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Photodisintegration of in three dimensional Faddeev approach. The 19th International IUPAP Conference on Few-Body Problems in Physics. S. Bayegan M. A. Shalchi M. R. Hadizadeh. University of Tehran. The 3N electromagnetic reactions. Nd capture. 3N Photodisintegrations.
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The 19th International IUPAP Conference on Few-Body Problems in Physics
M. A. Shalchi
M. R. Hadizadeh
University of Tehran
(Three body force is neglected)
Normally we can divide a free state to three sub states (Faddeev scheme).
Using permutation operator, P, we can rewrite the free state in term of the sub state in which nucleons 2 and 3 are in the sub system.
If we consider nucleons with their spins and isospins:
is anti symmetric under permutation of nucleons 2 and 3.
So is a fully anti symmetric state
We can use symmetry properties as follow:
Which is equal to:
And using the properties of permutation operator:
And definition of:
we obtain the main equation which is:
After that we can calculate the N-matrix elements by:
The 3D approach replaces the discrete angular momentum quantum numbers with continuous angle variables;
Therefore this non PW method is more efficient and applicable to the 3N and 4N scattering problems which consider higher energies, and consequently many PWs are needed to achieve convergence results.
Which are assumed to be normalized as follow:
Completeness relation is:
By multiplying these basic states on the left side of U integral equation we have:
And the matrix element becomes:
Effect of permutation operator on our basic states is
And the second term:
The first term in the integral equation contains current and triton binding energy as follow:
So it is important to evaluate this term:
The current which we use contains of single nucleon and two body current as follow (three body current has been neglected):
Because of symmetry properties we can write:
For single nucleon current:
the single nucleon current which we have used has two terms; convection current and spin current :
So the matrix element of this single nucleon current in our basis can be written as:
For the two body current, conservation of the momentum causes the following relation:
The two body current which has been used contain and exchange currents:
This current is obtained using continuity equation to the NN force AV18.
Different spin and isospin operator parts of this current can be treated as follow:
The spin operator parts of this current can be simplified as a form which is suitable for our basis and can be treated easily:
For evaluating the Triton wave function we need to make a relation between this wave function in our basic states to one which has been calculated before.
The wave function has been calculated in this basis:
If we introduce our spin parts of our basis as:
Then we can relate these two states with Clebsch–Gordan coefficients .
This is very important to mention that the spin of the nucleons is quantized in direction of the z axis which in the calculation of wave function it has been chosen to be to the direction of q. but we have to consider the z axis along the direction of incident photon Q.
So we should first rotate the spin of the nucleons in our basis to be settled in the direction of q axis. Then we should use Clebsch–Gordan coefficients to obtain the wave function in the calculated basis.
For numerical calculations, we need to write U as a function of real parameters which should be treated as grids in a suitable map.
has a singularity in which is deuteron binding energy.
By definition of:
We can rewrite integral equation as:
Two body t matrices can be related to the one which calculated in helicity basis: