Wigner-Eckart theorem

This is a theorem that connect a matrix element to a reduced matrix element that pull out the “magnetic sub-state”. We know that all quantum orbital with angular momentum are degenerate in to sub-states. A complete wave function looks like this , where is other quantum numbers, and for convenient, we will drop it in later discussion.

A matrix element for a spherical tensor operator with rank and -th component is

And the dependence can be pull out as

Here we used the 3-j symbol, and the term is called the reduced matrix element. This stored the essential physical information, and the 3-j symbol represents the coupling between difference -state, a kind of geometrical effect.

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In nuclear physics, we often encounter the matrix element. In fact, the total Hamiltonian is a rank 0 tensor, magnetic quadruple is a rank 2 tensor. And the matrix representation of the Hamiltonian requires the calculation of matrix element.

For the total Hamiltonian, the 3-j symbol becomes

For , it is particularly simple,

Thus, the Wigner-Eckart theorem becomes

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The advantage of the theorem is that, we can pick any convenient -state to calculate the matrix element, and use the theorem to have the reduced matrix element, and then, we can have all other matrix elements with different -states! for example, we can have the reduced matrix element from the state,

than, put back the reduced matrix element, and we get.

We can see, for rank 0 tensor, the diagonal elements are all degenerated, as we expected.

Sum rule of Spectroscopic factors

For transfer reaction, the wave function of nucleus B = A + 1 can be written as

where the due to anti-symetrization between the single nucleon and the core nucleus A, the Clebsh-Gordon coefficient is come from the angular coupling, and is the square root of the spectroscopic factor for orbital between nuclei B and A.

We can see that, to find out the spectroscopic factor, we can integrate out the core nucleus A

In the last step, we write the Clebsh-Gordon coefficient in Winger 3j-symbol. Using the Wigner-Eckart theorem

Thus,

using the identity

we have

replacing the reduced matrix element in the Wigner-Eckart theorem with previous result,

multiple the adjoint

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For A(d,p) B reaction, the nucleus A is in the ground state, we can sum all the and states, then sum all the , notice that

we have

recall

The last equality mean the number of nucleon in orbital in nucleus A. Thus, we obtain

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For B(p,d)A reaction, the nucleus B is in the ground state and we can sum all ,

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Thus, the adding A(d,p)B and removing A(p,d)C reaction from nucleus A, the sum of spectroscopic factors

 

 

3j, 6j, 9j-symbol for C++

Since I cannot find a odd c++ code ( < c++11 ) for the calculation of those j-symbol, I made one for myself!

The code just using the formula which you can find in wiki or here. So, the code is not optimized, but for simple calculation, I guess it does not matter. Also, I checked, in my pc, calculate 1000 9-j symbols only take 1 or 2 sec.

The program first build

1. factorial
2. Clebsch–Gordan coefficient
3. 3-j symbol
4. 6-j symbol
5. 9-j symbol

#include <stdlib.h>
#include <cmath>
#include <algorithm>

using namespace std;

double factorial(double n){
  if( n < 0 ) return -100.;
  return (n == 1. || n == 0.) ? 1. : factorial(n-1) * n ;
}

double CGcoeff(double J, double m, double J1, double m1, double J2, double m2){
  // (J1,m1) + (J2, m2) = (J, m)

  if( m != m1 + m2 ) return 0;

  double Jmin = abs(J1 - J2);
  double Jmax = J1+J2;

  if( J < Jmin || Jmax < J ) return 0;

  double s0 = (2*J+1.) * factorial(J+J1-J2) * factorial(J-J1+J2) * factorial(J1+J2-J) / factorial(J+J1+J2 + 1.);
  s0 = sqrt(s0);

  double s = factorial(J +m ) * factorial(J -m);
  double s1 = factorial(J1+m1) * factorial(J1-m1);
  double s2 = factorial(J2+m2) * factorial(J2-m2);
  s = sqrt(s * s1 * s2);

  //printf(" s0, s = %f , %f \n", s0, s);

  int kMax = min( min( J1+J2-J, J1 - m1), J2 + m2);

  double CG = 0.;
  for( int k = 0; k <= kMax; k++){
    double k1 = factorial(J1+J2-J-k);
    double k2 = factorial(J1-m1-k);
    double k3 = factorial(J2+m2-k);
    double k4 = factorial(J - J2 + m1 +k);
    double k5 = factorial(J - J1 - m2 +k);
    double temp = pow(-1, k) / (factorial(k) * k1 * k2 * k3 * k4 * k5);
    if( k1 == -100. || k2 == -100. || k3 == -100. || k4 == -100. || k5 == -100. ) temp = 0.;

    //printf(" %d | %f \n", k, temp);
    CG += temp;
  }

  return s0*s*CG;

}

double ThreeJSymbol(double J1, double m1, double J2, double m2, double J3, double m3){

  // ( J1 J2 J3 ) = (-1)^(J1-J2 - m3)/ sqrt(2*J3+1) * CGcoeff(J3, -m3, J1, m1, J2, m2) 
  // ( m1 m2 m3 )

  return pow(-1, J1 - J2 - m3)/sqrt(2*J3+1) * CGcoeff(J3, -m3, J1, m1, J2, m2);

}

double SixJSymbol(double J1, double J2, double J3, double J4, double J5, double J6){

  // coupling of j1, j2, j3 to J-J1
  // J1 = j1
  // J2 = j2
  // J3 = j12 = j1 + j2
  // J4 = j3
  // J5 = J = j1 + j2 + j3
  // J6 = j23 = j2 + j3

