# Setting up an FCC lattice in a simulation

• SchroedingersLion
In summary, ");generate_fcc(i, rows, columns, layer);}return 0;}In summary, to generate a Monte Carlo simulation of N hard spheres, one needs to set up the lattice as a fcc lattice. The lattice is composed of 3 lattice
SchroedingersLion
Greetings,

for a Monte Carlo simulation of N hard spheres, I want to set up the initial configuration as a fcc lattice. I am thinking about how to do it most efficiently.

Here is the lattice structure:

The 3 lattice vectors, each one pointing to a different type of the 12 nearest neighbors (see colors in picture), are given by
$$\vec a_{1}=\frac a 2 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \vec a_{2}=\frac a 2 \begin {pmatrix} 1 \\ 0 \\ 1 \end {pmatrix}, \vec a_{3}=\frac a 2 \begin {pmatrix} 1 \\ 1 \\ 0 \end {pmatrix}$$

How would you do this most easily in C++?

My first idea was to simply set up the nearest neighbors. First, I use the three lattice vectors separately to set up the first 3 nearest neighbors. Then I combine two of them to get the other 9. Then I combine all 3 to get the next-nearest neighbors.

But what then? I have to repeat these steps with different pre-factors in the linear combinations, but I don't know how to do it systematically in a loop.Regards,
SL

Ok, in case someone needs this as well, here is my routine:

Code:
#include <fstream>
#include <string>

//function that writes the coordinates of fcc lattice sites with given
//lattice constant, numbers of rows, columns and layers to an output file

using namespace std;

void generate_fcc(double a, int rows, int columns, int layers){

ofstream fcc("fcc_coordinates_a=" + to_string(a) + "_" +
to_string(rows) + "rows_" + to_string(columns) + "columns_" + to_string(layers) + "layers.dat");

//necessary initializations to get coordinates of 1st particle after incrementations at beginning of loops
double x0=-a, y0=-a, z0=-a, x1, y1, z1=-a/2;

// print coordinates, layer by layer, columns by columns, row by row
for (int i=0; i<layers; i++){
z0+=a;
z1+=a;

x0=-a;

for (int j=0; j<columns; j++){
x0+=a;
x1=x0 + a/2;

y0=-a;

for (int k=0; k<rows; k++){
y0+=a;
y1=y0 + a/2;

fcc << x0 << "\t" << y0 << "\t" << z0 << endl;
fcc << x1 << "\t" << y1 << "\t" << z0 << endl;
fcc << x0 << "\t" << y1 << "\t" << z1 << endl;
fcc << x1 << "\t" << y0 << "\t" << z1 << endl;
}
}
}

fcc.close();
}

## 1. What is an FCC lattice in a simulation?

An FCC (Face-Centered Cubic) lattice is a type of lattice structure commonly used in simulations to model the arrangement of atoms or molecules. It consists of a repeating pattern of atoms with each atom having 12 nearest neighbors, making it a highly symmetric and stable structure.

## 2. How do you set up an FCC lattice in a simulation?

To set up an FCC lattice in a simulation, you will first need to define the lattice parameters, such as the lattice constant and the number of unit cells. Then, you can use a software tool or code to generate the coordinates of the atoms in the lattice based on these parameters. Finally, you can import these coordinates into your simulation software and run the simulation.

## 3. What are the advantages of using an FCC lattice in a simulation?

An FCC lattice has many advantages in simulations, including its high symmetry and stability, which makes it a good representation of real-world structures. It also allows for efficient packing of atoms, making it a good choice for simulations involving dense materials. Additionally, many simulation software packages have built-in support for generating and working with FCC lattices.

## 4. Are there any limitations to using an FCC lattice in a simulation?

While an FCC lattice is a useful and commonly used structure in simulations, it may not be the best choice for all systems. For example, it may not accurately represent the behavior of materials with more complex or irregular structures. Additionally, the FCC lattice may not be suitable for simulations that require a high degree of precision or accuracy.

## 5. Can an FCC lattice be used for simulations in different dimensions?

Yes, an FCC lattice can be used for simulations in both 2D and 3D dimensions. In 2D simulations, the FCC lattice is often referred to as a hexagonal close-packed (HCP) lattice. The lattice parameters and arrangement of atoms will vary slightly between 2D and 3D simulations, but the overall structure and properties remain similar.

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