lecture 6: paramagnetism and elasticitywa14/camonly/statistical/lecture6.pdf · february 07 lecture...

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� Curie’s Law.

� Entangled polymers� Role of entropy in rubber elasticity.


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 quantised

1<Landé factor<21<Landé factor<2

( )[ ] 211+JJg Bµ

Bohr magneton = Bohr magneton = eme 2�

JmJgm JBJ ≤≤−= ,µµ

J = 3

mJ

-3 -2 -1 +1 +2+3

H


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=½, there are only 2 values of mJ.� Only two energy levels to consider, with energy

+/-µB.

� Calculate the expectation value of the moment from the weighting given by the BoltzmannDistribution,

( ) ( )�� −−=i

BBi

i kTUkTU expexpµµ

B

m = J -1/2 +1/2


Curie’s LawCurie’s 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

µµµµ µµµµ

tanh

eeee

=+−= −−

Curie’s LawCurie’s Law

kTBkTB

2µµµ <<

→

kT2µχ ∝

( )( )H

MHBχµ

µ+=+=

1o

o

= n <µ>= n <µ>

-2 -1 1 2µB kT/

tanh( )µB/kT

high- ,low- limit

TB


Paramagnetic saltsParamagnetic salts

�� ExperimentExperiment� Curves at different temperatures and fields

scale to lie on the curve given by Curie’s 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

<µ>/µB<µ>/µB

B/T (Tesla K-1)B/T (Tesla K-1)


Pierre CuriePierre Curieand Magnetismand Magnetism

�� Curie’s LawCurie’s Law� The subject of Pierre Curie’s doctoral thesis,

1895, the same year as his marriage to Marie.

� Ferromagnetic to paramagnetic transition at Tc.� Paramagnetism in salts ~ 1/T (Curie’s Law)� Diamagnetism is temperature independent.

� Died 1906, after a street accident.


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

kTBB

Z

BBZ

BBZ

µµβε

βµβµβµβµ

−=∂∂−=

−++=−+=

( )kTBkT

Bk

TC µµε 2

2sech�

�

��

�=∂

∂=


�� 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

lflS

TTTS

T

lfSTU

WQU

Tl

dd

dddddd

�

�� +

∂∂+

∂∂=

+=+=

TTU

ld

∂∂

ll

U

Td

∂∂

AA


�� 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 TS

TTU

∂∂=

∂∂

flS

Tl

U

TT

+∂∂=

∂∂

lT Tf

lS

lTS

TTlS

T∂∂+

∂∂+

∂∂∂=

∂∂∂ 22

Tl lS

Tf

∂∂−=

∂∂

BB

TlS

TTl

U∂∂

∂=∂∂

∂ 22

lT Tf

lS

lTS

TlT

U∂∂+

∂∂+

∂∂∂=

∂∂∂ 22

yzx

zyx

∂∂∂=

∂∂∂ 22


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: µ(T) = µo(1+β(T-To)).� Unstretched length: lo(T) = loo(1+α(T-To)).� β and α are ~10-5.

� Effect is due to the ∂U/∂l term. Entropy is unimportant.

TT lS

Tl

UTlf

∂∂−

∂∂=),(

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 Tf

Tl

UTlf

∂∂+

∂∂=),(

1<<∂∂

lTf

fT


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

lS

TTf

T

Tf

Tl

UTlf

fTf

T

∂∂≈

∂∂≈

∂∂+

∂∂=�

≈∂∂

),(

( )

( )221

ln

dddddd

oo

l

ol

llATT

CS

lllATTCSTlfUQ

−−��

��

�=

−−=−=

constantsconstantsconstantsconstants


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 U≠U(l).

( ) ( )( ) ( )

( ))ln()()ln()()2ln(2!!

!2ln

rNrNrNrNNnkrNrN

NkrS

−−−++−=

��

��

�

−+=

axmll

akT

NakTr

Nr

Nr

akT

rS

aT

lS

Tf

2

1ln1ln2

dd

21

dd

=≈

��

��

��

��

� −−��

��

� +=

��

��

�−=−=

Expand ln’s for small r/NExpand ln’s for small r/N

Note: T and l dependenceNote: T and l dependence


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.

lecture 6: paramagnetism and elasticitywa14/camonly/statistical/lecture6.pdf · february 07 lecture...

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