Take

(x_{j}, y_{j}, z_{j})

as fractional coordinates in the chosen unit cell. In the kinematic approximation the crystal basis structure factor is

F (hk l) = j \sum f_{j} (Q) exp [2 πi (h x_{j} + k y_{j} + l z_{j})], Q = \frac{4 π s i n θ}{λ} .

(1)

Equivalently, in Cartesian coordinates the phase is

G_{hk l} \cdot r_{j}

. Reversing every phase sign complex-conjugates the nonabsorbing amplitude and leaves its intensity unchanged. The atomic scattering factor is the Fourier transform of the atom's electron density, in electron scattering units:

f_{j} (Q) = \int ρ_{e, j} (r) e^{i Q \cdot r} d^{3} r .

(2)

At zero scattering angle the normal form factor counts electrons:

f_{j} (0) = Z_{j}

for a neutral atom, modified by the ionic charge for an ion. At increasing

Q

, different parts of the electron cloud scatter with different phases, reducing the coherent amplitude. Typical nonresonant atomic curves fall from their forward value toward small values at large angle. A localized cloud falls more slowly in reciprocal space. Anomalous scattering near an absorption edge adds energy-dependent complex terms and is excluded from this simple sketch.

For a fixed reflection and fixed instrumental and geometric corrections,

I_{hk l} \propto ∣ F (hk l) ∣^{2} .

(3)

The full amplitude also contains the lattice sum. At its reciprocal-lattice peak,

N_{c}

coherently illuminated identical cells give a peak amplitude

N_{c} F

; peak height and integrated peak intensity have different size dependences because the peak narrows with crystal size. Powder multiplicity, polarization, absorption, thermal motion and measurement geometry also affect observed intensities. The basis factor determines relative interference, not every experimental prefactor.