  //check triangle condition
  if( J3 < abs(J1 - J2 ) || J1 + J2 < J3 ) return 0; 
  if( J6 < abs(J2 - J4 ) || J2 + J4 < J6 ) return 0;
  if( J5 < abs(J1 - J6 ) || J1 + J6 < J5 ) return 0;
  if( J5 < abs(J3 - J4 ) || J3 + J4 < J5 ) return 0;

  double sixJ = 0;

  for( float m1 = -J1; m1 <= J1 ; m1 = m1 + 1){
    for( float m2 = -J2; m2 <= J2 ; m2 = m2 + 1){
      for( float m3 = -J3; m3 <= J3 ; m3 = m3 + 1){
        for( float m4 = -J4; m4 <= J4 ; m4 = m4 + 1){
          for( float m5 = -J5; m5 <= J5 ; m5 = m5 + 1){
            for( float m6 = -J6; m6 <= J6 ; m6 = m6 + 1){

              double f = (J1 - m1) + (J2 - m2) + (J3 - m3) + (J4 - m4) + (J5 - m5) + (J6 - m6);

              double a1 = ThreeJSymbol( J1, -m1, J2, -m2, J3, -m3); // J3 = j12 
              double a2 = ThreeJSymbol( J1, m1, J5, -m5, J6, m6); // J5 = j1 + (J6 = j23)
              double a3 = ThreeJSymbol( J4, m4, J2, m2, J6, -m6); // J6 = j23
              double a4 = ThreeJSymbol( J4, -m4, J5, m5, J3, m3); // J5 = j3 + j12

              double a = a1 * a2 * a3 * a4;
              //if( a != 0 ) printf("%4.1f %4.1f %4.1f %4.1f %4.1f %4.1f | %f \n", m1, m2, m3, m4, m5, m6, a);

              sixJ += pow(-1, f) * a1 * a2 * a3 * a4;

            }
          }
        }
      }
    }
  }

  return sixJ;
}

double NineJSymbol( double J1, double J2, double J3, double J4, double J5, double J6, double J7, double J8, double J9){

  double gMin = min( min (min( abs(J1 - J2 ), abs(J4 - J5)) , abs( J4 - J6 )) , abs(J7 - J8));
  double gMax = max( max (max( J1+J2, J4+J5), J3+J6), J7+J8);

  //printf(" gMin, gMax = %f %f \n", gMin, gMax);

  double nineJ = 0;
  for( float g = gMin; g <= gMax ; g = g + 1){
    double f = pow(-1, 2*g) * (2*g+1);
    double s1 = SixJSymbol(J1, J4, J7, J8, J9, g);
    if( s1 == 0 ) continue;
    double s2 = SixJSymbol(J2, J5, J8, J4, g, J6);
    if( s2 == 0 ) continue;
    double s3 = SixJSymbol(J3, J6, J9, g, J1, J2);
    if( s3 == 0 ) continue;
    nineJ += f* s1*s2*s3;
  }

  return nineJ;
}

Winger 6-j and 9-j symbol

The meaning of 3-j symbol is same as Clebsch-Gordan coefficient. So, we skip in here.

I am not going to construct the 6-j symbol from 3-j symbol. In here, I just state the meaning and usage in Mathematica.

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The 6-j symbol is the coupling between 3 angular momenta, .

There are 2 ways to couple these 3 angular momenta. First,

the other way is

The 6-j symbol is

We can see that there are 4 vector-sum must satisfy.

If we draw a line to connect these 4 vector-sum, we have:

In Mathematica, there is a build in function

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The 9-j symbol is the coupling between 4 angular momenta, .

The 9-j symbol can be used in coupling 2 nucleons, .

The 9-j symbol is

We can see, each row and column must satisfy the vector-sum.

Unfortunately, there is no build in function in Mathematica. The formula for 9-j symbol is

Where sum all possible value, which can be calculate using the 6 couplings inside the 3 6-j symbols.To check your result, the coupling between and to from a state is

 

 

 

Evaluation of mutual interaction

We are going to evaluation the integral

Recalling the multi-pole expansion,

and the basis

Set an averaged basis

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In the angular integrals, using Wigner 3-j symbol and the integral

we have

Thus,

The 3-j symbol restricted that

I guess it is the most simplified formula for the angular part.

The total integral is

The angular integral imposes condition for .

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I am not sure this is a correct way to treat the problem.

First, the averaged basis is still an energy eigen state. It is not the eigen state for the angular part. So, this averaging could introduces an error and we should reminded that this is an approximation. But in the perturbation view point, this averaged basis is still valid.

Second thing is, the sum

is not symmetry for exchange of in general. For example,

This is a very uprising result that the mutual interaction dependent on the magnetic quantum number. Thus, in detail, we should use as a basis.

Third, the sum is depend on . The mutual interaction require us to sum all possible .

Fourth, the coupling between 1s2p triplet state, the total spin is , total L is , and the total angular momentum can be  . In our treatment, we did not coupled the angular momentum in the calculation explicitly. In fact, in the integral of the spherical harmonic, the coordinate are integrated separately, and the coupling seem to be calculated implicitly. I am not sure how to couple two spherical harmonics with two coordinates.

Integration formulas of spherical harmonic

There are several important and useful formulas for the integration of spherical harmonic. We simplify the notation,

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The first one is the average of spherical harmonic.

The 2nd one is the orthonormal  condition.

The 3rd one is triplet integral, we use the product of spherical harmonic,

The 4th one is another triple integral,

Notice that

using Wigner 3-j symbol,

For other integral, we can use