Bramwell

Question 1

What is “Condensed Matter”?

Answer 1

Matter in a high-density state (i.e. solid, liquid, glass, colloid etc.) such that many-body (10^23) effects are important.

Question 2

What is “order”?

Answer 2

Typically characterised by restricted phase space, low entropy and discrete symmetries; crystalline state is the paradigm.

Question 3

What are “excitations”?

Answer 3

Excited quantum states reflect many-body nature: quasiparticle excitations in crystals are the paradigm.

Question 4

Why X-ray and neutron scattering ?

Answer 4

Our understanding of condensed matter is intimately bound up with these experimental techniques, which are among the most important in science.

Question 5

Why order and excitations in magnetism ?

Answer 5

Magnetic order and excitations are very diverse and illustrate all the general principles extremely well.

Question 6

X-rays are

Answer 6

EM radiation, wavelength ~1 Angstrom

Question 7

‘Thermal’ neutrons are

Answer 7

free quantum particles with wavelength ~1 Angstrom

Question 8

X-ray photon, p k E

Answer 8

p = hbar*k, k = 2*pi/lambda, E = hbar*k*c

Question 9

Thermal neutron, p k E

Answer 9

p = hbar*k, k = 2*pi/lambda, E = hbar^2*k^2/2m

Question 10

Reciprocal Lattice Vector G of a cubic lattice

Answer 10

G = 2pi/a (h k l)

Question 11

Reciprocal Lattice Vector G

Answer 11

By definition, G is a wavevector with the periodicity of the crystal, so

G=2pi/d_hkl

Question 12

Direct Space and Reciprocal Space

Answer 12

Question 13

orthorhombic structure

Answer 13

a=/=b=/=c, alpha=beta=gamma=90deg orthogonal

Question 14

Basis vectors of reciprocal lattice

Answer 14

Question 15

Sketch the basic experimental geometry of X-ray or neutron scattering in the ‘W’ (or ‘M’)- configuration. Appropriately annotate the different parts of your sketch

Answer 15

sample angle determines q (Wavevector)

analyser angle determines energy transfer deltaE = h*w_q

Question 16

Scattering triangle for X-ray or neutron scattering.

Answer 16

Scattering vector def Q = k’ - k

Question 17

Laue Condition

Answer 17

For a and Q to be parallel

Q.a=2pi*h

Where Q is the scattering vector

Question 18

Show that the the scattering triangle is isoscelese for elastic scattering. Hence show that |Q| = 4pi sin theta/lambda. Write down the Laue condition for a and Q parallel and use the result just derived to demonstrate its equivalence to Bragg’s law.

Answer 18

Question 19

First order (n = 1) Bragg reflection from an analyser crystal is used to measure the energy of a scattered X-ray or neutron.

Express how the energy of (i) the scattered photon and (ii) the scattered neutron depends on the scattering angle theta’ at the analyser crystal (d-spacing d_a).

Answer 19

Question 20

Explain why the neutrons produced by a reactor or spallation source are typically passed through a hydrogenous material and what the process is called.

Answer 20

Neutrons produced by a reactor or spallation source are very high in energy and so have wavelengths too short to be useful in probing condensed matter. Passing them through a hydrogenous material slows the neutrons through collisions (hydrogen has a large scattering cross section) and the neutrons eventually equilibrate with the material, with a much lower energy. The process is called moderation and the material is called the moderator.

Question 21

Argue that the Bragg peak becomes extremely intense and sharp in a real crystal

Answer 21

In real crystals, many atmoic planes are interfereing. This gives very sharp peaks surrounded by mostly destructive interference.

Question 22

Scattering Amplitude A

Answer 22

Question 23

Fourier Analysis

Answer 23

Question 24

Fourier Analysis of electron density

Answer 24

Question 25

Brillouin Zone

Answer 25

Wigner-Seitz unit cell of the reciprocal lattice

Question 26

Weigner-Seitz Cell

Answer 26

The Wigner–Seitz cell in the reciprocal space is known as the first Brillouin zone.

It is made by drawing planes normal to the segments joining nearest lattice points to a particular lattice point, through the midpoints of such segments.

