atoms arranged in a regular space lattice pattern

The cube is labeled Unit cell and the spheres at the corners are labeled Lattice points. In the first image, a cube with a sphere at each corner is shown. Pattern_points = lattice_generation(a1,a2) 1: A unit cell shows the locations of lattice points repeating in all directions. Return dash_trace(dotLine_a1,color),dash_trace(dotLine_a2,color)ĭef make_arrow(vec,text,color = 'Red',vec_zero =, text_shift = ): # Produces the dotted lines of the unit cellĭef dash_contour(a1,a2, vec_zero = np.array(), color='Red'):ĭotLine_a1 = np.transpose(np.array()+vec_zero)ĭotLine_a2 = np.transpose(np.array()+vec_zero) # Produces the lattice to be fed into plotly In the image below we show a 2D simple square lattice (panel A).īecause the environment of each point is the same, this set of points forms a lattice. With a i being the primitive lattice vectors.įor a n dimensional system, we have to define n linearly independent primitive lattice vectors in order to be able to map out the entire lattice.īut there exist multiple equivalent definitions of a lattice.Ī lattice is a set of points, where the environment of each point is the same. R = n 1 a 1 + n 2 a 2 + n 3 a 3, for n 1, n 2, n 3 ∈ Z. Such a framework is given by the concept of a lattice:Ī lattice is an infinite set of points defined by integer sums of a set of linearly independent primitive lattice vectors(will be explained later on). It is a geometrical pattern for the atomic arrangement in a crystal. To describe such periodic structures, we need a framework. Those atoms may be arranged in a regular, geometric pattern. Most of solid state physics deals with crystals, which are periodic multi-atomic structures. Therefore the upcoming two lectures will focus on developing, understanding, and applying this language and framework. It does however, require a new language and framework to fully describe such structures. The systems we considered were mainly 1D.īut most solids, such as crystals, are 3D structures.ĭescribing 3D systems is not much harder than describing 1D systems.

In the past few lectures, we derived some very important physical quantities for phonons and electrons, such as the effective mass and the dispersion relation. Each atom in the lattice has only six nearest neighbors in an octahedral arrangement. Determine the Miller planes of a given lattice The arrangement of the atoms in a solid that has a simple cubic unit cell was shown in part (a) in Figure 12.5.Determine the primitive, conventional, and Wigner-Seitz unit cells of a given lattice.The unit cells differ in their relative locations or orientations within the lattice, but they are all valid choices because repeating them in any direction fills the overall pattern of dots. Compute the volume filling factor given a crystal structure (ac) Three two-dimensional lattices illustrate the possible choices of the unit cell.Describe any crystal using crystallographic terminology, and interpret this terminology.General procedure to analyze a periodic structureĮxample: analyzing the graphene crystal structureĮxercise 2: The crystal structure of diamondĮxercise 3: Directions and spacings of Miller planes

Atoms arranged in a regular space lattice pattern