Para magnetism •The permanent magnetic moment results from the following contributions: • The spin or intrinsic moments of the electrons. •The orbital motion of the electrons. •The spin magnetic moment of the nucleus. •Paramagnetism is observed in: Metals •Atoms and molecules possessing an odd number of electrons, that is free sodium atoms, gaseous nitric oxide etc.
Classical Theory of Paramagnetism Langevin’s theory of Para magnetism: (a) In natural conditions (in the absence of external magnetic field)
Net dipole moment is zero μ Total =0
Sample
Langevin’s theory of Para magnetism contd.. (b) In the presence of external magnetic field: Dipole align themselves along the direction of external applied magnetic field .
In order to have to • have a state of lower energy. B •
•
the net magnetisation of the sample become non-zero and parallel to the magnetic field. For large magnetic field’s at low temp. the magnetisation become of the sample become constant. This T his implies that saturation is achieved and all the dipoles are aligned along the direction of Magnetic Field.
Langevin’s theory of Para magnetism contd.. The theoretical explanation for the same was given by Langevin’s on the basis of Boltzmann’s classical statistics.
acc. To this theory each atom consists of a permanent magnetic dipole moment μ. Therefore component of
μ
μ along the field direction is μ Cos θ
One atom case
B
Langevin’s theory of Para magnetism contd.. The magnetic potential energy of the atom in the magnetic field is W=-μ.B = - μB cos θ.---(1) then acc. To Boltzmann classical statistics, the number of atoms having P.E. in the range
W
W+ dW
dn = Ce
−
W kT
dW
is
Langevin’s theory of Para magnetism contd.. dn = Ce
−
W kT
dW
− − − − − − − − − ( 2)
substituting the value of W and dW µ B cosθ µ B sin θ d θ kT
dn = Ce
− − − − − −(3)
as W = − µ B cosθ dW = µ B sin θ d θ
Now total number of atoms of the specimen as a whole having energy range W to W + dW can be calculated by int egrating over all possible orientations of θ from 0toπ . π
∫ ∫
n = dn = Ce µ B cosθ / kT µ B sin θ d θ 0
− − − −(4)
Langevin’s theory of Para magnetism contd.. π
∫ ∫
n = dn = Ce µ B cosθ / kT µ B sin θ d θ this implies 0
C =
n
− − − − − − (5)
π
∫
e µ B cosθ / kT µ B sin θ d θ
0
as each atom contribute s µ cos θ along the direction of magnetic field thus total magnetisat ion ofthe sample in the presence of magnetic field is π
∫
M = n µ cos θ = Ce µ B cosθ / kT µ B sin θ µ cos θ d θ
−−−−
(6)
0
or
= π
n
∫
e µ B cosθ / kT µ B sin θ d θ
0
π
× ∫ e µ B cosθ / kT µ B sin θ µ cosθ d θ − − − − (7) 0
Langevin’s theory of Para magnetism contd.. M =
π
n π
∫
e µ B cosθ / kT µ B sin θ d θ
× ∫ e µ B cosθ / kT µ B sin θ µ cos θ d θ 0
0
π
∫
e µ B cosθ / kT sin θ cos θ d θ
or M = n µ 2 B
0
π
∫
e µ B cosθ / kT µ B sin θ d θ
0
µ B = x and cosθ = y so that − sin θ d θ = dy now let us write kT −1
∫
e xy ydy
thereforeM = n µ +−11
∫
e xy dy
as θ = 0, y = 1 and as θ = π y = −1
Langevin’s theory of Para magnetism contd.. −1
xy
∫
ye
e xy ydy
As M =
= n µ
n µ +−11
x
−1
−1
− ∫ dy 1
1
xy
e
∫
e xy dy
x
+1
e xy x
−1
1
−1
e e e 1 x 1 − x − x − x − x − x 2 [ ] − e + e − 2 [e − e x ] 1 = n µ x x or M = n µ 1 x e − x e x − [e − e − x ] − − x
x
xy
x
x
x
1
[e + e ] − [e − e − ] − 1 1 e e + x ⇒ M = n µ = n µ − − − = n µ coth x − [e − e ] x e − e x x
− x
x
x
x
x
x
x
x
x
Langevin’s theory of Para magnetism contd.. Now as component of magnetic dipole moment of each atom along B is μcosθ and when dipole moment is fully aligned along the direction of applied magnetic field then it is only μ as θ=00
M = n µ coth x −
coth x − 1 M = s x x
M = M s [ L( x)] ⇒
1
M M s
= [ L( x)]
Case I : For l arg e values of x = M M s
=1
µ B kT
confirms saturation.
