L03 Magnetism of Fine Particles SRM of Fine Particles...2013 SSRM 5/29/2013 1 Magnetism of Fine...
Transcript of L03 Magnetism of Fine Particles SRM of Fine Particles...2013 SSRM 5/29/2013 1 Magnetism of Fine...
2013 SSRM 5/29/2013
1
Magnetism of Fine Particles
Hysteresis Loops Magnetic DomainsEnergetics of a Ferrimagnetic ParticleS f M ti A i t
Fe3O4
Sources of Magnetic AnisotropyHysteresis in Single domain and Multidomain particlesDomain Calculations and MicromagnetismNéel SD Theory of Thermal Activated Magnetization
13x13 m
Magnetism of Fine Particles
ROCK = assemblage of fine grained ferrimagnetic/antiferromagnetic minerals dispersed within a matrix of diamagnetic/paramagnetic mineralsminerals dispersed within a matrix of diamagnetic/paramagnetic minerals
Ferromagnetism, Ferrimagnetism is based on exchange forces between spin magnetic moments of 3d electrons
Exchange forces are so great (103 T) that spins spontaneously aligned when Bapplied = 0 and T< Tc
In the lab (and nature) weak fields (B<< T) are all that are needed to magnetize or demagnetize samples
In some cases, very small fields (B~Bearth <10‐5 T) are all that are needed to change magnetization
Hysteresis Loop
Hysteresis ParametersSaturation Magnetization : Ms, JsSaturation Remanence: Mrs Jrsrs, rsCoercivity: HcCoercivity of Remanence: HcrInitial Susceptibility: k0
Composition dependent: MsConcentration dependent: Ms, k0Grain size/microstructure: Mrs, k0, Hc,Hcr
Dunlop and Özdemir, 2007
Ms Mrs, k0, Hc, Hcr0 as TTc
Coercivity
UnitsHc (A/M)Bc= 0Hc (Tesla)
Typical Coercivities in Rocks (1>100 mT)
Soft (low) components: < 10 mTHard (high) components: >100 mT
0.5
1.0Multi-Domain Loopd=3 mm
0.5
1.0Single-Domain Loopd=50 nm 0.10
0.15High Coercivity (>500 mT)
Fe2O2
-1.0
-0.5
0.0
-400 -200 0 200 400
M/M
sat
0H(mT) -1.0
-0.5
0.0
-400 -200 0 200 400M
/Msa
t
0H(mT)
-0.15
-0.10
-0.05
0.00
0.05
-4000 -2000 0 2000 4000
Magne
tization
0H (mT)Magnetite
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Magnetic Domains and Particle Size
Examples of SSD, SPM, and MD hysteresis loops for magnetite.
Vibrating Sample Magnetometer (VSM)
Domain state size
Single Domain (SSD) 50 nm (biogenic)
Superparamagnetic (SPM ) 10 nm (biogenic)
Multidomain (MD) 3 mm diameter sphere of single crystal
Magnetic DomainsFerrimagnetic materials are usually composed of small regions of uniform magnetization called magnetic domains separated by narrow (<1000 nm) transition regions of rapidly varying spin orientation called domain walls. Domains are small (1‐100’s micrometers), but much larger than atomic distances
The Origin of Domains (Kittel 1949)
m
m
J MM
The Origin of Domains (Kittel, 1949)
Single domain (SD) state is uniformly magnetized to Ms (A). In this configuration, uncompensated surface magnetic poles will form (M0)
Formation of surface or volume magnetic charges are an additional source of magnetic field (the demagnetizing field). The energy associated with the surface pole distribution is called the magnetostatic energy.
Kittel, C., Physical Theory of Ferromagnetic Domains, Rev. Mod. Phys, 21, 541‐583, 1949
Magnetic Domain Walls
Inside a domain wall, the spin moments rotate in direction from
1012‐1015 spins
103 spins 2007
M ltid i (MD) i t if l ti d
one domain to the other domain.
DW
Dunlop
and
Özdem
ir,
Bloch Wall: 180° spin rotation within plan of wall
Multidomain (MD) grains are not uniformly magnetized. While inside each domain the magnetization is Ms, the entire grain has a net magnetization M<<Ms
Magnetic grains seek a spin structure that minimizes its total energy for a given volume, shape, magnetic field, and temperature
Fe3O4
Domains imaged with Magnetic Force Microscopy
0m M
13x13 m
+,‐magnetic charges (poles) around edges of grains and inclusionPokhil and Moskowitz, 1997
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Magnetization process in SD GrainsRotation of MomentsResponse of a random assemblage of uniaxial single domain (SD) particles during hysteresis cycle
Tauxe, 2008
Single Domain BehaviorSD particles produced by magnetotactic bacterium strain MV‐1 d ~ 30‐100 nm
Frankel and Moskowitz, 2003
a) Demagnetized stateb) In the presence of a saturating field, c) Field lowered to +3 mTd) Remanent state, e) back field of ‐3 mT,
Magnetization process in MD grainsTranslation of domain walls
Inset shows detail of domain walls moving by small increments called Barkhausen jumps.
(Domain wall observations from Halgedahl and Fuller, J. Geophys. Res., 88, 6505‐6522, 1983)
Tauxe, 2008
Single grain of TM60
Halgedahl, S. Bitter patterns vs. hysteresis behavior in small particles of hematite.JGR, 100, 353‐364. 1995
Single grain of Hematite
Barkhausen jumps
TM60
Energetics of Ferrimagnetic ParticleWhat drives domain formation?
Exchange Energy (eex)l f hb
1 2
2
22 cos
ex ee J s se J S Coupling of nearest‐neighbor spins
Short‐range effectsPermits only a gradual change in spin orientation is small to minimize eex
2 cosex e
e J S
Anisotropy Energy (ea)Coupling of spins to crystallographic directions O2‐
O2‐
e‐Coupling of spins to crystallographic directionsEasy axis of MagnetizationCrystal structure or shape dependent
Magnetocrystalline anisotropyShape anisotropy O2‐
O2‐
O2‐
e‐e‐
Low energy High energy
3d orbitals
Energetics of a Ferrimagnetic ParticleWhat drives domain formation?
Magnetostatic Energy (em)Coupling of all spins due to long‐range dipole‐dipole interactions M ti fi ld d d b 1 di l ff t ll th di lMagnetic field produced by 1 dipole affects all other dipoles
High energy Low energy
External Field Energy (eh)Coupling of spins to external field
h ae m B
B m B m
‐mB +mBLow energy High energy
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Energetics of a Ferrimagnetic ParticleWhat drives domain formation?
Magnetoelastic Energy (ems): Coupling of spins to local stress directionsMagnetoelastic anisotropySt i dif MStrain can modify MMagnetic field can produce crystal strains as M rotatesMagnetostriction is the spontaneous deformation of crystal as M changes
Sources of Strain
MagnetostrictionNonuniform magnetization due to domain formationInternal stress due to crystal
Cullity, 1972
Internal stress due to crystal defectsExternal tectonic stress
Magnetic Domains (micromagnetic structure): Represents an equilibrium between short‐range and long‐range forces
Fe3O4
0t i t
e e e
13x13 m(110) Fe3O4
Fe3O4
3 4
(100) Titanomagnetite
http://www.ifwdresden.de/institutes/imw/sections/24/members/schaefer/magnetic‐domains/5
(100)‐oriented silicon iron crystal
Magnetic AnisotropyInfluence of:
crystal structure magnetocrystalline anisotropygrain shape shape anisotropystress stress anisotropystress stress anisotropyexchange coupling between two phases
Magnetic anisotropy strongly affects the shape of hysteresis loops and controls the coercivity and remanence. Anisotropy is also of considerable practical importance because it is exploited in the design of most magnetic materials of commercial importance.
