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2013 SSRM 5/29/2013
1
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
Why Study Paleomagnetism and Geomagnetism?
Geochronology
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Why Study Paleomagnetism and Geomagnetism?
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)
Why Study Paleomagnetism and Geomagnetism?
Regional Geology and Tectonics
Tectonostratigraphic terranes of the North American Cordillera. (From Butler, 1992)
Aeromagnetic Anomaly Map for Minnesota
Why Study Paleomagnetism and Geomagnetism?
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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Why Study Paleomagnetism and Geomagnetism?
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)
Why Study Paleomagnetism and Geomagnetism?
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
See slides at end for more details
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
2013 SSRM 5/29/2013
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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
( , , ) (cos )( cos sin )ll
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
( , , ) (cos )( cos sin )ll
m m ml l l
l m
aV r a P g m h mr
(Constable 2007)
1
0
( , , ) (cos )( cos sin )ll
m m ml l l
m
aV r a P g m h mr
l=2
l>1 terms
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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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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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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)
Langevin Theory of Paramagnetism
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
See slides at end for more details
Langevin Theory of Paramagnetism
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
Langevin Theory of Paramagnetism
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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29
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.
Orientation of dipole field based on 1990 IGRF Model (Blakely, 1995)
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:
Orientation of dipole field based on 1990 IGRF Model (Blakely, 1995)
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.
Langevin Theory of Paramagnetism
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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