International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 1.1, pp. 2-5
doi: 10.1107/97809553602060000572

Chapter 1.1. Summary of general formulae

E. Kocha

aInstitut für Mineralogie, Petrologie und Kristallographie, Universität Marburg, Hans-Meerwein-Strasse, D-35032 Marburg, Germany

In this chapter, general geometrical formulae are given that describe (i) the relations between the lattice parameters and unit cells in direct and in reciprocal space and (ii) the relations between lattice vectors, point rows and net planes, and allow (iii) the calculation of various angles in direct and in reciprocal space (including the Miller formulae).

In an ideal crystal structure, the arrangement of atoms is three-dimensionally periodic. This periodicity is usually described in terms of point lattices, vector lattices, and translation groups [cf. IT A (2005[link], Section 8.1.4[link] )].

1.1.1. General relations between direct and reciprocal lattices

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1.1.1.1. Primitive crystallographic bases

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The vectors a, b, c form a primitive crystallographic basis of the vector lattice L, if each translation vector [{\bf t}\in {\bf L}] may be expressed as [{\bf t}=u{\bf a}+v{\bf b}+w{\bf c}]with u, v, w being integers.

A primitive basis defines a primitive unit cell for a corresponding point lattice. Its volume V may be calculated as the mixed product (triple scalar product) of the three basis vectors: [\eqalignno{V&=({\bf abc})={\bf a}\times{\bf b}\cdot{\bf c}\cr &=\left[\left|\matrix{a^2&ab\cos\gamma&ac\cos\beta\cr ab\cos\gamma&b^2&bc\cos\alpha\cr ac\cos\beta&bc\cos\alpha&c^2\cr}\right|\right]^{1/2}\cr &=abc[1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma\cr &\quad+2\cos\alpha\cos\beta\cos\gamma]^{1/2}\cr &=2abc\bigg[\sin{\alpha+\beta+\gamma\over2}\sin{-\alpha+\beta+\gamma\over2}\cr &\quad\times\sin{\alpha-\beta+\gamma\over2}\sin{\alpha+\beta-\gamma\over2}\bigg]^{1/2}. &(1.1.1.1)}]Here a, b and c designate the lengths of the three basis vectors and [\alpha={\bf b}\wedge{\bf c}], [\beta={\bf c}\wedge{\bf a}] and [\gamma={\bf a}\wedge{\bf b}] the angles between them.