Without anisotropy there would be no remanence or magnetic memory
Anisotropy produces an energy barrier that an applied field must do work against to rotate M away from easy axis of magnetization
Magnetocrystalline anisotropyDirection cosines
[001]
2 2 2
1 2 31
1997
M
12
3
[100] [010]
Dunlop
and
Özdem
ir,
1
2
3
sin cossin sincos
[h.l.k] directionsCube edge: [100], [010], [001]Face diagonal: [110],[011],[101]Body diagonal: [111][100] 1=1, 2= 3=0
[111] 1= 2= 3 32 =1 =1/3 (57.1)[110] 1=2, 3=0 22 =1 =1/2 (45)
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Magnetocrystalline anisotropy
Cubic Anisotropy Energy:
K1,K2=magnetocrystalline anisotropy constantsdetermined from experiments (e g M H curves)
2 2 2 2 2 2 2 2 2
1 1 2 2 3 1 3 2 1 2 3v v( )
kE K K
determined from experiments (e.g. M‐H curves)v=volume
EK (K2=0) EK (K20)
EK[100] 1=1, 2=3=0 0 0
EK[111] 1= 2= 3=1/3
K1/3 K1/3 + K2/27
EK[110] = =0 K1/4 K1/4EK[110] 1=2, 3=0=1/2
K1/4 K1/4
Easy direction
Hard direction
examples
K1<0 EK[111] < EK[110] < EK[100] [111] [100] Fe3O4, Ni
K1>0 EK[100] < EK[100] < EK[111] [100] [111] TM60, Fe
Magnetocrystalline anisotropyAnisotropy Energy (Barrier): Amount of energy needed to rotated Ms from one easy axis to another easy axis (Depends on path)
1. Ek = EK[hard] – EK[easy]2. Ek = EK[medium] – EK[easy]
K1<0
K1 (J/m3) Path 1 Path 2
Fe3O4, Ti‐poor TM
‐1.3x104 ‐K1/3 ‐K1/12
TM60 0.21x104 K1/3 K1/4
Uniaxial Anisotropy: One easy axis of magnetization2iK
K1>0
2vsinK ue K
The energy is highest when the system is magnetized along a “hard” direction and lowest when magnetized along an “easy” direction.
Energy surfaces for magnetic anisotropy.
Ku>0
Harrison and Feinberg, 2009; Muxworthy, 2002
Cubic Magnetite (T>121 K)
Magnetocrystalline Anisotropy
4
K 0[100] easy axis
Temperature dependence of 2.5 T SIRM along [001] at 300 K
-1
-6
-11
Magnetite
Fletcher and O'Reilly (1974)Syono (1965)Kakol and Honig (1989)Aragon (1992)
K 1 (kJ m
‐3)
K1=0
[111] easy axis
TiTv
Isotropic point: Temperature where K1 =0 as it switches sign. For magnetite, Ti=130 K
-16100 200 300 400 500 600 700 800
Temperature (K)Verwey Transition and Isotropic point
Özdemir and Dunlop, 1999
Magnetocrystalline anisotropyMonoclinic Magnetite (T<121 K)
c‐axis : easy axisa‐axis: hard axis
Ka Magnetocrystalline anisotropy in monoclinic phase >10 x higher
KbKaaKbb K1
Özdemir and Dunlop, 1999Muxworthy, 2002
monoclinic phase >10 x higher than in cubic phase
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kol et a
l., 1991
Titanomagnetites
X=0.6Tc=465
K1 >0: [100] easy axisKak
Sahu and Moskowitz 1995Sahu and Moskowitz, 1995
4Jm
‐3)
m‐3)
Temperature Dependence of K1 and K2 anisotropy constants (T<300 K)
Titanomagnetites
K 1(104
Jm‐3)
K 1(104
Jm‐3) K 2
(104
K 2(104
Jm
TM0
TM05
TM19
TM28 TM55
TM41TM36
Kakol et al., 1991
Magnetocrystalline AnisotropyPhysical Origins
Single‐ion Anisotropy
Interaction of 3d electrons with the crystalline electric fields
Tetrahedral coord.A‐Sites
Octahedralcoord.B‐sites
yCoupling between spin and orbital momentsDifferent ions (Fe3+,Fe2+) in different coordination Total K=Sum of individual single‐ion anisotropies (+ or ‐)Analogous to Néel theory for M‐T curves
For exampleK(T)=KA(Fe3+)+KA(Fe2+)+KB(Fe3+)+KB (Fe2+)Can produce Isotropic points (K=0)
O2
‐
O2
‐O2
‐
O2
‐
O2
‐
e‐e‐
3d orbitals
Pair Model (Dipolar) Anisotropy
occurs in non‐cubic symmetriesdipoles coupled by ferromagnetic exchangeResults in a uniaxial (sin2) anisotropy
O’Reilly, 1984
Morin Transition: Spin‐flop transition
Hematite: Fe2O3
Magnetocrystalline single‐ion anisotropy dominates at low‐TEmin for spins along c‐axismin
positive
negative
Spin‐flop
O’Reilly, 1984; Özdemir et al, 2008
Competing anisotropies in corundum structure
Dipolar anisotropy dominates at high‐TEmin for spins in basal plane
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Mineral T (K) type First orderK1, Ku,Ka,Kb
Second OrderK2,Kaa,Kbb
Magnetocrystalline Anisotropy Constants for Common Magnetic Minerals
‐Fe 293 cubic +48 ‐1
Magnetite 293 cubic ‐13.5 ‐2.8
Magnetic 4.2 monoclinic +255 (a)+37 (b)
+18 (aa)+24 (bb)
TM55 293 cubic +1 +1
TM55 77 cubic +125 +0.8
Maghemite 293 cubic ‐4 65Maghemite 293 cubic ‐4.65
Greigite 293 cubic +20‐30
Pyrrhotite 293 uniaxial ~10‐30
Hematite 293 hexagonal ~0.1 (basal plane)
Data from various sources
Units for K (kJ/m)
MsMagnetostrictive Deformation
for Magnetite
Stress Anisotropy and Magnetostriction
Magnetostriction: Spontaneous change in dimensions (ℓ/ℓ) of a magnetic crystal when it is magnetized
= magnetostriction constant (= ℓ/ℓ , strain)
Ms
<001><0
<111>>0
>0 : expansion in direction of Ms<0: contraction in direction of Ms
Stress‐Induced Anisotropy: Stressing or straining a magnetic material in the presence of an applied field can produce a change in magnetization
Fabian and Heider, 1996
Magnetoelastic Anisotropy: Strain dependence of magnetocrystalline anisotropy For cubic symmetry ( and for a crystal that can deform freely)
'1 1
2 2911 12 100 44 1114 2
K K K
K c c c
111 , 100: magnetostriction constantsc11,c12,c44 : elastic constants (N/m2)
Stress Anisotropy and Magnetostriction
2 2 2 2 2 23 2100 1 1 2 2 3 3 111 1 2 1 2 2 3 2 3 3 1 3 12 3( ) 3 ( )
Cubic Symmetry: Saturation magnetostriction () in a direction i when M changes from 0 to Ms
i: direction cosines of strain; i: direction cosine of magnetization
Stress‐Induced Anisotropy: A uniaxial stress () can produce a unique easy axis of magnetization if the stress is sufficient to overcome all other anisotropies.
