Phase-Center Cube and Crystallographic Direction Notes

Phase-Center Cube (FCC) and Crystallographic Direction

• The speaker describes a cube with one atom on each side, which is consistent with a face-centered arrangement. In standard crystallography, this corresponds to a face-centered cubic (FCC) lattice where there is one atom on each face center of the cube.
• Purpose: to determine the crystallographic direction in 3D space between two lattice points (atoms A and B).
• The method uses a tail-to-head (tip-to-tail) construction to form the AB vector and then express its direction in Miller indices notation [uvw].

Crystallographic Direction in 3D Space

• Goal: Find the direction from atom A to atom B in the unit cell, expressed as a direction [uvw].
• Key idea: use the coordinates of A and B to build the displacement vector AB = rB − rA, then translate that displacement into direction components along the lattice vectors.

Step-by-Step Calculation

• Step 1: Determine the coordinates of point A and point B (the positions of atom A and atom B).

  - Let the fractional (within-cell) coordinates be

  $extbf{r}<em>A = (x</em>1, y<em>1, z</em>1)$
  $extbf{r}<em>B = (x</em>2, y<em>2, z</em>2)$

• Step 2: Form the displacement vector AB using the tail-to-head rule:

  • 
    oldsymbol{AB} = extbf{r}B - extbf{r}A = (x2 - x1,\, y2 - y1,\, z2 - z1) = (\Delta x, \Delta y, \Delta z)
  • In practice, this is the vector from A (tail) to B (head).

• Step 3: Convert the displacement into a crystallographic direction using the lattice constants a, b, c:

  • Let the lattice constants be a, b, c (along x, y, z respectively).
  • The direction components are
    $<br /> [u, v, w] = \left[ \frac{\Delta x}{a}, \frac{\Delta y}{b}, \frac{\Delta z}{c} \right]<br />$
  • If the lattice is cubic (a = b = c), this reduces to
    $<br /> [u, v, w] = \left[ \frac{\Delta x}{a}, \frac{\Delta y}{a}, \frac{\Delta z}{a} \right]<br />$
  • Equivalently, the direction is proportional to the displacement in fractional coordinates, often written as [Δx : Δy : Δz] in fractional units.

• Step 4: Interpretation notes

  • For a cubic lattice, the expression simplifies since all edges are equal.
  • The vector AB in Cartesian space is AB_cart = (a Δx, b Δy, c Δz).
  • In fractional coordinates, AB is represented by (Δx, Δy, Δz); the direction [u v w] gives how many lattice units along each axis the direction spans.

• Step 5: Practical tip

  • Use the tail-to-head technique to visualize AB, then convert using the corresponding lattice constants to obtain the Miller-style direction [uvw].

Mathematical Details and Notation

• Coordinates and vectors:
  • Let
    $extbf{r}<em>A = (x</em>1, y<em>1, z</em>1), \ extbf{r}<em>B = (x</em>2, y<em>2, z</em>2)$
  • Displacement:
    oldsymbol{AB} = (\Delta x, \Delta y, \Delta z) = (x2 - x1, y2 - y1, z2 - z1)
• Direction in lattice units:
  • General lattice:
    $[u, v, w] = \left[ \frac{\Delta x}{a}, \frac{\Delta y}{b}, \frac{\Delta z}{c} \right]$
  • Cubic lattice (a = b = c):
    $[u, v, w] = \left[ \frac{\Delta x}{a}, \frac{\Delta y}{a}, \frac{\Delta z}{a} \right]$
• Relation to fractional vs Cartesian:
  • Fractional (within-cell) displacement: (Δx, Δy, Δz)
  • Cartesian displacement: (a Δx, b Δy, c Δz)

Practical Considerations and Tips

• Start with clear coordinates for A and B in the unit cell reference frame.
• Use the tail-to-head method to form AB before converting to a direction.
• Keep track of whether you are using fractional coordinates or Cartesian coordinates; convert appropriately using the lattice constants a, b, c.
• In many crystallography problems, the cubic case is common, simplifying the conversion.
• The final direction is typically denoted as [uvw], with the components reflecting how far along each axis the B point is from A in lattice units.

Transcript Context and Remarks

• The speaker notes a casual interruption during the explanation, including non-technical comments. Those remarks do not contribute to the mathematical method but reflect a talk-style delivery.
• Core technical points are the identification of A and B, the AB vector, and the conversion to a crystallographic direction using lattice constants as shown above.