Editing Compton scattering (section)

=== Magnetic Compton scattering ===
Magnetic Compton scattering is an extension of the previously mentioned technique which involves the magnetisation of a crystal sample hit with high energy, circularly polarised photons. By measuring the scattered photons' energy and reversing the magnetisation of the sample, two different Compton profiles are generated (one for spin up momenta and one for spin down momenta). Taking the difference between these two profiles gives the magnetic Compton profile  (MCP), given by <math>J_{\text{mag}}(\mathbf{p}_z)</math> – a one-dimensional projection of the electron spin density.
<math display="block">J_{\text{mag}}(\mathbf{p}_z) = \frac{1}{\mu}\iint_{-\infty}^\infty (n_{\uparrow} (\mathbf{p}) - n_{\downarrow}(\mathbf{p})) d\mathbf{p}_x d\mathbf{p}_y</math>
where <math>\mu</math> is the number of spin-unpaired electrons in the system, <math>n_\uparrow(\mathbf{p})</math> and <math>n_\downarrow(\mathbf{p})</math> are the three-dimensional electron momentum distributions for the majority spin and minority spin electrons respectively.

Since this scattering process is [[Coherence (physics)|incoherent]] (there is no phase relationship between the scattered photons), the MCP is representative of the bulk properties of the sample and is a probe of the ground state. This means that the MCP is ideal for comparison with theoretical techniques such as [[density functional theory]].
The area under the MCP is directly proportional to the spin moment of the system and so, when combined with total moment measurements methods (such as [[SQUID]] magnetometry), can be used to isolate both the spin and orbital contributions to the total moment of a system.
The shape of the MCP also yields insight into the origin of the magnetism in the system.<ref name="Cooper2004">{{cite book|author=Malcolm Cooper|title=X-Ray Compton Scattering|url=https://books.google.com/books?id=m58jXIJDs3QC|access-date=4 March 2013|date=14 October 2004|publisher=[[OUP Oxford]]|isbn=978-0-19-850168-8}}</ref><ref>Barbiellini, B., Bansil, A. (2020). Scattering Techniques, Compton. Materials Science and Materials Engineering, Elsevier. https://doi.org/10.1016/B978-0-323-90800-9.00107-4
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