Isotropic Case: For a random polycrystalline materials=polycrystalline magnetostriction constant
2 23 i iK 100 111 s
325 5100 111s
Magnetostriction constants at 293 K(units =10‐6)
2 232 sin sins ue K Fe 21 ‐21 ‐7
Fe3O4 ‐19.5 +72.6 +35.8
TM60 +142.5 +95.4 +114
‐Fe2O3 ‐8.9
‐Fe2O3 8
Fe7S8 <10
= angle between M and stress
Easy axis when =0 and s>0 Tensional stress >0Compressive stress <0
Dunlop
and
Özdem
ir, 1997
Magnetostriction Constants in Titanomagnetites
Magnetostriction constants increase with Ti‐substitutionTi‐rich (x>0.5) titanomagnetites have high magnetostriction constants at low temperature
Moskowitz, 1992
Al‐substituted TM60
Özdemir and Moskowitz, 1993Klerk et al., 1977
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985
Mpa)
Titanomagnetites
Stress Anisotropy and Magnetostriction Magnetite
K/K1~0.1
Halged
ahl, 1985
Appe
l and
Soffel, 19
Stress (M
1988
Uniaxial stress can modify domain patterns in large MD particles
K/K1~1
Stress needed for magnetoelastic anisotropy energy (E) to equal magnetocrystalline anisotropy energy (EK) as a function of Ti composition (x)
Compressive strength for magnetite ~ 100 MPa
Moskowitz et a
l.,
Stress induced domain patterns in Ti‐rich titanomagnetite (Fe2.2Al0,1Mg0.1Ti0.6O4)
Scale bar =10 m
Temperature Dependence of Anisotropy Constants
Magnetostriction constants decrease with increasing temperature less rapidly than K1
Magnetoelastic anisotropy may thus be an important factor in remanence acquisition to higher T than magnetocrystalline anisotropy.
( 1)/2
1
1
( ) ( )(0) (0)
s
s
K T M TK M
Theoretical predictions:
l=4 for cubicl=2 for non cubic
Dunlop and Özdemir, 2007
For magnetite Theory (l=4): K1(T) ~Ms
10
Experimental: K1(T) ~Ms8‐9
Experimental: (T)~Ms2‐3
Ha
Shape AnisotropyInternal Demagnetizing Field (HD)
Apparent surface pole
M
Apparent surface pole distribution produces an internal field antiparallelto Magnetization
Uniformly magnetized ellipsoid: N DH M
N = demagnetizing factor (geometric factor)Nx+Ny+Nz=1 (or=4 in cgs units)
Sphere: Nx=Ny=Nz3N=1N=1/3
Demagnetizing Energy
Ha0mE m HM
Internal Field: N
i a D
i a
H = H +HH = H M
0 0 0
21
m i aE Nm vME NM
m H m H m M
M H 2120 0m aE v vNM M H
Magnetostatic EnergyZeeman Energy
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Shape Anisotropy
Long‐thin Needle Shaped Grain (elongated along z‐direction)
Nx=Ny=NN =N 0 N l d it f
Nx+Ny+Nz=1
M
MsMs
Nz=N0N=1/2
N pole density on surface
Axial Transverse
N0 N=1/2
Magnetostatic Energy
2 21 1( )E t N M M21( ) 0E i l N M
Ms
2 21 12 40 0( )mE transverse vN M vM 21
2 0( ) 0mE axial vN M
Easy direction Hard direction
Energy Barrier to moment rotation
2 20 0/ /( )
2 2m s sE N N vM NvM
Demagnetizing Factor
xx xy xz x
yx yy yz y
N N N MN N N N M
N N N M
DH M
zx zy zz zN N N M
Shape N
Sphere N= 1/3
Needle‐shaped grainProlate ellipsoid (a>>c,b)
N=1/2N = 0
Thin film, Thin Disk N = 0N=1
N transverse MN axial or in‐plane M
Demagnetizing Factors for Prolate and Oblate Spheroids
mir, 1997
1
,
2 1
x y z
z a x y b
N N N
N N N N N N
N N
Prolate Spheroid a
b
Dunlop
and
Özde
2 11 / 2(1 )
1 / 2(1 3 )
a
a
c a
N NN N
N N N N
N=1/2 N=0Typical value used in calculations:
Elongation= 1.5 (axial ratio, q=0.67)Na=0.232, N=0.15
2 212 0
212 0
( ) v ( )sind a b a s
u s
E N N N M
K NM
General Case: M at angle to z‐axis
Initial (low‐field) SusceptibilityAC fields ~ 10‐100 μT
Magnetic SusceptibilityInternal Demagnetizing field
Observed Susceptibility (0):
He
i eH H NM Inside grain
Observed Susceptibility (0):
0
( )(1 )
(1 )
i i i e
i e i
i
e i
M H H NMH M N
MH N
Intrinsic (“true”) susceptibility (i)measures the response of M to the internal field
Strongly magnetic materials (High Ms): Nχi>>1
01
(1 )i
iN N
Weakly dependent on grain shape/domain stateConcentration dependent
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Shape Anisotropy
Shape Anisotropy constant: 20
12u sK NM
demir, 2007
General Case 2v sina uE K Emax = Ku
To force M over anisotropy barrier an external field must be applied such that EH>Ea
What field do we need to flip M into opposite direction (from =0 to =180)?
Tauxe, 2008; Dun
lop and Özd
Equate Ea with equivalent field that keeps M aligned along easy axis
HaMs
e.a.
Ha Ms
<<1
Shape Anisotropy
20 cos sinT h a s a uE E E vM H vK
Anisotropy Field (Coercivity)
00 2Ts a u
dE vM H vKd
Solution: Set the derivative to zero and find Ha
212
2 2120
1 sin cos 1[1 ]T s a u
forE vM H vK
0
2 ua
s
K s
dKHM
H NM
Example: long thin needle of magnetiteN=1/2, Ms=480 kA/m
HK= 240 kA/m 0HK=0.3 T (300 mT)
0.5
1.0Single-Domain Loopd=50 nm
0 KKu=7.2x104 J/m3 (>K1)
Important source of anisotropy for materials with high Ms
Hematite (Ms=2kA/m, 0HK=1.2 mT)
-1.0
-0.5
0.0
-400 -200 0 200 400
M/M
sat
0H(mT)
Hysteresis in Single Domain ParticlesStoner‐Wohlfarth Model (1948)
Non‐interacting particlesCoherent rotation without thermal effectsU i i l i
v=volumeMs=saturation magnetizationKu=anisotropy constantH0=applied field
Uniaxial anisotropy
Total magnetic energy of particle consists of field interaction energy (EH) and the uniaxial shape anisotropy energy (Ea)
aH EEE ),(
0 0v cos( )H sE M H 0 02vsin
H s
a uE K 2
021 )( sabu MNNK
Stoner EC and Wohlfarth EP (1948) A mechanism of magnetic hysteresis in heterogeneous alloys. Philosophical Transactions of the Royal Society of London A 240: 599–642.
