2013 summer school for rock magnetism (ssrm) today’s … · 2013 ssrm 5/29/2013 1 2013 summer...

2013 SSRM 5/29/2013

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2013 Summer School for Rock Magnetism

1.4Fe

3O

4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250 300

Temperature (K)

(T)

SIRM

3 4

~ 1m

Monday ‐ Friday, June 3 – 7, Monday – Wednesday, June 10‐12

8:30 am – 10:00 am Lectures (Tate Hall (Physics Building), Room 143)10:00 am – 10:30 am Break10:30 am – 12:00 pm Lectures

2013 SUMMER SCHOOL FOR ROCK MAGNETISM (SSRM) SCHEDULE

10:30 am – 12:00 pm Lectures12:00 pm – 1:30 pm Lunch Break1:30 pm – 4:00 pm Laboratory /computer exercises (Shepherd Labs or Mechanical Engineering 314)

Evenings free to study, work on group project, or enjoy beautiful Minneapolis/St. Paul.

Saturday, June 810:00 am Depart for optional field trip6:00 pm Approximate return time

Sunday, June 12 Free day

Wednesday, June 151:30 pm Group presentations (Pillsbury Hall, Room 110)5:00 pm Group Dinner

Thursday, June 16: Check out of dormitory and transit to airport.

AM 8:30-12:00 PM 1:30-4:006/3 Mon Introduction

Physics Of MagnetismGeomagnetic Field

Magnetism of Solids I

How Instruments WorkLab Tour

2013 SUMMER SCHOOL FOR ROCK MAGNETISM (SSRM) SCHEDULE

6/4 Tu Magnetism of Solids IIMagnetic Mineralogy

Laboratory Measurements6/5 Wed Fine Particle MagnetismSuperparamagnetismMultidomain

behavior6/6T h Hysteresis Loops

FORC

6/7 Fri Low‐Temperature Magnetism Environmental MagnetismEnvironmental Magnetism

Overview of Midcontinent Rift System and Saturday Field Trip

6/10 Mon Remanent Magnetizations TRM, VRM, CRM, TVRM, DRM

6/11 Tu Paleointensity Anisotropy of Magnetic Susceptibility I

6/12 Wed AMS II Group Presentations

Today’s Schedule

• Overview of Rock and PaleomagnetismPh i f M ti (R i f El t ti )• Physics of Magnetism (Review of Electromagnetism)

• Overview of Geomagnetic Field• BREAK• Magnetism of Solids

2013 SSRM 5/29/2013

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Sir Harold Jeffreys(1891‐1989)

Annu. Rev. Earth Sci. vol 1., 1973Jeffreys, The Earth, 1970

What is Paleomagnetism and Geomagnetism?

• GeomagnetismStudy of the present day field origin of fieldStudy of the present day field, origin of field

• PaleomagnetismStudy of the remanent magnetization in naturally occurring magnetic minerals

• Rock MagnetismStudy of the physical/chemical basis of paleomagnetism and applications to geological and geophysical problems

Why Study Paleomagnetism?

An example From Mars

Viking on Mars (1975)Magnetic Dust

"They had a house of crystal pillars on the

Magnet array

planet Mars by the edge of an empty sea, and every morning you could see Mrs. K eating the golden fruits that grew from the crystal walls, or cleaning the house with handfuls of magnetic dust…"Ray Bradbury, The Martian Chronicles (1950)

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Martian Magnetic Anomalies

J.E.P. Connerney, 10.07 ‐ Planetary Magnetism, In: Editor‐in‐Chief: Gerald Schubert, Editor(s)‐in‐Chief, Treatise on Geophysics, Elsevier, Amsterdam, 2007, Pages 243‐280

MV1

Martian Magnetofossils???

Nanoparticles of magnetite as Nanoparticles of magnetite as potential biomarkers

(K.L. Thomas‐Keprta, GCA, 2000)

Why Study Paleomagnetism and Geomagnetism?

Geologic and Geophysical History of lithospheric plate motions


Geochronology

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Probe of Deep Earth Dynamics

(Glatzmaier and Roberts, Nature 377: 203‐209, 1995Glatzmaier and Olson, SA, 2005)

Record of polarity transition recorded at Steens Mountain (Figure from Tauxe, 2005)

Long‐term PaleointensityTauxe & Staudigel, GGG 2004

Paleointensity and Growth of Inner Core(Tarduno et a., 2006)


Regional Geology and Tectonics

Tectonostratigraphic terranes of the North American Cordillera. (From Butler, 1992)

Aeromagnetic Anomaly Map for Minnesota


Environmental Magnetism

(Paleoclimatic )reconstructions)

Verosub & Roberts, 1995

Chinese Loess

Egli, 200

5

Magnetic susceptibility records for two sites in the central Loess Plateau, Xifeng and Luochuan, and the deep‐sea oxygen isotope record (Maher, 2007)

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Geomicrobiology

L. Kirschvink, 2007

R.E. Kop

p, J.L

Magnetofossils in Marine sediments (S. Atlantic, Angola Basin, 50 ma)

Magnetotactic bacteria synthesize chains of magnetic nanoparticles that help them navigate in the geomagnetic field

Favre and Schuler, 2008

Possible oldest magnetofossils ~2 Gyr stromatolithic chert(Chang et at., 1989

ODP 738C

ODP 689D

FORC diagram is dominated by a sharp ridge centered on Bb = 0, which is indicative of non‐interacting SD magnetic particles

Why Study Paleomagnetism and Geomagnetism?Magnetism of Extraterrestrial Materials

Martian meteorite ALH 84001

Map of magnetic fields (in nT) measured with the SQUID microscope Weiss et al., 2008)

Total Field Anomaly Chicxulub impact

structure

Purucker and Whaler, 2007

Photograph of the pallasite Esquelshowing olivine crystals suspended in metal

Weiss, 2012 (Science 338, 897)


Magnetism of Fine Particles (nanoparticles)

7Du

nlop

& Ozdem

ir, 1997

005

Pokhil and Moskowitz, 1997

Feinberg et a

l., 20

Harrison et al., 2007

Vortex state

Williams and Wrig

ht, 1999

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What Makes Paleomagnetism Possible?

• SIGNAL: Planetary Magnetic Field (>3 Ga)– Fe is 4th most abundant element in crust– Fe has the property of permanent magnetism– Fe has the property of permanent magnetism

• Recording Media: Fe forms oxides and sulfides, some of which are magnetic minerals– Fe‐oxides are common accessory minerals in rocks, sediments, soils (<1% vol)

• Recording Processes: Earth’s magnetic field can be d d d i i l i lrecorded during various geological processes

Tauxe, 2008

What Makes Paleomagnetism Possible?

• Sensitive magnetometers are available to measure the weak magnetic signals in earth (and planetary) materials

DC SQUID U‐Channel Magnetometer and Shielded

Room

Vibrating Sample magnetometer

MPMS (Magnetic Property Measurement System)

How is Rock Magnetism Different from the Study of Magnetic Recording and Permanent Magnets?

