Magnetite Magnetic Susceptibilitty

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L , , ~ . r ' i . "

E L S E V IE R Physics of the Earth and Planetary Interiors 93 (1996) 239-256

PHYSICSOFTHE EARTHANDPLANETARYINTERIORS

M a g n e t i c s u s c e p t i b i l i t y a n d r e m a n e n t c o e r c i v e f o r c e i n g r o w nm a g n e t i t e c r y st als fr o m 0 . 1 / x m t o 6 m m

F r a n z H e i d e r * , A n k e Z i t z e ls b e r g e r , K a r l F a b i a nlnst itut f i ir A l lgemeine u nd Angew andte Geophysik , Ludw ig-Maximil ians-Universit i it , Theresienstrasse 41, 80333 M iincher~ G ermany

Received 11 August 1994; revision accepted 9 May 1995

Abstract

Initial susceptibility is frequently used as a palaeoclimatic indicator in sediments, but its grain size dependence isnot well established. We measured initial magnetic susceptibility X0 in grown and natural magnetite crystals rangingfrom 0.09/~m to 6 mm in grain size. Over these five decades of grain diameter, the presented initial susceptibilitiesare essentially independent of grain size with a mean value of 3.1 SI and a standard deviation of + 0.4 SI. Numericalresults of micromagnetic calculations for cylindrical particles in the size range 0. 06/zm < d < 0.120/zm agree wellwith the experimental data. Initial susceptibilities of grown synthetic and natural magnetite crystals larger than 80/~m can be explained with demagnetizing factors N = 1/ 3 and large intrinsic susceptibility (X i > 200) using therelation X0 = X i ( 1 + N X i ) . T h e observed number of magnetic domains in magnetite grains between 50 /~m and 1000/~m is too low for the required demagnetizing factor of about 0.33. In a lamellar domain model one needs a highernumber o f domains than those observed, to obtain a demagnetizing factor of 0.33. A simple lamellar stripe domainmodel without closure domains is therefore not a good approximation for large magnetite grains.Remanent coercive force of grown magnetite grains shows a weak dependence on grain diameter. The remanentcoercive force H e r decreases gradually from about 35 mT to 10 mT between 0.09/zm and 6 mm. A noticeable dropin Her occurs at a grain size of about 110 /zm, which is interpreted as the transition from pseudo-single-domain tomultidomain grains. The remanent coercive force of magnetite grains is not a sensitive indicator of grain size, unlikecoercive force or saturation remanent magnetization.

1 . I n t r o d u c t i o n

Initial susceptibility (X0) of lake sediments(Peck et al., 1994), marine sediments (Robinson,1986; Bloemendal and DeMenocal, 1989; Bloe-mendal et al., 1993) and loess sequences (Hellerand Liu, 1982; Kukla et al., 1988; Maher et al.,

* Corresponding author.

1994) can be successfully used as a palaeo climaticindicator. One assumption in these studies is thatmagnetic susceptibility is a direct measure of theconcentration of magnetic minerals, A potentialgrain size dependence of susceptibility is oftenneglected and only an enhancement of X0 owingto superparamagnetic magnetite grains is dis-cussed. Magnetic susceptibility is one of the basicobservables of rock magnetism and needs to bestudied to provide a better understandingof the magnetization of the Earth's crust. For

0031-9201/96/$15.00 1996 Elsevier Science B.V. All rights reservedS S D I 0031-9201(95)03071-9


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2 4 2 F. He ide r e t aL / Phy s ic s o f the E ar th and P lane tary In te r ior s 93 (1996) 2 39-2 56

were published previously for a subset of thesedispersed samples (Heider et al., 1987, 1992).Fourteen hydrothermal magnetite samples weredispersed between 1% and 2% by weight in CaF 2or A120 3 (d = 0.3 /xm), and four samples haddispersions between 4% and 10%. We compactedmost samples by hand with a brass piston toimmobilize the magnetite grains in the matrix.Hydrothermally grown as well as precipitatedsamples, which were pressed into cylinders with ahydraulic press, are denoted by a 'p' (e.g. AZlpor H12p in Table 1).A series of single crystals (SC1-SC15) between17 /xm and 224 txm in size were hand-pickedunder the microscope for magnetic hysteresismeasurements on individual hydrothermal mag-netite grains. Samples SC1-SC5 and SC9-SC13were picked from a batch of hydrothermally growngrains with smooth crystal faces. For a test of theinfluence of crystal perfection on magnetic prop-erties we selected Samples SC6"-SC8", SC14"and SC15 * from a batch of grains that had irreg-ular and rough surface features. The dimensionsof individual magnetite grains which were occa-sionally slightly elongated were determined underthe light microscope and from the saturationmagnetic moment. Using a saturation magnetiza-

tion of 4.8 105 A m - ~ we calculated a diameterfor each magnetite grain assuming it to be spheri-cal. For all particles there is good agreementbetween the observed dimensions and the calcu-lated diameters (second and third columns inTable 2).Samples S1 contained octahedral magnetitecrystals with a mean grain size of 335/zm. Thesemagnetites were grown in a borate flux by Smith(1988). We extracted the millimetre-size mag-netite octahedra (PJ1-PJ14) for this study from agreen schist found at the Pfitscher Joch in Aus-tria. These natural magnetites had Curie pointsof 558-567C, indicative of a small amount ofcation impurities in the inverse spinel lattice. Themean grain sizes of the natural octahedra rangedfrom 0.59 mm to 6 mm (Table 1).

3 . E x p e r i m e n t a l m e t h o d sWe measured initial volume susceptibility (SI)of the magnetites listed in Table 1 with an induc-tance bridge. This susceptibility bridge (Barting-ton MS2, Oxford, UK) utilizes an alternating

magnetic field of 0.1 mT at a frequency of460 Hz. Magnetite grains several hundredT a b l e 2S u m m a r y o f h y s t e r e s is p a r a m e t e r s o f s i n g l e m a g n e t i t e c r y s ta l s m e a s u r e d w i t h t h e a l t er n a t i n g g r a d i e n t f o rc e m a g n e t o m e t e rS a m p l e D i a m e t e r f r o m D i a m e t e r f r om M s M r~ M r s / M ~ Hc He , S t a n d a r d H c r / H ~

m i c r o s c o p e a s s u m i n g s p h e r e m a g n e t i c m o m e n t ( m T ) ( m T ) d e v i a ti o n( / z m ) ( t x m ) ( 1 0 - 1 2 A m 2 ) o f H e r