Hysteresis in Single Domain Particles
What’s the orientation of M for a given value of H0?What’s the flipping field or what value of H0 do you need to apply to switch the magnetization y pp y gand overcome the anisotropy barrier?
To solve this problem, need to solve the following energy minimization conditions for H
0),(
0),(
2
2
E
E
For φ=0 (applied field aligned with easy axis), the solution is s
uk M
KH
0
2
the solution is sM0
HK is the microscopic coercive force (or the anisotropy field). It is a critical field above which one of the energy minima (θ=0) becomes unstable and the particle moment undergoes an irreversible rotation to the other stable minima (θ=π)
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s
uk M
KH
0
2
(a) H0 aligned with easy axis (φ=0)
Hk = (Nb‐Na)MsMr/Ms=1.0
Ha Ms
(b) H0 perpendicular to easy axis (φ=π/2)Reversible magnetizationMr/Ms=0, Hc=0i=Ms/HK
(c) for all φ (randomly oriented array of SD particles)M /M =0 5
Ha
Ms
Mr/Ms=0.5Hc= 0.479 HK(φ=0) = 0.479 (Nb‐Na)Ms
Magnetocrystalline anisotropy
K1< 0, [111] easy axis Mr/Ms=0.87K1>0, [100] easy axis Mr/Ms=0.83
Ha
Dunlop and Ozdemir, 1997
Magnetization curves from Stoner‐Wohlfarth Model for various angles between the direction of magnetic field and easy axis
Microscopic Coercive Force for other
Jiles, 1991
0
20.479 uk
s
KHM
0.479( )k b a sH N N M
0
30.479ks
HM
types of Anisotropy
General uniaxial
Shape anisotropy
Magnetoelastic anisotropy0 s
SD particles have ideal hysteresis properties for paleomagnetic and technological applications
High stability against remagnetizationHigh remanence (Mr/Ms=0.5)
TEM image of a magnetic separate from a plagioclase crystal. (Tarduno et al., 2005)
Theoretical SD valuesRandom assembly of uniaxial particles
0
20.479 uc
KHM
Stoner‐Wohlfarth Model (1948)
0
0
20.524
0.5 1.09
s
ucr
s
r cr
s c
MKHM
M HM H
0.5
1.0Single-Domain Loopd=50 nm
Biogenic SD magnetite:
Dunlop and Özdemir, 1997
-1.0
-0.5
0.0
-400 -200 0 200 400
M/M
sat
0H(mT)
Mr/Ms = 0.48Hr/Hc = 1.12
freeze‐dried cells of magnetotactic bacterium strain MV‐1
Magnetization process in MD grain Wall Energy is a function of position in a crystal due to the crystal defects
Demagnetizing field is strong enough to derive walls backs
Wall translation Domain rotation Domain nucleation
Nucleation, displacement, denucleation of DW with changing field
enough to derive walls backs towards M~0
O’Reilly, 1984
Remanent state (Mr) is near zero with low Hc.Hysteresis Parameters for MD grains
Mr/Ms <0.1Hc(MD) << Hc(SD)
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0.0
0.5
1.0
M/M
sat
MD Loopd=3 mm
single crystal sphere
0.0
0.5
1.0
M/M
sat
Single-Domain Loopd=50 nm
Hysteresis in MD grain
-1.0
-0.5
-400 -200 0 200 400
0H(mT)
-1.0
-0.5
-400 -200 0 200 400
0H(mT)
Idealized MD hysteresis loop for magnetite
Balance between H0 and Hd
0 0H NM
Slope =1/N=2.7Sphere: 1/N=3
Dunlop and Özdemir. 1997
0
0
0
1
HMM
dMdH N
Domain Calculations
Wall Width (δw) and Wall Energy (w)
Bloch wall (180° spin rotation)Infinite medium (i.e., no crystal surfaces no surface poles)Infinite medium (i.e., no crystal surfaces no surface poles)
Exchange energy (Eex) between neighboring spins
Make angle between spins very small to keep Eex smallSpread spin rotation out over a large distance
Wants to make a wide wall δ=w
Anisotrop energ (E )
Domain wall
Anisotropy energy (Ea)
Spins no longer poin ng along easy axis when φ≠0Spread spin rotation over a short distance to avoid non easy directions
Wants to make a thin wall
Tauxe, 2008
Domain Calculations
The total wall energy will be the sum of exchange and anisotropy energies. A simple functional dependence on wall width (δ) is
KA
δw
γmin Ea
Eex
aw K
A= exchange constant ( Tc)Ka= general anisotropy constant
Minimum energy and equilibrium wall width is obtained by the condition
0 w
waw KAAK ,2
aK
More exact solution for uniaxial anisotropy
awaw K
AAK ,2
Exchange Constant (A)S=spin numbersz= no. of nearest neighborsa=lattice constantJe= exchange integral
2
3ezJ SAa
See slides at end for more details
Some numbers for magnetite
Constants Wall width and EnergyA = 1.3x10‐11 J/m K1= 13 k J/m3
δw = 100 nm w= 2.6 mJ/m2
More accurate estimate of anisotropy energy through wall (Stacy and Banerjee, 1975)K=0.115K1+0.021K2K=1.6 kJ/m3
δw = 280 nm w= 0.93 mJ/m2
Experimental MeasurementsOzdemir et al., 1993;Moskowitz et al., 1988, Foss et al., 1998
w=0.9 mJ/m2
δw = 180 nmδw = 200 nm
MFM image and magnetic profile of a 180° domain wall in (110) single crystal magnetite.
Half‐width = Domain widthDomain wall width ~ 200 nm (Foss et al., 1998)
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Domain Formation
Subdivision into finer and finer domains to reduce magnetostatic energy cannot go on indefinitely
High MSenergy
Smaller MSenergy
Smaller MS energyHigher wall energy No Ms energy
Wall energy
Domain Formation
Magnetostatic energy (long range effect)Wall energy (short‐range effect)
Domain walls require energy, so the more domains the higher the wall energy
7 domain grain of Fe3O4
Closure domains
Minimum Energy state: ET =EH + Eex +EK + ED +E
EH= external field energyEex= exchange energyEK= magnetocrystalline energyED= magnetostatic energyE= magnetoelastic energy
0TE 13x13 m
Single Domain (SD) – Two Domain (TD) Transition Size
What grain size (d) is the TD state (b) lower energy than the SD state (a)?
NSD is the demagnetizing factorNSD= 1/3 for a sphere or cube
SD State 20
2v
m SD s
SD m
E N M
E E
)(21 areawallEE wSDTD MD State
SDTD transition occurs at d0 when ESD=ETD
a2
00
sSD
w
MNa
d
, where a=6 for a sphere and a=4 for a cube
Some numbers for magnetite (at room temperature)
w= 1 mJ/m3, Ms=480 kA/m, NSD=1/3 d0 ~ 40 nm
more exact calculations give d0~ 80 nm
Effect on d0
elongated grains (N small)
increases
High Ms decreases
High T, low Ms increases
Image of SD BehaviorTop: chain of SD magnetite particles in Magnetospirillummagnetotacticum.
Bottom: Off‐axis Electron HolographicImage of particles (magnetosomes)
The contours provide a map of the localmagnetic field in the cell.
