# Reflection Conditions

If the conventional unit cell is different from the primitive one, Miller index must satisfy an additional condition. The crystal with spiral and glide symmetries also shows additional reflection conditions.

## Reflection Conditions for Bravais Lattices except Simple Lattice

### Body-Centered Lattice

In body-centered lattices, there are lattice points at 0, 0, 0 and 1/2, 1/2, 1/2 of each conventional unit cell. The crystal structure factor always contains
exp 2πi0+exp2πi(h+k+l)/2
= 1+(−1)h+k+l.
This is 2, if h+k+l is even, and 0 if odd. The diffraction takes place only when h+k+l is an even number.

### Face-Centered Lattice

In face-centered lattice, there are lattice points at 0, 0, 0; 0, 1/2, 1/2; 1/2, 0, 1/2; and 1/2, 1/2, 0 of each conventional unit cell. The crystal structure factor always contains
exp 2πi0+exp2πi(k+l)/2+exp2πi(h+l)/2 +exp2πi(h+k)/2
= 1+(−1)k+l+(−1)h+l+(−1)h+kup> This is 4, if h,k, and l are all even or all odd. Otherwise this becomes 0. The diffraction takes place only when h,k, and l are all even or all odd.

### Base-Centered Lattice

In the C base-centered lattice, there are lattice points at 0, 0, 0 and 1/2, 1/2, 0 of each conventional unit cell. The crystal structure factor always contains
exp 2πi0+exp2πi(h+k)/2
= 1+(−1)h+k.
This is 2 if h+k is even, and 0 if odd.
Likewise, the diffraction for the A(B) base-centered lattice takes place only when k+l (h+l) is even.

### Rhombohedral Lattice in Hexagonal Setting

Rhombohedral cells are often expressed by the hexagonal setting. One must take the a and b axes as the lattice points are located at 0, 0, 0; 2/3, 1/3, 1/3; 1/3, 2/3, 2/3 in each hexagonal unit cell. (Note that one must not rotate the primitive vectors by 60 degrees around the c axis.)
The crystal structure factor always contains
exp 2πi0+exp2πi(2h+k+l)/3+exp2πi(h+2k+2l)/3
= 1+2cos2(2h+k+l)π/3
This is 3, if 2h+k+l is a multiple of three. Otherwise this becomes 0.

## Reflection Conditions derived from Spiral Operations

If the crystal has a spiral symmetry, there appear additional reflection conditions for the diffraction with the scattering vector is parallel to the spiral axis.

### 61 Screw, 65 Screw

If the crystal has a 61 axis, identical atoms are located at x, y, z; x', y', z + 1/6; x'', y'', z+2/6; x''', y''', z + 3/6; and so on in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is a multiple of 6.
Exactly the same reflection condition is derived from the 65 screw symmetry.

### 62 Screw, 64 Screw

If the crystal has a 62 axis, identical atoms are located at x, y, z; x', y', z + 1/3; x'', y'', z+2/3; x''', y''', z; and so on in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is a multiple of 3.
Exacly the same reflection condition is derived from the 64 screw symmetry.

### 63 Screw

If the crystal has a 63 axis, identical atoms are located at x, y, z; x', y', z + 1/2; x'', y'', z; x''', y''', z+1/2; and so on in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is even.

### 41 Screw, 43 Screw

If the crystal has a 41 axis, identical atoms are located at x, y, z; x', y', z + 1/4; x'', y'', z+2/4; and x''', y''', z+3/4; in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is a multiple of 4.
Exacly the same reflection condition is derived from the 43 screw symmetry.

### 42 Screw

If the crystal has a 42 axis, identical atoms are located at x, y, z; x', y', z + 1/2; x'', y'', z; and x''', y''', z+1/2; in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is even.

### 31 Screw, 32 Screw

If the crystal has a 31 axis, identical atoms are located at x, y, z; x', y', z + 1/3; and x'', y'', z+2/3 in each unit cell. In the (00l) reflection, the crystal structure factor becomes 0 unless l is a multiple of 3.

### 21 Screw

The 21 spiral axis is not necessarily parallel to the z axis.
If the crystal has a 21 axis along the x axis, identical atoms are located at x, y, z and x+1/2, y', z' in each unit cell. In the (h00) reflection, the crystal structure factor becomes 0 unless h is a multiple of 2.
Similarly, 21 axis along the y axis makes the reflection condition for (0k0) of even k. 21 axis along the z axis makes the reflection condition for (00l) of even l.

## Reflection Conditions derived from Glide Operations

If the crystal has a glide symmetry, there appear additional reflection conditions for the diffraction with the scattering vector is parallel to the glide plane.

#### Example 1: b Glide Plane normal to a axis

An atom at x,y,z is transferred to 1/2−x,1/2+y,z by the b glide plane of x=1/4. The crystal structure factor for (0kl) reflection contains
exp 2πi(ky+lz)+exp2πi(ky+lz+k/2)
=[1+(−1)k]exp 2πi(ky+lz).
This is 0, unless k is even.

#### Example 2: n Glide Plane normal to b axis

An atom at x,y,z is transferred to 1/2+x,1/2−y,1/2+z by the n glide plane of y=1/4. The crystal structure factor for (h0l) reflection contains
exp 2πi(hx+lz)+exp2πi(hx+lz+h+l/2)
=[1+(−1)h+l]exp 2πi(hx+lz).
This is 0, unless h+l is even.

#### Example 3: d Glide Plane normal to a axis in Face-Centered Lattice

An atom at x,y,z is transferred to 1/4−x,1/4+y,1/4+z by the d glide plane of x=1/8. The crystal structure factor for (0kl) reflection contains
exp 2πi(ky+lz)+exp2πi(ky+lz+k+l/4)
=[1+(−1)(k+l)/2]exp 2πi(ky+lz).
This is 0, unless k+l is a multiple of 4. (Here, k+l is even because the lattice is face centered.)

#### Example 4: d Glide Plane normal to [1-10] axis in Body-Centered Tetragonal Lattice

An atom at x,y,z is transferred to 1/4+y,1/4+x,1/4+z by the d glide plane of x=y. The crystal structure factor for (hhl) reflection contains
exp 2πi(hx+hy+lz)+exp2πi(hx+hy+lz+h/2+l/4)
=[1+i2h+l]exp 2πi(hx+hy+lz).
This is 0, unless 2h+l is a multiple of 4. (Here, 2h+l is even for (hhl) because the lattice is body centered.)

Home

Taka-hisa Arima