X-ray Diffraction (Final Exam)

Miller indices notation for points

no brackets, yes commas, negatives allowed, fractions allowed

ex: 1,1,1 or 1,-1,1/2

Wat is the only miller indices type that is allowed to have fractions?

points

Do points have brackets in miller indices?

no

what is the only type of miller indices that allows negative numbers?

points

What type of miller indices use a bar to represent negatives?

directions, families of directions, planes and families of planes

Directions in miller indices

ex: [1 1 1]

no commas, yes brackets, negatives as a bar. no fractions

Families of directions in miller indices

<1 1 1>

same as directions, except different brackets

no commas, negatives as a bar, no fractions

planes (miller indices)

( 1 1 1 )

no commas, negative as a bar, no fractions

families of planes (miller indices)

{1 1 1}

no commas, negative as a bar, no fractions

what is the only type of miller indices that uses commas?

points

what if a plane intersects at the origin? (miller indices)

either shift the plane or shift the origin

what if a plane doesn't intersect an axis? (miller indices)

1. Write out intersection points

2. take the reciprocals

3. clear fractions (multiply by a factor)

4. format final answer

families of directions

symmetrically equivalent directions

What do families of directions include?

all changes in indices and all negatives

What do families of direction in a non-cubic unit cell look like?

take out any combination of indices that aren't equal (ex: tetragonal a=b but not c)

directions (miller indices)

labeled as a vector from the ending point to start

what if you move by a fractional length (directions, miller indices)

multiply by a factor

What if the origin of the vector is not placed at the origin of the unit cell? (directions, miller indices)

either:

1. create the vector by going from the end to start

2. shift the vector

3. shift the origin

diffraction

waves spread through or around openings or barriers if these openings/barriers are the same order magnitude as the wavelength of the wave

types of wave interactions

constructive and destructive interference

constructive interference

The interference that occurs when two waves combine to make a wave with a larger amplitude

are waves in constructive interference in phase or out of phase?

in phase

in a double slit diffraction experiment, does constructive interference appear dark or light?

light fringes

destructive interference

The interference that occurs when two waves combine to make a wave with a smaller amplitude (or a flat wave)

are waves in destructive interference in or out of phase?

out of phase

in a double slit diffraction experiment, does destructive interference appear dark or light?

dark fringes

double slit diffraction

2 wavefronts created at the slits interfere constructively and destructively to make a series of bright and dark fringes.

What changes the distance between light and dark fringes in a diffraction pattern?

line density

distance between the grating and wall

wavelength

lattice structure

Equation for multiple slit diffraction

nλ/d=y/D

n=order

d= spacing between slits

y= distances between fringes

D= distance from grating to screen

How does diffraction and crystal structure relate?

crystal lattice acts as the diffraction grating- the spaces between atoms are "slits" in the "diffraction grating"

What impacts the appearance of a diffraction pattern

structure of the material, wavelength, and distance to the diffraction pattern

why do we use x-ray in diffraction for crystal structures

x-ray wavelength is on the same order of magnitude as spacing between atoms in a crystal structure, whereas visible light is not (this is different from a diffraction grating experiment)

methods for diffraction of crystal structures (xrd)

laue and powder

laue method (xrd)

single crystal, gives a diffraction pattern

powder method (xrd)

many little crystals, sample is powdered, gives a diffraction spectrum

X-ray diffraction process

1. an arm with an x-ray rotates around the sample

2. the detector collects a signal from diffracted beam at every angle

3. signal appears when diffracted x-rays interfere constructively

How does constructive interference appear on an XRD spectrum?

large peaks or spikes, meaning that Bragg conditions are met

How does destructive interference appear on an XRD spectrum?

fuzzy, unclear small peaks (Bragg condition not met)

Bragg's law

describes which angles will result in constructive interference

Bragg's law equation

nλ=2dsinθ

n= order

λ=wavelength

d= spacing between gratings

θ= diffraction angle

What info does an XRD spectrum provide?

lattice parameter (calculate)

crystal structure (calculate)

types of atoms (use database)

Lattice parameter

separation between atoms

braggs law rearrganged for hkl

dhkl=nλ/2sinθ

most XRD spectrums use θ or 2θ?