μ B
Langevin’s theory of Para magnetism contd.. co incides Case-II For small values of x the curve is linear and coincides with the tangent to the curve at the origin. 1 = coth x − on exp anding coth x M s x 1 x x 3 1 x M = M s + − .... − = n µ x 3 x 3 45 ignoring higher values of x for small value of x µ B n µ 2 B n µ 2 µ 0 H M = n µ = = kT kT 3 3 3kT M
Langevin’s theory of Para magnetism contd..
now as M =
n µ 2 µ 0 H 3kT
therefore χ =
M
=
n µ 2 µ 0
=
M s2 µ 0
=
C
3kT 3nkT T H this is called as Curie' s Law and
C =
n µ 2 µ 0 3k 1
as χ ∝
T
=
M s2 µ 0 3nk
is called as Curie' s cons tan t .
Langevin’s theory of diamagnetism
The magnetic moment associated with the orbital motion of electron is related to the corresponding angular momentum L through the relation. ρ
e L µ = − 2m
y
r
In accordance with the Faraday F araday law
∫
Edl = −
ORBIT
d
∫
B.dS dt ORBIT
E ( 2π r )
=−
d
( B.π r )
dt 1 dB ∴ E = − r 2 dt
2
r
N B
S
Langevin’s theory of diamagnetism contd.. • The induced electric field exerts a torque = − eEr on electron, which in turn produces a change in angular momentum in accordance with the equation
=
dL
τ
dt
−eEr =
dL dt
dB dL 1 −e − r r = dt 2 dt The change in angular momentum ∆ L
of electron during the magnetic field changes from 0 to B is obtained by integrating above equation. Thus
∆ L =
1 2
er 2 B
Langevin’s theory of diamagnetism contd.. This change in angular momentum produces change in magnetic moment
e e 2 B 2 ∆ µ = − ∆ L = − r 2 m 4 m
∆ µ ATOM = −
e 2 B
Z
Where Z is the number of e in the atom
r ∑ 4m
2
i
i =1
=− Any arbitrary orbit radius
e 2 B 4m
ρ 2
x
2
=
y
2
=
Average radial distance of e in the atom
Z r 2 x 2
=
+ 2
z
y 2
=
+ 1 3
z 2
ρ 2
Spherical symmetrical
∴
r 2
=
x 2
+
y2
=
2 3
ρ 2
Langevin’s theory of diamagnetism contd..
• the expression for magnetic moment induced in atom µ ATOM
=−
Ze 2 B 2
=−
4m 3
Ze 2 B 6m
ρ 2
ρ 2
there are N atoms per unit volume, the magnetization induced in the specimen is 2 M = N µ ATOM
=−
NZe B 6m
ρ 2
The diamagnetic susceptibility of the specimen is µ 0 HNZe 2 2 χ = ρ =− 6mH H M
µ 0 NZe 2 2 ρ =− 6m
Quantum Theory of Paramagnetism
• Classical theory : The energy of a system is varied continuously. θ =All values of angle are possible. •Quantum theory : The change of energy is discrete i.e θ is discrete θ
= θ 1 ,θ 2 ,θ 3 ⋅ ⋅ ⋅
For each j only 2j+1 projections
.