Hard Disk: highly ordered magnetic system designed to carry maximum information

t t i ll t ibl

– Disordered system of irregular shaped particles with complex compositions, geometries, and crystal defects

– Magnetic minerals form only a small (<1%) fraction of most rocks

– Multiple magnetic phases occurring over a broad range of

content in smallest possible space

Rock: Not optimized for magnetic recording

Multiple magnetic phases occurring over a broad range of particle sizes

– Magnetic fields in Nature are weak

“Rock” = an assemblage of ferrimagnetic/antiferromagnetic phases in a paramagnetic/diamagnetic matrix

natural oxidized titanomagnetite (Krása et al., 2005)

Egli

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Physics of Magnetism

EM Review and Magnetic Parameters (Basic definitions)

• Magnetic fields, H‐field, B‐fieldCoulomb’s LawForce between current elementsGauss’s LawAmpére’s Law

• Magnetic Dipole Moment torque on a dipolemagnetic potential energyMagnetization B (H+M)B =0(H+M)

• Magnetic Units and Conversion between SI and cgs

Good Textbooks on MagnetismCoey, J.M.D., (2009) Magnetism and Magnetic MaterialsCullity, B.D., (1972) Introduction to Magnetic Materials

Physics of Magnetism

Two Views of Magnetism– Microscopic current loopsMicroscopic current loops– Magnetic poles (dipoles)

Both sources produce magnetic fields. All fields are produced by electrical currents associated with the motion of electron about atomic nuclei

Bar magnet

Current carrying solenoid

A Brief History of (Geo)magnetismMeet the Units

Karl Frederich Gauss1777‐1855 Michael Faraday

1791‐1867James Clark Maxwell

1831‐1879

Nikola Tesla1856 ‐ 1943

André‐Marie Ampére1775‐1836

Hans Christian Ørsted1775‐1851 emu

Electricity MagnetismHan Christian Oersted (1777‐1851)

Current flowing in a wire deflects a compass needle

Discovers magnetism due to

ndley, 2000

electric currents.

André‐Marie Ampère (1775‐1836)

Explains magnetism in terms of forces between electric

O’Han

of forces between electric currents

Wikimedia commons

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Michael Faraday (1791‐1867)

1831 Faraday’s Law

A i i i

Magnetism Electricity

andley, 2000

dB/dt

0 0

tEB J

BE

A time varying magnetic field induces an electric current in a coil

James Clerk Maxwell (1831‐1879)

1873 Maxwell’s Equations

O’Ha

Laws of Electrodynamics

0 0

0

0

c

B Jt

E

BThen there was light

Unified Electricity , Magnetism, and Optics

Wikimedia commons

Magnetic Induction: B‐Field

P1

P2r

Magnetic field patterns for various current distributionsCoulomb's Law for magnetic poles

1 22

p pF Kr

Jiles, 1991

B [Tesla, Weber/m2]

0=fundamental constant permeability of free space0= 4x10‐7 Henry/m (H/m)

B‐field is also called the magnetic flux density or magnetic induction.

0 22

1

( )4

magF pB rp r

Unlike electric charges which can be isolated, the two magneticpoles always come in a pair. When you break a bar magnet, twonew bar magnets are obtained, each with a North and SouthPole. In other words, magnetic “monopoles” do not exist inisolation. Note: 1 Henry is the

induction caused by a change in current of 1A/s1 H/m=Newton‐A‐2 (NA‐2)

Force on a current carry element

Instead of magnetic poles, currents due to the motion of electric charges are the actual sources of magnetic fields.

0 1 2

2I I lF

r

F I dl B Like the case with magnetic poles, the B‐field can be defined in terms of the current carry conductors but

A current‐carrying wire produces a magnetic field. In addition, when placed in a magnetic field, a wire carrying a current will experience a net force. Thus, two current‐carrying wires will exert a force on each other.

Ldefined in terms of the current carry conductors, but

here, current I2 creates a field that exerts a force on I1.

Note the units: [F]=[I][L][B] [B]= [F][I]‐1[L]‐1= NA‐1m‐1

The SI unit for B‐fields is called a Tesla: 1 NA‐1m‐1 = 1 Tesla (T)

Gauss’s Law for MagneticsIn electrostatics, charges act as sources or sinks of the electric field.

Magnetic poles can not be isolated

B

Magnetic poles can not be isolated meaning there is no magnetic equivalent of single charges.

Lines of B are continuous and have no start or end.

0B

0B ds (Differential form)

(Integral form)Magnetostatic version of Gauss’s Law:

Feynman, 1963

This equation is one of Maxwell’s Equations and states that the magnetic flux density or the B‐fieldthrough a closed surface does not diverge nor converge from any point.

0A 0A See Appendix A .3.5 (Tauxe, 2008) for more

details about vector calculus

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Ampére’s LawFor any closed loop path, the sum of the length elements times B‐field in the direction of the length element is equal to the permeability times the total electric current from all sources enclosed in the loop.

0 L

B dl I

I

dlB

electric current from all sources enclosed in the loop.

The differential form of Ampére’s Law can be obtained by applying Stokes’ theorem.

This vector theorem can change the line integral around path L to a surface integral of the curl of the vector over the surface S enclosed by L

O’Handley, 2000

0( )L S

B dl B ds I

The total current (I) flowing through S can be written in terms of a surface integral of the conduction current density J (Am‐2)

S

I J ds

Graphical illustration of Stokes’ theorem

Ampére’s Law

0( )

s s

B ds J ds

Ampére’s Law can be rewritten in the form

0( ) 0 s s

s

B J ds

This result is true for any choice of surface S or at any point in space. Hence we get the general result, and one of Maxwell’s EM equations

Feynman (1963)

0B J

This states that the curl (or circulation) of the B‐field is equal to the total current density.

Magnetic Dipole Moment

When a magnetic dipole (or circular current loop) is exposed to a uniform B‐field the net force is zero but there is an aligning torque that will cause the dipole to

Nm

( ) sin( )F r pd B m B

rotate (think of a compass needle)

Current Loops

S

Units [m]= [Am2]

iA Bm iA

Lowrie, 2007

Energy of Magnetic DipoleZeeman Energy

Bm

UThe magnetic potential energy is the work done by

the B‐field to rotate the magnetic dipole from an angle θ0to θ:

E

0 0

0sin (cos cos )E d mB d mB

cosE mB m B

-180 -90 0 90 180

angle θ0to θ:

Take E0=0 at θ0=90°, then dipole in an external field then has a potential energy of

B

cosE mB m B

Stable equilibrium whenm is aligned parallel to B Emin=‐mB.Unstable equilibrium when m and B are anti‐parallel Emax=+mB

E=0 E>0 E<0

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Magnetization

1M m

Surface poles Surface currents

iM mv

Unit: M [Am‐1]

Lowrie, 2007

Material M (kA/m)

