The reciprocal lattice

A three-dimensional lattice, reciprocal to the crystal lattice, is very useful in analyses of X-ray diffraction patterns it is called the reciprocal lattice. Earlier in this chapter the diffraction pattern of a series of regularly spaced slits was considered to be composed of an envelope profile, the diffraction pattern of a single slit, and sampling regions,  

TABLE 3.2. Some scattering factors for neutrons and X rays. 

Element Isotope X rays sinS/A = 0 X rays sinS/A = 0.5/A Neutrons 6(10 12 cui) Neutrons (normalized to -1.00/or Iff) 

The quantity b is the neutron coherent scattering amplitude. In the entry for Li, i = y/—l. The anomalous factor for Li involves a phase shift of about 8° [(0.025/0.18) = tan8°] relative to most other nuclei. 

Neutron scattering amplitudes normalized arbitrarily to the value -1.0 for in order to illustrate the small range of scattering amplitudes observed as compared with that observed in X-ray scattering (2.7 for neutrons versus 92 for X rays). 

Every direct lattice admits a geometric construction, the reciprocal lattice, by the prescription that the reciprocal lattice basis vectors (bi, hx, bs) obey the following important orthogonality rules relative to the direct lattice basis vectors (ai, ai, as)  

Like in direct space, any reciprocal lattice vector can be expressed as a linear combination of the basis vectors with integer coefficients such as 

Among all possible equivalent choices of a unit cell in the reciprocal lattice, one is particularly useful. It can be obtained by connecting one reciprocal lattice point to all its nearest neighbors and letting orthogonal planes pass through their midpoints. The volume within these planes is known as the first Brillouin zone. It includes all points that are closer to that reciprocal lattice point than to any other lattice point. 

Initially it may not be obvious why we proceed to describe the following operations, which define what is called the reciprocal lattice, but bear up for awhile and hopefully your patience will be rewarded. 

It is important in understanding electron (and x-ray) diffraction to know how waves interact with solids. The energy band diagrams described in Chapter 2 all refer to electron wave vectors and diffraction of the electron wave off the crystal lattice at the Brillouin zone boundary. To understand diffraction one has to consider the lattice from the perspective of a wave, not from the perspective of points. 

Any function, such as a function f(R) giving the probability of finding an atom at a given position R in real space, can be Fourier transformed to provide an indication of the amplitude f(k) of waves having wave vector k which, if summed or integrated, replicate f(R). Mathematically, the Fourier transform relationship is  

The amplitude function f(k) is similar to a display of the intensity of sound as a function of wavelength on a graphic equalizer or to the intensities of colors in a spectrum produced by passing a light beam through a prism. 

2° Twist ooooo ooooo ooooo ooooo ooooo 

The reciprocal lattice is defined by the set of points hb], kb2, and lb3 with the basis vectors defined by Equation 4.2. 

When dealing with the interactions of crystals with particles that can display wave-like properties, like photons, phonons or electrons, it is useful to introduce a reciprocal lattice associated with the real (or direct) crystal lattice. Let us consider a set of vectors R constituting a given 3D BL and a plane wave elk r. For special choices of k, it can be shown that k can also display the periodicity of a BL, known as the reciprocal lattice of the direct BL. For all R of the direct BL, the set of all wave vectors G belonging to the reciprocal lattice verify the relation 

The reciprocal lattice of a BL whose primitive unit cell is defined by three vectors ai, a2 and a3 is generated by three primitive vectors 

Many of the physical properties of crystals, as well as the geometry of the three-dimensional patterns of radiation diffracted by crystals, are most easily described by using the reciprocal lattice. Each reciprocal lattice point is associated with a set of crystal planes with Miller indices (hkl) and has coordinates hkl. The position of the hkl spot in the reciprocal lattice is closely related to the orientation of the (hkl) planes and to the spacing between these planes, dhki, called the interplanar spacing. Crystal structures and Bravais lattices, sometimes 

Let us come back to our task to find all possible diffraction peaks for a given crystal lattice. What are the possible scattering vectors that lead to constructive interference This question can be answered in an elegant way by defining the so-called reciprocal lattice If a, a2, and 23 are primitive vectors of the crystal lattice, we choose a new set of vectors according to 

2 Max von Laue, 1879 - 1960. German professor for physics. Nobel prize for physics 1914 for the discovery of X-ray diffraction in crystals. 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties  

Another property is that for every set of parallel lattice planes, there are reciprocal lattice vectors that are normal to these planes. The shortest one of these reciprocal lattice vectors is used to characterize the plane orientation. The components (h, k, l) of this vector are called Miller indices and the direction of the plane is denoted by (hkl) for the single plane or hkl for a set of planes (see Section 8.2.1). 