Question 27

Partial Differential Cross-section Def

Answer 27

Question 28

Differential Cross-section

Answer 28

Question 29

Total cross-section def

Answer 29

Question 30

Scattering Length def

Answer 30

Question 31

X-ray electronic charge scattering

Answer 31

Question 32

Neutron nuclear scattering

Answer 32

Question 33

Neutron magnetic scattering

Answer 33

Question 34

X-ray Diffraction Structure Factor, F_G

in terms of electron density

Answer 34

L2 p.30

Question 35

X-ray Diffraction Structure Factor for j atom basis/unit cell

Answer 35

j atoms in unit cell

f_j form factor, gives an ‘envelope’ to the scattering pattern that reduces intensity of Bragg peaks at higher scattering angle

rest is phase factor, determines intensities in absence of form factor

Question 36

X-ray Form Factor f_j

Answer 36

Question 37

X-ray form factor calculation

Answer 37

Question 38

Ewald Construction

Answer 38

Question 39

Neutron Nuclear Structure Factor F_G

Answer 39

Question 40

Magnetic Unit Cells

Answer 40

reflects the spin symmetry

Question 41

Neutron Magnetic Structure Factor F_{G_M}

Answer 41

form factor and spin-dependent factor (perp to scattering vector), and phase factor

Question 42

Magnetic Structure Factor example

NOTE choice of cell.

Answer 42

Question 43

Neutron Nuclear scattering question

Answer 43

PS2

Question 44

Neutron Scat Question

Answer 44

Question 45

Neutron Scat Q cont

The figure below shows classic powder neutron scattering data by Shull and Wollan. Comment on whether these patterns conform to the expectations discussed above.

Answer 45

As expected, the two patterns show the same peaks in the same positions but with very different intensities; the background is also much bigger in the case of NaH as expected

Question 46

Q. Is Neutron or X-ray scattering preferable for determining structure in the following:

i) diamond
ii) KCl

Answer 46

i) x-ray: higher resolution due to higher intensity of x-rays. No reason not to use x-rays.
ii) Neutrons: to get contrast. K+, Cl- isoelectric. To x-rays these ions would appear almost the same, relying on subtle differences in the form factor (which to first approx is the same).

Question 47

Q. Is Neutron or X-ray scattering preferable for determining structure in the following:

iii) CdCl₂
iv) ice

Answer 47

iii) x-rays: Cd is highly absorbing of neutrons
iv) neutrons: H has practically no electrons, x-ray scattering is off electrons. (Though modern x-rays synchotrons could still resolve H). BUT using neutrons still has the problem of incoherent scattering (high background), but could deuterate.

Question 48

What do the terms in this eq. represents, and where do they come from physically?

Answer 48

J - exchange constant, comes from overalp of the wavefunctions (think pauli principle)

D/E single ion anisotropy, comes from combination of crystal field effects with spin-orbit coupling

Question 49

Crystallography Methods (3)

Answer 49

1) Rotating (single) Crystal method: Diffracted beam is counted in the detector by rotating the crystal through the Bragg condition. (collecting over an angular range)
2) Laue method (single crystal): Uses white beam (broad spectrum of wavelengths). Different λs pick out different lattice planes G = (hkl) -> patterns of diffraction spots on detector
3) Powder Method: A powder of a crystalline solid is a mechanical mixture of crystallines at many different orientation. -> Diffraction occurs in Debye-Scherrer rings, so diffraction information is reduced to being one dimensional.

Time of flight techniques (neutrons) used for powder diff in pulsed neutron sources, Rietvield method for modelling powder patterns (fit func to scattering rather than integrate bragg peaks)

Question 50

‘define’ a crystal

Answer 50

A crystal is a state of matter with partial or discrete translational and rotational symmetry (when time-averaged and considered infinitely large). They have LOWER symmetry than time-averages liquids or gases, which has continuous symmetry.

Question 51

X-ray absorption

Answer 51

Question 52

Neutron absorption

Answer 52

Question 53

Nuclear incoherent background

Answer 53

Question 54

Advantages and disadvantages of neutron and x-ray scattering

Answer 54

Question 55

coherent and incoherent scattering cross sections

Answer 55

Question 56

Incoherent nuclear scattering - two types

Answer 56

i) isotopic incoherent
ii) nuclear spin incoherent

Question 57

Incoherent nuclear scattering

bbar and b²bar

Answer 57

Question 58

Incoherent nuclear scattering

scattering length b+/-

Answer 58

Question 59

scattering functions S

Answer 59

Question 60

correlation functions G

Answer 60

Question 61

magnetomechanical parallelism

Answer 61

Question 62

spin waves and magnons

Answer 62

Question 63

magnon dispersion measured by inelastic neutron scattering

Answer 63

Question 64

lots of magnetism stuff idc about

Question 65

1D approx to FM, magnon dispersion is

Answer 65

hbar*w = 4JS(1-cosqa)

Question 66

no. of modes with wavevector less than q

Answer 66

N = (1/8) * (4/3 pi q³) / (pi/L)³

eighth of sphere * vol of sphere / vol occupied by each mode!

Question 67

ferromagnetic order parameter

Answer 67

magnetisation M(T)/M(0)

Question 68

Bloch Law

Answer 68

M(T) = M(0) - CT^3/2