( m T )S C 1 2 0 1 7 3 4 . 6 0 . 0 2 6 8 3 . 8 2 0 . 7 2 . 9 5 . 4S C 2 3 0 4 0 3 1 5 6 0 . 0 0 7 4 1 . 4 3 1 . 7 4 . 8 2 2 . 6S C 3 3 5 4 0 3 3 4 6 0 . 0 0 5 2 0 . 9 1 1 . 4 2 . 4 1 2 . 6S C 4 4 0 5 0 3 4 6 9 0 . 0 0 6 8 0 . 8 2 3 . 6 2 . 6 2 9 . 5S C 5 5 0 7 0 5 2 2 3 1 0 . 0 0 6 5 0 . 9 2 2 . 4 2 . 6 2 4 . 8S C 6 * 5 5 6 0 5 4 3 7 3 0 . 0 0 9 4 1 .5 3 0 . 9 1 . 8 2 0 . 6S C 7 * 5 0 6 0 5 8 4 0 9 0 . 0 0 8 4 1 . 5 2 3 . 6 2 . 0 1 5 . 8S C 8 * 6 0 7 0 6 0 3 5 1 0 . 0 0 6 6 1 2 1 . 8 1 . 3 2 1 . 8S C 9 8 0 9 0 8 7 5 5 4 0 . 0 0 3 3 0 . 6 2 1 . 8 2 . 3 3 6 . 3S C 1 0 1 0 5 1 3 0 1 0 8 1 0 7 0 0 . 0 0 3 3 0 . 5 2 4 . 7 1 . 8 4 9 . 4S C l l 1 2 0 1 1 0 9 6 6 0 . 0 0 2 9 0 . 6 2 6 . 6 1 . 9 4 4 . 4S C 1 2 1 3 0 1 4 0 1 1 2 1 1 2 0 0 . 0 0 3 1 0 . 4 2 1 . 9 1 . 9 5 4 . 8S C 1 3 1 6 0 1 5 2 1 6 6 0 0 . 0 0 1 9 0 . 3 1 9 . 5 4 . 0 6 4 . 9S C 1 4 * 1 6 0 2 2 0 1 7 6 4 0 5 0 0 . 0 0 3 0 0 . 3 1 3 . 3 1 .1 4 4 . 2S C 1 5 * 2 8 0 2 2 4 1 3 1 0 0 0 . 0 0 4 7 0 . 6 1 6 . 3 0 . 6 2 7 . 2* S a m p l e s w i t h r o u g h a n d i r r e g u l a r s u r f a c e s .R e m a n e n t c o e rc i vi ty H e r w a s m e a s u r e d a b o u t t e n t i m e s p e r g r ai n .


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F. Heider et al . / Physics of the Earth and Planetary Interiors 93 (1996) 239 -256 243micrometers in size have coercive forces slightlyabove 0.1 mT (Table 2). Strictly speaking, oneshould not call the value measured with the sus-ceptibility bridge an initial susceptibility in thecase of large magnetite grains, as the coerciveforce is of the same order as the applied alternat-ing field.The hysteresis loops o f single magnetite parti-cles (SC1-SC15*) were measured with an alter-nating gradient force magnetometer (AGFM;Flanders, 1990) from Princeton MeasurementsCo. (Princeton, USA) at the Institute of RockMagnetism in Minneapolis. Magnetic field incre-ments ranged between 0.3 and 1 mT for differentgrains up to the maximum applied field of 0.6 T.Typical averaging times per measurement were

50-200 ms for the larger crystals and 1 s for thesmallest grains. We calibrated the AGFM forabsolute magnetization measurements with anickel foil of known magnetic moment. The fieldgradient at the sample position was 0.4/zT/zm-1.The 17/xm size magnetite was the smallest grainone could easily handle and mount withpetroleum jelly on the quartz sample holder. The17 /~m size crystal with a saturation magneticmoment M s = 1.3 1 0 - 9 Am 2 was the weakestsample we measured with the AGFM. Even forthis smallest grain (SC1) we could determine thecoercive force He, the remanent coercive forceHcr, and the saturation remanence Mrs.The remanent coercive force Her, also namedremanence coercivity, of all dispersed samples

l

r ~o lmsm.m, ie ~

r ~

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o

' ' ' " " I ' ' ' " ~ " I ' ~ ' ' " " I ' ' ' ' " " I0 . 0 1 0 . I I I 0 I 0 0

G r a i n s i z e [ l a m ]

Grown magnetites: H y d r o t h e r m a l ( n o t p r e s s e d ) H-s er i es[ ] H y d r o t h e r m a l ( p r e s se d ) )t P rec ip i t a t ed (no t p r es s ed) ~ . A Z - s e r i e sO Prec ip i t a t ed (p res s ed) )~ ) F l u x g r o w n o c t a h e d r a ( $ 1 ) N a t u r a l o c t a h e d r a ( PJ - s er i es )+ D u n l o p ( 1 9 8 6 a )O A m i n e t a l. ( 1 9 8 7 ) X u & M e r r i l l ( 1 9 8 7 ) M a h e r ( 1 9 8 8 ). . . . . . . . . . L ue r 1 9 8 3 ): e (1982)

~ X_ v

Crushed grains:I > R a h m a n e t a l. ( 1 9 7 3 )I I D ay e t a l. ( 1977)41. Ho dyeh (1986)

' ' ' " " l ' ' ' ' " " 11 0 0 0 1 0 0 0 0

F i g . 1 . E x p e r i m e n t a l l y d e t e r m i n e d i n i t ia l m a g n e t i c s u s c e p t ib i l i ty X 0 v s. a v e r a g e g r a i n s i z e f o r g r o w n m a g n e t i t e c r y st a ls . Ac o m p a r i s o n w i t h p u b l i s h e d d a t a f o r g r o w n m a g n e t i t e s sh o w s g e n e r a l l y g o o d a g r e e m e n t . C r u s h e d g r a i n s h a v e o c c a s i o n a ll y lo w e r X 0t h a n g r o w n c r y s t al s . L o w f i e l d s u s c e p t ib i l i ty sh o w s p r a c t i c a ll y n o g r a i n s i z e d e p e n d e n c e . T h e m e a n s u s c e p t i b il i ty f o r h y d r o t h e r m a l ,p r e c i p i t a t e d a n d n a t u r a l m a g n e t i t e s i s 3 . 1 S I w i t h a s t a n d a r d d e v i a t i o n o f 0 . 4 . T h e g r e y - s h a d e d a r e a s a r e p r e d i c t i o n s f o r X 0 f r o mt h e o r e t i c a l c a l c u l a ti o n s .