Why is it useful to introduce such a complicated set of vectors This becomes obvious when we look at the scalar product between a real space lattice vector R and a reciprocal lattice vector q. Expressing these vectors by the corresponding primitive vectors we can write  

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Figure Bl.8.2. Bragg s law. Wlien X = 2d sin 0, there is strong, constructive interference. (B) THE RECIPROCAL LATTICE...

Figure Bl.8.3. Ewald s reciprocal lattice construction for the solution of the Bragg equation. If Sj-s. is a vector of the reciprocal lattice, Bragg s law is satisfied for the corresponding planes. This occurs if a reciprocal lattice point lies on the surface of a sphere with radius 1/X whose centre is at -s..

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

The reciprocal lattices shown in figure B 1.21.3 and figure B 1.21.4 correspond directly to the diffraction patterns observed in FEED experiments each reciprocal-lattice vector produces one and only one diffraction spot on the FEED display. It is very convenient that the hemispherical geometry of the typical FEED screen images the reciprocal lattice without distortion for instance, for the square lattice one observes a simple square array of spots on the FEED display. 

Variation in energy for a tour (r-X-M-F) of the reciprocal lattice for a 2D square lattice of hydrogen 2S. (Figure adapted in part from Hoffmann R 1988. Solids and Surfaces A Chemist s View on Bonding in nded Structures. New York, VCH Publishers.)... 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

For a given structure, the values of S at which in-phase scattering occurs can be plotted these values make up the reciprocal lattice. The separation of the diffraction maxima is inversely proportional to the separation of the scatterers. In one dimension, the reciprocal lattice is a series of planes, perpendicular to the line of scatterers, spaced 2Jl/ apart. In two dimensions, the lattice is a 2D array of infinite rods perpendicular to the 2D plane. The rod spacings are equal to 2Jl/(atomic row spacings). In three dimensions, the lattice is a 3D lattice of points whose separation is inversely related to the separation of crystal planes. 

Figure 2 View looking down on the real-space mesh (a) and the corresponding view of the reciprocal-space mesh (b) for a crystal plane with a nonrectangular lattice. The reciprocal-space mesh resembles the real-space mesh, but rotated 90°. Note that the magnitude of the reciprocal lattice vectors is inversely related to the spacing of atomic rows.

Fig. 2.48. Direct and reciprocal lattices of a series of commonly occurring two-dimensional superlattices. Open circles correspond to the ideal (1x1) surface structure, whereas filled circles represent adatoms in the direct lattice and fractional-order spots in the reciprocal lattice [2.243],...

Pt2,V and Pt y have been investigated at 1393 K and 1224 K respectively and we have explored the [100] and [110] planes of the reciprocal lattice. The measured Intensities have been Interpreted in a Sparks and Borie approach with first order displacements parameters and using a model Including 29 a(/ ) for PfsV and 21 for PtsV. In figure 1 is displayed the intensity distribution due to SRO a q) in the [100] plane. As for PdjV, the diffuse intensity of Pt V is spread along the (100) axes with maxima at the (100) positions, whereas the ground state is built on (1 j 0) concentration wave ( >022 phase). 

The obtained static structure factors agree well with the experimental ones [4], all trends of the peak positions are reproduced correctly. There are only small deviations from the experiments (i) due to the pseudopotential (slighly too small bond lengths which correspond to slightly too large peak positions in the reciprocal lattice) and (ii) correct positions but a wrong trend in the heights of the prepeaks. For a detailed description see Ref. [7]. 

According to the above method, we rewrite the integration, in the irreducible BZ s ment, in terms of the abscissas, a, b and c, along either the reciprocal lattice primitive vectors or any oAer convenient set as follows... 

Since wavelengths smaller than twice the size of the lattice spacing are meaningless, we take the above integral over one Briiiouin zone of the reciprocal lattice i.e. we take —n/a [Pg.650]

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

Figure 4.26. Schematic drawing of the LEED experiment on a single costal with a unit cell given by vectors a-i and O2. The LEED pattern corresponds to the reciprocal lattice described by vectors and 02. ...

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