Reflection Conditions

Reflection Conditions for Bravais Lattices except Simple Lattice

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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+i^2h+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.)

Taka-hisa Arima