Transcript by Karl Fabian, AARCH workshop 3, Madrid, 2005

Thermoremanence
Karl Fabian
www.toepferstudio.at
University of Bremen,
Department of Geosciences
AARCH workshop, Madrid, 2005
The Earth's magnetic field
shields the biosphere from
solar irradiation
It is generated by a
buoyancy driven
magnetohydrodynamic
dynamo process in the
liquid outer core
www.toepferstudio.at
www.museum.upenn.edu
All pottery acquires TRM during production
Attic Red Figure Stamnos ca. 490 BC
Heracles fighting the Nemean Lion
All igneous rocks acquire a thermoremanence
within the external field during cooling
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Outline
Physical background
Néel's single domain theory
Cooling rate
Anisotropy
Multidomain
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
The physical origin of remanent magnetization
Electrons carry spin
magnetic moment
with
Hydrogen orbitals
In some crystals uncompensated
spins can order at low T to yield
net moment
Ordering requires energy gain by spin
coordination of electrons in overlapping
orbitals.
http://en.wikipedia.org/wiki/Electron
Only ions with uncompensated spins
in highly eccentric orbitals (e.g. 3d for Fe3+)
are possible sources of ferro-, ferri- or
antiferromagnetism.
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Magnetic spin ordering is a phase transition
ferromagnetic
paramagnetic
antiferromagnetic
ferrimagnetic
Critical temperature : Curie or Néel temperature
Ms (kA/m)
TC
Onset of ferrimagnetism
in magnetite below TC = 580° C
T (°C)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Magnetic energy of a spin ordered state (ferromagnet)
Continuum theory simplifies quantum mechanical description
Total Energy Etot is sum of three terms:
Magnetostatic energy
Anisotropy energy (magnetocrystalline or stress)
Exchange energy
Physical magnetization states correspond to minima of Etot
www.press.uillinois.edu/ epub/books/brown/ch7.html
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Micromagnetic modelling
Subdivision of particle into cells.
Numerical energy minimization gives physical structures
But: local vs. global minima
Energy barriers have to be overcome
to change magnetization structure
www.press.uillinois.edu/ epub/books/brown/ch7.html
global
minimum
local
minimum
This can occur by thermal activation
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
High dimensional energy landscape of a magnetic grain
Etot depends on each magnetization vector
www.press.uillinois.edu/ epub/books/brown/ch7.html
Exchange energy aligns neighboring spins
on a typical scale of lex
dim
Assuming nearly constant spin direction within volumes
of lex3 leads to a finite (but often large) intrinsic
dimension of a magnetic particle
T
AARCH, Madrid, 2005
Intrinsic dimension of 50 nm
magnetite cube in function of T
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Magnetization structure of remanence carriers
Magnetische Korngrößen
single domain particles (SD)
ideal storage media
pseudo single domain particles (PSD)
here: vortex structure
SD
superparamagnetic
PSD
MD
stable
multidomain particles (MD) :
magnetic domains separated
by domain walls
L
remanence in function of grain size
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Definition of thermoremanent magnetization (TRM)
Cooling
TC
Paramagnetic state:
Uncompensated spins
are disordered
Curie temperature
Phase transition
Ferromagnetic state:
Long range spin order
Pierre Curie
SP :
Superparamagnetic: instable net moment
TB
T
AARCH, Madrid, 2005
blocking
SD:
Stable single domain
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Definition of thermoremanent magnetization (TRM)
Cooling
TC
Paramagnetic state:
Uncompensated spins
are disordered
Curie temperature
SP :
Superparamagnetic
Pierre Curie
TB
Phase transition
Ferromagnetic state:
Long range spin order
blocking
SD:
Single domain
T
MD:
Multidomain
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Definition of thermoremanent magnetization (TRM)
Cooling
TC
Paramagnetic state:
Uncompensated spins
are disordered
Phase transition
Curie temperature
Ferromagnetic
state:
Si
Long range spin order
Pierre Curie
Sj
TB,1
TB,2
T
AARCH, Madrid, 2005
TB,3
MD:
Cascade of possible
domain configurations
and sub-blocking temperatures
Transdomain processes
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Néel's theory of single domain particles
Homogeneous magnetization structure in remanent state
During the reversal the magnetization is not necessarily
homogeneous
Energy barrier is related to microcoercivity
EB = ½ Ms(T) hc(T) V
Relation between energy barrier (EB) and thermal activation
