12
Martin Fally, Christian Pruner, Romano A. Rupp, and Gerhard
Krexner
Fakultat fur Physik, Institut fur Experimentalphysik, Universitat
Wien, A-1090 Wien, Osterreich http://nlp.exp.univie.ac.at,
[email protected], (V: 11th May 2005)
When the subject of photorefractive effects began with the
discovery of light- induced refractive-index inhomogeneities in
lithium niobate [1], neutron optics had already been established
for more than 20 years [2, 3]. Both of the fields have evolved
independently of each other into important branches of science and
industry. In 1990 those dynamic areas were linked by an experiment
in which cold neutrons were diffracted from a grating created by a
spatially inhomogeneous illumination of doped
polymethylmethacrylate (PMMA). A typical holographic two-wave
mixing setup was used to record a refractive-index pattern, a
grating, in PMMA that was reconstructed not only with light, as
usual, but also with neutrons [4]. Evidently, the illumination
induced refractive-index changes for both light and neutrons! In
analogy to light optics this phenomenon is called the
photo-neutron-refractive effect.
The chapter is organized as follows: Starting with a concise
explanation of the relevant concepts in neutron optics,
electrooptics, and photorefraction, as well as diffraction
phenomena, we introduce PMMA and the electrooptic crys- tal LiNbO3
as examples of photo-neutron-refractive materials. The main part is
concerned with neutron diffraction experiments performed on
deuterated PMMA (d-PMMA). It is shown that this type of experiment
can be useful in studying the polymerization process itself, when
serving simply as a neutron-optical el- ement, or when probing
fundamental properties of the neutron. The latter is in particular
true of electro neutron-optic LiNbO3, where the diffracted neutrons
are inherently exposed to extremely high electric fields due to the
light-induced charge transport. Corresponding experiments are
presented and future perspec- tives of photo-neutron-refractive
materials as well as their possible applications are discussed.
Finally, atomic-resolution neutron holography is introduced and
conducted experiments are presented.
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12.1 Basic Concepts
This part will provide the necessary prerequisites in
photorefraction, holography, neutron optics, and diffraction
physics to be able to understand the experiments and the obtained
results. First, we will present the technique for preparing the
gratings, introduce the equation of motion for neutron diffraction,
define the neutron-optical potential and the neutron-refractive
index, and finally discuss the relevant terms of the
neutron-optical potential, which is modulated by inhomo- geneous
illumination with light.
12.1.1 Holographic Gratings
Soon after the discovery of photorefraction [1], the technological
importance of the effect became clear, e.g., that such materials
can be used for information stor- age and as holographic memories
[5]. The big advantage over materials changing their absorption,
like standard photographic films, is that intensity losses do not
occur, and thus the whole volume rather than the surface may be
used for data storage. Important consequences when utilizing
thick-volume phase gratings in diffraction experiments are that a
sharp Bragg condition must be obeyed and that multiple scattering
effects must be taken into account, i.e., dynamical diffraction
theory has to be considered. Historically, the latter was
originally developed for X-rays by Darwin, Ewald, and von Laue at
the beginning of the twentieth cen- tury and extended in several
review articles (see, e.g., [6]). When lasers became available and
the technique of holography had become popular, Kogelnik rein-
vestigated the effects of coupled waves for light [7]. Finally,
when crystals of highest quality and thickness could be produced as
a consequence of semicon- ductor technology, Rauch and Petrascheck
performed this task for neutrons [8]. We will summarize the results
of this theory in so far as they are necessary to interpret our
experiments correctly.
As a first step, we will discuss the typical setup for the
preparation of light- induced refractive-index gratings, which is
sketched in Figure 12.1. Two coherent plane light waves interfere
in the photo-neutron-refractive material (two-wave mixing). In the
simplest case with waves of equal intensity and mutually
parallel
d
z
x
FI G U R E 12.1. Sketch of the setup for the preparation of
light-induced refractive- index gratings (hologram recording) and
the reconstruction with light or neutrons. I (i) denotes the
incoming intensity, for fur- ther abbreviations, see text.
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12. Neutron Physics with Photorefractive Materials 323
polarization states, the resulting modulation of the light pattern
is sinusoidal I (x) ∝ cos(K x) with an elementary grating
spacing
= 2π
2 sin (θ [e]) . (12.1)
Here 2θ [e] denotes the angle between the interfering beams in air
and λ0 the wavelength of light in vacuum. All angles are measured
in the medium un- less indicated by the superscript [e]. Typical
values for the grating spacing are 300 nm < < 2000 nm. In
general, this inhomogeneous illumination of the sample results in a
spatially dependent refractive-index change that can be ex- panded
in a Fourier series with a fundamental periodicity K :
n(x) = ∑
s
ns cos (sK x + φs). (12.2)
The actual pattern of course depends on the properties of the
material and the mechanism of photo-neutron-refraction. Usually in
electrooptic crystals a linear response, i.e., n(x) ∝ I (x),
ns>1 ≡ 0, is obtained though it may be nonlocal (φ1 = 0). This
is not always the case: e.g., the photorefractive effect in PMMA
depends on illumination time and intensity and is strongly
nonlinear as will be shown in Section 12.3.5.
Diffraction from such refractive-index gratings, which represent
thick holo- grams [9], is governed by the basic formulae of
dynamical diffraction theory. In our experiments we deal only with
nearly lossless dielectric gratings in trans- mission geometry. To
gain information about the material parameters, light or neutrons
are diffracted from those gratings in the vicinity of the
sth-order
Bragg angle [e] s = arcsin[sλ0/(2)]. In particular, the diffraction
efficiency
η(θ )s = Is/(Is + I0) is measured as a function of the deviation
from s , the so-called rocking curve, where Is is the sth
diffraction order. In the standard case where only zero-order
(forward diffraction) and first-order diffracted waves are present,
this reduces to the well-known definition [10]. For a monochromatic
plane wave in symmetric transmission geometry the diffraction
efficiency is described by [7]
ηs(θ, ν) = ν2 s sinc2
(√ ν2
. (12.5)
The thickness of the grating, which in the ideal case is identical
to the sam- ple’s thickness, is denoted by d. The parameters νs
(grating strength) and ξs (Off Bragg parameter) contain the
relevant material information. In particular, the light-induced
refractive-index change ns of order s can be probed and
determined
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324 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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by such a measurement. If any of those parameters appears without a
numerical subscript we assume the case of a linear response, i.e.,
only the first-order diffrac- tion is present and we simply speak
about diffraction efficiency η. Then n1, the first-order Fourier
coefficient in the expansion in (12.2), is frequently replaced by n
in the literature. Another useful measure is the integrated
diffraction efficiency (i.e., integrated reflectivity times
ν)
I (ν) := d
J0(u) du, (12.6)
with J0(u) the zeroth-order Bessel function. The practical
importance of I (ν) lies in the fact that this quantity is
independent of the lateral divergence of the beam and that it can
be accessed experimentally by performing rocking curves (see
Section 12.3.3).
12.1.2 Neutron Optics
The equation of motion for a field of nonrelativistic particles,
e.g., cold neutrons with a wavelength 5 A < λ < 50 A, is
Schrodinger’s equation. We restrict our considerations to coherent
elastic scattering (neutron optics) in condensed matter. Then the
coherent wave and the coherent scattering are described by a
one-body Schrodinger equation with the (time-independent)
neutron-optical potential V (x) [11], the energy eigenvalues E, and
m the neutron mass:
H(x) = E(x), (12.7)
H = − h2
2m ∇2 + V (x). (12.8)
For a general treatment of neutron optics see, e.g., [11, 12].
Inserting (12.8) in (12.7) leads to a Helmholtz-type equation, with
k0 being the magnitude of the vacuum wave vector, if we properly
define the refractive index for neutrons nN :
[∇2 + (nN (x)k0)2](x) = 0, (12.9)
nN (x) = √
E . (12.10)
It is evident that any change in the potential of matter results in
a refractive- index change for neutrons. In our particular context
the main task is to modify V (x) by illumination with light, i.e.,
to observe a photo-neutron-refractive ef- fect. Therefore it is
necessary to discuss the relevant terms of the neutron-optical
potential. Interactions between the neutron and condensed matter
may be classi- fied pragmatically into three groups according to
their magnitude [13, 14]: The strong interaction dominates in
nonmagnetic materials, whereas the electromag- netic neutron-atom
interaction is at least 2 to 3 orders smaller. The latter, however,
is important if high electric fields are applied and thus must be
taken into account (see Section 12.4).
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12. Neutron Physics with Photorefractive Materials 325
As a consequence of the strong interaction, the scattering
amplitude of cold neutrons is proportional to a constant and
independent of the scattering angle to first-order approximation,
because of the extremely short interaction length. Therefore the
nuclear potential is replaced by the Fermi pseudopotential. Dealing
with bulk matter, we are interested in the macroscopic optical
potential VN (x), which is simply given by a series of Fermi
pseudopotentials
VN (x) = 2πh2
m bρ(x). (12.11)
Summation is performed over the different nuclei j with scattering
length b j
at the corresponding sites y j . Here bρ(x) denotes the so-called
coherent scat- tering length density. Assuming that the internal
degrees of freedom of the atoms are statistically independent of
their positions, (12.11) can finally be simplified
VN (x) = 2πh2
m bρ(x). (12.12)
Here b is the mean bound scattering length averaged over a unit,
e.g., a unit cell in a regular crystal or a polymer unit, and ρ(x)
is the number density.
