Conventional Compton scattering studies did not involve magnetic properties of the scatter- ers. However, as has been indicated above, the magnetic term in the scattering cross section derived first by Lipps and Tolkoek made it possible to investigate the magnetic properties of the scatterers.
This type of experiment became feasible with the advent of the high energy synchrotron ra- diation sources, in spite of inherently small cross section involved. Compared to the charge scattering contribution, the magnetic term is smaller by a factor of~ω/m0c2, where m0c2 is
the rest mass energy of the electron and~ω is the incident photon energy. Therefore, these
signal connected with the cross section for magnetic scattering.
By alternately measuring the standard Compton profiles with opposite sample magnetiza- tion (or photon helicity), magnetic Compton scattering (MCS) provides a measure of the momentum distribution of the difference between the spin-up and spin-down electrons. To be sensitive to the magnetic electrons, MCS require circularly polarized photons that couple according to that term in the incoherent scattering cross section which arises from the charge and magnetic scattering interference.
The first magnetic Compton profile measurements were performed by Sakai and Ono [95] who obtained circularly polarized gamma rays from the cryogenically oriented radioactive source of 57Co. Although this pioneering experiment suffered from a low count rate, the
result clearly demonstrated the existence of the magnetic effect predicted theoretically by Platzman and Tzoar [87].
The interpretation of the measured Compton scattering cross section is in most cases made within the so-called impulse approximation (IA). In this approximation it is assumed that the electrons involved in the scattering can be treated like free and their binding can be only seen in the spread of their momenta. The physical meaning of this approximation can be seen as follows: the time the photon spends probing the electron distribution (∼ ~/ω) is so
short that the other electrons of the sample cannot relax to take into account the hole cre- ated by the recoiling electron until it has completely escaped from the system so that the scattering is essentially from single electrons which can be thought as free electrons. The potential energy of the electron distributionV(r)is considered constant for the duration of the collision and thus cancels out in the energy conservation equation.
The deviation from the impulse approximation has been the subject of several theoretical studies. Ribberfors [96] and Holm and Ribberfors [97] made thorough calculations using the incident photon energy, the atomic number and the scattering angle as parameters to find a possible deviation from the IA. It turned out that the IA works surprisingly well. Eisenberg and Platzman [98] studied the influence of the potential to the observed spectrum and expressed the errors in terms of (EB
Er)
2, where EB is the binding energy and Er is the
recoil energy of the photons. For small ratio, as it is the case for most experiments, the IA works well again.
Within this approximation, if one considers an incident photon with wave vector~k0 and a
scattered photon with the wave vector~k0 (see Fig. 3.1), the magnetic Compton cross section
for a solid can be written as:
· d2σ dΩdpz ¸ ↑ − · d2σ dΩdpz ¸ ↓ ≡ · d2σ dΩdpz ¸ ∆ =Pcr20 µ k02 k2 0 ¶ Ψ2(σ)Jmag(pz), (3.4)
wherePc is the degree of circular polarization andr0 is the classical electron radius andΨ2
is a geometrical factor, defined as:
Ψ2 =±σ(k0cosαcosϕ−k0cos(α−ϕ)) (cosϕ−1)
~c
m0c2
k
S
k’
k’-k
0B
α
φ
0Figure 3.1: The Compton scattering geometry.
In this formula,σ is the electron spin(±1),α is the angle between the incident photon (~k0)
and the magnetization direction (B~) andϕis the scattering angle. As can be seen from Eqs.
(3.4) and (3.5), the magnetic cross-section can be increased by making the angle α smaller
and φlarger. Jmag is the momentum distribution of the unpaired electrons projected along
the scattering vector (pz), also known as the magnetic Compton profile: Jmag(pz) =
Z Z ¡
n↑(~p)−n↓(~p)¢dp
xdpy . (3.6)
Here the electron momentum density for a given spin orientation is given byn↑(↓)(~p).
By constraining the sample magnetisation and the incident and scattered wave vector to be coplanar (see Fig. 3.1), the measured profile depends only upon the electron’s spin. The tra- ditional Compton scattering samples a projection of the electron momentum density, where the integrated area under the Compton profile obtained is proportional to the total number of electrons. The Compton profile is a one-dimensional projection of the electron momen- tum densityn(~p)along the scattering vectorpz:
J(pz) =
Z Z
n(~p)dpxdpy . (3.7)
The area under the profile is equal with the number of electrons in the Wigner-Seitz cell, i.e.
Z
In the magnetic scattering, we are interested only on those electrons which contribute to the spin moment, and which are, therefore, unpaired. If we consider the total electron momen- tum density to be composed of spin-up and spin-down electrons, we can write
n(~p) = n↑(~p) +n↓(~p). (3.9)
If a spin moment exists, this is given by the difference in occupancy of the spin-up and spin-down bands:
µspin =
Z £
n↑(~p)−n↓(~p)¤d3p . (3.10)
This difference can be measured in a magnetic Compton experiment due to the spin depen- dent term in the scattering cross section. The area under the magnetic Compton profile is equal to the spin moment per Wigner-Seitz cell:
Z +∞ −∞
Jmag(pz)dpz =µspin . (3.11)
On the basis of this derivation, we can consider the magnetic Compton scattering as an established technique for probing the spin-dependent momentum densities and band-struc- tures of magnetic solids. The orbital momentum is not measured, the magnetic Compton profile is solely sensitive to the spin moments of the scatterers [99, 90, 89, 87, 100] .
The shape of the magnetic Compton profile carries information about the localization of the electronic moment. This is a useful asset, as the localization of the moment and its interaction with the surrounding conduction electrons, e.g. via the s-d interaction, is the mechanism which drives the magnetic ordering.
The experiments of spin-polarized positron angular correlation also probe the spin density, but in this case the positron-electron correlation effects have to be considered. The positron doesn’t sample electrons from all states equally, whilst De Haas-van Alphen measurements are only sensitive to the electrons from the Fermi surface. Compared to these techniques, the value of the magnetic Compton scattering stems from its uniform sensitivity to the whole of electron momentum distribution.