Each vector lattice L and each primitive crystallographic basis a, b, c is uniquely related to a reciprocal vector lattice [{\bf L}^*] and a primitive reciprocal basis a*, b*, c*:[\matrix{&\left.\eqalign{{\bf a}^*&={{\bf b}\times{\bf c}\over V}\quad{\rm or\quad}{\bf a}^*\cdot{\bf b}={\bf a}^*\cdot{\bf c}=0,\quad{\bf a}^*\cdot{\bf a}=1\semi\cr {\bf b}^*&={{\bf c}\times{\bf a}\over V}\quad{\rm or\quad}{\bf b}^*\cdot{\bf a}={\bf b}^*\cdot{\bf c}=0,\quad{\bf b}^*\cdot{\bf b}=1\semi\cr {\bf c}^*&={{\bf a}\times{\bf b}\over V}\quad{\rm or\quad}{\bf c}^*\cdot{\bf a}={\bf c}^*\cdot{\bf b}=0,\quad{\bf c}^*\cdot{\bf c}\,=1.}\right\}\quad (1.1.1.2)\cr\cr &{\bf L}^*=\{{\bf r}^*|{\bf r}^*=h{\bf a}^*+k{\bf b}^*+l{\bf c}^*\ {\rm and}\ h,k,l\ {\rm integers}\}.\hfill}]The lengths [a^*], [b^*] and [c^*] of the reciprocal basis vectors and the angles [\alpha^*={\bf b}^*\wedge{\bf c}^*], [\beta^*={\bf c}^*\wedge{\bf a}^*] and [\gamma^*={\bf a}^*\wedge{\bf b}^*] are given by: [\left.\matrix{a^*=\displaystyle{bc\sin\alpha\over V},\quad b^*={ac\sin\beta\over V},\quad c^*={ab\sin\gamma\over V},\cr\noalign{\vskip5.5pt} \cos\alpha^*=\displaystyle{\cos\beta\cos\gamma-\cos\alpha\over\sin\beta\sin\gamma},\cr \noalign{\vskip5.5pt}\cos\beta^*=\displaystyle{\cos\alpha\cos\gamma-\cos\beta\over\sin\alpha\sin\gamma},\cr\noalign{\vskip5.5pt} \cos\gamma^*=\displaystyle{\cos\alpha\cos\beta-\cos\gamma\over\sin\alpha\sin\beta}.}\right\}\eqno (1.1.1.3)]a*, b*, c* define a primitive unit cell in a corresponding reciprocal point lattice. Its volume V* may be expressed by analogy with V [equation (1.1.1.1)[link]]: [\eqalignno{V^*&=({\bf a}^*{\bf b}^*{\bf c}^*)={\bf a}^*\times{\bf b}^*\cdot{\bf c}^*\cr &=\left[\left|\matrix{a^{*2}&a^*b^*\cos\gamma^*&a^*c^*\cos\beta^*\cr a^*b^*\cos\gamma^*&b^{*2}&b^*c^*\cos\alpha^*\cr a^*c^*\cos\beta^*&b^*c^*\cos\alpha^*&c^{*2}}\right|\right]^{1/2}\cr &=a^*b^*c^*[1-\cos^2\alpha^*-\cos^2\beta^*-\cos^2\gamma^*\cr &\quad+2\cos\alpha^*\cos\beta^*\cos\gamma^*]^{1/2}\cr &=2a^*b^*c^*\left[\sin{\alpha^*+\beta^*+\gamma^*\over2}\sin{-\alpha^*+\beta^*+\gamma^*\over2}\right.\cr &\quad\times\left.\sin{\alpha^*-\beta^*+\gamma^*\over2}\sin{\alpha^*+\beta^*-\gamma^*\over2}\right]^{1/2}.& (1.1.1.4)}]

In addition, the following equation holds: [VV^*=1.\eqno (1.1.1.5)]As all relations between direct and reciprocal lattices are symmetrical, one can calculate a, b, c from a*, b*, c*: [{\bf a}={{\bf b}^*\times{\bf c}^*\over V^*},\quad {\bf b}={{\bf c}^*\times{\bf a}^*\over V^*},\quad {\bf c}={{\bf a}^*\times{\bf b}^*\over V^*},\eqno (1.1.1.6)] [\left.\matrix{a=\displaystyle{b^*c^*\sin\alpha^*\over V^*},\cr\noalign{\vskip5.5pt}  b=\displaystyle{a^*c^*\sin\beta^*\over V^*},\cr\noalign{\vskip5.5pt}  c=\displaystyle{a^*b^*\sin\gamma^*\over V^*},\cr\noalign{\vskip5.5pt}  \cos\alpha=\displaystyle{\cos\beta^*\cos\gamma^*-\cos\alpha^*\over\sin\beta^*\sin \gamma^*},\cr\noalign{\vskip5.5pt}\cos\beta=\displaystyle{\cos\alpha^*\cos\gamma^*-\cos\beta^*\over \sin\alpha^*\sin\gamma^*},\cr \noalign{\vskip5.5pt}\cos\gamma=\displaystyle{\cos\alpha^*\cos\beta^*-\cos\gamma^*\over \sin\alpha^*\sin\beta^*}.}\right\}\eqno (1.1.1.7)]The unit-cell volumes V and V* may also be obtained from: [\eqalignno{V&=abc\sin\alpha^*\sin\beta\sin\gamma\cr &=abc\sin\alpha\sin\beta^*\sin\gamma\cr &=abc\sin\alpha\sin\beta\sin\gamma^*,& (1.1.1.8)}][\eqalignno{V^*&=a^*b^*c^*\sin\alpha\sin\beta^*\sin\gamma^*\cr &=a^*b^*c^*\sin\alpha^*\sin\beta\sin\gamma^*\cr &=a^*b^*c^*\sin\alpha^*\sin\beta^*\sin\gamma.& (1.1.1.9)}]

