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PPT – Unit 2: Part 2: Characterizing Arrangement of Atoms in a Nanostructure: Wide-angle x-ray diffraction Dr. Brian Grady-Lecturer bpgrady@ou.edu Read Material on Internet Relevant to X-ray Diffraction (Callister chapter) Read material on x-ray radiation PowerPoint presentation | free to download - id: 2164c4-ZDc1Z

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Unit 2 Part 2 Characterizing Arrangement of

Atoms in a Nanostructure Wide-angle x-ray

diffractionDr. Brian Grady-Lecturerbpgrady_at_ou.e

duRead Material on Internet Relevant to X-ray

Diffraction (Callister chapter)Read material on

x-ray radiation safety at http//ehs.uky.edu/class

es/analytical/anntrain.html

Outline (Blue already done, orange this lecture,

black will be done next lecture)

- Discrete Nanostructures
- Size, shape
- Size from microscopy
- Size from light scattering
- What type of atoms
- ICP/AAS for bulk concentration
- XPS for surface concentration
- Arrangement of Atoms
- Bonds-Raman,IR, NMR
- X-ray diffraction

What happens to x-rays when they hit a sample?

- Reflected (almost never)
- Pass through
- Absorbed
- Scattered
- Elastically (wavelength inwavelength out)
- Inelastically (wavelength in ? wavelength out)
- Development is for elastically scattered x-rays

Wide Angle X-ray Diffraction

- Wide-angle x-ray diffraction (WAXD), also will

see wide-angle x-ray scattering (WAXS) or simply

x-ray diffraction (XRD). These three

terminologies are equivalent, I will use all

three so you get used to seeing the three to mean

the same thing. - There is scattering at very small angles, termed

small-angle x-ray scattering (SAXS) however SAXS

is pretty specialized and many people only aware

of x-ray diffraction - You will be doing WAXS experiments on your

nanoparticles

WAXS What it measures

- Measure intensity as a function of scattering

angle (scattering angle2q) - Angles 5-120o

X-ray from source

2q

sample

detector

- Three things
- Peak intensity
- Peak position
- Peak width

Sample Geometry

2q2

Looking at area inside dotted box

2q1

2q

Normal Transmission

Pathlength of scattered beam through sample

depends on scattering angle (the higher the

scattering angle, i.e. 2q2, the longer the

pathlength)

q

q

q

2q

2q

2q

Symmetric Transmission

2q?20o

2q?45o

2q?90o

Pathlength of scattered beam through sample does

not depend on scattering angle

q

q

Reflection

q

q

q

q

2q?20o

2q?150o

2q?90o

What geometry used is determined by what

absorption is

- In general for every x-ray that is scattered,

about 106 are absorbed - With IR transmission, the sample must be

transparent to IR light (most light must go

through). There are ways around this if

necessary (diffuse reflectance, others) - With Raman, you typically work at 90o so you are

working in reflection (absorption not matter) - With XRD, if sample absorbs strongly then work in

reflectance, otherwise in symmetric transmission

Geometry

- In reflection
- Your sample must absorb all x-rays quickly (say

within 1-2 mm) or get distortions - No background subtraction necessary, no sample in

beam means practically no signal - Cant get to low angles because hard to align

sample - Must use this geometry for high angles

irrespective of absorption characteristics of

sample - In transmission
- Must subtract out background (spectra with no

sample in beam) - Optimal value of thickness for a given scattering

angle is where 36 of the x-rays are transmitted.

However, better to be too thin than too thick - Cant get to low angles because scattering near

main beam is too high

Calculation of Absorption Coefficient (m)

- Based on the atoms in a material (need atomic

compositions, i.e 20 C, 80 H etc.) - Higher the atomic number, the higher the

absorption - The higher the energy (lower wavelength) the

lower the absorption - Most x-ray scattering equipment have wavelength

1.54 A8038 eV. - Transmittancee-mt tthickness
- Optimal thickness for transmission, mt1.
- Program that does calculation, available on the

Internet

Peak Position-What it Means

Amorphous vs. Crystalline

- Amorphous means that individual atoms have no

long-range pattern - Long-range means 1-100 nm
- Examples of amorphous materials Liquids,

polymers, glass - Crystalline means that individual atoms have

long-range order - If you know where one atom is in space, then you

know where all other atoms are in space for a

long distance - Nearly all solids (other than polymers and glass)

are crystalline. - Can have materials that have both amorphous parts

and crystalline parts (termed semi-crystalline).

