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 PPT Presentation


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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


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Title: 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

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
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

WAXS What it measures
  • Measure intensity as a function of scattering
    angle (scattering angle2q)
  • Angles 5-120o

X-ray from source
  • Three things
  • Peak intensity
  • Peak position
  • Peak width

Sample Geometry
Looking at area inside dotted box
Normal Transmission
Pathlength of scattered beam through sample
depends on scattering angle (the higher the
scattering angle, i.e. 2q2, the longer the
Symmetric Transmission
Pathlength of scattered beam through sample does
not depend on scattering angle
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

  • 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
  • Must use this geometry for high angles
    irrespective of absorption characteristics of
  • 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
  • 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

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

Seven types of Unit Cells (also called crystal
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/
    for list of the 230 short hand notations.

  • Example of Short hand notation for silicon
  • 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
  • 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

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
  • 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,
  • 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)
Plane is parallel to z-axis
Y-intercept here
X-intercept here

intercepts x 1 y 1/2 z 1 reciprocal h
1/1 k 2/1 l 1/1 plane (121)
Z-intercept here
Y-intercept here
X-intercept here
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
  • 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

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

One important part of peak magnitudes related to
  • 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

Peak Broadening
  • Since nanoparticles are of limited size, the
    number of planes that are repeating will be
  • 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
  • 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
  • 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
  • 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
  • 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

Off Axis Distortion
Off Axis Distortion
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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