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These materials don't possess permanent dipole and hence the magnetic effects are smallSince the induced magnetic moment always oppose the applied field, the diamagnetic susceptibility is negative.
It is always temperature independent.
Diamagnetic materials usually repel the magnetic line of force.
Summary Diamagnetism
e.g.: Bi, H2O, CO2, Ge, Si etc.
Material Χ = (µr -1)
Cu -9.5x10-6
Al2O3 -5.0x10-6
Au -3.7x10-5
Ge -0.8x10-5
Si -0.3x10-5
Se -1.7x10-5
He -0.5x10-5
Para-magnetism
• B>B0 , is positive, μr>1, .
• The permanent magnetic moment results from the following contributions:
1. The spin or intrinsic moments of the electrons.
2. The orbital motion of the electrons.
3. The spin magnetic moment of the nucleus.
A form of magnetism which occurs only in the presence of externally applied magnetic field and materials are attracted to magnetic field.
Paramagnetism
• Paramagnetism is observed in:1. Metals2. Atoms and molecules possessing an odd number of
electrons, that is free sodium atoms, gaseous nitric oxide etc.
3. A few compounds having an even number of electrons (example Oxygen molecule)
4. Free atoms or ions having a partially filled inner shell e.g. rare earth and actinide elements, ions of some transition elements such as Mn2+ Manganese, platinum, tungsten, some members of rare earth group and ions formed by removing and adding electrons to basic atoms there by creating unpaired spins.
Basic assumptions of Langevin’s classical theory
• The theory considers the paramagnetic solids in terms of paramagnetic gas, in which each particle is assumed to bear a permanent magnetic moment .
• Mutual interaction between dipoles is assumed to be negligible.
• Orientation of permanent magnetic dipole moment.
Classical Theory of Paramagnetism
T
CCurie’s law :
)(
T
CCurie-Weiss law :
. Langavin’s analysis
. Temperature dependence of paramagnetism
- Paramagnetism has net magnetic moments:
• No field : M=0 • Field is applied, low Temp.
B• Field is applied, and High Temp.
B
T
Classical Theory of Paramagnetism
:μ(vector)
cosBEP
• The Magnetization(M) =The total magnetic moments per unit volume
0cos
0cos
sin
cossin
de
dena
a
n dnM 0 cos n a deKdnn 0 0
cos sin2 0cos cossin2 deK a
0
cossin
2d
nK
ea
B
. Langavin Function
• Let dn be the number of moments inclined at an angle between & +d•When no field is applied dn dA (solid angle)•When field is applied dn is proportional to dA x Boltzmann factor
.)sin(cos
exp2exp dkT
BK
kT
EKdAdn p
aan
aee
een
dxe
dxexn
aa
aa
ax
ax
1coth
1
11
11
ddx
x
sin
cos
aa
M
M 1coth
0
nM 0
Langavin function :
945
2
453)(
53 aaaaL
T
Ba
• Conclusions from Langavin function: 1. Saturation will occur if a is large enough. ⇒ Large B or low T is necessary 2. At small a, the magnetization M varies linearly with B.
. Relationship between Langavin theory and Curie’s lawFor small a, L(a)=a/3=B/3kT, M=M0 L(a)
.
.3
,
32
0
200
kTBforvalid
constCuriek
nCwhere
T
C
kT
n
B
M
HM
χ is called paramagnetic volume susceptibility, n is number density.
Mass susceptibility :
kgNumber
mNumber
N
N
mm /
/,
3
Molecular Susceptibility :
N should be replaced by NA
Understand some more formulae from S.O.Pillai – for Numerical
Note:
. Curie-Weiss law; Wiess Theory of Paramagnetism:
Lengevin’s theory failed to explain the complicated type of dependence of susceptibility upon temperature exhibited by many paramagnetic substances
e.g. Compressed & cooled gases
Solid salts & crystals etc.
Moreover, this theory does not throw light on the intimate relationship between para & ferromagnetic materials
To overcome these problems, the concept of intermolecular field was introduced by Weiss, on assuming the mutual influence of magnetic moments.
Molecular field (Hi) : The interaction of elementary moments with one another
Let this internal molecular field Hi be represented in terms of its own magnetization
Hi = λ M
λ - molecular field coefficient
Quantum Theory of Paramagnetism
. Quantum theory and Classical theory cosBEP
• Classical theory : The energy of a system is varied continuously. =All values of angle are possible. •Quantum theory : The change of energy is discrete i.e is discrete
321 ,,
L
ml=+l
ml = - l
B
• The component of in the direction of the applied field eff
BJH MgMJ : Quantum number associated with J J, J-1, J-2, ……-(J-2), -(J-1), -J ※J : Integer or half-integer of 1/2 ∞∼
• The effective moments( )eff
OeergJJmc
ehg
eff/)1(
4
BJJg )1(
Quantum Theory of Paramagnetism
. Brillouin function- The procedure of derivation is the same as Langavin, except that.
(1)
(2)
BJH Mg cos
J
JJMd )(0
HMgE BJP
∴ Boltzmann factor = ee kTHMgkTE BJP //
• M = (n atoms per unit volume) ×(The average magnetic moment resolved in the direction of the field
J
J
kTHg
J
J
kTHg
BJ
e
eM
BJ
BJ
M
Mgn
/
/
J
a
JJ
J
J
JngM aBJ 2
`coth
2
1
2
12coth
2
12 `
kT
H
kT
HgMa HBJ
`,
A special case
• If
kT
JJ
kTHgM
B
BJ
3
)1(gthen
If22
Χ = M/H
μ0
Quantum Theory of Paramagnetism
• When a` is small,
`),(3
)1`(`),(
aJBngJMJ
JaaJB
B
kT
Hn eff
3
2
T
C
AkT
N
H
M eff 3
2
Ak
NC eff
3
2X =?
For detail expressions follow S.O. Pillai book
Quantum Theory of Paramagnetism
J
a
Ja
J
J
J
J
M
M
2
`coth
2
1
2
12coth
2
12 `
0
Brillouin function, B(J,a)
1) If J = infinite, B(J,a`) = `
1`coth
aa
)(),( aLaJB 2) If J=1/2(only spin contribution),
`tanh a`)B(J,0
aM
M
. Paramagnetic materials ⇒atoms or ions which have a net magnetic moment because of noncancellation of the spin and orbital component.
• Salts of the transition elements - Incomplete inner shells - Magnetic moments due almost to spin(g 2) - This metal salts obey the Curie or Curie-Weiss law with a small Θ
• Salts and oxides of the rare earths• Rare-earth elements