# Interferometry

November 15, 2016 12:29 pmA short excursion into crystallography

Dynamical theory of diffraction

**Basic concept of neutron interferometry**

**Basic concept of neutron interferometry**

In a **neutron interferometer**^{1} a *monochromatic* beam of *massive* particles is split by amplitude division (*Mach–Zehnder type*) via *Bragg reflection, in triple Laue (LLL) *configuration*.* At the first plate of a silicon perfect crystal interferometer, which is cut from a single rod, the incoming neutron beam is split into two sub-beams. The *coherently superposed* sub-beams (*partial wave functions)* then passes through different regions of space before they are recombined at the third plate of the interferometer, where the interfere constructively and destructively.

Behind the first plate, acting as a beam splitter (due to *Laue* reflection – see below), the neutron’s wave function is found in a *coherent superposition* state of the transmitted and a reflected sub-beams, which can be written in terms of , where and are the transmission and reflection amplitudes ( for neglect able absorption since silicon has essentially zero absorption for thermal neutrons) with . Next an additional slab (*phase flag*), inducing an adjustable phase shift is inserted, which yields , where , with the thickness of the phase shifter plate , the neutron wavelength , the coherent scattering length and the particle density in the phase shifter plate. For instance, 0.15 mm of aluminium at 2 Å wavelength cause a phase shift of rad. This can be rewritten as , with As it is only the relative phase shift of the partial beams that counts for interferometry, it has become a widely used method to rotate a slab of matter that extends across both sub beams about an axis perpendicular to the interferometer. The thickness of the slab is therefore changed differently for the two partial beams, depending on the angle , as depicted above. Thus the phase difference is determined by the path difference, which yields , where is the Bragg angle and the rotation angle of the phase shifter slab. As seen in on the right side the function is almost linear for small . Thus an interferogram recorded versus remains sinusoid essentially. By rotating the plate, can be varied systematically. Leaving the interferometer at the third plate, where the sub-beams are recombines, the neutron’s final wave function on forward direction (O-direction) is given by . Thus the intensity in the O-detector is given by ,with . In similar manner we get in deviated (diffracted) direction (H-beam) , with . From particle conservation it follows immediately that . This behavior of the two intensities and is illustrated in the animation below (and for dimensions of the interferometer click **here**).

*A brief historical digression*

*A brief historical digression*

In 1964, when advances in semiconductor technology had allowed the production of large monolithic perfect crystal silicon ingots, U. Bonse and M. Hart invented a single-crystal interferometer for X-rays based on the effects of dynamical diffraction in perfect crystals. This type of interferometer was then applied to neutrons, resulting in the first interference fringes sighted in 1974 by Rauch, Treimer and Bonse at the rather small (250 kW) TRIGA reactor at the Atominstitut – TU Wien, Vienna, Austria. A picture of the first obtained interferogram is given below on the right hand side. The obtained interference demonstrates in impressive manner the wave-like nature of neutrons. Over the years numerous remarkable experiments on the fundamentals of quantum mechanics, have been carried out using neutron interferometry. Just to mention a few, there is the verification of the 4-spinor symmetry, followed by investigations of the influence of gravitation of the earth on the neutron’s wavefunction, and experiments on spin superposition, as well as topological phases. A detailed summary is given in the book *Neutron Interferometry: Lessons in Experimental Quantum Mechanics*, by Helmut Rauch and Samuel A. Werner ^{2}, which includes more than 40 neutron interferometry experiments along with their theoretical motivations and explanations.

*A short excursion into crystallography*

*A short excursion into crystallography*

*face centered*cubic crystal structure called

*The structure, including it’s chemical bonding, is schematically illustrated below left*

*diamond lattice.**atoms (yellow) are bonded to four others within the volume of the (cubic)*

*.*Four*.*

__conventional__unit cell*Six*atoms fall on the middle of each of the six cube faces, showing two bonds (light blue). Out of eight cube corners,

*four*atoms bond to an atom within the cube (dark blue) and

*four*are bonded outside of the cube (light brown), altogether

**18**Si atoms. The

*primitive cell*, depicted in the middle, consist of three

*lattice vectors*given by , , and . The lattice constant of silicon is roughly =5.43 Å.

