X-ray crystallography

Challenges for SCXRC (3)

• needs x-ray suitable crystals

• Only solid-state structure

• time-averaged structure may change during scanning

TRUE OR FALSE: X-ray crystallography cannot be applied to solids which are very fine or large crystals

FALSE: Powder X-ray diffraction

What are the uses of powder crystallography? (3)

• assess physical properties (e.g., phase transitions)

• analysis of mixtures (eg. impurities, components)

• grain size estimate

set up of SCXRC

In Å, what is the length of a) X-ray and b) atomic bond?

a) 0.5-2

b) 1-3

Diffraction

deflection of rays passing through narrow slits or next to opaque edges

the (electrons/neutrons/protons) of atoms (absorb/diffract) incident X-rays with an intensity (proportional/unrelated) to the electron density

the (electrons/neutrons/protons) of atoms (absorb/diffract) incident X-rays with an intensity (proportional/unrelated) to the electron density

Bragg’s Law: a) what it measures and b) the equation

a) if the incident and reflected d_hklsinθ is equal, there will be constructive interference and therefore diffraction

b) λ = 2d_hkl sinθ

where:

• λ= wavelength of incident radiation,

• d= spacing between the lattice planes

• θ= angle to the incident (and reflected) beam

Unit cell (definition)

The repeating structural motif in a crystal is the unit cell that gives a diffraction pattern with sharp peaks

Dimensions of a unit cell

axes lengths: a, b, c

angles: α, β, γ

TRUE OR FALSE: The unit cell describes the entire contents of the crystal by translational and rotational symmetry

FALSE: by translational symmetry only!

Name the Seven Crystal Systems and describe them in terms of axes lengths, angles, and minimum symmetry

CHaRred TOMaTo

The unit cell from which the lattice symmetry is constructed is…

the smallest unit cell with the highest symmetry

2D Bravais Lattice

an infinite array of points that looks the same when viewed from any point

What are the possible unit cell structures for 2D Bravais lattice?

What is the justification for the existence of a rectangular centered lattice?

The molecule at the center provides greater symmetry than the oblique lattice would

What are the centering types for Bravais lattice?

• Primitive (P), no centering

• Base Centered (C)

• Body Centered (I)

• Face Centered (F)

“C” is like “F” but on two opposing faces

What centering(s) is available for all 7 Bravais lattices?

• Primitive (P), no centering

• Base Centered (C)

• Body Centered (I)

• Face Centered (F)

• cubic: P, I, F

• hexagonal: P

• rhombohedral: P

• tetragonal: P, I

• orthorhombic: P, C, I, F

• monoclinic: P, C

• triclinic: P

Why is C centering not possible for a) cubic or b) tetragonal lattice?

a + b) a smaller tetragonal primitive lattice is created

What is the best unit cell for the below crystal and why?

The purple rhombus because of the planes of symmetry

What are the two types of symmetry elements and how are they differentiated?

point and translational; in point symmetry elements one point is always unchanged

What symmetry element(s) cannot exist in crystals of enantiomerically pure substances?

mirror plane ∴ rotation-inversion, glide plane, inversion center

point symmetry elements (4)

rotation axes (n)

mirror plane (m)

inversion centre (1‾)

Rotation Inversion centres

Rotation Inversion centres

Rotation followed by inversion

Screw axes

translates an object by a unit cell dimension multiplied by n/C along the direction of the axis, followed by a C-fold rotation

Symbol: C_n

imagine like a balloon deflating

Glide plane (+ types)

a reflection-translation operation:

Axial: a, b, c

Diagonal: n

diamond: d

What symmetry operation(s) result in systematic absence?

screw axis

glide plane

Symbol for a) rotation axes, b) mirror plane, c) inversion center, d) rotation inversion center.

C_n

m

1̅

n̅

Only C(__) axes allow space filling

Only C_2/3/4/6 axes allow space filling

Assign Miller indices

1. identify the origin; the plane should not pass through the origin

2. if the plane intersects the axes defined from the origin it has a value of 1

3. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero.

4. Take the reciprocals, clear fractions and reduce to lowest terms

How is the real crystal lattice translated to the reciprocal lattice?

Fourier transform:

TRUE OR FALSE: The reciprocal lattice has the same symmetry as the crystal lattice.

TRUE

distance of reciprocal lattice points from the origin

d* = h·a* + k·b* + l·c*

The direction of a*, b* and c* of the reciprocal lattice coincide with a, b, and c of the crystal lattice if…

α = β = γ = 90^o

structure factor of X-ray reflection: F(hkl) = |F(hkl)|· exp[iΦ(hkl)]

where:

• |F| is amplitude

• Φ is the phase/angle

What magnitude of temperature is used for crystallography and why?