1.1.1.2. Non-primitive crystallographic bases

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For certain lattice types, it is usual in crystallography to refer to a `conventional' crystallographic basis [{\bf a}_c,{\bf b}_c,{\bf c}_c] instead of a primitive basis a, b, c. In that case, [{\bf a}_c], [{\bf b}_c], and [{\bf c}_c] with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors [{\bf t}\in{\bf L}], [{\bf t}=t_1{\bf a}_c+t_2{\bf b}_c+t_3{\bf c}_c,]with at least two of the coefficients [t_1], [t_2], [t_3] being fractional.

Such a conventional basis defines a conventional or centred unit cell for a corresponding point lattice, the volume [V_c] of which may be calculated by analogy with V by substituting [{\bf a}_c,{\bf b}_c,{\bf c}_c] for a, b, and c in (1.1.1.1)[link].

If m designates the number of centring lattice vectors t with [0\le t_1,t_2,t_3\lt1], [V_c] may be expressed as a multiple of the primitive unit-cell volume V: [V_c=mV.\eqno (1.1.1.10)]With the aid of equations (1.1.1.2)[link] and (1.1.1.3)[link], the reciprocal basis [{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c] may be derived from [{\bf a}_c,{\bf b}_c,{\bf c}_c]. Again, each reciprocal-lattice vector [{\bf r}^*=h{\bf a}^*_c+k{\bf b}^*_c+l{\bf c}^*_c\in{\bf L}^*]is an integral linear combination of the reciprocal basis vectors, but in contrast to the use of a primitive basis only certain triplets h, k, l refer to reciprocal-lattice vectors.

Equation (1.1.1.5)[link] also relates [V_c] to [V^*_c], the reciprocal cell volume referred to [{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]. From this it follows that [V^*_c={1\over m}V^*.\eqno (1.1.1.11)]

Table 1.1.1.1[link] contains detailed information on `centred lattices' described with respect to conventional basis systems.

Table 1.1.1.1| top | pdf |
Direct and reciprocal lattices described with respect to conventional basis systems

Direct latticeReciprocal lattice
[{\bf a}_c, {\bf b}_c, {\bf c}_c][{\bf a}^*_c, {\bf b}^*_c, {\bf c}^*_c] 
Bravais letterCentring vectorsUnit-cell volume [V_c]Conditions for reciprocal-lattice vectors [h{\bf a}^*_c+k{\bf b}^*_c+l{\bf c}^*_c]Unit-cell volume [V^*_c]Bravais letter
A [{1\over2}{\bf b}_c+{1\over2}{\bf c}_c] 2V [k+l=2n] [{1\over2}V^*] A
B [{1\over2}{\bf a}_c+{1\over2}{\bf c}_c] 2V [h+l=2n] [{1\over2}V^*] B
C [{1\over2}{\bf a}_c+{1\over2}{\bf b}_c] 2V [h+k=2n] [{1\over2}V^*] C
I [{1\over2}{\bf a}_c+{1\over2}{\bf b}_c+{1\over2}{\bf c}_c] 2V [h+k+l=2n] [{1\over2}V^*] F
F [{1\over2}{\bf a}_c+{1\over2}{\bf b}_c,] [{1\over2}{\bf a}_c+{1\over2}{\bf c}_c,] [{1\over2}{\bf b}_c+{1\over2}{\bf c}_c] 4V [h+k=2n,] [h+l=2n,] [k+l=2n] [{1\over4}V^*] I
R [{1\over3}{\bf a}_c+{2\over3}{\bf b}_c+{2\over3}{\bf c}_c], [{2\over3}{\bf a}_c+{1\over3}{\bf b}_c+{1\over3}{\bf c}_c] 3V [-h+k+l=3n] [{1\over3}V^*] R

As a direct lattice and its corresponding reciprocal lattice do not necessarily belong to the same type of Bravais lattices [IT A (2005[link], Section 8.2.5[link] )], the Bravais letter of [{\bf L}^*] is given in the last column of Table 1.1.1.1[link]. Except for P lattices, a conventionally chosen basis for [{\bf L}^*] coincides neither with a*, b*, c* nor with [{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c]. This third basis, however, is not used in crystallography. The designation of scattering vectors and the indexing of Bragg reflections usually refers to [{\bf a}^*_c,{\bf b}^*_c,{\bf c}^*_c].

