Peculiarities of Laue Diffraction of Neutrons in Strongly Absorbing Crystals
Well-known Kato’s theory of the Laue diffraction of spherical x-ray waves is generalized to the case of the neutron diffraction in strongly absorbing crystals, taking into consideration both the potential and the resonant scattering of neutrons by nuclei as well as a realistic angular dispersion of incident neutrons. The saddle-point method is applied for estimation of the angular integrals, being more adequate in the case of strongly absorbing crystals than the usually used stationary-phase approximation. It is found that the intensity distribution of the diffracted and refracted beams along the basis of the Borrmann triangle significantly depends on the deviation of the neutron energy from the nuclear resonant level. When comparing our calculations with the Shull’s experimental data on neutron diffraction in silicon we regard also the role of finite width of the collimating and scanning slits.
The neutron scattering is one of the most powerful tools for investigation of the crystal structure and its magnetic properties (see, e.g., Refs. [1-6]). For interpretation of experimental data, obtained in very thin films, it is sufficient to apply the kinematical scattering theory, while for thick targets, where the multiple scattering of neutrons by atoms becomes significant, one has to use already its dynamical version. Such a dynamical scattering theory has been developed for the elastic diffraction in perfect crystals of both neutrons [7-9] and Mössbauer rays [10-13], treating them as plane waves. Its generalization to the case of inelastic diffraction at the crystals subject to external alternating fields was given in Refs. [14, 15].
In the case of two-wave diffraction it was predicted the effect of suppression of reactions and inelastic channels [8, 10], observed later in numerous Mössbauer diffraction experiments (see, e.g., the reviews [16, 17]) as well as in the neutron diffraction experiments [18, 19], studying the reaction in the cadmium sulphide crystal, abandoned with the nuclei Cd having the resonant level 0.178 eV.
Although there is close analogy of the suppression of nuclear reactions with the anomalous transmission of x-rays (the Borrmann effect) [20-25], the resonant nuclear scattering provides principally new character of the anomalous absorption of neutrons or -photons. Namely, in the scattering of x-rays by atomic electrons, when the Bragg condition is fulfilled, there is only partial suppression of inelastic scattering. At the same time, in the case of resonant nuclear scattering, choosing the corresponding geometry of the experiment, one can achieve complete suppression of the inelastic and reaction channels. This effect is explained by the dynamical scattering theory, which predicts that the energy exchange between the refracted and diffracted waves inside the crystal ensures splitting each of them into two waves with different wave vectors. One such wave is weakly absorbed and another strongly.
Notice once more that in Refs. [7-15] the electromagnetic waves and neutrons were described by plane waves. At the same time, in typical experiments on the Laue diffraction the incident waves first pass a narrow slit and only afterwards penetrate into the crystal (see Fig.1). In this case the incident neutrons are described already by a wave packet written as the integral over the angle. Moreover, both the refracted and diffracted waves travel inside the crystal within the region, confined by so-called Borrmann triangle [20-25]. The distribution of diffracted beam intensity along the basis of the Borrmann triangle oscillates due to interference of two waves, transmitting the crystal with different wave vectors. The same interference provides also the familiar Pendellösung effect [20-24].
Kato [26-29] developed a theory for such a diffraction, assuming the angular dispersion of incident x-ray waves to be much larger than a small diffraction interval of the order of several seconds of arc. Although Kato told about the diffraction of spherical waves emitted by a point-like source, in reality he considered the cylindrical waves, emitted by the thread-like source. More exactly, the collimating entrance slit has been regarded like the continuous line of such point emitters, stretched along the slit.
Shull [30-32] used the Kato’s theory in order to interpret the results of his Laue diffraction experiments of neutrons in perfect crystals of silicon and germanium. Measuring the period of interference oscillations for the diffracted neutron beam he determined with high precision the coherent scattering lengths of neutrons by the nuclei of silicon  and germanium . These experiments were repeated later by Abov and Elyutin [5, 6].
Worth noting also the papers [33-37], studying the Laue diffraction of neutrons in large silicon crystals at the Bragg angle close to . In these conditions the authors observed anomalous absorption and slowing down of neutrons. Moreover, they hoped to verify the equivalence principle of the inertial and gravitational masses on the example of such microscopic object as neutron and achieve higher accuracy .
Previously we analyzed the symmetric Laue diffraction of divergent neutron beams  in strongly absorbing crystals, taking into account both the potential and resonant neutron scattering by nuclei. Following Kato [26-29], we took the angular dispersion of the incident neutrons much exceeding the characteristic angular interval , where the diffraction proceeds. In other words, a real angular distribution of neutrons in this approximation was replaced by a constant. The analogous theory for the symmetric Laue diffraction of the Mössbauer radiation has been developed in .