The photo-neutron-refractive effect in PMMA is based on the fact
that the pho- topolymer has a higher number density than the
monomer MMA. By illuminating the photosensitized sample with a
sinusoidal light pattern, we also modulate the number density ρ(x)
= ρ + ρ(x) sinusoidally. Thus the neutron-optical poten- tial
reads
VN (x) = VN + VN (x) = 2πh2
m (bρ + bρ(x)), (12.13)
with the mean scattering length density per unit volume bρ . Next
we will consider the influence of a (static) electric field E(x) on
the
neutron-optical potential. From (12.11) it is evident that under
the application of an electric field the nuclear contribution of
the potential VN can be influenced by changing either at least one
of the scattering lengths or at least one of the partial number
densities δ(x − y j ) = ρ j (x). Changes of the scattering length
could be established by an influence of the electric field on the
nuclear polarizability [15]. Here, we discuss only its influence on
the number density, which is larger by orders of magnitude. Let us
assume an electrooptic crystal, which thus is also piezoelectric by
symmetry. Application of an electric field E(x) will hence lead to
a strained crystal, i.e., to a density variation ρ(x). Again
(12.13) is valid. The magnitude of the density modulation then
depends on the symmetry as well as the values of the compliance and
the piezoelectric tensor [16]. Moreover, a neutron moving in an
electric field E(x) with velocity v gives rise to an additional
contri- bution to the potential (Schwinger term, spin-orbit
coupling. Aharonov–Casher effect) due to its magnetic dipole moment
μ [17]. If an electric field is applied to
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a crystal, the corresponding term in the potential reads
VS(x) = − hε
mc2 μ · [E(x) × k]. (12.14)
Here, c is the velocity of light in free space and the static
dielectric constant ε
enhances the Schwinger term. Further terms in the expansion of the
potential that are linear in the electric field are the Foldy
effect [18, 19] VF (x) = − h|μ|
2mc2 [∇ · E(x)], and a tiny contribution due to a possibly existing
electric dipole moment d (EDM) of the neutron VEDM = d · E(x).
Thus, the total neutron-optical potential in the presence of a
static electric field amounts to V = VN + VS + VF + VEDM.
Finally, we estimate the neutron-refractive index for the materials
that will be discussed in next sections. If we take into account
only the leading terms in the potential, i.e., employing (12.11),
the neutron-refractive index is
nN = √
2π bρ. (12.15)
Note the quadratic dependence on the wavelength λ. The refractive
indices are for a typical cold neutron wavelength of λ = 20 A: 1 −
6.6 × 10−5 (PMMA). 1 − 4 × 10−4 (d-PMMA), and 1 − 2.6 ×
10−4(7LiNbO3). At this point it is worth emphasizing that in
photo-neutron-refractive materials we are dealing with changes of
the refractive index. This is quite a challenging task for
neutrons.
12.2 Materials
So far, two different types of photo-neutron-refractive effects
have been real- ized: changes of the optical potential V ∝ ρ
resulting from chemooptics in photopolymers and V ∝ E resulting
from electro (neutron) optics in crystals. Here, we will discuss
the basic mechanisms of photo-(neutron) refraction in such
materials, represented by PMMA and LiNbO3 respectively.
12.2.1 The Photo-Neutron-Refractive Mechanism in PMMA
The polymer PMMA is well known in everyday life as Plexiglas. The
basis for the occurrence of a photo-neutron-refractive effect in
PMMA is the large difference of number densities for the monomer
MMA and the polymer PMMA, ρMMA : ρPMMA = 0.8 : 1. A photoinduced
polymerization then allows us to modulate the density and hence the
neutron-refractive index by illumination. Moreover, the mechanical
and, in particular, the excellent optical properties have made PMMA
the favorite candidate not only for fundamental studies but also
for potential technical applications.
The polymerization process is performed in two steps: a thermal
pre- polymerization and a light-induced post-polymerization. To
polymerize the monomer MMA, a (C = C) double bond must be split.
This is established by free radicals, which are created by a
thermoinitiator. At elevated temperatures
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12. Neutron Physics with Photorefractive Materials 327
radicals are formed that react with MMA and start a chain reaction
resulting in the polymer. Termination processes at low temperatures
(T ≈ 325 K) leave us with a mixture of residual monomers solved in
the PMMA matrix. Those monomers now serve as a reservoir to restart
the polymerization by illumination. To sensitize the material,
photoinitiator, i.e., a substance [20] that decomposes into free
radicals under illumination, had been added prior to prepolymeriza-
tion. Thus the polymerization is restarted in the bright regions,
yielding a density modulation and, according to (12.13), a
neutron-refractive-index change. Typical recording intensities were
in the range of several hundred W/cm2. Exposure times between 2 and
60 seconds were employed. This parameter has turned out to be an
important quantity for the photorefractive response (see Section
12.3.5).
The photosensitivity of the doped PMMA system becomes manifest
primarily in light-induced absorption changes α(Q) [20]. Havermeyer
et al. proposed a model based on two relevant processes: the decay
of the photoinitiator into radicals and inert molecules [20, 21]
and a light-induced termination for the radicals. The first
mechanism, which is responsible for the polymerization, leads to a
permanent change of the refractive index: the photorefractive
effect. The solution of the rate equation, which nicely describes
the experimentally obtained results, yields for the exposure
dependence of the light-induced absorption changes
α(Q) = a1[1 − exp (−k1 Q)] + a2[exp (−k2 Q) − 1], (12.16)
with Q = I (i)t the exposure and ai , ki proportionality and rate
constants re- spectively. Via the Kramers–Kronig relations the
kinetics of the corresponding refractive-index change nL for light
can be obtained. However, it is impossible to discriminate between
the contributions of electronic polarizability changes and density
changes by employing light. On the other hand, neutron diffraction
is sen- sitive only to density variations, i.e., the
refractive-index change nN originates therefrom. Thus, by combining
light and neutron diffraction we can unravel the contributions (see
Section 12.3.3).
12.2.2 The Photo-Neutron-Refractive Mechanism in LiNbO3
LiNbO3 was the first photorefractive material to be discovered [1,
5]. Its pho- torefraction is based on the excitation, migration,
and trapping of charges when illuminated by coherent light
radiation. A space-charge density is building up, which, according
to Poisson’s equation, leads to a space-charge field Esc and via
the electrooptic effect to a refractive-index change n for light;
see, e.g., [10]. When illuminating the photorefractive sample with
a sinusoidal light pattern as described in Section 12.1.1, the
space-charge field in the stationary state is given by
Esc(x) = −E1 cos (K x + φ). (12.17)
Here E1 is the magnitude of the effective electric field, i.e., the
first coefficient in a Fourier series, which depends on the
recording mechanism. In LiNbO3:Fe the bulk photovoltaic effect is
dominant and thus E0 ≈ EPV and φ ≈ 0. Employing
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328 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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the linear electrooptic effect (Pockels effect), the
refractive-index change for light then is given by
nL (x) = −1
Lr Esc(x), (12.18)
with nL the refractive index for light and r the effective
electrooptic coefficient. It seems worth discussing the linear
electrooptic effect more accurately at this point. The electrooptic
tensor ri jk evolves from the expansion of the inverse di-
electric tensor (ε)−1 i j with a static electric field Ek .
Considering in addition the
elastic degrees of freedom, it is necessary to define which of the
thermodynamic variables, stress T or strain S, are to remain
constant when we take the derivatives. In an experiment, both of
these cases can be realized. Keeping the strain constant results in
the clamped linear electrooptic coefficient r S . The applied
electric field changes the refractive index directly, i.e., by
slightly modifying the electronic con- figuration. However, when
keeping the stress constant, the free linear electrooptic
coefficient rT is measured. Here, in addition, a contribution via
the piezoelectric coupling (di jk) in combination with the
elastooptic effect (pi jlm) must be con- sidered: r T
i jk = (rS i jk + pE
i jlmdklm
) Ek (12.19)
for the tensor of the optical indicatrix. In LiNbO3, rS contributes
about 90% of the polarizability to rT. This is plausible, since
light is quite sensitive to electronic changes but much less to
density variations.
The Electro Neutron-Optic Effect
(hk)2 V = rN E(x). (12.20)
Consequently, we call the proportionality constant rN the electro
neutron-optic coefficient (ENOC). By inserting (12.13) ff. into
(12.20) and comparing the cor- responding terms, we can identify
the following relations:
r S N = 2λ|μ|
hc2
N + λ2
12. Neutron Physics with Photorefractive Materials 329
TA B L E 12.1. Contributions to the ENOC for LiNbO3 in [fm/V] for
cold neutrons with λ = 2 nm and a grating spacing = 400 nm.
Contribution r T N − r S
N Schwinger-term Foldy-term EDM
[fm/V] 3 ±0.02 −2 × 10−6 < 10−9
These equations are valid for LiNbO3 (point group 3m) if the
grating vector K is parallel to the trigonal c-axis and the vectors
μ, E, k are mutually perpendicular. Moreover, the approximation
e333/C E
3333 ≈ d333 was made. The sign in front of the dielectric constant
in (12.21) is determined by the direction of the neutron spin μ : +
for parallel and − for antiparallel to E(x) × k. In another
geometry with μ⊥[E(x) × k] this term even vanishes. The complete
expressions are given by equation (12.12) in [16]. In comparison to
the electrooptic coefficient for light, the situation is reversed
in this case: r S
N r T N . This is again reasonable
and reflects the corresponding contributions to the neutron-optical
potential. We therefore suggest discontinuing the use of terms like
“primary” or “true” for r S and “secondary” electrooptic
coefficient for rT , which can be found in the literature. An
estimation of the various contributions to the electro
neutron-optic coefficient for LiNbO3 is summarized in Table 12.1.
In analogy to light optics and (12.18), the
neutron-refractive-index change induced by a holographically
created space-charge field Esc amounts to
nN (x) = 1
12.3 Experiments
12.3.1 Neutron Experimental Setup
The grating spacings of the holographically produced gratings are
on the order of several hundred nanometers. Since cold neutrons are
employed for the measure- ments, the corresponding Bragg angles are
a few tenths of a degree. Therefore, small-angle-neutron-scattering
facilities (SANS) are used for the experiments. In a typical
diffraction experiment, first the photoinduced neutron-refractive
grat- ings are adjusted to obey the Bragg condition for neutrons.
