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 2π*i*0+exp2π*i*(*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.

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 2π*i*0+exp2π*i*(*k+l*)/2+exp2π*i*(*h+l*)/2
+exp2π*i*(*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.
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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 2π*i*0+exp2π*i*(*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 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 2π*i*0+exp2π*i*(*2h+k+l*)/3+exp2π*i*(*h+2k+2l*)/3

= 1+2cos2(2*h+k+l*)π/3

This is 3, if 2*h+k+l* is a multiple of three. Otherwise this becomes 0.

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.

If the crystal has a 6_{1} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is a multiple of 6.

Exactly the same reflection condition is derived from the 6_{5} screw symmetry.

If the crystal has a 6_{2} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is a multiple of 3.

Exacly the same reflection condition is derived from the 6_{4} screw symmetry.

If the crystal has a 6_{3} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is even.

If the crystal has a 4_{1} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is a multiple of 4.

Exacly the same reflection condition is derived from the 4_{3} screw symmetry.

If the crystal has a 4_{2} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is even.

If the crystal has a 3_{1} 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 (00*l*) reflection, the crystal structure factor becomes 0 unless *l* is a multiple of 3.

The 2_{1} spiral axis is not necessarily parallel to the *z* axis.

If the crystal has a 2_{1} 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 (*h*00) reflection, the crystal structure factor becomes 0 unless *h* is a multiple of 2.

Similarly, 2_{1} axis along the *y* axis makes the reflection condition for (0*k*0) of even *k*.
2_{1} axis along the *z* axis makes the reflection condition for (00*l*) of even *l*.

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.

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 (0*k**l*) reflection contains

exp 2π*i*(*ky+lz*)+exp2π*i*(*ky+lz*+*k*/2)

=[1+(−1)^{k}]exp 2π*i*(*ky+lz*).

This is 0, unless *k* is even.

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 (*h*0*l*) reflection contains

exp 2π*i*(*hx+lz*)+exp2π*i*(*hx+lz*+*h+l*/2)

=[1+(−1)^{h+l}]exp 2π*i*(*hx+lz*).

This is 0, unless *h+l* is even.

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 (0*k**l*) reflection contains

exp 2π*i*(*ky+lz*)+exp2π*i*(*ky+lz*+*k+l*/4)

=[1+(−1)^{(k+l)/2}]exp 2π*i*(*ky+lz*).

This is 0, unless *k+l* is a multiple of 4.
(Here, *k+l* is even because the lattice is face centered.)

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 2π*i*(*hx+hy+lz*)+exp2π*i*(*hx+hy+lz*+*h*/2+*l*/4)

=[1+*i*^{2h+l}]exp 2π*i*(*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.)

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Taka-hisa Arima