Random Processes and Statistical Physics

Question 1

Difference between combinations and permutations

Answer 1

Combinations order isn’t important, permutations order is.

Question 2

Inclusion exclusion principle for 2 events.

Answer 2

P(AuB) = P(A) + P(B) - P(AnB)

Question 3

Probability of A given B.

Answer 3

P(A|B) = P(AnB)/P(B)

Question 4

Probability of A given B when A and B are independent.

Answer 4

P(A|B) = P(A) = P(AnB)/P(B)

implies

P(AnB) = P(A)P(B)

Question 5

What is a random variable?

Answer 5

A variable whose value depends on the outcomes of a random process.

Question 6

Example of a discrete random variable.

Answer 6

Roll two dice and add the scores.

Question 7

What is a continuous random variable.

Answer 7

Where the values a random variable can take are continuous.

Question 8

When finding the probability of a continuous random variable taking a specific value, what happens?

Answer 8

The probability of the variable exactly matching a specific value is vanishingly small, but there is a finite probability of the variable lying in a range.

Question 9

What is it when there is a probability per unit range for a continuous random variable?

Answer 9

The probability density function.

Question 10

What is the total area under the curve of a probability density function?

Answer 10

1 and is found by integrating.

Question 11

How do you find the mean from the probability density function?

Answer 11

Integrate the probability density function multiplied by the variable.

Question 12

Why is standard deviation advantageous over variance?

Answer 12

Standard deviation has the same units as the original variable.

Question 13

How to find the mode for a continuous random variable?

Answer 13

Differentiate the probability density function.

Question 14

How to find the median of a continuous random variable?

Answer 14

Intergrate between Q2 and infinity equal to 0.5 and solve for Q2.

Question 15

How to find the IQR of a continuous random variable?

Answer 15

Integrate between -infinity and Q3 equal to 0.75 and -infinity and Q1 equal to 0.25. Solve for Q3 and Q1 and IQR=Q3-Q1

Question 16

If you multiply a random variable by a scalar, what happens to the mean and variance?

Answer 16

Mean is multiplied by scalar

Variance is multiplied by scalar squared.

Question 17

What happens to the mean and variance when two random variables are added together?

Answer 17

The means are added

The variances are added.

Question 18

What is microstate?

Answer 18

Each unique configuration of the system where the state of each element is known.

Question 19

What is macrostate?

Answer 19

Multiple microstates can give the same macroscopic property.

Question 20

How to find the most probable state of a system?

Answer 20

Multiplicity is maximised.

Question 21

Expression for multiplicity.

Answer 21

Binomial distribution expression.

Question 22

Expression for entropy in terms of multiplicity.

Answer 22

S = klogW

Question 23

By considering the random walk in 1D, find an expression for the most probable end point.

Answer 23

Check derivation

Question 24

Compare the random walk in 1D to a Gaussian function.

Answer 24

Check comparison

Question 25

Model diffusion with random walk model deriving Fick's first law.

Answer 25

Check derivation.

Question 26

Derive Fick's second law of diffusion.

Answer 26

Check derivation

Question 27

How does rms change with dimension in diffusion?

Answer 27

Rms= N=2dDt d is no dimensions D is N/2t

Question 28

Derive an expression for heat flow between planes.

Answer 28

Check derivation

Question 29

What is the total length of a polymer chain known as?

Answer 29

Contour length.

Question 30

Equation for total length of polymer.

Answer 30

L=Nb

Question 31

How can end to end length of a polymer be expressed?

Answer 31

r = Σl

Question 32

Derive an expression for rms of a polymer chain of length N using random flight model.

Answer 32

Check derivation

Question 33

What is the characteristic ratio of a polymer?

Answer 33

Describes bond correlations as in real polymers orientations of bonds are not completely random.

Question 34

Typical values of characteristic ratios for polymers.

Answer 34

Cn>1

Question 35

What does the Kuhn model do?

Answer 35

Allows a real chain to be treated as an equivalent freely jointed chain split into Kuhn sgements which represent more than one real bond.

Question 36

Contour length for Kuhn model.

Answer 36

L = Nbcosψ = NkBk

Question 37

Derive an expression for the retractive force on a polymer chain when stretched.

