Structure and x-rays

Electromagnetic radiation transports energy and momentum and is quantized as photons.
Radiation is propagated at a finite velocity c and radiated energy is proportional to the square of the radiated electric field.

The connection between the wavelength λ and the frequency ν is given by c = νλ.

Energy E =  hc/ λ 

E [keV] = 12.39854 / λ [Å]

X-rays scatter from electrons and scattering experiments give information on the electron density variations in the sample. The spatial resolution depends on the experimental method. 

According to quantum mechanics, energy may be transferred, and the scattering photon can have lower energy than the incident one. Let the incident photon have a momentum of ħk and an energy E, and the scattered photon have a momentum of ħk' and an energy E'. The vectors k and k' are wavevectors.

The momentum transfer is 

ħq = ħk - ħk'.

The vector q is called the wave vector transfer or the scattering vector. It is usually expressed in units Å-1 (or nm-1).

Scattering is inelastic, when the energies E and E' are not the same.

In elastic scattering the energy of the incoming and scattered wave are the same.



Scattering from an electron

According to classical electrodynamics, the electric field exerts a force on the electronic charge, which then accelerates and radiates the scattered wave.  Electrons oscillate up and down with the same frequency as the electric and magnetic fields of the incident x-ray beam. Electron acts as a source of x-rays since accelerating charges  emit radiation. The scattering is elastic.


The electric field of an incident plane wave sets an electron in oscillation which then radiates a spherical wave.

The scattering vector

The scattering vector is  q = k´-k, where k and k' are the wave vectors of the incoming and outgoing wave.

Elastic scattering is assumed, thus |k| = |k'| = 2π/λ.

The length of the scattering vector is then

|q| = q = 2|k|sinθ  = 4π/λ sinθ,

where 2θ is the scattering angle.

The differential cross section of Thomson scattering

n The nicest X-ray detectors count single photons.  The measured intensity I is the number of photons per second recorded by the detector.

Let the intensity of the primary beam be I0 and its area A0. Let ΔΩ be the solid angle subtended by the detector at a distance R from the sample. The cross sectional area of the scattered beam is R2 ΔΩ.
 
The energy per unit area of the beam, E, is proportional to the square of the electric field Ein.

The intensity may be written as

I/I0 = |E|2 R2 ΔΩ /(|Ein|2 A0).

The scattered intensity is normalized by the incident flux I0/A0 and the solid angle ΔΩ.

The differential cross section dσ/dΩ is defined by

                (number of scattered photons per sec into ΔΩ)
dσ/dΩ = -------------------------------------------------
                                      (incident flux) ΔΩ

The differential cross section  is

   dσ/dΩ = I/(I0/A0 ΔΩ) = |Erad|2 R2/ |Ein|2 = r02 P

The polarization factor P takes into account the polarization of the incident electric field (Ein).

P = 1, synchrotron, vertical polarization P = cos2 2θ, synchrotron, horizontal polarization
P = (1 + cos2 2θ)/2,  unpolarized source.

The angle 2θ is the scattering angle and Erad is the scattered field.



The classical electron radius is r0 = e2 /(4πε0 mc2). It is also called Thomson scattering length. It's value is r0 ≈ 2.82 10-5 Å.



Scattering from two electrons

The scattering centers are located at  O and B. Denote a vector from the volume element O to the element B as r.  The distance between the elements is thus the length r.


The scattering vector q is defined as q = 2π/λ (k-k0).  (k, k0 unit vectors).

The  phase difference of the waves  is  2π/λ (k- k') · r = q · r, where q is the scattering vector.

Scattering from atom - classical description

Assume that the atom has Z electrons.  The electron density is denoted by ρ(r). The scattered field is a superposition of the contributions of different volume elements. Let the distance between scattering volume elements be r and  the wave vectors of the incident and scattered wave be  k and k'.

 The volume element d3r will contribute to the scattered field an amount 

-r0 ρ(r)d3r

with a phase factor of exp(iq·r). The total scattering length of the atom is

   -r0 f(q) = -r0 ∫ ρ(r) exp(iq·r) d3r.

