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    C H A P T E R 1 3

    C o l l e c t i o n s o f B i s t a b l e U n i t s

    In th i s c h a p t e r a n d i n th e fo l l o w i n g o n e , w e sh a l l e x p l o re t h e c o n se q u e n c e so f d e sc r i b i n g h y s t e r e s i s i n f e r ro m a g n e t i c sy s t e m s i n t e rm s o f c o l l e c t i o n so f b i s t a b le u n i t s o f t h e t y p e d i s c u s se d i n C h a p t e r 2 . D e sc r i p t io n s i n v o l v i n gd o u b l e -w e l l p o t e n t i a l s h a v e b e e n e n c o u n t e r e d s e v e ra l t i m e s i n t h e p re v i -o u s c h a p t e r s , w h e n d i s c u s s i n g c o h e re n t ro t a t i o n (C h a p t e r 8 ) , m a g n e t i cv i s c o s i ty (C h a p t e r 1 0), a n d c o e rc iv i ty m e c h a n i sm s (C h a p t e r 1 1 ). Th e r e a -so n i s t h a t b i s t a b i l i t y i s r o o t e d i n t h e v e ry c o n c e p t o f B a rk h a u se n j u m p .W h e n w e d e t e c t a j u m p , w e k n o w t h a t th e s y s t e m i s l e a v in g a m e t a st a b les t a t e i n f a v o r o f so m e o t h e r s t a t e o f l o w e r e n e rg y . Tw o e n e rg y t e rm s a r einvo lved in the even t , t he f ree energ y ch ang e AF f rom the in it ia l to thef in a l s t a te a n d t h e a m o u n t o f e n e rg y A E d i s s i p a t e d a s h e a t d u r i n g t h ej u m p . Th e m a g n e t i z a t i o n p ro c e s s is b a s i c a ll y a s e q u e n c e o f B a rk h a u se nj u m p s , a n d i f o n e w e re a b l e t o c o rr e c t ly k e e p t r a c k o f t h e e n e rg y c h a n g e sw h i l e t h e s e q u e n c e p r o c e e d s , o n e w o u l d h a v e a t h a n d q u i te a g e n e ra ld e sc r i p t i o n o f h y s t e r e si s . Th e m a i n d i f fi c u lt y b e h i n d t h is p i c t u re i s t h a t w ed o n o t k n o w t h e g e n e ra l p r i n c i p l e s c o n t ro l l in g t h e s t a ti s ti c s o f B a rk h a u se nj u m p s a n d t h o se d e t e r m i n i n g i n w h a t p r o p o r t io n t he w o r k p e r f o r m e d b ye x t e rn a l so u rc e s i s su b d i v i d e d i n t o s t o r e d a n d d i s s i p a t e d e n e rg y i n t h ej u m p se q u e n c e . Th e w o rk i n g h y p o t h e s i s t h a t w i l l b e e x p l o re d h e re i s t h a t,i n a sense to be m ad e m ore p rec i se in Sec t ion 13 .1 an d Sec t ion 13 .2 , t hep ro b a b i l i t y d i s t r i b u t i o n p(AE, AF) o f h a v i n g j u m p s c h a r a c t e ri z e d b y g i v e nA E a n d A F i s a n i n h e re n t p ro p e r t y o f t h e sy s t e m , w h i c h c a n b e p re sc r i b e di n a d v a n c e , b e fo re s e l e c t i n g a n y p a r t i c u l a r m a g n e t i z a t i o n c u rv e t o t r a -v e r s e . Th i s h y p o t h e s i s w a s a l r e a d y i m p l i c i t l y d o n e i n t h e t r e a t m e n t o fm agn e t i c v i scos i ty d i scu ssed in Sec tion 10.3 , bu t i n tha t case it s app l i ca t ionw a s l i m i t e d t o t h e sm a l l f ie l d a n d m a g n e t i z a t i o n c h a n g e s t y p i c a l o f t h e r -m a l r e l a x a t i o n p h e n o m e n a . H e re , t h e s a m e h y p o t h e s i s i s e l e v a t e d t o fu n -d a m e n t a l p o s t u l a t e fo r t h e d e sc r i p t i o n o f h y s t e r e s i s u n d e r c o m p l e t e lygenera l f i e ld h i s to r i e s .

    4 3 3

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    434 CHA PTER 13 Co llections of Bistable UnitsIn p r in c ip l e , t h e a s s u m p t io n o f th e e x i s t e n c e o f t h e p(AE, AF) d is t r ibu -t io n , t o b e c o n v in c in g , s h o u ld b e d e r iv e d b y s o m e a n a ly s i s o f t h e p ro p e r t i e s

    o f t h e mic ro ma g n e t i c f r e e e n e rg y a n d o f t h e s o lu t i o n s o f Bro wn ' s e q u a -t i o n s , wh e n th e p r e s e n c e o f s t r u c tu r a l d i s o rd e r g iv e s a r i c h s t a t i s t i c a le n s e m b l e o f m e t a s t a b l e s t a t e s a m o n g w h i c h t h e s y s t e m e v o l v e s d u r i n gth e ma g n e t i z a t i o n p ro c e s s . A t p r e s e n t , n o o n e i s a b l e t o c a r ry o u t s u c ha n a n a ly s i s u n d e r s u f fi c ie n t ly g e n e ra l c o n d i t i o n s a n d to p ro v e o r d i s p ro v eth e a s s u m p t io n . I t i s h e r e t h a t o n e c a n mo s t r e a d i ly a p p re c i a te t h e p o w e ra n d t h e e l e ga n c e o f t h e B o l t z m a n n - G i b b s d e s c r ip t i o n o f t h e r m o d y n a m i ce q u i l i b r iu m. I f t h e s y s t e m i s i n t h e rmo d y n a mic e q u i l i b r iu m, t h e n , n om a t t e r h o w c o m p l i c a t ed t h e f re e e n e r g y l a n d s c a p e i s, w e k n o w i n a d v a n c eth a t t h e p ro b a b i l i t y t o fi n d t h e s y s t e m in w h a te v e r s t a t e is c o n t ro l l e d b y i t se n e r g y t h r o u g h t h e c o r r e s p o n d i n g B o l t z m a n n w e i g h t . C o n v e r s e l y , w h e nm e t a s t a b i li t y a n d r a t e - i n d e p e n d e n t h y s t e r e s is d o m i n a t e , t h e p r o b a b i li t yo f v i s i t i n g c e r t a in s t a t e s d e p e n d s o n t h e f i e ld h i s to ry a n d th e t o p o lo g yo f t h e f r e e e n e rg y s u r f a c e , a n d t h e g e n e ra l p r in c ip l e s s t a t i n g wh a t t h i sp r o b a b i l it y m i g h t b e a re n o t y e t k n o w n i n d e ta il .Th e h y p o th e s i s o f th e e x i s te n c e o f t h e p (AE , AF) d i s t r i b u t io n i s a na t t e m p t t o t a ck l e th i s p ro b l e m, w h ic h i s o f i n t e r e st b e c a u s e o f t h e fo l l o win gpo in ts .

    ( i ) On e o b t a in s a u n i f i e d f r a me wo rk fo r t h e d e s c r ip t i o n o f r a t e -i n d e p e n d e n t h y s t e r e s i s , e d d y -c u r r e n t e f f e c t s , a n d ma g n e t i c v i s -c o s i t y . Th i s i s c o n c e p tu a l l y imp o r t a n t , b e c a u s e i t p e rmi t s o n eto a p p re c i a t e wh ic h a r e t h e g e n e ra l p h y s i c a l c o n n e c t io n s t o b ee x p e c t e d b e tw e e n th e s e v a r io u s a s p e c t s o f h y s t e re s i s. Th i s p o in tw i l l b e e l a b o ra t e d i n s o m e d e t a i l i n t h e n e x t c h a p t e r.( i i ) Qu a l i t a t i v e ly n e w p ro p e r t i e s e me rg e i n t h e d e s c r ip t i o n o f t h ema c ro s c o p ic h y s t e r e t ic b e h a v io r o f t h e s y s t e m. In p a r ti c u l a r, tw oo f t h e m , k n o w n a s t h e wiping-out (o r d e l e t i o n , o r r e tu rn -p o in tm e m o r y ) p r o p e r ty a n d t h e congruencyp r o p e rt y , p l a y a f u n d a m e n -ta l ro le , d i scu ssed in Sec t ion 13 .3 . The i r imp or tan ce l ie s in thef a c t t h a t , t o a c c e p t o r d i s c a rd t h e i n i t i a l wo rk in g h y p o th e s i s ,i t i s n o t n e c e s s a ry t o g o t h ro u g h th e d e t a i l e d a n a ly s i s o f t h em i c r o m a g n e t i c p r o b l e m , a s p r e v i o u s l y s u g g e s t e d . W i p i n g - o u ta n d c o n g ru e n c y a r e th e m a c ro s c o p ic f in g e rp r in t o f th e e x i s te n c eof the p (AE , AF) d i s t r ibu t ion . I f these p rop er t ie s a re ob eye d byth e s y s t e m, t h e n o n e k n o ws fo r s u r e t h a t t h e d i s t r i b u t io n e x i s t s ,e v e n i f o n e i s n o t a b l e t o p r e d i c t i t f ro m mic ro s c o p ic c o n s id e r a -t io n s . Th e s i t u a t io n h a s s o m e r e s e m b la n c e t o c la s si c al th e rm o d y -n a mic s , wh e re c e r t a in g e n e ra l c o n c lu s io n s o n t h e e x i s t e n c e o ft h e r m o d y n a m i c p o t e n ti a ls a re d e d u c e d f r o m p u r e l y m a c r o s co p i c

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    13.1 THE PREISACH APPROACH 435p r o p e r t i e s o f t h e r m o d y n a m i c s y s te m s , l e a v i n g t o s ta t is ti c al m e -c h a n i c s t h e t a s k o f p r o v i d i n g e x p l ic i t e x p r e s s i o n s f o r t h e s e p o t e n -t i a ls o n t h e b a s i s o f m i c r o s c o p i c m o d e l s . I n S e c t i o n 1 3 .4 , w e s h a l ld i s cu s s a p a r t i c u la r c a se , t h a t o f d o m a i n w a l l m o t i o n i n a r a n d o mp o t e n t i a l , w h e r e t h e s e g e n e r a l c o n s i d e r a t i o n s c a n b e t r a n s l a t e di n t o d e f i n i t e q u a n t i t a t i v e p r e d i c t i o n s .

