Factors that affect the diffracted intensity structure factor polarization factor Lorentz factor multiplicity factor temperature factor absorption factor

Single electron scattering The term is called the polarization factor

Single electron scattering The magnitude of the single electron scattering is very small Since there are ~1023 atoms/cm3, and even more electrons, the small magnitude of electron scattering can be detected (e2/mc) is known as the classical electron radius

The Compton effect Compton scattering occurs when an X-ray photon suffers an inelastic collision with a loosely-bound electron scattered X-ray photon hn1 hn2 mc2(g-1) q f incident X-ray photon relativistic recoil electron where

Scattering by an atom An atom with atomic number Z is surrounded with Z electrons, all of which can scatter X-rays Since an atom has a finite size, there will be a difference in the path traveled (i.e. a phase difference) in the X-rays scattered from different locations around the atom

Scattering by an atom The atomic scattering factor f is defined as: amplitude scattered by Z electrons in an atom amplitude scattered by a single electron f = The atomic scattering factor f = Z for any atom in the forward direction (2q = 0) Þ I(2q=0) =Z2 As q increases f decreases  functional dependence of the decrease depends on the details of the distribution of electrons around an atom (sometimes called the form factor) f is calculated from quantum mechanics

Scattering by an atom note that f is plotted as a function of sinq/l so that the same data can be used for different radiations

Scattering by a unit cell scattering by an single electron is modified by scattering by an collection of electrons is modified by scattering by an collection of atoms The structure factor is the amplitude scattered per unit structure, or the amplitude scattered by a unit cell

The structure factor amplitude scattered by all atoms in a unit cell amplitude scattered by a single electron Fhkl = The structure factor contains the information regarding the types (f) and locations (u, v, w) of atoms within a unit cell The intensity of a Bragg reflection is then proportional to the square of the structure factor: F may be complex, but “I” must be real!

The structure factor and complex numbers Fhkl can be (and often is) a complex number It is very useful to be able to do some mathematical manipulation of complex numbers Hint: don’t try to use the trigonometric form – using exponentials is much easier!

A simple example – primitive unit cell with one atom Let the atom be placed at 000 (remember that we can define the unit cell any way we want) In this case the structure factor is independent of h, k and l; it will decrease with f as sinq/l increases (higher-order reflections)

Base centered unit cell The bcc crystal structure has atoms at 000 and ½½0: the atoms are the same, so the atomic scattering factor can be factored out •If h and k are both odd or even, then F=2f and F2=4f2 • If h and k are mixed, F=0 and F2=0. • Example: – 111, 112, 113, and 021, 022, 023 have the same reflection – 011, 012, 013 and 101, 102, 103 have zero reflections

Body centered cubic unit cell The bcc crystal structure has atoms at 000 and ½½½: the atoms are the same, so the atomic scattering factor can be factored out So Fhkl = 2f for h+k+l = an even integer 0 for h+k+l = an odd integer (200), (400), (220)   (100), (111), (300)   “forbidden” reflections

CsCl unit cell The CsCl crystal structure is primitive cubic with a two-atom basis: Cs at 000 and Cl at ½½½: atoms are different, so we cannot factor out the atomic scattering factor fCs + fCl for h+k+l = an even integer fCs - fCl for h+k+l = an odd integer So Fhkl = (200), (400), (220)   (100), (111), (300)   if Cs and Cl were the same, this result would be the same as BCC

Face centered cubic unit cell The fcc crystal structure has atoms at 000, ½½0, ½0½ and 0½½: If h, k and l are all even or all odd numbers (“unmixed”), then the exponential terms all equal to +1  F = 4f If h, k and l are mixed even and odd, then two of the exponential terms will equal -1 while one will equal +1  F = 0 16f 2, h, k and l unmixed even and odd 0, h, k and l mixed even and odd