Then the diffracted and forward diffracted intensities are measured
as a function of time and/or of the deviation from the Bragg-angle
by rotating the sample. The diffracted and transmitted neutrons are
monitored with the help of a two-dimensional position- sensitive
detector. The setup is depicted schematically in Figure 12.2. Aside
from the neutron flux, which is defined by the available neutron
source, two important experimental parameters are the collimation
of the beam and the properties of the velocity selector. The first
determines the spread of angles θ impinging on the
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330 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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FI G U R E 12.2. Measurement setup for neutron diffraction from
light-induced neutron- refractive-index gratings. The sample is
placed on a rotation stage. Typical collimation lengths Lcoll are
about 15 to 40 m.
sample, the second the wavelength spread λ. The collimation can be
tuned by using slits along the neutron beam path. Typically a
rectangular entrance slit of width xn and a slit just in front of
the sample xs are used. To ensure a sufficiently collimated beam,
the distance Lcoll between the slits ranges from about 15 m up to
40 m. The wavelength distribution g0(λ) can also be adjusted to the
experimental needs and is typically of a triangular shape around a
central wavelength. How- ever, in practice, these parameters are
optimized according to minimum demand (coherence properties) on the
one hand and convenience (measuring time) on the other. The angular
(transverse momentum) distribution gtrans(θ ) forms a trape- zoid
with a base θb = |xn + xs |/Lcoll and a top θt = |xn − xs |/Lcoll.
Typical values for the spread are λ/λ ≈ 10% and θ = (θb + θt )/2
< 1 mrad.
12.3.2 The Early Experiments (History)
A photo-neutron-refractive material was realized for the first time
by Rupp et al. using a slab of PMMA. This PMMA matrix contained
residual monomer and a photoinitiator. The grating ( = 362 nm) was
prepared as described in Sec- tion 12.1.1 with a maximum
diffraction efficiency for λ = 1 nm neutrons in the range of ηB =
10−3% [4]. Interpreting (12.4), (12.13), and (12.15), the
diffraction efficiency ηB at the Bragg condition reads
ηB = sin2(ν) = sin2
) . (12.24)
In a follow-up publication the neutron wavelength, the sample
thickness, and the grating spacing were varied to reach maximum
diffraction efficiencies of about 0.05% [23]. The major limitation
of those early attempts was the use of protonated PMMA with its
high incoherent scattering cross section. Since then the deuterated
analogue d-PMMA has been used instead [24], considerably reducing
incoherent scattering and increasing the coherent scattering length
density. By optimizing the sample preparation, the polymerization,
and the exposure process, maximum diffraction efficiencies of up to
70% for λ = 2 nm neutrons are possible (Figure 12.3).
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12. Neutron Physics with Photorefractive Materials 331
-6 -4 -2 0
0 2 4 6
FI G U R E 12.3. Rocking curve ηN (θ ) for neutrons with wavelength
λ = 2 nm. Grating spacing = 380 nm, λ/λ = 0.1, Lcoll = 17.6 m, xs =
10 mm, xn = 55 mm.
These first successful experiments have opened up a wide field of
potential applications:
1. By following the kinetics of relaxation processes in
photopolymers, informa- tion on the complex phenomenon of
glass-forming processes and polymeriza- tion can be obtained
(materials scientific aspect; see Sections 12.3.3, 12.3.5).
2. By diffracting the neutrons in the presence of an electric
field, fundamental properties of the neutron itself are revealed
(pure physics; see Section 12.4).
3. Utilizing knowledge about how to produce gratings for cold
neutrons, neutron- optical devices can be designed. Mirrors, beam
splitters, lenses, and interfer- ometers are of outstanding
technological relevance these days (technological aspect; see
Sections 12.3.4, 12.3.6). Those instruments then can again be used
in turn to obtain material parameters (e.g., scattering lengths) or
insight into the foundations of quantum physics (e.g., EDM).
12.3.3 Temporal Evolution of the Polymerization
In order to obtain information about the kinetics of polymerization
it is im- portant to systematically improve the production process
of light-induced holo- graphic gratings in d-PMMA (see Section
12.2.1). In the photopolymer system d-PMMA/DMDPE the recorded
grating itself is used as a sensor to directly follow the
glass-forming process over large time scales. Up to the last few
years only diffraction experiments with light have been performed,
which has turned out to be a favorable technique because of its
simplicity. According to the Lorentz–Lorenz relation small
photoinduced changes of the refractive index for light nL may be
the result of changes either in the density ρ or the polarizability
of the material, nL ∝ (ρ + ρ). Neutron diffraction is a
complementary tech- nique in the sense that only density changes ρ
are probed. Combining (12.13) and (12.15) yields
nN = bλ2
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332 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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0 10 20 30 40 50 60 70 0
5
10
15
light λ0 = 543 nm
FI G U R E 12.4. Kinetics of the diffraction efficiency for
neutrons ηB;N (t) (left scale) and light ηB;L (t) (right scale) at
the exact Bragg-angle respectively. The measurement was performed
after restarting the polymerization by illumination with uv-light
for 10 seconds [25]. The reading wavelengths for neutrons were λ =
1.1 nm and λ = 543 nm for light respectively. Sample: d-PMMA.
Therefore light and neutron diffraction from photoinduced
refractive-index changes can serve as a tool to clarify several
aspects of the polymerization process in PMMA, in particular its
kinetics. Since the glass-forming processes are irreversible, soon
the demand for a facility was addressed that allowed for the
simultaneous performance of light- and neutron-optic experiments.
This led to the development and the design of HOLONS, which will be
introduced in Section 12.3.4 in detail.
The experimentally accessible quantity is the diffraction
efficiency, which is related to the physically relevant parameter n
in the ideal case by (12.3). Note that a unique inverse function
does not exist. The kinetics of the diffraction ef- ficiency ηB(t)
for light and neutrons that were recorded simultaneously reveal
another problem: they do not at all obey a sin2(ν) dependence.
Figure 12.4 shows the temporal evolution of ηB;N (t) and ηB;L (t)
for the first three days after pho- topolymerization had started.
The reasons for the discrepancy between theory and experiment are
inhomogeneities along the sample thickness and across the sample
area [26]. In the case of neutron diffraction, in addition the
wavelength and angular distribution are responsible. Fluctuations
in the refractive index change or of the grating spacing lead to a
decrease of the contrast between the maxima and min- ima of the
rocking curve. This has a huge influence at the exact Bragg
condition, where the extrema in the curve are smeared out until
complete disappearance. The fact that the measurements are
conducted with a partially coherent neutron beam calls for the
complete rocking curve to extract the refractive-index changes
unam- biguously. However, time-resolved experiments and
simultaneous light diffraction still need to be achieved. Therefore
the strategy is to measure rocking curves η(θ ) from time to time
(1 h for light, 10 h for neutrons) to ensure the correct absolute
value of n and to monitor ηB in the meantime. Rocking curves for
neutrons are shown in Figure 12.5 at certain times after
initializing the refractive-index change. When interpreting Figure
12.5, it is striking that the shape of the rocking curve changes
significantly during the exposure time of the gratings [27].
Starting with a trapezoidal shape that resembles the angular
distribution, we finally end up with a triangular or
Lorentzian-shaped curve. To explain this feature we have to account
for the partial coherence of the neutrons. Assuming a normalized
angular
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12. Neutron Physics with Photorefractive Materials 333
η N (θ) [%
5
10
15
20
25t [h] 2 15 22 40 47 63 117 240
θ [mrad]
FI G U R E 12.5. Rocking curve ηN (θ ) for neutrons at several
times after starting the photopolymerization pro- cess [25]. λ =
1.1 nm, Lcoll = 14.4 m, xn = 15 mm, xs = 2 mm, λ/λ = 0.23. Sample:
d-PMMA.
and wavelength distribution g0(k) and gtrans (θ ) as discussed in
Section 12.3.1, the measured rocking curve is the convolution with
η(ν, θ) from (12.3). The inte- grated diffraction efficiency Ipc(ν)
= ∫
g0(k) ∫
η(ν, θ)dθ ]dθdk,
however, is independent of the angular spread. This is important
since the inte- grated diffraction efficiency is accessible
experimentally by integrating the mea- sured rocking curve. By Iη
we will denote the experimentally determined value multiplied by d/
according to (12.6). Estimating the influence of the wave- length
distribution yields Ipc(ν) = I (ν) + (νλ/λ)2[J0(ν) − ν J1(2ν)]/6,
with J1(u) being the first-order Bessel function. For the
experimental reasons already discussed, it is important to relate
ηB with Ipc(ν), which is not possible in general. In the limit of
small ν as well as large ν, approximations can be found [25]. Ipc
can be solved for ν numerically and thus allows one to evaluate the
refractive-index change nN for neutrons. The temporal evolution of
the refractive-index changes follows power laws quite well for
light and neutrons. In fact, it is astonishing that several hours
after illumination with uv-light, which lasts for a few seconds
only, the refractive index evolves over time spans of weeks. The
first hours are gov- erned by an approximate
√ t-dependence of the changes for neutrons and light.
The reason for this kinetics is still under debate. Investigations
of samples with different grating spacings lead to similar values
for the exponent. This serves as a hint that diffusion does not
play a decisive role in the photopolymerization process.
