Shell Model study of 13C
The goal of this exercise is to compare structural information obtained in 12C(d,p)13C reaction with available theories. Using shell model code CoSMo available for download (www.volya.net CoSMo link) and preinstalled on lab computers students are expected to obtain spectra of 13C and 12C and compute spectroscopic factors using various model spaces with well established effective Hamiltonians.
The shell model code is installed and ready to use on lab computers, however it can be helpful and is highly recommended that you install it on your own machine. See download and installation information here.
The 13C nucleus is a p-shell nucleus, containing 6 protons and 7 neutrons, four of these nucleons occupy 0s shell and the remaining 4 protons and 5 neutrons are in the p shell.
Create xml description files for 13C in the p-shell
This can be accomplished by running
cosmoxml and selecting “p” for model space and "cki" for Hamiltonian (you can use other Hamiltonians but cki is among most well known). You can select defaults answer for all other questions (just press return). As a result 13C_cki.xml will be created. Review the structure of this file, see also xml format.
Create many-body states:
How many states did we get? Do you understand this number?
Create many-body Hamiltonian:
Diagonalize the Hamiltonian:
Analyze spins and isospins of egenstates :
Compare your output with experimental information about 13C, you can also compare this with 13N. Is there any difference in the answer if you run this shell model calculation for 13N? What would happen if one selects different Jz projection?
Repeat the same procedure to obtain ground state in 12C.
Compute spectroscopic factors
XSHLSF 13C_cki.xml –f 12C_cki.xml
The biggest drawback in the p-shell study is that positive parity states in 13C are missing. Why? Next we examine the same problem using psd valence space. We suggest psdmk effective Hamiltonian. The steps are the same, expect this interaction is designed to allow one particle being exited to sd shell. In order to create many-body states restricted by this principle we should create “rejection”. This can be done by selecting yes for rejection and answering questions as follows
Create Particle-Hole Rejection (use with XSHLMBSQRR)(y/n)[n]<> Create rejection (to be processed by Xsysmbs)(y/n)[n]y Orbital: 0p1 type= pninclude (y/n)[n]n Orbital: 0p3 type= pninclude (y/n)[n]n Orbital: 0d3 type= pninclude (y/n)[n]y neutron, proton, or both (n, p, or pn): [pn]pn Orbital: 0d5 type= pninclude (y/n)[n]y neutron, proton, or both (n, p, or pn): [pn]pn Orbital: 1s1 type= pninclude (y/n)[n]y neutron, proton, or both (n, p, or pn): [pn]pn The number of nucleons in selected orbits is in [0,24] Minimum N on selected orbits: 0 Maximum N on selected orbits: 1
As a result the xml created should have the following rejection tag
<rejection NMIN="0" NMAX="1"> <orbital name="0d3" n="0" l="2" j="3/2" index="3" type="pn"/> <orbital name="0d5" n="0" l="2" j="5/2" index="4" type="pn"/> <orbital name="1s1" n="1" l="0" j="1/2" index="5" type="pn"/> </rejection>
which limits the number of nucleons on sd shell to be between 0 and 1.
For 13C when you select negative parity what should be the number of states?
Next, finish this study by computing 12C, and states of both parities in 13C. Evaluate spectroscopic factors, compare theory and experiment.
Shell model offers a lot of opportunities for further studies , here are some ideas:
Examine modelspace spsdpf with wbp hamiltonian. This is a large model space and the Hamiltonian requires particle-hole restrictrictions. Thus, you can allow one particle in sd shell or one hole in 0s shell (one quantum of oscillator exitation). Is there substantial difference in results? From theoretical perspective, assuming harmonic oscillator basis placing symmetric limits allows to control the center of mass problem. Isolate the center of mass Hamiltonian in xml file, what are its eivenvalues?
Large scale study
Recently an unrestricted interaction psdu was developed (Phys Rev C. 83 021301, it is in psdpn space). This hamiltonian assumes full mixing, not limited by particle-hole exitations. It can be interesting to examine the above questions with this interaction. These calculations take substantial time (could be upto 20 hours on old pc), please to not run them on public computer systems.