Thermoremanence Karl Fabian www toepferstudio at University of

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. 3 d for Fe 3+) 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 (k. A/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/ch 7. 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/ch 7. html This can occur by thermal activation AARCH, Madrid, 2005 global minimum local minimum 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/ch 7. html Exchange energy aligns neighboring spins on a typical scale of lex dim Assuming nearly constant spin direction within volumes of lex 3 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 multidomain particles (MD) : magnetic domains separated by domain walls 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 Pierre Curie SP : Superparamagnetic: instable net moment TB T AARCH, Madrid, 2005 Phase transition Ferromagnetic state: Long range spin order 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 (k. BT) determines stability EB k. B T www. press. uillinois. edu/ epub/books/brown/ch 7. 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 m 0 Ms(T) V H cos Boltzmann statistics above the blocking temperature n+ n- = exp [ DE / k. BT ] = exp [2 m 0 Ms(T) V H cos / k. BT ] 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 / k. BT ] = exp [2 m 0 Ms(T) V H cos / k. BT ] n+ + n- = n( ) Net magnetization M= n+ - nn Ms(T) M = Ms(T) tanh [2 m 0 Ms(T) V H cos / k. BT ] Spherical averaging over isotropic ensemble yields for small H MTRM = m 0 Ms 2(T) V 3 k. BT 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 = m 0 Ms 2(T) V 3 k. BT H Generalization to MD equilibrium magnetization MTRM = m 0 Mr 2(T) V 3 k. BT H Mr 2(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 = m 0 Ms 2(T) V 3 k. BT 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 (T 0) = m 0 Ms 2(TB) V Ms (T 0) 3 k. BTB 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 (dim=1 is a significant simplification) Louis Néel 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 / k. BT determines, whether the particle carries a stable remanence, or behaves superparamagnetic (SP) at T. Relaxation time: Louis Néel t = t 0 exp( EB / k. BT ), EB EB AARCH, Madrid, 2005 t 0 ~ 10 -9 s k. B T 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 = t 0 exp( EB / k. BT ) rapidly increases to > 5000 Ma. The remanence then is geologically stable ! EB AARCH, Madrid, 2005 k. B T Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Describing an SD ensemble by its volume TB distribution MTRM (TB) = m 0 Ms (TB) V 3 k. BTB Ms (T 0) 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 T B < T 1< T B < T 2 T B > T 2 TB AARCH, Madrid, 2005 Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Thellier's laws of p. TRM for SD ensembles Linearity with field H Additivity + p. TRM(T 1, T 2) AARCH, Madrid, 2005 = p. TRM(T 2, T 3) p. TRM(T 1, T 3) Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Thellier's laws of p. TRM for SD ensembles Linearity with field H Additivity Independence: A p. TRM(T 1, T 2) is not influenced by heating and cooling below T 1 and is completely removed above T 2 This requires an additional fact: TB = TUB For each SD particle blocking and unblocking temperatures are equal EB AARCH, Madrid, 2005 k. B T EB k. B T Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Thellier's laws of p. TRM for SD ensembles Linearity with field H Additivity Both remain valid for MD ensembles Independence: A p. TRM(T 1, T 2) is not influenced by heating and cooling below T 1 and is completely removed above T 2 This requires an additional fact: TB = TUB For each SD particle blocking and unblocking temperatures are equal EB AARCH, Madrid, 2005 k. B T EB k. B T Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Thellier's laws of p. TRM for SD ensembles Linearity with field H Additivity Independence: A p. TRM(T 1, T 2) is not influenced by heating and cooling below T 1 and is completely removed above T 2 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 k. B T EB k. B T 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 This results in a two-dimensional extension of Néel's theory which can explain for many properties of MD TRM (Fabian, 2000; 2001) AARCH, Madrid, 2005 Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory A simple extension of Néel's SD model T 2 Intensity of remanence which blocks at T 1 and unblocks at higher T 2 T 1 AARCH, Madrid, 2005 Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Different types of partial TRM in MD samples p. TRM(T 1 , T 2) acquired by cooling from TC p. TRM*(T 1 , T 2) acquired from zero-field cooled state at T 0 after heating to T 1 p. TRM'(T 1 , T 2) acquired from zero-field cooled state at T 0 after heating to T'>T 1 (Shcherbakov et al. , 1993) AARCH, Madrid, 2005 Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory TRM tail of MD samples SD p. TRM(T 1 , T 0) is completely removed by second heating to T 1 AARCH, Madrid, 2005 MD Residual p. TRM(T 1 , T 0) tail remains often up to TC Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Decay of p. TRM(T 1 , T 2) during zero-field cycling below T 1 SD p. TRM(T 1 , T 2) remains constant AARCH, Madrid, 2005 MD Decay due to interaction and TUB<TB Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Additivity of p. TRM is valid p. TRM(T 1 , T 2) + p. TRM(T 2 , T 3) = p. TRM(T 1 , T 3) AARCH, Madrid, 2005 Karl Fabian, Universität Bremen