12.3.4 HOLONS
The measurements presented in the last section were performed using
a novel experimental facility at the Geesthacht Neutron Facility
(GeNF), which was de- signed for time-resolved simultaneous
diffraction experiments with light and neu- trons. In addition, it
allowed us to utilize a complete holographic setup while con-
ducting this type of experiment. The acronym HOLONS stands for
Holography and Neutron Scattering [28]. It basically consists of a
holographic optical setup including a vibration-isolated optical
table, an argon-ion laser (λ0 = 351 nm) for
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Mirrors, Beamsplitter
→ Detection unitCollimation line for neutrons: λ=1 - 1.4 nm
FI G U R E 12.6. Sketch of the HOLONS experiment at the SANS-2. The
light beams for recording the grating (λ0 = 351 nm), for
reconstructing (λ0 = 473 nm), and the neutron beam (λ = 1.1 nm) are
indicated by grey lines.
recording the gratings, and a diode-pumped solid-state laser (λ0 =
473 nm) for reading. In Figure 12.6 a sketch of the facility is
presented. The optical bench itself is placed on a metallic plate,
which can be transferred from the HOLONS cabin to the cold neutron
beamline SANS-2 by means of a crane. The HOLONS cabin itself is
situated in the guide hall of the GeNF. When performing an
experiment, a rotation of the whole optical bench (weight: 1.2
tons) with respect to the incident neutron beam is made possible
using air cushion feet and translation stages. Accu- racy amounts
to about ±0.01 deg over a range of ±2 deg. This technique ensures
that the sample can be positioned in the correct Bragg angle for
neutrons but keeps the light optical setup unchanged. In other
words, neutron rocking curves ηN (θ ) can be performed while one
simultaneously measures the diffraction efficiency ηB;L (t) for
light! This exactly meets our demand for clarifying the kinetics of
photorefraction in doped polymers. The photo-neutron-refractive
sample is fixed on another rotation stage with high accuracy
(±0.001 deg) in the common center of both rotation devices. Because
of the small Bragg angle for cold neutrons (1/10 deg), the neutron
beam impinges nearly perpendicularly onto the sample surface.
Therefore, the holographic recording geometry must be chosen in
asymmetric configuration so that the beam splitter does not block
the neutron beam. Thus the sample surface-normal is inclined with
respect to the axis beam splitter—sample but still remains the
bisector of the recording beams. Figure 12.7 is a photo- graph of
the HOLONS experiment at the SANS-2. Summarizing the benefits of
HOLONS, we would like to emphasize its importance for
improving and controlling the quality of photo-neutronrefractive
gratings; producing diffraction elements for calibration standards,
beam splitters, lenses,
and further neutron-optical devices on the basis of
photo-neutron-refractive materials;
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12. Neutron Physics with Photorefractive Materials 335
FI G U R E 12.7. Picture of the HOLONS experiment. To the left the
argon-ion laser for recording the gratings and to the lower right
the evacuated tube for the detection system can be seen. (1) marks
the end of the neutron collimation system, (2) the beam expansion
for the recording beams, (3) the photo- neutron-refractive sample
(d- PMMA). Compare to the beam diagram of the schematic in Fig- ure
12.6. Photograph courtesy of Geesthacht Research Center.
performing time-resolved simultaneous measurements of the
diffraction effi- ciencies for light and neutrons;
simultaneously recording holographic gratings and reconstructing
them by light and neutrons.
12.3.5 Higher Harmonics
Among the noteworthy characteristics of the photopolymerization is
its pro- nounced nonlinear response to light exposure. This can be
attributed to the growth and termination of the polymer chains in
d-PMMA. Illuminating the photosen- sitized sample with a sinusoidal
light pattern results in a refractive-index change which can be
expanded in a Fourier series according to (12.2) with nonvanishing
higher Fourier coefficients. Performing diffraction experiments,
this means that in addition to the (+1st, −1st) diffraction orders,
higher harmonics appear. When the photorefractive effect in PMMA
was studied by light, it turned out that the diffraction efficiency
of the harmonics can be tuned by the proper choice of exposure.
According to the model of the photorefractive effect in doped PMMA
(Section 12.2.1, [20]), the absorption changes and thus the
refractive-index changes depend exponentially on the exposure Q.
Only for very low exposure can the response be approximated to be
linear. For comparison of the measured data with the results of the
absorption model we used reasonable and appropriate parameters [20]
for the rate and proportionality constants in (12.16). Further, we
assumed that n ∝ α and I (x) = I (i)(1 + cos (K x)), and expanded I
(ν(Q)) into a Fourier series. The exposure dependencies of the
first and second Fourier coefficients then were compared to the
measured values Iη(Q). The experimen- tally obtained data and the
results of the model are presented in Figure 12.8. The constraint
to obey the Bragg condition limits the possibility of observing
diffraction from higher harmonics to 2/s > λ0, where s denotes
the diffraction order. Therefore, it was possible to detect the
second harmonic only with light of λ0 = 351 nm. This is
inconvenient and may even damage the refractive-index
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Iη
Exposure Q [Ws/cm2]
FI G U R E 12.8. Iη(Q) for a PMMA sample ( = 380 nm) exposed to
laser irradiation with λ0 = 351 nm for different time spans (2 s, 4
s, 8 s, 16 s) and an intensity of I (i) = 1700 W/m2. Rocking curves
were measured for the first-order Bragg peak (open triangle) and
the second-order Bragg peak (solid triangle) using light of
wavelengths λ0 = 543 nm and λ0 = 351 nm respectively. The solid
line represents the first Fourier coefficient of I (ν) employing
the proposed absorption model.
profile of the sample, since it is photosensitive in the uv region.
An elegant way of escaping this problem is to employ neutron
diffraction instead. After d-PMMA samples with high diffraction
efficiencies for neutrons had become available, Havermeyer et al.
detected the second harmonic for three samples with grating
spacings = 400, 250, 204 nm [29]. For the latter two samples it is
inherently impossible to detect the second diffraction order by
means of light. Nowadays higher harmonics up to the 4th diffraction
order have been observed, with a spacing /4 = 135 nm [21, 25].
Figure 12.9 shows the counting rate for four
0.0 2.5 5.0 7.5
/P ix
h
4K
3K
2K
1K
0K
FI G U R E 12.9. Counting rate along a horizontal line (= diagonal
line in the inset) of the detector matrix for four angular
positions: θ ≈ 1 (squares), θ ≈ 2 (circles), θ ≈ 3
(triangles), θ ≈ 4 (crosses). The full line represents an Off-Bragg
measurement at θ ≈ 61 [25]. Bragg peaks up to the fourth order are
clearly visible. The inset shows part of the detector matrix with
0K , 1K , 2K , 3K , 4K denoting the corresponding diffraction
orders and the sample in the Bragg position for the fourth order
(4K ) [21].
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12. Neutron Physics with Photorefractive Materials 337
angular positions θ ≈ 1,2,3,4 along a horizontal line of the
detector matrix. The controlled setting of the magnitude of higher
harmonics by tuning the exposure is important to facilitate the
development of neutron-optical elements (see, e.g., Section
12.3.6).
Before introducing gratings in d-PMMA based on the
photo-neutron-refractive effect as a standard neutron-optical
component, the question of lifetime arises. Since the fabrication
of gratings with diffraction efficiencies in the range of 10%–70%
reached a satisfactory level only a few years ago, the experience
has been rather limited. The oldest grating that is available and
still can be used was recorded on November 19, 1998. The grating
formation process was mon- itored for one week (see Section
12.3.3). In July 2000 and then in May 2002 rocking curves were
measured again. The integrated diffraction efficiency in- creased
for several weeks (followed also by light optical measurements)
until it reached a limit Iη ≈ 3.5–4.0, and then decreased to a
level of 60% (Iη ≈ 2) af- ter four years [30]. An additional
problem might have been that the sample was inhomogeneous and thus
the various measurements were not performed on the same area of the
sample. We estimate this error to be 20%, which corresponds to Iη =
0.4 [21].
12.3.6 The Neutron Interferometer
The extensive studies and experiments performed on the
photo-neutron-refractive effect in d-PMMA and described in the
previous sections served as a basis to set up a neutron
interferometer utilizing holographically produced gratings as beam
splitters and mirrors. In our view the interferometer may be
regarded as one of the most useful neutron-optical devices, since
it provides information about the wave function, i.e., amplitude
and phase, in contrast to standard scattering or diffraction
techniques, where only the intensity is measured.
Three decades ago, the successful development of an interferometer
for ther- mal neutrons [31] led to a boost in neutron optics,
opening up completely new experimental possibilities in applied
[32–34] and fundamental [35–38] physics, e.g., extremely accurate
measurements of scattering lengths [39, 40] or tests of quantum
mechanics [41–43]. For an excellent, complete, and recent review of
neutron interferometry see [44]. The neutron itself and its
behavior in various potentials, e.g., in a magnetic field [45], a
gravitational field [46, 47], or such of pure topological nature
[48–50], was studied as a model quantum-mechanical system by means
of interferometers. Some very recent and amazing results on quantum
states of the neutron in the gravitational field [51] or a
confinement- induced neutron phase [52] demonstrate the demand for
further research on those topics. Coherence and decoherence
effects, which are important in any experi- ment, were investigated
[37, 53, 54]. Based on the knowledge about the neutron and being
interested in fundamental questions of physics, several groups
started to elaborate interferometry with more complex quantum
objects like atoms [55–57] and molecules [58–60].
On the other hand, the materials-scientific aspect of neutron
interferometry has not yet reached a satisfactory level, i.e., it
has not yet become established as a
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standard technique. Thermal neutron interferometers are run at the
Institut Laue- Langevin (ILL) and the National Institute of
Standards and Technology [61], which can be used for precise
measurements of neutron scattering-lengths, but are naturally
limited in wavelength by the use of perfect silicon crystals as
beam splitters. Moreover, high expenditure (stabilization against
vibrations and thermal drift) was necessary to ensure phase
stability and contrast [62]. In the last few years, investigations
on biological materials and soft matter, e.g., complex organic
molecules, membranes, and bones, have met with more and more
interest, and the life science community has discovered neutron
scattering as a nondestruc- tive powerful method [63]. Because of
the large-scale structures, small-angle scattering with cold
neutrons is employed. The interferometer presented in this section
is a flexible and versatile device for the operation at any
SANS-facility.