Fe (Iron) 1700

Fe3O4 (Magnetite) 480

Fe2O3 (Hematite) 2

Fe7S8 (Pyrrhotite) 90

Uniform MagnetizationSurface Poles: Use of fictitious magnetic poles (or charges) to represent a uniformly magnetized rod is shown in the figure. The uniformly magnetized rod can be replaced by free space and two uniform surface charges on the ends of the rod with

ˆ

im

m

m pl pMv Al A

M n

and two uniform surface charges on the ends of the rod with area A represented by magnetic surface charge density, m:

Surface Currents: Uniformly magnetized rod can also be replaced by a uniform fictitious surface current (km, or current per unit length) and free space. Assume each “microloop” has the same area (A) and current (I)

ˆ

im

m

m I AM kv z A

k M n

The usual case is that the magnetization distribution in a material is not uniform

M changes discontinuously at the boundary between a magnetized substance and

Non‐Uniform Magnetization

m

m

J MM

air or vacuum (where M=0), or between two magnetized materials with differing properties

Amperian currents or magnetic charges are generated throughout the sample volume producing a volume current density (Jm) or a magnetic charge volume density (m)

Imaging surface charges withMagnetic Force Microscopy

m

The magnetic state of a body is accounted for only if all equivalent sources (surface and volume) are taken into account.

Pokhil and Moskowitz, 1997

++

+ +++ +- -

-

--

13x13 m

Magnetic Field: B‐Field and H‐fieldTwo sources of magnetic fields A magnetic field B0 is produced in some

region of free space by conduction (“real”) currents (Jc).

Conduction currents (Jc)

MagnetizationAmperian currents (Jm)

Within the magnetic material an addition field Bm is produced from the Amperian surface/volume currents (Jm).

The total field is now the sum of two contributions: B=B0+Bm

Ampére’s Law

0 ( )B H M

0 0

0

0

( ) ( )c m c

c

B J J J M

B M J

BH M

H =magnetic field strength (or intensity) with SI units Am‐1

Ampére s Law

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B, M and H for a magnet

Outside the magnet (In Free Space): (M=0), B=0H

0 ( )B H M

H –field lines appear to originate on the horizontal surfaces of the magnet

ˆ0

m M nH M

+ + +

‐ ‐ ‐Sources and sinksN and S poles

Coey, 2009

Inside the magnet B‐field and H‐field are different, and oppositely directedH is also oppositely directed to M, hence the name ‘demagnetizing field’

B‐field, form continuous closed loops 0B

Magnetic Field: B‐Field and H‐field

Note: There is really only one magnetic field; it simply has different magnitudes inside a magnetic material than its value in free space. All currents (conduction and Amperian) may contribute to B but only conduction currents may contribute to H.

In Free Space: (M=0), B=0H

This last relationship means that in free‐space (M=0), the B‐field and H‐field are identical (and parallel) but have different magnitudes and units.

At the surface of the Earth, the intensity of the geomagnetic field is:

B‐Field: 30,000‐60,000 x 10‐9 T or 30,000‐60,000 nanoTesla (nT)or 30‐60 microTesla (T))

H‐Field: 24‐48 Am‐1

Magnetic Susceptibility

Materials can acquire a component of magnetization in the presence of an external magnetic field. p g

For isotropic and linear materials, M will be parallel to H and given by: M=H,

where is a dimensionless quantity call magnetic susceptibility.

>0 for paramagnetic materials< 0 f di ti t i l

>0

< 0

< 0 for diamagnetic materials =0 for empty space

Lowrie,2007

Note the following terminology (not always followed )

Mass normalized susceptibility (m3/kg) Volume normalized susceptibility (dimensionless)

Magnetic SusceptibilityThe relationship between B, H, and M can be rewritten as

B = 0(H+M) = 0H(1+ ) = H

In ferromagnetic materials (“permanent magnets”) the relationship between M and H (also B and H) is neither usually linear nor

A linear relationship between H and M also implies one between B and H.

The quantity is the magnetic permeability and =0(1+ ) and is a measure of the ability of a material to convey magnetic flux.

H (also B and H) is neither usually linear nor isotropic.

It also depends on the pervious magnetic history of the material, a property called hysteresis or irreversibility. (O’Reilly, 1984).

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Magnetic Units and Conversions

Parameter SI Unit cgs Unit ConversionMagnetic Moment (m,) Am2 emu 1 Am2=103 emu

SI is the servant of man, not his masterP. Vigoureux (1971)

Magnetizationby volume (M,J)by mass ()

Am‐1

Am2 kg‐1emu cm‐3

emu g‐11 Am‐1= 10‐3 emu cm‐3

1 Am2 kg‐1 = emu g‐1

Magnetic field (H) Am‐1 Oersted (Oe) 1 Am‐1 =4x10‐3 Oe

Magnetic Induction (B) Tesla (T) Gauss (G) 1 T = 104 G

Susceptibility m/Hby volume (=M/H)*by mass ( =M/H)

m3

‐‐‐‐m3 kg‐1

emu Oe‐1emu cm‐3 Oe‐1emu g‐1 Oe‐1

1 m3= 106/4 emu Oe‐11 SI =1/4 emu cm‐3 Oe‐11 m3 kg‐1 = 103/4 emu g‐1 Oe‐1by ass ( / ) g g g 0 / e u g Oe

Permeability (= B/H)Permeability of free space (0)

H m‐1

H m‐1G Oe‐11

1 H m‐1 = 107/4 G Oe‐14x10‐7

1 Tesla (T) = kg A‐1 s‐1 1 Henry (H) = 1 Newton A‐2B= 0 (H + M ) SIB= (H+4M) cgs

*Even though k (susceptibility) is dimensionless, it is common to attach SI to the value to avoid confusion

The Geomagnetic FieldThe Signal

)(From Tauxe 2007)

inclination

(Constable, 2007)

Components of the (vector) Magnetic Field

cos cosNB X B I D 1tan

YD

2 2 2

cos sinsin

E

v

N E V

B Y B I DB Z B I

B B B B

Bh=Bcos(I)

Bv=

Bsi

n(I) B

B(units) =TeslaBearth~ 30‐60x10‐6 T (30‐60 µT)

1

2 2 2

tan

sin

DX

ZIX Y Z

Butler, 1992

B

Angle of DeclinationCompass direction 0≤D≤360

Angle of Inclination (Dip)‐90 ≤ I≤ 90

DeclinationIsogonic maps)

InclinationIsoclinic maps)

IGRF Model 2005

Total IntensityIsodynamic maps)

International Geomagnetic Reference Field

http://www.ngdc.noaa.gov/wist/magfield.jsp

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Earth as a Magnet

Geomagnetic Field at surface is similar to a magnetic Field points indipole

p

• Magnetic inclination is related to geographic latitude

• North magnetic pole ≈ aligned with rotation axis

Field points out

Magnetic Dipole

The magnetic field of the earth, to a first approximation, is similar to a field produced by a magnetic dipole at the center of the earth.

+pa

g p

0

4p pWx y

To calculate the field, start with the magnetic potential at point P:

-p

a

+p

-p

0 0

0 02 2

4 4

(2 )cos cos4 4

p p p y xWx y xy

lp mWr r

Magnetic potential

See slides at end for more details

0 0( / )0

B J M E tB

Magnetic Field (B) due to Magnetic Dipole

where J=current density ε0=permittivity of free space.

