Photodisintegration of in three dimensional faddeev approach
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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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Photodisintegration of in three dimensional Faddeev approach

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Photodisintegration of in three dimensional faddeev approach

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

The 3N electromagnetic reactions

Nd capture

3N Photodisintegrations


Photodisintegration of in three dimensional faddeev approach

To calculate N-matrix we need to follow these diagrams.

(Three body force is neglected)


Photodisintegration of in three dimensional faddeev approach

3

p

q

1

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

2


Photodisintegration of in three dimensional faddeev approach

By rewriting the free state in term of it’s sub states:

We can use symmetry properties as follow:

Then :

Which is equal to:

By introducing:

We have:


Photodisintegration of in three dimensional faddeev approach

By multiplying on the left side of the previous equation by

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:


Why do we use the three dimensional approach

Why do we use the Three-Dimensional approach?

The 3D approach replaces the discrete angular momentum quantum numbers with continuous angle variables;

  • consequently it considers automatically all PWs.

  • The number of equations in this non-truncated 3D representation is energy independent.

  • In higher energies the number of equations in 3D approach are very smaller than the number of equations in PW approach

    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.


Photodisintegration of in three dimensional faddeev approach

We use these basic sates:

Which are assumed to be normalized as follow:

Completeness relation is:


Photodisintegration of in three dimensional faddeev approach

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


Photodisintegration of in three dimensional faddeev approach

So the first term in calculating of the N- matrix can be written as:

Where

And the second term:


Photodisintegration of in three dimensional faddeev approach

Now we can write integral equation and N-matrix in our basic states as:


Photodisintegration of in three dimensional faddeev approach

The first term in the integral equation contains current and triton binding energy as follow:

So it is important to evaluate this term:

where


Photodisintegration of in three dimensional faddeev approach

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:


Photodisintegration of in three dimensional faddeev approach

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:


Photodisintegration of in three dimensional faddeev approach

For the two body current, conservation of the momentum causes the following relation:

Where:

And:


Photodisintegration of in three dimensional faddeev approach

The two body current which has been used contain and exchange currents:

This current is obtained using continuity equation to the NN force AV18.


Photodisintegration of in three dimensional faddeev approach

The exchange part can be written as:

Different spin and isospin operator parts of this current can be treated as follow:


Photodisintegration of in three dimensional faddeev approach

The exchange operator is:


Photodisintegration of in three dimensional faddeev approach

The spin operator parts of this current can be simplified as a form which is suitable for our basis and can be treated easily:


Photodisintegration of in three dimensional faddeev approach

Z

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:

Where:

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.

Y

x


Photodisintegration of in three dimensional faddeev approach

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.

Where

And finally:


Photodisintegration of in three dimensional faddeev approach

For numerical calculations, we need to write U as a function of real parameters which should be treated as grids in a suitable map.

For example:

In which:


Photodisintegration of in three dimensional faddeev approach

For two body t –matrix singularity problem We have:

has a singularity in which is deuteron binding energy.

By definition of:

We can rewrite integral equation as:


Photodisintegration of in three dimensional faddeev approach

Two body t matrices can be related to the one which calculated in helicity basis:

Where

And


Photodisintegration of in three dimensional faddeev approach

Two body t-matrix in helicity basis has been calculated before.

Where:


Photodisintegration of in three dimensional faddeev approach

Real part of t-matrix in this basis for s=0,t=0 and positive parity


Photodisintegration of in three dimensional faddeev approach

Real part of t-matrix in this basis for s=0,t=0 and positive parity


Photodisintegration of in three dimensional faddeev approach

Real part of t-matrix for s=1 t=1 and positive parity


Photodisintegration of in three dimensional faddeev approach

Imaginary part of t-matrix for s=1 t=1 and positive parity


Photodisintegration of in three dimensional faddeev approach

squared t-matrix in free particle states


Photodisintegration of in three dimensional faddeev approach

squared t-matrix in free particle states


Photodisintegration of in three dimensional faddeev approach

Past & current research projects:

  • Two body t-matrix calculation by using Bonn-B potential

  • Two body t-matrix calculation using AV18 potential

  • Two body t-matrix calculation using chiral potential

  • NN and 3N bound state calculations with chiral potential

    Future Plans:

  • cross section calculation of 3N photodisintegration and Nd capture by using

    • two-body current

    • three-body current

  • Electric and magnetic form factor calculations


Photodisintegration of in three dimensional faddeev approach

I Thank you for your attention


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