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(p.457) A20 Reciprocal Space and Reciprocal Lattice

Source:
Bonding, Structure and Solid-State Chemistry
Publisher:
Oxford University Press

A20.1 Reciprocal space

The reciprocal lattice concept was introduced by Ewald in 1913 [1]. He applied it some years later in the interpretation x-ray diffraction photographs [2], although the procedure was conceived independently a year earlier by Bernal [3].

In reciprocal space a lattice, the reciprocal lattice, is defined for each of the fourteen real space Bravais lattices; it is derived here first by the following construction, as applied to a monoclinic lattice in projection on the x, z plane.

A primitive unit cell is outlined by vectors a and c, with three families of planes indicated by their Miller indices; b is normal to the plane of projection. Lines are constructed from the origin O that are normal to the families of Bravais lattice planes (hkl) shown (Fig. A20.1a).

Fig. A20.1 (a) Mtice in projection on the x, z plane in real space, showing primitive unit cells with basic translations a and c, and traces of the (100), (101) and (001) families of planes; $β=∠Oa−Oc.$ (b) Monoclinic reciprocal lattice constructed from the lattice in (a) showing the lengths $a∗$ and $c∗$ of the reciprocal unit cell, in projection on the $x∗$, $z∗$ plane in reciprocal space; $β∗=∠Oa∗−Ob∗$.

[Structure Determination by X-ray Crystallography, 5th ed. Springer Science+Business Media, 2013d; reproduced by courtesy of Springer Science+Business Media, NY.]

Along each of these lines, reciprocal lattice points hkl are defined such that the distances to these points from the origin are the reciprocals of the corresponding interplanar spacings in Bravais space. Thus, in the figure, the families of planes (100), (101) and (001) give rise to reciprocal lattice points at distances from the origin that are proportional to $1/d(100),1/d(101)$ and $1/d(001),$ where $d(100)=OP,d(101)=OQ$ and $d(001)=OR.$ In general, a reciprocal length $d∗$ is given by

(A20.1)
$Display mathematics$

The vectors $d100∗,d010∗$ and $d001∗$ may be taken to define the translation vectors a$∗$, b$∗$ and c$∗$ of a unit cell in the reciprocal lattice (aka the reciprocal unit cell). The following equations for the monoclinic reciprocal lattice can now be determined from Fig. A20.1a:

(A20.2)
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But $d(100)$ is equal to $asinβ.$ Hence:

(A20.3)
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Similarly,

(A20.4)
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(p.458) but

(A20.5)
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because $d010∗$ is normal to the a, c plane. The unique $β∗$ angle is given by

(A20.6)
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The $a∗,c∗$ plane from this construction is shown in Fig. A20.1b. Furthermore,

(A20.7)
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and similarly for $b⋅b∗andc⋅c∗.$ For the mixed products:

(A20.8)
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(p.459) and similarly for all other such products. The relationships of Eqs. (A20.7) and (A20.8) apply to all crystal systems.

The reciprocal lattice is considered next in a general manner; knowledge of simple vector manipulations are assumed. In Fig. A20.2, the $z∗$-axis is normal to the plane a, b. Since $c⋅c∗=1=cc∗cos∠COR:$

(A20.9)
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Fig. A20.2 General, triclinic unit cell in real space; a, b and c are the unit cell vectors in real space, and a$∗$, b$∗$ and c$∗$ the corresponding reciprocal space unit cell vectors. Inter-axial angles in real space $α,β,γ,whereα=∠BOC,$ and $β,γ$ by cyclic permutation; in reciprocal space the angles are $α∗,β∗,γ∗,whereα∗=∠QOR,$ and $β∗,γ∗$ by cyclic permutation.

[Structure Determination by X-ray Crystallography, 5th$>$ ed. Springer Science+Business Media, 2013.]

But c$∗$ is normal to both a and b, so that it lies in the direction of their vector product. Hence:

(A20.10)
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where $η$ is a constant. Since $V=c∗⋅(a×b),$ the scalar product of Eq. (A20.10) and c is

$Display mathematics$

so that $η=1V;$ hence, from Eq. (A20.10):

(A20.11)
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and similarly for a$∗$ and b$∗$ by cyclic permutation. In scalar form, Eq. (A20.11) becomes:

$Display mathematics$

(p.460) and

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and similarly for $a∗$ and $b∗$ by cyclic permutation.

The angles are given by [1, 2]:

(A20.13)
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with $cosα∗andcosβ∗$ obtained by cyclic permutation.

It remains now to show that the reciprocal lattice points so constructed form a true lattice. The vector normal to the plane (hkl) is $h(b×c)+k(c×a)+l(a×b)$[4]. Dividing by V, and denoting the resulting vector $dhkl∗:$

(A20.14)
$Display mathematics$

Since h, k and l are integers, the vectors $dhkl∗$ drawn from the common origin form a lattice, the reciprocal lattice, with translation vectors a$∗$, b$∗$ and c$∗$ and interaxial angles:

(A20.15)
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It is a standard notation to denote reciprocal lattice points by the Miller indices of the family of planes in the Bravais lattice from which they were derived, but written without parentheses.

