Random Processes and Statistical Physics Flashcards
Difference between combinations and permutations
Combinations order isn’t important, permutations order is.
Inclusion exclusion principle for 2 events.
P(AuB) = P(A) + P(B) - P(AnB)
Probability of A given B.
P(A|B) = P(AnB)/P(B)
Probability of A given B when A and B are independent.
P(A|B) = P(A) = P(AnB)/P(B)
implies
P(AnB) = P(A)P(B)
What is a random variable?
A variable whose value depends on the outcomes of a random process.
Example of a discrete random variable.
Roll two dice and add the scores.
What is a continuous random variable.
Where the values a random variable can take are continuous.
When finding the probability of a continuous random variable taking a specific value, what happens?
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.
What is it when there is a probability per unit range for a continuous random variable?
The probability density function.
What is the total area under the curve of a probability density function?
1 and is found by integrating.
How do you find the mean from the probability density function?
Integrate the probability density function multiplied by the variable.
Why is standard deviation advantageous over variance?
Standard deviation has the same units as the original variable.
How to find the mode for a continuous random variable?
Differentiate the probability density function.
How to find the median of a continuous random variable?
Intergrate between Q2 and infinity equal to 0.5 and solve for Q2.
How to find the IQR of a continuous random variable?
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
If you multiply a random variable by a scalar, what happens to the mean and variance?
Mean is multiplied by scalar
Variance is multiplied by scalar squared.
What happens to the mean and variance when two random variables are added together?
The means are added
The variances are added.
What is microstate?
Each unique configuration of the system where the state of each element is known.
What is macrostate?
Multiple microstates can give the same macroscopic property.
How to find the most probable state of a system?
Multiplicity is maximised.
Expression for multiplicity.
Binomial distribution expression.
Expression for entropy in terms of multiplicity.
S = klogW
By considering the random walk in 1D, find an expression for the most probable end point.
Check derivation
Compare the random walk in 1D to a Gaussian function.
Check comparison
Model diffusion with random walk model deriving Fick’s first law.
Check derivation.
Derive Fick’s second law of diffusion.
Check derivation
How does rms change with dimension in diffusion?
Rms= N=2dDt
d is no dimensions
D is N/2t
Derive an expression for heat flow between planes.
Check derivation
What is the total length of a polymer chain known as?
Contour length.
Equation for total length of polymer.
L=Nb
How can end to end length of a polymer be expressed?
r = Σl
Derive an expression for rms of a polymer chain of length N using random flight model.
Check derivation
What is the characteristic ratio of a polymer?
Describes bond correlations as in real polymers orientations of bonds are not completely random.
Typical values of characteristic ratios for polymers.
Cn>1
What does the Kuhn model do?
Allows a real chain to be treated as an equivalent freely jointed chain split into Kuhn sgements which represent more than one real bond.
Contour length for Kuhn model.
L = Nbcosψ = NkBk
Derive an expression for the retractive force on a polymer chain when stretched.
Check derivation.
3 properties of ideal gases.
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.
Derive ideal gas equation from lattice model considering M sites containing N particles.
Check derivation
Derive the relation for kinetic energy and temperature by considering pressure.
Check derivation (A level)
Dalton’s law of gases.
In mixtures of gases each gas can be treated independently with partial pressures which sum to the total pressure.
Derive the distribution of the speed of the molecules in a gas.
Check derivation
Find an expression for most likely speed, knowing the distribution of molecular speeds in ideal gas.
Check derivation
Find an expression for rms speed, knowing the distribution of molecular speeds in ideal gas.
Check derivation
Find an expression for mean speed, knowing the distribution of molecular speeds in ideal gas.
Check derivation
Derive an expression for impingement rate using skewed cyclinder.
Check derivation
Derive an expression for flux onto a surface using skewed cylinder.
Check derivation
Derive an expression for pressure using momentum and skewed cylinder.
Check derivation
What is mean free path?