Answer 37

Check derivation.

Question 38

3 properties of ideal gases.

Answer 38

No intermolecular forces between gas molecules. Particle occupy negligible volume compared to volume of container they occupy. Interactions between the molecules and with the container walls are perfectly elastic collisions.

Question 39

Derive ideal gas equation from lattice model considering M sites containing N particles.

Answer 39

Check derivation

Question 40

Derive the relation for kinetic energy and temperature by considering pressure.

Answer 40

Check derivation (A level)

Question 41

Dalton's law of gases.

Answer 41

In mixtures of gases each gas can be treated independently with partial pressures which sum to the total pressure.

Question 42

Derive the distribution of the speed of the molecules in a gas.

Answer 42

Check derivation

Question 43

Find an expression for most likely speed, knowing the distribution of molecular speeds in ideal gas.

Answer 43

Check derivation

Question 44

Find an expression for rms speed, knowing the distribution of molecular speeds in ideal gas.

Answer 44

Check derivation

Question 45

Find an expression for mean speed, knowing the distribution of molecular speeds in ideal gas.

Answer 45

Check derivation

Question 46

Derive an expression for impingement rate using skewed cyclinder.

Answer 46

Check derivation

Question 47

Derive an expression for flux onto a surface using skewed cylinder.

Answer 47

Check derivation

Question 48

Derive an expression for pressure using momentum and skewed cylinder.

Answer 48

Check derivation

Question 49

What is mean free path?

Answer 49

The mean distance travelled between collisions

Question 50

Derive an expression for mean free path.

Answer 50

Check derivation

Question 51

Derive an expression for collisions rate per unit volume between molecules of same type.

Answer 51

Check derivation

Question 52

Derive an expression for collisions rate per unit volume between molecules of different type.

Answer 52

Check derivation.

Question 53

Derive an expression for effusion rate.

Answer 53

Check derivation

Question 54

Derive an expression for how pressure changes over the process of effusion and is dependent on mass and temperature.

Answer 54

Check derivation

Question 55

Give an expression for the ratio of effusion rates in a mixture of gases.

Answer 55

J1/J2 = N1/N2 sqrt(m2/m1)

Question 56

What is a Knudsen number?

Answer 56

A dimensionless constant that is the ratio of mfp to characteristic dimension of the deposition chamber, L

Question 57

What happens when Kn>>1?

Answer 57

Molecular flow

Question 58

What happens when Kn<<1?

Answer 58

Viscous flow.

Question 59

What is PVD?

Answer 59

Physical vapour deposition. | Source material vapourised and then condenses on a substrate.

Question 60

What kind of process is Molecular Beam Epitaxy (MBE)?

Answer 60

A form of PVD

Question 61

Derive an expression for flux per solid angle.

Answer 61

Check derivation

Question 62

Derive an expression for flux per unit area on a substrate.

Answer 62

Check derivation

Question 63

Turn flux per unit area into evapouration rate.

Answer 63

Check derivation.

Question 64

Derive an expression for flux from heated crucible.

Answer 64

Check derivation

Question 65

Describe E-beam evapouration

Answer 65

e- from heated filament accelerated through electrode then in B-field and impact crucible evapourating material. High local temperatures so reduces risk of crucible damage.

Question 66

Difference between planetary and semi-planetary geometry in terms of evapouration rate.

Answer 66

Planetary geometry has no angular dependence.

Question 67

Expression for evapouration rate in planetary geometry around crucible?

Answer 67

Check expression

Question 68

Describe set up for sputtering

Answer 68

High voltage between two plates Ions and electrons from random ionisation events accelerated towards cathode and anode. If voltage applied is sufficient these will ionise other gas molecules. The gas between the plates then conducts electricity forming a plasma known as a glow discharge. For plasma to self sustain, a certain pressure is required. To reduce operating pressure magnetron sputtering introduces magnetic field at target, trapping electrons in orbits and increasing no ionisation events.

Question 69

Describe the sputtering process

Answer 69

Ar+ iosn accelerated in E field to target and impact transferring momentum. Momentum transfer causes some atoms on surface to be released due to lower mass. The sputtered atoms travel towards the traget plate unaffected by the E field but do collide with atoms in gas.

