1. Chem E5225 – Electron Microscopy P
Elastic scattering

2. Particles and waves
The electron beam can be observed to be both as particles and as a beam. As particles the electrons have a scattering cross section and a differential scattering cross section. The electrons can interact with the nucleus and the electron cloud surrounding the atoms from the TEM specimens. The electrons also act with a wave nature because waves are created by atoms diffracting waves creating scattering centres. The strength of the wave scattering by an atom is determined by the atomic scattering amplitude. Solids create a more complicated diffraction but this is central to TEM

3. Particles and waves Mechanisms of elastic scattering
Elastic scattering around a single particle can occur in two ways
Elastic scattering around a single particle can occur in two ways which both involve coulomb forces. As shown in figure (1) the electron either interacts with the electron cloud which results in only a small angular deviation from the original path or penetrates electron cloud to the nucleus where a strong deviation can be seen the forces of which are applied by the nucleus. Both of these interactions are not truly elastic as both of these results in some small energy loss. The actuality is that the higher the angle of scattering of the electron the higher the chance of that electron having experiencing an inelastic interaction with the TEM specimen.

4. Particles and waves
The other method of elastic electron scattering happens when the electron wave interacts with the TEM specimen as a whole.
The other method of elastic electron scattering happens when the electron wave interacts with the TEM specimen as a whole. Using Huygen’s approach for the diffraction for visible light, it is thought that each atom in the specimen that interacts with the incident plane acts a source point for secondary wave point origins. Figure (2)This means that at low angles the scattering distribution is changed by the crystal structure.

5. Elastic Scattering from Isolated Atoms
The path of the electron can either interact with the electron field resulting in a small deviation from the original path or a strong deviation from an interaction with the nucleus.
The path of the electron can either interact with the electron field resulting in a small deviation from the original path or a strong deviation from an interaction with the nucleus. We can try to predict the scattering field of both of these types of interactions. Equation (1) gives the scattering field for an electron – electron interaction. Equation (2) gives the scattering field for an electron – nucleus interaction. Z is the number of electrons in the cloud, 𝜃 is the angle, e is the charge of the electron and V is the voltage of the accelerator.
𝑟 𝑒 = 𝑒 𝑉𝜃 (1)
𝑟 𝑛 = 𝑍𝑒 𝑉𝜃 (2)

6. The Rutherford cross section
This only concerns the electron – nucleus interactions.
𝜎 𝑟 (𝜃)= 𝑒 4 𝑍 (4 𝜋𝜀 0 𝐸 0 ) 2 𝑑 𝑜𝑚𝑒𝑔𝑎 𝑠𝑖𝑛 2 𝜃 2
This only concerns the electron – nucleus interactions. The high angle scattering is comparable to the backscattering of alpha particles from a thin film metal foil. Ernest Rutherford with the help of H.Geiger and E. marsden was able to deduce the existence of the nucleus and derived the expression (Equation (3)) for the differential cross section for the high angle scattering.

7. Modifications to the Rutherford cross section
There are many similar differential cross sections, the Rutherford cross section doesn’t take into account the screening effects the electron cloud.
𝜃 0 = 𝑍 𝐸 (4)
So far the equations 4 and 3 are non – relativistic, since relativistic are significant for electrons above 100 KeV
𝑎 0 = ℎ 2 𝜀 0 𝜋𝑚 0 𝑒 2 (5)
There are many similar differential cross sections, the Rutherford cross section doesn’t take into account the screening effects the electron cloud. This can make the nucleus appear less positive to the incoming electron. So the differential cross section is lowered and the scattering is lowered. This screening is only important when the incident electron passes far from the nucleus (this would mean less than 3 degrees scattering). To counteract the effect of screening we replace the term 𝑠𝑖𝑛 2 𝜃 2 from equation 3 with 𝑠𝑖𝑛 2 𝜃 2 + ( 𝜃 0 2 ) 2 . The 𝜃 0 is described by equation (4)
The important effect of this is that the cross section cannot go to infinity as the scattering angle goes to zero, this is an important limitation.
The screened Rutherford cross section is the one that is most widely used for TEM calculations, but has a limitation at high operating voltages ( KeV) and heavier elements (Z>30), it would scatter the electrons at high angles. So for different circumstances different cross section would need to be used.
To obtain the cross section over specific angle ranges, we can put in the correct values for the
Net result of adding the screening and the relativistic corrections is that the Rutherford cross section changes to equation (6).
𝜎 𝑟 (𝜃)= 𝑍 2 𝜆 𝑅 ( 𝜋 4 𝑎 0 2 ) 2 𝑑 Ω 𝑠𝑖𝑛 2 𝜃 2 ( 𝜃 0 4 ) (6)

8. Modifications to the Rutherford cross section
Figure 3 shows different beam energies and figure 4 shows the cross section for a three different elements.
Coherency of the Rutherford scattered electrons
High angle Rutherford scattered are incoherent, no phase relationship when considering the electrons as waves. This incoherent scattering can be used for very high resolution imaging of a crystalline structure. The high angle back scattered electrons can be used to form images of the beam entrance surface.

