Introduction

Daniel FitzGreen

Private: Neutron Diffraction of MnO

Introduction

Forms of Magnetism

Ferromagnetism is the tendency of magnetic dipole moments in materials to align parallel to each other. Ferromagnetism has been known for millennia, and used in technology in the form of compass needles, allowing ancient people to navigate the world. On the other hand, Antiferromagnetism is the tendency of magnetic dipole moments in materials to align antiparallel to each other. This phenomenon was discovered in MnO in 1949 by Clifford Shull and co-workers at Oak Ridge National Laboratory using the newly-developed neutron diffraction technique. Shull shared the 1994 Nobel Prize for Physics with McMaster’s Bertram Brockhouse (who was recognized for developing neutron spectroscopic studies of materials), in part for this discovery.

Antiferromagnetism, Ferromagnetism and Paramagnetism are qualitatively illustrated for an example where dipole magnetic moments are located at the vertices of a two dimensional square lattice in Fig. 1. Paramagnetism, is the case where the orientations of the magnetic dipoles, represented as arrows, are random, such that the sum of all the vector dipole moments would be zero. For Ferromagnetism, near neighbour magnetic dipole moments tend to align parallel to each other and the vector sum of all the magnetic dipole moments is a large number. In Antiferromagnetism, the magnetic dipole moments tend to align antiparallel to near-neighbours, such that, like Paramagnetism, the sum of the magnetic dipole moments is zero. However the magnetic dipole moments are highly ordered, organized into a structure where they rotate by pi as one translates from one dipole moment to the next along near neighbour bonds.

para, ferro, and antiferro magnetism — Figure 1: An illustration of the difference between Paramagnetism (a), Ferrogmagnetism (b), and Antiferromagnetism (c) for magnetic dipole moments (shown as little arrows) associated with certain ions (such as Mn2+) on a square lattice in two dimensions.

Paramagnetism it is the state that all magnets evolve to at high enough temperature, when entropy dominates over energy and the state of the dipoles is very disordered. Ferromagnetism and Antiferromagnetism are low entropy, low temperature states. They typically show up below some critical temperature that is either known as the Curie temperature for Ferromagnets, or the Neel temperature for Antiferromagnets.

Neutron Diffraction

We now know that Antiferromagnetic materials are much more common in nature than Ferromagnetic materials, but we didn’t know about them until 1949 because we had no way of measuring Antiferromagnetism directly. Intense beams of neutrons, which are sensitive to magnetism in solids, are needed for neutron diffraction experiments and they only come from nuclear reactors, which were invented as a byproduct of the Manhattan project during World War II.

Remember that matter can be expressed as a particle or as a wave with a wavelength of

(1) $\begin{equation*} \lambda = \cfrac{h}{mv}, \end{equation*}$

where $h$ is Planck’s constant and $v$ is the velocity of the particle.

One can select out specific wavelengths of neutrons using a diffraction grating (in this case, a silicon monochromator) and choosing the output angle according to the Bragg scattering law:

(2) $\begin{equation*} n\lambda = 2d \sin{\theta} \end{equation*}$

The resulting single-wavelength neutron beam is applied to the MnO, resulting in another diffraction. The measured angles of diffraction ( $\theta$ ) and known wavelength ( $\lambda$ ) are used to determine lattice spacing values ( $d$ ), which are used to determine the lattice structure. A room-temperature neutron diffraction pattern from MnO is shown in Figure 2.

MnO at room temp. — Figure 2: Neutron diffraction of MnO at room temperature.

To first order, the peaks can be described using a Lorentzian function:

(3) $\begin{equation*} L(x) = \cfrac{1}{\pi}\cfrac{\frac{1}{2}\Gamma}{(x-x_0)^2+(\frac{1}{2}\Gamma)^2}. \end{equation*}$

$L(x)$ is a normalized probability distribution function. When you fit your data, you will have to let Python determine the amplitude and you may consider if you need a variable to represent the background detection level.

Introduction

Forms of Magnetism

Neutron Diffraction

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