Some more reading sources

1
Some more reading sources

• The Carter and Williams book is on overnight loan
  in OLS
• Some other useful books
• Electron optics and the Electron Microscope
  QH212.E4ELE (be selective, its quite an old
  book)
• Experimental High-resolution electron microscopy
  QH212.E4SPE (his descriptions are good, but dont
  panic about the maths)
• Electron Optics and Electron Microscopy
  QH212.4HAW (another nice descriptive book, but
  not much depth to parts of it)
• High voltage electron microscopy QH212.E4INT
  (proceedings of the 3rd conference of HRTEM.
  Interesting to see how far the field has come in
  the 35 years since this conference. Just glance
  through for an appreciation)
• Introduction to Analytical Electron Microscopy
  TA417.23HRE (a very heavy read, but it has some
  good contents. Use only for clarification after
  consulting other texts)
• Electron Microscopy in the Study of Materials
  QH212.E4GRU
• This is an excellent, quick, clean walk through.
  But use it in conjunction with lecture notes.

2
A useful online resource

• The following website works you through much of
  the theory
• There are some small simulations to illustrate
  the concepts
• http//www.matter.org.uk/tem
• The page http//www.matter.org.uk/diffraction is
  also useful

3
Books and etiquette

• There is only one copy of each of the above
  mentioned books in the Biophy library
• Please dont hog them
• Especially the Carter and Williams and the Grundy
  books

4
Reminding and revisiting

• For a 300 kV TEM electrons have a wavelength of
  0.0708 nm
• Electrons are moving relativistically
• The source is either thermionic (big energy
  spread) or field emission (small energy spread)
• Lenses are electromagnets that are non-uniform
• The beam moves down the bore of the microscope
• Apertures can be used to select and control the
  spread of the beam

5
Reminding and revisiting

• Electrons interacting with the sample that we are
  interested in are only those that are forward
  scattered
• Elastically and incoherently and inelastically -gt
  Amplitude contrast
• Elastically and coherently -gt Phase contrast
• Magnification is achieved by changing the
  strength of the lenses

6
Basics of HRTEM Continued

• Aberration
• Astigmatism
• Specimen considerations

7
Aberration

• This term describes the distortion of the
  electron beam passing through the lenses
• Its cause is the imperfection of the lenses and
  the electron source as well as electron-electron
  interactions
• There are two significant forms of aberration in
  modern TEM
• Spherical aberration which is a geometric
  aberration
• Chromatic aberration which is an effect of an
  energy distribution
• Each of these carry a constant for a given
  machine, CS and CC respectively

8
Spherical aberration

• Consider the path of the electrons through the
  lens

9
Spherical aberration

• Those traveling further from the axis have to
  travel further
• Those traveling further from the axis experience
  a different magnetic field
• i.e. they experience a different focusing
  strength
• the focal length is different

10
CS

• The result is a disc of least confusion

11
Disc of least confusion

• This is a region where the beams that are
  overfocused are best matched with those that are
  underfocused
• The radius of the disc is given by
• Where M is the magnification, CS is the
  coefficient of spherical aberration and a is the
  semi-angle at which the beam is moving
• CS can be found by taking measurements of a known
  material and using the spread of the diffraction
  beams

12
Correction for spherical aberration

• Either in image analysis or physically
• If physically it is carried out using octopoles
  and hexapoles designed to have a negative CS
• Therefore those beams that are underfocused by
  the objective lens, become overfocused by the
  corrector
• This is elegant and works well
• However it means another set of lenses on the
  microscope
• So chromatic aberration becomes worse
• And aligning the microscope becomes even harder
• In image analysis, allowance is made for CS, much
  like instrumental broadening in applying the
  Sherrer equation in PXRD

13
Chromatic aberration

• There are three main sources of this
• Energy spread in electrons coming off the gun
• Electron-electron interactions at the gun
  cross-over
• Energy changes resulting from electron-electron
  interactions in the sample
• Chromatic aberration is the real limitation on
  the maximum resolution of the TEM

14
Chromatic aberration in optical lenses
15
Its effects

• The effect of chromatic aberration is to change
  the focal length of the microscope as given by
• Where CC is the coefficient of chromatic
  aberration for the TEM, dFC is the change in
  focal length from chromatic aberration, V is the
  accelerating voltage, I is the current in the
  objective lens and E is the spread in energy from
  electron-electron interactions.