The confinement of the magnetic flux within the magnetosomes shows that all the magnetite crystals SD magnetized approximately parallel to the axis of the chain
Dunin‐Borkowski et al, 1998, 2001
magnetosome (50 nm) displaying SD dipole‐like external magnetic field and uniformly magnetized interior
Harrison and Fienberg, 2009
Domain ModelsKittel Model: Semi‐Infinite Plate Model (Kittel, 1949)regular array of uniformly‐spaced, 180° DWs separating domains magnetized perpendicular to the plate surfaceUniaxial Anisotropy
Total energy = the sum of ED of the n domains d i D
and Ew of the n‐1 walls:n domainsn‐1 domain walls
2102
1/221/20
1( ) ( 1)
02
T SD s w
T SD seq
w
E LWDN M LW nn
dE N Mn Ddn
op and
Özdem
ir, 1997
ED= demagnetizing energyw= Bloch wall energyVolume=LDW, D=nd
Dunlo
slope=1/2
SD transition size (n=1)
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Domain CalculationsMagnetiteSD‐TD transition (d0) 40 nmWall width (w ) 200 nm
SD‐TD calculation assumes infinitely thin wallsWall calculation assumes no surfaces
No surface polesNo magnetostatic energy
In small particles (near d0), walls have finite thicknesses and produce surface poles due to rotation of spins within wall
Additional source of magnetostaticAdditional source of magnetostatic energy due to wall
To reduce this source of energy, wall width needs to be thinner
w
Theoretical Calculations for SD‐TD transitions in Magnetite and other Magnetic Minerals
(Dunlop and Özdemir, 1997)
Estimated critical sizes for SD‐TD behavior in magnetite as a function of particle elongation (Butler and Banerjee, 1975)
Theoretical Domain States for Magnetite
Bazylinski, 2000
nm) Single Domain
Two Domain
Metastable SD
1000
magnetosomes
part
icle
leng
ht (n
Superparamagentic
MV1 MS1
MC1MV4MV2
GS15
100
10
MSR1
AMB1
Confirmation of SD calculationsMagnetite particles produced by magnetotactic bacteria (MTB) fall within SD size range
Different strains of MTB
0.0 0.2 0.4 0.6 0.8 1.0
axial ratio (width/lenght)
10
SD Behavior (strain MV‐1)
Mr/Ms=0.5
SD (50 nm) magnetosome displaying dipole‐like external magnetic field and uniformly magnetized interior (Harrison and Fienberg, 2009)
Micromagnetic ModelingSubdivision of grain into LxMxN cells
In each cellMagnetization vector: M(x,y,z)fixed magnitude, variable direction (.)2 unknowns representing direction ofM(.)
ex K D s(E E E E E )T HvE dv
2 unknowns representing direction of M(.)
Total Energy
M(.) is varied independently in each cell until a local energy minima (LEM states) is found for a given set of constraints
Williams and Wright, 1999
g
Numerical solutionsLength scaleCell size () ~ exchange length (ℓex)
0
1 ~ 10nmexs
AM
Fe3O4:
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Micromagnetic EnergyEnergy Equation Material property
Zeeman energy H0 = External field
Demagnetizing N = Demagnetizing factor0 0H sE M H m
21E M Nenergy
Exchange energy A= exchange constant
Magnetocrystallineenergy
Ku= Uniaxial constK1= Cubic const
Magnetoelastic energy
202d sE M N m
2( )exE A m 2
2 2 2 2 2 21 1 2 2 3 3 1
sin
( )K u
K
E K
E K
2(cos 1)E K 3 sK
energy 2
For constant T < TcM(x,y,z) has fixed magnitude and only changes direction inside particlem(x,y,z)= is a unit vector in the local direction of magnetization
cos sin( , , )( , , ) sin sin
coss
M x y zm x y zM
Micromagnetic ModelingMore complicated spin states can be produced as constraints on magnetization directions are relaxed from 1‐D to 3‐D rotations producing more ways for flux reductions to be achieved
Local Energy MinimaLEM states
Flo er stateFlower stateMagnetostatic energy highExchange & anisotropy energy low
Vortex stateMagnetostatic energy lowExchange & anisotropy energy high
Williams and Wright, 1999
Micromagnetic Modeling
Flower +
Vortex
Energy density of a magnetite cube as a function of edge length d for an initial SD configuration at room temperature .
Increasing grain size SD structure collapses to a vortex structure at d0max = 96nm. Decreasing size vortex state becomes SD at d0min =64nm.
;Muxworthy et al., 2003
Exsolution and magnetic interactions between magnetite crystals
Magnetite‐Ulvospinel intergrowthFe (blue), Ti (red)
Off‐axis Electron Holography
3‐particlesupervortex
3‐particle flux‐coupled “AF” configuration
Magnetic field lines link neighboring magnetite crystals across intervening nonferrimagnetic ulvospinel
Isolated crystals have vortex (c) and SD (d) structuresInteracting crystals produce super‐vortex or flux‐coupled configurations
Harrison et al., 2002
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Closure domainsClosure domains
Micromagnetism
Micromagnetic model (2‐D) for 5 micron cube of magnetite showing edge closure domains (Xu et al., 1994)
MFM image of magnetite grain (image is 8x8 microns, Pokhil and Moskowitz, 1997)
Pseudo Single Domain (PSD) Behavior
PSD properties
Found in particle sizes above the SD size range but p gexhibit SD‐like properties
HC> “expected for MD grainsMR> “expected for MD grains
PSD particles have micromagnetic states intermediate between SD and MD
Carton describing the failure of classical domain theory to describe the grain size dependence of remanence above d0 for magnetite(Banerjee, 1977)
Room temperature saturationRoom temperature saturation remanence (Mrs) and Coercivity (Hc) as a function of grain size for magnetite (Dunlop and Özdemir 2007)
Rather than the rather abrupt decrease at the SD threshold predicted theoretically,p y,gradual decreases in properties are observed over several decades of grain diameter
Hysteresis parameters for dispersions of magnetite particles with different grain sizes (Dunlop and Özdemir, 1997)
SD
Day Plot
MDMr/Ms Hr/Hc
SD 0.5 1‐2
PSD 0.1‐0.5 2‐3
Diagnostic values for SD and (large) MD grains
PSD
MD <0.1 >3‐4
Ku, uniaxial Mr/Ms=0.5K1< 0, [111] easy axis Mr/Ms=0.87K1>0, [100] easy axis Mr/Ms=0.83
Theoretical SD values
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PSD Models
Theoretical stable size range for 1D local energy minima (LEM) states in magnetite. The ground state equilibrium size range is in black (Moon and Merrill, 1985)
(a) Nucleation failure in saturation remanence.
black (Moon and Merrill, 1985)
Metastable SD state
(b) back‐field nucleation of domain wall
(c‐e) domain wall displacement in increasing negative field
(Halgedahl and Fuller, 1980)
R li t TRM i t ft
LEM States
Replicate TRM experiments often produced different numbers of domains in the same particle.
Halgedahl, S.L., 1991. Magnetic domain patterns observed on synthetic Ti‐rich titanomagnetite as a function of temperature and in states of thermoremanent magnetization. J. Geophy. Res, 96: 3943–3972.