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244 F. He ide r e t a l . /Ph y s ic s o f the Ea r th an d P lane tary In te r ior s 93 (1996) 23 9-25 6

( T a b l e 1 ) w a s d e t e r m i n e d w i t h t h e b a c k - f i e ldme t h o d . I n c r e a s i n g f i e l d s w e r e a p p l i e d w i t h aso leno id an t ipara l l e l t o the sa tu ra t ion i so thermalr e m a n e n t m a g n e t i za t io n ( I R M s = Mrs) , w h i c h w a sacqu i red in a 1 T mag net i c f i e ld . Di rec t ion andi n t e n si t y o f t h e r e m a n e n t m a g n e t i z a t io n w e r em e a s u r e d w i t h a s p i n n e r m a g n e t o m e t e r . W e f o l -l o w t h e c o mmo n p r a c t i c e i n g e o p h y s i c s o f u s i n gt h e u n i t s o f ma g n e t i c i n d u c t i o n ( T e s l a ) f o r t h ema g n e t i c f i e l d H , i n s t e a d o f t h e c o r r e c t u n i t s ( Am - l ) .

4 . R e s u l t s an d i n t e r p r e t a t i on

4.1. Measurements of initial susceptibilityT h e g r o w n ma g n e t i t e g r a i n s i n v e s t i g a t e d i n

t h i s s t u d y c o v e r a b o u t f i v e d e c a d e s o f g r a i n d i a -m e t e r ( 0 . 0 9 / z m - 7 m m ) . T h e m o s t s u i t a b l e f o r mf o r p r e s e n t i n g t h e d a t a i s a s s e mi l o g a r it h mi c p l o t sof susce pt ibi l i ty vs . grain s ize (Fig. 1) . A t a f i rs tg l ance , t he in it i a l suscep t ib i l it y app ears near lyg r a i n s i z e i n d e p e n d e n t w h e n o n e p l o t s o u r d a t af r o m g r o w n g r a i n s t o g e t h e r w i t h s u s c e p t i b i l i t i e sf o r ma g n e t i t e f r o m t h e l i t e r a t u r e . Mo s t d a t ap o i n t s ( 8 4 % ) f a ll b e t w e e n t h e t w o h o r i z o n t a l c o n -t inuous l ines f rom 2 .4 to 4 .0 SI . The low-f i e ldsuscep t ib i l i t i es X0 o f p rec ip i t a t ed magnet i t es( A Z 1 - A Z 5 ) a r e s h o w n b y fi ll e d d ia m o n d s i n F ig .1 . There i s an increase in X0 f rom 2 .95 to 3 .74 SIa s g r a i n s i z e d e c r e a s e s f r o m 1 . 9 1 / z m t o 0 . 0 9 / ~m.F i v e p r e c i p i t a t e d s a mp l e s w i t h g r a i n s i z e s b e -t w e e n 1 . 9 1 t zm a n d 0 . 0 9 / x m ( o p e n d i a m o n d s i nFig . 1 ) which we re p resse d in to cy l indr i ca l pe l l e t sh a v e l o w e r s u s c e p t i b i l i t i e s t h a n t h e u n p r e s s e ds i s t e r s a mp l e s c o n t a i n i n g ma g n e t i t e o f i d e n t i c a lg ra in s i ze . In th i s g ra in -s i ze range the ag reementb e t w e e n o u r s u s c e p t i b i l i t i e s a n d t h e d a t a o fV e i t c h a n d Sc h mi d b a u e r ( 1 9 8 3 ) a n d D u n l o p(1986a) i s reasonab le , excep t tha t t he X0 va lueso f D u n l o p ( 1 9 8 6 a ) d e t e r m i n e d o n p r e c i p i t a t e dgra ins ( see F ig . 1 ) decrea se s l igh t ly wi th d ecrea s -i n g g r a i n s i z e d . V e i t c h a n d Sc h mi d b a u e r ( 1 9 8 3 )p r o d u c e d t h e i r ma g n e t i t e b y r e d u c i n g d i s p e r s e dh a e m a t i t e p a rt ic l e s i n a n H 2 - H 2 0 a t m o s p h e r e a t400C.

T h e ma g n e t i c s u s c e p t i b i l i t i e s o f c r u s h e d ma g -ne t i t es smal l e r t han 4 t zm in s i ze (da ta f romRahman e t a l . (1973) and Day e t a l . (1977) inF i g . 1 ) a r e l o w e r t h a n t h e v a l u e s o f g r o w n ma g -ne t i t es . The low suscep t ib i l i t i es a re caused byc r y st a l d e f e c t s in c r u s h e d g r a in s w h i c h o p p o s e t h er e v e r sa l o f ma g n e t i z a t i o n i n s i n g le - d o ma i n ( SD )g r a i n s o r i mp e d e d o ma i n w a l l mo v e me n t s i np s e u d o - s i n g l e - d o ma i n ( PSD ) g r a i n s . U n i a x i a l a n dhydros ta t i c p res su re above a c r i t i ca l t h resho ld(p > (2 x 107) - (8 107)pa) d ecre ase in i ti a l sus -cep t ib i l i t y in syn the t i c and na tu ra l magnet i t esamples (Kean e t a l . , 1976 ; Nulman e t a l . , 1978) .The smal l es t (1 /~m < d < 2 / zm) magnet i t e sam-p le o f Kean e t a l . (1976) has a reduced in i t i a lsuscep t ib i l i t y owing to g r ind ing and shows l i t t l es t res s sens i t iv i ty o f X0 compared wi th the i r l a rgermagnet i t e g ra ins . The two smal l es t s amples (25n m a n d 1 2 0 n m) o f O z d e mi r a n d Ba n e r j e e ( 1 98 2 )o u t o f t h e f o u r c o m m e r c i a l m a g n e t i te p o w d e r sa g r e e w e l l w i t h d a t a f r o m g r o w n ma g n e t i t e s ( s e eFig . 1 ) . The i r two l a rger magnet i t e samples (190nm and 1 /xm) have suscep t ib i l i t i es s imi l a r tot h o s e o f g r o u n d s a mp l e s .

