Photon Factory Activity Report 2004 #22 Part B (2005)
Electronic Structure of Condensed Matter

NE1A1/2002G188

Equivalence of LCW-folding and Schuelke’s method
Nobuhiro SHIOTANI*KEK-PF, Tsukuba, Ibaraki 305-0801, Japan
Introduction
Fermiology by high-resolution Compton scattering has
been developed to such that it can now provide the longsought experimental evidence for Fermi surface geometry
related phenomena in disordered alloys, metals andcompounds in high temperature phases.
To map out the Fermi surface from high resolution
Compton profiles, we have to take three steps: 1) wemeasure Compton profiles along n (n=15~30) different
crystalline directions, 2) we reconstruct three-dimensional
momentum density using one of established methods, 3)
we convert momentum density to occupation number
density using the so-called LCW-folding [1], and we lookfor edges of the Fermi surface in the reduced zone (thefirst Brillouin zone). There is another path, 2’) we
reconstruct the so-called B-function [2], 3’) we calculate
the occupation number density using Schuelke’s method[3, 4], and we look for edges of Fermi surface in the
reduced zone. Although Schuelke’s method is known tobe analogous to the LCW-folding, to the best of theauthor’s knowledge, equivalence of the two methods has
not explicitly been documented.
Equivalence of the two methods
The following is one way to show the equivalence ofthe two methods.
The so-called B-function is defined as a Fouriertransform of momentum density ρ(p),B(r ) =  ρ(p ) exp(− i p ⋅ r ) dp ,(1)

and inversely,ρ( p ) = (2 π )−3 B(r ) exp (i p ⋅ r ) d r .

We divide the integration over the whole space into thesum of integrations over the unit cells,

 d r    d r′ .
RicellThen ρ (p + G )iGi= C    d r ′ B (Ri + r ′) ×
GiRi cellexp (i ( p + Gi ) ⋅ (Ri + r ′))= C   d r ′ B(R i + r ′)×
Ricellexp (i p ⋅ (R i + r ′)) ⋅  exp (i G i ⋅ (R i + r ′))GiUsing the following identities,

exp (i Gi ⋅ Ri ) =1 , and  exp (i Gi ⋅ r ′) = C ′ δr ′, 0Gi

we get= C   d r ′ B (Ri + r ′) exp (i p ⋅ (Ri + r ′)) δ r ′, 0Ricell= C  B (Ri ) exp (i p ⋅ Ri ) .
RiFinally, we equate the above to the right hand side of Eq.
(2), ρ (p + G ) = C  B(R ) exp (i p ⋅ R )iGi(1’)

Giocc

iGi

Gik , p + Gi ,(2)b, kwhere Gi represents the reciprocal lattice vectors, and theconstant C collects the irrelevant prefactors. The righthand side is the occupation number density N(k) in therepeated zone scheme which is explicitly expressed by
the summation over Gi. Schuelke’s method is expressedas,C  B(Ri ) exp (i k ⋅ Ri ) = N (k ) ,Ri

where Ri represents the lattice translation vectors.
We rewrite the left hand side of Eq. (2) using Eq. (1’),

ρ (p +G ) = C   d r B (r)×iGiGi(3)iocc= C   δ k , p+G i

The LCW-folding is written as, ρ( p + G ) = C   δiRi.

b, kSince p is always reduced to k by the δ-function in theright hand, we replace p in the left hand side by k and wehave explicitly retrieved Schuelke’s method, Eq. (3).
References[1] D. G. Lock, V. H. C. Crisp, and R. N. West, J. Phys.
F3 (1973) 561.
[2] P. Pattison, W. Weyrich, and B. G. Williams, Solid
State Commun. 21 (1977) 967.
[3] W. Schuelke, phys. stat. sol. (b) 80 (1977) K67.
[4] W. Schuelke, G. Stutz, F. Wohlert, and A. Kaprolat,
Phys. Rev. B 54 (1996) 14381.
*Shiotani@post.kek.jp

exp (i (p + Gi ) ⋅ r) .

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