Space quantization
Only certain particular directions in space is possible. ⇒ It is expressed in terms of angular momentum rather than magnetic moments. moments.
• Spectroscopic spliting factor(g)
g:spectroscopic splitting factor orbital motion : g=1, spin motion : g=2
• Total angular momentum of atoms( J )
J = L + S
L
L : Total angular momentum of electrons S : Total angular momentum of spin • The effective moments( µ eff )
- The net magnetic moments of the atom µ eff
= g
eh
J
J ( J + 1)erg / Oe =
4π mc = − g J µ
B
S
=
J ( J
+
1)
=
L ( L
+
=
S ( S
+
1) 1)
h
2 h
2 h
2
Quantum Theory of Paramagnetism contd.. • The component of µ eff in the direction of the applied field = − g m J
µ
z
µ
B
mJ : Quantum number associated with J J, J-1, J-2, ……-(J-2), -(J-1), -J
• Measurement of g
g =1 +
J ( J +1) + S ( S +1) − L( L +1) 2 J ( J +1)
For each value of J , m j has 2J+1 orientations relative to the magnetic field. The potential energy of such a magnetic dipole in the presence of magnetic field is
cosθ W = − µ Z B cos
or
W = m j g µ B B
Quantum Theory of Paramagnetism contd.. Acc to Maxwell-Boltzmann distribution the number of atoms having a particular value of mj is thus proportional to exp
− m j g µ B B / kT
Considering a unit volume of the paramagnetic material containing a total of N atoms the magnetisation in the direction of the field is
average along B
M = N ×
M = N
mj = j
∑
of magnetic moment component per atom
− m j g µ B e − m g µ B / kT
mj = − j
j
e
B
− m j g µ B B / kT
m j g µ B B / kT ( m j g µ B B / kT ) 2 ... − − − m j 1 − mj = j 1! 2! = Ng µ B ∑ m j g µ B B / kT ( m j g µ B B / kT ) 2 mj = − j 1 − − − ... 1! 2!
Now consider the following two cases (i ) at normal flux densities and ordinary temperatur es
m j g µ B B kT
<< 1
and we can exp and the series exp ression of the above exp ression
Therefore, the above expression can be approximated approx imated as
m j g µ B B − m j 1 − kT mj = j M = Ng µ B ∑ m j g µ B B mj =− j 1 − kT mj =+ j
and
∑
m
2 j
=2
mj =+ j
∑
m 2 j
=
mj =0
mj =+ j
now as
mj =− j
3
g 2 µ 2 B B J ( J +1)(2 J +1) therefore M Now
χ
=
= N
µ 0 M
or χ = N µ 0
B
= N
2 J +1 2
=0
J ( J +1)(2 J +1)
mj =− j
kT
∑
m j
2
µ 0 g µ B
3kT
3
= N
g 2 µ 2 B B 3kT
J ( J +1)
J ( J +1)
2 2 P eff µ B
3kT where P eff is effective number of Bohr Magneton.
P eff
= g
J ( J +1)
THIS EXPRESSION IS IDENTICAL TO THE
CLASSICAL EXPRESSION
Quantum Theory of Paramagnetism contd.. (ii) At low temperature and strong fields
m j g µ B B kT
>1
and it is not possible to have a series expansion of the exponential terms. after algebraic manipulations we can write the M value as M = NgJ µ B B J ( x)
where x =
gJ µ B B
and B J ( x ) is
kT x 2 J + 1 2 J + 1 1 coth( ) x − coth( ) the Brillouin function defined as 2 J 2 J 2 J 2 J in this case B J ( x) ≈ 1 therefore M = NgJ µ B which implies state of saturation magnetisat ion
Weiss theory of ferromagnetism The theory of ferromagnetism put forward by Weiss is centered about the following two hypothesis: • A specimen of ferromagnetic material contains a number of small regions called domains which are spontaneously magnetized. The magnitude of spontaneous magnetization of the specimen as a whole is determined by the vector sum of the magnetic moments of individual domains. • The spontaneous magnetization of each domain is due to the presence of an exchange field, B E, which tends to produce a parallel alignment of the atomic dipoles. The field B E is assumed to be proportional to the magnetization M of each domain, i.e.,
B E = λ M
whereλ is a constant called the independent of temperature.