T h e s u s c e p t i b i l i t i e s f r o m SD a n d s u p e r p a r a -ma g n e t i c ( SP) p r e c i p i t a t e d ma g n e t i t e s ( n e w MTs e r i es ) o f M a h e r ( 1 9 88 ) s h o w a n i n c r e a s i n g t r e n dt o w a r d h i g h e r X 0 w i t h d e c r e a s i n g g r a i n d i a me t e r(see F ig . 1 ) . The h ighes t suscep t ib i l i t i es (5 .1 and5 . 4 ) f o r Ma h e r ' s s a mp l e s w e r e o b s e r v e d f o r ma g -n e t i t e s w i th m e a n g r a i n s iz e s o f 1 6 n m a n d 2 2 n m,s li g ht ly b e l o w t h e SP t o SD t r a n s it i o n o f 2 9 n m a t300 K (Dunlop , 1973) . Suscep t ib i l i t i es o f SP mag-ne t i t e g ra ins shou ld be a round 62 SI , i . e . 20 t imesh i g h e r t h a n t h o s e o f SD g r a in s , o w i n g t o t h ea s s i s t a n c e b y t h e r ma l f l u c t u a t i o n s ( S t e p h e n s o n ,1 9 7 1 ) . T h e e x p e r i me n t a l d a t a f o r ma g n e t i t e ( F i g .1 ) s h o w a mu c h s ma l l e r i n c r e a s e b y a b o u t a f a c t o ro f 1 . 7 f r o m SD t o SP p a r t i c l e s . T h e r e f o r e t h eq u e s t i o n a r i s e s w h e t h e r ma g n e t i t e g r a i n s i n t h es i z e r a n g e f r o m 1 6 n m t o 2 2 n m c a n b e s t a b l e SDpar t i c l es .T h e s u s c e p t i b i l i t i e s o f ma g n e t i t e s g r o w n b yA m i n e t a l. ( 1 98 7 ) w i t h t h e s a m e m e t h o d a s o u r ssca t t e r wide ly ( see F ig . 1 ). O ne poss ib le exp lana-t i on f o r t h e l a r g e v a r ia t i o n o f X 0 c o u l d b e t h e l o ws a t u r a t i o n m a g n e t i z a t i o n v a l u e s o f t h e s a m p l e s o fA m i n e t a l. T h e i r M s v a l u e s a r e i n s o m e c a s e s u p


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F. Heider et al. /P hys ics o f the Earth and Planetary Interiors 93 (1996) 239-256 247He showed that with increasing number of do-mains the multidomain demagnetizing factor Napproaches one-third, which is equal to the sin-gle-domain demagnetizing factor of a sphere. Xuand Merrill (1987) calculated a multidomain de-magnetizing factor for a cube with 20 domainswhich is 13% lower than the one-third obtainedfor an SD cube. The multidomain demagnetizingfactors that were calculated by Xu and Merrill(1987) for cubic magnetite grains with zero netmagnetic moment and 2 -8 domains are very simi-lar to the earlier results of Dunlop (1983). InTable 3 we employ the demagnetizing factors ofXu and Merrill (1987) because they are availablefor up to 100 lamellar domains.Now one only has to know the number ofdomains which are typically observed in mag-netite grains of a certain size. It is well estab-lished from domain observations with the Bittermethod and the magneto-optical Kerr effect thatgrains of fixed size can occur in a multiplicity ofdomain states (Soffel, 1971; Halgedahl and Fuller,1983). We used experimental observations onmagnetite (Heider, 1988; Worm et al., 1991; Hei-der & Hoffmann, 1992; GeiB, et al., 1995) toobtain an upper and lower limit for the numberof domains n which are found in grains of aspecific diameter. Reasonable upper and lowerbounds for n in magnetite grains around 10/zmdiameter are nmax = 10 and nmin = 3 (Table 3).An example of a three-domain particle with 7 ~mdiameter is shown in Fig. 2(a). The number ofdomains observed in magnetites around 79 /zmdiameter varies between six and 16. An averagedomain width for millimetre-size magnetite crys-tals is 30/~m (Hanss, 1964; Williams et al., 1992;()zdemir and Dunlop, 1993). The mul tidomaindemagnetizing factor for a 1 mm size particlewith 33 domains is N = 0.32, which is very closeto the single-domain demagnetizing factor ofone-third.

The maximum initial susceptibilities X0,ma~ inTable 3 were obtained from Eq. (1) using theappropriate intrinsic susceptibility and the de-magnetizing factor Nmin for the lowest observednumber of domains nrain for a given grain size.The X0,m~, values for hydrothermal ly grown mag-netite crystals with 11.7, 79 and 1000/zm average

grain diameter are plotted as filled circles on theupper edge of the dark shaded triangle in Fig. 1.The minimum initial susceptibilities of the 11.7and 79 /~m size samples (Table 3), which werecalculated analogous to X0,m~, lie on the lowerside of the dark shaded triangle (which is strictlyspeaking a pentagon). Hence, this dark shadedtriangle represents the area between the theoreti-cally determined maximum and minimum boundsof susceptibility. There is good agreement be-tween the theoretical estimates of X0 and theexperimental dat a from the two 11.7 /zm sizesamples (filled squares in Fig. 1). The experimen-tally determined susceptibilities X0 from mosthydrothermal magnetite samples (not pressed)between 50 ~m and 1000/~m plot below the darkshaded triangle except for Sample Hll (d = 356/zm). We can find agreement between the calcu-lated and measured susceptibilities for the sam-ples between 50 and 1000/~m only, if we assumean infinite number of domains (i.e. N = 1/3).The discrepancy between experimentally de-termined initial susceptibilities and the valuescalculated from Eq. (1) for large MD grains leavesus with two possible explanations. Either the esti-mates of intrinsic susceptibilities (Xi--4.5 104A m - l / H e ) are too high or the demagnetizingfactors Nm~x for Samples H13 and H19 are toolow. The second explanation is more likely, as wewill show in the following discussion. The lamel-lar domain model which was assumed for thecalculation of demagnetizing factors (Dunlop,1983; Xu and Merrill, 1987) is not applicable tolarge magnetite grains. Lamellar stripe domainsare primarily observed in materials with highuniaxial anisotropy. Cubic materials with lowmagneto-crystalline anisotropy, such as mag-netite, try to avoid magnetic surface charges. Themagnetostatic energy is reduced by flux closure asmuch as possible. The qual ity parameter Q =K 1 / K d with K d = 1/2/x0 M2 is a measure for therelative proportions of magnetocrystalline aniso-tropy energy and demagnetizing energy (Hubert,1988; Jiles, 1991). In the case of magnetite Q =0.09, which implies that the dominating demagne-tizing energy E d has to be minimized by thealignment of magnetization with the grain bound-aries. The formation of a magnetization vortex


