Rock Magnetism

Solid State Physics

Paleomagnetism

Petrology Mineralogy

MAGNETISM OF ROCKS MAGNETISM OF ROCKS AND MINERALSAND MINERALS

How do rocks record paleomagnetic information?

OutlineOutline

Basics of magnetism (today)

Magnetic minerals

Magnetization processes in rocks

aleksey.smirnov@yale.edu

Basics of magnetismBasics of magnetism

At a conference on magnetism in Leiden, 1920 (from Physics Today)

A. Einstein

P. Ehrenfest

P. Langevin

H. Onnes

P. Weiss

Everything should be made as simple as possible.

But not simpler.

S S

SSN

N N

N

The field of a force – a property of the space in which the force acts

Magnetic field

attraction

repulsion

Magnetic field (force lines)

Magnetic field is not a central field (no free magnetic charges)

SN

F

Magnetic field definitions

B – magnetic induction

H – magnetic intensityTwo quantities describing a magnetic field

In vacuum:

B = H

B = µ0H

(cgs: centimeter, gram, second)

(Système Internationale, SI)

µ0 = 4π · 10-7 N A-2 - the permeability of free space (the permeability constant)

Magnetic induction (B) units

B

qv

FL

FL = q(v X B)

SI: Tesla (T) [N A-1 m-1]

cgs: Gauss (G) [dyne-1/2 cm-1]

1 γ (gamma) =10-5 Gauss

Lorentz force (FL )1 Tesla =104 Gauss

Tesla Gauss

[µ0]

[B]

Magnetic intensity (H) units

SI:

cgs: Ørsted (Oe)

1 A/m = 4π/103 Oersted

B = µ0H , hence H = B/µ0

[H] =

Ampere

Ørsted

A=N A-1 m-1

N A-2 = m

Magnetic moment (M)

No free magnetic poles can exist, hence the dipole field is the simplest configuration

Real source of magnetism is moving electrical charges (electrical currents)

Thin bar magnet (dipole)

Electric current loop

Uniformly magnetized sphere

I

Magnetic moment (M) units

m

m = AIn

[m] = Am2SI:

cgs: [m] = emu

1 Am2 =103 emu

A – area, I – current, n – unit vector

Emu

θ

Interaction with magnetic field

m = AInm = pd

+p

-p

dτ = m B sinθ

B

θ

aligning torque:

Magnetic field of a current loop (dipole)

Baxial =2µ0 m4πz3

z

decreases as the cube of distance

m

=AI

The Earth as a big magnet

MEarth ≈ 8∙1022 Am2

Earth magnetic field at the surface:

≈ 5 ∙ 10-5 T (0.5 G)

Magnetic fields in the universe

Sun surface: ~10-4 T (~10 G)

Sun spot: 10-2 - 10-1 T (~102-103 G)

At Earth’s orbit: ≈ 5∙10-9 T (~10-5 G)

Neutron Star: ~108 T (~1012 G)

Magnetar: ~1011 T (~1015 G) (strongest known field)

Galactic field: ~10-10 - 10-9 T (~10-6 – 10-5 G)

Filling a free space with matter…

Rigorous consideration requires quantum-mechanical approach… We go simple…

e-nucleus

Orbital magnetic moment

Morbital Mspin

Spin magnetic moment

Bohr magneton:

µB = 9.274 ∙ 10-24 Am2

MAGNETIZATION AND THE MAGNETIC FIELD INSIDE A MATERIAL

Atomic moment = orbital

moment + spin moment

A m2

m3

mi

mi

mi

mi

mi

mi

mi

mi mi

mi

mi

mimi

mimi

mi

mimi

mi

volume = V

Magnetization - the magnetic moment per unit volume

M = mtotal /V

Net magnetic moment of a volume V:

imtotal = ∑ mi

[ M ] = =

MAGNETIZATION AND THE MAGNETIC FIELD INSIDE A MATERIAL

SI:

cgs: emu / cm3

1 A m-1 =103 emu/cm3

Am

B = µo (H + M)

B = µo H – free space (M = 0)

In a magnetizable material the induction (B) has two sources:

1. Magnetizing field H (external sources)

2. Set of internal atomic moment, causing magnetization M

MAGNETIZATION AND THE MAGNETIC FIELD INSIDE A MATERIAL

Magnetic susceptibility

M = κ H

If M and H are parallel and the material is isotropic:

κ – magnetic susceptibility (dimensionless in SI)