    (iii) T h e c o m p a r i s o n w i t h e x p e r i m e n t s s h o w s t h a t t h e a p p r o a c h is a b l et o s a t i s f a c t o r i l y r e p r o d u c e m a n y f e a t u r e s o f o b s e r v e d h y s t e r e s i sp h e n o m e n a , a n d y i e ld s u s e f u l p h e n o m e n o l o g i c a l d e s c r i p ti o n s o fm a n y s i t u a t i o n s o f i n t e r e s t t o a p p l i c a t i o n s .

    1 3 .1 T H E P R E I S A C H A P P R O A C HI n t h i s s e c t i o n , w e r e c o n s i d e r t h e b i s t a b l e s y s t e m s d i s c u s s e d i n C h a p t e r2 a n d w e u s e t h e m a s b u i l d i n g b l o c k s f o r t h e g e n e r a l c h a r a c t e r i z a t i o n o fh y s t e re s i s . T h e i d e a o f d e s c r i b i n g a g e n e r i c h y s t e r e t ic s y s t e m a s t h e s u p e r -p o s i t i o n o f m a n y b i s t a b l e u n i t s i s i n e v i t a b l y a s s o c i a t e d w i t h t h e n a m e o fF . P r e i s a c h , w h o g a v e t h i s i d e a a n e l e g a n t a n d i l l u m i n a t i n g g r a p h i c a lr e p r e s e n t a t i o n . W e s h a l l d i s c u s s h o w t h e P r e i s a c h a p p r o a c h f i t s i n w i t ht h e g e n e r a l i n t e r p r e t a t i o n o f t h e m a g n e t i z a t i o n p r o c e s s a s a s e q u e n c e o fB a r k h a u s e n i n s ta b i li ti e s, a n d o n t h i s b a s i s w e s h a l l i n t r o d u c e t h e Preisachdistr ibution, w h i c h r e p r e s e n t s i n m o r e p r e c i s e t e r m s t h e p (A E , A F) d i s t r ib u -t i o n m e n t i o n e d a t th e b e g i n n i n g o f t h i s c h a p t e r. T h e t r e a t m e n t w i l l c o n c e r ns c a la r q u a n t i t i e s o n ly . W e i m a g i n e t h a t t h e m a g n e t i c f i el d is a p p l i e d a l o n ga fi x ed d i r e ct io n , a n d w e c o n c e n t r a te o n t h e c o m p o n e n t o f t h e m a g n e t i cm o m e n t o f t h e s y s t e m a l o n g t h e s a m e d i r e c t i o n . I n a d d i t i o n , w e w i l li g n o r e t h e p o s s i b l e e x i s t e n c e o f r a t e - d e p e n d e n t d i s s i p a t i o n m e c h a n i s m sa s w e l l a s o f t h e r m a l r e l a x a t i o n p h e n o m e n a . T h i s m e a n s t h a t w e a r ei n t h e z e r o - t e m p e r a t u r e l i m i t o f r a t e - i n d e p e n d e n t h y s t e r e s i s . T h e r m a lr e l a x a t i o n w i l l b e d i s c u s s e d i n S e c t i o n 1 4 . 2 .

    1 3 .1 .1 P r e i s a c h u n i t sT h e s i m p l e s t e n e r g y p r o f il e g i v i n g a c le a r re p r e s e n t a t i o n o f a B a r k h a u s e ni n s t a b i l i t y i s s h o w n i n F i g . 1 3 . 1 . W e s h a l l u s e t h e t e r m Preisach uni t tor e fe r to a s y s t e m c h a r a c t e r i z e d b y a n e n e r g y p r o f il e o f t h i s k in d . T h e s t a tev a r i a b l e m r e p r e s e n t s s o m e c h a r a c t e r i s t i c m a g n e t i c m o m e n t i n v o l v e d i nt h e p r o b l e m , t o b e i d e n t i f i e d c a s e b y c a s e . T h e s y s t e m i s c o n f i n e d t o t h ein te rva l - A m _< m 0 ). Th e f r ee ene rg y o f the u n i t , Fp (m) ,

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    436 CH APT ER 13 Col lec tions of B is tab le Uni t s

    H = O

    H > O

    ~ o h ~ m ~ I ~t ~ o h u A m =

    . . . . . . . . . . . . . . -laoh~Am

    l~oHAmi II~ T ~ II / // i/ i /I~oHA I'"

    m, |, |

    5 :,- Am 0 AmFIGURE 13 .1 . To p : Ene rgy prof i le Fp(m) charac te r iz ing a Pre isach uni t . Bot tom:c o r r e spond ing G ibbs e ne r gy Gp(m;H), as def ined by Eq. (13.1) .

    i s d e t e r m i n e d b y it s v a l u e s a t t h e p o i n t s - A m , 0 , + A m . It is c o n v e n i e n tt o e x p r e s s t h e s e v a l u e s i n t h e f o r m - p , o h u A m , i zo h c A m , i z o h u A m , w h e r e h ua n d h e a r e a p p r o p r i a t e f i e l d s . T h e t h r e e p o i n t s a r e c o n n e c t e d b y s t r a i g h ts e g m e n t s , w h i c h g i v e s t h e t w o c h a r a c t e r i s t i c e n e r g y g r a d i e n t s , hu - hef o r m > 0 , a n d h u + h e f o r m < 0 . F i n a ll y , w e a s s u m e t h a t t h e u n i t isc o u p l e d l i n e a rl y t o th e m a g n e t i c f i e ld H , i n th e s e n s e t h a t t h e s t a b i li t y o ft h e u n i t i s c o n t r o l l e d b y t h e G i b b s e n e r g y

    G p ( m ; H ) = F p ( m ) - p , o H m (13.1)T h e p r e s e n c e o f t h e f i e ld g i v e s r is e t o a G p p r o f il e o f t h e s a m e f o r m a st h e F p o n e , w i t h (h u - H ) i n t h e p l a c e o f h u ( s e e F i g . 1 3 . 1 ) . O t h e r m o r er e al is t ic e n e r g y p r o f i l es m i g h t b e c o n s i d e r e d , p r o v i d e d t h e d i s t in c t i v ef e a t u r e s o f t h e d e s c r i p t i o n a r e p r e s e r v e d , n a m e l y 1 :

    1These two conditions are n ot indepen dent if one interprets a Preisach unit as a two-levelsystem with energy levels +AG,,, and considers its thermody nam ic equilibrium properties.The situation is p erfectly analog ous to spin-lA paramag netism, discu ssed in Section 5.1.1.Sim ilarly to Eq. (5.7), one has Z = 2 ch(hGu/kBT). The average mom ent of the un it is then

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    1 3.1 T H E P R E I SA C H A P P R O A C H 4 3 7

    (i) t h e p o s i t i o n , + A m , o f t h e t w o e n e r g y m i n i m a i s i n d e p e n d e n t o ft h e f i e l d H ;

    (ii) t h e h e i g h t o f t h e e n e r g y b a r r i e rs s e p a r a t i n g t h e t w o m i n i m a d e -p e n d s l in e a r ly o n H .

    D e p e n d i n g o n th e re l a ti v e v a l u e s o f h u , h c, a n d H , G p ( m ; H ) m a y e x h ib i to n e o r t w o l o c a l m i n i m a . W h e n H >-- h u + h e, o n l y on e m i n i m u m e xis ts ,a t m = + A m . A n a l o g o u s l y , w h e n H 0 . W e s h a l l n o t c o n s i d e r t h e c a s e h e 0 i n t o t h e t h r e e r e g i o n s s h o w n i n F i g. 13 .2 .

    F I G U R E 1 3 . 2 . P re is ach p l an e w i t h i n d i ca t io n o f r eg i o n s wh e re t h e en e rg y G p ( m ; H )o f F ig . 1 3 . 1 (Eq . 1 3 .1 ) h a s o n e o r t w o m i n i m a . T h e b o u n d a ry o f r eg io n I I is t h eb i fu rca t i o n s e t wh e re B a rk h au s en j u m p s m ay o ccu r . C o m p are w i t h F i g . 2 . 9 ./z0 = - k B T 3 1 n ZI OH = - ( 3A G u lO H ) t h ( A G u / k B T ). Th erefo re, /z0Am = - 3 AGulOHwhich sh ow s that h G u is a line ar function of H if A m is constant, that is, hG u = tao(hu -H)h m, in agree me nt with E q. (13.1).