If the differences with respect to the coefficients of direct- and reciprocal-lattice vectors are disregarded, all other relations discussed in Part 1 are equally true for primitive bases and for conventional bases.

1.1.2. Lattice vectors, point rows, and net planes

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The length t of a vector [{\bf t}=u{\bf a}+v{\bf b}+w{\bf c}] is given by [\eqalignno{t^2&=u^2{\bf a}^2+v^2{\bf b}^2+w^2{\bf c}^2+2uvab\cos\gamma \cr&\quad+2uwac\cos\beta+2vwbc\cos\alpha.&(1.1.2.1)}]Accordingly, the length [r^*] of a reciprocal-lattice vector [{\bf r}^*= h{\bf a}^*+k{\bf b}^*+l{\bf c}^*] may be calculated from [\eqalignno{r^{*2}&=h^2a^{*2}+k^2b^{*2}+l^2c^{*2}+2hka^*b^*\cos\gamma^*\cr&\quad+2hla^*c^*\cos\beta^*+2klb^*c^*\cos\alpha^*.&(1.1.2.2)}]If the coefficients u, v, w of a vector [{\bf t}\in{\bf L}] are coprime, [uvw] symbolizes the direction parallel to t. In particular, [uvw] is used to designate a crystal edge, a zone axis, or a point row with that direction.

The integer coefficients h, k, l of a vector [{\bf r}^*\in{\bf L}^*] are also the coordinates of a point of the corresponding reciprocal lattice and designate the Bragg reflection with scattering vector r*. If h, k, l are coprime, the direction parallel to r* is symbolized by [[hkl]^*].

Each vector r* is perpendicular to a family of equidistant parallel nets within a corresponding direct point lattice. If the coefficients h, k, l of r* are coprime, the symbol (hkl) describes that family of nets. The distance d(hkl) between two neighbouring nets is given by [d(hkl)=r^{*-1}.\eqno (1.1.2.3)]Parallel to such a family of nets, there may be a face or a cleavage plane of a crystal.

The net planes (hkl) obey the equation [hx+ky+lz=n\quad(n={\rm integer}).\eqno (1.1.2.4)]Different values of n distinguish between the individual nets of the family; x, y, z are the coordinates of points on the net planes (not necessarily of lattice points). They are expressed in units a, b, and c, respectively.

Similarly, each vector [{\bf t}\in{\bf L}] with coprime coefficients u, v, w is perpendicular to a family of equidistant parallel nets within a corresponding reciprocal point lattice. This family of nets may be symbolized [(uvw)^*]. The distance [d^*(uvw)] between two neighbouring nets can be calculated from [d^*(uvw)=t^{-1}.\eqno (1.1.2.5)]A layer line on a rotation pattern or a Weissenberg photograph with rotation axis [uvw] corresponds to one such net of the family [(uvw)^*] of the reciprocal lattice.

The nets [(uvw)^*] obey the equation [uh+vk+wl=n \quad (n={\rm integer}).\eqno (1.1.2.6)]Equations (1.1.2.6)[link] and (1.1.2.4)[link] are essentially the same, but may be interpreted differently. Again, n distinguishes between the individual nets out of the family [(uvw)^*]. h, k, l are the coordinates of the reciprocal-lattice points, expressed in units [a^*], [b^*], [c^*], respectively.

A family of nets (hkl) and a point row with direction [uvw] out of the same point lattice are parallel if and only if the following equation is satisfied: [hu+kv+lw=0.\eqno (1.1.2.7)]

This equation is called the `zone equation' because it must also hold if a face (hkl) of a crystal belongs to a zone [uvw].