Maxwell’s equations for magnetic fields are

0

00,

BB where B H

B VFirst Equation implies that B can be derived as a gradient of a

In source free regions (J=0, M=0, and E/t =0, i.e., an electromagnetic vacuum) and when there are no magnetic monopoles then

B V

Outside the core (and up to ~50 km height) these conditions are appropriate for modeling the geomagnetic field.

First Equation implies that B can be derived as a gradient of a scalar potential function (recall vector identity )0

2 0V V Second Equation shows that magnetic potential is a solutions to Laplace’s equation

Magnetic Dipole FieldThe radial and tangential components of the dipole magnetic field at point P can be determined from the negative of the derivative of the potential in that direction: m

( , )( , ) 1 ( , )

B V rV r V rB r

r r

02

cos4

mVr

m

03

03

2cos4

sin4

rmB

rmB

r

There are no sources and sinks, onlycontinuous lines of force. (.B = 0)

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Magnetic Dipole Field

30

60

90

1.8

2.0

-90

-60

-30

0

30

-90 -60 -30 0 30 60 901.0

1.2

1.4

1.6

-90 -60 -30 0 30 60 90latitude latitude

1/22 2 1 2 203 1 3cos

4rMB B Br

tan I BrB

2cot 2 tan

Magnetic PolesMagnetic North Pole where the

magnetic field is straight down (I = +90).

Geomagnetic North Pole where

Tauxe, 2008

NP

GMP

GP

Geomagnetic North Pole where the axis of the best tilting dipole pierces the surface.

Geographic North Pole

Butle

r, 1992

GMP

Virtual Geomagnetic Pole (VGP). Geocentric dipole which would give rise to the observed

ti fi ld di ti t i l tit d (λ) Bmagnetic field direction at a given latitude (λ) and longitude (φ)

Paleomagnetic Pole. Ancient pole position averaged over 106‐108 years

Magnetic field of the Earth measured at the surface comes from three sources

External Field

Main field generated by dynamo action in the outer core

External Field~1‐2%

Crustal Field~1‐2%

External field generated in space in the magnetosphere

Crustal field from remnant magnetization

(Constable, 2007)

Internal Core Field~98% of Field

External field varies with time scales of minutes to days Crustal field on varies over geological time scales

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Crustal Magnetic FieldNGDC‐720: Bz at Earth’s Surface

Permanent (remanent) magnetization only possible above the Curie depthDirection of remnant magnetization depends on main field direction at time rocks became magnetized

http://www.ngdc.noaa.gov/geomag/EMM/emm.shtml

Mathematical separation of the geomagnetic field into internal and external contributions

2 0 VThe geomagnetic potential, V, satisfies Laplace's Equation:

In spherical coordinates, this equation is n.ht

ml

22 2

2 2 2 2 2

1 1 1sin 0sin sin

V V VV rr r r r r

p , q

General solution to Laplace’s equation

( 1)( ) ( ) ( )l

l l mV r Q r G r Y

esea

rch/

proj

ects

/pro

ject

8/in

dex.

en

.

,

l m

l m

QG

( , ) mlY Spherical Harmonic Functions

degree l and order m

. ,0 0

( , , ) ( ) ( , )l m l m ll m

V r Q r G r Y

External field terms (weighting factors)

Internal field terms (weighting factors)

ww

w.un

istu

ttgar

t.de/

gi/re


Mathematical separation of the geomagnetic field into internal and external contributions

Internal Field Contribution (simplified)

0 00,0 1,0 1 2,0 2

2 3

( , ) ( , )( , , )

G G Y G YV r

( 1),

0 0

( , , ) ( ) ( , )l

l ml m l

l m

V r G r Y

Let’s look at the first few terms for l=0,1,2 and m=0

2 3( , , )r r r

monopole dipole quadrupole l=0 l=1 l=2

Important Point: l=1 m=0 gives the magnetic potential of a magnetic dipole

http://www.ipap.jp/jpsj/news/jpsj‐nc_26.htm

Spherical Harmonic Description of Geomagnetic Field: Internal Contribution

1l

In geomagnetism, the general form of spherical harmonics expansion for internal sources is given as

a = 6371 km (mean radius of the Earth, Earth’s surface when r=a)(r, θ,φ) are geographic coordinatesgml,, hml are the Gauss coefficients ( in units of nT) and are evaluated from observations

1

1 0

( , , ) (cos )( cos sin )ll

m m ml l l

l m

aV r a P g m h mr

Pml are normalized Schmidt functions, which l t d t th i t d L d

Normalized Schmidt surface harmonics (Blakely, 1995)

are related to the associated Legendre Polynomials (l=degree, m=order)

Why Normalization? magnitude of terms independent of (l,m)

Good reference: Blakely, R.J, (1995) Potential Theory in Gravity and Magnetic Applications

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Dipole Quadrupole01g 0

Octupole

1g 02g

Examples of surface harmonics and associated patterns for global inclination for axial fields (Tauxe, 2008)

p03g

g/h l m 1990.0 1995.0 2000.0 2005.0g 1 0 ‐29775 ‐29692 ‐29619.4 ‐29556.8g 1 1 ‐1848 ‐1784 ‐1728.2 ‐1671.8h 1 1 5406 5306 5186.1 5080.0g 2 0 ‐2131 ‐2200 ‐2267.7 ‐2340.5g 2 1 3059 3070 3068.4 3047.0h 2 1 ‐2279 ‐2366 ‐2481.6 ‐2594.9g 2 2 1686 1681 1670.9 1656.9h 2 2 ‐373 ‐413 ‐458.0 ‐516.7g 3 0 1314 1335 1339.6 1335.7g 3 1 ‐2239 ‐2267 ‐2288.0 ‐2305.3h

International Geomagnetic Reference Field10th generation (2005)

The most important observations from these data

No g00 (no magnetic monopole term)

Dominance of the l=1 (dipole) terms The c.no

aa.gov/IAG

A/vm

od/ig

rf.html

h 3 1 ‐284 ‐262 ‐227.6 ‐200.4g 3 2 1248 1249 1252.1 1246.8h 3 2 293 302 293.4 269.3g 3 3 802 759 714.5 674.4h 3 3 ‐352 ‐427 ‐491.1 ‐524.5

The l>1 are the nondipole terms and are

Dominance of the l=1 (dipole) terms. The geomagnetic field is very nearly that of an inclined dipole

g01 is negative. This simple means that the dipole moment along the z‐axis is pointing along the –z axis (towards the South Pole)

from

http://www.ngdc

much smaller than the l=1 (dipole term)

The Gauss coeff. change from one model year to the next. Part of this change is due to improved instrumentation, but most of it is real time variation and is an expression of the secular variation.

