Problem:Scattering from an antiferromagnet

This problem is a simple illustration of magnetic diffraction, as calculated in section Magnetic diffraction.

We consider the two-dimensional square lattice with lattice constant \(d\) and antiferromagnetic interactions between nearest neighbours.

In the previous problem, we showed that this system is antiferromagnetic with alternating spin directions between nearest neighbours, as illustrated in this figure (middle) from the Elastic magnetic scattering page.

Question 1

Argue that the square 4-atom cell with side length \(2 d\) is a valid magnetic unit cell for the system and that the reciprocal lattice then becomes a square lattice with side length \(\pi/d\).

Question 2

We define as usual the reciprocal lattice coordinates, \((h,k)\), in terms of the nuclear reciprocal lattice vectors, \(|{\bf a}^*| = |{\bf b}^*| = 2 \pi/d\). Show that the points of the 4-atom magnetic reciprocal lattice can be reached by integer and half-integer values of \(h\) and \(k\).

Question 3

In the nuclear Brillouin zone, there are 4 different magnetic reciprocal lattice points: \((0,0)\), \((0,1/2)\), \((1/2,0)\), and \((1/2, 1/2)\). Calculate the magnetic scattering structure factor for these 4 reflections when the spins are all pointing out of the plane (the \(l\)-direction). Draw a map of \(q\)-space, showing the allowed nuclear and magnetic reflections, respectively.

Question 4

Compare to the value of the ordering \(Q\)-vector, \({\bf Q} = (1/2 1/2)\), found in the previous problem.