Silicon’s *crystallographic structure*, as mentioned before, is *diamond cubic*, which is in the Fdm space group, following the face-centered cubic *Bravais lattice*. The *conventional unit cell, *depicted above right*,* consists of *two* basis atoms and may be thought of as *two* inter-penetrating *face centered cubic lattices*, with a basis of two identical Si atoms associated with each lattice point one displaced from the other by a translation of () along a* space diagonal *(indicated by the dotted red line). We start with the gray colored Si atom sitting in the origin of the cell and the second is the red colored Si atom. The seven remaining corners are colored in dark blue (note that in this illustration the colors have *different* meanings). In light blue we have the face-centered atoms of the original conventional cell and in yellow the face-centered atoms of the shifted cell (whose origin is indicated by the red coordinate system).

In our experiments we only use the {2,2,0} reflection, which is illustrated below left and the right (top-view). The lattice separations is calculated as Å. The reflected wavelength is given by Bragg’s law as . Hence, the maximum wavelength is found for *back-reflection* () yielding Å.

On the left side of the image below you can see an image of the {111} reflection planes with Å, which is significantly larger compared to the {220} reflection. On the right side we have the {113} reflection with Å (note that the unit cell has been rotated by 90 degree for illustrative purpose).

Atoms colored in brown do not belong to the particular reflection planes in the figure above. Other reflections, apart from {2,2,0}, {1,1,0}, {111} and {113}, that are also used in neutron optics are formed by the {115}, {117}, {331}, {335}, and {551} lattice planes.

**Dynamical theory of diffraction**

**Dynamical theory of diffraction**

Compared to the *geometrical* theory, the *dynamical* theory of diffraction accounts for the spatial periodicity of the interaction potential in perfect single crystals. Thereby the Schrödinger equation has to be solved taking this potential into account ^{3} ^{4}. When neutrons illuminate a perfect crystal under *near-Bragg* orientation conditions, the dynamical theory of diffraction predicts a coherent splitting of the incident wave into four components, with two travelling wave components passing within the crystal in the Bragg direction and two components in the forward (incident) direction. P. Ewald has shown in his historic investigation of the dynamical theory of *X-ray diffraction*, which in turn characterizes neutron diffraction as well, that this splitting will result in a periodic beating of *radiation density* travelling in either the *Bragg* or forward direction at different depths in the crystal, this feature being described as a *Pendellösung* structure.

**Single crystal plate:** The neutron wave function inside the crystal is deduced by solving the stationary Schrödinger equation for the time-independent crystal potential , where the potential is given by the sum over all nuclear scattering centres located at and with being the mass of the neutron. The vector of lattice position is given by , where denotes the vector from origin to elementary cell and is the vector from origin of elementary cell to position of atom in cell expressed as . Here are the lattice vectors of an elementary cell and are rational numbers between 0 and 1. Since the crystal potential is periodic it can be transformed into reciprocal space by calculating the Fourier transformation of the crystal potential as follows: , where denotes the *lattice factor* and the *structure factor*.

The *lattice factor* is given by , where Number of elementary cells and is a reciprocal lattice vectors with , where are the *Miller indices*, where , and with being the volume of elementary cell given by .

The *structure factor* is calculated using the definition and as . As a simple example, we consider a body-centered cubic crystal system, with and which yields . Now we consider the (2,2,0)-reflexion of the face-centered cubic lattice of a silicon single crystal, which is (is explained above in detail) of diamond lattice type. Then for (e.g. silicon in -direction) and for all odd/even (e.g. silicon in -direction).

Thus we get . If is the number of scattering centers (or atoms) per elementary cell, then is the number of scattering centers per unit volume. Calculating , the structure factor is always equal to and we get and . Plugging in the numbers from the table below

we get , eV. For thermal neutrons with energy of about eV the ratio between the potential and the energy yields . The *inverse* Fourier transformation of the potential is given by , which describes the periodicity of the potential in the crystal lattice.

