In condensed matter physics and crystallography, the static structure factor (or structure factor for short) is a mathematical description of how a material scatters incident radiation. The structure factor is a critical tool in the interpretation of scattering patterns (interference patterns) obtained in X-ray, electron and neutron diffraction experiments. Confusingly, there are two different mathematical expressions in use, both called 'structure factor'. One is usually written S ( q ) {\displaystyle S(\mathbf {q} )} ; it is more generally valid, and relates the observed diffracted intensity per atom to that produced by a single scattering unit. The other is usually written F {\displaystyle F} or F h k ℓ {\displaystyle F_{hk\ell }} and is only valid for systems with long-range positional order — crystals. This expression relates the amplitude and phase of the beam diffracted by the ( h k ℓ ) {\displaystyle (hk\ell )} planes of the crystal ( ( h k ℓ ) {\displaystyle (hk\ell )} are the Miller indices of the planes) to that produced by a single scattering unit at the vertices of the primitive unit cell. F h k ℓ {\displaystyle F_{hk\ell }} is not a special case of S ( q ) {\displaystyle S(\mathbf {q} )} ; S ( q ) {\displaystyle S(\mathbf {q} )} gives the scattering intensity, but F h k ℓ {\displaystyle F_{hk\ell }} gives the amplitude. It is the modulus squared | F h k ℓ | 2 {\displaystyle |F_{hk\ell }|^{2}} that gives the scattering intensity. F h k ℓ {\displaystyle F_{hk\ell }} is defined for a perfect crystal, and is used in crystallography, while S ( q ) {\displaystyle S(\mathbf {q} )} is most useful for disordered systems. For partially ordered systems such as crystalline polymers there is obviously overlap, and experts will switch from one expression to the other as needed. The static structure factor is measured without resolving the energy of scattered photons/electrons/neutrons. Energy-resolved measurements yield the dynamic structure factor.
Derivation of S(q) Consider the scattering of a beam of wavelength λ {\displaystyle \lambda } by an assembly of N {\displaystyle N} particles or atoms stationary at positions R j , j = 1 , … , N {\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N} . Assume that the scattering is weak, so that the amplitude of the incident beam is constant throughout the sample volume (Born approximation), and absorption, refraction and multiple scattering can be neglected (kinematic diffraction). The direction of any scattered wave is defined by its scattering vector q {\displaystyle \mathbf {q} } . This vector is q = k s − k o {\displaystyle \mathbf {q} =\mathbf {k_{s}} -\mathbf {k_{o}} } , where k s {\displaystyle \mathbf {k_{s}} } and k o {\displaystyle \mathbf {k_{o}} } ( | k s | = | k 0 | = 2 π / λ {\displaystyle |\mathbf {k_{s}} |=|\mathbf {k_{0}} |=2\pi /\lambda } ) are the scattered and incident beam wavevectors, and θ {\displaystyle \theta } is the angle between them. For elastic scattering, | k s | = | k o | {\displaystyle |\mathbf {k} _{s}|=|\mathbf {k_{o}} |} and q = | q | = 4 π λ sin ( θ / 2 ) {\displaystyle q=|\mathbf {q} |={{\frac {4\pi }{\lambda }}\sin(\theta /2)}} , limiting the possible range of q {\displaystyle \mathbf {q} } (see Ewald sphere). The amplitude and phase of this scattered wave will be the vector sum of the scattered waves from all the atoms Ψ s ( q ) = ∑ j = 1 N f j e − i q ⋅ R j {\displaystyle \Psi _{s}(\mathbf {q} )=\sum _{j=1}^{N}f_{j}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}} For an assembly of atoms, f j {\displaystyle f_{j}} is the atomic form factor of the j {\displaystyle j} -th atom. The scattered intensity is obtained by multiplying this function by its complex conjugate
The structure factor is defined as this intensity normalized by 1 / ∑ j = 1 N f j 2 {\displaystyle 1/\sum _{j=1}^{N}f_{j}^{2}}
If all the atoms are identical, then Equation (1) becomes I ( q ) = f 2 ∑ j = 1 N ∑ k = 1 N e − i q ⋅ ( R j − R k ) {\displaystyle I(\mathbf {q} )=f^{2}\sum _{j=1}^{N}\sum _{k=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot (\mathbf {R} _{j}-\mathbf {R} _{k})}} and ∑ j = 1 N f j 2 = N f 2 {\displaystyle \sum _{j=1}^{N}f_{j}^{2}=Nf^{2}} so