Physics Néel theory Cooling rate Anisotropy MD theory Additivity of p. TRM* is not valid p. TRM*(T 1 , T 2) + p. TRM*(T 2 , T 3) AARCH, Madrid, 2005 p. TRM*(T 1 , T 3) 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) tp. TRM(T 1 , T 2) (T) + tp. TRM(T 2 , T 3) (T) = tp. TRM(T 1 , T 3) (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 Blocking temperature depends on cooling rate: Slow cooling leads to lower TB Lower TB implies higher remanence MTRM (T 0) = AARCH, Madrid, 2005 m 0 Ms (TB) V 3 k. BTB a 0 ~ 46 K/min fast cooling slow cooling (Papusoi, 1972) Ms (T 0) 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 Metastable state with high remanence (SD) is frozen in only during fast cooling through TB Cooling rate dependence of MD magnetite TRM fast cooling slow cooling 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 a 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 a from T to T 0 during time t = (T-T 0) / a Calculate expected number N (T, a) of reversals across the energy barrier TB is defined by N (TB, a) = 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 T 0 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 10 nm SD AARCH, Madrid, 2005 No generally adopted theory multidomain MD PSD high remanence 100 nm low remanence 1 mm MD 10 mm 100 mm 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 / k. T ] 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(Z 0) in state Z 0 after zero-field cooling r(Z) S-6 S-5 S-4 S-3 S-2 S-1 S 0 S 1 S 2 S 3 S 4 S 5 S 6 Asymmetric probability density r(Z 0) in state ZH after cooling in field H<0 r(Z) S-6 S-5 S-4 AARCH, Madrid, 2005 S-3 S-2 S-1 S 0 S 1 S 2 S 3 S 4 S 5 S 6 Karl Fabian, Universität Bremen

Physics Néel theory During the process according to Cooling rate Anisotropy MD theory Z Z´ the probability distribution r changes 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 p. TRM (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 S 0 S 1 S 4 S 3 S-6 S 0 S-1 S-4 S-3 S 6 p( P, H) = p( P , 0) + Dp H -P m(P) p( P AARCH, Madrid, 2005 p(-P, H) = p( P - Dp H , H) + 0) m(-P) p(-P , H) , = Karl Fabian, Universität Bremen 2

Physics Néel theory Cooling rate Anisotropy MD theory Complex repeated thermal magnetization processes (tp. TRM*) are correctly predicted by the statistical theory (Fabian & Shcherbakov, 2004) An important process in paleointensity determination is acquisition and deletion of a p. TRM* A = [T 0, 0] [T 1, H] D = [T 0, 0] [T 1, 0] [T 0, H] [T 0, 0] After the combined process P = A D there remains a previously not understood residual remanence tp. TRM*(T 1) The statistical theory explains this tp. TRM*(T 1) and predicts for the iterative process Pk = P P P . . . P that tp. TRM*(T 1) 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 12 B (natural rock) T 1 = 400°C p. TRM* + tp. TRM* Remanence 4800 700 tp. TRM* 4600 4400 500 300 4200 p. TRM* 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) T 1 = 400°C p. TRM* + tp. TRM* Remanence tp. TRM* 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