*Basic equations*: Using the so-called *Bloch ansatz* , where is a wave vector inside the crystal and is the amplitude denoted (in analogy to the potential) as . Hence, the amplitude , which is defined in the reciprocal space, has to be determined. The wave-function is now given by . The basic equations of the *dynamical theory of diffraction* are therefore given by – a finite set of coupled equations for the amplitudes , representing equivalent waves in the crystal whose wave vectors differ by respectively. This system of equations cannot be solved generally, nevertheless, the respective amplitudes are expected to be large, if the wave vector of the reflected beam is near a reciprocal lattice point. Note that since the aim of this page is to explain the neutron interferometer only the *Laue* case is considered from this point on.

*One-beam approximation (beam refraction):* Here only the contribution is considered, where it is assumed that the neutron-nucleus interaction is small and that consequently the difference between the wave vector within the crystal and the vacuum wave vector is negligible. The reflected beam can safely be neglected if the optical potential is small compared to the kinetic energy of the incident neutron. A schematic illustration is given below. In the case of is given by the number of scattering centers in the elementary cell (no periodic structures are taken into account). Since is the number of elementary cells per volume unit, is given by . For a thermal beam with a wavelength of Å the kinetic energy is meV (m/s), which is orders of magnitude larger than typical values of , for example for silicon neV. Thus the corresponding equation is given by , with and . According to the energy conservation, for the free neutron with momentum , of magnitude , the kinetic energy is given by . Hence , since one can write , which yields . In analogy to light optics the refractive index is defined as . In general the wave vector changes both direction and magnitude – the neutron wave is refracted, which is depicted in the illustration below on the left side. Note, that for most materials () this leads to a refractive index slightly smaller than one. However, the neutron index of refraction is typically very close to one, e.g. for silicon . Despite the additional potential energy the kinetic energy is lowered. The one-beam approximation is valid for a perfect crystal far off any Bragg condition, as well as for homogeneous non-crystalline media. The direction of the beam is calculated according to the continuity of the tangential component , where we get for the perpendicular component we get . Using , where is the surface normal, this can be written as . Now one can calculate the phase shift induced by a phase shifter of thickness as , with .

*Two-beam approximation:* In this case, two lattice points, namely and one single have to be taken into account. This approximation is justified only if all the other lattice points are far away from Bragg refraction. We shall see that the refracted intensity decreases very quickly in close to a lattice point. Hence, from the basic equations from above can be concluded that two equations have to be solved simultaneously:

As we have seen in the one-beam approximation in the mean crystal potential is almost equal . Thus we now expect and also to be close to . Consequently we can make the following *ansatz*: and , where and are introduced as *excitation errors*, which are in the range of . The vectors themselves are given by for the forward beam and for the diffracted beam, which is depicted in the schematic illustration above. Note that here only occurs, but and are *not* independent due to . Finally we get (for *Laue* configuration) with , where describes the deviation from the exact Bragg angle . Hence, within the framework of the dynamical theory of diffraction, the exact Bragg equation is denoted as , with all these vectors belong to the *interior* of the crystal. In usual notation a Bragg reflection is obtained when the Bragg condition is fulfilled, i.e., with being the lattice plane distance defined the miller indices () and the lattice constant . The thermal neutron wave length is typically in the range of 1.9 to 2.7 Å, which corresponds to energies of 22 to 11 meV and Bragg angles of 30 to 45 degrees for silicon (220)-net plane reflection (m). Close to a Bragg condition two reciprocal lattice points lie near the *Ewald sphere* and therefore two amplitudes of the system of coupled equations, have to be taken into account. Returning to the basic equations of the two-beam approximation, we insert these results and obtain the following homogenous equation system:

from which we get , since , with the two solutions , where for silicon (2,2,0) reflection. Due to the two possible values we have two wave fields in the forward direction as well as two wave fields in the diffracted direction inside the crystal. Now we can define the ratio of the amplitudes . The corresponding wave vectors are given by . hence, two wave functions arise, which are superpositions of the forward and diffracted directions themselves. However, the most interesting are the wave functions that are outside in the forward (O) and diffracted (H) direction given by

Now we have to take a look at the boundary conditions. Putting the coordinate origin on the crystal surface and the -axis parallel to , provides the boundary condition of the front side of the plate and accounts for the boundary condition on the back side, where denotes the thickness of the crystal plate. For stake of simplicity the incoming wave is represented by a plane wave written as . The *continuity condition* of the wave function and the wave function at the front side of the crystal yields . Since there is no reflected wave at the front side in the Laue case we also get . Combining these equation yields and . The forward and diffracted beam behind the crystal are therefore given by

where and are complex crystal functions, accounting for *transmission* and *reflection*. The squared absolute values and are of interest because they are the ratios of intensity, which are diffracted behind the crystal.