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    438 CH APT ER 13 Collections of B istable Un its

    R eg io n h hu >- H + h c . I f t h e r e p r e s e n t a t i v e p o i n t l i e s i n t h is r e g i o n , t h eu n i t i s c e r t a i n ly in t h e s ta t e m = - A m , w h i c h w e c a ll t h e ( - ) s ta t e.R eg io n I I: H - hc < hu < H + h c . I f t h e r e p r e s e n t a t i v e p o i n t l i e s i n t h i sr e g i o n , b o t h t h e s ta t e s m = - A m a n d m = + A m a r e l o c a ll y s ta b le . W h i c hs t a t e t h e u n i t o c c u p i e s w i l l d e p e n d o n p a s t h i s t o r y .R eg io n I I I: h u

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    13.1 THE PREISACH APPROAC H 439j u m p o c c u rs , th e e n e r g y G p ( m ; H ) s u d d e n l y d e c r e a se s b y 2 / ~ o h c A m . T hi si s t h e a m o u n t o f e n e r g y d i s s i p a t e d a s h e a t i n t h e j u m p . O n t h e o t h e rh a n d , a t t h e j u m p t h e f r e e e n e r g y F p ( m ) c h a n g e s b y t h e a m o u n t A F p =2 p , o h u A m . W e see t ha t h e a n d h u a r e app r op r i a t e coo r d i na t e s t o keep t r acko f d i s s i pa t ed and s t o r ed ene r gy . I n s ubs equen t s ec t i ons , w e s ha l l s ome-t i mes u s e t he s qua r e l oop s w i t ch i ng f i e l d s ,

    = hu + hc (13.2)/3 = h u - hr

    i n s t e a d o f h c a n d h u , as a l t e r na t i ve coo r d i na t e s i n t he P r e i s ach p l ane .I n t he l i t e r a t u r e on t he P r e i s ach mode l , t he s t a r t i ng po i n t i s u s ua l l yt he i dea t ha t a s y s t em s hou l d be des c r i bed by a co l l ec t i on o f e l emen t a r ys qua r e l oops o f t he k i nd s ho w n i n F ig . 13 .3 . T h i s is m a t hem a t i ca l l y app r o -

    p r i a t e , beca us e t he e l em en t a r y l oops a r e ac t ua l ly a p i c to r i a l r ep r e s en t a t i ono f e l e m e n t a r y h y s t e r e s i s o p e r a t o r s t h a t c a n b e s u p e r p o s e d , t o b u i l d u phys t e r e s i s non l i nea r i t i e s w i t h a mor e compl ex s t r uc t u r e . H ow eve r , t hel o o p a p p r o a c h m a y g i v e t h e im p r e s s io n t h a t , i n o r d e r to a p p l y t h e m e t h o dto a spec i fi c phys ic a l s i tua t ion , one n ee ds p hy s ica l en t i t i es , l ike for ins tancemagne t i c pa r t i c l e s , i nd i v i dua l l y cha r ac t e r i zed by a hys t e r e s i s l oop . T h i si s no t t he ca s e. T he bas ic phy s i ca l a s pec t o f t he ap p r oa ch i s t he i n t r oduc t i ono f t he t w o va r i ab l e s, h e a n d hu , des c r i b i ng s t o r ed ve r s us d i s s i pa t ed ene r gy .T h i s des c r i p t i on a l r eady app l i e s a t the l eve l o f a s ing l e B a r k haus e n j ump ,a n d d o e s n o t r e q u i r e t h e i n tr o d u c t i o n o f t h e i d e a th a t i n d i v i d u a l e l e m e n -t a r y hys t e r e s i s l oops s hou l d ex i s t a s s uch f r om t he beg i nn i ng .

    1 3 .1 .2 C o l l e c t i o n s o f P r e i s a c h u n i t sT he gene r a l p i c t u r e o f t he magne t i za t i on p r oces s a s a s equence o f B a r k -h a u s e n j u m p s h a s it s P r e is a c h c o u n t e r p a r t i n t h e a s s u m p t i o n t h a t a g i v e ns ys t em can be des c r i bed i n te r m s o f a co ll ec ti on o f P r e i s ach un i t s, r an do m l yd i s t r i bu t ed i n t he P r e i sach p l ane , s w i t ch i ng u nd e r t he ac t ion o f t he ex t e r na lf ie ld . T he s t ab i li ty r u l e s d i s cus s ed i n the p r ev i ous s ec t ion w i l l de t e r m i net he s t a t e o f t he va r i ous un i t s p r e s en t i n t he P r e i s ach p l ane . W e a l r eadyknow t ha t a l l po i n t s i n r eg i on I a r e ce r t a i n l y i n t he ( - ) s t a t e , and po i n t sin r eg ion I I I a re in the (+ ) s t a te . The s t a te o f the po in t s in r eg ion I I i si n s t ead a p r i o r i unde t e r mi ned . I n p r i nc i p l e , i n o r de r t o de f i ne t he g l oba ls t a t e o f t he s y s t em unde r a g i ven f i e l d H , one s hou l d s pec i f y , f o r eachun i t i n r eg i on I I, i f i t is in t he ( + ) o r ( - ) s t at e . A ny s ubd i v i s i on o f r eg i onI I i n t o ( + ) a nd ( - ) s ub r eg i ons , o f t he t ype s ho w n i n F ig . 13.4 , i s a ccep t ab le ,and r ep r e s en t s a pos s i b l e m e t a s t a b l e s t a t e f o r t he s y s tem . T he t o ta l nu m be r

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    440 CH APT ER 13 Collect ions of Bistable Units

    FIGURE 13.4 . Pre i sach p lane re presenta t ion of metas tab le s ta te in which Prei sachuni t s are gener ica l ly d i s t r ibuted over (+) or ( - ) admiss ib le s ta tes .o f d i ff e r e n t m e t a s t a b l e s t a te s is e q u a l t o 2 N, w h e r e N i s th e n u m b e r o fP r e i s a c h u n i t s p r e s e n t i n r e g i o n I I .

    W e a r e n o t g o i n g t o c o n s i d e r a l l t h e m e t a s t a b l e s t a t e s t h a t c o u l di n p r i n c i p l e e x i s t . W e a r e m a i n l y i n t e r e s t e d i n t h e c o n n e c t i o n b e t w e e nm e t a s t a b l e s t a t e s a n d f i e l d h i s t o r i e s a n d w e s h a l l t h u s f o c u s a t t e n t i o n o nt h e r o l e o f t h e f i e ld i n s e l e c t in g p a r t i c u l a r s u b c l a s s e s o f m e t a s t a b l e s t a te s .L e t u s i m a g i n e t h a t t h e f i e l d h a s i n i t i a l l y a l a r g e n e g a t i v e v a l u e . T h e n a l lu n i t s w i l l b e i n t h e i r ( - ) s t a t e . W e t h e n i n c r e a s e t h e f i e l d u p t o t h e p o i n tH = H 1 , a s s h o w n i n F i g . 1 3 . 5 . A c c o r d i n g t o t h e s t a b i l i t y r u l e s p r e v i o u s l yd i s c u s s e d , t h e m o t i o n o f t h e m e t a s t a b i l i t y c o n e o f F ig . 13 .2 is s u c h t h a ta l l t h e p o i n t s b e l o w t h e l i n e h u = H I - h c w i l l s w i t c h f r o m ( - ) t o ( + ) . I fn o w w e d e c r e a s e t h e f ie l d f r o m H I t o H 2, o t h e r s w i t c h i n g e v e n t s , t h i st i m e f r o m ( + ) t o ( - ) , w i l l o c cu r, w h i c h c a n b e d e t e r m i n e d a g a i n b yf o l l o w i n g t h e e v o l u t i o n o f th e m e t a s t a b i li t y c o n e . T h e s a m e a n a l y s is c a nb e r e p e a t e d a f t e r e a c h f i e l d r e v e r s a l . A s s h o w n i n F i g . 1 3 . 5 , t h e r e s u l t i st h a t a n y a r b i t r a r y f ie ld h i s to r y e n d i n g w i t h t h e f i el d v a l u e H w i l l g e n e r a t ea p a r t i t i o n o f t h e P r e i sa c h p l a n e i n o n l y o n e ( + ) a n d o n e ( - ) r e g i o n ,s e p a r a t e d b y a l in e , b( hc ), w h i c h o r i g i n a t e s f r o m t h e p o i n t h e = 0, h u =H , is a ll c o n t a i n e d i n t h e m e t a s t a b i l i t y c o n e o f F ig . 13 .2 , a n d i s m a d e u po f a c h a i n o f s e g m e n t s o f a l t e r n a t i n g s l o p e d b / d h c = + 1 a n d d b / d h c =- 1 . R e m a r k a b l y , t h e s h a p e o f t h e b ( h c ) l in e d o e s n o t d e p e n d o n th e d e t a i l e dt i m e b e h a v i o r o f t h e H ( t ) f i e l d h i s t o r y , b u t o n l y o n t h e s e q u e n c e o f f i e l d

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    13.1 THE PREISACH APPROACH 441

    FIGURE 13 .5. Preisach plan e represe ntation (right) of the m etastable state gener-ated b y a part icular f ie ld history (left) s tart ing from a large negat ive f ie ld.

    r e ve r s a l p o i n t s ( se e F ig . 13 . 5 ) . T h i s r e s u l t r e f le c ts t he r a t e - i n de pe nd e n tc h a r a c t e r o f th e d e s c r ip t io n . T h e s y s t e m s t o r e s i n f o r m a t i o n o n p a s t f i e ldh i s t o r y i n a s e l e c t i ve w a y . A c c o r d i ng l y , d i f f e r e n t f i e l d h i s t o r i e s m a y g i ver i s e t o a n i de n t i c a l f i na l s t a t e . T h i s a s pe c t w i l l be f u r t he r d i s c us s e d i nS e c ti o n 1 3 .3 .1 , w h e r e i ts r e la t io n t o t h e w i p i n g - o u t p r o p e r t y w i l l b e c o n s i d -e r e d .

    W e h a v e d e s c r i b e d t h e b(hc) l in e a s a c ha i n o f s e g m e n t s o f a l t e r na t i ngs l ope d b / d h c = + 1 , w i t h n o r e s t r i c ti ons a s t o t he s i z e o f t he i nd i v i d ua ls e g m e n t s o f t h e c h a in . W e c a n g e n e r a t e b(hc) l in e s t ha t c on t a i n a r b i t r a r i l yf in e m i x t u r e s o f s e g m e n t s o f a l te r n a t i n g s l o p e b y a p p l y i n g a n o s c i ll a ti n gf ie ld h i st o ry , i n w h i c h t h e m e a n v a l u e a n d t h e a m p l i t u d e o f t h e o s c il la ti o nc h a n g e s l o w l y , o v e r t i m e s m u c h l o n g e r t h a n o n e o s c i l l a t i o n p e r i o d ( s e eF i g. 13 . 6 ) . A pa r t i c u l a r l y i n t e r e s t i ng e x a m pl e i s t he demagne t i zed s ta te ,a c h i e v e d b y a p p l y i n g a f i e l d o s c i l l a t i n g a r o u n d z e r o , w i t h a m p l i t u d es l ow l y de c r e a s i ng f r om i n f i n i t y t o z e ro . T h i s h i s t o r y ge n e r a t e s a f i na l b (hc)l ine w i t h a f ine s a w t oo t h s t r uc t u r e a r b i t r a r i ly c l o s e to b ( hc ) = 0 . I n o r de rt o be a b l e t o de a l w i t h f i e l d h i s t o r i e s o f t h i s k i nd , w e s ha l l e x t e nd t hes p a c e o f a d m i s s i b l e s t at e f u n c t i o n s b(hc) t o a ll c o n t i n u o u s l in e s w i t h s lo p e- 1

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    442 CH APT ER 13 Collections of Bistable Units

    'hu g~a~ric~ b(hc)~ emagnetizedstate

    FIGURE 13 .6. States genera ted by oscillating field histories in w hich the m ea nvalue and the am pli tude of the f ie ld oscil la t ion slowly va ry in time. In par t icular ,the demagnetized state is generated by a f ie ld osci l la t ing around zero, with anampli tude slowly decreasing from inf ini ty to zero.