Some polymers - Liquid crystals know where atoms are in one

dimension or in two dimensions, but not all three - XRD Mostly useful for crystalline (or

semi-crystalline) materials

Unit Cells

- Can always draw a geometric shape around groups

of atoms and that geometric shape will repeat

over and over - Atoms will always be at the same place within the

geometric shape - There are only 7 of these geometric shapes

possible that are unique - The geometric shapes are cubic-like, so you have

three unit cell lengths a,b,c (corresponding to

x,y,z).

Seven types of Unit Cells (also called crystal

classes)

In hexagonal, g is the angle between the two

sides of equal length (a and b) In monoclinic, g

is the angle between a and c In triclinic, a is

the angle between b and c, g is the angle between

a and b, and b is between a and c.

Rhombohedral (also called trigonal)

Building Unit Cells into structure

Where are atoms in unit cell?

- People use techniques that are complicated to

determine type of unit cell (cubic, orthorhombic

etc), unit cell parameters (i.e. lengths of

sides, angles) and where atoms are in unit cell

from x-ray patterns (however, the x-ray pattern

contains all the necessary information!) - There is a short-hand notation that converts the

type of unit cell and different positions.

Involves symmetry. - See http//cars9.uchicago.edu/ravel/software/doc/

Atoms/Atoms/node17.htmlsecspace-group-symbols-1

for list of the 230 short hand notations.

- Example of Short hand notation for silicon

dioxide - Space group P 32 2 1 a 4.9134 c 5.4052
- Si is at 0.46987 0.0 0.0
- O is at 0.4141 0.2681 0.1188

Peak Positions in X-ray pattern are related to

locations of planes of atoms

- On the next slide, you will see a line of atoms.

Think of this as a plane, with the plane in and

out of the slide - Atoms that repeat form a series of repeating

planes for a cubic unit cell there are repeating

planes along each side. A given crystal

structure has many planes like this.

Superposition of two parallel X-ray beams results

in directions with constructive interference and

with destructive interference

- path difference SQ QT but, sin q

SQ/d then, SQ QT 2 d sin q - constructive interferenceboth rays in-phase,

thereforepath difference multiple of l - Braggs Law (note that n1 almost always, higher

order reflection are typically very weak) - 2 d sin q n l

Braggs Law and WAXS

- Remember, WAXS spectra are a series of peaks at

different scattering angles. - The scattering angles are determined by Braggs

law. - In theory, there are an infinite number of planes

in any unit cell in order for a plane to have a

reflection or peak in an XRD spectra, there must

be atoms in the plane - This is a necessary, but not sufficient condition

- If no peak is occurring and there are atoms in

the plane, then this is the same as saying the

peak has a magnitude of zero - Not going to discuss generally how you determine

specific peak magnitudes, will talk some aspects

though

Scattering angle2q

Braggs Law 2 d sin q n l Note difference

between scattering angle and Braggs law (a

factor of 2!)

Shorthand Notation for Planes

- Miller indices (hkl) h relates to x, k to y and

l to z. - The following pages show how you determine h,k

and l for a given plane however this is not

something you will be asked to do.

- Obtaining Miller Indices of a Plane
- 1. Identify coordinate intercepts (i.e. places

where plane crosses axis) x1, y1, z1 - If plane is parallel to axis, intercept is

infinity - If plane passes through 0,0,0 move origin to

another equivalent point in lattice - 2. Take the reciprocal intercepts
- h1 1/x1, k1 1/y1, l1 1/z1
- 3. Clear fractions, but DO NOT reduce to lowest

integer values h, k,

l - 4. Cite plane in parentheses again with bars over

negative values (h k l)

intercepts x 1 y 1 z ? reciprocal h

1/1 k 1/1 l 1/? 0 plane (110)

z

Plane is parallel to z-axis

110

y

Y-intercept here

X-intercept here

x

intercepts x 1 y 1/2 z 1 reciprocal h

1/1 k 2/1 l 1/1 plane (121)

z

Z-intercept here

121

Y-intercept here

y

X-intercept here

x

Distance between planes

Distance between planes

Recap for Position

- Braggs law relates scattering angle to distance

between planes - Distance between planes given by the (hkl) of the

given plane and a formula - Atoms must be in a given plane in order to have

scattering intensity.