κ is a measure of the ease with which the material can be magnetized

Magnetic permeability

B = µo(H + M) = µoH (1 + κ) = µoµH

µ = 1 + κ - magnetic permeability

M = κ H

µ is a measure of the ability of a material to convey a magnetic flux

MAGNETIC UNITS AND CONVERSIONS

Magnetic properties of materials

Pauli’s exclusion principle: each possible electron orbit can be occupied by up to two electrons with opposite spins

e- e-

me me

e-

me

∑ mspin = 0 ∑ mspin ≠ 0

Diamagnetism

M

H

κ < 0

Magnetization develops in the direction opposite to the applied magnetic field

• Exists in all materials (but observable when electron spins are paired)

• Diamagnetic κ (and magnetization) is reversible

• Diamagnetic κ is temperature-independent

H M

Quartz (SiO2) - (13-17) · 10-6

Calcite (CaCO3) - (8-39) · 10-6

Graphite (C) - (80-200) · 10-6

Halite (NaCl) - (10-16) · 10-6

Sphalerite (ZnS) - (0.77-19) · 10-6

Examples of diamagnetic mineralsκ (SI)Mineral

Data from Hunt et al (1995)

the partial alignment of permanent atomic magnetic moments by a magnetic field

M

H

κ > 0

Paramagnetism

• One or more electron spins is unpaired (the atomic net moment is not zero)

• Paramagnetic κ (and magnetization) is reversible

• Very large H or very low T is required to align all the moments (saturation)

• Paramagnetic κ is temperature-dependent

H = 0, M = 0 H > 0, M > 0

H

Thermal energy dominates

Paramagnetism: Temperature dependence

κ

T T

1/κ κ-1 ~ T

κ-1 ~ (T – θ)κ =

CT

The constant C is material-specific

θ

κ = CT - θ

The Curie-Weiss law

θ – the paramagnetic Curie temperature (near 0 K for most paramagnetic solids)

Examples of paramagnetic minerals

Olivine (Fe,Mg)2SiO4 1.6 · 10-3

Montmorillonite (clay) 0.34 ·10-3

Siderite (FeCO3) 1.3-11.0 · 10-3

Serpentinite 3.1-75.0 · 10-3 (Mg3Si2O5(OH)4)

Chromite (FeCr2O4) 3-120 · 10-3

Data from Hunt et al (1995)

κ (SI)Mineral

FerromagnetismAtomic magnetic moments are always aligned (even for H = 0)

due to exchange interaction (quantum-mechanical effect)

M ≠ 0

Conditions for ferromagnetism:

1) Non-compensated spin moments

2) Positive exchange interaction (i.e. co-directed spins)

Ferromagnetic elements:

• Iron (Fe) (κ = 3900000)

• Nickel (Ni)

• Cobalt (Co)

• Gadolinium (Gd)

Spontaneous magnetization

H = 0

FerromagnetismExchange interaction (Eex) decreases with temperature

Spontaneous magnetization, Ms

T

Ferromagnetism (Eex > kT)

Paramagnetism (Eex < kT)

Tc

Tc – the ferromagnetic Curie temperature (material-specific)

Ferromagnetism: Magnetic hysteresis

M

H

Ms – Saturation magnetizationMrs

HcHc – Coercive force (the field needed to bring the magnetization back to zero)

Mrs – Saturation remanent magnetization

Ms

Ferromagnetism (magnetic hysteresis)

M

HHcr

Ms – Saturation magnetizationMrs

Hc – Coercive force (the field needed to bring the magnetization Ms back to zero)

Mrs – Saturation remanent magnetization

Hcr – Coercivity of remanence

(the field needed to bring Mrs to zero)

AntiferromagnetismNegative exchange interaction (anti-parallel spin moments)

M = 0Antiferromagnetic elements:

• Chromium (Cr)

• Manganese (Mn)

Conditions for antiferromagnetism:

1) Non-compensated spin moments

2) Negative exchange interaction (i.e. anti-parallel spins)

Non-perfect antiferromagnetism

spin-canted antiferromagnetism

defect antiferromagnetism

M

M

Eg., Hematite (Fe2O3)

Ferrimagnetism

Ferrimagnets (ferrites) behave similar to ferromagnets

M

Super-exchange interaction

Eg., Magnetite (Fe3O4)

5µB 6µB

O2-Fe2+ Fe3+

Summary

Ferromagnetism Antiferromagnetism

Non-perfect Antiferromagnetism Ferrimagnetism

important for rock and paleomagnetism

Diamagnetism

Paramagnetism

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Magnetic mineralsRock magnetizations