Micromagnetic Simulation of Hysteresis
0.1 m cube of magnetiteA. Initial state is a negative
Flower state (F)B. Flower state (F) jumps to
BB. Flower state (F) jumps to
vortex state (V)
Vortex state is LEM state for successive hysteresis cycles
C. Reversible spin rotation in outer most cells of vortex (MD‐like process)
F
DC
D
D. Discontinuous vortex reversal of vortex core (SD‐like process)
V
FA
Williams and Dunlop, (1995)
Magnetic Relaxation and Blocking
• Natural Remanent Magnetization• Néel SD Theory of Thermal Activated Magnetization
• Relaxation times and Magnetic Blocking• Superparamagnetism
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Pioneers of Rock MagnetismL. Néel
1949 Théorie du traênage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites. Annales de Géophysique 5: 99–136.1950 Some theoretical aspects of rock magnetism. Phil.
L. Néel (1905‐2000)
Mag. Suppl., 4, 191‐243.
Fundamental study of the magnetism of single‐domain grains Explain quantitatively the experimental results of E. Thellier on baked clays and lavas
Developed theory of magnetic relaxation and the effect of thermal fluctuations on magnetization
thermoremanent magnetization (total and partial)thermoremanent magnetization (total and partial)isothermal remanence & viscous remanenceDemagnetization by heating in zero field and by alternating fields
Discovered superparamagnetism and calculated the lower limit of magnetic stability of single‐domain iron (~16 nm) EMILE THELLIER (1904‐1987)
Magnetic RelaxationHysteresis (isothermal effect): field assisted magnetization transition over anisotropy barrier
Magnetic relaxation (thermal fluctuations) : time‐assisted magnetization transition over anisotropy Du
nlop
and
Özdem
ir, 1997
barrier
Example 1: Apply large Ha then reduce Ha=0
When Ha=0, E1=E2Equal energy, unequally populated states
N number of identical particles with uniaxial anisotropy
E ample 2 D ti l th l H <H
n+=N n‐=0
E1 E2Example 2: Demagnetize sample, then apply Ha<Hc1 2
n+=N/2 n‐=N/2
E1 E2
HaInitially both states have E1=E2 and equally populatedWhen Ha is applied, E1<E2Unequal energy, equally populated sates
Magnetic states will approach (relax) towards equilibrium state
Magnetic Relaxation
Thermal activation (random thermal vibrations) can lead to over barrier transitions
nt magne
tization
/0( ) t
r rM t M e
Remanent magnetization of an assembly of SD grains will decay with time (Ha=0) towards equilibrium state where Mr=0
= relaxation timet/ << 1, Mr(t)= Mr0t/ >>1, Mr(t)0
Mr(t), reman
en
Paleomagnetic record carried by magnetic grains represent non‐equilibrium states, but the approach to equilibrium can be slow enough (> 1Ga) that even geologic time is sufficient to reach equilibrium)
Dunlop and Özdemir, 1997
Thermal activation
At low T magnetic moment of particle trapped in one of the potential wells
Particle magnetic moment is “blocked”
Easy Axis
Particle magnetic moment is “blocked”
At higher T, thermal energy can flip
Low T
g , gy pmagnetic moment between the wells
Results in rapid fluctuation of momentParticle moment becomes “unblocked”
High T
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Magnetic RelaxationN‐n particles
N=total number of particlesN=+
Probability of a thermal vibration of amplitude E occurring in a given time interval (dt) will have a relaxation time for transition between state 2 state 1:
2112
n particles
state 2 state 1:
212 1
0
1 exp Ef kT
0= frequency factor (~109 Hz)2
sn NM MN
Rate Equation:
21 12
dn N n ndt
/0
12 21
12 21
12 21
( ) ( )
1 1 1
teq eq
eq s
M t M M M e
M M
Solution:Meq is the thermal equilibrium magnetization at t=∞ for a given temperature and field
Dunlop and Özdemir, 1997
Magnetic Relaxation
/0( ) ( ) t
eq eqM t M M M e
General Equation
Magnetization versus time for (a) saturation remanence in zero field, (b) zero initial magnetization in a field, (c) magnetization l d i ti ll l fi ld (f T 2008)
(a) zero‐field decayHa=0, M0=Mr, t=0
Meq=0, t
(b) demagnetized stateM0=0, t=0MMeq , t
placed in an antiparallel field (from Tauxe, 2008)
(c) Antiparallel fieldH=‐Ha, M‐Meq , t
Equilibrium MagnetizationTwo‐state particle (spin up or spin down)
E1= ‐0vMsH0
H0
121 2
0 0
1 1exp eEf kT f
21 12E E
E2= +0vMsH00
H0=0 H00
12 21
12 21
tanh( )
t h
eq s s se eM M M Me e
EM M
212 1
0 0
1 1exp eEf kT f
v=particle volume
0 0
tanh
tanh
eq s
seq s
M MkT
vM HM M
kT
Thermal equilibrium magnetizationH0=0, Meq=0
One SD grain has no M=0 state for Ha=0 except when T>TcEnsemble of SD grains can have M=0 states
Néel SD Theory of Thermal Activated MagnetizationTheory that provides theoretical basis for understanding time‐dependent magnetization and stable magnetization
Single domain particles with uniaxial anisotropyIdentical non interacting particlesIdentical, non‐interacting particles
Reversals between states in which moments are either parallel (=0, state 1) or antiparallel (=,state 2) to a weak Ha
20 0
2 20max
cos( ) sin
1 /2
s u
K s a K
E vM H K v
E H M H H
max
12 max
21 max
2( 0)( )
K s a K
E E EE E E
Dunlop and Özdemir, 1997
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Néel SD Theory of Thermal Activated Magnetization
0
0
2
0
10 : exp2
10 : exp 1
s Ka
as K
vM HHf kT
HvM HH
Néel Relaxation Times
0
0 : exp 12a
K
Hf kT H
sKsus
uK NMHNMK
MK
H ,2
,2 0
0
Relaxation time is a strong function of temperature, applied field (Ha), volume, and shape (HK) of the grains.
12 21 21 12E E
One of the energy barriers becomes much larger than the other
State favored by Ha “drains” the less favored state and MeqMs
Néel Relaxation Times0
0 0
1 10 : exp exp2
s K ua
vM H vKHf kT f kT
vKu/kT (sec)
25 100
60 1017 (~ 4.5 Ga)
Example: SD Fe3O4, HK= 20 mT, v=/6D3
D (nm) ( )
A small change in volume (T=const) can increase relaxation times by a factor of 1015!
v3v, 100 s109 yrsd1.4d (40% increase in d)
Laboratory time
Geologic time
D (nm) (sec)
28 100
33 1017
t=0, H=Hsat t=t1, H=0
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Superparamagnetism
HHH=0
Unblocked particles that respond to a field are known as superparamagnetic
Relaxation Times
Relaxation time in magnetite ellipsoids as a function of grain width in nanometers (all length to width ratios of 1.3:1.)
Tauxe 2008
SuperparamagnetismIf volume is small enough or temperature high enough, thermal fluctuations will be sufficient to overcome the anisotropy barrier and flip the magnetization, even when Ha=0
Superparamagnetism (SPM): For a group of SPM particles each with magnetic moment, M, the net magnetization in H0=0 (T>0 K) will average to zero or follow the equilibrium magnetization (H00)
This means that Hc=0 and Mr=0, but Ms0.