Two (non-parallel) nets [(h_1k_1l_1)] and [(h_2k_2l_2)] intersect in a point row with direction [uvw] if the indices satisfy the condition [u:v:w=\left|k_1l_1\atop k_2l_2\right|:\left|l_1h_1\atop l_2h_2\right|:\left|h_1k_1\atop h_2k_2\right|.\eqno (1.1.2.8)]The same condition must be satisfied for a zone axis [uvw] defined by the crystal faces [(h_1k_1l_1)] and [(h_2k_2l_2)].

Three nets [(h_1k_1l_1)], [(h_2k_2l_2)], and [(h_3k_3l_3)] intersect in parallel rows, or three faces with these indices belong to one zone if [\left|\matrix{h_1k_1l_1\cr h_2k_2l_2\cr h_3k_3l_3}\right|=0.\eqno (1.1.2.9)]Two (non-parallel) point rows [[u_1v_1w_1]] and [[u_2v_2w_2]] in the direct lattice are parallel to a family of nets (hkl) if [h:k:l=\left|v_1w_1\atop v_2w_2\right|:\left|w_1u_1\atop w_2u_2\right|:\left|u_1v_1\atop u_2v_2\right|.\eqno (1.1.2.10)]The same condition holds for a face (hkl) belonging to two zones [[u_1v_1w_1]] and [[u_2v_2w_2]].

Three point rows [[u_1v_1w_1]], [[u_2v_2w_2]], and [[u_3v_3w_3]] are parallel to a net (hkl), or three zones of a crystal with these indices have a common face (hkl) if [\left|\matrix{u_1v_1w_1\cr u_2v_2w_2\cr u_3v_3w_3}\right|=0.\eqno (1.1.2.11)]A net (hkl) is perpendicular to a point row [uvw] if [\eqalignno{&{a\over h}(au+bv\cos\gamma+cw\cos\beta)\cr &\quad={b\over k}(au\cos\gamma+bv+cw\cos\alpha)\cr &\quad={c\over l}(au\cos\beta+bv\cos\alpha+cw).& (1.1.2.12)}]

1.1.3. Angles in direct and reciprocal space

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The angles between the normal of a crystal face and the basis vectors a, b, c are called the direction angles of that face. They may be calculated as angles between the corresponding reciprocal-lattice vector r* and the basis vectors [\lambda={\bf r}^*\wedge{\bf a}], [\mu={\bf r}^*\wedge{\bf b}] and [\nu={\bf r}^*\wedge{\bf c}]: [\left.\matrix{\cos\lambda=\displaystyle{h\over a}d(hkl),\quad\cos\mu={k\over b}d(hkl),\cr \noalign{\vskip5.5pt}\cos\nu=\displaystyle{l\over c}d(hkl).}\right\}\eqno (1.1.3.1)]The three equations can be combined to give [ \left.\matrix{&\quad&\displaystyle a:b:c={h\over\cos\lambda}:{k\over\cos\mu}:{l\over\cos\nu}\cr {\hskip-32pt{\rm or}}\cr &&h:k:l=a\cos\lambda:b\cos\mu: c\cos\nu.}\right\}\eqno (1.1.3.2)]The first formula gives the ratios between a, b, and c, if for any face of the crystal the indices (hkl) and the direction angles λ, μ, and ν are known. Once the axial ratios are known, the indices of any other face can be obtained from its direction angles by using the second formula.

