Pysics

18. What is reciprocal lattice? Sketch Reciprocal lattice vectors of a hexagonal crystal and indicate their length.

R: Reciprocal lattice is any vector that satisfies exp(iG·R)=1 where R is the direct lattice

G=ha*+kb*+lc* where hkl are the miller indeces and a, b, c are reciprocal vector. 

Properties:

G·R=2*pi*m

a·a*=2pi

a·b*=0

19. Explain Laue conditions and their experimental implications

R: Scattering condition associated with diffraction grating. It relates incoming to outcoming elastic scattering waves in a crystal. 

R*(k-k’)=2*pi*m

G=k’-k 

-interference will occur provided that the change in wave vector is a vector of reciprocal lattice.
- which is also what we know as crystal momentum. so this means that interference being constructed is the same as saying momentum is conserved.

Experimentally:

x-ray diffraction: Incident k vector leads to diffraction peaks if the tip of the wave vector lies on kspce. However, for a fixed wavelength and incident k and fixed direction, we dont see any diffraction, so we need to either vary k or the wavelength or the distance. 

*From the positions of reflection we obtain the lattice parameters in reciprocal space.

*From intensities of reflection we obtain the atomic positions.

20. Explain concept of Ewald Sphere and its implications in Scattering experiment

R: Is a geometric construction to visualize the crystal structure from observed peaks.

k and k’ define a sphere that crosses G. Where the sphere crosses G we obtain a reflection. Radius of sphere is determined by the wavelength.  since k=2pi/lambda.

Implications:

 different peaks only observed if G lies on surface of sphere

pos of refl give lattice parameters (a*, b*, c*)

intensities of refl give the atomic positions 

21. How are lattice planes related to reciprocal lattice

each reciprocal lattice vector corresponds corresponds to a family of planes:

G=2pi/d (d is interplanar distance)

for a cubic lattice we have:

d=a/Sqrt(h2+k2+l2)

22. What is the difference between diffraction experiments on powders and single crystals

Powder: (1D pattern) we cant orient this sample individually. We obtain rings of intensities which are imprints of diffraction cones. Bc this has too much info, we reduce the info to diffraction peaks.

Single: (3D pattern): Easy to manipulate. Diffraction pattern is image of Recip Latt. Pts show different intensity (they dep. on hlk.). Measure G -> measure Intens. -> obtain atom positions.

23. Elucidate extinction conditions in a diffraction experiment

Are given by the structure factor: F=0 means there is an extinction of the scattered wave. F=/= 0 means there is an intensity showing, so the wave is scattered. The symmetry of the crystal requires that there is no reflection on G on certain family of planes.

24. Explain the diff between elastic and inelastic scattering. Give an example of one experiment of each type and explain which property of crystal it measures

Elastic: No energy transfer. Experiment: x-ray diffraction is used to determine the lattice parameters of our crystal

Inelastic: Energy transfer. Inelastic x-ray scattering. Measures the phonon spectra/ disp by measuring intensity as function of angle and energy transfer.

Xray: scattered particles are electrons. Scatter strongly from heavy atoms (scatt factor f prop to Z). Cant differentiate between atoms with similar atomic number or light atoms. 

Neutron: Scatters neutrons inside nucleus. They are good for analyzing light elements or atoms with similar Z.

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25. Difference between x-ray and neutron diffraction

Ionic - due to oppositely charged particles attracting eachother. Ex: Na+ Cl-

Covalent - localized electrons are shared between atoms (overlap of orbitals) Ex: H-H

Metallic - delocalized electrons are shared over the whole crystal ex: Sodium

Van d. Waals - due to dipole-dipole interaction between non charged units (can be molecules) Ex: Ar - Ar

Hydrogen - Bond between H+ and negatively charged ions Ex: H2O

27. Introduce main types of chemical bonds. Give an example of a solid for each type of bonding.

We expect to see diffraction peaks. From this we obtain the interplane distance (d) which we can relate to the miller indices and obtain the lattice parameters.

(why monochrom. Light? We need to change at least one parameter either the wavelength or the angle of incidence, so if we keep the same wavelength, then we need to rotate our crystal to get diff angle. Bragg peaks are recorded whenever circle intersects ewalds sphere)

​26. You scatter monochromatic xrays on a crystal. What do you expect to see? How do you use this info.