Orientation of dipole field based on 1990 IGRF Model (Blakely, 1995)

Maps of geomagnetic field 2005

intensity (T) inclination

(From Tauxe 2007)

declination1

1 0


m m ml l l

l m

aV r a P g m h mr

Total field, B (uT)

Non dipole field (uT)

1

1 0


m m ml l l

l m

aV r a P g m h mr

(Constable 2007)

1

0


m m ml l l

m

aV r a P g m h mr

l=2

l>1 terms

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17

Uniqueness

2 ( , , ) 0, ( , , ) ( , , )V r B r V r

Spherical Harmonic Analysis of GMFSolution: Mulipole expansion

Other Models that can be given in terms of SHA

axial dipole displaced from center (eccentric dipole)sum of axial dipoles distributed at core‐mantle boundarysum of axial dipoles distributed at core mantle boundary

CMB

Time variations of Geomagnetic Field

Kilohertz‐Femtohertz

Constable, 2003Constable, 2007

Time Variations in GMF

• Most of surface field (~99%) is generated in liquid outer core– Flow is influenced by rotation of Earth and geometry of inner core

– Flow produces secular variation in magnetic field

• Crustal magnetic sources makes a• Crustal magnetic sources makes a small, static contribution

• External field (outside of solid earth)

Interactions of charged particles and BE

Olson et al., 2007

Time Variations in GMF

• Field averaged over 105‐106 years is a geocentric axial dipole (GAD)axial dipole (GAD)

• Fluctuates in strength and direction (105‐106 years) but maintains constant polarity for long periods (106‐108 years)

• Bipolar, +B, ‐B states• Changes in Reversal Frequency (107‐108 years)

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Secular Variation of Geomagnetic Field

• Historic record of geomagnetic field direction at Greenwich, England.

• Change in Declination in Minnesota (1900‐2005)

Time variation 5

6

7

8

IGRF version 10Twin Cities

n (E

ast)

Time variationinternal motions of km’s/year

1

2

3

4

1880 1900 1920 1940 1960 1980 2000 2020

Dec

linat

ion

YearButler, 1992

Paleosecular Variation (PSV)

Paleosecular variation is a fundamental property of the geomagnetic field

Records of PSV

Historical recordsArchaeomagnetic records (< few 1000 yrs)Lake sediment records ( 10,000’s years)Dated lava flows ( 0‐5 ma)

But

ler,

1992

Dated lava flows ( 0 5 ma)

Geomagnetic polarity timescale from marine magnetic anomalies for 0–160 Ma.

Lowrie, 2007

Early Mesozoic, Paleozoic, Precambrian

Exposed stratigraphic sections on landGPTS much less refined

On average field spends about 50% of its time in each polarity

1‐2% in transitional state

Butler, 1992

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Glatzmaier andGlatzmaier and Coe, 2007

Sedimentary Records of polarity transition for the last 12 My (Laj, 2011)

Time‐Averaged Field (TAF)Geocentric Axial Dipole

2005 Geomagnetic Field

Time averaged intensity of the geomagnetic field (0‐5 Ma)

Taux

e, 2

008

[Data from Hatakeyama andKono. 2002.]

Although the field is not perfectly GAD, the flux lobes seen in the historical field are nearly erased.

Basic Observation to Explain:First Order Features of Geomagnetic Field

a. Longevity of field (~ 3 Ga)energy source must be long‐lived

b. Time averaged field is Geocentric Axial Dipole (GAD)rotation plays an important role

c. Intensity of field (at surface) 50‐100 T for most of geologic time (>2 Ga)

d. Polarity reversals exist & random in time

P l it t iti ti l ~ 104e. Polarity transition time scale ~ 104 yr

f. Two stable states (Normal and Reverse)equations should work for B and ‐B

R. Blakely

Great Moments in Geophysics:Gary Discovers His Dynamo Model

Origins of the Geomagnetic Field1. Remanent Magnetization of Crust?

Too weak2. Remanent Magnetization of Mantle or Core?

Too hot, Cannot exist (T>Curie Temperature)Too hot, Cannot exist (T>Curie Temperature)3. Primordial Magnetic Field?

equatorial current flowToo old, Subject to diffusion (time<104 yr)

3 Basic Conditions for generating Planetary magnetic FieldGeodynamo action

in liquid outer coreLarge volume of electrically conductive fluid

Thermal/chemical convection (energy source)

Rotation (fluid motion)

in liquid outer coreCurrent must be maintained

3D magnetic field from Glatzmaier‐Roberts geodynamo model

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Transition from Geomagnetic Field to Electron Spin

Earth Electron

Coey. http://esm.neel.cnrs.fr/2009/program.html

Dipole Moment =7.75 x1022 Am2 Dipole Moment =9.27 x10-24 Am2

Magnetism and Magnetic Materials

• Magnetic moments of electronsorbital magnetic momentBohr magnetonelectron spin moment

• Electron states of free atomprincipal quantum number (n)orbital angular momentum quantum number (l)spin quantum number (s)

i i l ltransition metal elementsmagnetic quantum numbersPauli exclusion principle and Hund's rulesTotal magnetic moment calculations

Coey, 2009

“A Physicist View”Magnetic Periodic Table

of Elements

YELLOW: ferromagnetic at room temperatureGREEN: widely encountered in magnetic compoundsPINK: of some importance in magnetismBLUE: of little importance in magnetismGRAY: of little or no importance in magnetismRED: magnetically inert

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Sources of Magnetism in Atoms

Electron spinElectron orbit

Nuclear spin

The magnetic properties of solids derive essentially from the magnetism of their electrons.

Nuclei also possess magnetic moments, but they are ~1000 times smaller.

Magnetic Moments of ElectronsOrbital Magnetic Moment: The orbital motion of an electron around a nucleus is like a current loop. This motion produces the smallest unit of magnetic moment allowed and is called the Bohr magneton (B).

h

Po

chargetime 2

2orbit

vI er-ev = (I)(Area)=m r2 r

e

-e = 2m

orbit om P

Orbital angular momentum: Po = mevr

From Quantum Mechanics: Po must be an integral multipleof = h/2: Po = , where h is Planck’s constant and is theorbital quantum number ( =0, 1, 2, 3, ...)

orbite

e h = m2 m

B = 9.27x10‐24 Am2

e = electron charge 1.602176565(35)×10−19 Cme = mass of an electron 9.10938188 × 10‐31 kgh = Planck’s constant 6.626068 × 10‐34 m2 kg / s

Magnetic Moments of ElectronsElectron Spin: Observations of atomic spectra lead to the discovery that electrons "spin" about own axis as well as orbit about nucleus.

Quantum Mechanics: Ps= sh, where s=spin quantum number (s = ± 1/2)

B

e

e = - = 2 sm

spin sm P

Coey ,2009

Spin is a consequence of relativistic quantum mechanics

Magnetic Moments of Electrons

e 2

Two contributions to the electron moment:

Be

e= - = 2 sm

spin sm P

B

e

e= - = l2 m

orbit om P

e

e= g 2 m

tot totm P

For Spin:

For Orbit:

General Case:

where g is called the Lande‐splitting factor: g=1 for orbital motion only g=2 for spin motion only

Ps is twice as effective as Po in producing a magnetic moment. Reasons for these observations are not simple.

