X-ray diffraction

• Meet in the LGRT lab• Again, will hand in worksheet, not a formal lab report

• Revision exercise – hand in by April 17th class.

Diffraction summarized

• The 6 lattice parameters (a,b,c,) of a crystal determine the position of x-ray diffraction peaks.

• The contents of the cell (atom types and positions) determine the relative intensity of the diffraction peaks.

• If a diffraction peak can be identified with a Miller Index, the unit cell on the phase can usually be determined.

Miller indices

• Choose origin• Pick 1st plane away from origin

in +a, +b direction• Find intercepts in fractional

coordinates (“none = ∞”).

• h = 1/(a-intercept)• k = 1/(b-intercept)• l = 1/(c-intercept)

• If no intercept, index = 0

• Plane equation:ha + kb + lc = 1

Miller index, hkl

(2-10)

(110) (100)

(150)

• Same set of planes will be described if all 3 Miller indices are inverted (2-10) ≡ (-210)

• Minus sign → bar 10)2(0)1(2

Crystals facets correspond to Miller indices

• Haüy, 1784

• Crystals (like calcite) are made of miniscule identical subunits

ba

(-120)

Woolfson

• Facets can be described by low-order Miller indices

(-120)

(-110)(-100)

(-1-10)

(-2-10)

(3-40)

(340)

Miller indices

• For a lattice of known dimensions, the Miller indices can be used to calculate the d-spacing between hkl planes (a,b,c = lattice parameters).

• This d-spacing will determine where powder diffraction peaks are observed

Miller index, hkl

(2-10)

(-110) (100)

(150)2

2

2

2

2

2

2

1cl

bk

ah

d

Origin at orange dot

Distance between planes (Miller indices)

Bragg’s law derivation(angle of incidence = angle of diffraction)

• x = d sin • extra distance = 2d sin = n• n = 2d sin

d

x

d

x x

a b

= 2d sin (for x-ray diffraction)

Powder diffractometer

KI (CsCl-type) x-ray pattern

Crystals, diffraction, and Miller indices

(100)

(010)

(001)

(0-10)

Coherent scattering from a row of atoms

Will only happen when emission from all atoms is simultaneously stimulated.(Solid lines represent spatial regions where phase = 0 deg).

a

Laue condition – vector description• Extra distance = a cos - a cos • Extra distance = -(a·S0) + (a·S)• Extra distance = (a·s)

a

S

S0

a

• Will have diffraction when: a cos - a cos = h (h = integer)• Will have diffraction when: a·s = h (h = integer)

s = (S – S0)/

v1

v2

cos = v1·v2

a

Laue condition – 2D• Must have coherent scattering from ALL ATOMS in the lattice, no just from one row.• If color indicates phase of radiation scattered from each lattice point when observed at a distant site P, we see

that scattering from rows is in phase while columns is out of phase, making net scattering from all 35 points incoherent and therefore NOT observable

S

S0

b

• Will have diffraction when: a·s = h (h = integer)• Will have diffraction when: b·s = k (k = integer)• Will have diffraction when: c·s = l (l = integer)

Every lattice point related by translational symmetry will scattering in phase when conditions are met

Single crystal diffraction

• Used to solve molecular structure– Co(MIMT)2(NO3)2 example

• Data in simple format – hkl labels + intensity + error

0 0 1 0.00 0.10 0 0 2 42.60 1.40 0 0 3 1.10 0.30 0 0 4 100.30 2.50 0 0 5 -0.30 0.50 0 0 6 822.30 16.70 0 0 7 -0.40 0.50 0 0 8 656.40 13.90 0 0 9 1.00 0.80 0 0 10 73.40 3.00 0 0 11 0.00 1.40 0 0 12 4.70 1.60 0 0 13 1.00 1.70 0 2 1 611.40 14.40 0 -2 -1 613.90 12.10 0 2 2 443.90 8.90 0 -2 -2 443.00 8.70 0 2 3 59.90 1.50 0 -2 -3 56.90 1.50 0 2 4 55.20 1.60 0 -2 -4 51.80 1.50

h=2

h=1

h=0

h=-1

l=6l=7l=8

h=2

l=9l=10

3D view of reciprocal lattice

Massa

(Lines connecting dots are unecessary)

Relation between direct and reciprocal cells

Stout and Jensen

Direct vs. reciprocal lattice

direct or real

reciprocal

Lattice vectors a, b, c

Vector to a lattice point: d = ua + vb + wc

Lattice planes (hkl)

Lattice vectors a*, b*, c*

Vector to a reciprocal lattice point: d* = ha* + kb* + lc*

Each such vector is normal to the real space plane (hkl)

Length of each vector d* = 1/d-spacing (distance between hkl planes)

Hexagonal lattice

changes from 120 to 60 deg

ab

b*a*

direct or real reciprocal

Distances between planes (d*)

direct or real

reciprocal

Vector to a reciprocal lattice point: d* = ha* + kb* + lc*

Length of each vector d* = 1/d-spacing (distance between hkl planes)

|d*| = (d*·d*)1/2

=(ha* + kb* + lc*) (ha* + kb* + lc*)

=(ha*)2 + (kb*)2 + (lc*)2 + 2(ha*) (kb*) + 2(ha*) (lc*) + 2(kb*) (lc*)

= h2a2* + k2b*2 + l2c*2 + 2klb*c*cos* + 2lhc*a*cos* + 2hka*cos*

• Density = mass / volume

• Mass of unit cell = (# Ca)(mCa) + (# Ru)(mRu) + (# O)(mO)

• Volume of unit cell = abc = (3.7950 Å)3

• mamuNA = mgrams

Density of CaRuO3

VNnMV

NnM

AA

/

• M = molar mass (g/mol)

• n = # f.u. per cell• NA = Avagadro’s #

(1/mol)• V = cell volume