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248 F. Heider et al . / Physics of the Earth and Planetary Interiors 93 (1996) 23 9-256(see Section 4.2.2) in small magnetite grains (d120 nm) is the direct outcome from three-dimen-sional micromagnetic calculations (e.g. Thomsonet al., 1994). In larger grains the anisotropy en-ergy term favours the formation of closure do-mains, in which case the magnetization tries topoint along an easy direction. A lamellar domainstructure in magnetite with magnetization point-ing at right angles to two surfaces of a cube isenergetically very unfavourable, owing to the re-suiting large magnetic surface charges.Evidence for the existence of closure domainsin magnetite comes from Bitter pattern observa-tions (e.g. Heider et al., 1988; Geil3, 1993;Ozdemir & Dunlop, 1993). Two magnetite grains,labeled A (d = 8/xm) and B (d = 4/xm), from aglass-ceramic sample have closure domains atboth sides where the 180 domain walls approachthe edges of the particle (Fig. 2(b)). Anotheroption for magnetite grains to reduce the demag-netizing energy is the formation of domains on asmall scale near some edges of the grain. Asubdivision into smaller domains near the edgesis in principle comparable with the echelon pat-terns observed in iron (Dillon, 1963). An examplefor such a complicated domain structure withreduced stray-field energy E d is shown in Fig.2(a) for the case of a hydrothermally grown mag-netite grain with d = 20/zm. There are numerousexamples of magnetite particles without closuredomains (e.g. Worm et al., 1991; Heider andHoffmann, 1992), but at least as frequently oneobserves closure domains at the grain edges orcomplicated domain structures (Figs. 2(a) and2(b)).The demagnetizing factors obtained from thelamellar domain models (Dunlop, 1983; Xu andMerrill, 1987) for a given number of domains aretherefore too low to explain, with Eq. (1), theexperimentally determined susceptibilities. Therelatively low number of domains which is ob-served, for example, in 79/~m size grains (Table3 : 6 ~ 50) and N --- 1/3 we calculate lower initialsusceptibilities (X0 --- 3), which are in good agree-ment with the experimental data between 50 and1000/~m (Fig. 1).Ground magnetite powders should have highdislocation density and many other lattice imper-fections which impede domain wall motion. Thehigh coercive forces of the crushed Samples Day1 and Day 2 (Day et al., 1977) correspond to lowintrinsic susceptibilities according to X/= 4.5 104 A m- ~ / H c (Table 3). The crushed magnetitesamples (Day 1 and Day 2) have Xi values aboutan order of magnitude lower than hydrothermallygrown samples. The maximum initial susceptibili-ties one obtains from Eq. (1) for the two crushedsamples are plotted as filled circles along theupper edge of the light shaded polygon in Fig. 1at 9.3 and 131/xm. Accordingly, the X0,min resultsfor a maximum number of domains (10 and 16)plot on the lower edge of the light shaded poly-gon. The right corner of this light shaded polygonat 1000/zm grain diameter (Fig. 1) is formed bythe same X0 as the right corner of the darkshaded triangle above. The light shaded polygon,which was calculated based on two coercive forcevalues from the samples of Day et al. (1977), is ingood accord with the experimentally determinedX0 values from the same set of samples (Fig. 1). Apossible explanation for the better agreement be-tween the lamellar domain model and the experi-mental susceptibilities of crushed magnetitescould be a stress-induced anisotropy. It is con-ceivable that the application of high stress duringthe grinding process produces in some grains auniaxial anisotropy. In this case, the lamellardomain model forms a more suitable approxima-tion to the domain structure than the above-de-scribed cubic anisotropy with effective flux clo-sure.The lamellar domain model does not seemwell suited to explain the experimental A'0 data ofgrown submicron magnetites. The calculated ini-tial susceptibilities for Samples AZ4p and H1(4.92 ~>g0,mi, >i 4.24 in Table 3 and two filledcircles in the shaded rectangle in Fig. 1) are


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F. Heider et aL /Physics of the Earth and P lanetary Interiors 93 (1996) 239-256 249s u b s ta n t ia l ly h i g h e r t h a n t h e e x p e r i m e n t a l l y o b -s e r v e d v a lu e s a t 0 . 7 5 / ~ m ( f il le d s q u a r e a n d o p e nd i a m o n d i n F ig . 1) . T h e d i s c r e p a n c y b e t w e e nl a m e l l a r d o m a i n t h e o r y a n d e x p e r i m e n t is p r o b a -b l y d u e t o t h e e x i s t e n c e o f a d d i t i o n a l d o m a i ns t a t e s ( e . g . v o r t e x ) i n t h e s u b m i c r o n g r a i n - s i z er a n g e .4 . 2 . 2 C y l i n d r ic a l m o d e l f o r s u b m i c r o n - s i z e g r a in s

F o r p a r t i c le s in th e S D - P S D t r a n si t io n r a n g ei t i s d i f f i c u l t t o o b t a i n e s t i m a t e s f o r t h e i n t r i n s i cs u s c e p t i b i l i t y X ; a n d t h e d e m a g n e t i z i n g f a c t o r N .T h i s d i f f i c u l ty i s p a r t i a l l y d u e t o t h e n o n - l a m e l l a rd o m a i n s t a t e s ( e. g . v o r t e x ) w h i c h o c c u r i n t h a ts iz e r a n g e . F o r t u n a t e l y , w e c a n u s e t h e f u l l m i c r o -m a g n e t i c d e s c r i p t i o n t o c a l c u l a t e t h e m a g n e t i cd o m a i n s t r u c t u r e o f t h e s e s m a l l P S D g r a i n s. T h ea d v a n t a g e o f m i c r o m a g n e t i c c a l cu l a ti o n s o v e r t h eu s e o f E q . ( 1 ) i s t h a t t h e y a r e b a s e d o n e x p e r i -m e n t a l l y d e t e r m i n e d m a t e r ia l p a r a m e t e r s s u c h a ss a t u r a t i o n m a g n e t i z a t i o n , m a g n e t o c r y s t a l l i n e a n -i s o t r o p y C O n s t a n t a n d e x c h a n g e C O n s t a n t . S u c ht h r e e - d i m e n s i o n a l f i n i t e e l e m e n t c a l c u l a t i o n sh a v e b e e n m a d e f o r p a r t i c l e s a f e w 1 0 0 n m i ns i z e , t o i n v e s t i g a t e d o m a i n s t r u c t u r e s a n d t r a n s i -t i o n m e c h a n i s m s f o r P S D p a r t i c l e s . S t a r t i n g f r o mt h e S D t o v o r t e x tr a n s it i o n m o d e w h i c h h a s b e e no b s e r v e d i n t h e 3 D c a l c u l a t i o n s ( E n k i n a n dW i l l i a m s , 1 9 9 4 ) a s i m p l i f i e d m a t h e m a t i c a l m o d e lw a s d e v e l o p e d . W i t h t h i s m o d e l w e o b t a i n e s t i -m a t e s f o r i n i t i a l s u s c e p t i b i l i t y i n m a g n e t i t e p a r t i -c le s a r o u n d 1 0 0 n m i n si ze , w h e r e b o t h S D a n dv o r t e x m a g n e t i z a t i o n s t a t e c a n o c c u r . H e r e w ep r e s e n t a n o u t l i n e o f t h e t h e o r y ; a d e t a i l e dd e r i v a t i o n o f t h e f o r m u l a e w i l l b e g i v e n in af o r t h c o m i n g p a p e r .