9. The atomic scattering Factor
An aspect of the wave approach is the atomic scattering factor which can be related to the differential elastic cross section
𝑓 𝜃 2 = 𝑑𝜎 𝜃 𝑑Ω (8)
𝑓 𝜃 is complimentary to the Rutherford differential cross section analysis, most notably low angle elastic scattering where the Rutherford approach is lacking
Where Lambda is the wavelength, E0 is the beam energy, Fx is the scattering factor for X-rays. Here the screening term has been dropped. The most used scattering factor for TEM is from the classic work by Doyle and Turner 1968.
The angular variation can be for a singlular atom can be shown graphically.
Picture
Eqution (9) contains expressions of both elastic nuclear scattering (Z term)

10. The origin of F(θ) 𝜑= 𝜑 0 𝑒 2𝜋𝑖𝑘𝑟 (10)
We can describe the incident beam as a wave of amplitude
𝜑= 𝜑 0 𝑒 2𝜋𝑖𝑘𝑟 (10)
When the incident plane wave is scattered by the atom a spherical wave is created around that point with a different amplitude, but it keeps the same phase
In which The f(0) is the amplitude. But we need to know F(0), we could use Schrodinger equation, but usually a simple approximating is enough. The scattering from equation 10 an 11 can be used to describe Figure 6.
The K1 is a wave propagation parameter for the incident plane and k is the scalar for the spherical waves. The 90 degrees phase change for the scattered phase can be understood if the wave is initially and once it passes through the specimen it will be Theta total and after scattering the phase is increased and can be expressed:
Now we start to consider what happens when we stack atoms together regularly in a crystal structure. The Structure factor F(ϴ) is a measure of the amplitude of scattered by a unit cell of a crystal structure. It also has length. We can define F(ϴ) as a sum of f(ϴ)terms from all the atoms in the unit cell multiplied by a phase facto. This phase factor takes into account the difference in phase from the waves scattered from different atoms on different planes but parallel ones. The scattering angle between the incoming and the scattered electrons:
The amplitude of the screening is affected by the type of atom and the position of the atom in the cell. It certain situations the model predicts at some points the scattering will be zero.

11. Simple Diffraction Concepts
We can use diffraction to determine the spacing of the planes in crystals. The positions of the diffracted electrons are determined by the size and shape of the unit cell within the specimen, the intensities of the diffracted electrons are directly related to the structure of the atoms and size and type.
For an arrangement of atoms in a specimen, they can be arranged randomly the amplitude of diffraction is stronger at some angles rather than others, so we would see bright rings on the TEM output figure (7). If the specimen is crystalline then the diffracting intensity is at specific angles figure (8).
Figure (7) an amorphus specimens scattering profile, figure (8) a cycstalline structure.
Interference of electron waves, creation of the direct and diffracted waves.
When a new wave is created from a nucleus ad when there are many present in the surrounding area the waves can interact with one another, this happens even with the thinest specimen. As a rule the waves are thought to reinforce each other when in phase and cancel each other when out of phase.

12. Diffraction Equations
The Path difference between scattered waves is AB-CD. B and C are the atoms In 2 dimensions the path difference is:
𝑎 𝑐𝑜𝑠 𝜃 1 −𝑐𝑜𝑠 𝜃 2 =ℎ𝜆
And for 3 dimensions two more Laue equations :
𝑏 𝑐𝑜𝑠 𝜃 3 −𝑐𝑜𝑠 𝜃 4 =𝑘𝜆
𝑐 𝑐𝑜𝑠 𝜃 5 −𝑐𝑜𝑠 𝜃 6 =𝑟𝜆
The idea of using diffraction to probe the atomic structure is credited to Von Laue 1913, his idea was that shorter electromagnetic rays than light would cause diffraction or interference within a crystal. The idea was tested experimentally by irradiating copper sulphate crystals.
Von Laue used light optics approach to say that if the path difference between waves is a whole integer of wavelengths apart then they are in phase. Figure (9).

13. Diffraction Equations
usually simpler approach is Used in TEM, that waves are reflected off atomic planes (bragg and bragg 1913)
Similar to the Von Laue approach the bragg approach says that the scattering centers must have a path difference equal to an whole integer of wavelengths to remain in phase. So this means in TEM the path difference between reflected electron waves from the upper and lower planes in figure () is AB+CD. Thus if hkl planes are spaced d apart the incident and reflecting electron waves are the same angle and AB and BC are also the same and equal to dsinθB and the total difference between the two paths is 2dsinθB. From this we can get bragg’s law
𝑛𝜆=2𝑑𝑠𝑖𝑛 𝜃 𝐵
Bragg’s Law