The first perfect-crystal neutron interferometer successfully run
was designed, by Rauch et al. [31]. It consists of three equally
spaced parallel slabs that are produced by cutting two wide grooves
in a large, perfect silicon crystal in the so-called LLL geometry
(triple Laue case) [44]. Dynamical diffraction from the (220)
reflection with a lattice constant of = 0.19 nm is used to split
the in- coming neutron beam and finally to recombine the subbeams
as sketched in Figure 12.10. Such an interferometer can be properly
run with thermal neutrons, e.g., for wavelengths less than λ ≤ 0.6
nm. Interferometers for very cold and ultracold neutrons are based
on different techniques: the gratings are created by sputter
etching ( ≈ μm range) in the LLL geometry [47, 64, 65], by
photolithog- raphy ( ≈ 20 μm) in reflection geometry [66], or
reflection from multilayers is applied [67]. To close the gap
between thermal neutrons and very cold neutrons Schellhorn et al.
[68] constructed a prototype interferometer in the LLL geom- etry
built of “artificial” gratings employing the
photo-neutron-refractive effect of d-PMMA as described in the
previous sections. They succeeded in demon- strating that the
arrangement of the three gratings acts as an interferometer. A
larger and hence more sensitive interferometer was constructed by
Pruner et al.
H-beam: ΨI t+ΨII
ΨII
FI G U R E 12.10. Sketch of an LLL interferometer for cold
neutrons. It is based on holo- graphically generated density
gratings G1, G2, and G3 in slabs of deuterated PMMA. Rotating a
phase flag by an angle , a phase difference is generated between
beams I
and I I . After diffraction from G3 the reflected and transmitted
amplitudes I r + I I
t and I
t + I I r add up to the 0-beam and H-beam, respectively.
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12. Neutron Physics with Photorefractive Materials 339
FI G U R E 12.11. Interferometer for cold neutrons based on
gratings in photo-neutron- refractive d-PMMA. G1–G3 denote the
gratings, L = 150 mm, = 380 nm.
[69], a photograph of which is shown in Figure 12.11. The crucial
point for a successful operation of any LLL interferometer is the
extremely accurate mutual alignment of the gratings. Thus perfect
silicon crystal interferometers rely on the quality of the silicon
crystal. The cutting and etching of a monolithic crystal ensures
the desired accuracy over the full distance. For the very cold
neutron interferometer, the alignment of the phase gratings is
performed dynamically by tilting and translating each grating,
which is controlled by operating three auxil- iary laser-light
interferometers additionally [47]. Thermal and acoustic isolation
for both types of interferometers is mandatory. The methodology to
overcome such problems and to construct an interferometer for cold
neutrons is as follows: (1) Three photosensitive d-PMMA samples are
prepared and mounted on a lin- ear translation stage. (2) The first
photo-neutron-refractive slab is exposed to the interference
pattern of the holographic two-wave mixing setup. (3) Successively
the second and third slabs are moved, i.e., nominally translated,
to the position of the sinusoidal interference pattern. Prior to
exposure the motion is corrected for deviations from the ideal
translation (pitch, yaw, and roll). The latter is the decisive step
during the production of the interferometer, since demand on ac-
curacy is an absolute requirement. To control the accuracy of the
translation and to correct deviations, an optical system was used
in combination with piezo- driven stages. The roll angle, which is
the most critical parameter, is controlled by a polarization optic
method that is independent of the distance of translation.
Therefore in principle, interferometers of any length may be
fabricated using this technique.
The advantages of this attempt over other techniques are striking:
The grating spacing is easily tailored to the required value within
the range 250 nm < <
10 μm. Moreover, the sinusoidally modulated light pattern
creates—if properly prepared (see Section 12.3.5)—a sinusoidal
grating. Therefore only+1st and−1st diffraction orders occur. In
addition, the adjustment of the three photopolymer samples is done
during the recording of the grating once and for ever. Finally, the
whole interferometer has turned out to be very stable, compact, and
robust
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340 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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despite the high demand on accuracy. Moreover, it can be
transferred to any neutron source and then set up ready to run
within a few hours.
This type of interferometer is useful for many investigations with
cold neu- trons. The most interesting are determination of the
scattering length for var- ious materials in the low-energy regime,
probing the coherence properties of cold neutrons, and gaining
phase information in large-scale structure investi- gations. The
knowledge of the coherence properties is particularly important for
scattering, diffraction, spin echo, and reflection experiments
[70–74]. e.g., reflectometry with polarized neutrons where the
coherence volume plays an essential role and is a decisive quantity
only roughly estimated [75–77] or even more often simply not
considered. To access the absolute value of the normalized
correlation function (1)(x − x′, t − t ′) (coherence function)
exper- imentally, the visibility v of the interference fringes is
measured. The phase difference is thereby varied by rotating a
phase flag through both beam paths as sketched in Figure 12.10. The
visibility (contrast) is defined as [44] v = (Imax − Imin)/(Imax +
Imin) = m|(1)(x − x′, t − t ′)|, with m being the mod- ulation.
Then an interference pattern of the form
I0,H (x) = A0,H ± v cos ( bρλ(x − xE ) − 3
) , (12.26)
lcπ
]2 }
, (12.27)
is expected if a Gaussian coherence function is assumed. Here x and
xE are the geometric path difference due to the rotation of the
flag and the initial path difference of the empty interferometer
respectively, lc is the coherence length, and 3 the relative phase
impressed by G3. The parameters A0.H and m are functions of the
diffraction efficiencies η1,2,3 of the gratings. In addition,
contrast is reduced because of inhomogeneities of the phase flag,
thickness variations, and beam attenuation effects [37, 78].
A typical interference pattern for a wavelength of λ = 2.6 nm and a
collimation of xn = 5 mm, xs = 1 mm, Lcoll = 19 m is presented in
Figure 12.12. All three above-mentioned purposes of neutron
interferometry can be demonstrated on the basis of this
measurement: (1) The coherent scattering length of the phase flag
is given by the periodicity of the fringes (real part of the
coherence function). (2) The fringe visibility (i.e., the
envelope), which is continuously reduced upon larger phase shifts,
constitutes the absolute value of the coherence function for the
actual neutron beam at the instrument D22 (ILL). (3) The relative
shift of the fringes with respect to the envelope comprises the
relative phase 3 induced by G3. By replacing the latter by a small
angle scattering object, the corresponding phase can be measured.
This possibility is of utmost interest for solving phase-sensitive
large-scale structure problems [79].
The present approach exhibits several inherent features that will
open up new possibilities in neutron physics. The ultimate goal is
to create a novel device for neutrons that can easily be
implemented and run at any beam line for cold neutrons. The
principal future application of such an instrument will be the
investigation
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12. Neutron Physics with Photorefractive Materials 341
-20 -15 -10 -5 5 10 15 20
8
10
12
16
18
20
3/bρλ
H-beam
Δx [μm]
0 xE
FI G U R E 12.12. Interference fringes obtained by rotating a
sapphire phase flag around an axis perpendicular to the plane of
incidence. Squares and triangles show the measured intensity of the
0-beam and H-beam, respectively, as a function of the geometric
path difference x(). At xE (indicated by the solid black vertical
line) the envelope reaches its maximum, since any path difference
of the beams is compensated by rotation of the phase flag. The
maximum of the interference fringes, however, deviates from that
value by 3/λbρ (dashed gray vertical line).
of mesoscopic structures and their kinetics in the fields of
condensed matter physics and engineering, chemistry, and biology. A
summary of the cold neutron interferometer’s future perspectives
and its possible impact on physics and/or materials science can be
found in [30]. It is supposed that the flexibility, the low costs,
and the excellent properties of the interferometer composed of
gratings created by the photo-neutron-refractive effect will
promote the development of novel small angle scattering techniques
with cold neutrons. In addition, due to its different design and
operating wavelength range, this type of interferometer can
contribute constructively to unsolved problems, e.g., about the
consistency of the measured gravitational phase shift with theory
[47, 80–84].
12.4 Electro Neutron-Optics
In this section we will present a standard electrooptic material
(LiNbO3) exhibiting a photo-neutron-refractive effect of a
completely different nature. The correspond- ing mechanism has
already been briefly discussed in Section 12.2.2 and is based on
the creation of space-charge fields that modulate the
neutron-refractive index.
The reason why the scientific literature has not referred to any
measurement of ENOCs up to now is that the neutron-refractive-index
changes due to an applied electric field are rather tiny (cf. Table
12.1). Therefore, any standard technique
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(refraction, total reflection, interferometry, diffraction from a
crystal lattice) must fail in detecting an electro neutron-optic
effect. It has been estimated [16] that typical changes are on the
order of 10−10. Hence, a different approach was used to tackle the
problem: The photo-neutron-refractive sample was illuminated with a
sinusoidal light interference pattern, thus reaching space-charge
fields |E(x)| of up to 100 kV/cm, a magnitude that exceeds by far
the values usually achieved. As a consequence, the
neutron-refractive index is modulated via the neutron electrooptic
effect according to (12.23). Then neutrons are diffracted from
these holographically recorded gratings. Thus the
neutron-refractive-index changes un- der the influence of
(spatially modulated) electric fields with the extraordinary
sensitivity required can be detected.
The experiment for the determination of the ENOC is performed in
three steps: the recording of the grating by a two-wave mixing
technique, measuring the rockinge curve for light, and subsequently
measuring for neutrons. In the first step the space-charge field is
created; then its magnitude Esc is calculated according to (12.18),
since the refractive index and the electrooptic coefficient for
light are known. Then the sample is transferred to the neutron beam
line. From the measured angular dependence of the diffraction
efficiency for neutrons ηN (θ ) the neutron-refractive-index change
can be calculated and in turn the ENOC r T
N using (12.23). This conceptually simple investigation goes hand
in hand with many difficulties, ranging from sample preparation via
the narrow Bragg angles to delicate adjustment and the extremely
small diffraction efficiencies [30].
