testLattice.cpp

This example demonstrates the use of Lattice class

1. Initializing a lattice

     Eigen::Matrix3d A;
     A << 0.5, 0.5, 0, 0.5, 0.0, 0.5, 0.0, 0.5, 0.5;
     Lattice<3> L(A);
    
     std::cout << "Lattice basis:" << std::endl;
     std::cout << L.latticeBasis << std::endl;
     std::cout << "Reciprocal lattice basis:" << std::endl;
     std::cout << L.reciprocalBasis << std::endl;

2. Initializing a lattice vector

     std::cout << "Creating a zero lattice vector u:" << std::endl;
     LatticeVector<3> u(L);
     std::cout << u << std::endl;

   1. Modifying its integer coordinates

        std::cout << "Modifying u:" << std::endl;
        u << 2, 1, 3;
        std::cout << u << std::endl;
        std::cout << "Cartesian coordinates of u:" << std::endl;
        std::cout << u.cartesian() << std::endl;
        std::cout << "Recover lattice vector from its cartesian components:";
        std::cout << L.latticeVector(u.cartesian()) << std::endl;

   2. Initializing using Cartesian coordinates

        std::cout << "Create a lattice vector using cartesian coordinates: 1.5,2,3.5"
                  << std::endl;
        Eigen::Vector3d vCoords;
        vCoords << 1.5, 2, 3.5;
        LatticeVector<3> v(vCoords, L);
        std::cout << "Integer coordinates of v:" << std::endl;
        std::cout << v << std::endl;

   3. Initializing using Cartesian coordinates may fail if they don't describe a lattice point

        try {
          // This fails
          Eigen::Vector3d wCoords;
          wCoords << 1.8, 2, 3.5;
          LatticeVector<3> w(wCoords, L);
        } catch (std::runtime_error &e) {
          std::cout << e.what() << std::endl;
        }

3. Lattice vector algebra

     v << 1, 1, 2;
     LatticeVector<3> w = 2 * u + 6 * v;
     std::cout << "Integer coordinates of w:" << std::endl;
     std::cout << w << std::endl;
     std::cout << "Cartesian coordinates of w:" << std::endl;
     std::cout << w.cartesian() << std::endl;

4. Lattice direction from a lattice vector

     std::cout << "Lattice direction along w:" << std::endl;
     LatticeDirection<3> wDir(w);
     std::cout << wDir << std::endl;

5. Creating a reciprocal lattice vector

     std::cout << "Creating a reciprocal lattice vector r:" << std::endl;
     ReciprocalLatticeVector<3> r(L);
     r << 3, 6, 9;
     std::cout << r << std::endl;
     std::cout << "Cartesian coordinates of the reciprocal lattice vector r:"
               << std::endl;
     std::cout << r.cartesian() << std::endl;

6. Reciprocal lattice direction from a reciprocal lattice vector

     std::cout << "Reciprocal lattice direction along r:" << std::endl;
     ReciprocalLatticeDirection<3> s(r);
     std::cout << s << std::endl;

7. Get reciprocal lattice vector from a reciprocal lattice direction. This may be required to access functions that accept a vector as an input.

     std::cout << "Recover the reciprocal lattice vector from the reciprocal "
                  "lattice direction s:"
               << std::endl;
     ReciprocalLatticeVector<3> t = s.reciprocalLatticeVector();
     std::cout << t << std::endl;

8. Get a lattice direction along a given cartesian vector. CAUTION: This will fail if the cartesian vector is not a lattice vector.

     LatticeDirection<3> latticeDirectionAlong_s =
         L.latticeDirection(s.cartesian());
     std::cout << "Lattice direction along s = " << std::endl;
     std::cout << latticeDirectionAlong_s << std::endl;

9. Get the stacking along a lattice plane

     std::cout << "Stacking of the lattice plane described by s = " << s.stacking()
               << std::endl;

10. The cross product of two lattice vectors is a reciprocal lattice direction. Similarly, the cross product of two reciprocal lattice vectors is a lattice direction. The cross product is enabled only for dim=3

      ReciprocalLatticeDirection<3> uxv(u.cross(v));
      std::cout << "The cross product of the lattice vectors u and v is a lattice "
                   "direction: "
                << uxv << std::endl;
     
      ReciprocalLatticeVector<3> q(L);
      q << 7, 9, 11;
      std::cout << "Reciprocal lattice vector q = " << std::endl;
      std::cout << q << std::endl;
      LatticeDirection<3> qxr(q.cross(r));
      std::cout << "The cross product of two reciprocal lattice vectors q and r is "
                   "a lattice direction: "
                << qxr << std::endl;

11. Output a configuration of lattice points bounded by three lattice vectors (named boxVectors) of a 3D lattice.

      std::vector<LatticeVector<3>> boxVectors;
      boxVectors.push_back(LatticeVector<3>(u, L));
      boxVectors.push_back(LatticeVector<3>(v, L));
      boxVectors.push_back(
          LatticeVector<3>(latticeDirectionAlong_s.latticeVector(), L));
      std::cout << "Outputting a configuration of lattice points bounded by three "
                   "box vectors: "
                << std::endl;
      for (const auto &vector : boxVectors)
        std::cout << vector.cartesian().transpose() << std::endl;
      L.box(boxVectors, "lattice.txt");