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Quantum numbers

Principal Quantum number, n: n determines the size of the orbit (mean distance to nucleus) and the energy of orbit.

n=1, 2, 3, ....., K, L, M, etc., electron shells2n2 electrons in each shell

Orbital Angular Momentum Quantum number, l: l gives a f th t i it f th l t bit

l = 0, 1, 2, 3, ..(n‐1),d f l tmeasure of the eccentricity of the electron orbit s,p,d,f electrons

4l+2 electrons in eachorbital shell

Magnetic Quantum Numbers, lz, sz (H 0): In a magnetic field(applied or internal) the orientations of spin and orbital angularmomenta are affected such that the projections of the angularmomenta along H (z‐axis) are quantized.

lz= l, l‐1, l‐2.....0.......‐(l‐1), ‐lsz= ‐1/2, +1/2

Orbital energy equation for electrons in QM is derived from Schrödinger’s Wave EquationOrbital energy equation for electrons in QM is derived from Schrödinger s Wave Equation

Electrons described by Wave functions (probability of finding an electron at point, r,,)

Wave functions given in terms of Spherical Harmonic Functions (n,l,lz)

Schematic of surfaces of equal energy of the first three orbital shells(Tauxe 2007)

As atomic number of an element increases, the number of electrons increases and these electrons occupy orbitals starting with the lowest energy level and working outward.

between shells energy is usually lower for lower n valuewithin shell energy is lowest for lowest l valuegynotation 3p6 (n=3, l=1, 6 electrons)

3d Transition Elements (potassium‐zinc)

electrons fill orbitals according to increasing l, within a given shell (1s, 2s, 2p, 3s, 3p) until potassium (K)

Instead of filling 3d orbitals, electrons fill lower energy 4s levels first

4s electrons take part in chemical bonding3d electrons responsible for magnetism

(Tauxe, 2008)

Magnetic Quantum Numbers, lz, sz (H 0)When there is a magnetic field (applied or internal) present the orientations of spin and orbital angular momenta are influenced by B such that the projections of the angular momenta along B are quantized.

lz= l, l‐1, l‐2.....0.......‐(l‐1), ‐lsz= ‐1/2, +1/2

H0, there are 4 quantum numbers (n, l, lz, sz)

Example: d electron (l=2)

can take 5 orbital orientations with (l =2 1 0 1 2) and

Vector model of atom

Zeeman Splitting:can take 5 orbital orientations with (lz=2, 1, 0, ‐1, ‐2) and the spin can be parallel or antiparallel to the field

Pauli Exclusion Principle: each electron orbit defined by the four quantum numbers can contain only one electron

Dunlop and Özdemir. 1997

Zeeman Splitting: Energy of the system is changed when placed in magnetic fieldSplits into 2l+1 states

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23

Example: Electron states of the n=3 (M) shell

0 1 2

(Chikazumi, 1964)

Number of 2 6 10electrons

Hund's Rules: Occupancy of Available Electron States

The net orbital and spin angular momentum will tell ushow many Bohr magnetons the atom will have.

Hund's rules determine the most stable spin (s ) andHund s rules determine the most stable spin (s ) andorbital (l ) configurations in the ground state

(1) electrons will fill levels so as to maximize atomic spin while being consistent with Pauli principle. i= S s

Example: Ti (3d2 + 4s2)two 3d electrons, szi can be either +½, ‐½

S = 1 ½ ½

½½

S = ‐1 ‐½, ‐½0 ‐½, ½+1 ½, ½ <‐‐‐‐maximum S

(2) orbitals values fill levels to maximize L while beingconsistent with rule (1) and the Pauli principle.

iL = l,

Magnetic Moments of Electrons

How do we combined S and L to find the total magnetic moment?

(iii) Total atomic angular momentum J (spin + orbit) is equal(iii) Total atomic angular momentum J (spin + orbit) is equalL‐S electron shell is less than half fullL+S electron shell is more than half fullS electron shell is exactly half full (L=0)

2 ( 1) ,

( 1)spin Bm S S

m L L

The total magnetic moment (called the effective moment) is the vector sum of the spin and orbital contributions and( 1)orbit Bm L L

( 1)( 1) ( 1) ( 1)1

2 ( 1)

total Bm g J JJ J S S L Lg

J J

spin and orbital contributions and according to rules of quantum mechanics

The equation for g is called the Landé equation

Isolated Atoms vs. SolidsThe values of J (and hence magnetic moment) in a solid can be different than in a free or isolated atom.

Orbital Quenching: In transition metals (3d electrons) and their compounds, the orbital contribution averages to zero (a 2 ( 1eff Bm S S

( 1)eff Bm g J J

their compounds, the orbital contribution averages to zero (a process called orbital quenching) and the ionic moments are spin only (L=0, J=S)

(eff B

0

0 2B

B

m gJm S

Spatial quantization: The component (mz) of the effective moment (associated with J, spin‐only(S), orbit‐only(L)) along H to have only certain values and the maximum values (m0) are

Which moment (effective or saturated) is measured depends on the magnetic method and experimental conditions (high field, internal field, low temperature, etc.) and material being studied

B

( 1)eff

B

m

g J J

o

B

m

gJ

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24

Ti with two 3d electrons (3d2) will fill the l=+2 and +1 levels

Example 1: Ti

Electron states: L=3, S=1, J=(L‐S)=2, g=2/3

2( 1) 6 1 63g J J Eff ti t

½ ½

3( 1) 6 1.63B B Bg J J

( 1) 2 2 2.83B B Bg S S

2 2B BS

Effective moment

Spin‐only moment

maximum moment

Example 2: Fe2+

Fe2+ with six 3d electrons (3d6) will filll=+2,+1,0,‐1,‐2,+2 levels

Electron states½ ½ ½ ½ ½½

Effective moment = 32( 1) 20B Bg J J

Spin‐only moment= ( 1) 2 6B Bg S S

L=2, S=2, J=(L+S)=4, g=3/2 ½ ½ ½ ½ ½‐½

Spin only moment ( )B Bg

maximum moment= 2 4B BS (spin‐only)

Example 3

Ar atom has 3p6 (filled orbital)

L=0, S=0, J=0 No atomic momentAr is diamagnetic

Minerals: Very few atoms have filled electron orbitals, so most free atoms should have atomic magnetic moments. But what happens when atoms bond to form molecules and crystalline solids?

Example: NaCl (Na+1Cl‐1)

Na+1: 2p6 ‐ filled orbital, no ionic magnetic momentCl‐1: 3p6 ‐ filled orbital, no ionic magnetic moment

NaCl is a diamagnetic solid

In general, both ionic and covalent bounding tend to yield filled electron orbitals and no molecular magnetic moment is allowed.