The function  f  is called the atomic scattering factor. Its value at origin is Z and it decreases as q increases.

The atomic scattering factor, which is different for each element, is defined by

         amplitude of the wave scattered by a atom
f  =  -----------------------------------------------
         amplitude of the wave scattered by an electron

It gives the efficiency of scattering of a given atom in a given direction. Sometimes f is called the atomic form factor.

On the basis of definition f(0) ≈ Z.



Basic x-ray based methods for structural studies on molecular scale

Bragg law

Consider reflection of x-rays from parallel lattice planes. Constructive interference occurs when the path difference of the waves is some integer times the wavelength. According to the Bragg law the lattice spacing d and the scattering angle 2θ corresponding to a diffraction peak are related as

2d sin θ = λ,


where λ is the wavelength.

Instead of the scattering angle it is more convenient to use the magnitude of the scattering vector q. The scattering vector is the difference of the wave vectors of the incoming and scattering waves. Bragg law in terms of q:

d = 2π/q.

The scattering angle and q are related as q = 4π/λ sin θ.

Isotropic crystalline powder


Diffraction pattern contains rings. These are called Debye rings.

Then it is sufficient to integrate the pattern over the azimulthal angle and  examine only intensity as a function of the magnitude of the scattering vector q.

Example. Powder pattern of CsCl, which has BCC structure.  The y-axis is the relative intensity and x-axis the scattering angle.

 

 

 

 

 

  

X-Ray scattering intensity in terms of electron density

Especially in the case of small angle scattering the scattering intensity may be generally interpreted as follows.
The scattering amplitude A is proportional to the Fourier transform of the electron density ρ(x) and can be written as

 ∫ ρ(x) exp(-i q·x) d3x.

The intensity is the square of the amplitude: I = A*A.

I = ∫ ρ(x) exp(-i q·x) d3x  ∫ ρ(y) exp(i q·y) d3y

  = ∫∫ ρ(x) ρ(y) exp(-i q·(x-y)) d3x d3y

Substitute z = x-y

I(q) = ∫∫ ρ(z+y) ρ(y) exp(-i q·z) d3z d3y

       = ∫∫ ρ(z+y) ρ(y) d3y exp(-i q·z) d3z

The function

C(z)= ∫ ρ(z+y) ρ(y) d3y

is called the autocorrelation function of the electron density.

The intensity is the Fourier transform of the autocorrelation function: I(q) = ∫  C(z) exp(-i q·z) d3z

About correlation and autocorrelation

Let f and g be functions of one variable. Their correlation function is the integral

C(x) = ∫ f(x+y) g(y) dy.

Compare it to the convolution integral

C(x) = ∫ f(x-y) g(y) dy.

The correlation function of a function f  itself, is called the autocorrelation function

C(x) = ∫ f(x+y) f(y) dy.

The autocorrelation has high values where both f(x) and f(x+y) have high values. It represents the correlation of f and translated f.

Example. Intensity of a spherical particle or a molecule at very low resolution. Let one consider scattering from an uniform sphere with the electron density ρ and the radius a. The scattering amplitude is

F(q) = ∫0a 4 π2 ρ sin qr dr.

This integral evaluates to

F(q) = 4/3 π a3 ρ 3 (sin x –x cos x)/x3, where x = qa.

This assumption can be used at low q-values (SAXS).

Further information

Jan Skov Pedersen, Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least
squares fitting. Advances in Colloid and Interface Science 70, 1997, 171-210.

Fitting small-angle scattering curves SASfit  http://kur.web.psi.ch/sans1/SANSSoft/sasfit.html

EMBL saxs group Solution saxs data analysis programs

The amplitude of a molecule

The scattering amplitude of a molecule  

F = ∑n fn(q) exp(iq·rn),  

where rn and fn(q) are the position and scattering factor of the atom n, respectively.
The intensity is FF*.

Amplitude of a crystal

In the case of crystal, it is convenient to write the electron density as

 ρ(x) = ∑lj δ(x-Rl -rj).

where Rl refer to the lattice points and rj to the positions of atoms in the unit cell.

