BL Practice: BL08W, Cheiron2012

Spin moment determination of ferro- or ferrimagnetic
materials using magnetic Compton scattering
Masayoshi Itou and Yoshiharu Sakurai (JASRI/SPring-8)
1. Introduction to Compton scattering and magnetic Compton scattering(1) Compton Scattering [1]Most students remember learning the Compton effect, i.e. the effect of Compton scattering, from an
introductory textbook on quantum ideas. The Compton effect, the increase in the wavelength, i.e. thedecrease in the energy, of X-rays and gamma-rays on being scattered by material objects, clearlydemonstrates that X-rays and gamma-rays have a particle nature and these particles, the photons,
behave like materials particles in collisions with electrons in material objects. Considering acollision between a photon and an electron, conservation laws for energy and momentum lead to asimple result for the energy shift (ΔE) of a photon by Compton scattering,ΔE =ℏ2 𝐪2 ℏ𝐊 ∙ 𝐩+2mm(1)where p is the electron momentum before scattering, q the scattering vector, m the electron mass.
The first term corresponds to the fixed shift in energy only depending on the scattering geometry,
and the second term is a Doppler shift that depends on the component of the electron momentum palong the scattering vector q. Therefore, the Compton-scattered X-ray line is broadened by moving
electrons (see Fig.1). Under an impulse approximation, the line shape is in proportion to theso-called Compton profile, J(pz),J(pz ) = � n�pX , py , pz �dpX dpy(2)

where n(px, py, pz) is the probability distribution of the electron momenta in material object and the z
direction is taken along the scattering vector q. The Compton profile can be normalized to the total
number of electrons N in the material object,� J(pz )𝑑pz = N(3)

Fig.1: X-ray energy spectrum from CeRh3B2 measured by a Ge-SSD with ascattering angle of 168 degrees. The incident X-ray energy is 175 keV at
Elastic Peak and the Compton scattered X-ray line at 105 keV is broadened
by Doppler-shift in collision between the photons and moving electrons.

(2) Magnetic Compton Scattering [2]When the X-rays are circularly polarized and the material object is a ferro- or ferromagnetic material,
the Compton-scattered X-ray line shape is proportional to the scattering cross section,

d2 σ= 𝐶 J(pz ) + 𝐶𝑚𝑎𝑔 P𝑐 𝐒 ∙ (𝐤 cos 𝜃 + 𝐤′) J𝑚𝑎𝑔 (pz )
dΩdω

(4)

where Pc is the degree of circular polarization of X-rays, S the spin direction, k (k’) the wavevectorof incident (scattered) x-rays, θ the scattering angle. C and Cmag are constants. The first term
corresponds to the eq.(2) and the second term is the magnetic term which includes the magnetic
Compton profile Jmag(pz),J𝑚𝑎𝑔 (pz ) = ��n𝑢𝑝 (𝐩) − n𝑑𝑜𝑤𝑛 (𝐩)�dpx dpy(5)𝐩 = �𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 �where nup(p) and ndown(p) are the probability distribution of the electron momenta for up-spin
electrons and down-spin electrons in a ferromagnetic material. The magnetic Compton profile can benormalized to the difference in the numbers of up-spin and down-spin electrons, i.e spin moment
μspin (μB),

� J𝑚𝑎𝑔 (pz )𝑑pz = � n𝑢𝑝 (𝐩)dpx dpy dpz − � n𝑑𝑜𝑤𝑛 (𝐩)dpx dpy dpz= N𝑢𝑝 − N𝑑𝑜𝑤𝑛= µ𝑠𝑝𝑖𝑛 (µ𝐵 )(6)The second term in eq.(4) can be isolated by changing the degree of circular polarization of incident
X-rays Pc or changing the spin direction S in a ferromagnet. Usually, the spin direction is changed byexternal magnetic field for isolating the magnetic term. Figure 2 shows spectra of Compton scattered
X-rays from a polycrystalline Fe with different spin directions (Positive and Negative), together withthe difference between the Positive and Negative spectra (Difference), i.e. magnetic Compton
profile.

Fig.2: Spectra of Compton scattered X-rays from Fe with positive andnegative spin directions, together with the difference spectrum (magneticCompton profile) between them.

2. Spin moment determination [3]In order to determine the absolute spin moment in a sample, the magnetic
effect R is defined by the following equation,

𝐼+ − 𝐼−
𝑅= +𝐼 + 𝐼−(7)

where I+ and I- are the integrated intensity of Compton scattered X-ray spectra measured when thespin direction, i.e. the direction of external magnetic field, is parallel (+) and anti-paralell (-) to thescattering vector. On the other, using eqs.(3), (4) and (6),𝐼 + − 𝐼 − = 2 𝐶𝑚𝑎𝑔 P𝑐 𝐒 ∙ (𝐤 cos 𝜃 + 𝐤′) � J𝑚𝑎𝑔 (pz )𝑑pz= 2 𝐶𝑚𝑎𝑔 P𝑐 𝐒 ∙ (𝐤 cos 𝜃 + 𝐤′) µ𝑠𝑝𝑖𝑛(8)

𝐼 + + 𝐼 − = 2 𝐶 � J(pz )𝑑pz = 2 𝐶 𝑁and thus the magnetic effect R is given by,

𝑅 = 𝐴�

µ𝑠𝑝𝑖𝑛
�
𝑁(9)

where μspin is the spin moment and N the total number of electrons, again. Once the coefficient A isgiven, the spin moment can be determined from the experimentally determined magnetic effect R
through eq.(9) since the total number of electrons N is known. The coefficient A is usuallydetermined by measurement on a well-characterized ferromagnet, such as Fe, in which its spin
moment is already known. Figure 3 shows the relationship between the experimentally obtained Rand μspin/N for Fe, Co, Ni, LSMO and SRO in which their spin moments are determined by othermethods.

Fig.3: Relationship between the magnetic effect R and μspin /N for Fe, Co,Ni, LSMO and SRO in which their spin moments are determined by othermethods. Magnetic Effect,

3. Experimental Setup: Magnetic Compton Scattering Spectrometer
Figure 4 show the magnetic Compton scattering spectrometer at the BL08Wbeamline. Thespectrometer consists of a 3T-superconducting magnet, a sample cryo-cooler and a 10-element Ge
SSD. The superconducting magnet can reverse the magnetic field in few seconds between -2.5 T and+2.5T and the cryo-cooler can hold the sample at a temperature between 5 K and R.T. The incident
X-rays are elliptically polarized and their energy is 175 keV. Compton scattered X-rays from asample are detected by the 10-element Ge SSD at a scattering angle of 178 degrees.

Fig.4: Magnetic Compton scattering spectrometer at BL08W 10-element

4. PracticeIn the beamline practice (BL08W), we measure three ferromagnetic samples, Fe, Ni and SmAl2.
Firstly, we measure polycrystalline samples of Fe and Ni at room temperature. The aim of thisexperiment is to see that the magnitude of magnetic effect depends on the value of spin moment. Thespin moments on Fe and Ni are 2.083 μB and 0.512 μB from other measurements.
Secondly, we measure a single crystal of SmAl2, at 10 K. The aim is to see that the magnetic effectis negative due to the presence of significant orbital moment.

5. References[1] see, X-ray Compton Scattering, eds M. J. Cooper et al., Oxford University Press
2004.
[2] N. Sakai, J. Appl. Crystallogr. 29 (1996) 81.