The experiment to search for a neutron electrooptic effect was
suggested by Rupp. In [85] he gave an overview of tentative
standard (light) electrooptic materi- als including estimations for
the magnitudes of their ENOCs. A few years later, the first
experimental evidence of the effect was reported [16]. To extract
the neutron- refractive-index change (12.6) and finally the ENOC
using (12.23), a complete rocking curve was measured in a
subsequent experiment using a 7LiNbO3:Fe sam- ple [86]. The results
are shown in Figure 12.13. Neutrons at each of the angular
positions were accumulated for 1 h. Note that the maximum
diffraction efficiency ηB = 5 × 10−5 is 4 to 5 orders of magnitude
lower in contrast to typical ones for d-PMMA samples. Based on the
data presented above and the space-charge field determined from
light-optical measurements, the ENOC of LiNbO3 is estimated
-3 -2 -1 0 2 3
5x10-3
4x10-3
3x10-3
2x10-3
1x10-3
0
) [%
]
FI G U R E 12.13. Angular dependence of the diffraction efficiency
for cold neutrons. The latter are diffracted from a grating induced
by the elec- tro neutronoptic effect in 7LiNbO3:Fe. +1st and −1st
diffraction orders are visible.
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12. Neutron Physics with Photorefractive Materials 343
0 2500 5000 7500
]
FI G U R E 12.14. ηB;N (t) during illu- mination with white light;
t = 0 in- dicates the start of the illumination. Counts were
collected for 20 min; the data points are an average over this time
span.
to be r T N ≈ 3 ± 2 fm/V. To prove that the gratings originate from
a light-induced
space-charge field and the electro neutron-optic effect, in
addition the diffraction efficiency ηB;N (t) at the Bragg position
was measured during illumination with white light. The resulting
time dependence is depicted in Figure 12.14. The decay of the
diffraction efficiency gives evidence that the charge carriers
composing the space-charge field are electrons and that the
observed effect is indeed due to the coupling of the
neutron-refractive-index change to that field via electro
neutron-optics (cf. (12.23)).
Up to this point one might assume that the electro neutron-optic
effect is of pure fundamental interest. However, it will be shown
that the method proposed can be used, in contrast, to solve applied
physical questions as well.
One major disadvantage of electrooptic photorefractive materials is
that photo- induced changes basically are volatile, i.e., they
vanish upon illumination with homogeneous light. Therefore, it is
evident that fixing mechanisms have to be developed for long-term
applications. Several attempts to solve the problem are reported in
the literature [87–90]. Among them the most common technique for
LiNbO3, the material in question, is called thermal fixing (cf.
Chapter 12 in the first volume of this series [91]). As discussed
in Section 2.2, holographic record- ing in electrooptic crystals
means that a light-induced space-charge density of electrons
modulates the refractive index via the electrooptic effect. By
increas- ing the temperature to about 400–450 K, positively charged
ions become mobile and neutralize the effect of the electronic
space-charge pattern, i.e., the (light) refractive-index grating
disappears (Esc = 0). However, density gratings of ions and
electrons with equal amplitudes remain. After returning to ambient
temper- atures, homogeneous illumination partially redistributes
the electrons, whereas the ions are insensitive to illumination
because of their mass. Thus two gratings, an ionic and an
electronic one, appear that slightly differ in their amplitudes.
This yields again a space-charge density and a refractive-index
grating via the electrooptic effect, which exactly mimics the
primary grating: it is fixed. This technologically important
process depends significantly on the species of ions, the
temperature dependence of their mobility, on their concentration,
and vari- ous other parameters. Spectros copic investigations have
revealed that the ionic
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344 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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0.0
0.5
1.0
1.5
2.0
θ [mrad]
0 -5]
FI G U R E 12.15. Rocking curves performed on the hydrated sample.
Diffraction from the combined grating (left) and from the pure
ionic density grating (right graph) [94].
density grating can usually be traced back to hydrogen ions [92,
93]. Surpris- ingly, thermal fixing also works if crystals are
dehydrated. In the latter case the question arises which ion will
be responsible for the compensation process. Employing the
knowledge gained from measuring the ENOC, neutron diffrac- tion
from two different LiNbO3 samples (hydrated, dehydrated) in two
states (Esc = 0, Esc = 0) was performed to determine the species of
ions [94]. The conclusions that can be drawn from the four
measurements are the following: If Esc = 0, then the
photo-neutron-refractive effect is due only to ionic density grat-
ings, i.e., nion = −λ2/(2π )bcion . For the hydrated sample it was
supposed that the whole contribution originates from hydrogen ions,
for the dehydrated sample from the unknown ion species. If Esc = 0,
the photo-neutron-refractive effect comprises two terms, the ionic
density grating as before and the electro neutron- optic (ENO) part
according to (12.23), n = nion + nENO. Evaluating the
neutron-refractive-index changes on the basis of the rocking curves
measured and as we know the concentration of ions, the coherent
scattering length of the ion species responsible for the
diffraction from the density grating can be calcu- lated. In Figure
12.15 the angular dependence of the diffraction efficiency for the
hydrated sample is shown. The evaluation for this sample in case of
Esc = 0 yields nion = 9.2 × 10−10, which yields |bc| = 3.6 fm for
the compensating ion. This ties in nicely with the assumption that
hydrogen (bc = −3.74 fm) is responsible for the compensation
process. Moreover, by adding or subtracting nion from the total
neutron-refractive-index change as evaluated for Esc = 0, the ENOC
for LiNbO3 can be estimated as rN = (2.4 − 3.4) ± 0.5 fm/V.
Equivalent measure- ments and evaluations for the dehydrated sample
have shown that any other effect is more than an order of magnitude
smaller. Therefore, only the upper limit for nion < 6.5 × 10−10
can be estimated, which in turn yields |bc| ≤ 1.92 fm for the
compensating ion. This indicates that thermal fixing in dehydrated
LiNbO3
may be attributed to the motion of Li-ions (bc = −1.9 fm).
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12. Neutron Physics with Photorefractive Materials 345
12.5 Neutron Holography
Holographic techniques are widely applied in the wavelength range
of visible light, while Gabor’s original idea, dating back to 1948
[95], aimed at the construction of a lensless microscope with
atomic resolution. This goal could be achieved only during the last
fifteen years by introducing successively electron, X-ray, and γ
-ray holography, largely supported by the increasing availability
of synchrotron radiation (for reviews see, e.g., [96, 97]). Neutron
holography was not considered feasible for various reasons, and
only very recently a detailed analysis showed how to transfer and
adapt well-known holographic conceptions to the neutron case [98].
Neutron holography is not specifically related to photorefractive
materials. However, it takes a novel approach in linking neutron
physics, holography, and structural studies of matter, and
therefore a brief introduction to the field appears justified in
the present context.
Being a holographic imaging technique, neutron holography is based
on record- ing of the interference pattern of two coherent waves
emitted by the same source. The first wave reaches the detector
directly and serves as the reference wave; the second one is
scattered by the object of interest (object wave) and subsequently
interferes with the reference wave. Atomic resolution holography is
usually done in one of two complementary experimental setups. The
first one, known as inside source holography (ISH), puts the source
inside the sample while the detector recording the interference
pattern is situated at a larger distance. In the second approach,
named inside detector holography (IDH), the positions of source and
detector are interchanged.
Neutron ISH (NISH), can be realized by taking advantage of the
extraordinarily large (≈80 barns) incoherent elastic neutron
scattering cross section of hydrogen nuclei permitting them to
serve as pointlike sources of spherical neutron waves within the
sample. In an incoherent scattering process a spherical neutron
wave is generated at the site of the hydrogen nucleus, which
propagates towards the detector either directly, forming the
reference wave, or after having been scattered coherently by other
nuclei in the sample [98, 99]. We consider a single crystalline
sample of a substance containing hydrogen and located in a
monochromatic neu- tron beam of intensity I0. If the origin of the
coordinate system is chosen at the position of a source nucleus, it
can be shown that the intensity I observed with a detector at a
radius vector R is given by
I(R) = I0
r j exp[i(r j k − r j k)]. (12.29)
Here, a j stands for the wave amplitude with wave vector k
scattered off a nucleus j, while b j and r j denote the neutron
scattering length and the position vector
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346 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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of the nucleus j, respectively. The summation is taken, in
principle. over a large number of nuclei in the sample, yet it is
found that under typical experimental conditions only nuclei in the
immediate neighborhood of the source will contribute significantly.
The first and the third terms in the above equation are due to the
reference and the object beams, respectively, while the second term
represents the interference between these two. It contains the
phase information and depends on the relative orientation of sample
and detector (but in no way depends on the incident neutron beam).
The amplitude of this intensity modulation is typically ≈ 10−3 of
the reference beam, so that the requirements concerning counting
statistics and detector stability are quite demanding. A hologram
of the arrangement of nuclei forming the local environment of the
source nucleus can be recorded by scanning the interference pattern
over a sufficiently large solid angle. The real space positions of
the nuclei can be reconstructed from the hologram by a proper
mathematical procedure based essentially on the Helmholtz–Kirchhoff
formula. The first NISH experiment was done by a Canadian group at
Chalk River [100] on a sample of a mineral called simpsonite, which
contains substantial amounts of hydrogen. Though the precision of
the atomic positions obtained was rather limited, this work
unambiguously demonstrated the feasibility of the technique. A
second NISH study was performed recently at the Laboratoire Leon
Brillouin (LLB) in Saclay [101]. A hologram of a palladium single
crystal charged with ≈70 at% hydrogen (a well-known metal–hydrogen
system) was successfully recorded. The reconstruction of the atomic
arrangement around the hydrogen nuclei resulted in the positions of
the six Pd atoms forming an octahedron around the hydrogen site.
Another paper of the Canadian group, discussing the observation of
Kossel and Kikuchi lines in neutron incoherent elastic scattering
experiments and their relevance for neutron holography, was
published recently [102].
