Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Basic computations
In introductory classes in crystallography and material science we find topics such as crystal lattice and -systems, unit cell and Miller indices. In this tutorial we will familiarise ourselves with functionalities of this package by working with these basic topics. See the box below for an overview of relevant functions.
The package must be loaded:
In[1]:=
<<MaXrd`
calculates the Bragg angle for a given wavelength and reflection | |
calculates the theoretical density of a crystal | |
returns the metric G | |
calculates d hkl | |
returns the lattice parameters of a crystal |
Relevant functions for basic crystallographic computations.
Unit cell volume
The unit cell volume can be found by taking the square-root of the determinant of the metric matrix.
First we find the metric matrix with :
CrystalMetric
In[2]:=
G=
["Silicon"];
 | GetCrystalMetric |
The unit cell volume of silicon is found by taking the square-root of the determinant (units: ):
3
Å
In[3]:=
Sqrt@Det@G
Out[3]=
160.181
Another formula for the volume is given by . Let us use this to find the volume of the corundum unit cell.
V=abc
1-α-β-γ+2cosαcosβcosγ
2
cos
2
cos
2
cos
We need the lattice parameters of corundum. One way to do this is by using the function:
In[4]:=
lattice=Thread{a,b,c,α,β,γ}
["Corundum","Units"True]
| GetLatticeParameters |
Out[4]=
a,b,c,α,β,γ
4.76094
Å
4.76094
Å
12.9662
Å
90
°
90
°
120
°
Substituting values into the volume formula:
In[5]:=
V=abc
1---+2Cos[α]Cos[β]Cos[γ]
/.lattice2
Cos[α]
2
Cos[β]
2
Cos[γ]
Out[5]=
254.524
3
Å
Conversion from crystal to Cartesian coordinates
The two atoms, and , have fractional coordinates:
Al
O
In[6]:=
coords=
"Corundum","AtomData",All,"FractionalCoordinates"
| $CrystalData |
Out[6]=
{0.,0.,0.352105},0.30626,0.30626,
1
4
We want to use the standard orthogonal unit vectors instead of the crystallographic vectors. In introductory books we may find the following transformation matrix:
In[7]:=
P=
["CrystallographicToCartesian"];P//MatrixForm
| $TransformationMatrices |
Out[7]//MatrixForm=
a | bCos[γ] | cCos[β] |
0 | bSin[γ] | c(Cos[α]-Cos[β]Cos[γ])Csc[γ] |
0 | 0 | c 1- 2 Cos[β] 2 (Cos[α]-Cos[β]Cos[γ]) 2 Csc[γ] |
Next we want to insert the lattice parameters for this particular crystal.
In[8]:=
lattice=Thread{a,b,c,α,β,γ}
["Corundum","Units"True]
| GetLatticeParameters |
Out[8]=
a,b,c,α,β,γ
4.76094
Å
4.76094
Å
12.9662
Å
90
°
90
°
120
°
Then we overwrite the matrix with these values:
P
In[9]:=
P=QuantityMagnitude[P/.lattice];P//MatrixForm
Out[10]//MatrixForm=
4.76094 | -2.38047 | 0. |
0 | 4.12309 | 0. |
0 | 0 | 12.9662 |
The new coordinates for the atom:
Al
In[11]:=
P.coords〚1〛
Out[11]=
{0.,0.,4.56546}
The new coordinates for the atom:
O
In[12]:=
P.coords〚2〛
Out[12]=
{0.729043,1.26274,3.24155}
Alternatively, we can perform the computations as follows:
In[13]:=
P.#&/@coords
Out[13]=
{{0.,0.,4.56546},{0.729043,1.26274,3.24155}}
Distance between two points (bond distances)
In the previous section we found the Cartesian coordinates of two atoms:
In[14]:=
xyzAlCart={0,0,4.56546};xyzOCart={0.729043,1.26274,3.24155};
Since we are now using an orthonormal basis, the distance between them is found with the Pythagorean theorem:
In[16]:=
dist=Sqrt@Total[]
2
(xyzAlCart-xyzOCart)
Out[16]=
1.96946
alternatively by using:
In[17]:=
dist=EuclideanDistance[xyzAlCart,xyzOCart]
Out[17]=
1.96946
A more direct approach is to use the metric matrix directly:
The metric matrix:
In[18]:=
G=
["Corundum"]
| GetCrystalMetric |
Out[18]=
{{22.6665,-11.3333,0.},{-11.3333,22.6665,0.},{0.,0.,168.122}}
The coordinates (in the crystal system):
In[19]:=
{xyzAl,xyzO}=
"Corundum","AtomData",All,"FractionalCoordinates"
| $CrystalData |
Out[19]=
{0.,0.,0.352105},0.30626,0.30626,
1
4
The distance between them:
In[20]:=
dist=
d.G.d
/.dxyzAl-xyzOOut[20]=
1.96946
""