Weiss – field constant and
is
The effective magnetic field on an atom or ion becomes
Beff = B + B E = B + λ M Consider a ferromagnetic solid containing N atoms per unit volume each having a total angular momentum quantum number J.
M = NgJ µ B B J ( x ) B J ( x )
where
=
2 J + 1 2 J
− − − (1)
( 2 J + 1) x 1 x coth − coth 2 J 2 J 2 J
x ≡ gJ µ B Beff k B T
= gJ µ B ( B + λ M )
k B T
In case of spontaneous magnetization, B=0 so above equation becomes
x ≡ gJ µ B λ M k BT
∴
M ( T )
=
xk BT
λ
− − − ( 2)
As T → 0 or x → ∞, B J ( x ) → 1; the magnetic moments align themselves parallel to the field and the magnetization M becomes the saturation magnetization, M S ( 0 ). Thus, we get
M S ( 0 )
= NgJ µ B − − − (3)
From equations (2) & (3), we get
M ( T ) M S ( 0 )
=
xkT 2
2
2 B
λ Ng J µ
− − − (4)
From equations (1) & (3), we get
M ( T ) M S ( 0 )
= B J ( x ) − − − (5)
M ( T )
T>Tc
T=Tc
T
M S ( 0 ) Plot of eqn. (4)
Plot of eqn. (5)
x
0
Graphical solution of the simultaneous equations (4) and (5). A point of intersection determines the spontaneous magnetization Ms(T) at a given temperature. No solution exists for T>Tc.
M/M s 1
0
T/T c
1
Spontaneous magnetization versus temperature for T< T
paramagnetic region For T>Tc, the spontaneous magnetization is zero and an external field will have to be applied to produce some magnetization. This field should, however, be weak enough to avoid the saturation state. In such a state, we find that x<<1 and we can write
J + 1 B J ( x ) ≅ x 3 J So, expression (1) becomes
M = Ng µ B ( J + 1) x 3 x =
gJ µ B ( B + λ M ) k B T
Ng 2 µ B J ( J + 1) 2
Thus,
M =
( B + λ M )
Which gives
χ
µ 0 M =
µ 0T C / λ
=
B
T
C =
where
−
T C
=
T
T C
−
T C
µ 0 T C λ
λ Ng µ B J ( J + 1) 2
and
C =
2
3k B
The expression (6) is called Curie-Weiss law,
---(6)
Weiss theory is a good phenomenological theory of magnetism, But does not explain source of large Weiss field.
Heisenberg and Dirac showed later that ferromagnetism is a quantum mechanical effect that fundamentally arises from Coulomb (electric) interaction.
Magnetic hysteresis • Hysteresis is well known in ferromagnetic materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized.
The relationship between magnetic field strength (H) and magnetic flux density (B) is not linear in such materials. If the relationship between the two is plotted for increasing levels of field strength, it will follow a curve up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation.
From the hysteresis loop, a number of primary magnetic properties of a material can be determined. Retentivity - A measure m easure of the residual flux density corresponding to the saturation induction of a magnetic material. In other words, it is a material's ability to retain a certain amount of residual magnetic field when the magnetizing force is removed after achieving saturation. (The value of B of B at point b on the hysteresis curve.) c urve.) Coercive Force - The amount of reverse magnetic field which must be applied to a magnetic m agnetic material to make the magnetic flux return to zero. (The (T he value of H of H at point c on the hysteresis curve.)
Hard and Soft Magnets
