Lattice Metric and Plane Geometry

Miller indices say which lattice direction or plane family is meant. The lattice metric supplies the lengths and angles that turn those indices into physical geometry. It determines the length of \([uvw]\), the normal to \((hkl)\) and the spacing between neighbouring \((hkl)\) planes.

MTEX stores the metric in the crystalSymmetry together with the point group and crystal reference frame.

cs = crystalSymmetry('12/m1',[5.2 6.3 7.1],...
  [90 106 90]*degree,'X||a','mineral','Example monoclinic crystal')

cs = crystalSymmetry (⊙c*→b)
 
  mineral           : Example monoclinic crystal
  symmetry          : 12/m1                     
  elements          : 4                         
  a, b, c           : 5.2, 6.3, 7.1             
  alpha, beta, gamma: 90°, 106°, 90°            
  reference frame   : X||a, Y||b, Z||c*

Reading the Lattice Parameters

The three axis lengths and the three interaxial angles are available as abc and abg. Their units are the units used to define the lattice, commonly Angstrom for crystallographic data.

cs.abc

ans =
    5.2000    6.3000    7.1000

cs.abg ./ degree

ans =
    90   106    90

The Unit Cell

The unit cell is the parallelepiped spanned by \(\vec a\), \(\vec b\) and \(\vec c\). Repeating it by integer multiples of these three vectors fills the lattice. A crystalSymmetry stores the geometry of this cell and its point symmetry, but not the atoms or other motif repeated with it.

The Seven Crystal Systems

A crystal system restricts which lattice parameters may differ. MTEX records the system in cs.lattice and checks the required equal axis lengths when a crystalSymmetry is constructed.

Trigonal and hexagonal crystals share the same conventional hexagonal metric in MTEX; their threefold and sixfold point symmetries distinguish the systems.

The monoclinic point-group symbol also states which axis is unique. The pages on reference frames and axis alignment explain how those lattice axes are placed in MTEX's Cartesian crystal frame.

cs.lattice

ans = 
  latticeType enumeration
    monoclinic

The Reciprocal Basis

Lattice planes are most naturally described in the basis dual to the direct-lattice vectors. If

\[V=\vec a\cdot(\vec b\times\vec c),\]

then

\[\vec a^*=\frac{\vec b\times\vec c}{V},\qquad \vec b^*=\frac{\vec c\times\vec a}{V},\qquad \vec c^*=\frac{\vec a\times\vec b}{V}.\]

The direct and reciprocal bases are dual: each direct axis has dot product one with its matching reciprocal axis and zero with the other two. MTEX provides them as axes and axesDual.

directBasis = cs.axes;
reciprocalBasis = cs.axesDual;

cellVolume = abs(det(directBasis))

cellVolume =
  223.5856

The volume is in the cube of the lattice-parameter unit. The defining duality is seen directly in the matrix of pairwise dot products.

dot_outer(directBasis,reciprocalBasis,'noSymmetry')

ans =
    1.0000         0         0
    0.0000    1.0000         0
    0.0000    0.0000    1.0000

The result is the identity matrix. Reciprocal axes have inverse-length units, and MTEX uses the crystallographic convention without a factor \(2\pi\). Direct and reciprocal axes are parallel in an orthogonal lattice; in a monoclinic or triclinic lattice they generally are not.

The schematic shows the direct basis at the lower left and the reciprocal basis from a second lattice point. The red points are translation-equivalent positions; the blue points illustrate a repeated motif. Notice in particular that \(\vec a\) and \(\vec a^*\) are not parallel.

Lengths of Lattice Directions

A direct-lattice direction is a real vector

\[\vec m=u\vec a+v\vec b+w\vec c.\]

Its norm is therefore a physical length, not just a plotting radius. For example, \([101]\) spans one \(\vec a\) and one \(\vec c\).

m = Miller(1,0,1,cs,'uvw')

norm(m)

m = Miller (Example monoclinic crystal)
  u v w
  1 0 1
ans =
    7.5563

The result is in the same units as cs.abc. Multiplying all indices by two leaves the geometric direction unchanged, up to rounding, but doubles the vector length.

angle(m,Miller(2,0,2,cs,'uvw')) ./ degree

ans =
   8.5377e-07

norm(Miller(2,0,2,cs,'uvw')) ./ norm(m)

ans =
     2

Use normalize when only the direction matters. Keep the original magnitude when the lattice translation or Burgers-vector length is part of the calculation.

Interplanar Spacing

The normal of \((hkl)\) is a reciprocal-lattice vector. MTEX uses the crystallographic convention without a factor \(2\pi\), so its length is the inverse of the plane spacing:

\[ d_{hkl}=\frac{1}{\lVert\vec n_{hkl}\rVert}. \]

h = Miller(1,0,0,cs)

d100 = dspacing(h)

h = Miller (Example monoclinic crystal)
  h k l
  1 0 0
d100 =
    4.9986

The same command works for a list. For a cubic lattice with parameter \(a=3.6\), the familiar result is \(d_{hkl}=a/\sqrt{h^2+k^2+l^2}\).

csCubic = crystalSymmetry('m-3m',[3.6 3.6 3.6]);
hCubic = Miller({1,0,0},{1,1,0},{1,1,1},csCubic);

dspacing(hCubic)

ans =
    3.6000    2.5456    2.0785

What a Crystal Symmetry Does Not Store

A crystalSymmetry contains the point symmetry, lattice metric and frame convention. It does not retain the atomic basis, Wyckoff positions or the translational parts of a space group. In particular, the 14 Bravais lattices distinguish translational centring, whereas crystalSymmetry retains the associated crystal system and point group. A CIF or space-group symbol can provide the lattice parameters and point group, but MTEX reduces the space group to that point group. Structure factors and systematic absences therefore require information outside this geometry model.

Next

Operations uses the metric to test whether a direction lies in a plane and to compute zone axes and multiplicities. Miller Indices introduces direct and reciprocal notation. Reference System explains how the lattice is embedded in a Cartesian crystal frame.