Question 70

Ethreshold for sputtering equation.

Answer 70

Check equation

Question 71

Sputter Yield equation

Answer 71

no. sputtered atoms/no. incident ions

Question 72

Compare sputtering and evapouration.

Answer 72

Evapouration: Kn>>1 leads to line of sight deposition. Sputtering: Kn<<1 leads to improved step coverage. Evapouration: Hard to get stoichiometric compounds. Sputtering: Composite targets or reactive sputtering can achieve deposition of compounds Evapouration: Highly pure deposition without gas inclusion. Sputtering: Better film adhesion due to more energetic atoms. Evapouration: Low cost and normally more efficient use of material. Sputtering: Higher defect density due to gas inclusion and e- bombardment.

Question 73

Derive a flux for a general transport property.

Answer 73

Check derivation

Question 74

List flux laws for heat flow, viscocity and diffusion.

Answer 74

Check

Question 75

List kinetic expressions for thermal conductivity, viscocity and diffusivity.

Answer 75

Check

Question 76

Expression for rates in differential form:

Answer 76

d[A]/dt = -k[A]

Question 77

Obtain an rate equation from differential form of rate equation.

Answer 77

Check

Question 78

Rete constant follows what kind of equation?

Answer 78

Arrhenius

Question 79

Factors that influence rate of an elementary bimolecular reaction.

Answer 79

Encounter rate Energy requirement Steric requirement

Question 80

Derive an expression for encounter rate based on collsion rate.

Answer 80

Check derivation

Question 81

State an expression for energy requirement where fraction of molecules exceeds activation energy.

Answer 81

Check Maxwell-Boltzmann distribution of molecular speeds

Question 82

How are stetric factors added to collision theory?

Answer 82

With the inclusion of a factor P, P is usually <1.

Question 83

From collision theory, state expressions for rate constant and rate of reaction.

Answer 83

Check expressions

Question 84

What happens in CVD?

Answer 84

Precursors are supplied and react with the surface of the heated substrate forming a thin film of desired material.

Question 85

What regime does CVD typically take place in?

Answer 85

Viscous flow where Kn<<1

Question 86

What kind of flow takes place over the substrate in CVD and how is it achieved?

Answer 86

Laminar flow by low velocities.

Question 87

On a substrate surface in CVD a conc gradient forms, what can this be thought of in terms of?

Answer 87

Flux (mass).

Question 88

Derive an expression for thin film growth in CVD.

Answer 88

Check expression.

Question 89

Derive Clausius Equation of State.

Answer 89

Check derivation

Question 90

Give Lennard-Jones potential expression, and state variables.

Answer 90

Check expression.

Question 91

Derive VdW equation of state from Clausius equation of state.

Answer 91

Check derivation

Question 92

How does thermal conductivity deviate from expected relationship at high and low pressure?

Answer 92

Expected κ proportional to sqrt(T) and κ independent of pressure. At high pressure, molecules interact At low pressure, molecules don't collide Kn>>1 κ doesn't quite follow sqrt(T) as hard sphere model was assumed. By modelling with soft sphere κ proportional to 1/sqrt(T) which is more accurate.

Question 93

State equipartition principle.

Answer 93

In a system in thermal equilibrium at temperature T, the mean energy associated with each degree of freedom is the same and equal to 1/2 kT.

Question 94

What degrees of freedom are there for a monatomic gas?

Answer 94

Three translational.

Question 95

What is can a degree of freedom be considered as for the equipartition principle?

Answer 95

Each squared term entering into the equation for total energy of the particle.

Question 96

What are the energy types that contribute to the degrees of freedom for a diatomic gas?

Answer 96

Kinetic Rotational Vibrational

Question 97

How many degrees of freedom does each vibrational mode have?

Answer 97

2, a potential and kinetic energy.

Question 98

Heat capacity equation.

Answer 98

Heat capacity = dU/dT

Question 99

Derive expressions for the molar heat capacities of a monatomic gas (Cp and Cv)

Answer 99

Check

Question 100

What happens to energy in diatomic molecules as temperature is reduced?

Answer 100

Certain degrees of freedom are frozen out.

Question 101

Sketch a graph of Cv against Temperature, showing degrees of freedom.

Answer 101

Check graph.