16
Correction for Chromatic aberration

• In optics this can be done using a corrector lens
• In TEM no such lens can be built

17
Reducing chromatic aberration

• Using a source that produces electrons with the
  smallest possible energy distribution
• Operating in geometry as close to parallel beam
  as possible, limiting electron-electron
  interactions

18
Aberration

• It is a balance in correction between chromatic
  and spherical aberration
• If spherical aberration is corrected too much,
  chromatic aberration becomes worse
• If it is not corrected, then the two sum to cause
  deterioration in image resolution

19
Astigmatism

• Arises from the non-uniformity of the magnetic
  field in specific directions
• So the x-direction field being stronger than the
  y-direction field
• Its effect is to create streaking on the image
• It is found that it is possible to get the image
  in focus in the x, but not the y, or vice versa
• The easiest way to observe it is to take the
  power spectrum, i.e. the FT, of an image of the
  amorphous support
• If the rings that result are symmetrical, there
  is no astigmatism
• If they are oval in shape there is some
  astigmatism
• If they form a cross shape, there is chronic
  astigmatism

20
Correcting astigmatism

• Unlike aberration astigmatism is easily corrected
  by simply changing the potential and current the
  electrons experience
• As octapoles and quadrapoles are used, the x and
  y can be controlled separately
• Therefore the strength of the field is varied for
  each of these and the astigmatism is corrected

21
Specimen considerations

• There are many thoughts that need to go into
  specimens for the TEM
• Thickness
• Dispersion
• Charging
• Beam damage
• Etc

22
Thickness

• One of the features of the TEM is the large depth
  of field and depth of focus
• Depth of field means that the image can be
  collected through the entire sample at the same
  time
• Depth of focus means that all of the image is in
  focus at the same time through the entire depth
• However the thicker the sample, the worse
  scattering effects become
• If the sample is too thin, amplitude contrast
  vanishes almost entirely and phase contrast can
  also be hard to pick out from the background

23
Dispersion

• If the sample is a powder, it needs to be well
  dispersed on the support
• Clumps of sample are hard to analyse
• It also relates to the projection problem are
  you looking at one particle or two?

24
Charging and beam damage

• These are considerations in choosing whether TEM
  is a method that can really be used to study the
  sample.

25
Specimen preparation

• For powders
• A suspension is made in ethanol, acetone or
  chloroform (or another solvent with low
  vaporisation temperature)
• This is sonicated in the hope of dispersing the
  particles
• A drop of the suspension is loaded on a support
  and dried before being mounted in the TEM
• For solids
• A thin slice is cut and loaded onto the support

26
That concludes the introduction. You are now
familiar with the basic operation of the TEM, now
for the science behind the images
27
Electron diffraction

• We consider the electron beam to be a plane wave
  of the form
• Where ?0 is the amplitude and 2pik.r is the phase
• Upon interacting with an atom the wave is
  scattered
• The resulting wave, the scattered wave, is
  described by
• Where f(?) is the atomic scattering amplitude

28
Scattering amplitude

• Electrons are scattered by the potential in the
  specimen as described by
• Where K is a constant, S is the vector describing
  the beam, r is the vector describing the atomic
  position and ?(r) is the potential
• This is just like the scattering factor for
  X-rays
• However it depends on the atomic mass to
  approximately Z1/3
• So TEM is much more sensitive to light elements
  than X-rays, yet still sensitive to heavy
  elements
• Therefore electron diffraction is the best method
  to find light atoms

29
Ewald sphere

• The Ewald sphere is a mathematical construct
  relating to the wavelength of incident radiation
• For X-rays, with a cubic lattice of 4Å for X-rays
  a typical Ewald sphere is

30
Ewald sphere for TEM

• The sphere radius is approximately 40Å-1

31
Results of Ewald sphere

• If we have a fixed detector located on the optic
  axis, the specimen will have to be rotated by 27
  before the sphere will contact the 200 point for
  the X-ray case
• The system is also limited to sampling only those
  beams and reciprocal lattice points that lie
  within it
• In electron diffraction, multiple beams will lie
  on the Ewald sphere at the same time and without
  any rotation
• The result is that with a single electron
  diffraction image, far more of the sample is
  observed

32
Laue circles
33
More about the Ewald sphere

• Laue zones

34
Measuring d spacings directly

• This means that d-spacings can be found directly
  if the camera length is known using the fact that
  for small 2?, sin? is approximately ?.
• The camera length, L, is known
• t can be measured
• Hence and b can be found

