magneto chem

7/30/2019 Magneto Chem

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MAGNETISM A Few BasicsA. HISTORY

Magnetite or lodestone (FeIIOFeIII2O3)mineral in its natural state often has a strong attraction for iron and steel...mined in magnesiabut Pliny says it was named for the discoverer, a shepherd, magnes, "the nails of whose shoes

and the tip of whose staff stuck fast in a magnetick field while he pastured his flocks."

Lodestone appears in Greek writings by 800 B.C.First magnetic technological invention compass: in China, sometime between 2637 B.C. and

1100 A.D. (!)others say it was introduced to China in 1200s by Italians or Arabs

Early theories and experiments on magnetism were developed by the great minds of all times Gilbert, Descartes, Coulomb, Poisson, Green, Oertsted, Ampere, Davy, Freesnel, Faraday,Maxwell...

But: Quantum Mechanics is necessary to develop a viable model, and the model continues toevolve: MAGNETISM IS NOT A SOLVED PROBLEM

B. BASICSAll materials respond to an external magnetic field (H). The magnetization (M) of a sample isproportional to H:

M = H. The proportionality constant, , is the magnetic susceptibility

C. TYPES OF MAGNETISM - see chartThe Big five1. Diamagnetism

electrons in closed shells cause a material to be repelled by H ( < 0).Typically, m = -1x10-6 emu/molusually a very small contribution to expexp = dia + para + Paulidia diamagnetic term due to closed-shell (core) electrons. ALL materials have this. For our

systems, which will probably have a very small , the diamagnetic correction is non-trivial can be obtained from atom/group additivities or the Curie plot (see below).

dia is temperature and field independent.para is due to unpaired electrons (= p)Pauli is typically seen in metals and other conductors - due to mixing excited states that are not

thermally populated into the ground (singlet) state - temperature independent. Presumably notimportant in our systems.

ALL OTHER TYPES OF MAGNETISM REQUIRE UNPAIRED SPINS (MOMENTS) THEINTERACTIONS AMONG SPINS DETERMINES THE TYPE OF MAGNETISM

2. Paramagnetismrandomly oriented, rapidly reorienting moments; no permanent, spontaneous magnetic moment

(M = 0 if H = 0)spins are non-interacting (non-cooperative)all other forms of magnetism have a critical temperature, TC, below which there is some

interaction of moments. Above TC paramagnetp varies with T empirically according to the Curie Law:

p

= CT

C = Curie constant

or the Curie-Weiss Law:


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2

p

= CT = Weiss constant

is indicative ofintermolecularinteractions among the moments> 0 - ferromagnetic interactions (NOT ferromagnetism)< 0 - antiferromagnetic interactions (NOT antiferromagnetism)

DIAMAGNETISM INCIPIENTFERROMAGNETISM

METAMAGNETISM

ANTI-FERROMAGNETISM

FERRIMAGNETISM

FERROMAGNETISM

SUPER-PARAMAGNETISM

SPERIMAGNETISM

PARAMAGNETISMARAMAGNETISM

SPEROMAGNETISM ASPEROMAGNETISM

SPIN GLASS HELIMAGNETISM

MICTOMAGNETISM

Using the Curie Plot:

exp

= CT + T (T = dia + Pauli = temperature independent contribution)

exp

= CT

+T

(at high temperature if is small)


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3

Plot exp vs 1/T

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2 0.3 0.4 0.5

para

ferro

antiferro

ParamagneticSusce

ptibility

1/ Temperature (1/ K)

slope = C; intercept = T; exp - T = p...then plot 1/p vs. T

0

10

20

30

40

50

0 5 10 15 20

= 0

= +2

= -2

ParamagneticSusce

ptibility

1/ Temperature (1/ K)

slope = 1/C; intercept = /CA magnetometer measures M which gives . Do the first plot to get T. Then at all T and H, p

= exp - T

The magnetization plotFor an atom with spin ms, its energy in H is:

E(ms) = m

sg

BH = -msgB


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4

B is Bohr magneton (= 9.27x10-24J/T); g is the Lande constant (= 2.0023192778)Consider N particles, with ni in ith level

ni

i = N

P i =

ni

N =

exp

i

kT

exp

i

kT

i

M = N i

i P

i

now, consider two spin states (ms = 1/2),

M = N

msg

B( )expm

sg

BH

kT

ms=1/2

+1/2

expm

sg

BH

kT

ms=1/2

+1/2

=Ng

B

2

expg

BH

2kT

exp

gB

H

2kT

expg

BH

2kT

+ exp

gB

H

2kT

=Ng

B

2tanh

gB

H

2kT

since tanh(y) = (ey-e-y)/(ey+e-y)

if y


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a) For x >> 1, BS

x( ) = 1S

S+ 12( )

12

= 1

x =g

BH

kT>> 1 means H

T>> k

gB

= 1.38x1023 J / K

2x9.27x10 24 J/ T= 0.7T / K

b) For x


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6

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

S=1/2S=1S=3/2S=2S=5/2S=3S=7/2S=4S=9/2S=5

M

molar

(B)

H/T (Tesla/Kelvin)

The effective magnetic momentdirectly relates to number of spins:

eff= g[S(S+1)]1/2BS eff/B1/2 1.731 2.832 4.89

related to molar susceptibility (m = emu/mol)

m

=N

eff2

3kT= C'

T

Therefore,

eff= 3kC'

N= 2.82 C'

eff = 2.824 mT3. Ferromagnetismmoments throughout a material in 3-D tend to align parallelcan lead to a spontaneous permanent M (in absence of H)but, in a macroscopic system, it is energetically favorable for spins to segregate into regions

calledDOMAINS- domains need not be aligned with each other

A Domain

may or may not have spontaneous Mapplication of H causes aligned domains to grow at the expense of misaligned

domains...alignment persists when H is removed


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7Ferromagnetism is a critical phenomenon, involving a phase transition that occurs at a critical

temperature, Tc = Curie Temperature. Above Tc paramagnetTC is directly proportional to S(S+1)It takes energy to move domain walls - hysteresis:

MR = Remnant Magnetization (M at Happ = 0)

Msat = Saturation Magnetization (Msat= NgBS)

HC = Coercive Field (Happ required to flip M)

"Hard" magnetic material = high Coercivity"Soft" magnetic material = low Coercivity

Electromagnets High MR and Low HC

Electromagnetic Relays High Msat, Low MR, and Low HC

Magnetic Recording Materials High MR and relatively High HC

Permanent Magnets High MR and High HC

(-) (+)

(+)

(-)

Sample Magnetization

Applied Field MR

2HC

M

M

HappHapp

Msat

M = Happ

Msat = gBS

Remanence: Magnetization of sample after H is removedCoercive field: Field required to flip M (+M to -M)

4. Antiferromagnetismspins tend to align antiparallel in 3-Dno spontaneous Mno permanent Mcritical temperature: TN (Neel Temperature)

above TN paramagnet

5. Ferrimagnetismrequires two chemically distinct species with different momentsthey couple antiferomagnetically:no M; critical T = TC (Curie Temperature)

bulk behavior very similar to ferromagnetismMagnetite is a ferrimagnet

6. Other types


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DIAMAGNETISM

INCIPIENTFERROMAGNETISM

METAMAGNETISM

Low T; temporary alignment of momentsof ions & itinerant electrons

Field-induced transition to magnetic

state;must be below Neel temperature

ANTI-FERROMAGNETISM

FERRIMAGNETISM

FERROMAGNETISM

SUPER-

PARAMAGNETISM

SPERIMAGNETISM

PARAMAGNETISM

Small particle size allows singledomain fluctuation; destroyed by

cooling below blocking temperature

Two species frozen as in speromagnet;

often net magnetication

SPEROMAGNETISM ASPEROMAGNETISM

SPIN GLASSHELIMAGNETISM

Local moments licked into randomorientation;no net magnetiztion

Magnetic ions cooled in a nonmagnetic host;usually dilute; RKKY coupling only (?)