W e c o n s i d e r a c y l i n d r i c al p a r t i c l e o f h e i g h t ha n d r a d i u s R w h i c h i s s y m m e t r i c w i t h r e s p e c t t ot h e z - ax i s. T h e d i r e c t i o n o f t h e m a g n e t i z a t i o nu n i t v e c t o r a t t h e p o i n t ( x , y , z ) i s g i v e n b ym ( x , y , z )

= - [ sin ~o , - sin ~or r , COS (2 )w h e r e r=( x2+y 2) 1/2 a n d m d o e s n o t d e p e n du p o n z . T h i s d e f i n e s a s t r u c t u r e w h i c h i s s y m m e t -r ic w it h r e s p e c t t o r o t a t i o n a r o u n d a n d t r a n sl a -

, . ol R = 2 0 n m

A O O0 . 8 O o

0 0 0 g i g g o O

O O O O O o o o o 0 R = 2 6 . 5 n m

A R = 4 4 n mA A A

0 . 0 ' I ' ' I ' U ' - T0 2 0 4 0 6 0 8 0 10 0

R e l a t i v e d i s t a n c e f r o m z - a x i s r / R [ % ]Fig. 3. Radial dependen ce of the z-comp onent of m agnetiza-tion parallel to the cy linder axis for three different structures.In the single dom ain state the magn etization points uniformlyalong the cylinder axis (0 ) o, A, vortices with flattenedmagnetization toward the edg e of the cylinder. The normal-ized radius r / R = 0 correspond s to the cylinder axis andr / R = 100% represents the ou ter surface of the cy linder. Theradius R increases fro m the SD to the vo rtex configuration.

0 .6

0 .4o

0.2

t i o n a l o n g t h e z - ax i s. T h e f u n c t i o n ~p d e s c r i b e st h e a n g l e b e t w e e n t h e d i r e c t io n o f m a g n e t i z a t i o na n d t h e z - a x is a t a p o i n t w i t h d i s t a n c e r f r o m t h ec e n t r e . W e o b t a i n a p u r e S D s t a te f o r ~ p - 0w h e r e t h e z - c o m p o n e n t o f m a g n e t i z a t i o n m z - - 1( d i a m o n d s i n Fi g . 3 ) , w h e r e a s a f u n c t i o n ~ w h i c hv a r i e s f r o m ~ p(0 ) = 0 i n t h e c e n t r e o f t h e c y l i n d e ro f ~ p( 1) = 7 r / 2 a t t h e b o u n d a r y r e s u l t s i n a v o r t e xc o n f i g u r a t i o n ( t r i a n g l e s i n F i g . 3 ) . I n t h a t w a y ,e v e r y f u n c t i o n ~p d e f i n e s a u n i q u e m a g n e t i z a t i o ns t r u c t u r e m w i t h d i v ( m ) = 0 i n t h e i n t e r i o r , w h i c his a n i m p o r t a n t p r o p e r t y o f s o f t m a g n e t i c m a t e r i-a ls ( Q


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2 5 0 F. Heider et at/Phys ics of the Earth and Planetary Interiors 93 (1996) 239-256where A is the exchange constant (for magnetite1.33 X 10 -11 J m -1) and p = r / R . This energyterm will diverge whenever sin[~o(0)] e 0, whichimplies that the magnetization in the centre ofthe vortex always has to point along the z-axis.During a complete magnetization reversal themagnetization along the central line has to beinverted. In real particles this magnetization re-versal is achieved by moving the vortex centreoutside the grain (Enkin and Williams, 1994).Because of the imposed symmetry of our simplestructure, such reversal processes cannot be mod-elled.(2) The magnetocrystalline anisotropy energyis given for the two extreme cases where thez-axis is either an easy or a hard direction ofmagnetization for the cubic magnetite structure.We obtain

2 "/'/"E~l(~) = K I R h ~ f o l p [ 9 + 7 cos{2~(p)}]xsin 2 ~(p ) do (4a)

2 71"E ~ ' l J ( ~ ) = K~R h - f f fo l p [ 3 s i n 4 ~ ( p )+4 C OS @ ( p ) ] dp (4b)

where g I is the first anisotropy constant (formagneti te K~ = - 1.25 104 J m-3).(3) The energy owing to an external magneticfield can be calculated asEH( ~) - 2~rlzoM,HzR2h o lp cos q~(p) dp

( 5 )where M s is the saturation magnetization and ~0the permeability of free space. Again, the magne-tization processes are constrained by the symme-try of our configuration, because only the z-com-ponent of the external field H z affects the en-ergy. In reality, the application of horizontal fieldsleads to a movement of the vortex centre whichcannot be achieved with the present simple model.For the computation of susceptibilities we aretherefore restricted to the application of fields inthe z-direction, which implies that our resultsshould be understood as susceptibilities for thisspecial transition mode.