Diamond cubic unit cell The diamond cubic structure is fcc with a two atom basis (000 and ¼¼¼) Eight atoms per unit cell: 000, ½½0, ½0½, 0½½ ¼¼¼, ¾¾¼, ¼¾¾, ¾¼¾ The structure factor can be written: This could get very messy; however, there is a trick…

“Factoring the structure factor” We can re-write the structure factor as following [fcc term] the two-atom basis modulates the fcc structure factor, but if the fcc term is zero, Fhkl is zero no matter what the two-atom term is! 4f for h, k and l unmixed 0 for h, k and l mixed

Dealing with a complex structure factor Now we multiply by the fcc term (42 =16 for hkl unmixed and zero for hkl mixed even and odd) and we have the following conditions on h, k and l when (h + k + l) is a even multiple of 2 when (h + k + l) is an odd multiple of 2 when (h + k + l) is odd h, k and l mixed

Dealing with a complex structure factor Neglecting the fcc term the diamond cubic structure factor is The exponential term is complex, but remember that we’re interested in FF*, so we multiply by the complex conjugate:

Multiple atom unit cells -- NaCl NaCl is fcc with a two atom basis: Na+ at 000 ½½0 ½0½ 0½½ fcc translations Cl- at ½½½ 00½ 0½0 00½ fcc translations We can immediately factor out the common fcc term: so the structure factor becomes: h, k and l mixed h, k and l unmixed (h + k + l) even (h + k + l) odd

The structure factor for the hexagonal close-packed structure The hcp structure is primitive hexagonal with a two-atom basis: 000 and 1/3 2/3 1/2 :

The structure factor for the hexagonal close-packed structure h + 2k l 3m odd 0 3m even 4f 2 3m1 odd 3f 2 3m1 even f 2

The multiplicity factor The multiplicity factor arises from the fact that in general there will be several sets of hkl-planes having different orientations in a crystal but with the same d and F2 values Evaluated by finding the number of variations in position and sign in h, k and l and have planes with the same d and F2 The value depends on hkl and crystal symmetry For the highest cubic symmetry we have: p100 = 6 p110 = 12 p111=8

The Lorentz factor

The Lorentz factor In a polycrystalline sample, the diffracted intensity is spread over a cone with a base radius of Rsin2qB so the relative intensity per unit length of line is proportional to 1/sin2qB

The Lorentz-polarization factor The combination of geometric corrections are lumped together into a single Lorentz-polarization (LP) factor: The effect of the LP factor is to decrease the intensity at intermediate angles and increase the intensity in the forward and backwards directions

Thermal Motion

The temperature factor As atoms vibrate about their equilibrium positions in a crystal, the electron density is spread out over a larger volume This causes the atomic scattering factor to decrease with sinq/l (or |S| = 4psinq/l )more rapidly than it would normally:

The temperature factor The temperature factor is given by: where the thermal factor B is related to the mean square displacement of the atomic vibration: This is incorporated into the atomic scattering factor:

The absorption factor Angle-dependent absorption within the sample itself will modify the observed intensity Absorption corrections are complicated in the Debye-Scherrer geometry tables of absorption correction factors are given for different geometries in the International Tables for Crystallography

The absorption factor For a flat diffractometer sample in a symmetric geometry, the absorption factor is independent of angle Let A be the cross-section area of the incident beam; the intensity diffracted at the Bragg angle by an infinitesimal thickness dx at depth x:  then integrate from x = 0:

Expressions for the integrated intensity in a powder pattern The relative integrated intensities in the lines in an X-ray powder diffraction experiment are given by: approximate (but suitable for this class) exact, for diffractometer exact, for Debye- Scherrer camera

Complicating factors in intensity measurements Preferred orientation we explicitly assume that the polycrystalline sample contains randomly oriented grains the presence of preferred orientation can cause enormous headaches careful sample preparation is a must! Extinction the simple theory assumes the sample is “ideally imperfect”: small crystals, single diffraction events large perfect crystals (“ideally perfect”) diffract more weakly than ideally-imperfect crystals extinction can be minimized by grinding, liquid N2, etc.