energy (kBT) determines stability
EB
kBT
www.press.uillinois.edu/ epub/books/brown/ch7.html
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM of SD particles
Unblocked equilibrium magnetization for a two-state-system
H cos 

n-
H
n+
DE = 2 m0 Ms(T) V H cos 
Boltzmann statistics above the blocking temperature
n+
n-
= exp [ DE / kBT ] = exp [2 m0 Ms(T) V H cos  / kBT ]
n+ + n- = n()
(Néel, 1949)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM of SD particles
Boltzmann statistics above the blocking temperature
n+
n-
= exp [ DE / kBT ] = exp [2 m0 Ms(T) V H cos  / kBT ]
n+ + n- = n()
Net magnetization
M=
n+ - n n
Ms(T)
M = Ms(T) tanh [2 m0 Ms(T) V H cos  / kBT ]
Spherical averaging over isotropic ensemble yields for small H
MTRM =
m0 Ms2(T) V
3 kBT
H
(Stacey & Banerjee, 1974)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM of SD particles : Extension to MD remanence
SD equilibrium magnetization at high T
MTRM =
m0 Ms2(T) V
3 kB T
H
Generalization to MD equilibrium magnetization
MTRM =
m0 Mr2(T) V
3 kBT
H
Mr2(T) = weighed average squared
magnetization of all LEM at T
(Fabian, 2000)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM of SD particles : From equilibrium to room temperature
Equilibrium magnetization at high T
MTRM =
m0 Ms2(T) V
3 kB T
H
At some temperature TB this remanence is blocked (frozen)
Further cooling to room temperature only changes the remanence
by the change of Ms(T). At room temperature the TRM therefore is
MTRM (T0) =
m0 Ms2(TB) V Ms (T0)
3 kBTB
Ms (TB)
H
Equilibrium at TB Temperature
variation of Ms
(Stacey & Banerjee, 1974)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Stability of thermoremanence: The blocking process
Assumption:
Remagnetisation by coherent rotation of all spins
1-dimensional energy landscape
Louis Néel
(dim=1 is a significant simplification)
EB
EB = ½ Ms(T) hc(T) V
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Stability of thermoremanence: The blocking process
The ratio EB / kBT determines, whether the particle carries
a stable remanence, or behaves superparamagnetic (SP) at T.
Relaxation time:
Louis Néel
t = t0 exp( EB / kBT ),
EB
EB
AARCH, Madrid, 2005
t0 ~ 10-9 s
kBT
kBT
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Blocking temperature
Within a tiny interval DT the particle
changes from superparamagnetic to
stable remanence.
The critical temperature of this interval
is the blocking temperature TB
EB
Below TB the relaxation time
t = t0 exp( EB / kBT )
rapidly increases to > 5000 Ma.
The remanence then is geologically
stable !
EB
AARCH, Madrid, 2005
kBT
kBT
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Describing an SD ensemble by its volume TB distribution
MTRM (TB) =
m0 Ms (TB) V
3 k BT B
Ms (T0) H
Ms(TB) / TB for magnetite
V (TB)
TB
TB
Full thermoremanence of SD ensemble in constant field H
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Partial thermoremanence of an SD ensemble
H
zero
field
H(TB) =
{
0
H
0
for
TB < T1
T 1 < TB < T 2
TB > T2
TB
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Thellier's laws of pTRM for SD ensembles
Linearity with field H
Additivity
+
pTRM(T1,T2)
AARCH, Madrid, 2005
=
pTRM(T2,T3)
pTRM(T1,T3)
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Thellier's laws of pTRM for SD ensembles
Linearity with field H
Additivity
Independence:
A pTRM(T1,T2) is not influenced by heating and cooling
below T1 and is completely removed above T2
This requires an additional fact:
TB = TUB
For each SD particle blocking and unblocking temperatures
are equal
EB
AARCH, Madrid, 2005
kBT
EB
kBT
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Thellier's laws of pTRM for SD ensembles
Linearity with field H
Both remain valid for MD ensembles
Additivity
Independence:
A pTRM(T1,T2) is not influenced by heating and cooling
below T1 and is completely removed above T2
This requires an additional fact:
TB = TUB
For each SD particle blocking and unblocking temperatures
are equal
EB
AARCH, Madrid, 2005
kBT
EB
kBT
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Thellier's laws of pTRM for SD ensembles
Linearity with field H
Additivity
Independence:
A pTRM(T1,T2) is not influenced by heating and cooling
below T1 and is completely removed above T2
This requires an additional fact:
TB = TUB
Not valid for
MD ensembles
For each SD particle blocking and unblocking temperatures
are equal
EB
AARCH, Madrid, 2005
kBT
EB
kBT
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
A simple extension of Néel's SD model
Consider an ensemble of 'particles' with TB  TUB
Dm = c( TB, TUB) H
TB
TUB
This results in a two-dimensional extension of Néel's theory
