Two State Paramagnet

Question 1

The Two Level Paramagnet

Description

Answer 1

• paramagnetic salts with one unpaired electron have only two possible quantum states
• the magnetic moment of the electron can be either +μ or -μ
• in an external magnetic field the spins prefer to line up with the field, but entropy randomises the spins to increase multiplicity

Question 2

The Two Level Paramagnet

Microstates

Answer 2

-a microstate is specified if the directions of all spins of the unpaired electron in every atom is specified

Question 3

The Two Level Paramagnet

Macrostates

Answer 3

-a macrostate is specified by the total number of dipoles that point up (Nu) and the total number of dipoles that point down (Nd)

Question 4

The Two Level Paramagnet

N

Answer 4

N = Nu + Nd

-the total number of dipoles in the body

Question 5

The Two Level Paramagnet

μ

Answer 5

-the magnetic moment of an individual dipole

Question 6

The Two Level Paramagnet

Energy Per Dipole in the Field

Answer 6

-for dipoles lined up parallel to |B:
-μB
-for dipoles lined up antiparallel to |B:
+μB

Question 7

The Two Level Paramagnet

Total Magnetic Moment

Answer 7

|M = |μ( Nu - Nd )
-and since Nd=N-Nu,
|M = |μ (2Nu - N)

Question 8

The Two Level Paramagnet

Energy of a Macrostate

Answer 8

E = μB (Nd - Nu) = μB (N-2Nu)

Question 9

How many microstates do two spins have?

Answer 9

-two spins have 4 possible microstates:
uu, ud, du, dd
-this corresponds to three macrostates
-for the two microstates ud and du, M=0
-for the microstate uu, M=2μ
-for the microstate dd, M=-2μ

Question 10

How many macrostates do two spins have?

Answer 10

• the macrostates can be classified by their moment M, and their multiplicity
• there are three macrostates, M= 0, +2μ, -2μ
• for M=0, Ω=2
• for M=+2μ , Ω=1
• for M=-2μ , Ω=1

Question 11

How many microstates and macrostates do three spins etc. have?

Answer 11

• the number of microstates is given by 2^N where N is the number of spins (so for three spins there are 8 microstates)
• the number of macrostates is (N+1), so for three spins there are 4 macrostates
• the multiplicity of each macrostate is given by the lines of pascals triangle so for N=3, the multiplicities are 1, 3, 3, 1

Question 12

The Two Level Paramagnet

Calculating Number of Microstates, Probability of Being in a Particular Microstate, Multiplicity

Answer 12

number of equally probable microstates = 2^N
probability of being in a particular microstate = 1/(2^N)
multiplicity of a two-state paramagnet:
Ω(N,Nu) = N! / (Nu! Nd!)
= N! / (Nu! (N-Nu)!)

Question 13

The Two Level Paramagnet

Effect of B

Answer 13

-for B=0 the energy of each macrostate is the same

Question 14

How does the energy of a state effect its probability?

Answer 14

-the higher energy the state is the lower the probability it will be occupied

Question 15

Boltzmann Factor

General Definition

Answer 15

• a Boltzmann factor is an exponential function

- the Boltzmann factor gives the fraction of atoms in one energy state compared to another

Question 16

Boltzmann Factor

Nu / Nd

Answer 16

Nu/Nd = exp (2μB/kT)

Question 17

Boltzmann Distribution

General Definition

Answer 17

-gives the distribution between states of different energies

Question 18

Boltzmann Factor

Nd

Answer 18

Nd = exp (-μB/kT)

Question 19

Boltzmann Factor

Nu

Answer 19

Nu = exp (μB/kT)

Question 20

Boltzmann Factor

N

Answer 20

N = exp (μB/kT) + exp (-μB/kT)

Question 21

Boltzmann Factor

General Formula

Answer 21

-the boltzmann factor for a particular energy state is given by:

exp (-ε/kT)

-where ε is the energy of the state

Question 22

Single Particle Partition Function

General Definition

Answer 22

• it is the Boltzmann factor associated with each state added together
• in principle a partition function can be written for any system provided all of the energy states are known
• the number of terms in the partition function is equal to the number of energy states in the system
• the symbol for partition function is Z1

Question 23

The Two Level Paramagnet

The Single Particle Partition Function

Answer 23

Z1 = exp(-μB/kT) + exp(μB/kT)

Question 24

Occupational Probability of a Two-State Paramagnet With Temperature

Answer 24

-at lower temperatures, the lower energy state is
occupied
-as temperature increases, the two states become more equally occupied
-at very high temperatures the two states become equally populated
-these trend are seen in all systems not just the two state

Question 25

Internal Energy of the Two-State Paramagnet With Temperature

Answer 25

• asymptotically approaches zero
• increases towards a maximum value
• there is a temperature at which the maximum rate of change of states occurs
• heat capacity would follow the gradient of this graph

Question 26

Two-State Paramagnet

Entropy Equation

Answer 26

S = Nk [ ln(2cosh(μB/kT) - μB/kT tanh(μB/kT) ]

Question 27

Two-State Paramagnet

Entropy as a Function of Temperature

Answer 27

• follows the same shape as internal energy
• lower magnetic field results in higher entropy at every temperature
• because the field causes alignment so a weaker field means more disorder and therefore higher entropy

Question 28

Two-State Paramagnet

Heat Capacity and Temperature

Answer 28

• at low temperatures heat capacity is small because kT«2μB, thermal fluctuations are rare and it is hard for the system to absorb thermal energy, this is universal for systems with energy quantisation
• at high temperatures, Nu~Nd and it is hard for the system to absorb thermal energy, this is not universal behaviour, it occurs if the system’s energy spectrum occupies a finite interval of energies

Question 29

Two-State Paramagnet

Internal Energy Equation (and Magnetic Moment)

Answer 29

E = -NμB tanh(μB/kT) = -MB

-where M = Nμ tanh(μB/kT) is the total magnetic moment

Question 30

Magnetic Moment - High Temperature Limit

Curie’s Law

Answer 30

M = Nμ tanh(μB/kT)
-for x«1 , tanh(x) -> x
-so in the high temperature limit T->∞ => 1/T->0 :
M = Nμ * μB/kT = Nμ²B/kT
-increasing B orders spin and increase M
-increasing T disorders spin and decreases M