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Milds # File:Reciprocal monoclinic lattice.png

## Summary

 Description The Fourier transform of a monoclinic lattice with real-space vectors a = (1, 0, 0), b = (0, 1, 0), c = (0.5, 0, 0.8). (These vectors are arbitrary.) A 12×12×12 lattice of delta functions was used. The reciprocal lattice vectors are marked on in black. Date 31 October 2019 Source Own work Author GKFX

## Generating code

Data points for this file were created with the following code:

```#include <complex.h>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>

typedef struct {
double x, y, z;
} vector;

static inline vector mult_sv(double scalar, vector vec) {
vector r = { scalar * vec.x, scalar * vec.y, scalar * vec.z };
return r;
}

static inline vector add_vvv(vector v1, vector v2, vector v3) {
vector r = {
v1.x + v2.x + v3.x,
v1.y + v2.y + v3.y,
v1.z + v2.z + v3.z };
return r;
}

static inline double dot_vv(vector v1, vector v2) {
return v1.x*v2.x + v1.y*v2.y + v1.z*v2.z;
}

static inline int cube(int n) { return n*n*n; }

vector a = {1.0, 0.0, 0.0};
vector b = {0.0, 1.0, 0.0};
vector c = {0.5, 0.0, 0.8};
double meshStepLen = 10.0/64.0;
int nPerSide = 12;
int meshSideN = 100;

/*
* The Fourier transform of δ(x - a, y - b, z - c) is
*     exp(i(aX + bY + cZ))
*     --------------------
*          (2√2)(√π)³
* using Mathematica's default definition of FT.
*/
double complex FTDD(vector realDDPos, vector reciprPos) {
return cexp(I * dot_vv(realDDPos, reciprPos)) /
(M_SQRT2 * M_2_SQRTPI * M_PI * M_PI);
}

int main() {
// Make progress bar work.
setvbuf (stdout, NULL, _IONBF, BUFSIZ);

vector *directLattice = (vector*) malloc(cube(nPerSide)*sizeof(vector));
{
vector *directLatticeTmp = directLattice;
for (int i = 0; i < nPerSide; i++) {
for (int j = 0; j < nPerSide; j++) {
for (int k = 0; k < nPerSide; k++) {
directLatticeTmp = add_vvv(mult_sv(i, a), mult_sv(j, b), mult_sv(k, c));
directLatticeTmp++;
}
}
}
}

double complex *reciprocalLattice = (double complex*) malloc(cube(meshSideN)*sizeof(double complex));
for (int i = 0; i < cube(meshSideN); i++) {
reciprocalLattice[i] = 0.0;
}

{
double complex *reciprocalLatticeTmp = reciprocalLattice;

for (int i = 0; i < meshSideN; i++) {
putchar('.');
for (int j = 0; j < meshSideN; j++) {
for (int k = 0; k < meshSideN; k++) {
vector imagPoint = { i*meshStepLen, j*meshStepLen, k*meshStepLen };
for (int l = 0; l < cube(nPerSide); l++) {
reciprocalLatticeTmp += FTDD(directLattice[l], imagPoint);
}
reciprocalLatticeTmp++;
}
}
}
}
printf(" Created complex lattice\n");

double *magData = (double*) malloc(cube(meshSideN)*sizeof(double));
for (int i = 0; i < cube(meshSideN); i++) {
magData[i] = cabs(reciprocalLattice[i]);
}
FILE *datafile = fopen("rlattice.bin", "wb");
if (!datafile) {
perror(0);
goto free_rlattice;
}
fwrite(magData, sizeof(double)*cube(meshSideN), 1, datafile);
fclose(datafile);

printf("\nDone.\n");

free_rlattice:
free(reciprocalLattice);
free(directLattice);
return 0;
}
```

It was then plotted in Mathematica.

## Licensing

I, the copyright holder of this work, hereby publish it under the following license:  This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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#### original creation by uploader

Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.