Here the parameter is is proportional to the crystal thickness with and is referred to as *Pendellösung length, *simply because the intensity oscillates between the forward and diffracted directions, depending on the crystal thickness , if the Bragg condition is perfectly fulfilled (), which is illustrated above. The *parameter* is a dimensionless parameterization of the deviation from the Bragg angle given by . Note that this is only correct for the special silicon (2,2,0)-reflection, where we have . So the intensity also oscillates as a function of the deviation from the Bragg angle and these oscillations become narrower the thicker the crystal is. The intensities, dependent of the deviation from the Bragg angle for the O-beam on the left and the H-beam on the right for different thicknesses of the crystal plate, are plotted below. For a silicon crystal of thickness mm and thermal neutrons of a wavelength Å the spacing between two intensity minima is less than 5 seconds of arc, which due to facts like beam divergence and wavelength distribution cannot be experimentally resolved any more.

The reflected wave vector (inside the crystal) is given by , where the direction of the diffracted beam behind the crystal occurs, which is , with , a *correction vector* perpendicular to the surface. In addition, we introduce the quantity , which will be used for the wave functions in O and H direction. In forward (O) direction we have , being the incoming wave and with . For the reflected direction we get , which can be further evaluated as with , the quantity denotes the *back* of the crystal. Note that the index O here denote that initial direction of the the wave front is the O-direction, and t and r denotes whether it is *reflected* or *transmitted.*

**The complete interferometer:** If we apply the calculations from above to the second and third interferometer blade, the incident wave may also be in direction, which is depicted here:

, , , with and finally . Thus we distinguish waves have direction O and H instead of reflected and transmitted. By applying these to the second and third interferometer plate all wave functions of the LLL interferometer depicted below can be calculated. Note that factors and also depend the crystal thickness , as well as on the position of the crystal plate’s back. For the general case we assume different thicknesses for each blade and we them with the subscripts for beam** S**plitter, **M**irrow and **A**nalyzer. We start with beam path I, where we get behind the beam splitter** S** , and behind the mirror **M** . Behind the beam analyzer **A** we have and . All together the wavefunctions of beam path I are given by and . Likewise we can calculate the wavefunctions in beam path II, where we get behind the beam splitter **S** , and behind the mirror **M **. Behind the beam analyzer **A** the wavefunctions read ** ** and . All together we get and .

Thus, the *focusing condition* for an ideally perfect interferometer is given by , which just means that the three plates of the interferometer have to be *equal* and they have to be *equispaced. *Consequently in the forward direction, the two wavefunctions are equal expressed as . Both wave functions are reflected twice and transmitted once (r*rt*). Both wavefunctions have e*qual amplitudes* and *equal phases, *which is the most important result concerning the perfect Laue interferometer. The final wavefunctions in the O and H beam are the *superpositions* of both path contributions:

Finally, the last step is to compute the *intensities* in O and H beam. First we to calculate the intensity in O- direction in respect to the incoming wave, which yields , since .

1. H. Rauch, W. Treimer, and U. Bonse,* Phys. Lett.* **47A**, 369-371 (1974). ↩

2. H. Rauch and S.A. Werner, *Neutron Interferometry*, Clarendon Press Oxford (2000). ↩

3. H. Rauch and D. Petrascheck, Grundlagen für ein Laue Neutroneninterferometer – Teil 1 & 2 (1976).↩

1. H. Rauch, W. Treimer, and U. Bonse,* Phys. Lett.* **47A**, 369-371 (1974). ↩

4. M. Suda, *Quantum Interferometry in Phase Space*, Springer (2006). ↩