Xray DIFFRACTION

XRD, ED

Unit cell

Smallest repetitive volume which contains complete lattice pattern of a crystal

There are _______ crystal systems and ______ Bravais lattices

7; 14

How to find crystallographic directions

1. reposition vector if necessary to pass through origin

2. read off projects in terms of unit cell dimensions a, b, c

3. adjust to smallest integer values

4. adjust like [uvw]

Miller indices 

Reciprocals of three axial intercepts for a plane, cleared of fractions and common multiples - all parallel planes have same miller indices 

Find miller indices

1. read off intercepts of plane with axes in terms of a, b, c

2. take reciprocals of intercepts

3. put fractions on smallest common denominator

4. enclose in parentheses, no commas (hkl)

Miller indices of this?

(1 1 0) (from intercepts 1 1 infinity)

If the intercepts are (1 ½ ¾ ) what is the miller indices?

(634)

Basic idea of Fourier Transformations

Any function can be expressed as linear combination of waves

For periodic signals, only waves representing the function’s __________ can contribute 

periodicity 

Fourier series is used when

Function is periodic and it repeats itself after every T (period) seconds

Periodic function can be represented using only integer multiples of a 

base frequency 

Fourier series to discrete set of frequencies

Fourier series can be represented with periodic signals, and a formula can be used to determine “how much” of each frequency component (of the periodic function) is needed

Fourier transform used when

function is not periodic or is defined on infinite domain

When to use fourier transform

When function does not repeat —> must use all possible frequencies to represent it (so get integral instead of sum)

1D fourier transformation

between time and frequency

or v and f

2d/3d fourier transformation

transfer between r and k=2piv

For a 2D fourier, a longer wavelength means what and a smaller wavelength means what?

longer —> smaller spatial frequency

shorter —> larger spatial frequency

Fourier-based image processing - low pass

remove high spatial frequencies, noise

Fourier-based image processing - high pass

remove low spatial frequencies, enhance edges and small features

Longer wavelength = _____ spatial frequency, shorter wavelength = ______ spatial frequency

smaller

larger

Reciprocal lattices

vector in reciprocal space that describes the periodicity of a crystal, representing wave vectors for plane waves in crystals

Incoming X rays and Crystal planes

1. Incoming X-rays diffract (bounce off of) crystal planes 

   1. Reflections must be in phase for detectable signal, delL = n * lambda (n = 1, 2, 3, etc.) 

Measurement of critical angle allows for

1.  computation of planar spacing (space between crystal planes) 

Ewald sphere

1. reciprocal space of all the possible points where planes (reflections) could satisfy the Bragg equation

How are X-ray sources generated for Xray Diffraction

1. Can be generated by a similar method as for X-ray spectroscopy techniques 

   1. Can change these values to see if the X-ray beam has enough energy to knock electrons 

Hull-Debye-Scherrer Method

Uses near-parallel incident beam of X-rays with sufficient cross-section to hit powder sample

X-Ray diffractometer (XRD)

X-rays produced within X-ray tube, X-rays hit the sample, detectors are able to detect the diffraction - Bragg’s law describes this diffraction pattern

Cons with XRD

size and angle precision - angle off by 1% - 100 atoms off 

The residual ________ and _________ on diffraction peak position and shape

stress, strain

Wide Angle XD

Photographic technique, produces image, uses monochromatic X-ray beam

Wide Angle Scattering used on what types of samples

Amorphous

Larger objects have __________ scattering angles

Smaller objects have __________ scattering angles

smaller

larger

Small Angle XRD

X-ray sample is closer to small angle sample

Can get nm level resolution

X ray beam closer to _____ sample than ______sample 

SAXS

WAXS