the lattice points. Calculate:
(1) the partition function;
(2) the internal energy;
(3) the heat capacity;
(4) the entropy;
(5) the magnetic moment.

Options:
A. $$
\frac{\operatorname{s i n} [ 2 ( s+1 ) x ]} {\operatorname{s i n} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t} x-( s+1 ) \mathrm{c o t} \big[ 2 ( s+1 ) x \big] \rangle
$$
$$
N k x^{2} \langle\sec^{2} x-( s+1 )^{2} \sec^{2} [ 2 ( s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n s i n} [ 2 ( s+1 ) x ]-\mathrm{l n s i n} x+x \mathrm{c o t} x-( s+1 ) x \mathrm{c o t} [ 2 ( s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{( s+1 ) \mathrm{c o t} \big[ 2 ( s+1 ) x \big]-\mathrm{c o t} x \}
$$
B. $$
\frac{\operatorname{c o s h} [ ( 2 s+1 ) x ]} {\operatorname{c o s h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{t a n h} x-\mathrm{t a n h} [ ( 2 s+1 ) x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{s e c h}^{2} x-( 2 s+1 )^{2} \mathrm{s e c h}^{2} [ ( 2 s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n c o s h} [ ( 2 s+1 ) x ]-\mathrm{l n c o s h} x+x \mathrm{t a n h} x-( 2 s+1 ) x \mathrm{t a n h} [ ( 2 s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{( 2 s+1 ) \mathrm{t a n h} [ ( 2 s+1 ) x ]-\mathrm{t a n h} x \}
$$
C. $$
\frac{\operatorname{c o s h} [ 2 s x ]} {\operatorname{c o s h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{t a n h} x-( 2 s ) \mathrm{t a n h} [ 2 s x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{s e c h}^{2} x-( 2 s )^{2} \mathrm{s e c h}^{2} [ 2 s x ] \rangle
$$
$$
N k \{\mathrm{l n c o s h} [ 2 s x ]-\mathrm{l n c o s h} x+x \mathrm{t a n h} x-( 2 s ) x \mathrm{t a n h} [ 2 s x ] \}
$$
$$
N \mu_{\mathrm{B}} \{2 s \mathrm{t a n h} [ 2 s x ]-\mathrm{t a n h} x \}
$$
D. $$
\frac{\operatorname{s i n h} [ 3 ( s+1 ) x ]} {\operatorname{s i n h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t h} x-3 ( s+1 ) \mathrm{c o t h} [ 3 ( s+1 ) x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{c s c h}^{2} x-3^{2} ( s+1 )^{2} \mathrm{c s c h}^{2} [ 3 ( s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n s i n h} [ 3 ( s+1 ) x ]-\mathrm{l n s i n h} x+x \mathrm{c o t h} x-3 ( s+1 ) x \mathrm{c o t h} [ 3 ( s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{3 ( s+1 ) \mathrm{c o t h} \big[ 3 ( s+1 ) x \big]-\mathrm{c o t h} x \}
$$
E. $$
\frac{\operatorname{s i n h} [ 3 s x ]} {\operatorname{s i n h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t h} x-3 s \mathrm{c o t h} [ 3 s x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{c s c h}^{2} x-3^{2} s^{2} \mathrm{c s c h}^{2} [ 3 s x ] \rangle
$$
$$
N k \{\mathrm{l n s i n h} [ 3 s x ]-\mathrm{l n s i n h} x+x \mathrm{c o t h} x-3 s x \mathrm{c o t h} [ 3 s x ] \}
$$
$$
N \mu_{\mathrm{B}} \{3 s \mathrm{c o t h} [ 3 s x ]-\mathrm{c o t h} x \}
$$
F. $$
\frac{\operatorname{s i n} [ 2 s x ]} {\operatorname{s i n} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t} x-2 s \mathrm{c o t} [ 2 s x ] \rangle
$$
$$
N k x^{2} \langle\csc^{2} x-( 2 s )^{2} \csc^{2} [ 2 s x ] \rangle
$$
$$
N k \{\mathrm{l n s i n} [ 2 s x ]-\mathrm{l n s i n} x+x \mathrm{c o t} x-2 s x \mathrm{c o t} [ 2 s x ] \}
$$
$$
N \mu_{\mathrm{B}} \{2 s \mathrm{c o t} [ 2 s x ]-\mathrm{c o t} x \}
$$
G. $$
\frac{\operatorname{s i n h} [ ( 2 s+1 ) x ]} {\operatorname{s i n h} x}
$$
$$
N \mu_{\mathrm{B}} H \{ \mathrm{c o t h} x-( 2 s+1 ) \mathrm{c o t h} \big[ ( 2 s+1 ) x \big] \}
$$
$$
N k x^{2} \{ \mathrm{c s c h}^{2} x-( 2 s+1 )^{2} \mathrm{c s c h}^{2} [ ( 2 s+1 ) x ] \}
$$
$$
N k \{\mathrm{l n s i n h} [ ( 2 s+1 ) x ]-\mathrm{l n s i n h} x+x \mathrm{c o t h} x-( 2 s+1 ) x \mathrm{c o t h} [ ( 2 s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{( 2 s+1 ) \mathrm{c o t h} \big[ ( 2 s+1 ) x \big]-\mathrm{c o t h} x \}
$$
H. $$
\frac{\operatorname{c o s h} [ 2 ( s+1 ) x ]} {\operatorname{c o s h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{t a n h} x-( s+1 ) \mathrm{t a n h} [ 2 ( s+1 ) x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{s e c h}^{2} x-( s+1 )^{2} \mathrm{s e c h}^{2} [ 2 ( s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n c o s h} [ 2 ( s+1 ) x ]-\mathrm{l n c o s h} x+x \mathrm{t a n h} x-( s+1 ) x \mathrm{t a n h} [ 2 ( s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{( s+1 ) \mathrm{t a n h} [ 2 ( s+1 ) x ]-\mathrm{t a n h} x \}
$$
I. $$
\frac{\operatorname{s i n} [ ( 2 s+1 ) x ]} {\operatorname{s i n} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t} x-( 2 s+1 ) \mathrm{c o t} [ ( 2 s+1 ) x ] \rangle
$$
$$
N k x^{2} \langle\mathrm{c s c}^{2} x-( 2 s+1 )^{2} \mathrm{c s c}^{2} [ ( 2 s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n s i n} [ ( 2 s+1 ) x ]-\mathrm{l n s i n} x+x \mathrm{c o t} x-( 2 s+1 ) x \mathrm{c o t} [ ( 2 s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{\mathrm{c o t} x - ( 2 s+1 ) \mathrm{c o t} \big[ ( 2 s+1 ) x \big] \}
$$
J. $$
\frac{\operatorname{s i n h} [ ( s+1 ) x ]} {\operatorname{s i n h} x}
$$
$$
N \mu_{\mathrm{B}} H \langle\mathrm{c o t h} x-( s+1 ) \mathrm{c o t h} \big[ ( s+1 ) x \big] \rangle
$$
$$
N k x^{2} \langle\mathrm{c s c h}^{2} x-( s+1 )^{2} \mathrm{c s c h}^{2} [ ( s+1 ) x ] \rangle
$$
$$
N k \{\mathrm{l n s i n h} [ ( s+1 ) x ]-\mathrm{l n s i n h} x+x \mathrm{c o t h} x-( s+1 ) x \mathrm{c o t h} [ ( s+1 ) x ] \}
$$
$$
N \mu_{\mathrm{B}} \{( s+1 ) \mathrm{c o t h} \big[ ( s+1 ) x \big]-\mathrm{c o t h} x \}
$$
Answer: G