2θ (so don't forget to divide in Bragg's law)

How to find lattice parameter (a)

dhkl=a/√(h)²+(k)²+(l)²

for cubic systems (FCC, BCC, SC)

when you increase θ in bragg's law

↑sinθ, ↓dhkl

in 2nd order diffraction, n=

2

in the direct beam of diffraction, n=

0 (0th order)

on an XRD spectrum, as 2θ (x-axis) increases,

h²+k²+l² increases

what is the y-axis label on an XRD spectrum

intensity

how does grain structure relate to Bragg's law?

size of the beam (spot size) is ≥ 2 orders of magnitude great than the size of crystallites

structure factor

presence or absence of peaks in XRD

what does strucutre factor represent

the relative positions of atoms

how many atoms are in the unit cell

types of atoms in the unit cell

how does strucutre factor consider types of atoms

form factor f

how does structure factor consider how many atoms are in a unit cell?

sum all atoms

how does structure factor consider where atoms are located?

x,y,z coordinates

form factor

scattering power of an atom

phase difference

2π(hx+ky+lz)

structure factor equation

Fhkl=fe^((2πi)(hx+ky+lz))

see also sum of all atoms version (can't write in quizlet, but its a summation over N cells)

equation for a wave (used in structure factor with amplitude f)

Ae^i(phi)

Simple cubic geometry

1 atom per unit cell

body centered cubic geometry (bcc)

2 atoms per unit cell

BCC selection rules (structure factor)

if h+k+l= even, F=2f

If h+k+l is odd, F=0 (no peak)

Face Centered cubic (FCC)

4 atoms per unit cell

FCC selection rules (structure factor)

h,k,l all odd or all even, F=4f

h,k,l mixed odd/even, F=0

Strucuture factor useful equations

e^evenπi= 1

e^oddπi= -1

what factors impact peak intensity

form factor f

experimental geometry

absorption

temperature

multiplicity

intensity equation

Ihkl=|Fhkl|²×Phkl×Lp(θ)×A(θ)e^(-2m)

what happens to intensity when 2θ increases

intensity decreases

Intensity and structure factor relation

i(intensity)α|Fhkl|²

When scattering in the forward direction (single atom)

no destructive interference (no path length difference)

total intensity= 2x the intensity from a single electron

when scattering in any other direction (θ) but forward

destructive interference

total intensity< total intensity in forward direction

form factor relation to amplitude

f(sinθ/λ)α amplitude scatter by entire atom/ amplitude scattered by a single atom

form factor is a function of

sinθ/λ

Lorentz polarization factor

instrument geometry (no relation to material itself)

Lorentz polarization factor equation

Lpθ=(1+cos²2θ)/(sin²θcosθ)

absorption in XRD

x-rays get absorbed when they go through a material

absorption relation to intensity

need to multiply the intensity of scattered x-rays by absorption factor A

absorption equation

A=1/2µ

Intensity equation with linear absorption coeff

I=Ioe^(µ(x))

µ(x)= linear absorption coefficient

Io= initial intensity

i= intensity

when an atom at high temp vibrates,

the electron cloud becomes more diffuse

why is temperature related to intensity

form factor of some temperature multiplied by a damping factor

Form factor of some temp

Ft=foe^(-BTs²)

B= Debye-Waller factor (typically listed)

T= temp

s= sinθ/λ

fo= form factor at absolute zero

every atom scatters ___ strongly than it would at absolute zero

less

can polymers be at all crystalline?

yes, a certain degree

Can XRD still give info even if something is not crystalline

yes

how do crystalline peaks appear in XRD

sharp and clear

do crystalline peaks have long or short range order?

long range order

do amorphous peaks show up broad or sharp?

broad

do amorphous humps have long or short range order

short range order

long range order

describes the existence of a regular repeating arrangement of atoms, ions, or molecules within a crystalline region of a material.

short range order

The local arrangement of nearest-neighbor atoms or ions around a centrally located atom or ion.

percent crystallinity

Area of the crystalline peaks ÷ total area

what info does polymer XRD give (little to no crystallinity)

percent crystallinity

average d-spacing

average d-spacing in a spectrum

distance from to the top of a peak (2theta vs intensity)

How to do XRD in biological samples

1. Crystallize the sample

2. collect diffraction pattern

3. use FTIR to convert the diffraction pattern into an electron density map

4. create an atomic model

Crystallizing a biological sample for XRD

sample is suspended in small cryoloop and frozen in liquid nitrogen, often takes many attempts

drop diffusion

a technique for crystallizing in a biological sample, put a drop of protein solution on a glass slide over a well with precipitation

collecting a diffraction pattern for biological samples XRD

use a high intensity x-ray, and then cryoloop is rotated 360 degrees and diffraction pattern is recorded at every angle

why do we need a high intensity x-ray to analyze biological samples with XRD?

because proteins are light and scatter weakly