Another useful simplification is if the material is isotropic, like a powder or a simple liquid. In that case, the intensity depends on q = | q | {\displaystyle q=|\mathbf {q} |} and r j k = | r j − r k | {\displaystyle r_{jk}=|\mathbf {r} _{j}-\mathbf {r} _{k}|} . In three dimensions, Equation (2) then simplifies to the Debye scattering equation:
An alternative derivation gives good insight, but uses Fourier transforms and convolution. To be general, consider a scalar (real) quantity ϕ ( r ) {\displaystyle \phi (\mathbf {r} )} defined in a volume V {\displaystyle V} ; this may correspond, for instance, to a mass or charge distribution or to the refractive index of an inhomogeneous medium. If the scalar function is integrable, we can write its Fourier transform as ψ ( q ) = ∫ V ϕ ( r ) exp ( − i q ⋅ r ) d r {\displaystyle \textstyle \psi (\mathbf {q} )=\int _{V}\phi (\mathbf {r} )\exp(-i\mathbf {q} \cdot \mathbf {r} )\,\mathrm {d} \mathbf {r} } . In the Born approximation the amplitude of the scattered wave corresponding to the scattering vector q {\displaystyle \mathbf {q} } is proportional to the Fourier transform ψ ( q ) {\displaystyle \textstyle \psi (\mathbf {q} )} . When the system under study is composed of a number N {\displaystyle N} of identical constituents (atoms, molecules, colloidal particles, etc.) each of which has a distribution of mass or charge f ( r ) {\displaystyle f(\mathbf {r} )} then the total distribution can be considered the convolution of this function with a set of delta functions.
with R j , j = 1 , … , N {\displaystyle \textstyle \mathbf {R} _{j},j=1,\,\ldots ,\,N} the particle positions as before. Using the property that the Fourier transform of a convolution product is simply the product of the Fourier transforms of the two factors, we have ψ ( q ) = f ( q ) × ∑ j = 1 N exp ( − i q ⋅ R j ) {\displaystyle \textstyle \psi (\mathbf {q} )=f(\mathbf {q} )\times \sum _{j=1}^{N}\exp(-i\mathbf {q} \cdot \mathbf {R} _{j})} , so that:
This is clearly the same as Equation (1) with all particles identical, except that here f {\displaystyle f} is shown explicitly as a function of q {\displaystyle \mathbf {q} } . In general, the particle positions are not fixed and the measurement takes place over a finite exposure time and with a macroscopic sample (much larger than the interparticle distance). The experimentally accessible intensity is thus an averaged one ⟨ I ( q ) ⟩ {\displaystyle \textstyle \langle I(\mathbf {q} )\rangle } ; we need not specify whether ⟨ ⋅ ⟩ {\displaystyle \langle \cdot \rangle } denotes a time or ensemble average. To take this into account we can rewrite Equation (3) as:
Perfect crystals In a crystal, the constitutive particles are arranged periodically, with translational symmetry forming a lattice. The crystal structure can be described as a Bravais lattice with a group of atoms, called the basis, placed at every lattice point; that is, [crystal structure] = [lattice] ∗ {\displaystyle \ast } [basis]. If the lattice is infinite and completely regular, the system is a perfect crystal. For such a system, only a set of specific values for q {\displaystyle \mathbf {q} } can give scattering, and the scattering amplitude for all other values is zero. This set of values forms a lattice, called the reciprocal lattice, which is the Fourier transform of the real-space crystal lattice. In principle the scattering factor S ( q ) {\displaystyle S(\mathbf {q} )} can be used to determine the scattering from a perfect crystal; in the simple case when the basis is a single atom at the origin (and again neglecting all thermal motion, so that there is no need for averaging) all the atoms have identical environments. Equation (1) can be written as
I ( q ) = f 2 | ∑ j = 1 N e − i q ⋅ R j | 2 {\displaystyle I(\mathbf {q} )=f^{2}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}} and S ( q ) = 1 N | ∑ j = 1 N e − i q ⋅ R j | 2 {\displaystyle S(\mathbf {q} )={\frac {1}{N}}\left|\sum _{j=1}^{N}\mathrm {e} ^{-i\mathbf {q} \cdot \mathbf {R} _{j}}\right|^{2}} . The structure factor is then simply the squared modulus of the Fourier transform of the lattice, and shows the directions in which scattering can have non-zero intensity. At these values of q {\displaystyle \mathbf {q} } the wave from every lattice point is in phase. The value of the structure factor is the same for all these reciprocal lattice points, and the intensity varies only due to changes in f {\displaystyle f} with q {\displaystyle \mathbf {q} } .
