Download Section 5.3 - 1 5.3 Paramagnetism • Paramagnetism originates from

5.3
Paramagnetism
•
Paramagnetism originates from the angular momenta of unpaired electrons...
i.e., randomly oriented, rapidly reorienting moments.
•
In a paramagnetic material, there is no permanent, spontaneous magnetic moment.
i.e, if H = 0, then M = 0
•
The electron spins (or more correctly, the magnetic moments) are non-cooperative.
•
All forms of magnetism, other than diamagnetism and paramagnetism, are governed
by a critical temperature (TC, TN) below which there is some cooperativity of the
magnetic moments. Above these critical temperatures → PARAMAGNET!
•
χpara varies with T
.... let’s look at this more closely now ...
(NOTE: in the text below, χ refers to molar paramagnetic susceptibility)
VII.
The molar magnetic susceptibility χ is a bulk measurable quantity that can be
related to the average magnetization per mole (m),
i.e., average moment per molecule:
!
mN A
χ= !
H
VIII.
where NA is Avogadro’s number
For a collection of identical molecules, each will have its own magnetic moment
(µ) and will be free to orient itself in an applied magnetic field H. Some
alignment will occur, but the thermal motion of the molecules will interfere
preventing “perfect” alignment … so the average molar magnetization m is
related to the individual molecular magnetic moment vectors µ by both applied
field H and by temperature T (proven by Langevin):
!
! µ2H
m=
3kT
where k is the Boltzmann constant
k = 0.69503877 cm-1 K-1
Section 5.3 - 1
IX.
If we combine the last two equations, we get the Curie law –
“magnetic susceptibility is inversely proportional to the absolute temperature”.
N Aµ 2 C
χ=
=
3kT
T
where C is the Curie constant
C = NAµ2/3k
This equation directly relates the molar magnetic susceptibility to the magnetic
moment of each molecule!!
It predicts a linear relationship between paramagnetic susceptibility and T-1,
with χpara = 0 as T → ∞
The Curie Law is the magnetic analog to the ideal gas law.
(Just as intermolecular interactions cause deviations from the ideal gas law,
interactions between unpaired electrons cause deviations from the Curie Law,
especially at low T.)
It is not surprising, then, that true Curie magnetism is not often observed, but
many paramagnetic materials have near Curie behaviour...so the Curie Law
provides a good model for many systems.
X.
It is convenient to express µ in units of Bohr magnetons (β), where one Bohr
magneton is 4.66864374 x 10-5 cm-1 G-1 and is the fundamental quantum of
magnetic moment … so we write µ = µeff β.
1
µeff
XI.
2
⎛ 3kTχ ⎞
⎟ ≈ 8χT
= ⎜⎜
2 ⎟
⎝ N A β ⎠
Now, all that remains is to relate µeff to the total spin angular momentum quantum
number S, or more correctly, to the total angular momentum quantum number J.
µ eff = g J (J + 1)
µ eff = g S (S + 1)
spin only case
spin-orbit coupling case
Note that g can be estimated by:
g = 1+
S ( S + 1) − L( L + 1) + J ( J + 1)
2 J ( J + 1)
Section 5.3 - 2
HOMEWORK:
• Prove to yourself that when L = 0, g = 2.
• Find the spin-only value of µeff in units of Bohr magnetons for a Ni2+ free ion.
• Derive an equation to calculate χ from S in the spin-only case.
• Find the value of χT in units of cm3 K mol-1 for the Ni2+ free ion (spin-only case).
• Using an approximation that has already been given above, show that this “quick”
calculation is valid: χT ≈ [S(S+1)]/2
S
1/2
1
3/2
2
5/2
3
7/2
Values of (4/g2)χT and µeff as a function of total spin S
(4/g2)χT
µeff (spin only, for g = 2)
3
-1
(cm K mol )
(units of Bohr magnetons)
0.375
1.73
1.000
2.83
1.876
3.87
3.001
4.90
4.377
5.92
6.002
6.93
7.878
7.94
The above account has given us an easy step-by-step method for relating the measured
magnetic susceptibility to the value of S (or the value of J) for a molecular paramagnetic
material that has near Curie behaviour.
In order to understand more complicated paramagnetic behaviour, we have to delve a
little deeper by deriving the Van Vleck equation, and showing that the Curie Law is a
special case that can be obtained from this equation.
... but first things first...
What is the origin of g ?
•
Consider an atom with one unpaired electron that is spherically distributed about the
nucleus such that there is no orbital angular momentum (L = 0).
•
The magnetic moment of the electron can align with an applied
external field in either a “parallel” or “anti-parallel” orientation,
depending on the two possible quantum states (ms = +1/2, -1/2)
Lande constant g = 2.0023192778 for a free electron
"↑
µ
= −mS g β
!n
↓
Magnetic moment of an electron in quantum state n
Section 5.3 - 3
•
g is a coefficient that must be used in order to account for the fact that the ratio of the
magnetic moment of an electron and the spin angular momentum differ from the
classical value (i.e., g = 1)
... in other words to offset the fact that ms is not a unit vector!