Overall View

- The location of peaks and their magnitude from a

crystalline material is enough to determine the

crystal structure! It is simply a case of

guessing a structure, then seeing if the

calculated pattern matches the actual pattern.

Look for peak magnitude and position matches.

Commercial software that does this

http//www.accelrys.com/products/datasheets/rplus.

pdf - Another approach is to have a large database of

powder diffraction patterns then see if your

diffraction pattern matches the diffraction

pattern of what is in the database (or some

linear combination). We have that capability at

OU

Peak Magnitudes

Peak Magnitudes

- The determination of how theoretically a given

crystal structure determines peak positions and

magnitude is too complicated for this discussion. - Commercial software that calculates diffraction

patterns from crystal structure

www.crystalmaker.com

One important part of peak magnitudes related to

nanoparticles

- A perfect crystal with a perfect machine will

have an infinitely thin peak - The width of a peak in relation to its height is

related to three things - Instrument resolution, temperature, how many

planes are repeating for a given vibration

vs.

Peak Broadening

- Since nanoparticles are of limited size, the

number of planes that are repeating will be

limited - This leads to peak broadening, and a reduction in

peak height - Turning this around, you can determine the size

of crystals in a nanoparticle (which presumably

is almost equivalent to the nanoparticle size) by

analyzing the broadness of XRD peaks - Many nanoparticles are single crystals
- What does single crystal mean

Single Crystals vs. Polycrystals

- A single crystal means that all crystal planes

are oriented the same way - A polycrystal means that all the material in a

sample is crystalline, but the crystals are

oriented many different ways (the x-ray pattern

from a randomly oriented polycrystalline sample

is termed a powder pattern)

Determination of Size in Nanoparticles using XRD

- Assumption Each nanoparticle is a single

crystal - Orientation of nanoparticle is random hence

scattering pattern is identical to that of a

scattering pattern from a randomly oriented

crystalline powder - What about surface atoms these atoms might be

amorphous - Does the nanoparticle have the same shape as a

unit cell, i.e. no amorphous surface atoms or is

there a transition near the surface from

crystalline arrangement to amorphous arrangement.

- Hence, this is not a stand-alone method, better

to use this in conjunction with some other sizing

method - More exact method using XRD to determine

nanoparticle size is given in an incomprehensible

paper by Giannini et.al. PHYSICAL REVIEW B 72,

035412 2005

Determination of Size Using XRD

- Often ignore temperature effects
- Atoms vibrate which moves atoms from their

average positions which creates peak broadening - Not necessarily a very good idea, but going down

to very low temperatures to eliminate this

contribution is a problem - Measure single crystal, usually silicon (or

polycrystalline crystal with very large crystal

size) in order to determine peak width of

perfect crystal to determine instrument

broadening contribution - Quantify broadness of a peak in terms of the

half-width of a peak at half-height (b)

Size Determination via XRD (cont.)

- Use Scherrer Equation D is crystallite

(nanoparticle) size for a given hkl, l is the

x-ray wavelength and bhkl the peak width - K typically set at 0.94
- To correct for instrument broadening, use the

following formula (Bhkl is the measured half

width for the sample, b is the measured half

width for the perfect crystal)

Our Experiments

Overall Description

- Working in a geometry close to symmetric

transmission - Using area detector, so the center of the

detector is at symmetric transmission, but rest

of detector off a bit. Use 2 detector positions

to get good angular range. - Off geometry does cause some working at low

sample absorptions so distortion is negligible - Sample will be stuck to scotch tape, so we will

need to subtract scotch tape spectrum. - This subtraction is done automatically by Excel

program on computer

Detector

Detector

q1

q2

Off Axis Distortion

2q2

Off Axis Distortion

2q1

Position 1

Position 2

X-ray safety

- The equipment you are using is totally shielded,

i.e. there is no radiation beyond background. - You cannot do something to change this, if you

do, the x-rays will automatically turn off. - You will be in the presence of the operator (the

operator is the only one that will turn on and

off x-rays, and open and close shutters and

hence is the only one operating the x-rays), and

hence you do not need radiation badges. - Still, I am having you read an on-line training

session on x-ray radiation safety and doing a

homework question

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