Two State Paramagnet Flashcards
The Two Level Paramagnet
Description
- 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
The Two Level Paramagnet
Microstates
-a microstate is specified if the directions of all spins of the unpaired electron in every atom is specified
The Two Level Paramagnet
Macrostates
-a macrostate is specified by the total number of dipoles that point up (Nu) and the total number of dipoles that point down (Nd)
The Two Level Paramagnet
N
N = Nu + Nd
-the total number of dipoles in the body
The Two Level Paramagnet
μ
-the magnetic moment of an individual dipole
The Two Level Paramagnet
Energy Per Dipole in the Field
-for dipoles lined up parallel to |B:
-μB
-for dipoles lined up antiparallel to |B:
+μB
The Two Level Paramagnet
Total Magnetic Moment
|M = |μ( Nu - Nd )
-and since Nd=N-Nu,
|M = |μ (2Nu - N)
The Two Level Paramagnet
Energy of a Macrostate
E = μB (Nd - Nu) = μB (N-2Nu)
How many microstates do two spins have?
-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
How many macrostates do two spins have?
- 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
How many microstates and macrostates do three spins etc. have?
- 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
The Two Level Paramagnet
Calculating Number of Microstates, Probability of Being in a Particular Microstate, Multiplicity
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)!)
The Two Level Paramagnet
Effect of B
-for B=0 the energy of each macrostate is the same
How does the energy of a state effect its probability?
-the higher energy the state is the lower the probability it will be occupied
Boltzmann Factor
General Definition
- a Boltzmann factor is an exponential function
- the Boltzmann factor gives the fraction of atoms in one energy state compared to another
Boltzmann Factor
Nu / Nd
Nu/Nd = exp (2μB/kT)
Boltzmann Distribution
General Definition
-gives the distribution between states of different energies
Boltzmann Factor
Nd
Nd = exp (-μB/kT)
Boltzmann Factor
Nu
Nu = exp (μB/kT)
Boltzmann Factor
N
N = exp (μB/kT) + exp (-μB/kT)
Boltzmann Factor
General Formula
-the boltzmann factor for a particular energy state is given by:
exp (-ε/kT)
-where ε is the energy of the state
Single Particle Partition Function
General Definition
- 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
The Two Level Paramagnet
The Single Particle Partition Function
Z1 = exp(-μB/kT) + exp(μB/kT)
Occupational Probability of a Two-State Paramagnet With Temperature
-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
Internal Energy of the Two-State Paramagnet With Temperature
- 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
Two-State Paramagnet
Entropy Equation
S = Nk [ ln(2cosh(μB/kT) - μB/kT tanh(μB/kT) ]
Two-State Paramagnet
Entropy as a Function of Temperature
- 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
Two-State Paramagnet
Heat Capacity and Temperature
- 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
Two-State Paramagnet
Internal Energy Equation (and Magnetic Moment)
E = -NμB tanh(μB/kT) = -MB
-where M = Nμ tanh(μB/kT) is the total magnetic moment
Magnetic Moment - High Temperature Limit
Curie’s Law
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
