CRYSTAL — Geometry of Crystals

Metric Tensor

Computes G (real→reciprocal) and F (reciprocal→real) metric tensors for any crystal system, plus unit cell volume. Ref: Crystallography (Bhadeshia, 2018), Ch. 9, p. 142.

Crystal System

Lattice Type

a Å

b Å

c Å

α °

β °

γ °

D-Spacing Calculator

Generates an ordered list of interplanar spacings d(hkl), accounting for systematic absences from the lattice type.

Crystal System

Lattice Type

a Å

b Å

c Å

α °

β °

γ °

Max Miller index search range

Sort by

Angle Between Vectors

Calculates the angle between two lattice vectors. Real space [uvw] and reciprocal space (hkl) can be freely mixed.

Crystal System

a Å

b Å

c Å

α °

β °

γ °

VECTOR 1

h / u

k / v

l / w

Real [uvw]

Reciprocal (hkl)

VECTOR 2

h / u

k / v

l / w

Real [uvw]

Reciprocal (hkl)

Real ↔ Reciprocal Space Conversion

Converts vector components between real and reciprocal space using the metric tensor (F = G⁻¹). A real-space direction [uvw] is parallel to the reciprocal-space plane normal (hkl) when h·F·h = u·G·u and they share the same zone axis. Enter a vector and choose the conversion direction.

Crystal System

a Å

b Å

c Å

α °

β °

γ °

Component 1

Component 2

Component 3

Real [uvw] → find (hkl)

Reciprocal (hkl) → find [uvw]

Axis-Angle Pairs (Cubic)

Calculates the rotation axis [uvw] and angle θ relating two cubic crystals, given two pairs of parallel vectors. Also generates all 23 symmetry-equivalent descriptions. Ref: Geometry of Crystals, Bhadeshia, pp. 18–21, 79 (Example 25). MAP_CRYSTAL_ROTAT

           2B       2A      1B
             (     |     )
              (    |    )
               (   |   )
                (  |  )
                 ( | )
                  (──────────── 1A

Vectors 1A & 2A from Crystal A · Vectors 1B & 2B from Crystal B · Angle between 2A & 2B required

Crystal A

Vector 1A direction [h k l]

Vector 2A direction [h k l]

Crystal B

Vector 1B parallel to 1A

Vector 2B related to 2A by angle φ

Acute angle between 2A and 2B degrees

Four Index Notation

Converts between three-index and four-index notation for the hexagonal system. Plane normals use Miller-Bravais (hkil) where i = −(h+k). Directions use Weber notation [uvtw] where t = −(u+v)/3. Ref: Bhadeshia, Microstructural Characterisation, 1988, Appendix 2. MAP_CRYSTAL_NOTAT1/2

Four Index → Three Index

Vector type

Real [uvtw]

Reciprocal (hkil)

u (h)

v (k)

t (i)

w (l)

Three Index → Four Index

Vector type

Real [uvw]

Reciprocal (hkl)

u (h)

v (k)

w (l)

Formulae (Bhadeshia 1988, Appendix 2):
Real [uvw] → [uvtw]: u₁=(2u−v)/3, v₁=(2v−u)/3, t=−(u+v)/3, w₁=w
Real [uvtw] → [uvw]: u₁=u−t, v₁=v−t, w₁=w
Reciprocal (hkl) → (hkil): h₁=h, k₁=k, i=−(h+k), l₁=l
Reciprocal (hkil) → (hkl): h₁=h, k₁=k, l₁=l (drop i)

Vector Magnitude

Computes the magnitude (Å) of a real-space lattice vector [uvw] via the metric tensor scalar product u·G·u.

Crystal System

a Å

b Å

c Å

α °

β °

γ °

u

v

w

Step 1 — Load Diffraction Pattern & Select Diffraction Spots

Upload a diffraction pattern image (JPG, PNG, BMP, TIF). Detected spots are circled automatically with their estimated radii. Step 1: click the central beam (transmitted beam) to set the pattern centre. Step 2: click two or more diffraction spots — sub-pixel centroids are computed automatically from the intensity distribution.

⬆

DROP IMAGE HERE

or click to browse — JPG · PNG · BMP · TIF

Pattern Image

Click the central beam to set the pattern centre

Spot sensitivity: 1.2

Step 2 — Pattern Solver

Camera Constant

Known — enter d-spacings

Unknown — enter ratio only

Common Steel & Related Phases — select to load

Crystal System

Lattice Type assume P if unknown

a Å

b Å

c Å

α °

β °

γ °

Measured Spot Data

Camera constant λL mm·Å — optional

Search Options

Tolerance fraction

Typical 0.03 (3%)

Max search index

h,k,l = 0…N (typical: 3)

Suppress equivalent solutions

Unique zone axes (slower)

Prefer low indices tie-break

Prefer simpler (lower hkl) solutions when match % is close

Coordinate Transformation Matrix

Computes the (J₂/J₁) transformation matrix relating two crystals from two pairs of parallel vectors. Useful for orientation relationships. Ref: Crystallography (Bhadeshia, 2018), Ch. 7, pp. 109–113

CRYSTAL 1 — Unit Cell

a₁ Å

b₁ Å

c₁ Å

α₁ °

β₁ °

γ₁ °

CRYSTAL 2 — Unit Cell

a₂ Å

b₂ Å

c₂ Å

α₂ °

β₂ °

γ₂ °

PAIR A — Crystal 1 ∥ Crystal 2

Crystal 1, Vector A

h

k

l

Real

Reciprocal

Crystal 2, Vector A

h

k

l

Real

Reciprocal

PAIR B — Crystal 1 ∥ Crystal 2

Crystal 1, Vector B

h

k

l

Real

Reciprocal

Crystal 2, Vector B

h

k

l

Real

Reciprocal