35
Finding the third parameter

• If we are looking down the (001) zone axis, then
  we are looking down the c axis, so we can measure
  a and b directly from the diffraction pattern
• c can be found by using the FOLZ (if the crystal
  is large enough for it to be observed)

90??
90??
?L?c
?
ZOLZ
FOLZ
r1
36
Lattice parameter relations
V 2abcsin(s)sin(sa)sin(sb)sin(sg)
½ where s ½(a b g)
37
Intensity of diffraction spot

• In the case of a discrete crystal, the structure
  factor can be calculated using the relation
• The intensity of the spot for a particular hkl is
  then given by the modulus of the structure factor
• Note that the spot also carries a phase term from
  the imaginary component
• In recording the image, we lose this phase
  information
• However to solve the structure we need phase
  information back again

38
Image of effect of discrete number

• When dealing with a finite crystal, the intensity
  of each diffraction spot is no longer a delta
  function, but is given by the summing of the
  phase changes from each scattering point.
• The result is that the intensity is given by

39
Discrete crystal sizes and spot intensity

• For 7 atoms For 30 atoms

40
Dynamical scattering

• It is important to note that what is taking place
  in this system is that the electrons are
  scattered and interact with each other as wave
  objects
• This is subtly different to treating the lattice
  as a phase grating
• The reason for this difference is the very small
  wavelength of the electrons
• Consider the path of a single electron
• It interacts with an atom and scatters
  elastically, incurring a phase change
• It then can either pass through the sample with
  no further interaction, or it can interact with
  another atom and incur an additional phase change
  after another scattering event
• This leads to the idea of dynamical scattering
• The kinematic scattering that we use for PXRD is
  not completely valid

41
Dynamical scattering
42
Mathematics of dynamical scattering

• After layer 1 Ap A Ad B
• After layer 2 Ap (A2 B2)exp(i?) Ad (AB
  BA) exp(i?)
• 2AB exp(i?)
• where exp(i?) is the phase factor from
  propagation between the layers.
• The same process will be repeated at every
  subsequent layer of atoms with both beams losing
  and gaining amplitude from diffraction and
  transmission respectively. After the third
  layer, the amplitudes will be given by
• Ap (A3 3AB2) exp(2i?) Ad (3A2B B3)
  exp(2i?),
• progressing at the fourth layer to
• Ap (A4 6A2B2 B4) exp(3i?) Ad (4A3B
  4AB3) exp(3i?)

43
Dynamical scattering continued

• After n layers, the separate amplitudes may be
  written as
• Ap ½(A B)n (A - B)n exp(n-1)i?
• Ad ½(A B)n - (A - B)n exp(n-1)i?
• But AP2 Ad2 1
• So to reconcile this we must have
• A cos(?) B isin(?)

44
Wave formulation

• ?O(x,y) expi??(x,y)?z
•  
• This may then be expanded as with any exponential
  
•  
• ?O(x,y) 1 i??(x,y)?z - ?2?2(x,y)?z2/2 -
  i?3?3(x,y)?z3/6 .......
•  
• If the total scattering of this single plane of
  atoms is relatively small, we may neglect all but
  the first term in ??(x,y)?z. Consequently the
  diffracted amplitude emerging from this plane of
  atoms can be written as
• ?D(h,k) ?(0,0) i?FT(h,k)?z?(h,k)

45
And after some maths and n layers

• AP cos?FT(h1,k1)t exp(n-1)i?
• And
• Ad isin?FT(h1,k1)t exp(n-1)i?

46
What does this mean?

• Consider the (h20) and (-h30) beams for a system
  with a 21 axis of symmetry parallel to y
• This relates atoms at (x,y,z) to (-x, ½y, -z)
• It can be shown using the structure factor that
  the (0k0) spots are systematically absent
• However the summation of the two beams above
  gives (h-h,23,0) (050)
• So the spot is observed

47
Effect of multiple scattering

• It means that it is not always possible to
  determine symmetry directly from SAED as the
  systematic absences may be broken.
• There are ways around this, that we will touch on
  later

48
Relate this to the thickness effect and edge
fringes

• From the dependence of the intensity function on
  the number of atoms that it is summed across
  think back to the variation in intensity in the
  image.
• You may start to get an appreciation of how the
  thickness effect and shape of the crystal can
  influence the relative contrast observed in the
  TEM.

49
Diffraction and apertures

• If we place and aperture in the beam, we can
  select only certain spots of the diffraction
  pattern

50
Meaning of n-beam images