As in speromagnetism, but some orientation;net magnetization

Crystalline asperomagnet

MICTOMAGNETISMCluster glass; local correlations dominant

D. SPECIAL TOPICS1. Magnetooptical Disks

a. ReadingMagnetic materials rotate plane polarized lightin transmission: Faraday effectin reflection: Kerr effect (used in devices)Direction of rotation depends on polarization of M

b. Writingactually a thermo-magnetic process since a high-powered laser is used to heat the sample1) Curie point writing-heat above TC, then cool in applied field2) Threshold (or compensation point) writing - if material has a temperature dependentcoercive field heat to point where H > Hc, then on cooling H < Hc

c. MaterialsTypically rare-earth transition metal alloys (RE-TM) RE: Gd, Tb; TM: Fe, Co


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9Many factors to tune: Tc, Hc, Magnetic anisotropy, reflectivity, Kerr rotation, stability, etc.

2. Spin Glasses one of the more intriguing new forms of magnetismmay be a new state of matter, maybe notprototypical SG is an alloy, with 1-10% of a paramagnetic impurity (e.g., Fe or Mn) in adiamagnetic metal (e.g. Cu or Au)Spin coupling is mediated through the conduction electrons through the RKKY interaction(Ruderman, Kittel, Kasuya, Yosida) - oscillatory in nature

This leads to frustration:

A

B

C

ferromagnetic coupling

antiferromagnetic coupling

spin center

frustrated

unfrustrated

unfrustrated frustrated

A critical consequence of Frustration is a highly degenerate ground state - there areMANY equally probable, equally acceptable combinations of spins

Therefore, even at 0 Kelvin, many states are populated - novel thermodynamic andmagnetic properties

Experimentally, the hallmark of a SG is a cusp in the AC susceptibility - defines a criticaltemperature - TSG

Heat capacity measurements do not reveal the some critical temperatureSG do show remanence and hysteresis effectsSG theory has been adopted to other problems, including:

combinatorial optimization (traveling salesman) problemsneural networkspre-biotic evolution

E. MISCELLANY1. Units and fundamental constants

N = 6.022 x 1023 mol-1

g = 2.0023

k = 1.3807 x 10-23 J/K

B = 9.274 x 10-24 J/TNB2/k = 0.375 emu K/molgB/k = 1.3449 K/TG = g1/2 cm-1/2 s-1

T = 104 G

T2 = 10 J/emu2. Saturation behavior of S-only paramagnets

M = Msat BS(gBH/kT)BS(x) = (1/S)((S + 1/2)x coth(S + 1/2)x - (x/2) coth(x/2))


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10

Msat = NgBS= 11.18245 SJ/molT = 1.118245 x 104 S emuG/mol3. Curie Law for paramagnets

M = H

T = C =Ng2

B2 S(S+1)

3k= 0.125g2S(S+1) emuK/mol

eff= (g2S(S+1)1/2 = (8T)1/2

4. Relation between saturation and Curie behaviors

C emu K( )

Msat

emu G( )=

Ng2B2 S(S+1)

3kNgB

S=

gB

(S +1)

3k=

2.00239.274x10 28 J/ G

31.3807x10 23 J / K(S +1) = 4.4831x10 5 K

G(S +1)

F. MAGNETIC DIMERS THE HEISENBERG-DIRAC-VAN VLECK HAMILTONIAN1. Introduction

In section C.2 we examined primary magnetochemistry relationships for paramagnets

species with non-interacting spins. The magnetic properties of such materials can be describedby the Curie-Weiss law (for small values of H/T), or more generally by:

M = NgBSBS

However, in the next simplest case a molecule with two exchange-coupled unpairedelectrons the magnetic properties cannot usually be described by this relation except at verylow temperatures. This will be the case when the exchange coupling creates thermally accessiblestates of different spin multiplicity. Such is the case for most organic biradicals.