    FIGURE 13.7 . State of a Preisach system gene rated by an arbi trary f ie ld historyending with the f ie ld value H.t h e n t h e s t a t e o f a P r e i s a c h s y s t e m i s d e s c r i b e d b y t h e s t a t e l i n e b ( h c ) , hc- O, p o s s e s s i n g t h e f o l l o w i n g p r o p e r t i e s :

    (i ) H - h e ~ b ( hc ) ~ H + he ;( ii ) - 1

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    13.1 THE PREISACH APPROA CH 443f ro m th e s u m o f t h e c o n t r i b u t io n s c o m in g f ro m a ll t h e u n i t s o f th e c o l le c -t ion . Le t us cons ide r the s ta t i s t i ca l d i s t r ibu t ion o f the un i t r ep resen ta t ivep o in t s i n t h e P re i s a c h p l a n e . L e t n(h c, hu) dhcdhu b e t h e n u m b e r o f P r e is a c hu n i t s p r e s e n t i n t h e s m a l l e l e m e n t dhcdhu, n o r m a l i z e d a s

    f =h c _~on(hc,hu) dh u = N (13.3)Th e n th e a v e ra g e ma g n e t i z a t i o n o f t h e s y s t e m in t h e s t a t e s p e c i f i e d b y acer ta in b(hc) l ine w il l be

    1 J_ ~ Am(hr (13.4)= ~ d h c- fb(~c)Am(hc, hu)n(hc, h. ) dhu ]

    w h e r e V is t h e s y s t e m v o l u m e . B e c au s e th i s e q u a t i o n i n v o l v e s t h e p r o d u c tAm(hc,hu)n(hc, h,) on ly , w e c an recas t the fo rm ula t io n in te rm s o f the e f fec -t iv e d i s t r i b u t io n

    p (hc,hu) = Am(h c'hu )n(hc'h u) dhr _ooP hc,hu) dh u = 1N ( a m )

    w h e r e < A m > is t h e a v e r a g e m a g n e t ic m o m e n t o f a s in g l e un i t,l f ~ f ~Am} = ~ dhc Am(hc,hu)n(hc, hu) dh ,

    W e o b ta in

    (13.5)

    (13.6)

    s l r s + ]= M s dh c p (hc,hu) dhu - p (hc,hu) dhu0 L - -oo b(hc) (13.7)w h e r eN { A m )Ms - V (13.8)

    Eq u a t io n (1 3 .7 ) s u m m a r i z e s t h e fu n d a m e n ta l a s p e c t s o f t h e P re i sa c h a p -p r o a c h . T h e e q u a t i o n i n v o l v e s t w o f u n c t i o n s , t h e b o u n d a r y l i n e b(hc),wh ic h id e n t i f i e s t h e s t a t e o f t h e s y s t e m, a n d th e fu n c t io n p(hc,hu), w h i c hs u mma r i z e s a l l t h e mic ro s t ru c tu r a l a n d ma g n e t i c f e a tu r e s a f f e c t i n g t h ema g n e t i z a t i o n p ro c e s s . Th e e q u a t io n r e p re s e n t s t h e c o n s t i t u t i v e l a w o fth e ma te r i a l . Th e c o n s t i t u t i v e l a w i s h i s to ry -d e p e n d e n t t h ro u g h th e s t a t eline b(hc), w h i c h i s d e t e r m i n e d b y t h e f i e l d h i s t o r y t h r o u g h t h e r u l e sd i s c u s s e d i n S e c t io n 1 3 .1 .2 . Ac c o rd in g ly , t h e d e p e n d e n c e o f M o n H wi l l

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    444 CH APTE R 13 Co llect ions of Bistable Unitsd i s p l a y h y s t e r e s is . N o t e t h a t t h i s d e p e n d e n c e is c h a r a c te r i z e d b y nonlocalm e m o r y ( S e c ti on 2 .1 .3 ), be c a us e t h e f ie l d a nd m a g ne t i z a t i on va l u e s a r e no ts u f f ic i e n t t o i d e n t if y t h e s t a t e o f t h e s y s t e m . T h e k n o w l e d g e o f t h e e n t i r eb(hc) l i ne i s necessa ry .T h e f u n c t i o n p(hc,hu) i s c a l l ed t he Preisach distribution. I t e xp r e s s e s i np r e c is e t e r m s t h e p(AE,AF) d i s tr i b u ti o n m e n t i o n e d a t t h e b e g i n n i n g o f t h ec ha p t e r , a s h e an d h u a r e p r o p o r t i o n a l i n d e e d t o t h e e n e r g y d i s s i p a t e d o rs t o re d i n t h e s y s t e m a f te r a B a r k h a u s e n ju m p . T h e w a y w e h a v e i n t r o d u c e dt h e d i s t r i b u t i o n m a k e s c le a r th e d r a s ti c s i m p l if i c a ti o n o n w h i c h i t is b a s e d .I n f ac t, i m a g i n e a n i d e a l e x p e r i m e n t , i n w h i c h w e a r e a b le t o m e a s u r e a llt h e d e t a i ls o f t h e e n e r g y v a r i a t i o n s o c c u r r i n g a l o n g a c er t ai n m a g n e t i z a t i o nc u r v e , a n d f r o m t h i s w e r e c o n s t r u c t t h e P r e i s a c h d i s t r i b u t i o n a s s o c i a t e dw i t h t h a t c u r v e . I t m i g h t s e e m t h a t , if w e r e p e a t t h e e x p e r i m e n t f o rd i f f e r e n t m a g n e t i z a t i o n c u r v e s a n d d i f f e r e n t f i e l d h i s t o r i e s , w e s h o u l do b t a i n a d i ff e r e n t d i s t r i b u t i o n e a c h t im e . T h e P r e is a c h a p p r o a c h a m o u n t st o a s s u m i n g t h a t t h i s i s n o t t h e c a s e , a n d t h a t w e s h a l l o b t a i n t h e s a m ed i s t r i b u t i o n e a c h t i m e . I n g e n e r a l , t h e P r e i s a c h a p p r o a c h i s e x p e c t e d t or e p r e s e n t a n a p p r o x i m a t e , n o t r i g o r o u s , d e s c r i p t i o n o f r e al sy s t e m s , a n di t is o f b a s i c im p o r t a n c e t o d e v e l o p c r it e ri a b y w h i c h o n e c a n e s t i m a t eh o w a c c u r a t e t h e a p p r o x i m a t i o n m a y b e i n s p ec if ic c as es . T h i s a s p e c t w i l lbe d i s c u s s e d i n S e c t ion 13 .3 , S e c t ion 13 .4 , a n d S e c t ion 14.1 . F o r t he m om e n t ,w e s h a l l s i m p l y a s s u m e t h a t t h e P r e i s a c h d e s c r i p t i o n i s a c c e p t a b l e a n dt h a t a P r e i s ac h d i s t r i b u t i o n h a s b e e n i d e n t if ie d .O n c e p(hc,hu) is k n o w n , w e c a n u s e E q . ( 13 . 7) t o c a lc u l at e t h e m a g -n e t i z a t i o n o f th e s y s t e m c o n s e q u e n t t o w h a t e v e r f i e ld h is to r y . S o m e q u a l i-t a ti v e a s p e c t s o f t h e r e s u l t w i l l d e p e n d o n t h e f o l l o w i n g g e n e r a l p r o p e r t i e so f p(hc,hu).Symmetry proper t i es . F e r r o m a g n e t i c m a t e r i a l s a re u s u a l l y c h a r a c t e r iz e d b yt h e f a c t t h a t , g i v e n a g e n e r i c i n p u t - o u t p u t h i s t o r y {H( t ) ,M ( t) } ,{ - H ( t ) , - M ( t ) } i s a l s o a n a d m i s s i b l e h i s t o r y f o r t h e s y s t e m . W h e n t h i s i st he ca se , p(hc,hu) i s a n e v e n f u n c t i o n o f h u,

    P (hc,hu) = p (hc, - hu) (13.9)a n d E q . (1 3 .7 ) c a n b e w r i t t e n i n t h e s i m p l e r f o r m

    fbo(hC)M = 2 M s f dh c p (h c,h u) dh u0(13.10)

    I n t h i s c a s e , a l i m i t e d po r t i on o f t he P r e i s a c h p l a ne c on t r i bu t e s t o t hev a l u e o f M (F ig . 1 3 .8 ). W e s h a l l a s s u m e t h a t t h e s y m m e t r y e x p r e s s e d b yE q . ( 13 . 9 ) i s a l w a y s obe ye d .