The second approach (neutron IDH = NIDH) can be realized using
strongly neutron-absorbing isotopes acting as pointlike detectors
in the sample. A single crystalline sample is put in a beam of
plane monochromatic neutron waves, which can arrive at the detector
nuclei either directly or after having been scattered co- herently
by other nuclei in the sample. The interference of these two
contributions entails a modulation of the probability amplitudes of
the neutron waves at the sites of the detector nuclei and,
consequently, also the neutron absorption probability. An
absorption process in a (properly chosen) detector nucleus creates
an excited state that emits γ -radiation upon its transition to the
ground state. It is the in- tensity of this γ -radiation that is
actually registered in such an experiment and serves as a measure
for the neutron density at the sites of the detector nuclei. Since
the interference pattern depends on the orientation of the sample
in the beam, a hologram can be recorded by rotating the sample
through a sufficient number of orientations. The only NIDH
experiment performed so far was done at the ILL in Grenoble on a
single crystal of lead alloyed with a small amount of cadmium, a
strongly neutron-absorbing element [103]. The γ -rays emitted by
the cadmium nuclei were detected by two scintillation detectors,
and a hologram was recorded by measuring the intensity of the γ
-radiation for about 2000 different orientations of the sample with
respect to the detectors. The hologram and the reconstructed
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12. Neutron Physics with Photorefractive Materials 347
FI G U R E 12.16. Recorded hologram of Pb99.74Cd0.26 as a function
of the rotation angles χ, φ.
positions of the lead atoms surrounding the cadmium sites are shown
in Figures 12.16, 12.17. The holograms recorded in the three
above-mentioned experiments contain each on the order of 1000
points with an angular resolution (pixel size) of ≈3 × 3 degrees.
The neutron wavelength used is about 0.1 nm. Taken together, these
studies have demonstrated convincingly that neutron holography can
indeed be performed by applying the concepts presented in
[98].
12.5.1 Outlook
Electron and X-ray holography have undergone a rapid development
during the last decade. Likewise, neutron holography offers the
potential for a number of interesting applications. However, a
number of technical problems, some of which are briefly listed
below, have still to be overcome before neutron holography can be
performed routinely [104]:
Presently, the recording of a neutron hologram requires about one
week of beam time. For NISH experiments it is obvious that in
principle, this time could be reduced to one hour or even less by
the use of properly calibrated multidetectors permitting one to
record a large number of pixels of the hologram
simultaneously.
In contrast to NISH, in NIDH experiments the pixels of the hologram
can be measured only one after the other. Nevertheless, the count
rates could be increased by one order of magnitude by using larger
arrays of γ -detectors.
FI G U R E 12.17. Reconstructed fcc lattice of twelve
nearest-neighbor lead nuclei forming a sphere around a cadmium
probe nucleus (not shown). The squares indicate different lattice
planes.
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348 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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The signal-to-noise ratio in NIDH experiments is highly unfavorable
due to a high background of γ -radiation, which originates from
various sources that are notoriously difficult to eliminate. The
excellent energy resolution of germanium detectors would allow one
to single out the characteristic γ -radiation emitted from the
detector nuclei and to discriminate background radiation. The use
of appropriate arrays of detectors including Compton suppression is
under consideration.
Neutron holography is a local technique in the sense that it images
primarily the local environment of the incoherently scattering
(NISH) or neutron-absorbing (NIDH) probe nuclei. Due to the unique
properties of hydrogen, possible ap- plications of NISH are
suggested in the investigation of the local structure of
hydrogen-containing substances like metal–hydrogen systems [105]
and many inorganic as well as organic compounds. In addition,
simple estimates show that there are several more elements/isotopes
exhibiting sufficiently large in- coherent scattering cross
sections to be serious candidates for NISH. Likewise, there is
quite a number of neutron-absorbing nuclei that could be applied in
NIDH. In all, there is also a large class of
non-hydrogen-containing materi- als in the investigation of which
neutron holography may serve as a poten- tially useful tool. It
will take a few more years and a number of technical im- provements
to see whether neutron holography can be established as a standard
technique.
12.6 Outlook and Summary
Some of the novel aspects opened up by the use of
photo-neutron-refractive samples have already been discussed in the
relevant previous sections (see, e.g., Section 12.3.6). Here, we
will briefly suggest additional future experiments that promise
great potential for cold neutron physics and materials science.
Moreover, new tentative photo-neutron-refractive materials are
suggested.
A straightforward experiment to be conducted is an Aharonov–Casher
(AC) type diffraction experiment [85]. Using the same experimental
configuration as in the determination of the ENOC but employing
polarized neutrons, the value of r S according to (12.14) and
(12.21) will depend on the mutual orientation of the neutron spin,
the space-charge field, and the propagation direction. Using
(12.21) and (12.23) the AC contribution of the ENOC to the
diffraction efficiency is η(AC) = (|μ|πεEd/hc2)2, and thus
independent of the neutron wavelength λ
and the grating spacing . Moreover, for further experiments it is
noteworthy that the AC effect is suppressed by choosing μE.
An interesting aspect of the Foldy contribution (second term in
(12.21)) is its appearance as a nonlocal effect. The corresponding
neutron-refractive-index modulation is phase-shifted by π/2 with
respect to that of the density modulation (12.22). Thus, measuring
the phase shift, e.g., by means of an interferometric technique, it
might be possible to determine its contribution to the ENOC.
Because
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12. Neutron Physics with Photorefractive Materials 349
of the small diffraction angles, the Foldy contribution, however,
is three to four orders of magnitude smaller than the AC
effect.
The measurement of the ENOC also provides a possibility for
detecting an EDM of the neutron. Novel techniques have been
developed, suggested and employed for this purpose [106–109]. Under
illumination, huge electric space-charge fields build up in the
electro neutron-optic crystal that cannot be reached in vacuum. A
combination of the interferometric technique and the use of those
electric fields is a tentative method to lower the limit for the
existence of an EDM.
Further promising photo-neutron-refractive materials are
photosensitive poly- mers, which nowadays are available in large
numbers, provided that they exhibit a considerable density ratio
ρpolymer/ρmonomer, e.g., [110–114].
Considering materials that are photo-neutron-refractive via the
electro neutron- optic effect, one must admit that the effects will
be small when compared to the photopolymer systems. However, the
advantage of these materials is that they are well established in
light optics and respond with linear neutron-refractive- index
changes to illumination. Moreover, the magnitude of the
refractive-index changes depends on the ENOC and the space-charge
field. Estimations for various common light electrooptic materials
are included in [85].
Similar to thermal fixing in LiNbO3, ionic (=density) gratings may
be created in any photorefractive sample with high ionic
conductivity. Promising material for this type of effect is LiIO3,
which is known as a quasi one-dimensional ionic conductor at
ambient temperatures. Moreover, photorefractive properties have
been proven to exist at temperatures below 180 K [115–117]. Since
the neutron- refractive-index changes are proportional to the
concentration modulation cion
of the specific ion, we expect that at lower temperatures the ionic
grating must be very efficient in diffracting neutrons.
Another very interesting possibility to observe light-induced
neutron- refractive-index changes is the use of photomagnetic
samples and polarized neu- trons, which was suggested by Rupp
[118]. The fact that the bound scattering length in general is
spin-dependent will be of utmost importance. Provided that the
nuclear spins are or can be (re)oriented, extremely efficient
diffraction of polar- ized neutrons from light-induced gratings can
be expected. Transparent magnetic borates, e.g., FeBO3 are also
highly promising materials.
The development of useful devices for neutron optics is based on
the simplest optical element: the grating. The combination of
gratings and/or the modification of the light-optical setup allows
the preparation of new devices. A single grating can be used as a
monochromator. By creating more complex neutron-refractive- index
profiles, for example lenses for cold and ultracold neutrons can be
designed. Only recently, Oku performed this task for cold neutrons
based on compound refractive Fresnel lenses, which consisted of
about 50 elements [119, 120]. The fabrication of such a lens by
holographic means seams much simpler and cheaper.
A range of phenomena related to light-induced changes of the
refractive index for cold neutrons was presented. This
photo-neutron-refractive effect was defined in terms analogous to
those of light optics. We demonstrated that the prepara- tion of
gratings based on this effect by light-optical means and the
diffraction
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350 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
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of cold neutrons from that grating is possible. Combined neutron
and light- optical experiments are useful in retrieving information
on the photorefractive and photo-neutron-refractive nonlinear
mechanism. This knowledge in turn promotes the production of
neutron-optical elements, e.g., diffraction gratings, lenses, or
even an interferometer. The design, setup, and successful operation
of such a neutron interferometer was presented. Employing this
device, coherent scatter- ing lengths at cold neutron wavelengths
can be precisely determined. More- over, the coherence properties
of the neutrons are revealed by interferometric measurements.
Operation of the interferometer with partially coherent waves al-
lows one to retrieve phase and intensity of small-angle diffraction
or scattering objects.
Fundamental properties of the neutron that are revealed in high
electric fields can be probed by diffraction from gratings that
originate from a space-charge field through the electro
neutron-optic effect. The latter gives rise to a change of the
neutron-refractive index under the application of electric fields.
Experiments demonstrate the feasibility of this approach and
suggest a novel way to search for a possible electric dipole moment
of the neutron.
Atomic-resolution neutron holography was introduced, the chapter
finishing with planned experiments and a discussion of their impact
on the foundations of physics.
Acknowledgments. We are indebted to Dr. F. Havermeyer for providing
us with unpublished data. Measurements presented here have been
performed at the Research Center Geesthacht (Germany), the Institut
Laue-Langevin (Grenoble, France), the Paul Scherrer Institut
(Villigen, Switzerland), and the Laboratoire Leon Brillouin
(Saclay, France). Financial support by the grant FWF P-14614- PHY
and the Austrian ministry bm:bwk (infrastructure for HOLONS at the
GeNF) is acknowledged.
References
1. A. Ashkin, G.D. Boyd, J.M. Dziedzic, et al: Apl. Phys. Lett. 9,
72 (1966). 2. E. Fermi, L. Marshall: Phys. Rev. 71, 666 (1947). 3.
W.H. Zinn: Phys. Rev. 71, 752 (1947). 4. R.A. Rupp, J. Hehmann, R.
Matuall, et al: Phys. Rev. Lett. 64, 301 (1990). 5. F.S. Chen. J.T.
la Macchia, D.B. Fraser: Appl. Phys. Lett. 13, 223 (1968). 6. B.W.