The Heisenberg-Dirac-Van Vleck (HDVV) Hamiltonian, eq 1, is an empirical operator thatmodels interaction (coupling) of unpaired electrons. The form of the Hamiltonian suggests that

coupling arises through interaction of spin angular momentum operators. The magnitude of theinteraction depends on the interaction parameter, Jij (called the exchange parameter). The Jijterm embodies all of the interactions that determine the ground state spin preference. Theproduct of spin operators is a reasonable component since interaction of species containingunpaired electrons results in either a decrease (antiferromagnetic coupling) or increase(ferromagnetic coupling) in spin angular momentum. For example, two doublets can couple toyield a singlet and a triplet, the relative energies of which depend on Jij. By definition, forferromagnetic (high-spin) coupling Jij > 0, while for antiferromagnetic coupling, Jij < 0.

Hij

= 2Jij

SiS

j(1)

The product of spin angular momentum operators may be expressed in terms of component

and product (total) spin angular momentum operators,

STot2 = S

i+ S

j

2

= Si2 + S

j2 + 2S

iS

j(2)

therefore, SiS

j= 1

2S

Tot2 S

i2 S

j2

(3)

Since the eigenvalue ofS2 is S S +1( ) , the energy of the state with spin STot

resulting from

interaction of species with spins Si

and Sj

is given by,


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11

ETot

= Jij

STot

STot

+1( ) S i S i +1( ) Sj Sj +1( )[ ] (4)

Thus for two doublets (2 x multiplicity 2 = 4 states), coupling results in a triplet and a singlet(multiplicity 3 and multiplicity 1 = 4 states). In the case of ferromagnetic coupling (J > 0), theenergy of the triplet state (S = 1) is,

ET = J 11 +1( ) 12

12 +1( ) 12 12 +1( )[ ]= J 2 34 34[ ] = J2 (6)

The energy of the singlet state (S = 0) is,

ES

= J 0 0 +1( ) 12

12

+1( ) 1212

+1( )[ ] = J 0 34 34[ ] =

3J

2(7)

The singlet triplet gap, EST, is given by,

EST

= ES

ET

= 3J2

J2( ) = 2J (8)

The energy level diagram for ferromagnetic coupling is shown below.

S = 1/2S = 1/2

Triplet, S = 1

Singlet, S = 0

EST

The energy level diagram for antiferromagnetic coupling is shown below.

S = 1/2S = 1/2

Triplet, S = 1

Singlet, S = 0

EST

Note that by definition of the Hamiltonian, J< 0 describes antiferromagnetic coupling and J> 0. Had we defined the Hamiltonian as Hij = 2J ijSiSj , then J < 0 would be for

ferromagnetically coupled spins. Therefore, when reading the literature, always note the form ofthe Hamiltonian to be sure of the nature of the type of exchange coupling.

2. Measuring Jby Electron Paramagnetic Resonance SpectroscopyThe intensity of the EPR signal is directly proportional to the paramagnetism which can be

given by the Curie Law,

IEPR

= CT

(9)


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12

Although the paramagnetism of the triplet follows the Curie Law, the concentration of tripletis temperature dependent and is given by the Boltzmann distribution,

nT

nT

+ nS

= TRIPLET[ ] rel =

3exp

T

RT

3exp

T

RT

+ exp

S

RT

(10)

Multiplication of the numerator and denominator by exp

S

RT

, and letting T - S = EST gives,

TRIPLET[ ] rel =

3expE

ST

RT

1 + 3expE

ST

RT

(11)

Thus, the EPR signal intensity is given by,

IEPR

= CT

TRIPLET[ ] rel =CT

3expE

ST

RT

1 +3expE

ST

RT

(12)

A plot of EPR signal intensity versus 1/T for different singlet-triplet gaps is shown below. Notethat very little change in the slope of the plot occurs as the triplet becomes more stable.

0

0.05

0.1

0.15

0.2

0.25

0

IEPR

0.1275 0.255 0.3825 0.51

E = 50 cal

E = 20 cal

E = 10 cal

1/T (1/K)

EPR Signal Intensity vs 1/ T for Several Values of ETS

E = 5 cal

E = -5 calE = -10 cal

E = -20 cal

E = -50 cal

3. Measuring Jby Magnetometry1

Van Vleck derived a general equation for calculating paramagnetic susceptibility as a

function of temperature.2 Use of the equation requires knowledge of the energy levels, whichwe calculate with the HDVV Hamiltonian (see previous section). Van Vleck began the


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13derivation by letting the energy levels of the system be developed as a power series expansion inthe applied field.