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    13.1 THE PREISACH APPROACH 445

    FIGURE 13 .8 Regions of integra tion con tributing to Eq. (13.7) w h en p(hc,hu) hasthe s ym m etry of Eq. (13.9), w ith s ign of the co ntribution. The result is Eq. (13.10).Factorization properties. The P r e i s a c h de s c r i p t i on ge t s g r e a t l y s i m p l i f i e dw h e n e v e r t h e P r e i s a c h d i s t r ib u t i o n s p l it s in t o t h e p r o d u c t o f fa c to r s. O fa ll th e f a c t o r iz a t io n s c o m p a t i b l e w i t h t h e s y m m e t r y o f Eq . ( 13 .9 ), t w o a r eo f pa r t i c u l a r i n t e r e s t . I n one o f t he m , p(hc,hu) t a k e s t h e f o r m

    p(hc,hu) = f ( ~ ) f ( - ~ ) = f(hc + hu)f(hc- hu) (13.11)w he r e a a n d 1~ a r e t he s w i t c h i ng f i el d s de f i ne d by E q . ( 13. 2) . T h i s f a c to r i z a -t i o n s h o u l d h o l d w h e n t h e d i s t r i b u t i o n f(x) of t he i ns t ab i l i t y f i e l ds a tw h i c h B a r k h a u s e n j u m p s o c c u r i s t h e i m p o r t a n t q u a n t i t y , a n d t h e r e i ss ta t is ti c al i n d e p e n d e n c e b e t w e e n t h e f ie ld s a t w h i c h t h e ju m p s t a k e p la c e ,i n a g i v e n v o l u m e o f t h e m a t e r i a l , u n d e r i n c r e a s i n g o r d e c r e a s i n g f i e l d .I n t h e p a r t ic u l a r c a se w h e r e f(x) i s L o r e n t z i a n , one ob t a i n s t he d i s t r i bu t i on

    2p(hc, hu) = (13.12)l r ~ H 0 ~- + a r c t g

    1

    I n t h e o t h e r o n e , p(h~,hu) t a k e s t h e f o r mp(hc,hu) = f(hc)g(hu) (13.13)

    w h e r e g(hu) i s a n e v e n f un c t i on o f hu. T h i s t yp e o f f a c t o r i z a ti on i s e xpe c t e di n c as e s w h e r e w e h a v e i n d i v i d u a l p h y s i c a l u n i ts , li ke m a g n e t i c p a r ti c le s ,a n d w e a t t e m p t a d e s c r i p t i o n o f a n a s s e m b l y o f s u c h u n i t s . I n t h i s c a s e ,t h e u n i ts , w h e n c o n s i d e r e d o n e a t a t i m e s e p a r a t e d f r o m t h e o t h e r s , p o s s e s sa s y m m e t r i c l o o p , a n d t h e f u n c t i o n f(h~) i s r e l a t e d t o t he d i s t r i bu t i on o ft he c oe r c i v i t i e s o f i nd i v i dua l un i t s . I n t e r a c t i on e f f e c t s a r e de s c r i be d by

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    446 CH APTE R 13 Col lec t ions of B is tab le Uni t s

    g(hu). A G a u s s i a n f u n c t io n i s o f te n a g o o d a p p r o x i m a t i o n t o g(hu). Th ef a c t o r i z a t i o n i n t r o d u c e d i n E q . ( 1 3 . 1 3 ) a m o u n t s t o a s s u m i n g t h a t t h e r e i sn o i n t e r f e r e n c e b e t w e e n l o c a l c o e r c i v i t y a n d i n t e r a c t i o n e f fe c ts . A l o g n o r -m a l flhc) a n d a G a u s s i a n g(hu) l e a d t o a P r e i s a c h d i s t r i b u t i o n o f t h e f o r m] _ n 2 ( h c / H o ) ] (1 H ~ e x p - e x p -P (hc'hu ) 2r H2 h---~ 2 ~ J 2Cr2uH2J (13 .14)A d i s t r i b u t i o n t h a t is a l so o f te n e m p l o y e d i s a G a u s s i a n - G a u s s i a n d i s t r i b u -t i o n o f t h e f o r m

    1 2p(hc'hu) = 2r H2 e r f ( c r c V ~ ) + 1 ( 13 .15 )

    e x p [ - 0 2 ] e x p ;2 ~ H 2 2 ~ H 2Jw h e r e e rf (x ) i s t h e e r r o r f u n c t i o n . 2 F i g u r e s 1 3 .9 a n d 1 3 .1 0 g i v e a g r a p h i c a lr e p r e s e n t a t i o n o f t h e m a i n d i f f e re n c e s b e t w e e n E q . ( 13 .1 2 ) a n d E q . ( 1 3. 14 ).Rev ersible contributions. W e m e n t i o n e d a t t h e b e g i n n i n g o f t h e s e ct io n t h a tt h e re m a y b e P r e is a c h u n i t s th a t d i s p l a y o n e e n e r g y m i n i m u m u n d e r a llc i r c u m s t a n c e s . U n i t s o f t h is k i n d g i v e a c o n t r i b u t i o n t o th e m a g n e t i z a t i o nt h a t i s a s i n g l e - v a l u e d f u n c t i o n o f t h e f ie l d , M r e v ( H ) . T h i s f u n c t i o n d e -s c r i b e s r e v e r s i b l e p r o c e s s e s , i n t h e s e n s e t h a t t h e y a r e i n d e p e n d e n t o f

    hu/H0 20

    0.2

    0hc / H0

    hu/H0 2

    0.2P0.1

    03 4 hc/HO 3 4

    FIGURE 13 .9. Ex am ple of Pre isach distr ibut io ns. Lef t: Facto r ized-Lo rentz ian dis-tr ib ut io n (Eq. 13.12), crC = 1 . Right: Lo gno rm al-Gau ss ian d is t r ibu t io n (Eq. 13 .14),~rc = 1, ~ru = 1.f x2 e x p ( - - U 2 ) du.T h e a s s u m e d n o r m a l i z a t i o n i s e r f (x ) = ~ 0

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    13 .1 THE PREISACH APPROA CH 447i 3I

    20 1

    - , . . . . . - 3 /

    0 1 2 3 4 0 1 2 3 4hc / H0 hc / H0

    FIGURE 13.10 . C on tou r plot of Preisach distributio ns of Fig. 13.9. Left: Factorized-Loren tzian distrib ution (Eq. 13.12), crC= 1. Right : Lognorm al-Gaussian d istr ibut ion(Eq . 13.14 ), cr -- 1, cru = 1.

    f i e l d h i s t o r y , a n d c a n s i m p l y b e a d d e d t o E q . ( 1 3 . 1 0 ) . A c o n v e n i e n t w a yt o r e p r e s e n t c o n t r i b u t i o n s o f t h i s n a t u r e i s t o i n c l u d e i n t h e P r e i s a c hd i s t r i b u t i o n a s i n g u l a r p a r t c o n c e n t r a t e d a l o n g t h e h e = 0 axis :

    p(hc,hu) = Pirr(hc,hu) + Prev(hu)8(hc) (13.16)where Pirr i s a s s o c i a t e d w i t h t h e b i s t a b l e P r e i s a c h u n i t s s o f a r d i s c u s s e da n d Prev(hu) is a n e v e n f u n c t i o n o f h u, p r o p o r t i o n a l t o t h e r e v e rs i b le s u s c e p -t i b il i ty Xrev = d M r e v / d H . W h e n t h e P r e i s a c h d i s t r i b u t i o n h a s t h e s t r u c t u r eof Eq . (13 .16) , Eq . (13 .10) becomes

    M = 2M s f~dhc "b(hc) h h " HJ0 Pirr( c , u )dhu + M s fo Prev(hu) dhu (13.17)I n c a r r y i n g o u t t h e i n t e g r a t i o n o v e r t h e r e v e r s i b l e c o n t r i b u t i o n , w e h a v ea s s u m e d t h a t t h e 8(he) s i n g u l a r i t y , b e i n g l o c a t e d a t t h e b o u n d a r y o f t h er eg io n o f i n t e g r a t io n , co n t r ib u te s 189 to th e in t eg r a l o v e r h e. Th e s e c o n dt e r m o f E q . (1 3.1 7) r e p r e s e n t s t h e r e v e r s i b l e m a g n e t i z a t i o n Mrev(H). W ed i s c u s s e d i n S e c t i o n 1 0 . 1 . 2 t h a t , i n g e n e r a l , t h e r e v e r s i b l e s u s c e p t i b i l i t yX rev r e s u l t s f r o m e l a s t i c d i s t o r t i o n s o f t h e d o m a i n s t r u c t u r e a r o u n d t h es t a b l e s t a t e o c c u p i e d a t a p a r t i c u l a r m o m e n t , a n d a s s u c h i t i s e x p e c t e dt o b e s t a t e - d e p e n d e n t . I n th e s p i r i t o f t h e a p p r o a c h d e v e l o p e d s o f ar, t h eq u a n t i t y c h a r a c t e r i z i n g t h e s t a te o f t h e s y s t e m is th e b(hc) l i n e , so wee x p e c t t h a t X rev a n d Mrev s h o u l d b e e x p r e s s i b l e a s f u n c t io n a l s o f t h e b(hc)

