Question

The deuteron is a bound state of a neutron and a proton in which the two spins are coupled with a resultant total angular momentum S = 1. By absorbing a gamma ray of more than 2.2 MeV the deuteron may disintegrate into a free neutron and a free proton.

(a) Write a wave function for the final state in the reaction \(\gamma\) + D → n + p using plane waves and being sure to include properly the spin coordinates for the two particles. Assume that the interaction with the gamma ray is via electric dipole coupling.

(b) Suppose the neutron and the proton are to be detected far apart from each other after the disintegration of the deuteron. Looking at this in the center-of-mass system, what correlations will be found in time and space, and in spin? Assume that the target consists of unpolarized deuterons. (You may use the following definition of spin correlation: If a proton is detected with spin "up", what is the probability that the corresponding deuteron will also be detected with spin "up"?)

Answer 1

(a) The ground state deuteron ³S₁ has positive parity. The electric dipole transition requires a change of parity between the initial and final states. Hence the parity of the free (n, p) system must have parity -1. Assume that the wave function of (n, p) can be written as \(\Psi\)(n, p) \(\sim\) \(\psi\)(r_n, r_p)x(n, p). For x = x₁³, after the nucleons are interchanged the wave function becomes \(\Psi\)(p, n) = (-1)^l\(\Psi\)(n, p). For x = x₀¹, after the nucleons are interchanged, the wave function becomes Q(p, n) = (-1)^l+1\(\Psi\)(n, p).

A fermion system must be antisymmetric with respect to interchange of any two particles, which means that for the former case, l = 1, 3,. . . , and for the latter case, l = 0, 2, 4,. . . , and so the parities are -1(l = odd) and +1(l = even) respectively. Considering the requirement we see that only states with x = x₁³, i.e. spin triplet states, are possible. Further, S = 1, L = 1,3,. . . , and so J = 0, 1,2,. . . . As the deuterons are unpolarized, its spin wave function has the same probability of being x₁₁, x₁₀ or x_1-1. Therefore, after the transition (n, p) can be represented by the product of a plane wave and the average spin wave function:

(b) The correlation of time and space is manifested in conservation of energy and conservation of momentum. In the center-of-mass coordinates, if the energy of the proton is measured to be E_p, the energy of the neutron is E_n = E_cm - E_p; if the momentum of the proton is p, the momentum of the neutron is -p. Let a be the spin function for "up" spin, and \(\beta\) be that for "down” spin. Then x₁₁ = α(n)α(p), x_1-1 = \(\beta\)(n)\(\beta\)(p), x₁₀ = \(\frac1{\sqrt 2} [\alpha(n) \beta (p) + \alpha(p) + \beta(n)]\), and the spin wave function is

Thus, if the spin of p is detected to be up, we have

Hence the probability that the spin state of n is also up is