(4) The calculation of the demagnetizing en-ergy is always the main obstacle if one tries toformulate a realistic model for magnetizationstructures. In our case, the calculation is simpli-fied by the property that div (m)= 0 inside theparticle, which implies that the demagnetizingfield is generated by surface charges only. Thesurface charges on the curved surface vanish ow-ing to the cylindrical symmetry of the investigatedmagnetization structures and of the grain. Conse-quently, we only have to deal with the top andbottom surfaces of the cylinder, which leads tothe following term for the demagnetizing energy:

2 3 1 PE(~) = 4zr/xMs R fo fo cos ~o(p)cos q~(p ' )s ( h , p , p ') dp dp (6)

where [ 1 lS(Iz , p , p ') =p'F -~, -~, 1;pp'

( / / , 2 - I - f l , 2 q _ 0 2 ) 1 / 2[3 1 ( 2pp' ) 2

XF ~, ~, 1; / . ~ 2 _ b p , 2 + p 2The F(a, b, c; x) are hypergeometric functionswhich in this case can be expressed in terms ofcomplete elliptic integrals. For given values of h,R and H z we have to determine a function ~0(r)which minimizes the total energy functionalE(q~) =EA(q~ +Er(~o ) +EH(~O ) +Eo(qQ (7)This is done numerically by discretizing the unitinterval and using a conjugate gradient methodfor the minimization of the energy in the result-ing finite dimensional space. We determined theinitial susceptibility of particles above the singledomain threshold size d o by first minimizing hemagnetization structure in zero field and thenwith an applied field parallel to the cylinder axisHz= 0.i mT. From the difference in net magneti-zations divided byHz one can calculate the initialsusceptibility X0 for cylinders around the single-domain to vortex transition size. This was done


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F. Heider et al. /Phy sics of the Ea rth and Planetary Interiors 93 (1996) 239 -256 251f o r a n u m b e r o f p ar t ic l e s w it h d i a m e t e r s d = 2 Rr a n g i n g f r o m 5 0 n m t o 1 2 0 n m a n d f o r e l o n g a -t i o n s i n t h e z - d i r e c t i o n q = h / 2 R b e t w e e n o n ea n d t w o f o r b o t h c a s e s o f th e a n i s o t r o p y o r ie n t a -t i o n . Fi g . 4 s h o w s t h e t h e o r e t i c a l l y d e t e r m i n e di n it i al s u s c e p t i b il i ti e s a s a f u n c t i o n o f d i a m e t e r d .M o n o t o n i c a l l y i n c r e a s i n g c u r v e s o f f i l l e d c i r c l e sa r e f o r t h e h a r d a x is o f m a g n e t i z a t i o n [ 1 00 ] p o i n t -i n g a l o n g t h e c y l i n d e r a x is . S i m i la r ly , t h e d e c r e a s -i n g c u r v e s ( o p e n d i a m o n d s i n F ig . 4 ) r e p r e s e n tt h e e a s y a x i s [ 1 1 1 ] p o i n t i n g a l o n g t h e c y l i n d e rz -a x is . In t h e S D a n d l o w e r P S D r e g i o n b e t w e e n5 0 n m a n d 1 2 0 n m t h e c a l c u l a t e d i n i t i a l s u s c e p t i -b i l it y v a r i e s b e t w e e n 2 a n d 4 S I. T h e s e t h e o r e t i -c a ll y d e t e r m i n e d X 0 v a l u e s a r e r e p r e s e n t e d b yt h e s h a d e d r e c t a n g l e in F ig . 1 . T h e a g r e e m e n tw i t h e x p e r i m e n t a l l y d e t e r m i n e d s u s c e p t i b il i ti e s o ng r o w n m a g n e t i t e s a m p l e s i s v e r y g o o d c o n s i d e r i n gt h e c o n s t r a i n ts o f th e m o d e l . T h r e e - d i m e n s i o n a lm i c r o m a g n e t i c c a l c u l a t i o n s a r e i n p r o g r e s s t om o d e l t h e o b s e r v e d s u s c e p t i b i l it i e s ( F ig . 1 ) in t h eg r a i n s i z e r a n g e f r o m 0 . 1 / z m t o 2 ~ m .

g r a in s j u s t a b o v e t h e S P - S D b o u n d a r y . T h e v is -c o u s f r a c ti o n a m o n g t h e s e S D g r a in s , w i th a p p r o -p r i a t e r e l a x a t i o n t i m e s , r e v e r s e s e a s i ly , w h i c h r e -su l t s i n t h e l o w H cr v a l u e s b e t w e e n 1 0 a n d 2 0m T . S e c o n d , s o m e o f t h e g r a in s w i t h d i a m e t e r sl e ss t h a n 1 0 0 n m m a y n o t b e t i g h t ly p a c k e d i n t h en o n - m a g n e t i c m a t r ix a n d r o t a t e u p o n a p p l ic a t io no f r e v e r s e f ie l d s a r o u n d 1 0 m T . T h i r d , c l u s t e r s o fs u p e r p a r a m a g n e t i c g r a i n s m a y a c t c o l l e c ti v e l y v i am a g n e t o s t a t i c i n t e r a c t io n a n d b e a b l e t o c a r r y ar e m a n e n t m a g n e t i z a t io n t h a t i s s o f t e r t h a n a n S Dr e m a n e n c e .

I n t h e g r a i n s i z e r a n g e f r o m 0 . 0 3 / z m t o a p -p r o x i m a t e l y 0 . 1 5 / z m a l a r g e f r a c ti o n o f a s a m p l eo c c u r s in t h e s i n g le d o m a i n s t a te . A b o v e 0 . 1 5 / z mg r a i n s i z e t h e s i n g l e d o m a i n s t a t e i s e n e r g e t i c a l l yn o l o n g e r s t a b l e , a c c o r d i n g t o m i c r o m a g n e t i c c a l -c u l a t i o n s f o r m a g n e t i t e g r a i n s w i t h e l o n g a t i o n sq ~< 1 .5 (N ewe l l e t a l ., 1993; F ab ian e t a l . , 1995) .A s s y n t h e t ic a l l y g r o w n m a g n e t i t e s i n th e s u b m i -c r o n g r a i n s i z e r a n g e a r e n o t a l w a y s p e r f e c t l y