The scattering amplitude is

 F(q) = Σm fm(q) exp(i q·rm) Σn exp(i q·Rn)

where the 1. term is the unit cell structure factor and 2. term is the lattice sum.

Diffraction maxima are obtained when

q·Rn = 2π · integer.

Debye formula

Consider the system of N scatterers, which may be atoms or larger spherically symmetric particles. Let the positions of the scatterers be rn. The intensity is proportional to

    I(q) = <| Σ fn exp(iq·rn)|2 > = < ΣΣ fnfm* exp(iq·rnm)>,

where

rnm = rn - rm 

is the relative position of the scattering centres n and m. The brackets <> mean the thermal average of the positions. Here fm is the scattering lenght for scatterer m. If atoms, fm  is atomic scattering factor.

Elastic scattering of monatomic system

Let the number of atoms be N. The intensity may be written as

   I(q) = f2 < Σn≠m exp(iq·rn)>+ N f2  

Elastic scattering of a unit of several kinds of scattering centers

Assuming that the material is isotropic and by averaging over all directions the famous Debye formula 

 I(q) ≈ ΣΣ fi fj* sin(q rij)/(q rij),

where  fi is the atomic scattering factor, rij  is the distance of atoms i and j and  q is the magnitude of the scattering vector.

The function

   S(q) = (1/N) < Σn,m exp(iq·rn) > = H(q) + 1

is called the static structure factor. S(q) is dimensionless and approaches 1 when q approaches infinity.

The pair distribution function g(r) is proportional to the probability of finding an atom at a position r relative to a reference atom taken to be at the origin. it is obtained from the total structure factor S by

   S(q) - 1 = ρ0 ∫ [g(r) - 1] exp(iq·r) dr

and

   g(r) - 1 = 1/[ρ0 (2π)3] ∫ [S(q)-1] exp(-iq·r) dq.

Here ρ0 is the atomic number density.

Example. Intensity of a chain of atoms

Example. Intensity if a molecule at very low resolution.

Scattering from an uniform sphere with the electron density ρ and the radius a. The scattering amplitude is

F(q) = ∫0a  4 π2 ρ sin qr dr.

This integral evaluates to

F(q) = 4/3 π a3 ρ 3 (sin x –x cos x)/x3,

where x = qa.  This assumption can be used at low q-values (SAXS).

Intensity of a sphere with R = 100 Å.

Partial structure factor for liquids and amorphous materials

The partial structure factor ρab(r) presents the average atomic density of b-type of atoms at the distance r from an a-type of atom. This function is related to the so called partial structure factor Sab by

Pab(r) = 4π r2 ρab(r) = 4π r2 wb ρ0 + 2r/π ∫ qSab(q) sin qr dr

where ρ0 is the average atomic density.

Intensity in terms of partial structure factors is

I (E,q) = Σ wa f*a(E,q) fa(E,q) +  ΣΣ wa f*a(E,q) fb(E,q) Sab(q)

where wa is atomic fraction of component a, fa(E,q) atomic scattering factor and E is energy.

Total structure factor from x-ray scattering experiment

Coherently scattered intensity I(q) is defined as

S(q) = (I(q) - <f2>) / < f>2,

where <f2> = Σ wn fn2 and <f> = Σ wn fn.

For derivation see for instance C.N.J. Wagner. Direct methods for determination of atomic scale structure of amorphous solids (x-ray, electron and neutron scattering). J. Non-Cryst. Solids (1978) 31, 1-40.



RDF from intensity

The experimental coherently scattered intensity is denoed by I(k). The total structure factor is defined as

S(k) = (I(k) - <f2>)/< f>2,

where <f2> = Σwn fn2 and <f> = Σwn fn

The Fourier sine transform of S(k) is called the difference radial distribution function

dRDF = 2r /π ∫ k S(k) sin kr dk

The (total) radial distribution function is

RDF = 4π r2 ρ + dRDF

The pair correlation function PCF is obtained from RDF as

PCF = RDF / 4π r2 ρ,

where ρ is the atomic density.

