How to write FCC lattice vector with MATLAB?

Description;

T (Inverse lattice vector), a, b, c are atoms vector. T= n1*a + n2*b+ n3*c a= a/2*(i+j) b= a/2*(j+k) c= a/2*(i+k)

-3<n1<3 -3<n2<3 -3<n3<3

 I want to calculate T vector by the using of above equations.

Answer 1

[Karan Singh]

Hi @Anil Hallac,

Below is an example MATLAB code snippet that computes the FCC lattice vector T. The vectors a, b, and c are defined using the given formulas. Each is scaled by half the lattice constant a0. The integer parameters are set from –2 to 2. In the end three nested loops iterate over all combinations of n1, n2, and n3. For each combination, the T vector is computed and stored in the array "T_all".

% Define the lattice constant (change as needed)
a0 = 1;  

% Define the FCC basis vectors
a = (a0/2) * [1, 1, 0];   % a = a0/2*(i + j)
b = (a0/2) * [0, 1, 1];   % b = a0/2*(j + k)
c = (a0/2) * [1, 0, 1];   % c = a0/2*(i + k)

% Define the range for n1, n2, n3 (here from -2 to 2 as an example)
n_range = -2:2;

% Preallocate an array to store T vectors (optional)
% There will be length(n_range)^3 vectors (here 5^3 = 125)
T_all = zeros(length(n_range)^3, 3);
count = 1;

% Loop over all integer combinations of n1, n2, n3
for n1 = n_range
    for n2 = n_range
        for n3 = n_range
            T = n1*a + n2*b + n3*c;  % Compute T vector
            T_all(count, :) = T;     % Store in the matrix (each row is a T vector)
            count = count + 1;
        end
    end
end

% Display all calculated T vectors
disp('Calculated FCC lattice vectors T:');

Calculated FCC lattice vectors T:

disp(T_all);

   -2.0000   -2.0000   -2.0000
   -1.5000   -2.0000   -1.5000
   -1.0000   -2.0000   -1.0000
   -0.5000   -2.0000   -0.5000
         0   -2.0000         0
   -2.0000   -1.5000   -1.5000
   -1.5000   -1.5000   -1.0000
   -1.0000   -1.5000   -0.5000
   -0.5000   -1.5000         0
         0   -1.5000    0.5000
   -2.0000   -1.0000   -1.0000
   -1.5000   -1.0000   -0.5000
   -1.0000   -1.0000         0
   -0.5000   -1.0000    0.5000
         0   -1.0000    1.0000
   -2.0000   -0.5000   -0.5000
   -1.5000   -0.5000         0
   -1.0000   -0.5000    0.5000
   -0.5000   -0.5000    1.0000
         0   -0.5000    1.5000
   -2.0000         0         0
   -1.5000         0    0.5000
   -1.0000         0    1.0000
   -0.5000         0    1.5000
         0         0    2.0000
   -1.5000   -1.5000   -2.0000
   -1.0000   -1.5000   -1.5000
   -0.5000   -1.5000   -1.0000
         0   -1.5000   -0.5000
    0.5000   -1.5000         0
   -1.5000   -1.0000   -1.5000
   -1.0000   -1.0000   -1.0000
   -0.5000   -1.0000   -0.5000
         0   -1.0000         0
    0.5000   -1.0000    0.5000
   -1.5000   -0.5000   -1.0000
   -1.0000   -0.5000   -0.5000
   -0.5000   -0.5000         0
         0   -0.5000    0.5000
    0.5000   -0.5000    1.0000
   -1.5000         0   -0.5000
   -1.0000         0         0
   -0.5000         0    0.5000
         0         0    1.0000
    0.5000         0    1.5000
   -1.5000    0.5000         0
   -1.0000    0.5000    0.5000
   -0.5000    0.5000    1.0000
         0    0.5000    1.5000
    0.5000    0.5000    2.0000
   -1.0000   -1.0000   -2.0000
   -0.5000   -1.0000   -1.5000
         0   -1.0000   -1.0000
    0.5000   -1.0000   -0.5000
    1.0000   -1.0000         0
   -1.0000   -0.5000   -1.5000
   -0.5000   -0.5000   -1.0000
         0   -0.5000   -0.5000
    0.5000   -0.5000         0
    1.0000   -0.5000    0.5000
   -1.0000         0   -1.0000
   -0.5000         0   -0.5000
         0         0         0
    0.5000         0    0.5000
    1.0000         0    1.0000
   -1.0000    0.5000   -0.5000
   -0.5000    0.5000         0
         0    0.5000    0.5000
    0.5000    0.5000    1.0000
    1.0000    0.5000    1.5000
   -1.0000    1.0000         0
   -0.5000    1.0000    0.5000
         0    1.0000    1.0000
    0.5000    1.0000    1.5000
    1.0000    1.0000    2.0000
   -0.5000   -0.5000   -2.0000
         0   -0.5000   -1.5000
    0.5000   -0.5000   -1.0000
    1.0000   -0.5000   -0.5000
    1.5000   -0.5000         0
   -0.5000         0   -1.5000
         0         0   -1.0000
    0.5000         0   -0.5000
    1.0000         0         0
    1.5000         0    0.5000
   -0.5000    0.5000   -1.0000
         0    0.5000   -0.5000
    0.5000    0.5000         0
    1.0000    0.5000    0.5000
    1.5000    0.5000    1.0000
   -0.5000    1.0000   -0.5000
         0    1.0000         0
    0.5000    1.0000    0.5000
    1.0000    1.0000    1.0000
    1.5000    1.0000    1.5000
   -0.5000    1.5000         0
         0    1.5000    0.5000
    0.5000    1.5000    1.0000
    1.0000    1.5000    1.5000
    1.5000    1.5000    2.0000
         0         0   -2.0000
    0.5000         0   -1.5000
    1.0000         0   -1.0000
    1.5000         0   -0.5000
    2.0000         0         0
         0    0.5000   -1.5000
    0.5000    0.5000   -1.0000
    1.0000    0.5000   -0.5000
    1.5000    0.5000         0
    2.0000    0.5000    0.5000
         0    1.0000   -1.0000
    0.5000    1.0000   -0.5000
    1.0000    1.0000         0
    1.5000    1.0000    0.5000
    2.0000    1.0000    1.0000
         0    1.5000   -0.5000
    0.5000    1.5000         0
    1.0000    1.5000    0.5000
    1.5000    1.5000    1.0000
    2.0000    1.5000    1.5000
         0    2.0000         0
    0.5000    2.0000    0.5000
    1.0000    2.0000    1.0000
    1.5000    2.0000    1.5000
    2.0000    2.0000    2.0000

Karan