neutron physics with photorefractive materials

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 crystal 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 element, 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 perspectives 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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SVNY276-Gunter-v3 October 27, 2006 7:44
322 Martin Fally, Christian Pruner. Romano A. Rupp, and Gerhard Krexner
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 inhomogeneous 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 storage 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 further 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 expanded 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 holograms [9], is governed by the basic formulae of dynamical diffraction theory. In our experiments we deal only with nearly lossless dielectric gratings in transmission 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 sample’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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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 diffraction 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 effect. 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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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 scattering 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 photopolymer 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 potential 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 contribution 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 realized: 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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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 respectively. 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 sensitive 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 photorefraction 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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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 configuration. 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 considered: 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 corresponding terms, we can identify the following relations:
r S N = 2λ|μ|
hc2
N + λ2
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 measurements, 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 gratings 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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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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-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, information on the complex phenomenon of glass-forming processes and polymerization 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 interferometers 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 important to systematically improve the production process of light-induced holographic 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 technique in the sense that only density changes ρ are probed. Combining (12.13) and (12.15) yields
nN = bλ2
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 efficiency 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 photopolymerization 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 unambiguously. 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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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 process [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 integrated diffraction efficiency Ipc(ν) = ∫
g0(k) ∫
η(ν, θ)dθ ]dθdk,
however, is independent of the angular spread. This is important since the integrated diffraction efficiency is accessible experimentally by integrating the measured rocking curve. By Iη we will denote the experimentally determined value multiplied by d/ according to (12.6). Estimating the influence of the wavelength 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 governed 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 designed for time-resolved simultaneous diffraction experiments with light and neutrons. 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 photograph 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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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 efficiencies 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 photosensitized 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 experimentally 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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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 monitored 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 increased 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) after 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 thermal 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 experiment, 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 incoming 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 geometry 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 holographically 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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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 linear 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 accuracy 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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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 neutrons. The most interesting are determination of the scattering length for various materials in the low-energy regime, probing the coherence properties of cold neutrons, and gaining phase information in large-scale structure investigations. 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) experimentally, 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 modulation. 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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-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 corresponding 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 under 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 materials 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 sample [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 electro neutronoptic effect in 7LiNbO3:Fe. +1st and −1st diffraction orders are visible.
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0 2500 5000 7500
]
FI G U R E 12.14. ηB;N (t) during illumination with white light; t = 0 indicates 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 photoinduced 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 recording in electrooptic crystals means that a light-induced space-charge density of electrons modulates the refractive index via the electrooptic effect. By increasing 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 temperatures, 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 various other parameters. Spectros copic investigations have revealed that the ionic
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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 diffraction 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 gratings, 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 calculated. 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 measurements 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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12.5 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 recording 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 neutron 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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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 coherently 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 intensity 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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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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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 applications 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 incoherent 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 materials 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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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 polymers, 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 neutrons, 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 polarized 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 preparation of gratings based on this effect by light-optical means and the diffraction
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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 scattering 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 allows 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.
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