4 .3 . R e m a n e n t c o e r c iv e f o r c eT h e r e m a n e n t c o e r c i v e f o r c e s H er a r e p l o t t e d

a g a i n s e m i l o g a r i t h m i c a l ly v e r s u s g r a i n s i ze f o rm a g n e t i t e s b e t w e e n 1 2 n m a n d 6 . 8 m m ( F ig . 5) .M o s t ncr v a l u e s i n F ig . 5 w e r e d e t e r m i n e d w i t ht h e c o n v e n t i o n a l d . c . b a c k - f i e l d m e t h o d . O n l y t h ed a t a o f M a h e r ( 1 9 8 8 ) w e r e o b t a i n e d f r o m t h e' i n t e r s e c ti o n m e t h o d ' . A d e t e r m i n a t i o n o f H e r o nt h e s e s a m p l e s w i t h t h e d . c . b a c k - f i e l d m e t h o dr e s u l t e d in 8 - 1 8 % h i g h e r v a l u e s ( B . M a h e r , p e r -s o n a l c o m m u n i c a t i o n , 1 9 9 4 ) . T h e i n t e r s e c t i o n o fI R M a c q u i s i t i o n a n d a . f . d e m a g n e t i z a t i o n c u r v e sg i v e s a n e s t i m a t e f o r H er w h i c h i s o f t e n c l o s e t ot h e H c , d e t e r m i n e d w i t h t h e d . c . b a c k - f i e l dm e t h o d ( C i s o w s k i , 1 9 8 1 ; D a n k e r s , 1 9 8 1 ; D u n l o p ,1 9 8 6 b ) . T h e s m a l l e s t m a g n e t i t e s i n F i g . 5 a r ef r o m M a h e r ( 1 9 8 8 ) a n d p o s s e s s r e l a t i v e l y l o wr e m a n e n t c o e r c i v e f o r c e s. O n e w o u l d e x p e c t t h a tt h e s m a l l e s t s a m p l e s o f M a h e r ( 1 9 8 8 ) h a v e a l a r g ef r a c t io n o f s u p e r p a r a m a g n e t i c g r a in s , a s t h e i rm e a n g r a i n s iz e is l e ss t h a n 3 0 n m . T h e r e a r et h r e e p o s s i b l e e x p l a n a t i o n s f o r t h e l o w H cr v a l u e so f t h e s e s m a l l m a g n e t i t e s . F i rs t , th e r e m a n e n tm a g n e t i z a t i o n o f t h e s e s a m p l e s c a n b e c a r r i ed b y

gh

q = 2 00q - 1 7 5 0 O 00 O _ t o o e e o o o o q = 2

q = ~5 * * * o.~ 1 ? ' '%'* '~ ' 'o 'o 'o 'o-o. .q = 1 "2 50 O ~ _ . e ~ O ~ . . L _ - " v g ~ q = 1 .7 5 0 0 8 o " m l O ~ O ~ , " ~ e ~ l O ~ o o o q = 1 .5

q = l O ~ o e e e ~ ; ; e l e o O e ~ t ~ l ~ , l l . l ~ 9 9 G ,= l . 2 5 6 - . . : * . -._ -: * * . * * * * * * *

O o - v O o ~ s A ~ A A " O e O O O O O o o o c I = I J v v g v g o 0 0 0 0 0 0 0 0 0 0 0 --

' I ' i ' I ' I4 0 6 0 8 0 1 0 0 1 2 0

D i a m e t e r [ n m ]Fig. 4. Theo reticallycalculated initial susceptibilities from twomagnetization states w ith a one-dimensional cylindricalmicro-magnetic model. Each sym bol is the result of two completeenergy miuimizations, one fo r H = 0 m T and the sec ond forH = 0.1 roT. The diameter of the cylinder varied from 50 to120 um and elongation q varied from one to two. The twosets of curves are for easy direction (~) and hard direction ofmagnetization (o) p ointing along the cylinder axis. The mod elresults ( 2 < X 0 < 4 ) agree wel l w i t h t h e experimen tal resultssumm arized in Fig. 1.


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25 2 F. Heider e t al . / P hysics o f the Earth an d Planetary Interiors 93 (1996) 2 39 -25 6equidimensional, it is realistic that the SD sizeinterval is extended owing to grains with elonga-tions between 1.0 and 1.5. In this 'SD grain sizerange' there is a large scatter of He, values. Theremanence coercivities for grown magnetites(Dunlop, 1986a; Levi and Merrill, 1978; Ozdemirand Banerjee, 1982; precipitated grains, thisstudy) range from 22 to 40 mT. Submicron mag-netites with He, values above 40 mT were pro-duced by reduction of haematite (Schmidbauerand Schembera, 1987; Veitch and Schmidbauer,1983) or by grinding (Day et al., 1977), or are ofbiogenic origin (Moskowitz et al., 1989).

Grown magnetites above 0.2/zm in size exhibita very slowly decreasing remanent coercive forcewith increasing grain diameter up to about 110/zm. Based on the ncr results we redefine thePSD grain size range as the interval from 0.2 tzm

to roughly 110 /zm. A changeover from PSD tomultidomain behaviour at about 100 ~m wasfound from a.f. demagnetization properties ofsaturation IRM and thermoremanent magnetiza-tion (Lowrie-Fuller test in Heider et al. (1992)).The coercive force H C of grown magnetit es de-creases by a factor of 30 between 0.2/~m and 110/zm (Heider et al., 1987), whereas ncr data varyirregularly between 15 and 35 mT (Fig. 5). Espe-cially noteworthy are the relatively high ncr val-ues for grown grains between 10 and 110 /zm.Whether the lower Her values between 1 and 10/~m (Fig. 5) are an inherent property of magnetitegrains in this size interval or whether they aredue to our lack of data is at present not clear.Crushed magnetites (e.g. Day et al., 1977) havegenerally high remanent coercivities for d < 10/zm, owing to the known co ntami natio n with small

60

5 0L ..

~. 411. i i

e ~

~ 311o

= 20ell

10

0

Precipitated grains (pressed)1 Flux grown octahedra~r X Natural octahedra (green schist)Hi-Dunlop (1986a)V


15/18


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25 4 F. Heider e t al. / P hysics o f the Earth a nd Plan etary Inter iors 93 (1996) 2 39 -25 6

stable magnetization, is not supported by our /-/~rresults. Large (100 /zm size) magnetite grainswith remanent coercivity/-/~r = 25 mT (Hcr /H c =50) would constitute the lower coercivity fractionduring a.f. demagnetization of a rock, but never-theless be able to carry part of the stable rema-nent magnetization.

AcknowledgementsThanks are due to Anita H6fer for measure-ments of the magnetite crystals from the PfitscherJoch. We benefited from discussions with mem-bers of the rock- and palaeomagnetism group atthe Institut fiir Allgemeine und AngewandteGeophysik in Miinchen. We are grateful to Dr.

H.-U. Worm for donating the glass-ceramic mag-netite sample, and to Christoph GeiB for con-tributing the Bitter image in Fig. 2(b). Commentson the manuscript by Drs. D.J. Dunlop, J.G. Kingand H.-U. Worm are appreciated. F.H. wishes tothank the Institute for Rock Magnetism in Min-neapolis and its members for the use of theAGFM and for their hospitality. The IRM isfunded by the Keck Foundation, the NationalScience Foundation and the University of Min-nesota. This research was supported by a grantfrom the Deutsche Forschungsgemeinschaft.

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