The size below which the anisotropy barrier to magnetization changes can easily be surmounted by thermal energy on a specified (usually laboratory) timescale is called the superparamagnetic (SPM) threshold dtimescale is called the superparamagnetic (SPM) threshold ds
ds d0
SPMUnstable
magnetizationStable SD Non‐uniform
magnetization
~20 nm ~80 nmFe3O4 (T=300K)
Superparamagnetism Critical relaxation time (s)Relaxation time leads to the concept of blockingPaleomagnetic recorders must have relaxation times on the order of geologic time
<s unstable magnetization (SPM)
00
, v v , constant2v ln( ), 100
s s
s s ss K
TkT f s
M H
magnetization decays with time
>s stable, remanent magnetization
Estimated critical sizes for SD‐TD behavior in magnetite as a function of particle elongation (Butler and Banerjee, 1975)
vs
M=MeqMr=0
Mr=0.5MsMr
unstable stable
Blocking volume (very sharp transition)
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Blocking and Superparamagnetism
SD grain size population f(v,HK)Néel Diagram
00
2vH ln( )K ss
kT fM
H0=0, Constant T depends on v and HK
Butle
r, 1992
Increase time, sweep out more area and unblock remanence
Blocking Temperature
Relaxation time: Strong function of temperature and timescale of observations
01 1KvM H vK 0( ) ( )K T NM T0
0 0
1 1exp exp2
s K uvM H vKf kT f kT
KU is a function of temperature
0( ) ( ),2u sK T NM T
0
0
v ( ) ( )2 ln( )
s KB
M T H TTk f t
Blocking Temperature (v=constant)
T >TB unstable magnetization (SPM)magnetization decays with time
T < TB stable, remanent magnetization
Important: Stable SD with > 1Ga at 300 K can be SPM at elevated temperatures close to the Curie temperature
STABLE Blocked
T T T1 3 4
All points to the left of any curve H(t,he,T) correspond to grains with t<. e
Néel Diagram
SUPERPARA-MAGNETIC Unblocked
Tempe
rature
or tim
e
H(t,h,T)
g
Likewise, all points to the right have t>.
Coercivity (HK)
volum
00
2vH ln( )K ss
kT fM
Grains with small v and Hc will be superparamagnetic. As temperature, or time, isincreased, more and more grains with larger v and HK will become progressivelyunblocked, until at T=TB when all grains become unblocked.
K
-1.0
-0.5
0.0
0.5
1.0
-400 -200 0 200 400
M/M
sat
SuperparamagneticLoopd=10 nm
Ms
MrT> Tc: Thermal energy dominateskT > Exchange energy
T<TC: Exchange energy dominatesKv<<kT
T<TBk
kT > Kv
0H(mT)
TB Tc
Kv> kT
Blocking Temperature: critical temperature (Tb<Tc) where remanence is lost (T>Tb, t <tb)
Between Tb and Tc, grains are ferrimagnetically ordered but are superparamagnetic (remanence decays quickly to zero)
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Superparamagnetism
0.03
Unblocked and Superparamagnetic0.04
Blocked and Stable SD
Two‐state particle: Meq= Mstanh ()All orientations: Meq=MsL()
0 0svM HkT
-0.03
-0.02
-0.01
0.00
0.01
0.02
-600 -400 -200 0 200 400 600
M (A
m2 /k
g)
p p gT=300 K
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
-600 -400 -200 0 200 400 600M
(Am
2 /kg)
H ( T)
Blocked and Stable SDT=25 K
kT>Kuv
Mr/Ms ~0.5Hc~100 mT
Mr=0Hc=0
H (mT) H (mT)Temperature Dependence of Hysteresis (submarine basaltic glass) SPM grains are ferromagnetically (ferrimagnetically)
ordered (exchanged coupled) below Tc
The remanence (time‐averaged magnetization) for a group of SPM particles is zero ( disordered)
2 10-4
Blocking and Superparamagnetism
T>TB , is frequency independent indicating thermodynamic equilibrium, (superparamagnetic state)
Superparamagnetic behavior depends on the time scale of observation (the choice for )
viscous
5 10-5
1 10-4
xdc10 Hz110 Hz1 kHz10 Khz
in-p
hase
sus
cept
ibili
ty
Blocking Temperature
(superparamagnetic state).
T <TB, is frequency‐dependent,indicating a nonequilibrium state (blocked state).
Since TB depends on the measurement frequency, the peak occurs at different temperatures for different
SPM
SSD
0 100
0 50 100 150 200Temperature (K)
magnetite, <d> 8 nm
Blocking Temperature
Blocking temperature increases with frequency of measurement
pfrequencies.
SummaryHysteresis and Domain State
0.0
0.5
1.0
M/M
sat
Multi-Domain Loopd=3 mm
0.0
0.5
1.0
M/M
sat
Single-Domain Loopd=50 nm
0.0
0.5
1.0
M/M
sat
SuperparamagneticLoopd=10 nm
Low Hc
-1.0
-0.5
-400 -200 0 200 400
0H(mT)
-1.0
-0.5
-400 -200 0 200 400
0H(mT)
-1.0
-0.5
-400 -200 0 200 400
0H(mT)
Multidomain behaviorDomain wall translation
Single Domain BehaviorMoment rotation over anisotropy barrier
SuperparamagnetismThermal fluctuations can flip the magnetization
Relaxation time
Low Hc and Mr
High Hc and Mr
Hc=0Mr=0
Ha Ms Ha Ms
Tauxe, 2008
0
2001 exp 1
2s K
fb K
HVM Hk T H
Relaxation time
0= frequency factor (~109 Hz)0
2 uK
s
KHM
Ku= anisotropy “constant”Crystalline, shape or stress contributions
Flipping field
SPM
SD
Unstable Stable
MD
PSD
civi
ty H
c
The magnetic behavior can be subdivided on the basis of grain size into
SPM superparamagnetic SD single domain PSD pseudo single domain
Particle diameter d
d0ds
Coe
rc
0 03 0 08 20 mMagnetite
PSD pseudo‐single domain MD multidomain
The maximum coercivity for a given material occurs within its SD range.
For larger grain sizes, coercivity decreases as the grain subdivides 0.03 0.08 20 m
0.03 15 ?Hematite
decreases as the grain subdivides into domains.
For smaller grain sizes, coercivity again decreases, but this time due to the randomizing effects of thermal energy.