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    448 CH APT ER 13 Co llection s of Bistable Unitsfunc t ion . In th i s re spec t , t he desc r ip t ion g iven by Eq . (13 .17) i s a f i r s ta p p r o x i m a t i o n , d e p e n d e n t o n H -~ b(h c = 0 ) on ly . Genera l i za t ions invo lv -i n g t h e w h o l e b(hc) l i ne a re poss ib l e , bu t wi l l no t be cons ide red . Note , a sa f i n a l r e m a rk , t h a t t h e s i n g u l a r p a r t o f Eq . (1 3 .1 6 ) m a y s i m p l y r e p re se n ta c o n v e n i e n t a p p ro x i m a t i o n t o a d i s t r i b u t i o n o f b i s t a b le u n i t s w i t h ad i v e r g e n c e f o r h e --~ O. I n o t h e r w o r d s , l a r g e p o p u l a t i o n s o f v e r y s m a l le n e r g y b a r r i e r s b e h a v e i n a q u a s i - r e v e r s i b l e w a y t h a t m a y b e h a r d l yd i s t i n g u i sh a b l e f ro m a p u re l y r e v e r s i b le b e h a v i o r , i f t h e e x t e rn a l f i el d isn o t sm a l l e n o u g h t o r e so l v e f i n e m a g n e t i z a t i o n d e t a i l s .Field and temperature dependence. A c c o rd i n g t o t h e p a t h fo l l o w e d t o i n t ro -d u c e p(hc,hu), t h is f u n c ti o n s h o u l d b e f ie l d - in d e p e n d e n t , a l t h o u g h h a v i n gso m e e x p l ic i t f i e l d -d e p e n d e n c e m a y b e u se fu l t o d e a l w i t h e n e rg y b a r r i e r st h a t a r e n o n l i n e a r fu n c t i o n s o f t h e f i e l d . O n t h e o t h e r h a n d , p(hc,hu) isc e r ta i n ly t e m p e r a t u r e - d e p e n d e n t , b e c a u s e t h e e n e r g y o f m e t a s t a b le s t at e sa n d t h u s t h e e n e rg y p ro fi le s o f t h e i n d i v i d u a l P re i s a c h u n i ts w i l l e v e n t u -a l ly b e d e t e r m i n e d b y e x c h a n g e , a n i so t ro p y , o r m a g n e t o s t a t i c e n e rg y c o n -t r i b u t io n s , e a c h o f w h i c h h a s a d e fi n it e , d i s ti n c t d e p e n d e n c e o nt e m p e ra t u re . A s a g e n e ra l r u l e , p(hc,hu) i s e x p e c t e d t o sh r i n k a s a w h o l ea s th e C u r i e p o i n t is a p p ro a c h e d . Th e se a sp e c t s w i l l n o t b e d i s c u s se d a n yfu r t h e r . W e sh a l l f o c u s o n t r a n s fo rm a t i o n s a t c o n s t a n t t e m p e ra t u re o n l y .

    F i g u r e 1 3 .1 1 s h o w s s o m e m a g n e t i z a t i o n c u r v e s c a l c u la t e d t h r o u g hEq . (13 .10) , when the Pre i sach d i s t r ibu t ion i s g iven by Eq . (13 .12) o r Eq .(13 .14) . In each ca l cu la t ion , t he sys t em i s i n i t i a l ly in the demagne t i zeds t a t e , d e f i n e d b y b(hc) = 0 . A n osc i l l a ting f ie ld o f i nc reas ing am pl i tu de i st h e n a p p l i e d , a n d t h e ru l e s d i s c u s se d i n S e c t i o n 1 3 .1 .2 a r e u se d t o d e t e r -m i n e h o w t h e b(hc) s t a t e l i ne i s mod i f i ed by the f i e ld h i s to ry . Once b(hc)i s k n o w n , i t c a n b e i n se r t e d i n t o Eq . (1 3 .10 ) , t o c a l cu l a te t h e c o r r e sp o n d i n gm a g n e t i z a t i o n .

    1 3.2 E N E R G E T I C A S P E C T ST h e f u n d a m e n t a l o u t c o m e o f t h e P r e is a c h a p p r o a c h d e v e l o p e d i n t h eprev ious sec t ion i s Eq . (13 .10) . Th i s equa t ion rep resen t s t he cons t i t u t ivel a w o f a n y e l e m e n t a r y v o l u m e o f m a t e r i a l l a rg e e n o u g h t o g i v e ri se t om a n y B a rk h a u se n ju m p s , o r , e q u i v a l e n tl y , to c o n t a i n m a n y P re i s a c h u n i ts .Th e c o n s t i t u t i v e l a w d e sc r i b e s a n i n h e re n t l y o u t -o f - e q u i l i b r i u m s i t u a t i o nb e c a u se m a n y s t a t e s , t h a t i s m a n y d i f f e r e n t b(hc) l in e s , m a y b e a s so c i a t e dw i t h t h e s a m e f i e l d v a l u e , w h e r e a s t h e t h e r m o d y n a m i c e q u i l i b r i u m s t a t eu n d e r g i v e n f i e l d a n d t e m p e r a t u r e i s u n i q u e . F r o m t h e t h e r m o d y n a m i c

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    13 .2 ENERGE TIC ASPECTS 449

    1 . 0

    0 . 5

    ~ 0 . 0-0.5-I.0

    / ,_J//

    Y1.o0.5o.o

    -0.5-I .o

    L

    i . //

    - d . o - 2 ' . s 0 . 0 2 1 5 5 1 0H / H o

    FIGURE 13.11. Hy steresis loops an d ini tia l mag netizat ion curve calculated f romEq. (13.10). Top : Facto rized-Loren tzian distribu tion (Eq . 13.12), o" = 1. Bottom:Logn orm al-Ga ussia n distribu tion (Eq . 13.14), cr = 1, cru = 1.v i e w p o i n t , t h e f a ct th a t h y s t e r e s i s i s a l r e a d y p r e s e n t i n t h e m a t e r i a l c o n s t i-t u t i v e l a w m e a n s t h a t t h e h y p o t h e s i s o f l o ca l e q u i l i b r i u m o f n o n e q u i l i b -r i u m t h e r m o d y n a m i c s c a n n o t b e a p p l i e d ( s e e S e c t i o n 4 . 2 . 3 ) . I r r e v e r s i b l ep r o c e ss e s t a k e p la c e in e a c h e l e m e n t a r y v o l u m e , a n d w e n e e d s o m e r u l ep r e d i c t i n g h o w m u c h e n e r g y is r e v e r s i b l y s t o r e d o r i r r e v e r s ib l y d i s s i p a t e di n a g i v e n l o c a l t r a n s f o r m a t i o n . T h i s is t h e p o i n t a d d r e s s e d i n t h i s se c t io n .

    N o t e t h a t w e c o u l d d i s c u s s e n e r g y t r a n s f o r m a t i o n s a t t h e l ev e l o f t h ei n d i v i d u a l P r e is a c h u n i ts a n d t h e n t a k e a p p r o p r i a t e a v e r a ge s . W e a l re a d yr e a c h e d s o m e c o n c l u s i o n a l o n g t h i s d i r ec t io n , w h e n w e c l a r if ie d t h e p h y s i -c a l m e a n i n g o f t h e h e a n d h , f i e ld s i n S e c t io n 1 3 .1 .1 . H o w e v e r , w e p r e f e rt o c a r r y o u t t h e a n a l y s i s b y d i r e c t l y a s s u m i n g E q . ( 1 3 . 1 0 ) a s t h e s t a r t i n gp o i n t , w i t h n o f u r t h e r s p e c i f ic a t io n o f t h e s y s t e m i n t e r n a l s t r u c t u r e , ino r d e r t o s h o w h o w c e r t a i n e n e r g y r e l a t i o n s n a t u r a l l y e m e r g e f r o m t h em a t h e m a t i c a l s t ru c t u r e o f t h e a s s u m e d p h e n o m e n o l o g i c a l d e s c r ip t i o n o fh y s t e r e s i s .

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    450 CH APT ER 13 Collections of Bistable Un its1 3.2 .1 M e a n - f i e l d t e r m sTO k e e p t h e d i s c u s s i o n s i m p l e , w e s h a l l a s s u m e t h a t t h e s y s t e m i s h o m o g e -n e o u s , a n d t h a t t h e i rr e v e r si b le p r o c e s s e s g o i n g o n a r e s p a c e - i n d e p e n d e n t .A s m e n t i o n e d i n S ec ti o n 4.2 .3 , t h is r e q u i r e m e n t c a n b e o n l y a p p r o x i m a t e l yf u l f i l l e d i n a r e a l m a gne t i c m a t e r i a l , b e c a us e t he p r e s e nc e o f m a gne t i cd o m a i n s a n d o f B a r k h a u s e n i n s t ab i li ti e s in e v i t a b l y le a d s t o s p a t ia l i n h o -m o g e n e i t ie s i n th e m a g n e t i z a t io n p r o c e s s a n d i n e n e rg y d i s s ip a ti o n . H o w -e ve r, w e a s s u m e l o o k i n g a t th e p r o b l e m o n a sc ale c o ar se e n o u g h to m a k et h e s e e f f e c t s o f s e c o n d a r y i m p o r t a n c e . O n t h i s s c a l e , t h e m a g n e t i z a t i o nM o f a g i v e n e l e m e n t a r y v o l u m e z~V o f t h e b o d y is th e r e s u l t o f a n a v e r a g eo v e r m a n y d o m a i n s , w h i c h h i d e s i n t h e h y s t e r e t i c c o n s t i t u t i v e l a w , E q .( 13 . 10 ) , t he f i ne - s c a l e de t a i l s o f t he m a gne t i z a t i on p r oc e s s . A c c o r d i ng l y ,M i s n o l o n g e r a v e c t o r o f c o n s t a n t m o d u l u s a n d v a r i a b le o r i e n t a ti o n , a si n t h e m i c r o m a g n e t i c d e s c r i p t i o n , b u t r a t h e r a s c a l a r q u a n t i t y t h a t c a nt a k e a n y v a l u e f r o m p o s i t i v e t o n e g a t i v e s a t u r a t i o n .