Note: Free Na atom (3s1) is paramagnetic

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Transition Metal Solids

Example: NiS (Ni+2S‐2) Ni=3d8 + 4s2

S‐2: 3p6 ‐ filled orbital, no ionic magnetic momentNi+2: 3d8 ‐ unfilled orbitalNi : 3d ‐ unfilled orbital

By using Hund's rules for 3d8 configuration: L=3 and S=1 NiS has a molecular spin‐only magnetic moment of 2.83B ( 22)

Important: The unfilled 3d orbitals in transition elements are the source of magnetic moment in these solids and are responsible for the paramagnetism and ferromagnetism in rock forming minerals.

General: Atoms and Molecules possessing an add number of electrons will have a total spin moment 0.

Transition MetalsUnpaired electrons and paramagnetism are usually associated with the presence of either transition metal or lanthanide (actinide) ions.

2 ( 1B S S Ion No. of 3d

electronsJ L S g mobserved( 1Bg J J

In many transition metal compounds the surrounding anions/ligands quench the orbital angular momentum and one needs only to take into account the spin only moment.

Cr2+ ,Mn3+ 4 0 2 2 -- 0 4.82 4.90

Mn2+, Fe3+ 5 5/2 0 5/2 2 5.91 5.82 5.91

Fe2+ 6 4 2 2 3/2 6.71 5.36 4.90

Co2+ 7 9/2 3 3/2 4/5 6.63 4.90 3.88Observed moments are consistent with orbital quenching and spin only moments.

•ParamagnetismField and temperature dependenceCurie Law

Magnetism of Solids

Langevin Theory

•FerromagnetismField and temperature dependenceCurie temperatureWeiss Theory

•Exchange Forces and Magnetic Ordering in Minerals

AntiferromagnetismFerrimagnetism

Dunlop and Özdemir, 1997; Coey, 2009

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Classification of Magnetic Materials3 Main Types

1 kg of SiO2M=‐0.0005 Am2/kg

T=300 K, B=1 T, H=80,000 A/M

1 kg of Fe2SiO4M=0.1 Am2/kg

1 kg of Fe3O4M=92Am2/kg

>0

< 0

Diamagnetism Paramagnetism Ferromagnetism

R. EgliDiamagnetism

Diamagnetism is very weak and usually masked by paramagnetism and ferromagnetism

Diamagnetic materials do not contain unpaired electrons so spin and orbital atomic moments cancel each otherso spin and orbital atomic moments cancel each other

An external field can modify electron orbitals producing a small induced magnetization opposite to the applied field

Diamagnetic materials are pushed away from strong fields (magnetic levitation)

Mineral (x10‐8 m3/kg)

Quartz (SiO2) ‐0.62

Calcite (CaCO3) ‐0.48

Water ‐0.90

Paramagnetism1 T1

T

B

Magnetic Energy: Application of field causes alignment of moments Em=Bcosθ

Thermal Energy: Randomizes moment E kT k B lt ’ t t (k 1 38 10 23 2 k 2 K 1)

Curie’s Law

ET= kT, k=Boltzmann’s constant (k=1.38× 10‐23 m2 kg s‐2 K‐1)

B,T Em ET Em/ETB=0.01 T, T=300K 4.6x10‐25 J 4.1x10‐21 J 10‐4

B=1T, T=10K 4.6x10‐23 J 1.38x10‐22 J 0.3

ExampleFayalite (=5B)

Recall the energy of a dipole in a field E B

cossin

E BdE B d

B

Langevin Theory of Paramagnetism

B

We wish to find the number dn of moments inclined at an angle between and + d to the field B.

Transitions between energy levels are produced by thermal perturbations so each energy level is populated with a probability that is determined by the Boltzmann distribution

Egli

Proportional to surface area ,dA, multiplied by the Boltzmann factor

n0 = normalizing constant, exp(‐E/kT) is the Boltzmann factorn= total number of dipoles per unit volume with moment

/0 2 sin exp( cos / )E kT

odn n dAe n d B kT

Cullity2 sindA d

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Component of in field direction:

total magnetization:

Magnetization (M)


0

cos

cosM dn

B

0

0 00

2 exp( cos / )sinn

n dn n B kT d

Proportionality factor (n0) determined by:

Substitute everything back in 0

exp( cos / )cos sin

exp( cos / )sin

n B kT dM

B kT d

0

exp( cos / )sinB kT d

Solution:

L() is the Langevin Function

1 1coth( ) ( )M e e Ln e e



Ms= maximum magnetization when all

O’Handley, 2000

( , ) ( ) ( )sM B T N L M L

0.6

0.8

1.0

L(

)

Langevin Function

sdipole moments are aligned with B

(i) B=0, T; 0, L()=0, M=0(ii) B, T0; , L()=1, M=Ms

(iii) <<1, H is small, T is large

3

0.0

0.2

0.4

0 5 10 15 20

L

BkT

( ) , , g

2

( ) ...3 45 3

L


M B

Curie‐Law of Paramagnetism

1T

Keeping the first term in the expansion of L(), which dominates when H is small and T is high

02

0

3

13

sM kTB H

NM CH kT T

1 T

1/C

C= Curie ConstantAt room temperature

Mineral (x10‐8 m3/kg)

Illite (clay) 15

Biotites 67‐98

Siderite (FeCO3) 123

Fayalite (Fe2SiO4) 126Due to the presence of 3d transition metals Mostly Fe+2,Fe+3Unpaired, non‐interacting 3d electrons

Magnetization Curves (M‐H‐T) for dysprosium oxide (Dy2O3)Saturation effects at low T (3 K) and high B (6 T)Linear behavior at high T( >100K)

Egli

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Quantum Theory of ParamagnetismTwo‐state particle (spin up or spin down)

s=+1/2 n+ , E1=‐B

s=‐1/2 n‐ , E2=+B

B =gBJ, J=1/2, g=2N=n++ n‐ (# of spins)

Energy partition function

Magnetization

Net spin imbalance Spins per unit vol.

1 2

1 2 1 2

/ /

/ / / /

E kT E kT

E kT E kT E kT E kT

n e n eN e e N e e

( ) , /vn nM N N V

volume

1 2

1 2

/ /

/ / tanh( ),

1, tanh( )1, tanh( ) 1

E kT E kT

v sE kT E kTe e BM N Me e kT

20 vN CkT T

Curie's Law

volume

Quantum Theory of Paramagnetism

2J+1 discrete spin states components in direction of B

j (j 1) 0 (j 1) j

O’Handley, 2000

Brillouin Function

2 1 2 1 1( ) coth coth ,2 2

BJ

g JBJ JBJ J J J kT

Magnetization ( )v B jM N g JB

mz= ‐j, ‐(j‐1),…0,….(j‐1), j

2

20

( 1)1: ( )3

3

j

v eff

J JBJ

NM CH kT T

Susceptibility

( 1)eff Bg J J

Curie's Law

The response is the same as the classical (Langevin) case but with eff instead of

Quantum Theory of ParamagnetismBrillouin Function

1/2 ( ) tanh( )( ) ( )

BB L

Special Cases

J=1/21.0

J=5/2

J=

Experimental Data

Brillouin Function

0 2

0.4

0.6

0.8

Bj(

)

