Magnets, Metals and Superconductors

kitefleaUrban and Civil

Nov 15, 2013 (3 years and 4 months ago)

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Magnets, Metals and
Superconductors

Tutorial 1

Dr. Abbie Mclaughlin

G24a

1.

Determine the ground state configuration and predict the
effective magnetic moment for the following Ln
3+
ions.





Gd
3+
, Er
3+

L

0

1

2

3

4

5

6

7

Symbol

S

P

D

F

G

H

I

K

m
j

3

2

1

0

-
1

-
2

3

)
1
(
2


S
S
eff

Gd
3+

= f
7

S = 7/2, L = 0, J = 0

i.e. is spin only!


eff

= 7.94

Term symbol =
8
S
0

m
j

3

2

1

0

-
1

-
2

3

Er
3+

= f
11

The term state symbol is written
2S+1
L
J.

Hund’s Rules:

For less than half
-
filled shells,
the smallest
J

term lies lowest;
for more than half filled shells
the largest J lies lowest.

S = 3/2

L =

m
j

= 6, J = 15/2, 13/2,
11/2, 9/2

Term symbol =
4
I
15/2

g
j
= 6/5
µ
eff

= 9.58


L

0

1

2

3

4

5

6

7

Symbol

S

P

D

F

G

H

I

K

)
1
(
2
)
1
(
)
1
(
2
3






J
J
L
L
S
S
g
)
1
(


J
J
g
J
eff

2 Exam question 1 (2004) (b).

The gradient =1/C.
This can be used to determine S from the
equation:

C =
Ng
2
µ
B
2
S(S+1)/3k

The value of


can be determined from the 1/


vs T plot.

This gives an indication of the strength and nature of the
interactions between neighbouring molecules.

b) Ferromagnetic exchange: S
T

= nS














nS = 10 x 2 = 20.














=41

B



)
21
(
20
2

eff

3a. 2 cyclic
-
[Fe(OMe)(OAc)]
10


Fe
2+

d
6

S = 2 n = 10

a) Antiferromagnetic exchange S
T

= 0 or ½ depending on
whether there is and even or odd number of electrons. There
are 10 antiferromagnetic S = 2 ions, S
T

= 0

M
B
nS
nS
eff
.
)
1
(
2



c) Non interacting (high temperature limit).

S = 2, n = 10






















= 15.5
µ
B
.




)]
1
(
[
2


S
S
n
eff

)]
3
(
2
[
10
2

eff

3b) [Cu
3
OCl
4
(Mepy)
4
]

Cu
2+

d
9
, S = 1/2, n = 3

a) Antiferromagnetic exchange: S
T

= 0 or ½ depending on
whether there is and even or odd number of electrons. There
are 3 antiferromagnetic S = 1/2 ions, S
T

= ½.

)
2
/
3
(
2
/
1
2

eff

µ
eff

= 1.73 µ
B

c) Non interacting (high temperature limit).

S = 1/2, n = 3




























= 3.00
µ
B



b)Ferromagnetic exchange: ST = nS

nS = 3 x 1/2 = 3/2.















= 3.87
µ
B
.


M
B
nS
nS
eff
.
)
1
(
2



)]
1
(
[
2


S
S
n
eff

)]
2
/
3
(
2
/
1
[
3
2

eff

)
2
/
5
(
2
/
3
2

eff

4. Determine

eff

per mole of Cu
2
(OAc)
4
.2H
2
O. Apply a
diamagnetic correction to


and redetermine

eff
.
Does it make
a difference?

T
N
T
k
B
eff




828
.
2
.
3
2


298
10
2
.
1
828
.
2
3




=1.7
µ
B

per mole of dimer

Diamagnetic correction (for dimer)

Cu =
-
11 X 10
-
6
, OAc =
-
30 X 10
-
6
, H
2
O =
-
13 X 10
-
6

Overall correction =
-
22
-
120
-
26 (X 10
-
6
) =
-
168 X 10
-
6


M

=

dia

+

para



para

=

M

-


dia

=1.2 x 10
-
3

+ 168 X 10
-
6

= 1.368 x 10
-
3
.

298
10
368
.
1
828
.
2
3





eff

= 1.806
µ
B

per mole of dimer

Uncorrected = 1.7
µ
B

= per mole of dimer 0.1 difference.
It’s important to correct if you want to be accurate.


2 Exam question 1 (2004) (a).

What is meant by Curie behaviour? Give reasons why
paramagnetic materials may deviate from Curie behaviour and
explain what additional information can be extracted from such
deviations.

The magnetic susceptibility,


(M/H) is dependent on 1/T.

=C/T.

As the temperature increases the increase in thermal energy
gives rise to greater randomisation of the spin orientation and
hence a smaller induced magnetisation.

The Curie constant
C
, comprises a series of fundamental
constants and S, the spin quantum number. Thus from a plot of
1/


Vs T the value of S can be determined form the gradient.

Paramagnetic materials may deviate from Curie behaviour if:

a) there are local ferromagnetic or antiferromagnetic
interactions between spins. The materials can then be
described as Curie Weiss paramagnets.



= C/(T
-

)


When


> 0 it indicates ferromagnetic interactions; if


= 0 we
have ideal Curie behaviour and if


< 0 then it indicates
antiferromagnetic interactions.



can be determined from a plot of 1/


vs T, which should be
linear with an intercept on the T axis equal to

. The larger the
value of


the greater the interaction between spins on
neighbouring molecules.

Paramagnetic materials may deviate from Curie behaviour if:

b) If the material shows Van Vleck behaviour.

This occurs when there is
thermal population of excited states
whose magnetic behaviour is different to that of the ground
state.

For example Eu
3+
. The ground state term is
7
F
0

hence the
predicted

µ
eff

is 0 B.M. Observed values are typically in the
range 3.3
-
3.5 B.M. at room temperature, although the value
decreases upon cooling.

In the case of Eu
3+

the separation of the ground state
7
F
0

and
the first excited state is
ca
. 300cm
-
1
. At room temperature there
is enough thermal energy for the
7
F
1

state to be partially
populated.

On cooling the
7
F
1

state becomes depopulated and the
magnetic moment approaches 0 B.M. as T approaches 0 K
when all the ions are in the
7
F
0

state.

However
a second effect (temperature independent
paramagnetism, TIP) is required to rationalize the data
satisfactorily.