paramagnetism and elasticity
DESCRIPTION
This document explain about paramagnetism and elasticty in physicsTRANSCRIPT
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February 07 Lecture 6 1
Lecture 6:Lecture 6:ParamagnetismParamagnetism and elasticityand elasticity
Applications of statistical methodsApplications of statistical methods
Aims:Aims: Spin paramagnetism: Paramagnetic salts Curies Law.
Entangled polymers Role of entropy in rubber elasticity.
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February 07 Lecture 6 2
ParamagnetismParamagnetism
Spin systemsSpin systems Some atoms in salts have a permanent
magnetic moment. Example: Gd2(SO4) 3.8H2O, where the Gd3+
ions have a spin moment, S=7/2. General case:General case: Angular-momentum, quantum number, J, gives
paramagnetic moment
Component of magnetic moment is quantised1
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February 07 Lecture 6 3
Spin Spin paramagnetismparamagnetism
Simplest system: a spinSimplest system: a spin--paramagnet.paramagnet. In this case there is no orbital, angular
momentum so J=S. Since S=, there are only 2 values of mJ. Only two energy levels to consider, with energy
+/-B.
Calculate the expectation value of the moment from the weighting given by the BoltzmannDistribution,
( ) ( ) =i
BBi
i kTUkTU expexp
B
m = J -1/2 +1/2
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February 07 Lecture 6 4
Curies LawCuries Law
Spin Spin paramagnetismparamagnetism There are 2 states and, hence, 2 terms in each
summation. Average moment, at temperature, T.
In the limit of high-temperature and/orlow field
The magnetic susceptibility can be measured
( ) ( ){ } ( ) ( ){ }( )kTB
kTBkTBkTBkTB
tanheeee
=
+=
Curies LawCuries Law
kTBkTB
2
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February 07 Lecture 6 5
Paramagnetic saltsParamagnetic salts
ExperimentExperiment Curves at different temperatures and fields
scale to lie on the curve given by Curies Law.
Waldram Theory of Thermodynamics Ch 15, p187
Gd3+, S=7/2Gd3+, S=7/2
Cr3+, S=3/2Cr3+, S=3/2
Fe3+, S=5/2Fe3+, S=5/2
/B/B
B/T (Tesla K-1)B/T (Tesla K-1)
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February 07 Lecture 6 6
Pierre CuriePierre Curieand Magnetismand Magnetism
Curies LawCuries Law The subject of Pierre Curies doctoral thesis,
1895, the same year as his marriage to Marie.
Ferromagnetic to paramagnetic transition at Tc. Paramagnetism in salts ~ 1/T (Curies Law) Diamagnetism is temperature independent.
Died 1906, after a street accident.
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February 07 Lecture 6 7
22--state system: heat capacitystate system: heat capacity
Thermal properties of a 2Thermal properties of a 2--state systemstate system Thermal Energy
Heat Capacity
Note the drop at both high and low temperature. An exception to the rule that systems tend to
the classical, equipartition limit at high T
( ) ( )( ) ( )( )
)/tanh(ln
2exp1lnlnexpexp
kTBBZ
BBZBBZ
=
=++=
+=
( )kTBkT
BkT
C 22
sech
=
=
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February 07 Lecture 6 8
Classical treatment:Classical treatment: Any stretched string (metal or rubber)
Length, l; Tension f(T,l). Tension, f, and other thermodynamic quantities
depend on l and T. Start from the First Law:Start from the First Law:
Entropic contribution to Entropic contribution to elasticityelasticity
lflSTT
TST
lfSTUWQU
Tldd
dddddd
+
+
=
+=
+=
TTU
ld
ll
UT
d
AA
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February 07 Lecture 6 9
Classical analysis continuedClassical analysis continued Need to relate entropy and tension. From previous results, differentiating gives
We have derived a Maxwell relation, which connects the entropy to measurable quantites.
Maxwell relationMaxwell relation
ll TST
TU
=
flST
lU
TT
+
=
lT Tf
lS
lTST
TlST
+
+
=
22
Tl lS
Tf
=
BB
TlST
TlU
= 22
lT Tf
lS
lTST
lTU
+
+
= 22
yzx
zyx
= 22
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February 07 Lecture 6 10
Elasticity: general caseElasticity: general caseand metal wireand metal wire
What determines the tension?What determines the tension? f may be a function of l and T. Eq. A gives
Using B we get
Metal wire:Metal wire: Elastic modulus: (T) = o(1+(T-To)). Unstretched length: lo(T) = loo(1+(T-To)). and are ~10-5.
Effect is due to the U/l term. Entropy is unimportant.
TT lST
lUTlf
=),(
Direct contribution tointernal energy.
For example throughthe stretching of
intermolecular bonds.
Direct contribution tointernal energy.
For example throughthe stretching of
intermolecular bonds.
Entropic contribution.For example through
the ordering ofintermolecular bonds
Entropic contribution.For example through
the ordering ofintermolecular bonds
lT TfT
lUTlf
+
=),(
1
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February 07 Lecture 6 11
Rubber elasticityRubber elasticity
Rubber:Rubber: Generally have large elastic strain. In simple cases
From which,
Tension in rubber is an effect of entropy.
Band shortens on heating (at constant f). Band heats on sudden stretching (constant S) Entropy decreases on stretching (molecules
unfold).
)(),( ollATTlf
Tl
lT
l
lST
TfT
TfT
lUTlf
fTfT
+
=
),(
( )( )2
21ln
dddddd
oo
l
ol
llATTCS
lllATTCSTlfUQ
=
=
=
constantsconstantsconstantsconstants
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February 07 Lecture 6 12
Rubber elasticityRubber elasticity
11--D statistical modelD statistical model Take molecules to have 2N links, of length, a. Each link points right or left.
N+r point Right; N-r point Left. Length of the stretched molecule, l = 2ra. Entropy (from k ln g)
We know TdS = dU - fdl and UU(l).
( ) ( )( ) ( )( ))ln()()ln()()2ln(2
!!!2ln
rNrNrNrNNnkrNrN
NkrS
++=
+=
axmll
a
kTNakTr
Nr
Nr
a
kTr
Sa
TlSTf
2
1ln1ln2
dd
21
dd
=
+=
==
Expand lns for small r/NExpand lns for small r/N
Note: T and l dependenceNote: T and l dependence
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February 07 Lecture 6 13
Elasticity in rubberElasticity in rubber
Molecular modelMolecular model Without strain With strain
ExperimentExperiment X-ray diffraction from un-strained and strained
samples of rubber.
Note the diffraction spots showing enhanced order in the strained sample.