E n = E n0 + HEn

(1) + H2E n(2) + H3E n

(3) +... (13)

The term which is independent of H is the zero-order term (and has units of J, the exchangeparameter), while the term linear in H is the first-order Zeeman term, and the term that scales as

H2

is the second-order Zeeman term, and so on. Neglecting Zero-field splitting, and assumingisotropic g, the total magnetic moment, M, for the system is:

M = Nn exp E n kBT( )n

exp E n kBT( )n(14)

Where n is the magnetic moment of state n (n = - En/H). Now,

exp

E nkBT( ) = exp

En0 + HE

n(1) + H2E

n(2) +...( )

kBT

= expE n

0

kBT

expHE n

(1)

kBT

... (15)

Ignoring H2 and higher-order exponentials, and recalling that for small x,

exp(x) 1 xThen,

expE n

kBT( ) = expE n

0

kBT

exp

HE n(1)

kBT

... 1

HEn(1)

kBT

exp

E n0

kBT

(16)

(This assumes that first-order Zeeman splitting is much smaller than exchange coupling.) and,

n = E nH

= E n(1) 2HE n

(2) +... (17)

From the approximation and substitution we obtain,

M = N

En(1) 2HE

n(2)( )

n 1

HEn(1)

kBT

exp

En0

kBT

1HEn

(1)

kBT

exp

En0

kBT

n

(18)

If we limit the derivation to paramagnetic substances, such that M=0 at H=0, then

E n(1)

expE n

0

kBT

n = 0 (19)

Retaining only terms linear in H, and ignoring second-order energy terms, we obtain,


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14

M = NH

En(1)( )

2

kBTn exp

E n0

kBT

expE

n0

kBT

n

(20)

Since =M/H, then

= N

En(1)( )2

kBTn exp

E n0

kBT

expE

n0

kBT

n

(21)

Recall that,

E n(1) = msg B and ms

2

S

+S = 1

3S S +1( ) 2S +1( ) (22)

and adding degeneracy terms gives,

=Ng2

B2

3kBT

S S +1( ) 2S +1( )S exp ES kBT

2S +1( )expE

SkBT

S

(23)

Plugging in the constants gives,

=0.125g2emuK / mol

T

S S+ 1( ) 2S +1( )S expE

S kBT

2S +1( )expE

SkBT

S

(24)

Where ES is the energy of the exchange-coupled spin states determined using the HDVVHamiltonian. The energies of the states (in units ofJ) are relative to the lowest state, which istaken as zero. The denominator can be modified according to the Curie-Weiss law to give:

= 0.125g2

emuK/molT( )

S S+1( ) 2S+ 1( )S

exp ESkBT

2S+1( )expE

SkBT

S

(25)

Below are plots of T vs T for = -0.01 K and -500 cm-1J +500 cm-1.


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15

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500

-500-250-150-50-10-5+5+10+50+150+250+500

paraT(emuK

/mol)

Temperature (Kelvin)

Below are plots of vs T for = -0.01 K and -500 cm-1J +500 cm-1. Note that forferromagnetic coupling, increases asymtotically as the temperature is lowered, while for

antiferromagnetic coupling, exhibits a maximum at Tmax.

0

0.001

0.002

0.003

0.004

0.005

0 500 1000 1500 2000 2500

-500-250-150-50-10-5+5+10+50+150+250+500

para

(emu/mol)

Temperature (Kelvin)


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16The exchange coupling parameter is related to Tmax by:

|J| = 0.8 kTmax (26)

Thus, a maximum in the plot of vs. T is the signature of antiferromagnetic coupling.

References1)Kahn, O.Molecular Magnetism; VCH: New York, 1993.

2)Van Vleck, J. H. The Theory of Electric and Magnetic Susceptibilities; Oxford UniversityPress: Oxford, 1932.

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