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

February 18, 2011

GoLuckyRyan papers reading , , , , , Leave a comment

DOI: 10.1103/PhysRev.76.1288

Angular Distribution of Neutrons from the Photo-Disintegration of

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.

Green function (Dirac notation)

February 18, 2011

GoLuckyRyan theory , Leave a comment

The Green function is a very genius way to find the particular solution for an in-homogeneous equation.

L \psi(x) = f(x)

L  is called the operator of the equation. A Green function G(x,x') is a 2D function, such that

L G(x,x') = \delta ( x-x')

Many beginner will feel uncomfortable and wonder why the Green function depends on 2 variables rather 1. Just relax, the operator only affect the x part, leave x’ unchanged, so, beginner can always treat x’ as a constant, a parameter, like the slope in linear equation. For those who touched multi-variable function, x’ is just another dimension. anyway, if we integrate it with the function f(x),

\int { L G(x,x') f(x') dx'} = \int {\delta(x-x') f(x') dx' } = f(x) = L \psi(x)

since the operator only act on the x part, not x’, so L can be pulled out, then,

\psi(x) = \int { G(x,x') f(x') dx' }

which is our solution! But sadly, we have to find the green function rather then the solution! Ok, lets find the Green function.

to find the Green function, we have to solve the eigen equation of the operator.

L \alpha_n(x) = \lambda_n \alpha_n(x)

the eigen functions must be orthonormal ( the normality of the eigen function can be done by divided a constant) :

\int { \alpha_n^* (x) \alpha_m(x) dx } = \delta_{nm}

which this, the eigen function space must span every function. thus, the Green function can be spanned by:

G(x,x') = \sum { \alpha_n(x) c_n }

by using the orthonormal properties. the coefficient can be found and the Green function is:

G(x,x') = \sum { \alpha_n^*(x') \frac {1} {\lambda_n} \alpha_n(x) }

thus the particular solution is

\psi(x) = \sum { \left( \int {\alpha_n^*(x') f(x') dx'} \right) \frac {1} {\lambda_n} \alpha_n(x) }

If we use the eigenfunction expansion on the solution, this is the same result. However, the Green Function provided an “once and for all ” method to solve any function. and this often give a theoretical insight. For example, in solving static electric potential.

when we sub the eigen function expansion  of the Green function back into the operator. we will have 1 more discovery.

L G(x,x') = \sum { \alpha_n^*(x') \alpha_n(x) } = \delta(x-x')

the delta function is equal to the sum of the eigenfunctions!

OK. the stuff on above is very common and textbook or wiki. now i am going to show how this Green function stated in Dirac notation of ket and bra.

for an operator A, which is hermitian, so that it has complete eigenket.

A \left| \psi \right> = \left| f \right>

and the eigenkets are

A \left| \alpha_n \right> = \left| \alpha_n \right> \lambda_n

thus, the solution is

\left| \psi \right> = \sum{ \left| \alpha_n \right> \frac { 1} { \lambda_n} \left< \alpha_n | f \right> }

we can compare with the above formalism, and see the Green function in Dirac notation is:

G = \sum{ \left| \alpha_n \right> \frac { 1} { \lambda_n} \left< \alpha_n \right| }

which is rather an operator then a function, so, we call it Green Operator. if we notices that the operator A is related to the eigenket by

A = \sum{ \left| \alpha_n \right> \lambda_n \left< \alpha_n \right| }

Thus, the Green operator is just the inverse of the operator A!

By using the Dirac notation, we have a more understanding on the Green function. and more easy approach! Dirac rocks again by great invention of notation!