Batterman, H. Cole: Rev. Mod. Phys. 36, 681 (1964). 7. H. Kogelnik:
Bell Syst. Tech. J. 48, 2909 (1969). 8. H. Rauch, D. Petrascheck:
In: Neutron Diffraction, Topics in Current Physics, vol. 6,
ed. H. Dachs, chap. 9, pp. 303–351 (Springer-Verlag, Berlin
Heidelberg New York, 1978).
9. T.K. Gaylord, M.G. Moharam: Appl. Opt. 20, 3271 (1981). 10. P.
Gunter, J.-P. Huignard: In: Photorefractive Materials and Their
Applications I,
Topics in Applied Physics, vol. 61, eds. P. Gunter, J.-P. Huignard,
chap. 2, pp. 7–70 (Springer-Verlag, Berlin Heidelberg New York,
1987).
P1: OTE/SPH P2: OTE
12. Neutron Physics with Photorefractive Materials 351
11. V.F. Sears: Neutron Optics, Neutron Scattering in Condensed
Matter, vol. 3 (Oxford University Press, New York–Oxford,
1989).
12. A.G. Klein, S.A. Werner: Rep. Prog. Phys. 46, 259 (1983). 13.
C.G. Shull: Trans. Am. Crystallogr. Assoc. 3, 1 (1967). 14. V.
Sears: Phys. Rep. 141, 281 (1986). 15. M. Forte: J. Phys. G 9, 745
(1983). 16. F. Havermeyer, R.A. Rupp, P. May: Appl. Phys. B 68, 995
(1999). 17. J. Schwinger: Phys. Rev. 73, 407 (1948). 18. L.L.
Foldy: Phys. Rev. 83, 688 (1951). 19. L.L. Foldy: Phys. Rev. 87,
688 (1952). 20. F. Havermeyer, C. Pruner, R.A. Rupp, et al: Appl.
Phys. B 72, 201 (2001). 21. C. Pruner: Untersuchung
lichtinduzierter Strukturen in PMMA mit holographis-
chen Methoden und Neutronenstreuung. Master’s thesis, Universitat
Wien (1998). In German.
22. M. Jazbinsek, M. Zgonik: Appl. Phys. B 74, 407 (2002). 23. R.
Matull, R. Rupp, J. Hehmann, et al: Z. Phys. B 81, 365 (1990). 24.
R. Matull, P. Eschkotter, R. Rupp, et al: Europhys. Lett. 15, 133
(1991). 25. F. Havermeyer: Licht- und Neutronenbeugung an
Holographischen Gittern. Ph.D.
thesis, Universitat Osnabruck (2000). In German. 26. S. Breer:
Konstruktion und Aufbau eines LLL-Interferometers. Master’s thesis,
Univ.
Osnabruck (1995). In German. 27. F. Havermeyer, R.A. Rupp, D.W.
Schubert, et al: Physica B 276–278, 330 (2000). 28. R.A. Rupp, F.
Havermeyer, J. Vollbrandt, et al: SPIE Proc. 3491, 310 (1998). 29.
F. Havermeyer, S.F. Lyuksyutov, R.A. Rupp, et al: Phys. Rev.Lett.
80, 3272 (1998). 30. M. Fally: Appl. Phys. B 75, 405 (2002). 31. H.
Rauch, W. Treimer, U. Bonse: Phys. Lett. A 47, 369 (1974). 32. M.
Schlenker, W. Bauspiess, W. Graeff, et al: J. Magn. Magn. Mater.
15–18, 1507
(1980). 33. H. Rauch, E. Seidl: Nucl. Instrum. Methods A 255, 32
(1987). 34. W.E. Wallace, D.L. Jacobson, M. Arif, et al: Appl.
Phys. Lett. 74, 469 (1999). 35. C. Shull, D.K. Atwood, J. Arthur,
et al: Phys. Rev. Lett. 44, 765 (1980). 36. H. Rauch, A. Zeilinger:
Hadronic Journal 4, 1280 (1981). 37. H. Kaiser, S.A. Werner, E.A.
George: Phys. Rev. Lett. 50, 560 (1983). 38. H. Rauch: Journal de
Physique 45, 197 (1984). 39. A. Boeuf, U. Bonse, R. Caciuffo, et
al: Acta Crystallogr. Sect. B 41, 81 (1985). 40. A. loffe, M. Arif,
D.L. Jacobson, et al: Phys. Rev. Lett. 82, 2322 (1999). 41. S.A.
Werner: Phys Today p. 24 (1980). 42. D. Greenberger: Rev. Mod.
Phys. 55, 875 (1983). 43. H. Rauch: Science 262, 1384 (1993). 44.
H. Rauch, S.A. Werner: Coherence Properties, Neutron Scattering in
Condensed
Matter, vol. 12 (Oxford University Press, New York–Oxford, 2000).
45. S.A. Werner, R. Colella, A.W. Overhauser, et al: Phys. Rev.
Lett. 35, 1053 (1975). 46. R. Colella, A.W. Overhauser, S.A.
Werner: Phys. Rev. Lett. 34, 1472 (1975). 47. G. van der Zouw, M.
Weber, J. Felber, et al: Nucl. Instrum. Methods A 440, 568
(2000). 48. A. Cimmino, G.I. Opat, A.G. Klein, et al: Phys. Rev.
Lett. 63, 380 (1989). 49. B.E. Allman, A. Cimmino, A.G. Klein, et
al: Phys. Rev. Lett. 68, 2409 (1992). 50. A.G. Wagh, V.C. Rakhecha,
J. Summhammer, et al: Phys. Rev. Lett. 78, 755 (1997).
P1: OTE/SPH P2: OTE
SVNY276-Gunter-v3 October 27, 2006 7:44
352 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard
Krexner
51. V.V. Nesvizhevsky, H.G. Borner, A.K. Petukhov, et al: Nature
(London) 415, 297 (2002).
52. H. Rauch, H. Lemmel, M. Baron, et al: Nature (London) 417, 630
(2002). 53. A.G. Klein, G.I. Opat, W.A. Hamilton: Phys. Rev. Lett.
50, 563 (1983). 54. H. Rauch, H. Wolwitsch, H. Kaiser, et al: Phys.
Rev. A 53, 902 (1996). 55. Y.B. Ovchinnikov. J.H. Muller, M.R.
Doery, et al: Phys. Rev. Lett. 83, 284 (1999). 56. S. Inouye, T.
Pfau, S. Gupta, et al: Nature (London) 402, 641 (1999). 57. J. M.
McGuirk, M.J. Snadden, M.A. Kasevich: Phys. Rev. Lett. 85, 4498
(2000). 58. B. Brezger, L. Hackermuller, S. Uttenthaler, et al:
Phys. Rev. Lett. 88, 100404 (2002). 59. L. Hackermuller, S.
Uttenthaler, K. Hornberger, et al: Phys. Rev. Lett. 91,
090408
(2003). 60. L. Hackermuller, K. Hornberger, B. Brezger, et al:
Nature (London) 427, 711 (2004). 61. D.L. Jacobson, M. Arif, L.
Bergmann, et al: SPIE Proc. 3767, 328 (1999). 62. G. Kroupa, G.
Bruckner, O. Bolik, et al: Nucl. Instrum. Methods A 440, 604
(2000). 63. P. Fratzl, O. Paris: In: Neutron Scattering in
Biology—Techniques and Applications,
eds. J. Fitter, T. Gutberlet, J. Katsaras, pp. 205–221, Springer
Biological Physics Series (Springer, Heidelberg, 2006).
64. J. Summhammer, A. Zeilinger: Physica B 174, 396 (1991). 65. M.
Gruber, K. Eder, A. Zeilinger, et al: Phys. Lett. A 140, 363
(1989). 66. A. Ioffe, G.M. Drabkin, Y. Turkevich: Sov. Phys. JETP
Lett. 33, 374 (1981). 67. H. Funahashi, T. Ebisawa, T. Haseyama, et
al: Phys. Rev. A 54, 649 (1996). 68. U. Schellhorn, R.A. Rupp, S.
Breer, et al: Physica B 234–236, 1068 (1997). 69. C. Pruner, M.
Fally, R.A. Rupp, et al: Nucl. Instrum. Meth. A 560, 598 (2006).
70. K.I. Goldman, H. Kepa, T.M. Giebultowicz, et al: J. Appl. Phys.
81, 5309 (1997). 71. T. Keller, W. Besenbock. J. Felber. et al:
Physica B 324–236, 1120 (1997). 72. S.K. Sinha, M. Tolan, A.
Gibaud: Phys. Rev. B 57, 2740 (1998). 73. N. Bernhoeft, A. Hiess.
S. Langridge, et al: Phys. Rev. Lett. 81, 3419 (1998). 74. N.
Bernhoeft: Acta Crystallogr. Sect. A 55, 274 (1999). 75. H. Zabel.
R. Siebrecht. A. Schreyer: Physica B 276–278, 17 (2000). 76. H.
Kepa, J. Kutner-Pielaszek, A. Twardowski, et al: Phys. Rev. B 64,
121302(R)
(2001). 77. K.V. O’Donovan. J.A. Borchers, C.F. Majkrzak, et al:
Phys. Rev. Lett. 88, 067201
(2002). 78. R. Clothier, H. Kaiser, S.A. Werner: Phys. Rev. A 44,
5357 (1991). 79. X.S. Ling, S.R. Park, B.A. McClain, et al: Phys.
Rev. Lett. 86, 712 (2001). 80. U. Bonse, T. Wroblewski: Phys. Rev.
D 30, 1214 (1984). 81. S.A. Werner, H. Kaiser, M. Arif, et al:
Physica B & C 151, 22 (1988). 82. K.C. Littrell, B.E. Allman,
S.A. Werner: Phys. Rev. A 56, 1767 (1997). 83. S.A. Werner: J.
Phys. Soc. Jpn. 65, 51 (1996). 84. K.C. Littrell, B