Units The units of the structure-factor amplitude depend on the incident radiation. For X-ray crystallography they are multiples of the unit of scattering by a single electron (2.82 × 10 − 15 {\displaystyle \times 10^{-15}} m); for neutron scattering by atomic nuclei the unit of scattering length of 10 − 14 {\displaystyle 10^{-14}} m is commonly used. The above discussion uses the wave vectors | k | = 2 π / λ {\displaystyle |\mathbf {k} |=2\pi /\lambda } and | q | = 4 π sin θ / λ {\displaystyle |\mathbf {q} |=4\pi \sin \theta /\lambda } . However, crystallography often uses wave vectors | s | = 1 / λ {\displaystyle |\mathbf {s} |=1/\lambda } and | g | = 2 sin θ / λ {\displaystyle |\mathbf {g} |=2\sin \theta /\lambda } . Therefore, when comparing equations from different sources, the factor 2 π {\displaystyle 2\pi } may appear and disappear, and care to maintain consistent quantities is required to get correct numerical results.
Definition of Fhkl In crystallography, the basis and lattice are treated separately. For a perfect crystal the lattice gives the reciprocal lattice, which determines the positions (angles) of diffracted beams, and the basis gives the structure factor F h k l {\displaystyle F_{hkl}} which determines the amplitude and phase of the diffracted beams:
where the sum is over all atoms in the unit cell, x j , y j , z j {\displaystyle x_{j},y_{j},z_{j}} are the positional coordinates of the j {\displaystyle j} -th atom, and f j {\displaystyle f_{j}} is the scattering factor of the j {\displaystyle j} -th atom. The coordinates x j , y j , z j {\displaystyle x_{j},y_{j},z_{j}} have the directions and dimensions of the lattice vectors a , b , c {\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} } . That is, (0,0,0) is at the lattice point, the origin of position in the unit cell; (1,0,0) is at the next lattice point along a {\displaystyle \mathbf {a} } and (1/2, 1/2, 1/2) is at the body center of the unit cell. ( h k l ) {\displaystyle (hkl)} defines a reciprocal lattice point at ( h a ∗ , k b ∗ , l c ∗ ) {\displaystyle (h\mathbf {a^{*}} ,k\mathbf {b^{*}} ,l\mathbf {c^{*}} )} which corresponds to the real-space plane defined by the Miller indices ( h k l ) {\displaystyle (hkl)} (see Bragg's law).
F h k ℓ {\displaystyle F_{hk\ell }} is the vector sum of waves from all atoms within the unit cell. An atom at any lattice point has the reference phase angle zero for all h k ℓ {\displaystyle hk\ell } since then ( h x j + k y j + ℓ z j ) {\displaystyle (hx_{j}+ky_{j}+\ell z_{j})} is always an integer. A wave scattered from an atom at (1/2, 0, 0) will be in phase if h {\displaystyle h} is even, out of phase if h {\displaystyle h} is odd. Again an alternative view using convolution can be helpful. Since [crystal structure] = [lattice] ∗ {\displaystyle \ast } [basis], F {\displaystyle {\mathcal {F}}} [crystal structure] = F {\displaystyle {\mathcal {F}}} [lattice] × F {\displaystyle \times {\mathcal {F}}} [basis]; that is, scattering ∝ {\displaystyle \propto } [reciprocal lattice] × {\displaystyle \times } [structure factor].