In the present paper we study general case of the Laue diffraction, when the reflecting crystal planes are canted at arbitrary angle with respect to the crystal surface. For the first time the angular distribution of incident neutrons is included into consideration with dispersion , which may be of the order of the diffraction angular range . For calculation of the angular integrals, which determine the neutron wave packet inside a thick crystal, we use the saddle-point method. It is more adequate for strongly absorbing crystals than the stationary-phase method, which has been used previously in the theory of x-rays diffraction in weakly absorbing crystals . Note that in the Kato’s approximation, when the function is replaced by a constant, these angular integrals are calculated exactly . However, in our approach due to the factor we need estimations of the integrals by means of the saddle-point approximation for thick enough crystals.
We consider typical experimental situation (see, e.g., [30-32]), when the neutrons pass first through the entrance slit cut in a shield and then move inside the crystal within the Borrmann triangle. We assume that all the neutrons fall perpendicularly to the collimating slit, which in turn is parallel to the reflecting crystal planes. First we shall regard the slit as a thread-like emitter. In this case the neutron waves have a cylindrical symmetry with the symmetry axis along the slit. And in the fourth section we analyze the role of finite width of both slits used in experiments.
2 Basic formulas
Let the incident neutrons at be described by the initial wave packet
where are the wave vectors of neutrons, is their energy and the mass. The product is interpreted as a probability of finding the neutron with the wave vector lying in the interval beside . For brevity, we omit the spin factor, which does not affect the coherent scattering by nonmagnetic crystal with unpolarized nuclei.
We introduce the right-hand coordinate frame with the origin on the entrance surface in the middle of the collimating slit. The axis is directed inside the crystal perpendicularly to its surface and the axis along the slit (see Fig.1). One introduces also the frame with the axis parallel to the reflecting crystal planes and axis coinciding with . It is obtained from by rotation through the angle around the axis . Here for clockwise rotation and otherwise. The angle between the neutron wave vector and the axis is , the incidence angle on the reflecting planes is and the Bragg angle
Besides, we introduce the angles and between the axis and the sides of the Borrmann triangle, corresponding to transmitting and diffracted rays ( and ).
Let the divergent beam of neutrons move in the plane perpendicularly to the slit and be spread over the angle . Then , while the components of along the axes are
The wave function (1) may be rewritten as
where characterizes the energy distribution of incident neutrons, the function
describes neutrons with fixed energy . We approximate the angular distribution by the Gaussian function with maximum at the angle close to the Bragg angle :
denotes the mean-square angular dispersion of neutrons. Usually , that enables us to spread the integration limits over from to . If much exceeds the diffraction interval , then can be replaced by a constant.
The energy distribution is also described by the Gaussian function:
We shall consider scattering of the neutron wave by atomic nuclei in the crystal, ignoring influence of the electrons. Then the coherent scattering of neutrons by an elementary cell of the crystal from the state to is determined by the amplitude
where is the scattering wave vector, the radius-vector defines equilibrium position of the th atom within the elementary cell, is the coherent scattering amplitude of low-energy neutrons by the th nucleus:
where is the coherent scattering length of neutrons by the th nucleus, is the square root of the Debye-Waller factor, is the coherent resonant scattering amplitude. In vicinity of an isolated resonance it is given by
where is the probability of finding the resonant isotope in the th site, is the spin of the ground state of the initial nucleus and the spin of the excited state of the compound nucleus, is the energy of the resonant level, is the total width, consisting of the partial widths for neutron , radiation and possibly fission decay channels, is the displacement of the th nucleus from the equilibrium position, and are sets of phonon numbers in the initial and final states of the crystal, are the phonon frequencies, the brackets denote averaging over the initial states of the crystal lattice.
The sum in Eq.(10) can be transformed to the integral:
where is the Lamb-Mössbauer factor,
is the mass of the th atom, is the number of elementary cells, is the average number of phonons of the th normal vibration with polarization .
In the framework of the Debye model of the crystal with one atom per the elementary cell, ignoring anisotropy of vibrations, one can rewrite this expression as (see also )
where and are the initial and final momenta of neutrons, is the Debye temperature, is the maximal frequency of phonons.
In such Debye approach the parameter is given by 
where represents the recoil energy of the nucleus with mass .
3 The wave function
According to collision theory  every plane wave of the wave packet (4) is scattered independently of each other, giving rise to the wave function . Therefore the neutron wave packet, born by the incident wave Eq. (4) takes the form
In the two-wave case the wave inside the crystal as , where is the crystal thickness, consists of the refracted wave with the wave vector and the diffracted one with the wave vector , where denotes a reciprocal lattice vector. The components of the vectors and along the entrance surface coincide. Therefore the vectors with can be written as
where is the unit vector along the axis .
As a consequence, the wave function inside the crystal transforms to
with given by
Following  we introduce the notations
The parameters are related by
The means the refractive index for the incident wave .
Recall that are the angles between the vectors and the axis . The angle indicates deviation from the exact Bragg condition corresponding to . For neutrons with fixed energy 
The corrections to the wave numbers in the medium are related with the parameters by
The scattering matrix is defined by the expression
where stands for the volume of the elementary cell.
Henceforth the root with sign plus is associated with and minus with .
It is more convenient to express them in terms of new deviation parameter
where the angular shift
Note that is already a complex number.
