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\sect{The calculation of $Z(T)$ for AB and AA molecules}

A diatomic molecule ${AB}$ can be regrded as two atoms 
that are connected by a spring of length ${r_0}$, 
and vibration frequency $\omega_{0}$. 
The total angular momentum is ${\ell=0,1,2,...}$.
The masses of the atoms are $m_A$ and $m_B$, 
and they have spins $S_A$ and $S_B$. 

\Dn

(a) Explian what are the conditions that allow 
to ignore all the excited vibrational levels, 
so you can treat the molecule as a rigid body ("rotor").
 
\Dn

(b) Calculate the partition function 
of the diatomic molecule, assuming that 
it is like a classical rigid rotor.
Define the condition on the temperature 
for this approximation to hold. 

\Dn 

(c) Calculate the partition function 
of the diatomic molecule, if the temperature 
is very low, taking only the ${\ell=0,1}$ 
states into account.

\Dn

(d) How the previous answers are modified for an $AA$ 
molecule that is composed of two identical spin0 atoms?


\Dn

(e) How the previous answers are modified for an $AA$
molecule that is composed of two identical spin 1/2 atoms?
What is the probability to find 
the spin configuration in a triplet state?
Relate to the two limits in (b) and (c).