2 ( 1B S S Ion mobs

Cr2+ ,Mn3+ 0 4.82 4.90

Mn2+, Fe3+ 5.91 5.82 5.91

Fe2+ 6.71 5.36 4.90

Co2+ 6.63 4.90 3.88

( 1Bg J J

Morrish, 1980

0.0

0.2

0 2 4 6 8 10

Example: Curie Law of ParamagnetismC

T

3.50E-06

4.00E-06

Olivine: (Mg0 9Fe0 1)2SiO4

1.00E-06

1.50E-06

2.00E-06

2.50E-06

3.00E-06

m

3 /kg

)

( g0.9 0.1)2 4

0.00E+00

5.00E-07

0 50 100 150 200 250 300

Temperature (K)

Experiment: Moment vs. Temperature at constant field (2.5T)

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N=6.03x1023/mole, kb= 1.38x10‐23 J/KMw (olivine)=147x10‐3 kg

1 1 TC C

2

0

3eff

w b

NC

M k

Example: Curie Law of Paramagnetism

y = 17873x + 63581

4.00E+06

5.00E+06

6.00E+06

)

0=4x10‐7 H/mB(Bohr magneton )=9.27x10‐24 Am2

From data fit: c=5.63x10‐5 (m3kg‐1K)‐/C=63231, =‐3.5 K

0.00E+00

1.00E+06

2.00E+06

3.00E+06

0 50 100 150 200 250 300

1/

kg/m

3 )

Temperature (K)Magnetic moment for olivine:

Olivine: (Mg0.9Fe0.1)2SiO4

0

3 w beff

M k CN

eff(ol)=2.1x10‐23 Am2= 2.3 B

Magnetic moment of Fe2+ in olivine: (Fe0.1Mg0.9)2SiO4

There are 0.2 moles of Fe2+ per mole of olivine; therefore, the Curie constant changes to

200 2 NC 2 3 1( ) ( )bM k CF l

Example: Curie Law of Paramagnetism

Quantum Mechanical Predictions for paramagnetic moment of Fe2+ (3d6)Electron states: L=2, S=2, J=(L+S)=4, g=3/2

00.23 w b

CM k

2

0

3 1( ) ( )0.2 0.2

w bM k CFe olN

eff(Fe2+) = 4.7x10‐23 Am2 = 5.1B

3Effective moment ( 1) 20 6 71g J J

The observed moment for Fe2+ (5.1B) is close to the predicted spin‐only moment (4.9B) and is consistent with orbital quenching of angular momentum in transition metals.

2Effective moment ( 1) 20 6.71

Spin-only moment ( 1) 2 6 4.9B B B

B B B

g J J

g S S

Paramagnetism and Mineral Magnetism

Paramagnetic minerals cannot carry a magnetic remanence (M=0 when B=0)

Depending on mineral symmetry, paramagnetic susceptibility in rock‐forming minerals can be anisotropic ( is a function of crystal direction)

1 11 1 12 2 13 3

2 21 1 22 2 23 3

(1,2,3) [ ] (1,2,3)M HM H H HM H H H

Anisotropy of Magnetic Susceptibility (AMS)

Can be used as a petrofabric tool to reconstruct: deformation and strain historypaleocurrent directions in sedimentary environments

3 31 1 32 2 33 3M H H H p ydirection of magma injection

Mathematical Details and Extra Slides

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Magnetic Potential due to Magnetic DipoleMathematical Details

Use Law of Cosine

2 2

2 2

22 cos 1 cos

22 cos 1 cos

lx r l rl r

rl

y r l rl r

+p

2

1, 0l

a

2 cos 1 cosy r l rl rr

1/22 cos , (1 )lLetr

r>>l, <<1, 1/2 12(1 ) 1

( )

Some algebra and simplification

-p

’

,r

coscos

x r ly r lr l

22 2 2

2

2 cos

1 cos

y x l

lxy r r

r

0 02 2

02

(2 )cos cos4 4

ˆ ˆ4

lp mWr r

m r rW rr r

0 0

4 4pp p y x

Wx y xy

Gauss Coefficients

1. Each l term is equivalent to a multipole of 2l poles at the Earth’s center (l=1 is a dipole, l=2 is a quadrupole, etc.)

2. The m=0 terms are due to the +z component of each respective multipole, and is the component aligned along the rotation axis of the Earththe rotation axis of the Earth.

3. The 0<ml terms are due to components of multipoles that are inclined to the rotation axis

4. The way the Gauss coefficient were set up using the partially normalized Schmidt functions, the magnitude of each coefficient tell us the relative importance of the various multipole terms.


1

1sin

rVBr

VBr

VBr

Once the geomagnetic potential is obtained the field components can be determined from

Good reference: Blakely, R.J, (1995) Potential Theory in Gravity and Magnetic Applications

Spherical Harmonic Description of Earth's Magnetic Field

Internal Field

Let’s look at the first few terms and attach some physical significance to themphysical significance to them

.....)sincos)(()sincos)(()(

)sincos)(()(

22

22

22

312

12

12

302

02

3

11

11

11

201

01

2

hgPrahgP

ragP

ra

hgPragP

ra

a

l=1, dipole term22

cossincossincos

sin)(,cos)(

)sincos)(()(

112

3112

3012

3

11

01

11

11

11

201

01

2

hrag

rag

ra

PP

hgPraagP

raa

Geocentric Dipole (l=1)

3 3 30 1 11 1 12 2 2cos sin cos sin cos

a a ag g hr r r

01

11

11

cos

sin cos

sin sin

g

g

h

dipole aligned along the spin axis (+ z axis)

equatorial dipole pointing along 0, 0 (+x axis)

equatorial dipole pointing along 90 E at 0 latitude (+y axis)

03

13

13 444 ahaa

( it h i T l i A 2)

Dipole moments

01

0

11

0

11

0

,, gmhmgm zyx (units: g,h in Tesla, m in Am2)

30 2 1 2 1 21 1 1

0

4 ( ) ( ) ( )

am g g h

2013 SSRM 5/29/2013

31

Geocentric Dipole (l=1)Orientation of Geomagnetic Pole Intersection of the extension of the + end of the inclined geocentric dipole with the Earth’s surface:

i d f h h l

Orientation of dipole field based on

01 1 1

2 2 20 1 11 1 1

cos ( / ) coszgm m

g g h

11 1 1xm g

Latitude of the north pole:

Longitude of north pole:


1 1 12 2 2 21 1

1 1

cos cosx

x y

gm m g h

If the l>1 terms were all zero (field perfectly dipolar), the geomagnetic north pole would coincide with the position of the north magnetic pole.


0

0

exp( cos / )cos sin

exp( cos / )sin

n B kT dM

B kT d

Good References: Cullity, B.D., Introduction to Magnetic

Materials (1972)Appendix A (Lisa , 2008)

Make the following substitutions:

cos cos , sin , , Ms

B Bx dx d nkT kT

1 x

xs

xe dxMM

e dx

Solution:

L() is the Langevin Function

e dx

1 1coth( ) ( )s

M e e LM e e

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