µn = −(!
± 12 ) gβ
↓
The “unit” vector of ms is NOT in fact unity!
•
For a long time it was thought that g for electron spin was exactly 2 (as would follow
from the above equation), but it has now been found, both experimentally and
theoretically, that the value for a free electron is 2(1.001145 ± 0.000013)
•
g can deviate dramatically from ≈ 2, depending on how “free” the unpaired electron is
in material. For those of you familiar with EPR spectroscopy, you will know that the
g-value is akin to chemical shift in NMR spectroscopy. The more delocalized the
electron is in a molecular orbital, the closer the g-value is to ≈ 2.
•
Furthermore, g may not have the same value in all spatial directions. In other words,
the atom or molecule in question may be magnetically anisotropic. Solid state or
frozen solution EPR spectroscopy can be used to determine the values of gx, gy and
gz. In a room temperature solution (particularly of an organic radical) the solute is
tumbling in solution rapidly enough that an average g-value is usually observed:
gavg2 = (gx2 + gy2 + gz2)/3
In most cases, magnetic susceptibility data are recorded on polycrystalline samples,
so an average g-value is observed for these measurements as well.
Deriving the Van Vleck equation
•
Molar paramagnetic susceptibility (χ) characterizes the way in which an applied
magnetic field H interacts with the angular momentum associated with the thermally
populated states of a molecule.
•
In CLASSICAL MECHANICS, when a sample is perturbed by an external magnetic
field, its magnetization is related to its energy variation with applied field:
M=
•
− ∂E
∂H
In QUANTUM MECHANICS, we can define a microscopic magnetization µn for
each energy level En (n = 1, 2, 3, ... ) as:
− ∂En
µn =
∂H
Section 5.3 - 4
•
The macroscopic molar magnetization M is then obtained by summing the
microscopic magnetizations weighted according to the Boltzmann distribution law:
M = NA
⎛ − ∂En ⎞
⎟ exp (− En kT )
∂H ⎠
∑ exp(− En kT )
∑ ⎜⎝
n
n
This expression may be considered as the fundamental expression in
molecular magnetism. It does not lean on any approximations!
•
However, it is difficult to apply this equation because it requires the knowledge of
En = f(H) variations for all thermally populated states in order to calculate the
derivatives ∂En/∂H.
•
In 1932, Van Vleck proposed a simplification of the above general equation based on
a few approximations:
1. It is legitimate to expand the energies En according to an increasing power series
in H.
Energy of level n in zero field (H = 0)
↓
!
( 0)
(1)
( 2)
En = En + En H + En H 2 + #
"
"
↑
↑
First and Second Zeeman coefficients (i.e., energies dependent upon the
magnitude of the applied magnetic field)
Note that using µn = -∂En/∂H, we get the Van Vleck expression for microscopic
magnetization:
µn = − En (1) − 2 En ( 2 ) H + !
2. The second approximation is that H/kT << 1 so that
⎛ − En ( 0 ) ⎞⎛ En (1) H ⎞
⎛ − En ⎞
⎟⎜1 −
⎟
exp⎜
⎟ = exp⎜⎜
⎟⎜
kT ⎟⎠
⎝ kT ⎠
⎝ kT ⎠⎝
Q. When is the V.V. eq’n not valid?
Section 5.3 - 5
So where do these two approximations get us?
We are starting with our fundamental expression:
∑ µ exp (− E kT )
∑ exp (− E kT )
n
M = NA
n
n
n
n
where µn is the magnetic moment of state n
µn = -∂En/∂H
•
Let’s look at the term exp(-En/kT) ...
(
)
⎡ − En ( 0) + HEn (1) + H 2 En ( 2 ) + ! ⎤
− En ⎞
⎛
exp⎜
= exp ⎢
⎥
kT ⎟⎠
⎝
kT
⎣
⎦
⎛ − En ( 0) ⎞ ⎛ − HEn (1) ⎞ ⎛ − H 2 En ( 2 ) ⎞
⎟ exp⎜
⎟ exp⎜
⎟!
= exp⎜⎜
⎟ ⎜
⎟ ⎜
⎟
kT
⎝ kT ⎠ ⎝ kT ⎠ ⎝
⎠
For the sake of simplicity, we will ignore the H2 and higher order potentials...
Why can we do this?
•
Recall:
for small x,
exp(-x) ≈ 1 – x
Then, ignoring H2 and higher order exponentials...