A20.2 Reciprocal lattice and Ewald’s construction

The Bragg construction for diffraction (Appendix A3) requires a consideration of the distribution of planes, which is not always easy to visualize. An x-ray photograph of a crystal is a picture of its reciprocal lattice (Fig. A20.3). It is a regular array of spots governed in position by the space group symmetry of the crystal, and in intensity by the nature and positions of the atoms in the crystal structure.

Fig. A20.3 X-ray precession photograph of the 0kl layer of reciprocal space for a crystal of a cubic perovskite, space group $Pm3¯m.$ The distances between adjacent spots are the reciprocals of the translation distances b and c in real space; they are equal in this example because this perovskite is a cubic crystal. No systematic absences of spots arise because of the complete absence of translational symmetry in the structure.1 Streaks attached to the more intense spots indicate a lack of complete monochromatization of the incident beam; the crystal effectively selects the wavelengths from those available in the incident x-ray spectrum that satisfy the Bragg equation for hkl reflections, as in a Laue photograph [5].

The Ewald sphere construction [2, 45] discusses the diffraction record in terms of the geometry of the recorded x-ray spot pattern (Fig. A20.4). A sphere of radius unity in reciprocal space is constructed on the x-ray beam as the diameter AQ, where Q is the origin of the reciprocal lattice: the crystal under consideration is at the centre C of the sphere. From the construction, AQ = 2 and $∠APQ=90∘.$ Thus, $QP=AQsinθhkl=2sinθhkl$, which, from Eqs. (A3.2) and (A20.1) is $λ/dhkl.$ Thus, P is the reciprocal lattice point corresponding to the hkl family of (p.461) planes and CP is the direction of the reflected x-ray beam. Hence, an x-ray reflection from a family of planes (hkl) occurs when the reciprocal lattice point hkl intersects the Ewald sphere (aka sphere of reflection), and the direction of the reflected beam is from the crystal C through the point P(hkl). Note that although the term x-ray reflection is used, following the Bragg equation, the process is one of diffraction or, more specifically, of combined diffraction and interference processes.

Fig. A20.4 Interpretation of crystal diffraction maxima by the Ewald sphere (aka sphere of reflection) construction, assuming without loss of generality the zero-layer $x∗$, $y∗$, 0 of the reciprocal lattice: C crystal, AQ incident x-ray beam and θ‎‎hkl its glancing angle of incidence, Q origin of reciprocal lattice, P reciprocal lattice point hkl intersecting Ewald sphere and giving rise to a reflected beam in the direction CP.

[Structure Determination by X-ray Crystallography, 5th ed. Springer Science+Business Media 2013; reproduced by courtesy of Springer Science+Business Media, NY.]

Another illustration of diffraction through the Ewald sphere concept is shown by Figs. A20.5 and A20.6; the annotations correspond to those of Fig. A20.4. The radius of the Ewald sphere in a practical context is set at $1/λ.$ A reflection arises at the point Phkl on the Ewald sphere if $QP=CP−CQ,$ that is, $QP=(1/λ)(s−s0),$ where $s$ and $s0$ are unit vectors in the reflected and incident x-ray beams respectively. Other reciprocal lattice points, such as S, that intersect the Ewald sphere will also give rise to reflections, the direction being CS for the point S. The Ewald sphere is rotated about the origin Q of the reciprocal lattice (in practice the crystal is rotated, taking the reciprocal lattice with it). As reciprocal lattice points intersect successively the Ewald sphere, the corresponding reflections are obtained.

Fig. A20.5 Ewald sphere: reflection is obtained along the direction $|$CP$|$ as the reciprocal lattice point P intersects the Ewald sphere; the annotations correspond to those in Fig. A20.4.

Fig. A20.6 Observable reflections, those for which $|QP|≤λ$/2, lie within the limiting sphere, radius $2/λ$ (= diameter of Ewald sphere). The theoretical maximum d-value $(sinθ=1)$ for Cu $Kα$ x-radiation is ca. 12.97 nm−1.

(p.462) Only those reflections are observable for which $|QP|≤λ/2,$ the diameter of the Ewald sphere. These reflections lie within a sphere, centre Q and radius $2/λ$ that is termed the limiting sphere, of which the circle in the figure is one section. From the Bragg equation, it is clear that if $λ>dmax(sinθ>1)$ there can be no reflection. The limiting sphere represents the possible limit of resolution of crystal diffraction. Thus, for Cu $Kα$ the maximum radius of the limiting sphere is ca. 12.97 nm−1.

Exercise A6

A crystal has the unit-cell dimensions a = 0.1230 nm, b = 0.4560 nm and c = 0.7890 nm, and angles $α=90.00∘,β=95.00∘$ and $γ=105.00∘.$ Calculate the constants of the reciprocal unit cell and the volumes of the two unit cells. (Answer at end of Appendices.)

References A20

Bibliography references:

[1] Ewald PP. Z. Phys. 1913; 144: 465.

[2] Ewald PP. Math. Ann. 1921; 56: 615.

[3] Hodgkin DMC. Biogr. Mems. Fell. R. Soc. 1980; 26: 28.

[4] Ladd M and Palmer R. Structure Determination by X-ray Crystallography, 5th ed. Springer Science+Business Media, 2013.

[5] Ladd M. Symmetry of Crystals and Molecules. Oxford University Press, 2014.

Notes:

(1) Symmetry dependent absences are regular, governed by the space group of the crystal.