    A s t h e p r o c e s s e s r e s p o n s i b l e f o r h y s t e r e s i s p r o c e e d , t h e f r e e e n e r g yo f t h e s y s t e m i s m o d i f i e d . L e t u s t r e a t t h e s e m o d i f i c a t i o n s b y a m e a n -f i e l d a p p r o x i m a t i o n , i n w h i c h t h e t o t a l f r e e e n e r g y i s d e c o m p o s e d i n t o am e a n - f i e l d p a r t , d e s c r i b i n g t h e l a r g e -s c a le p r o p e r t i e s o f t h e s y s t e m a n de x p r e s s ib l e a s a fu n c t i o n o f t h e m a g n e t i z a t i o n M a n d t h e t e m p e r a t u r e Ton l y , a nd a f i ne - s c a l e c on t r i bu t i on , c ons e que n t t o t he e x i s t e nc e o f t hem e t a s t a b l e s t a t e s t r uc t u r e r e s pons i b l e f o r hys t e r e s i s . T he m e a n - f i e l d f r e ee n e r g y c h a r a c t e r i z e s t h e i n f l u e n c e o f t h e r e s t o f t h e s y s t e m o n a g i v e ne l e m e n t a r y v o l u m e . I n f ac t, le t f M F (M ,T ) b e t h e m e a n - f i e ld H e l m h o l t zf re e e n e r g y p e r u n i t v o l u m e . I f n o h y s t e r e s is w e r e p r e s e n t , t h e n , a c c o r d i n gt o E q . ( 4. 42 ) , o n e w o u l d h a v e / z 0 H a = [OfMF/OM]T. T h i s e q u a l i t y c a n b ew r i t t e n i n t h e f o r m

    H a + HMF = 0 (13.18)w h e r e

    HM f(M ,T) = 1 [3/M r ] (13.19)# o k O M TE q u a t i o n ( 13 .1 8 ) e x p r e s s e s t h e b a l a n c e b e t w e e n t h e a p p l i e d f ie ld a n d t h ei n t e r n a l f o r c e s d e s c r i b e d b y t h e f r e e e n e r g y g r a d i e n t HM F. The p r e s e n c eo f h y s t e r e s i s m a k e s t h i s b a la n c e i m p e r f e c t. T h e f in e s t r u c t u r e o f m e t a s t a b l es t at e s a c c es s ib l e to t h e s y s t e m i n t r o d u c e s a d d i t i o n a l f o rc e s th a t m u s t b ei n c l u d e d i n t h e b al a n ce . E q u a t i o n ( 13 .1 8 ) t h e n b e c o m e s

    H a + HMF = H (13.20)

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    13.2 ENERGE TIC ASPECTS 451w her e t he f i e l d H r ep r e s en t s t he ef fec t ive f ie ld i nduc i ng t r ans i t i ons f r omone m e t a s t ab l e s t a te t o t he n ex t . 3 I n t he P r e i sach app r oach , H i s thef ie l d ac t i ng i n t he P r e is ach p l ane , acc o r d i ng t o t he r u l e s d i s cu s s ed i n t hep r ev i ous s ec t i on . W e s ee t ha t H does no t co i nc i de w i t h t he app l i ed f i e l dH a, bu t a l s o i nc l udes t he mean - f i e l d t e r m HMr_, des c r i b i ng t he i n f l uenceo f t h e r e st o f t h e s y s t e m o n a g i v e n e l e m e n t a r y v o l u m e .The ro le o f e f fec t ive f i e lds , as opposed to tha t o f the ex te rna l f i e ld ,w a s a l r e a d y c o n s i d e r e d i n S e c ti o n 1 0 . 3 , w h e n w e d i s c u s s e d m a g n e t i cv i s cos i t y . D ec i d i ng w h i ch pa r t o f t he t o t a l f r ee ene r gy s hou l d p l ay t her o le o f m ean - f i e l d en e r gy fM F i s no t i m m ed i a t e l y obv i ous and r equ i r e st he deve l opmen t o f s pec i f i c mode l s o f t he magne t i za t i on p r oces s . Fo rexam pl e , exchan ge an d d i po l a r i n t e r ac t ions a r e expec t ed t o g i ve a con t ri -bu t i on i n pa r t i c le a s s em bl i e s and po l yc r ys t a l l ine m a t e r ia l s , bu t t he quan t i -t a t i ve e s t i ma t e o f t he i r i mpor t ance i s d i f f i cu l t and i s o f t en l umped i nu n k n o w n p h e n o m e n o l o g ic a l m e a n -f ie ld p a r a m e t e r s to b e e v a l u a te d apos t e r i o r i by compar i s on w i t h expe r i men t s . W e s ha l l b r i e f l y d i s cus s t w ocas es t ha t a r e amenab l e t o a quan t i t a t i ve e s t i ma t e and t ha t a r e expec t edt o p l a y a ro le u n d e r m o s t c ir c u m s ta n c e s: m a g n e t o s t a t ic e n e r g y a n d d o m a i nw a l l ene r gy .M agne tos ta t i c e nergy. I f t he bo dy i s o f e l l ip s o i da l s ha pe , a nd N i s i ts dem ag-ne t i z i ng f ac t o r , t hen t he magne t os t a t i c ene r gy dens i t y , a s s umi ng un i f o r mm a g n e t i z a t i o n , i s

    #0IM F = y N M 2 ( 1 3 . 2 1 )In th i s case , HMF i s the d em ag ne t iz in g f ie ld ,

    HM F = - N M (13.22)A s s ug ge s t ed b y E q . ( 11 .58 ) , one m ay expec t t ha t d i po l a r i n t e r ac ti onsu n d e r m o r e c o m p l e x c o n d i t i o n s s h o u l d b e d e s c r i b a b l e b y a m e a n - f i e l dt e r m o f t he f o r m

    HM F = N e f f M (13.23)w h e r e N ed i s a n a p p r o p r i a t e p h e n o m e n o l o g i c a l p a r a m e t e r .Domain wal l energy . A t l o w f i e l d s , a r o u n d z e r o a v e r a g e m a g n e t i z a t i o n ,s o m e c o m p l i c a te d d o m a i n s t ru c t u r e i s u s u a l l y p r e s e n t , w h i c h e n t a il s su b -s t an t ia l w a l l ene r gy . A r o ug h e s t i ma t e o f t h i s ene r gy con t r i bu t i on c an be3 I n p r i n c i p l e , t h e n o t a t i o n Hey w o u l d b e m o r e ap p r o p r i a t e , t o r e ca l l t h e f a c t t h a t t h ef i e l d d e f i n ed b y E q . ( 1 3 . 2 0 ) i s n o t o f p u r e l y e l e c t r o m ag n e t i c n a t u r e . H o w ev e r , t o av o i du n n e c e s s a r i l y h e a v y n o t a t io n s , w e s h a l l s im p l y u s e t h e s y m b o l H w h e n e v e r t h e r e w i ll b en o r i s k o f am b i g u i t y .

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    452 CH APT ER 13 Collections of Bistable Un itsa t t e m p t e d b y c o n s i d e r i n g t h a t t h e a m o u n t o f i n te r fa c e su r f a c e p r e s e n ti n a f i n e l y - d i s p e r s e d t w o - p h a s e s y s t e m i s p r o p o r t i o n a l t o t h e c o n t a c tp r o b a b i l it y o f t h e t w o p h a s e s , o r e q u i v a l e n t l y to th e p r o d u c t o f t h e v o l u m ef r a c ti o n s o f t h e tw o p h a s e s . I f t h e t w o p h a s e s c o r r e s p o n d t o d o m a i n s o fo p p o s i t e m a g n e t i z a t i o n , t h e i r v o l u m e f r a c t i o n s a r e g i v e n b y t h e r a t i o s(M s - M ) / 2 M s a n d (M s + M ) / 2 M s, w h i c h m e a n s t h a t o n e e x p e c t s

    kw 2 ~'w (13.24)lI n E q . ( 13.24) , Y w i s t he w a l l s u r f a c e t e n s i o n a n d l i s t he e s t i m a t e d a ve r a ged o m a i n s iz e a r o u n d z e ro m a g n e t iz a t io n . T h e m e a n f ie ld a s s o ci at e d w i t hEq. (13.24) i s the n

    kwHMF "" ~ 2 M (13.25)T h e w a l l e n e r g y t e r m , E q . ( 1 3.2 4 ), f a v o r s n e a r l y s a t u r a t e d s ta t es , w i t h n od o m a i n s , c o n t r a r y t o t h e d e m a g n e t i z i n g f i e l d t e r m , E q . ( 1 3 . 2 1 ) , w h i c hi n s te a d f a v o r s s ta te s o f z e ro m a g n e t i z a ti o n , w i t h m a n y d o m a i n s . O n et h e r e f o r e e x p e c t s a c o m p e t i t i o n t h a t m a y l e a d t o a m e a n - f i e l d b e h a v i o rw i t h o n e o r tw o e n e r g y m i n i m a . A n e x t r e m e e x a m p l e , w h e r e w a l l e n e r g yd o m i n a t e s a n d m a g n e t o s t a t ic e f f ec ts h a v e a n e g l ig i b l e ro le , is t h e p i c tu r e -f r a m e e ne r gy o f F i g . 10 . 13 .

    I f t h e m e a n - f i e ld e n e r g y w e r e t h e o n l y r e l e v a n t o n e , t h e n t h e p r e s e n c eo f t w o m i n i m a i n fMFw o u l d g i v e r i s e t o s o m e m a c r o s c o p i c i n s t a b i l i t yw i t h a g l oba l m a g ne t i z a t i o n r e v e r s a l a t a c e r t a i n c r it ic a l fi el d . I n f a c t, th i sw a s t h e i d e a o n w h i c h w e b a s e d t h e e s t i m a t e s o f c o e r c i v i t y d i s c u s s e d i nS e c t i on 11 .3 .2 . A s w e s ha l l s e e, ho w e ve r , s t r uc t u r a l d i s o r de r , de s c r i be d byt h e P r e i s ac h d i s t r i b u t io n , m a y b r e a k a p a r t t h e m a c r o s c o p i c i n s t a b i li ty i n t oa r a n d o m s e q u e n c e o f s m a l l B a r k h a u s e n j u m p s . T o d is c u s s t h is p o ss ib i li ty ,w e n e e d f i r s t t o c l a r i f y t h e n a t u r e o f t h e e n e r g y c o n t r i b u t i o n t h a t m u s tb e a d d e d t o t h e m e a n - f i e l d p a r t , i n o r d e r t o t a k e a c c o u n t o f f in e - sc a leh y s t e re s is m e c h a n i s m s .