Then the parameters take simple form
and , defined by Eq.(26), may be written as
means the Pendellösung distance in the case of weakly absorbing crystals (see, e.g., ).
For the Laue diffraction () the amplitudes of the waves satisfy the following boundary condition at :
Being expressed in terms of the variable , they take the form
Here has a sense of the characteristic diffraction range, corresponding to variation of from zero to unity.
According to (3) the maximal amplitude of the diffracted neutron wave is achieved at , i.e. if
From here we see that can be interpreted as the Bragg angle shifted by .
Assuming that the incident beam is oriented along the corrected Bragg angle, , we rewrite the angular distribution as
where the mean-square width
From definition of it follows also that
The neutron intensity distribution over the basis of the Borrmann triangle is usually analyzed with the aid of the scanning slit, located on the rear surface and directed along the axis z. Let be the radius vector of any point inside this slit, while be the coordinate of the midpoint E on the side AB of the Borrmann triangle. Following  we determine the reduced coordinate of this point as
where is the length of the line segment AB, is the coordinate of the point relative to the midpoint E. The definition (44) is equivalent to
where is the angle between the reflecting planes and the direct line CS, connecting the entrance slit and the point (see Fig. 1).
It remains now to expand the plane wave in the point in powers of . Keeping the linear terms in the expansion of and introducing the notation we get
With the help of relations
which follow from the equalities
we find that
Substitution of this formula into (47) after some trigonometric manipulations gives
Taking also into account Eq. (34), we are led to the following expression for the waves in the exit slit:
where we used the abbreviation
one finds the integral representation for the wave function in the point :
where the integration path in the complex plane is a direct line, defined by the condition Im
For a crystal, whose thickness , the large parameter allows us to estimate the integral over with the aid of the saddle-point method (see, e.g., ). Here we assume that as well as are smooth functions. First from the equation one finds the saddle points:
Since the integrand in (57) is an analytical function one can deform the integration contour on the complex plane . This contour should cross the th saddle point along the line which indicates a steepest decent of the function . Along such a line Im const and the function Re is maximal in the point . These requirements are satisfied if the line is directed with respect to the real axis at the angle
where the second derivative of in the saddle point equals
Evaluating the integral (57) with the aid of the saddle-point method, one has
where the amplitudes in the saddle points are
while the angular factor
and for the diffracted those
and the amplitudes are
The corresponding intensities of the monochromatic neutron beams are determined by
Introducing the notation
we get the following intensity distribution of the refracted beam through the basis of the Borrmann triangle :
while for the diffracted beam distribution one has
and are the absorption coefficients for neutrons incident far from the Bragg condition , no diffraction), but having the wave vectors and , respectively. They are determined by
where is the total cross section for scattering and absorption of neutrons by an elementary cell, which have in the initial state the wave vector . In accordance with the optical theorem 
Outside the Borrmann triangle in close vicinity to the points these intensities are
Outside the Borrmann triangle the intensity of the diffracted neutrons (77) in strongly absorbing crystal first quickly falls down and then begins to grow with increasing due to the hyperbolic sine. But in this region our approach is not valid, since the departure becomes too large.
In order to illustrate the role of the resonant scattering we have done numerical calculations for the symmetric Laue diffraction in an isotropic crystal containing single resonant nucleus in every unit cell. We neglected the potential scattering amplitude compared to the resonant one. The latter was described by Eq. (16) with the Debye-Waller factor . In this approximation the absorption coefficient, depending on the detuning of the resonance , is written as
where is the resonant value of the absorption coefficient, given by
As to the functions and , they are determined by the following expressions:
Intensities of the diffracted and refracted beams as functions of , calculated in units of with and are presented in Figs. 2, 3 for and in Figs. 4, 5 for more thick crystal with .
4 Averaged intensities
Remind that up to now we dealt with the waves ejected from the thread-like source with the coordinates and changing from to . And now we shall analyze the role of finite width of the entrance slit, regarding it as a sum of the parallel thread-like sources, spread over the interval , which corresponds to variation of the coordinate in the interval of the width . In the case of symmetric diffraction .
The neutron wave in any point of the scanning slit is a superposition of the waves emitted by every such thread and afterwards passing the crystal region, confined by their own Borrmann triangle. The resulting waves in the point will be
The corresponding integral intensity
Besides, when the scanning slit has the width , the intensity should be integrated over from to , where denotes the coordinate of the slit’s middle. So the flux of neutrons per unit time, emerging from the scanning slit in the th direction, is determined by
where is the velocity bound to the average energy :
We compared our results with the data of Shull , who had observed the symmetric Laue diffraction of neutrons at (111) planes of silicon crystal. It has the diamond structure with the crystal constant Å and contains 8 atoms in the elementary cell . A spacing of the adjacent (111) planes equals . The corresponding scattering amplitudes are
where the scattering vector and is an integer. We took .
In numerical calculations of the intensity we used the following experimental parameters: the wave length Å, crystal thickness cm, the coherent scattering length cm and the factor . The calculated dependence of