Question

In a recent classic table-top experiment, a monochromatic neutron beam (λ = 1.445 Å) was split by Bragg reflection at point A of an interferometer into two beams which were recombined (after another reflection) at point D (see Fig.). One beam passes through a region of transverse magnetic field of strength B for a distance 1. Assume that the two paths from A to D are identical except for the region of the field.

Find the explicit expressions for the dependence of the intensity at point D on B, 1 and the neutron wavelength, with the neutron polarized either parallel or anti-parallel to the magnetic field.

Answer 1

This is a problem on spinor interference. Consider a neutron in the beam. There is a magnetic field B in the region where the Schrodinger equation for the (uncharged) neutron is

Supposing B to be constant and uniform, we have

where t₀, t₁ are respectively the instants when the neutron enters and leaves the magnetic field.

Write \(\psi\)(t) = \(\psi\)(r, t)\(\psi\)(s, t), where\(\psi\)(r, t) and \(\psi\)(s, t) are respectively the space and spin parts of \(\psi\). Then

which is the same as the wave function of a free particle, and

The interference arises from the action of B on the spin wave function. As \(\psi\)(r, t) is the wave function of a free particle, we have t₁ - t₀ = l/v = ml/hk and

where \(k = \frac{2\pi}\lambda = \frac {mv} h\) is the wave number of the neutron. The intensity of the interference of the two beams at D is then proportional to

and \(\sigma.B =\pm \sigma B \)  depending on whether σ is parallel or anti-parallel to B, we have

Therefore, the interference intensity at \(D \propto \cos^2(\pi \mu ml \lambda B/h^2)\), where µ is the intrinsic magnetic moment of the neutron (µ < 0).