Dear Silvia

I am sorry for late response. The following is my quick answer.
If you find more questions, please ask me.

The odd-even spin dependence (the signature dependence) of
the level density itself is natural in the case of Pb196, which
is somehow similar to Hg192 discussed in the papers with Yoshida.

This is because there is little signature splitting 
in the single-particle orbits near the Fermi surface,
which are mostly high-Omega oribtals in the A~190 nuclei.
The yrast states with signature alpha=0 (even spins) is
thus energetically favored while the lowest energy states
with signature alpha=1 (odd spins) are unfavoured since
it has to involve at least a one-particle-one-hole excitation.
The negative parity states also have to involve 1p1h excitation,
and hence they are also unfavoured. 
Thus, among the four combiations of signature and parity,
one particular set (alpha=0, and pi=+) is energetically
lower. In a sense, there is only one "yrast" instead of four yrasts. 
These are clearly seen in your table listing the lowest energy 
for each spin and parity as far as the spins are not very high.

Then the problem is how to extract the level density parameter
and how to parameterize the level density with the Fermi gas formula.

In the Yoshida-Matsuo-Shimizu paper NPA696, we were not very
precise in the extraction and fitting. The prescription is described 
in p.100. Though the statements are not fully selfcontained, I remember
(though not perfectly clear) that we picked up only a representative
spin and signature, made a fit, and extracted the level density.
We took a particular set of spin and parity which corresponds to the
"yrast", i.e. alpha=0 and pi=+. (The spin could be 20 or 30, but
I do not remember). We used the extracted level density parameter
a for all spins and parities by neglecting the spin dependence.

I guess that your fitting procedure is probably like as follows.
For (alpha,pi)=(0,+) states, you simply take all the states
including the yrast, and made a fit. For (alpha,pi)=(0,-) states, 
you take all the (alpha,pi)=(0,-) states, but in this case
you should have refereed to the yrast energy of (0,+) in evaluating
the excitation energy of the (0,-). Right? I think there is no 
problem so far. 
But for the odd spin states, i.e. (1,+) and  (1,-) states, 
you may have evaluated the excitation energies with respect to
the lowest states among the selected (1,+) and  (1,-) states.
If this is the case, it is the reason why you get a large values of
the level density parameters for the odd spins. As I pointed out 
above, the lowest energy states of (1,+) and  (1,-) states are
not true "yrast". If you like to improve this, we should rather 
refer to the true "yrast" in evaluating the energy. Namely you 
may interpolate the "yrast" state energies at spins e.g 30+ and 32+ 
(4.881 and 4.257) to 'define'
a reference yrast energy at spin 31, (4.881+4.257)/2=4.569. 
If you evaluate the excitation energies of the 31+ and 31- states 
with respect to this reference yrast energy 4.569 instead of 5.193, 
I think you will get  a reasonable level density parameters also 
for odd spins.

>   32     4.881     4.881     5.520
>   31     5.193     5.193     5.227
>   30     4.257     4.257     4.926

I can propose a simpler approach. 
The spin and parity dependece could be neglected. 
Namely, you may use the same level density parameter
a=13.60 extracted from 30+ states also for
30-, 31+, 31- states. Note however also in this case 
you should use the intepolated reference yrast energy 
4.569 for 31+- states when you use the Fermi gas
formula of the level density.

>      30   13.60   13.51
>      31   17.56   17.63

Is this clear enough?

Ciao
Masayuki Matsuo


