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