which can explain for many properties of MD TRM
M Néel's theory
TUB
TB =TUB
TB = TUB
SD
AARCH, Madrid, 2005
T0
MD
TB
(Fabian, 2000; 2001)
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
A simple extension of Néel's SD model
M Néel's theory
TUB
TB =TUB T2
Intensity of remanence which
blocks at T1 and unblocks at
higher T2
TB = TUB
SD
AARCH, Madrid, 2005
T0
T1
MD
TB
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Different types of partial TRM in MD samples
TUB
TUB
T'
T1
H
T0 T2 T1 TB
pTRM(T1 ,T2)
acquired by
cooling from TC
TUB
H
T0 T2 T1 TB
H
T0 T2 T1 TB
pTRM*(T1 ,T2)
acquired from
zero-field cooled
state at T0 after
heating to T1
pTRM'(T1 ,T2)
acquired from
zero-field cooled
state at T0 after
heating to T'>T1
(Shcherbakov et al., 1993)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM tail of MD samples
TUB
m
MD
T'
SD
T0
T1
T'
SD
pTRM(T1 ,T0)
is completely
removed by second
heating to T1
AARCH, Madrid, 2005
T
T0
T1
TB
MD
Residual pTRM(T1 ,T0) tail
remains often up to TC
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Decay of pTRM(T1 ,T2) during zero-field cycling below T1
TUB
m
Heff H
SD
MD
T'
T' T1
T2
SD
pTRM(T1 ,T2)
remains constant
AARCH, Madrid, 2005
T
T0 T2 T1 TB
MD
Decay due to interaction
and TUB<TB
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Additivity of pTRM is valid
+
T3 T2 T1
=
T3 T2 T1
T3 T2 T1
pTRM(T1 ,T2) + pTRM(T2 ,T3) = pTRM(T1 ,T3)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Additivity of pTRM* is not valid
+
T3 T2 T1
=
T3 T2 T1
pTRM*(T1 ,T2) + pTRM*(T2 ,T3)
AARCH, Madrid, 2005
T3 T2 T1
 pTRM*(T1 ,T3)
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Additivity of magnetization tails
Relation of Shcherbakova et al.
(Shcherbakova et al., 2000)
+
T3 T2 T1
=
T3 T2 T1
T3 T2 T1
tpTRM(T1 ,T2) (T) + tpTRM(T2 ,T3) (T) = tpTRM(T1 ,T3) (T)
This model is not a physical theory of MD TRM !
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM intensity depends on cooling rate:
general scenario
Cooling rate dependence of SD hematite TRM
SD TRM corresponds to frozen
equilibrium remanence at TB
16
14
Blocking temperature depends
on cooling rate: Slow cooling
leads to lower TB
DM / M0 [%]
12
10
8
6
4
2
Lower TB implies higher
remanence
0 ~ 46 K/min
0
1
fast cooling
MTRM (T0) =
AARCH, Madrid, 2005
m0 Ms (TB) V
3 kBTB
2
3
ln ( / 
4
5
slow cooling
(Papusoi, 1972)
Ms (T0) H
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
TRM intensity depends on cooling rate:
scenario for metastable SD particles and MD
During slow cooling the particle
more likely relaxes into lower
remanence (e.g. vortex) state
In this case slow cooling
reduces TRM
AARCH, Madrid, 2005
Cooling rate dependence of MD magnetite TRM
fast cooling
slow cooling
ln ( / 
0
-1
DM / M0 [%]
Metastable state with high
remanence (SD) is frozen in
only during fast cooling
through TB
1
2
3
4
5
-2
-3
-4
-5
-6
0 ~ 46 K/min
(Papusoi, 1974)
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
A precise definition of SD blocking temperature
Assume a particle cooling with constant rate  from
T to T0 during time t = (T-T0) / 
Calculate expected number N (T, )
of reversals across the energy barrier
TB is defined by N (TB, ) = 1
or : Below TB we expect less than 1 reversal
(Stacey & Banerjee, 1974)
(Winklhofer et al., 1997)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Time-temperature curves for SD cooling
Ideal curves
and
micromagnetic
calculations
(Pullaiah et al., 1975,
Winklhofer et al., 1997)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
For MD particles a unique blocking temperature
cannot be defined
SD blocking
TC
TB
T0
TC
TB
T0
PSD + MD
Several blocking
events at different
temperatures fix
parts of the final TRM
(Fabian, 2003)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Previous theory deals with isotropic particle ensemble
Possible origin of anisotropy in pottery
Preferential alignment of magnetic grains
during moulding
www.toepferstudio.at
Higher degree of alignment by potter's wheel
www.toepferstudio.at
(Rogers et al. , 1979)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Correction for anisotropy
Measurement of symmetric TRM anisotropy tensor K=(kij)
Measure direction and intensity with field in z-direction
Unit vector in direction of paleofield
Correction factor of paleofield intensity
(Veitch et al. , 1984)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Remanence carriers in natural rocks rarely are
single domain particles
Neél's theory
single domain
SD
superparamagnetic
no remanence
10nm
SD
AARCH, Madrid, 2005
No generally adopted theory
multidomain MD
PSD
high
remanence
100nm
low
remanence
1mm
MD
10mm
100mm