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Blocking Parameters and Types of Remanence
Magnetic Relaxation and Blocking
T(emperature) Thermoremanent magnetization (TRM)
v(olume) Chemical remanent magnetization (CRM)
t(ime) Viscous remanent magnetization (VRM)
H(field) Isothermal remanent magnetization (IRM)
0
0
2
0
0
10 : exp2
10 : exp 12
s Ka
as Ka
K
vM HHf kT
HvM HHf kT H
Progressive Demagnetization Techniques
Néel Theory2
0
0
1 exp 12
as K
K
HvM Hf kT H
0 K
Low stability component(s) (short , low Hcr and Tb): “easily” removed first by demagnetization (secondary components)
High stability component (long , high Hcr and Tb): most resistant to demagnetization (usually primary component )
( )Characteristic Remanent Magnetization (ChRM): The highest‐stability component of NRM that is isolated by partial demagnetization
Main methodsUse magnetic fields: Alternating magnetic field(AF) demagnetizationUse temperature: Thermal demagnetization
Thermal DemagnetizationHeating samples to elevated temperatures (T<Tc)Cooling to room‐temperature in zero magnetic field
All grains TB<Tdemag become unblocked)B demag
Grains with short relaxation times (low TB) are likely to pick up secondary NRMs (unstable NRM)
These grains get demagnetized first before grains with long relaxation times (stable NRM)
Thermal Demagnetization Furnaces
Mu‐metal shielding to reduce external fields to near zero during heating and cooling
Heating areaCooling area
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Alternating Field Demagnetization
mir, 1997Room‐temperature method
No heating involved, no chemical alteration
Dunlop
and
Özdem
2
0
0
1 exp 12
AFs K
K
HvM Hf kT H
If an AF field of peak value 10 mT is applied and amplitude is smoothly reduced to zero grains withamplitude is smoothly reduced to zero, grains with 0<Hc<10 mT are randomized
Process is repeated to higher and higher AF fields
Low coercivity components = unstable NRM (low reliability)
AF Demagnetization Mu‐metal shields to attenuate DC field (e.g. Earth’s magnetic field) Solenoid for
Ac field
Alt ti fi ld ti d i ith H HAlternating fields remagnetized grains with Hc < HAF
Half are magnetized “up” and half “down” = zero net moment
Typical AF demagnetizes have peak fields 100‐200 mT
The motion of the blocking curves under the influence of h do not
When the field is not zero, theblocking curves are asymptotic tothe particular value of HAF
AF Demagnetization
T
SUPERPARA-MAGNETIC Unblocked
hh1 2
STABLE Blocked
under the influence of h do not reproduce the motion due to temperature or time.
Particles of small v and large Hc may require very large AC fields to become unblocked.
However, the same grains can be
V
HK
unblocked by mild heating.
AF demagnetization is often effective in removing secondary NRM and isolating ChRM in rocks with (titano)magnetite as the dominant ferrimagnetic mineral.
Secondary NRM carried by MD grains (hc <20 mT)ChRM carried by SD or PSD grains with higher hc .
Extra Slides
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TM60
Effects of Stress on Domain Structure
TM0
10mMoskowitz et al., 1988
13x13 m
Pokhil and Moskowitz, 1997
(100) TM60
Stress can override the intrinsic magnetocrystalline anisotropy and produce a (local) uniaxial anisotropy
Anisotropy Energy > Magnetostatic Energy
Example Response of domain structure in Al‐substituted TM62 to compressive stress
Complex patterns (A) shrink (15 5
Lamellar domains (B) (nearly parallel to compressive axis) on the left of the picture were driven out by 15.5 Mpa and re‐formed almost normal to the stress by 25.5 MPa.
L ll d i (C) t t
Complex patterns (A) shrink (15.5 Mpa) and disappear by 22.5 Mpa)
A
B C
Lamellar domains (C) rotate away from compressive axis at 15.5 Mpa and some expand by 25.5 Mpa
Appel and Soffel, 1985
MD Hysteresis: Domain nucleation and translation
The switching behavior of a magnetite particle imaged during the descending and H0during the descending and ascending branches of a hysteresis cycle between ±500 Oe along the in‐plane [110] direction (vertical).
The sequence of images begins with remanent state (a) after application and removal of +500 Oe field.
0
10x10‐m particle patterned from 250 nm thick magnetite film grown on (110) Mg0 (Pan et al., 2002)Magnetic Force Microscopy (MFM)
Bloch WallsAnisotropy energy
2
( )
( ) sin ( ) :uniaxialK
u
E g
g K
The wall energy is given by the sum of the exchange and anisotropy energies, integrated over the thickness of the wall
Bloch wall energy
2
wall ( )ex KdE E A g dxdx
Need to solve for (x) (spin rotation function)=0, x‐ =, x+
Bloch wall lying in the yz plane, with the direction of magnetization rotating about the x‐axis while remaining parallel to the yz plane. x = position in the wall = direction of the local magnetization
Cullity, 1972
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Torque resulting from the exchange energy
Torque resulting from the
Domain Wall StructureMinimize wall energy wall 0ex KE E
q ganisotropy energy
At equilibrium, these torques must be equal and opposite, giving zero net torque
Multiplying by d/dx and integrating over x,
Exchange and anisotropy energies are equal everywhere in the wall
Where the anisotropy energy is the highest, the rate of change of magnetization angle d/dx is the greatest
Cullity, 1972
Domain Wall Structure
Uniaxial Anisotropy
Solution
Cullity, 1972
Bloch Walls
ln tan2
cos tanh
u
u
AxK
K z
Solution (Landau and Lifshitz, 1933)
cos tanh zA
wAK
Effective wall thickness = Wall thickness with a constant value of d/dx equal to that at the center of the wall.
Thickness of the domain wall is formally infinite.
Cullity, 1972
Variation of magnetization direction through a 180° domain wall. Dashed line shows definition of wall width.
w
Domain Wall Energy2
wall ( )ex KdE E A g dxdx
Since the two terms are everywhere equal wall 2 ( )g dx
0 0
( )
( )2 ( )( )wall
ddx Ag
gA d A g dg
For uniaxial anisotropy:
00
( ) sin
sin ( cos )
4
u
wall u u
wall u
g K
AK d AK
AK
Cullity, 1972
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SD‐Non SD Transition Sizes
Dunlop and Özdemir, 2007
Domain ModelsLandau‐Lifshitz Model: Closure Domain ModelDemagnetizing energy =0
Total energy = the sum of Wall Energy and the Magnetoelastic Energy
1/2 1/21/2 1/2111 449 4 wcn D d D
D
ahl, 1987
111 44
,4 9eq eq
w
n D d Dc
d
D=nd
Halged
d (m)
Experimental Results: MagnetiteÖzdemir and Dunlop, 1993
Closure domains along in (110) magnetite single crystal(Özdemir et al.1995)
D1/2 (m)1/2
From slope, W=0.91x10‐3 J/m3
2w AK
From w , determine exchange constantA=1.28x10‐11 J/M
Experimental Results: Magnetite
contain fewer domains than equilibrium theorypredicts.
deq~D1/2
SD transition sizen=1, d~0.1 m
Dunlop and Özdemir, 2007
Micromagnetic Simulation of Hysteresis
0.1 m cube of magnetiteA. Initial state is a negative
Flower state (F)B. Flower state (F) jumps to
BB. Flower state (F) jumps to
vortex state (V)
Vortex state is LEM state for successive hysteresis cycles
C. Reversible spin rotation in outer most cells of vortex (MD‐like process)
F
DC
D
D. Discontinuous vortex reversal of vortex core (SD‐like process)
V
FA
Williams and Dunlop, (1995)
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Micromagnetic Models: Complex Shapes
SEM image of a magnetite dendrite in archaeological
Octahedron Dendrite Six independent octahedra
Large Octahedron: ramp‐like (MD) loop , Mr/Ms=0.07Dendrite, Small Octahedra: SD‐like loops, Mr/Ms=0.83
dendrite in archaeological copper slag from Israel. Shaar, et al., 2009.
Averaged hysteresis curves formed from fields applied along the [100], [110], and [111] directions for the three octahedral model structures
All three structures: Hc~15 mT
Williams et al., 2009