The mean distance travelled between collisions
Derive an expression for mean free path.
Check derivation
Derive an expression for collisions rate per unit volume between molecules of same type.
Check derivation
Derive an expression for collisions rate per unit volume between molecules of different type.
Check derivation.
Derive an expression for effusion rate.
Check derivation
Derive an expression for how pressure changes over the process of effusion and is dependent on mass and temperature.
Check derivation
Give an expression for the ratio of effusion rates in a mixture of gases.
J1/J2 = N1/N2 sqrt(m2/m1)
What is a Knudsen number?
A dimensionless constant that is the ratio of mfp to characteristic dimension of the deposition chamber, L
What happens when Kn»1?
Molecular flow
What happens when Kn«1?
Viscous flow.
What is PVD?
Physical vapour deposition.
Source material vapourised and then condenses on a substrate.
What kind of process is Molecular Beam Epitaxy (MBE)?
A form of PVD
Derive an expression for flux per solid angle.
Check derivation
Derive an expression for flux per unit area on a substrate.
Check derivation
Turn flux per unit area into evapouration rate.
Check derivation.
Derive an expression for flux from heated crucible.
Check derivation
Describe E-beam evapouration
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.
Difference between planetary and semi-planetary geometry in terms of evapouration rate.
Planetary geometry has no angular dependence.
Expression for evapouration rate in planetary geometry around crucible?
Check expression
Describe set up for sputtering
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.
Describe the sputtering process
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.
Ethreshold for sputtering equation.
Check equation
Sputter Yield equation
no. sputtered atoms/no. incident ions
Compare sputtering and evapouration.
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.
Derive a flux for a general transport property.
Check derivation
List flux laws for heat flow, viscocity and diffusion.
Check
List kinetic expressions for thermal conductivity, viscocity and diffusivity.
Check
Expression for rates in differential form:
d[A]/dt = -k[A]
Obtain an rate equation from differential form of rate equation.
Check
Rete constant follows what kind of equation?
Arrhenius
Factors that influence rate of an elementary bimolecular reaction.
Encounter rate
Energy requirement
Steric requirement
Derive an expression for encounter rate based on collsion rate.
Check derivation
State an expression for energy requirement where fraction of molecules exceeds activation energy.
Check Maxwell-Boltzmann distribution of molecular speeds
How are stetric factors added to collision theory?
With the inclusion of a factor P, P is usually <1.
From collision theory, state expressions for rate constant and rate of reaction.
Check expressions
What happens in CVD?
Precursors are supplied and react with the surface of the heated substrate forming a thin film of desired material.
What regime does CVD typically take place in?
Viscous flow where Kn«1
What kind of flow takes place over the substrate in CVD and how is it achieved?
Laminar flow by low velocities.
On a substrate surface in CVD a conc gradient forms, what can this be thought of in terms of?
Flux (mass).
Derive an expression for thin film growth in CVD.
Check expression.
Derive Clausius Equation of State.
Check derivation
Give Lennard-Jones potential expression, and state variables.
Check expression.
Derive VdW equation of state from Clausius equation of state.
Check derivation
How does thermal conductivity deviate from expected relationship at high and low pressure?
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.
State equipartition principle.
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.
What degrees of freedom are there for a monatomic gas?
Three translational.
What is can a degree of freedom be considered as for the equipartition principle?
Each squared term entering into the equation for total energy of the particle.
What are the energy types that contribute to the degrees of freedom for a diatomic gas?
Kinetic
Rotational
Vibrational
How many degrees of freedom does each vibrational mode have?
2, a potential and kinetic energy.
Heat capacity equation.
Heat capacity = dU/dT
Derive expressions for the molar heat capacities of a monatomic gas (Cp and Cv)
Check
What happens to energy in diatomic molecules as temperature is reduced?
Certain degrees of freedom are frozen out.
Sketch a graph of Cv against Temperature, showing degrees of freedom.
Check graph.
