Dynamical diffraction of neutrons and transition from beam splitter to phase shifter case
Hartmut Lemmel*
Atominstitut der Österreichischen Universitäten, 1020 Wien, Austria
Received 15 June 2007; revised manuscript received 6 August 2007; published 30 October 2007
This paper presents formulas for the transmission and the reflection of neutrons on a perfect crystal blade in
symmetric Laue geometry. While the standard formulas are valid either for the situation very close to the Bragg
condition or far off the Bragg condition index of refraction model the formulas presented here smoothly
cover the whole range of transition. The paper concludes with experimental considerations.
DOI: 10.1103/PhysRevB.76.144305 PACS numbers: 61.12.-q, 03.75.Dg
I. INTRODUCTION
If neutrons are sent through a beam splitter made of a
perfect crystal in Laue geometry Fig. 1a, a very distinct
pattern in phase and intensity can be observed, known as the
Pendellösung oscillations. The pattern depends on the angle
alignment between the beam and the beam splitter on an
arcsecond scale. This was demonstrated experimentally by
Shull
1,2
and is well understood with the theory of dynamical
diffraction.
3–8
On the other hand, if the crystal blade is ro-
tated far off the Bragg condition, it behaves similar to a
noncrystalline material Fig. 1b and can be used as a phase
shifter. By rotating the blade up to a few degrees the optical
path length is changed due to the index of refraction. This
way the phases of interferometry experiments are controlled.
In the recent years more interest has arisen concerning the
transition range between the beam splitter case and the phase
shifter case. In particular, experiments have been proposed to
measure the neutron electron scattering length by precision
measurements of the phase and intensity oscillations around
the Bragg condition.
9–11
However, the standard solution of
the theory of dynamical diffraction does not cover the tran-
sition range, as it contains approximations which restrict its
validity to the arcsecond scale around the Bragg angle. De-
pending on the experimental setup and the desired precision,
the standard formulas may not be sufficient.
In this paper I derive formulas covering the whole angular
range for the symmetrical Laue geometry Sec. III. Before, I
briefly summarize the standard formulas and their limitations
Sec. II. Finally, I discuss experimental considerations con-
cerning the rotation of a Laue crystal through the Bragg con-
dition Sec. IV.
II. THE STANDARD FORMULAS
A. The beam splitter case
The theory of dynamical diffraction of neutrons is con-
tained in several text books and articles.
4–7
Here I give only
a sketch of the derivation of the beam splitter formulas and
summarize the results for the transmission factor t and the
reflection factor r. The geometry for t and r is illustrated in
Fig. 1a.
The neutron wave function r
inside the crystal is cal-
culated by solving the stationary Schrödinger equation on the
periodic crystal potential. After transforming the Schrödinger
equation into the reciprocal space and using the Bloch ansatz
r
= expiK
r
H
u
H
expiH
r
1
with the sum running over all reciprocal lattice sites, we get
an infinite system of coupled equations for the amplitudes u
H
2
2m
|K
+ H
|
2
- E u
H
=-
H
'
V
H
'
-H
u
H
'
2
with the neutron energy E, the neutron mass m, and the Fou-
rier components V
H
of the potential Vr
=
H
V
H
expiH
r
. In
the case of normal Bragg reflection only two reciprocal lat-
tice points lie near the Ewald sphere and the two-beam ap-
proximation applies. All but two amplitudes can be set to
zero and the infinite system of equations is reduced to two
coupled equations for the unknown amplitudes u and u
H
which refer to the forward and reflected direction, respec-
tively,
2
2m
K
2
- E u =- V
0
u - V
-H
u
H
, 3a
2
2m
K
H
2
- E u
H
=- V
H
u - V
0
u
H
. 3b
K and K
H
denote the absolute values of the forward and the
reflected wave vector inside the crystal. The reflected vector
is given by K
H
= K
+ H
with H
denoting the reciprocal lattice
vector of the reflecting planes.
The potentials V
0
and V
±H
are given by the crystal prop-
erties. The general formula is
x
z
0
z
D
e
i
k r
e
i
k r
r e
i
k
H
r
t
a) b)
0
z
D
e
i
k r
e
i
k r
θ θ
t
ph
FIG. 1. Sketch of a beam splitter in symmetrical Laue geometry
a and an ordinary phase shifter b. The transmission and reflec-
tion factors are denoted by t and r not to be confused with the
vector r
.
PHYSICAL REVIEW B 76, 144305 2007
1098-0121/2007/7614/1443059 ©2007 The American Physical Society 144305-1