⎛ − En ( 0 ) ⎞
⎛ − HEn (1) ⎞
⎛ − En ( 0 ) ⎞⎛ HEn (1) ⎞
⎟ exp⎜
⎟ ≈ exp⎜
⎟⎜
⎟
exp⎜⎜
⎟
⎜ kT ⎟
⎜ kT ⎟⎜1 − kT ⎟
⎝ kT ⎠
⎝
⎠
⎝
⎠⎝
⎠
NOTE: This assumes that the first order Zeeman splitting is much
smaller than exchange coupling (we will look at this soon...)
•
The magnetic moment of state n, ignoring H2 and higher order terms is ...
µn =
− ∂En
(1)
(2)
≅ − En − 2 HEn
∂H
Section 5.3 - 6
Substituting these approximations into our fundamental expression, we get ...
M=
•
⎞ ⎛ − En ( 0) ⎞
⎟ exp⎜
⎟
⎟ ⎜ kT ⎟
⎝
⎠ ⎝
⎠
(1)
( 0)
⎛ HE ⎞ ⎛ − E ⎞
∑n ⎜⎜1 − kTn ⎟⎟ exp⎜⎜ kTn ⎟⎟
⎝
⎠ ⎝
⎠
(
(1)
N A ∑ − En − 2HEn
n
( 2)
)⎛⎜⎜1 − HEkT
(1)
n
In zero field (H = 0), the magnetization vanishes (M = 0)
i.e.
∑E
n
n
(1)
⎛ − En ( 0) ⎞
⎟ = 0
exp⎜⎜
⎟
⎝ kT ⎠
prove this to yourself!
NOTE: by definition, this excludes materials exhibiting spontaneous
magnetization ... we are only considering paramagnetic species.
•
If we combine these two equations, retaining only linear terms in H, and we divide
both sides by H, we get ...
⎛ En (1) 2
⎛ − En ( 0 ) ⎞
( 2 ) ⎞
⎜
⎟
⎟
N A ∑ ⎜
− 2 En ⎟ exp⎜⎜
⎟
kT
kT
M
n ⎝
⎠
⎝
⎠
=χ=
( 0)
H
⎛ − E ⎞
∑n exp⎜⎜ kTn ⎟⎟
⎝
⎠
Van Vleck Equation
•
Now, all we need to know are the quantities: En(0), En(1) and En(2)
•
From a theoretical standpoint, the Van Vleck equation can be used when the
eigenvalues En(0) and eigenfunctions |n> of the Hamiltonian in zero-field are known.
•
Then En(1) and En(2) can be calculated through perturbation theory as:
En(1) = n Hˆ ZE n
E
(2)
n
where Σ' means the summation runs over the levels m with Em(0) ≠ En(0).
2
n Hˆ ZE m
ĤZE is the Zeeman operator, which accounts for the interaction between
= ∑ ' ( 0)
( 0)
(
E
−
E
)
m
the magnetic field and the electronic angular momenta:
n
m
ĤZE = βΣi(Ii+gesi)·H
where Ii and si are the orbital and spin momenta operators of electron i, respectively.
Section 5.3 - 7
Okay, so this is still not very helpful in terms of applying the Van Vleck equation, but at
least we can start to see a few useful things ...
•
When all energies En are linear in H, the second-order Zeeman coefficients En(2)
vanish and we get a simplified version of the Van Vleck equation:
2
χ=
N A ∑n En(1) exp(− En( 0) kT )
kT ∑exp(− En( 0) kT )
n
•
We can now have a closer look at Temperature Independent Paramagnetism (TIP):
If the only thermally populated state of a molecule is a spin singlet (S = 0) without
first-order angular momentum, then the paramagnetic susceptibility is intuitively
expected to be zero, and the measured susceptibility will be negative.
i.e.
χpara = 0
therefore χexp = χdia
However, in some cases this is not the full story...and this can be accounted for by
looking at the Van Vleck equation.
Let the energy E0(0) of the ground state be the energy origin (i.e. equal to zero).
Since this state has no angular momentum – because it is diamagnetic – ...
E0(1) = 0
and
χpara = -2NAE0(2)
from the V.V. equation
Or
0 Hˆ m
χ = −2 N A ∑ ( 0) ZE ( 0)
m≠0 ( E0 − Em )
2
So...the diamagnetic ground state may couple with excited states through the Zeeman
perturbation – provided that they are close enough in energy.
χ is positive since all denominators are negative, and temperature independent. This
contribution is called temperature independent paramagnetism.
TIP is usually small, often on the same order of magnitude as the diamagnetic
contribution to the magnetic susceptibility, but of opposite sign.
For example, it is estimated to be about
60 x 10-6 cm3 mol-1 for copper(II) mononuclear species
100 x 10-6 cm3 mol-1 for nickel(II) mononuclear species
200 x 10-6 cm3 mol-1 for octahedral [Co(NH3)6]3+ possessing a 1A1g ground state
Section 5.3 - 8