Examples of Fhkl in 3-D
Body-centered cubic (BCC) For the body-centered cubic Bravais lattice (cI), we use the points ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} and ( 1 2 , 1 2 , 1 2 ) {\displaystyle ({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}})} which leads us to
F h k ℓ = ∑ j f j e − 2 π i ( h x j + k y j + ℓ z j ) = f [ 1 + ( e − i π ) h + k + ℓ ] = f [ 1 + ( − 1 ) h + k + ℓ ] {\displaystyle F_{hk\ell }=\sum _{j}f_{j}e^{-2\pi i(hx_{j}+ky_{j}+\ell z_{j})}=f\left[1+\left(e^{-i\pi }\right)^{h+k+\ell }\right]=f\left[1+(-1)^{h+k+\ell }\right]}
and hence
F h k ℓ = { 2 f , h + k + ℓ = even 0 , h + k + ℓ = odd {\displaystyle F_{hk\ell }={\begin{cases}2f,&h+k+\ell ={\text{even}}\\0,&h+k+\ell ={\text{odd}}\end{cases}}}
Face-centered cubic (FCC) The FCC lattice is a Bravais lattice, and its Fourier transform is a body-centered cubic lattice. However to obtain F h k ℓ {\displaystyle F_{hk\ell }} without this shortcut, consider an FCC crystal with one atom at each lattice point as a primitive or simple cubic with a basis of 4 atoms, at the origin x j , y j , z j = ( 0 , 0 , 0 ) {\displaystyle x_{j},y_{j},z_{j}=(0,0,0)} and at the three adjacent face centers, x j , y j , z j = ( 1 2 , 1 2 , 0 ) {\displaystyle x_{j},y_{j},z_{j}=\left({\frac {1}{2}},{\frac {1}{2}},0\right)} , ( 0 , 1 2 , 1 2 ) {\displaystyle \left(0,{\frac {1}{2}},{\frac {1}{2}}\right)} and ( 1 2 , 0 , 1 2 ) {\displaystyle \left({\frac {1}{2}},0,{\frac {1}{2}}\right)} . Equation (8) becomes
F h k ℓ = f ∑ j = 1 4 e [ − 2 π i ( h x j + k y j + ℓ z j ) ] = f [ 1 + e [ − i π ( h + k ) ] + e [ − i π ( k + ℓ ) ] + e [ − i π ( h + ℓ ) ] ] = f [ 1 + ( − 1 ) h + k + ( − 1 ) k + ℓ + ( − 1 ) h + ℓ ] {\displaystyle F_{hk\ell }=f\sum _{j=1}^{4}\mathrm {e} ^{[-2\pi i(hx_{j}+ky_{j}+\ell z_{j})]}=f\left[1+\mathrm {e} ^{[-i\pi (h+k)]}+\mathrm {e} ^{[-i\pi (k+\ell )]}+\mathrm {e} ^{[-i\pi (h+\ell )]}\right]=f\left[1+(-1)^{h+k}+(-1)^{k+\ell }+(-1)^{h+\ell }\right]}
with the result
F h k ℓ = { 4 f , h , k , ℓ all even or all odd 0 , h , k , ℓ mixed parity {\displaystyle F_{hk\ell }={\begin{cases}4f,&h,k,\ell \ \ {\mbox{all even or all odd}}\\0,&h,k,\ell \ \ {\mbox{mixed parity}}\end{cases}}}
The most intense diffraction peak from a material that crystallizes in the FCC structure is typically the (111). Films of FCC materials like gold tend to grow in a (111) orientation with a triangular surface symmetry. A zero diffracted intensity for a group of diffracted beams (here, h , k , ℓ {\displaystyle h,k,\ell } of mixed parity) is called a systematic absence.
Diamond crystal structure The diamond cubic crystal structure occurs for example in diamond (carbon), tin, and most semiconductors. There are 8 atoms in the cubic unit cell. We can consider the structure as a simple cubic with a basis of 8 atoms, at positions
x j , y j , z j = ( 0 , 0 , 0 ) ( 1 2 , 1 2 , 0 ) ( 0 , 1 2 , 1 2 ) ( 1 2 , 0 , 1 2 ) ( 1 4 , 1 4 , 1 4 ) ( 3 4 , 3 4 , 1 4 ) ( 1 4 , 3 4 ,