Ambatiello, Fabian, Hoffmann, 1999
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Three steps to extend Neél‘s SD theory
to MD remanence carriers
Statistical physics of multidomain remanence
Micromagnetic calculation of
energy barriers in multidomain particles
Including temporal change:
Thermo-viscous magnetization processes
(laboratory vs. geological time scale)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
New theory of multidomain thermoremanence based
on non-equilibrium statistical physics
600°C
Non-equilibrium domain states Si
Inhomogeneous Markov chain of state
transitions Si  Sj during heating or
cooling
Si
T
Sj
Transition probabilities M ij depend
upon energy barriers between states
M ij ~ exp [ - DEij / kT ]
Transitions respect fundamental time
inversion symmetry
AARCH, Madrid, 2005
20°C
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Probability density of magnetization states
Symmetric probability density r(Z0) in state Z0 after zero-field cooling
r(Z)
S-6
S-5 S-4
S-3
S-2
S-1
S0
S1
S2
S3
S4
S5
S6
Asymmetric probability density r(Z0) in state ZH after cooling in field H<0
r(Z)
S-6
AARCH, Madrid, 2005
S-5 S-4
S-3
S-2
S-1
S0
S1
S2
S3
S4
S5
S6
Karl Fabian, Universität Bremen
Physics
Néel theory
During the process Z
according to
Cooling rate
Anisotropy
MD theory
 Z´ the probability distribution r changes
r(Z') = M(Z
Z') r(Z)
For pure heating and cooling processes the probability
distribution r develops according to
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
The statistical theory is compatible with all observed
properties of MD TRM
Linearity of TRM in weak fields (Thellier‘s 1. law)
Additivity of pTRM (Thellier‘s 3. law)
All new experimentally found additivity laws can be derived
The statistical descripion solves previous inconsistencies in the notions
of blocking and unblocking temperatures
P
H
S0
S1
S4
S3
S-6
S0
S-1
S-4
S-3
S6
p( P,
H) = p( P
,
0) + Dp H
-P
m(P) p( P
AARCH, Madrid, 2005
p(-P,
H) = p( P
0) -p(-P
Dp H , H)
, H) + ,
m(-P)
=
Karl Fabian, Universität Bremen
2
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Complex repeated thermal magnetization processes (tpTRM*)
are correctly predicted by the statistical theory
(Fabian & Shcherbakov, 2004)
An important process in paleointensity determination is
acquisition and deletion of a pTRM*
A = [T0, 0]
[T1, 0]
[T1, H]
D = [T0, 0]
[T1, 0]
[T0, 0]
[T0, H]
[T0, 0]
After the combined process P = AD there remains a
previously not understood residual remanence tpTRM*(T1)
The statistical theory explains this tpTRM*(T1)
and predicts for the iterative process
Pk = P
P
P
...
P
that tpTRM*(T1) k increases rapidly for small k and approaches
a limit value which corresponds to an eigenstate of M(P).
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Sample 12B (natural rock) T1 = 400°C
pTRM* + tpTRM*
Remanence
4800
700
tpTRM*
4600
4400
500
300
4200
pTRM*
100
1 2 3 4 5 6 7 8 9 10 11 12
Iteration k
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Sample B (oxidized synthetic magnetite ) (< 60 mm) T1 = 400°C
4400
1200
Remanence
pTRM* + tpTRM*
tpTRM*
4000
800
3600
400
pTRM*
3200
0
1
2
3
4
5
6
7
8
9
10
Iteration k
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
For the first time a physical theory of MD TRM
conforms with all known experimental facts
Extension to thermo-viscous magnetization and
paleointensity determination is in progress
Detailed evaluation requires micromagnetic
calculation of energy barriers
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Calculation of energy barriers by micromagnetic modelling
Transition probability depends on the energy
barrier between adjacent magnetization states
Energy barriers correspond to saddle points
in high dimensional configuration space
We developed a new iterative nudged elastic band technique
together with action minimization to find the optimal transition
paths in micromagnetic models
(Fabian & Shcherbakov, in prep.)
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
e [red. units]
An optimal vortex rotation in magnetite (~90 nm)
distance [rel. units]
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
An optimal vortex rotation in magnetite
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen
Physics
Néel theory
Cooling rate
Anisotropy
MD theory
Some important open problems
www.toepferstudio.at
Complete theory of thermo-viscous magnetization (SD & MD)
Are PSD particles reliable TRM carriers ?
Is extrapolation of VRM from laboratory to geological time possible ?
Best method of paleointensity determination
Physical relation between TRM and TMRM
AARCH, Madrid, 2005
Karl Fabian, Universität Bremen