Coherently scattered intensity from measured intensity:

The figure above shows that in order to determine the contribution of Compton scattering in the intensity curve reliably the intensity should be measured to at least to q of about 8 1/Å.

Information on crystal structures of materials

Crystallinity

The crystallinity means the volume fraction of crystalline material in a sample. 

The diffraction pattern of a sample is presented as a sum of intensities of crystalline and amophous components. The crystallinity is the integral of the intensiy of  the crystalline component divided by the total intensity.

Ruland Acta Cryst 1961, 14, 1180-1185


Vonk. J. Appl. Cryst. 1973, 6, 148-152
Disorder function is included: D(s) = exp(-kq2)

Vonk. J. Appl. Cryst. 1983, 16, 274-276
Different electron densities of the crystalline and amorphous component

Crystallite size

The width of the diffraction peak gives an estimate for the crystallite size L via Scherrer equation

B(2θ) = 0.94 λ / (L cosθ),

where B is the FWHM of the reflection

More about SAXS 

If the unit cell is large enough, the diffraction peaks appear at SAXS angle range. Then SAXS may be used to determine the crystal structure. This is often the case with polymeric materials.

Example.
Dynamics of structural transformations between the lamellar and inverse bicontinuous cubic lyotropic phases. C.E. Conn (a), O. Ces (a), X. Mulet (a), S. Finet (b), R. Winter (c), J.M. Seddon (a) and R.H. Templer (a) Phys. Rev. Lett., 96, 108102 (2006).
http://www.esrf.eu/UsersAndScience/Publications/Highlights/2006/SCM/SCM1/

SAXS may be used to study both particular and non-particular two-phase systems and solutions as well. A conventional use of SAXS is to determine the shape and size of particles, e.g. proteins in solution. Then the intensity is interpreted in terms of models. The radius of gyration of particles in obtained directly from the shape of the intensity curve (Guinier law).


At large q-values the intensity obeys often a power law:

Particles in solution

A two-phase system

The SAXS intensity follow a power law
     I ≈ 1/qa.
This can be interpreted as arising from fractal structures, if the characteristic length scale R of a fractal satisfies the condition Rq >>1.


e.g. Teixeira. J.Appl. Cryst. 21 1988, 781-785 and SAXS text books

Feigin LA, Svergun DI. Structure analysis by small-angle X-ray and neutron scattering. http://www.embl-hamburg.de/ExternalInfo/Research/Sax/reprints/feigin_svergun_1987.pdf
Glatter O, Kratky O (1982). Small Angle X-ray Scattering. http://physchem.kfunigraz.ac.at/sm/

Example. Starch structure in potato revealed by WAXS and SAXS experiments

WAXS intensity of potato slice. The intensity patterns include contributions of crystalline and amorphous starch and water.

 

Starch crystal structure (output of PowderCell)

SAXS intensity of potato slice during drying. At the beginning of the measurement, the potato sample has a lamellar diffraction maximum and
an approximate power law exponent of  -1.6. After 2 h of drying the lamellar peak collapses fairly rapidly and the intensity of scattering in the
low angles increases, giving an increase in the apparent power law to -3.0. Further drying increases the overall intensity of scattering dramatically,
indicating that structures containing large scattering contrast (presumably voids) emerge in the sample. The apparent power law also increases to -3.8.




T. Väänänen et al. X-Ray scattering study on three different potato cultivars during winter storage. Carbohydrate Polymers, 54 (2003) 499-507

GISAXS

G Renaud, R Lazzari, F leroy. Probing surface and interface morphology with Grazing Incidence Small Angle X-Ray Scattering. Surface Science Reports 64 (2009) 255380
http://staff.chess.cornell.edu/%7Esmilgies/gisaxs/GISAXS.php

Anomalous scattering

Rhe aim of ASAXS experiments is to find the partial structure factors (PSF) of a system with more than one component. A tunable source of radiation is needed, so there experiments are done at synchrotron facilities.

http://hasylab.desy.de/facilities/doris_iii/beamlines/b1/asaxs/index_eng.html

 References