Similarly, the angles between a direct-lattice vector t and the reciprocal basis vectors [\lambda^*={\bf t}\wedge{\bf a}^*], [\mu^*={\bf t}\wedge{\bf b}^*] and [\nu^*={\bf t}\wedge{\bf c}^*] are given by [ \left.\matrix{\displaystyle\cos\lambda^*={u\over a^*}d^*(uvw),\quad\cos\mu^*={v\over b^*}d^*(uvw),\cr\noalign{\vskip5.5pt}\displaystyle\cos\nu^*={w\over c^*}d^*(uvw).}\right\}\eqno (1.1.3.3)]The angle [\psi] between two direct-lattice vectors [{\bf t}_1] and [{\bf t}_2] or between two corresponding point rows [[u_1v_1w_1]] and [[u_2v_2w_2]] may be derived from the scalar product [\eqalignno{{\bf t}_1\cdot{\bf t}_2&=u_1u_2a^2+v_1v_2b^2+w_1w_2c^2+(u_1v_2+u_2v_1)ab\cos\gamma\cr &\quad+(u_1w_2+u_2w_1)ac\cos\beta+(v_1w_2+v_2w_1)bc\cos\alpha\cr&& (1.1.3.4)}]as [\cos\psi={{\bf t}_1\cdot{\bf t}_2\over t_1t_2}.\eqno (1.1.3.5)]Analogously, the angle [\varphi] between two reciprocal-lattice vectors [{\bf r}^*_1] and [{\bf r}^*_2] or between two corresponding point rows [[h_1k_1l_1]^*] and [[h_2k_2l_2]^*] or between the normals of two corresponding crystal faces [(h_1k_1l_1)] and [(h_2k_2l_2)] may be calculated as [\cos\varphi={{\bf r}^*_1\cdot{\bf r}^*_2\over r_1r_2}\eqno (1.1.3.6)]with [\eqalignno{{\bf r}^*_1\cdot{\bf r}^*_2&=h_1h_2a^{*2}+k_1k_2b^{*2}+l_1l_2c^{*2}\cr &\quad+(h_1k_2+h_2k_1)a^*b^*\cos\gamma^*\cr &\quad+(h_1l_2+h_2l_1)a^*c^*\cos\beta^*\cr &\quad+(k_1l_2+k_2l_1)b^*c^*\cos\alpha^*. & (1.1.3.7)}]

Finally, the angle [\omega] between a first direction [uvw] of the direct lattice and a second direction [hkl] of the reciprocal lattice may also be derived from the scalar product of the corresponding vectors t and r*. [\cos\omega={{\bf t}\cdot{\bf r}^*\over tr^*}={uh+vk+wl\over tr^*}.\eqno (1.1.3.8)]

1.1.4. The Miller formulae

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Consider four faces of a crystal that belong to the same zone in consecutive order: [(h_1k_1l_1)], [(h_2k_2l_2)], [(h_3k_3l_3)], and [(h_4k_4l_4)]. The angles between the ith and the jth face normals are designated [\varphi_{ij}]. Then the Miller formulae relate the indices of these faces to the angles [\varphi_{ij}]: [{\sin\varphi_{12}\sin\varphi_{43}\over\sin\varphi_{13}\sin\varphi_{42}}={u_{12}u_{43}\over u_{13}u_{42}}={v_{12}v_{43}\over v_{13}v_{42}}={w_{12}w_{43}\over w_{13}w_{42}}\eqno (1.1.4.1)]with [u_{ij}=\left|k_il_i\atop k_jl_j\right|,\quad v_{ij}=\left|l_ih_i\atop l_jh_j\right|,\quad w_{ij}=\left|h_ik_i\atop h_jk_j\right|.]If all angles between the face normals and also the indices for three of the faces are known, the indices of the fourth face may be calculated. Equation (1.1.4.1)[link] cannot be used if two of the faces are parallel.

From the definition of [u_{ij}], [v_{ij}], and [w_{ij}], it follows that all fractions in (1.1.4.1)[link] are rational: [{\sin\varphi_{12}\sin\varphi_{43}\over \sin\varphi_{13}\sin\varphi_{42}}={p\over q}\quad{\rm with\ }p,q{\ \rm integers}.]Therefore, (1.1.4.1)[link] may be rearranged to [p \cot\varphi_{12}-q\cot\varphi_{13}=(\, p-q)\cot\varphi_{14}.\eqno (1.1.4.2)]This equation allows the determination of one angle if two of the angles and the indices of all four faces are known.

References

International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.








































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