BL Practice: BL08W, Cheiron2008
Fermi Surface and Electron Correlation Effects Probed by Compton Scattering
Yoshiharu Sakurai
Japan Synchrotron Radiation Research Institute
1. Introduction to Compton scattering [1]Most of physics students remember having learned the Compton effect, i.e. theeffect of Compton scattering, from an introductory lecture or textbook on quantum ideas.
The Compton effect, the increase in the wavelength, i.e. the decrease in the energy, ofX-rays or gamma-rays on being scattered by material objects, obviously demonstrates
that X-rays and gamma-rays have a particle nature and these particles, the photons,
behave like materials particles in collision with electrons in material objects.
Considering a collision between a photon and an electron, the conservation laws forenergy and momentum lead to a simple result for the energy shift (∆E) of the photon by
Compton scattering,
2 2q
q·p+,
12mm
where p=(px,py.pz) is the electron momentum before scattering, q the scattering vector,
m the electron mass. The first term is the fixed shift in energy only depending on thescattering geometry, and the second one is a Doppler shift that depends on the∆E=component of the electron momentum along the scattering vector q.
Summing up all events of Compton scattering with moving electrons in amaterial object, the Compton-scattered X-ray line appears as a broadening one (seeFig.1), and the line shape is converted into the so-called Compton profile, J(pz), under
the impulse approximation,
J pn

dp dp ,

2

where n(p) is the electron momentum density, the probability distribution of the electron
momenta, in a 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 thematerial object,J p dp

N.

3

5e+7Compton Scattered
X-raysCounts4e+7

Fluorescent
X-rays3e+7

CeRh3B2

2e+7Elastic
Peak

1e+70050

100

150

200

X-ray Energy (keV)

Fig.1: X-ray energy spectrum from CeRh3B2 measured by a Ge solid state detector witha scattering angle of 168 degrees. The incident X-ray energy is 175 keV at the Elastic
Peak and the Compton Scattered X-rays line at 105 keV is broadened by the Doppler
shift in collision between photons and moving electrons.

2. Fermi surface and electron correlation effects by Compton scattering
The defining property of a metallic ground state is the “Fermi surface,” whichmarks the boundary the electron occupied and unoccupied states in momentum space(see Fig.2), and thus the electron momentum density, n(p), reflects the shape of Fermi
surfaces and delineates the electron correlation effects in momentum space. The
Compton profile contains the signatures of underlying Fermi surfaces, since it is given
as a double integration of n(p) (see eq.(2)) [2].
In order to map out the Fermi surface, one need to know the three-dimensional
n dp . In themomentum density, n(p), or two-dimensional one, n p , pCompton experiment, one can obtain the three- or two-dimensional momentum density
using a topographic reconstruction from experimental J(pz)’s measured along different
crystal directions.
The advantage of Compton scattering over other probes for Fermiology are thatit do not need a nearly lattice-defect-free crystal or clean surface or ultra high vacuum
or very low temperature. For example, one can map out the Fermi surface of adisordered alloy in gas atmosphere at room temperature. Even with these advantages
Compton scattering had not been a workable probe for more than half a century,because the method requires high energy (~100 keV) photons with high flux (~1×1013
photons/s) that can be available only at large-scale synchrotron radiation facilities suchas SPring-8 and ESRF. Fermiology by high-resolution Compton scattering has now
been developed to such that it can now provide information about Fermi surface ofvarious materials [3-7].

Fig.2: The Fermi surface is defined as a boundary between electron occupied andunoccupied states in momentum space. The thin line shows the electron momentum
density of the free electron model, in which a step-like discontinuity appears at theFermi surface in the momentum density profile. When the interaction between electrons
is turned on (interacting electrons), some of the weights in the density are moved from p< pF to p > pF , although the discontinuity remains at the Fermi surface. The differencein n(p) between the free and interacting electrons shows the electron correlation effects
on the electron momentum density.

3. An examples of Fermi surface mapping: NixAl1-x (x=0.625) [4]
The Fermi surface of the shape-memory alloy Ni0.62Al0.38 was determined using theCompton scattering technique. A large area of this Fermi surface can be made to nest
with other areas by translation through a vector of

0.18[1,1,0](2/a), whichcorresponds to the wave vector associated with martensitic precursor phenomena suchas phonon softening and diffuse streaking in electron diffraction patterns. This results
provides a piece of evidence that these phenomena are driven by the enhancedelectron-lattice coupling due to the Fermi surface nestingFig.3: Fermi surface and its nesting feature in Ni0.62Al0.38 determined by Compton
scattering.

4. PracticeIn this beamline practice, we measure the Compton profiles of Al and V. Al is a simple
metal with a Fermi surface of nearly free electron, whose shape is a sphere in an
extended zone. V is a 3d metal with complicated Fermi surfaces because of large
contribution of 3d electrons to the Fermi surfaces.
The aim of this practice is to learn the X-ray spectrometer [8-10] (see Fig.4) for
Compton scattering experiment and to find the difference in Compton profiles between
the nearly free electron and complicated 3d electron Fermi surfaces.

Ge-SSD

Cauchois-type
Analyzer

Position Sensitive
Detector (PSD)

Fig.4: X-ray spectrometer for Compton scattering experiments at BL08W.

References[1] See, e.g., M. J. Cooper, Rep. Prog. Phys. 48, 415 (1985).
[2] Y. Sakurai, Y. Tanaka, A. Bansil, S. Kaprzyk, A. T. Stewart, Y. Nagashima, T. Hyodo,
S. Nanao, H. Kawata, and N. Shiotani, Phys. Rev. Lett. 74, 2252 (1995).
[3] S. Mizusaki, T. Miyatake, N. Sato, I. Yamamoto, M. Yamaguchi, M. Itou, andY. Sakurai , Phys. Rev. B 73, 113101 (2006).
[4] S. B. Dugdale, R. J. Watts, J. Laverock, Zs. Major, M. A. Alam, M. Samsel-Czekala,
G. Kontrym-Sznajd, Y. Sakurai, M. Itou, and D. Fort, Phys. Rev. Lett. 96, 046406(2006).
[5] N. Hiraoka, T. Buslaps, V. Honkimäki, J. Ahmad, and H. Uwe, Phys. Rev. B 75,121101(R) (2007).
[6] N. Hiraoka, T. Buslaps, V. Honkimäki, T. Nomura, M. Itou, Y. Sakurai, Z. Q. Mao,and Y. Maeno, Phys. Rev. B 74, 100501(R) (2006).
[7] J. Laverock, S. B. Dugdale, J. A. Duffy, J. Wooldridge, G. Balakrishnan, M. R. Lees,G. –q. Zheng, D. Chen, C. T. Lin, A. Andrekczuk, M. Itou, and Y. Sakurai, Phys.
Rev. B 76, 052509 (2007).
[8] N. Hiraoka, M. Itou, T. Ohata, M. Mizumaki, Y. Sakurai and N. Sakai,J. Synchrotron Rad. 8, 26 (2001).
[9] M. Itou and Y. Sakurai, AIP Conference Proceedings, 705, 901 (2004).
[10] Y. Sakurai and M. Itou, J. Phys. Chem. Solids, 65, 2061 (2004).