## Angular distribution of Neutrons from the Photo-Disintegration of the Deuteron

this paper was written on 1949. at that time, deuteron just discovered 20 years. this paper presents a method on detecting the diffraction cross section of the neutron from a disintegrated deuteron by gamma ray of energy 2.76MeV. and by this, they found the photo-magnetic to photo-electric cross section ration. the ratio is 0.295 ± 0.036.

the photo-electric dipole transition and photo-magnetic dipole transition can both be induced by the gamma ray. Photo carry 1 angular momentum, the absorption of photon will excited the spherical ground state $^1S$ into $^3P$. the 2 mechanisms of the disintegrations results 2 angular distributions of the neutrons. by examine the angular distribution, they find out the ratio.

the photo-magnetic cross section is isotropic and the photo-electric cross section is follow a of a $sin^2$ distribution. the average intensity of neutron detected on a angle is:

$I(\gamma ) = \int_{\gamma_1}^{\gamma_2} {(a + b sin^2(\gamma)) sin(\gamma) d\gamma } / \int_{\gamma_1}^{\gamma_2} {sin(\gamma) d\gamma }$

where a is the contribution from the photo-magnetic interaction and b is from photo-electric interaction. and $\gamma_1$ and $\gamma_2$ are the angle span by the finite size of the target and detector. the integration is straight forward and result is:

$I(\gamma) = a+b( 1 - 1/3 ( cos^2(\gamma_1) + cos(\gamma_1) cos(\gamma_2) + cos^2 ( \gamma_2) )$

and the author guided us to use the ration of 2 angle to find the ration of a and b. and the ration of a and b is related to the probability of the magnetic to the electric effect by

$a/b = 2/3 \tau$

. and the photo-magnetic to photo-electric cross section ratio is:

$\tau/(\tau+1)$

the detector was described in detail on 4 paragraphs. basically, it is a cylindrical linear detector base on the reaction $B^{10} ( n,\alpha)Li^7$. it was surrounded by paraffin to slow down fast nuetrons.

on the target, which is heavy water, $D_2 O$, they use an extraordinary copper toriod or donut shape container. it is based on 3 principles:

• The internal scattering of neutron
• Departure from point source
• The angular opening of the γ – ray source

they place the γ – ray source along the axis of the toriod, move it along to create different scattering angle.

they tested the internal scattering of the inside the toriod and found that it is nothing, the toriod shape does not have significant internal scattering.

they test the reflection of neutron form surrounding, base on the deviation from the inverse-square law. and finally, they hang up there equipment about 27meters from the ground and 30 meters from buildings walls. (their apparatus’s size is around 2 meters. They measured 45, 60, 75 and 90 degree intensity with 5 degree angular opening for each.

## Deuteron

The deuteron is the nucleus that contains 1 proton and 1 neutron. The spin and isospin of proton and neutron are the same, both are equal to half.  It is the only stable state for 2 nucleons. Deuteron provides an unique aspect to study the inter nuclear force. The strong force are believed to be charge independent. Thus, the strong force can be more easily to study on deuteron due to the absent of other force or eliminate from the Coulomb force, which is understood very much.

The mass of deuteron is 1876.1244MeV. The binding energy is then 2.2245MeV. It was determined by the slow neutron capture of a proton. The emitted gamma ray is approximately equal to the binding energy and the deuteron mass was calculated.

Deuteron has no excited state. It is because any excitation will easily to make the system break apart.

The parity is positive from experiment. If we separate the deuteron wavefunction into 3 parts. The proton wavefunction, neutron wavefunction and the orbital wavefunction. Under the only force, the strong force in this system, proton and neutron are the same nucleon with different state. Thus, the parity are the same for proton and neutron. So, the product of these 2 wavefunction always has positive parity. The total parity then is solely given by the angular orbital.

Any orbital wave function can be represented by the spherical harmonic, $Y(l,m)$.

The parity transform is changing it to

$Y(l,m) \rightarrow (-1)^l Y(l,m)$

So, the experimental face of positive parity fixed the angular momentum must be even.

Ok, we just predicted the possible angular momentum from parity.

The experimental fact on spin is 1. Since J = L + S, and the value of J can take every integer from |L-S| to L + S. and L must be even. The spin of proton and neutron is 1/2. Thus the possible S is 0 or 1 ( we are using L-S coupling scheme ). J = 1 = L + S , that tell us S must be odd to give out 1 for an even L. Thus S=1. So, the only possible L is 0 and 1. Thus, the possible state of deuteron is (L,S) = (0,1) or (2,1). Therefore, a deuteron is a mixed state, if without any further argument.

Now, 2 out of 3 parts of the wave function symmetry were determined by symmetry argument. The isospin can now be fixed by the 2 fermions state must be antisymmetry. The spatial state symmetry is even by L = 0 or 2. And for the state (L , S) = ( 0, 1 ), the spin state is symmetric. Thus, the isospin must be antisymmetric. Since the algebra for isospin and spin are the same. We use T = 0 for the isospin. Thus a complete wavefunction is ( L , S , T ) = ( 0 , 1, 0 ). For the other possible state (L , S) = ( 2 , 1 ) , we can use same argument for isospin state. And for the degenerated state with Ms = +1, 0, -1. By the symmetry of the raising and lowering ladder operator, they all preserved the symmetry. Thus, the Ms = 0 state can only be the + state.

So, we now have 2 possible states of deuteron. If the hamiltonian is commute with L^2 and  S^2, both L and L is a good quantum number and those states are eigen state. And the deuteron ground state must be one of them.