12. Create a 2D lattice and output its lattice points bounded by two lattice vectors (named boxVectors) of a 2D lattice.

      Eigen::Matrix2d B;
      B << 1.0, 0.0, 0.5, 1.0;
      Lattice<2> L2(B);
      LatticeVector<2> b1(L2);
      b1 << 1, 13;
      LatticeVector<2> b2(L2);
      b2 << 2, 7;
      std::cout << "Outputting a configuration of lattice points bounded by two "
                   "box vectors: "
                << std::endl;
      std::cout << b1.cartesian().transpose() << std::endl;
      std::cout << b2.cartesian().transpose() << std::endl;
      L2.box(std::vector<LatticeVector<2>>{-1 * b1, b2}, "lattice2.txt");

Full code:

#include "../../include/Lattices/LatticeModule.h"
#include <fstream>
#include <vector>
 
using namespace oILAB;
int main() {
  Eigen::Matrix3d A;
  A << 0.5, 0.5, 0, 0.5, 0.0, 0.5, 0.0, 0.5, 0.5;
  Lattice<3> L(A);
 
  std::cout << "Lattice basis:" << std::endl;
  std::cout << L.latticeBasis << std::endl;
  std::cout << "Reciprocal lattice basis:" << std::endl;
  std::cout << L.reciprocalBasis << std::endl;
  std::cout << "Creating a zero lattice vector u:" << std::endl;
  LatticeVector<3> u(L);
  std::cout << u << std::endl;
  std::cout << "Modifying u:" << std::endl;
  u << 2, 1, 3;
  std::cout << u << std::endl;
  std::cout << "Cartesian coordinates of u:" << std::endl;
  std::cout << u.cartesian() << std::endl;
  std::cout << "Recover lattice vector from its cartesian components:";
  std::cout << L.latticeVector(u.cartesian()) << std::endl;
  std::cout << "Create a lattice vector using cartesian coordinates: 1.5,2,3.5"
            << std::endl;
  Eigen::Vector3d vCoords;
  vCoords << 1.5, 2, 3.5;
  LatticeVector<3> v(vCoords, L);
  std::cout << "Integer coordinates of v:" << std::endl;
  std::cout << v << std::endl;
  try {
    // This fails
    Eigen::Vector3d wCoords;
    wCoords << 1.8, 2, 3.5;
    LatticeVector<3> w(wCoords, L);
  } catch (std::runtime_error &e) {
    std::cout << e.what() << std::endl;
  }
  v << 1, 1, 2;
  LatticeVector<3> w = 2 * u + 6 * v;
  std::cout << "Integer coordinates of w:" << std::endl;
  std::cout << w << std::endl;
  std::cout << "Cartesian coordinates of w:" << std::endl;
  std::cout << w.cartesian() << std::endl;
  std::cout << "Lattice direction along w:" << std::endl;
  LatticeDirection<3> wDir(w);
  std::cout << wDir << std::endl;
  std::cout << "Creating a reciprocal lattice vector r:" << std::endl;
  ReciprocalLatticeVector<3> r(L);
  r << 3, 6, 9;
  std::cout << r << std::endl;
  std::cout << "Cartesian coordinates of the reciprocal lattice vector r:"
            << std::endl;
  std::cout << r.cartesian() << std::endl;
  std::cout << "Reciprocal lattice direction along r:" << std::endl;
  ReciprocalLatticeDirection<3> s(r);
  std::cout << s << std::endl;
  std::cout << "Recover the reciprocal lattice vector from the reciprocal "
               "lattice direction s:"
            << std::endl;
  ReciprocalLatticeVector<3> t = s.reciprocalLatticeVector();
  std::cout << t << std::endl;
  LatticeDirection<3> latticeDirectionAlong_s =
      L.latticeDirection(s.cartesian());
  std::cout << "Lattice direction along s = " << std::endl;
  std::cout << latticeDirectionAlong_s << std::endl;
  std::cout << "Stacking of the lattice plane described by s = " << s.stacking()
            << std::endl;
  ReciprocalLatticeDirection<3> uxv(u.cross(v));
  std::cout << "The cross product of the lattice vectors u and v is a lattice "
               "direction: "
            << uxv << std::endl;
 
  ReciprocalLatticeVector<3> q(L);
  q << 7, 9, 11;
  std::cout << "Reciprocal lattice vector q = " << std::endl;
  std::cout << q << std::endl;
  LatticeDirection<3> qxr(q.cross(r));
  std::cout << "The cross product of two reciprocal lattice vectors q and r is "
               "a lattice direction: "
            << qxr << std::endl;
  std::vector<LatticeVector<3>> boxVectors;
  boxVectors.push_back(LatticeVector<3>(u, L));
  boxVectors.push_back(LatticeVector<3>(v, L));
  boxVectors.push_back(
      LatticeVector<3>(latticeDirectionAlong_s.latticeVector(), L));
  std::cout << "Outputting a configuration of lattice points bounded by three "
               "box vectors: "
            << std::endl;
  for (const auto &vector : boxVectors)
    std::cout << vector.cartesian().transpose() << std::endl;
  L.box(boxVectors, "lattice.txt");
  Eigen::Matrix2d B;
  B << 1.0, 0.0, 0.5, 1.0;
  Lattice<2> L2(B);
  LatticeVector<2> b1(L2);
  b1 << 1, 13;
  LatticeVector<2> b2(L2);
  b2 << 2, 7;
  std::cout << "Outputting a configuration of lattice points bounded by two "
               "box vectors: "
            << std::endl;
  std::cout << b1.cartesian().transpose() << std::endl;
  std::cout << b2.cartesian().transpose() << std::endl;
  L2.box(std::vector<LatticeVector<2>>{-1 * b1, b2}, "lattice2.txt");
  return 0;
}