Low to avoid smearing of diffraction spots due to thermal vibration

What are the wavelength features of the x-rays in SCXRC?

single λ Κα line radiation

What elements are used for the X-ray tubes giving the radiation for a) small and b) large unit cells?

a) Mo (small λ)

b) Cu or Cr (larger λ)

Do the X-ray tubes giving the radiation for small or large unit cells give a more easily interpretable result and why?

small because shorter λ of Mo is higher energy therefore less radiation is able to be absorbed by the crystal, so less corrections are needed

What are the features of an ideal x-ray crystal? (3)

even dimensions

single crystal

50-200 microns (size of beam)

General process of data collection in SCXRC (5 steps)

1. select crystal

2. mount under cold atmosphere

3. collect pre-experiment to plan data collection

4. check for diffraction spots that are spherical, clear and distinct

5. collect complete dataset

TRUE OR FALSE: more symmetrical space groups need less frames to reconstruct from

TRUE

Phase problem

In X-ray diffraction experiments, we collect only the diffraction magnitudes, and not the phases. Unfortunately the phases contain the bulk of the structural information

What conditions must the electron density map display for accurate atoms? (3)

electron density should be positive everywhere

electron density should be concentrated spherically around atoms

atoms must be placed feasible distances apart

What are the four methods available for solving crystallography data? Give a short description of each.

• direct method: manually assign atoms based on known information/ electron density

• using models for existing isomorphous structures to refine data

• Patterson synthesis: each diffracted atom gives an equal and opposite sign vector to the others, giving overall relative position of the atoms to each-other

• Intrinsic phasing: Solves structure in lowest symmetry (P1) and then determines the space group

In Patterson maps, each peak has a size proportional to…

atomic mass

Problems with Patterson Syntheses (3)

• Many possible solutions due to the periodic nature of a crystal, especially if there is more than one heavy metal

• Problems when atoms are close to special positions of a unit cell.

• Pseudo (false extra)-symmetry

a) From what values are Fourier difference maps created and b) what are they for?

a) |F_obs| - |F_calc|

b) finding missing/ incorrect atoms

Electron Density Maps for a) F_obs and b) F_calc are constructed from what?

a) experimental |F| together with phases calculated from some structural model

b) phases and amplitudes that are calculated from a structural model

What do a) Q peaks, b) troughs, c) large R values mean?

a) missing atoms

b) incorrectly assigned atoms

c) bad fit between data and model

What are the ideal values for a) R₁, b) wR₁, c) GooF?

a) <5%

b) <10%

c) 0.8 to 1.2

What is the difference between a) constraints and b) restraints?

a) exact mathematical relationship to ensure not all parameters are independently/freely refined

b) approximate targets (like additional experimental observations)

What are some shortcomings of XRC? (4)

• time averaged structure as opposed to snapshot

• structure may change in different conditions/ states

• single crystals may not be representative of bulk sample

• lightweight atoms cannot be seen

TRUE OR FALSE: racemic mixture unit cells contain both enantiomers within a single crystal

TRUE, BUT the enantiomers can crystallize into separate crystals too, although this is rarer

Space groups with only rotation and screw axes are (__) molecules

Space groups with only rotation and screw axes are chiral molecules

In PXRC, what parameters affect peak a) position (3), b) intensity (2), c) shape (4)?

a) unit cell dimensions, absorption, wavelength

b) atomic dimensions, preferred orientation of crystal

c) grain size, strain, stress, purity

What does the X-axis of a PXRD diagram show?

d-spacing in θ or 2θ

What problem was encountered in scanning this sample?

Nothing, its just powder diffraction method!

What are the Miller indices of this unit cell?

(101^-)

orthorombic d spacing formula

Indexing PXRD Patterns (5 steps)

1. find the 2θs from x-axis

2. calculate sin²θ. If you have 2θ, divide it.

3. calculate ratio = sin²θ_n /sin²θ₁

4. IF NEEDED multiply ratios to get valid numbers

5. assign Miller indices that (squared individually) add up to ratio (h² + k² + l²)

When do ratios need to be modified in indexing?

• if the values are not very close to integers

• if the values cannot be given by three squared values (eg. 7, 15, 23)

Ascertaining lattice parameter formula

4a² = λ² (h² + k² + l²)_n / sin²θ_n

1. divide by 4

2. square root

3. a = d in Angstrom

systematic absence

the intensity of the reflection from a set of Miller planes is zero

TRUE OR FALSE: systematic absences occur for all space groups except body centered

FALSE: observed for non-primitive lattices

For a systematic absence to be observed, (fewer/equal/greater) number of diffracted beams must be out of phase by (__)

For a systematic absence to be observed, equal numbers of diffracted beams must be out of phase by 𝜆/2

What indicates the presence of systematic absences for a) Face centered and b) Body centered cubics?

a) h+k+l is a multiple of 2

b) h, k & l are either all odd or even

What contributes to peak intensity in PXRC? (5)

• higher mass atoms are more likely to appear (atom identity)

• coordinates of atom in unit cell

• symmetry of space group

• if the crystal is in preferred orientation

• how many diffraction planes contribute to a peak

How does a) symmetry of space group and b) if the crystal is in preferred orientation, c) number of diffraction planes affect peak intensity for powders?

a) higher symmetry makes fewer peaks

b) preferred orientation gives greater peaks

c) more planes means sharp, more defined peaks

Peak width can be affected by…? (4)

sample size

instrumental parameters

stacking faults

defects in the crystal structure.

Scherrer Equation to estimate crystal size from analysis of the peak width

β_hkl = Kλ / L_hklcosθ 

• K: constant based on crystal shape

• L: width of diffraction peak

• β_hkl: vertical thickness of crystal face