    13 . 2 . 2 F r e e e n e r g y an d e n t r op y p r od u c t i onI n o r d e r t o d i s c u s s t h e c o n s e q u e n c e s o f t h e j o i n t p r e s e n c e o f m e a n - f i e l de f fe c ts , de s c r i be d by f M r , a n d f i ne -s c a l e p r oc e s s e s , de s c r i be d by E q . ( 13 .10 ),l et u s c o n s i d e r a n i n f i n i te s i m a l tr a n s f o r m a t i o n , w h e r e M v a r i e s b y d M a t

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    13 .2 ENERGE TIC ASPECTS 453

    c o n s t a n t t e m p e r a t u r e . A c c o r d i n g t o E q . (4.5 8), th e w o r k p e r u n i t v o l u m e 4p e r f o r m e d b y e x t e r n a l s o ur c es , 81_, = ~ oH adM , l e a d s p a r t l y t o a c h a n g e i nt h e s y s t e m f r ee energy a n d p a r t l y t o a production o f en t ropy

    lzoHadM = df + TrOis (13.26)w h e r e f a n d 8 is a r e t h e f r e e e n e r g y a n d t h e e n t r o p y p r o d u c t i o n p e r u n i tv o l u m e . T h e v a r i a ti o n d M c a n b e c a l c u l a t e d f r o m E q . (1 3 .1 0 ), c o n s i d e r i n gt h a t i t w i l l c o r r e s p o n d t o s o m e v a r i a t i o n 8b(hc) o f t h e b(hc) s t a t e l i n e . Bytak in g in to acco u n t Eq . ( 1 3 .1 9) an d Eq . ( 1 3 .2 0 ) , we f in d

    H ~ dM = - H M ~ d M + H d M (13.27)= 1 [aSMF] d M + 2 M s f ~ 17 6 ] d h c11"o [ OM J T 0

    A c c o r d i n g t o t h e r u l e s d i s c u s s e d i n S e c t i o n 1 3 . 1 . 2 , a c h a n g e 8b(hc) in b(hc)c a n o n l y o cc u r w h e n b(hc) t o u c h e s t h e m e t a s t a b i l i t y c o n e o f F ig . 13 .7 , t h a tis , w h e n b(hc) = H - hc o r b(hc) = H + hc. In t h e f o r m e r c a s e , 8b(hc) > O,w h e r e a s 8b(hc) < 0 in t h e l a t te r . By ex p r e ss in g H in Eq . (1 3.2 7) a s H =b(hc) + hc, d e p e n d i n g o n t h e s i g n o f 8b(hc), w e o b t a i n

    Had M = 1 [3 f M F] d M + 2M s foo b (hc)8b (hc) p [hc ,b (hc)] dh c/.Zo L OM J T 0+ 2M s hcSb(hc) p[hc, b(hc)] dhc (13.28)o

    E q u a t io n (1 3 .2 8 ) h a s a s t r u c tu r e id en t i ca l t o Eq . (1 3.2 6) . Th e f i r s t tw o t e r m sr e p r e s e n t t h e v a r i a t i o n a t c o n s t a n t t e m p e r a t u r e o f t h e s y s t e m f re e e n e r g yf i f o n e t a k e s f o r f t h e e x p r e s s i o n

    f = SM~+ 2/a, M s dhc hu p(hc, hu) dhu (13.29)T h e t h i r d t e r m g i v e s i n s t e a d t h e e n t r o p y p r o d u c t i o n ,

    T Sis = 2 /~oM s foo h cSb (hc) P [hc,b (hc)] dh co (13.30)E q u a t i o n ( 1 3 . 2 9 ) e x p r e s s e s t h e t o t a l f r e e e n e r g y o f t h e s y s t e m . I n a d d i t i o nt o t h e m e a n - f i e l d p a r t , w e h a v e t h e c o n t r i b u t i o n c o m i n g f r o m t h e P r e is a c h

    4Equation (4.58) app lies to a ma gnetic bo dy as a w hole and involves the total magneticmo m ent m of the body. Accordingly, in the p resen t analysis M = m /V, w here V is the bod yvolume. Ho wever, given the a ssum ed hom ogeneity of the body, M is also the m agnetizationof individual elem entary volumes.

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    454 CH APT ER 13 Collections of Bistable Units

    s t r u c t u r e o f m e t a s t a b l e s ta t es . I n a g r e e m e n t w i t h t h e f a c t t h a t t h e s t a t e o ft h e s y s t e m i s i d e n t i f ie d b y t h e l i n e b(hc), t h i s t e r m i s a f u n c t i o n a l o f b(hc).E q u a t i o n ( 1 3 . 3 0 ) g i v e s i n s t e a d t h e a m o u n t o f e n e r g y d i s s i p a t e d a s h e a ti n t h e i n f i n i t e s i m a l t r a n s f o r m a t i o n . I t i s a q u a n t i t y a l w a y s p o s i t i v e , a sr e q u i r e d b y i t s p h y s i c a l m e a n i n g . I n a d d i t i o n , E q . ( 1 3 . 2 6 ) a n d E q . ( 1 3 . 2 9 )s h o w t h a t t h e i n t e g r a l o f E q . (1 3.3 0) o v e r a c y c l ic fi e ld h i s t o r y i s e q u a l t ot h e a r e a o f t h e h y s t e r e s i s l o o p .

    E q u a t i o n ( 1 3 . 2 9 ) a n d E q . ( 1 3 . 3 0 ) d e t e r m i n e i n w h a t p r o p o r t i o n t h ee x t e r n a l w o r k m a d e o n a P r e i s a c h s y s t e m i s s u b d i v i d e d i n t o s t o r e d a n dd i s s i p a t e d e n e rg y . If w e g o b a c k t o th e l e v e l o f t h e i n d i v i d u a l P r e i s a c hu n i t s , t h e s e e x p r e s s i o n s h a v e a n a t u r a l i n t e r p r e t a t io n . I n fa c t, a s d i s c u s s e din Sect ion 13 .1 .1 , 2#ohcAm g i v e s t h e e n e r g y d i s s i p a t e d b y a s i n g l e u n i tw h e n i t m a k e s a B a r k h a u s e n j u m p . E q u a t i o n ( 1 3 . 3 0 ) r e s u l t s f r o m t h es u m o f m a n y s u c h c o n t r i b u t i o n s o v e r a l l u n i t s s w i t c h i n g i n t h e g i v e nt r a n s f o r m a t i o n . A n a l o g o u s l y , 2/~ohuAm i s t h e d i f f e r e n c e i n t h e e n e r g i e sFp(+Am) o f t h e t w o e n e r g y m i n i m a a c c e s s i b l e t o a g i v e n u n i t ( F i g . 1 3 . 1 ) .B y ta k i n g i n to a c c o u n t t h e s y m m e t r y p r o p e r t i e s o f t h e P r e is a c h d i s t ri b u -t i o n ( E q. (1 3 .9 )) , o n e c a n s h o w t h a t t h e s e c o n d t e r m o f E q . (1 3.2 9) r e p r e s e n t st h e t o t a l Fp i n c r e a s e b r o u g h t a b o u t b y a l l t h o s e u n i t s t h a t o c c u p y t h e i rm e t a s t a b le m i n i m u m o f h i g h e r Fp e n e rg y . I n a g r e e m e n t w i t h t h i s i n t e r p r e -t a t i o n , f >- fM E f o r a n y s t a t e b(hc), f = fM ~ o n l y w h e n t h e s y s t e m i s i nth e demagnetized state, b(hc) = O. In th i s s t a t e , eac h un i t oc cup ies the s t a t eo f m i n i m u m Fp e n e r g y .

    1 3.2 .3 M i n i m u m e n e r g y s t a t esG i v e n t h e f re e e n e r g y f o f E q . (1 3.2 9), w e c a n i n t r o d u c e t h e c o r r e s p o n d i n gG i b b s e n e r g y s ; , , , , c l~ f -- [d,o H a M - - f M F q - 2 # o M s (hu - - H a ) p ( h c , h u ) d h u

    (13.31)A c c o r d i n g ~o E q . (1 3.2 6), w e h a v e t h a t , a t c o n s t a n t t e m p e r a t u r e ,

    dg = - # o M d H a - TSis (13.32)I t i s a l w a y s t r u e t h a t TSis >- 0, so dg ~ 0 i n a n y t r a n s f o r m a t i o n u n d e rg i v e n f ie ld a n d t e m p e r a t u r e . T h e s t a te o f t h e r m o d y n a m i c e q u i l i b r iu m ist h e s ta t e w h e r e g r e a c h e s its a b s o lu t e m i n i m u m .

    T h e e n e r g y g o f E q . ( 1 3 . 3 1 ) r e p r e s e n t s t h e f o r m t a k e n b y t h e Landaufree energy o f t h e s y s t e m ( s e e S e c t i o n 2 . 1 . 4 ) i n t h e P r e i s a c h a p p r o a c h . I nm i c r o m a g n e t i c s , t h i s e n e r g y is g i v e n b y a fu n c t i o n a l o f t h e m a g n e t i z a t i o n

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    13.2 ENERGETIC ASPECTS 455d i s t r i bu t i on i n t he bod y (E q. 6 .10 ) . T h r ou gh t he a s s um pt i on s d i s cus s edi n the p r ev i ous s ec ti on , t he des c r i p t i on i n r ea l s pace has been t r ans f o r m edi n t o an app r ox i m a t e l y e qu i va l en t de s c r i p t i on i n a s u i t ab l e fi e ld s pace , t heP r e i s ach p l ane . T he s pa t i a l p r ope r t i e s o f t he f r ee ene r gy s u r f ace , and i np a r t i c u l a r t h e p r e s e n c e o f m a n y m e t a s t a b l e m i n i m a , a r e n o l o n g e r e x p l ic -i tl y v i s ib l e i n the P r e i s ach des c r i p t i on . T he y hav e bee n t r an s l a t ed i n t o t her u l e s con t r o l l i ng t he s t ab i l i t y o f P r e i s ach un i t s and t hey a r e i nd i r ec t l ydes c r i bed by t he s t a te l i ne b (hc) an d by i ts de pe nd enc e on f i e ld h i st o ry .I n t h i s s ens e , each s t a t e des c r i bed by a g i ven b(hc) l i ne co r r e s ponds t o al oca l m i n i m um o f t he i n i ti a l m i c r omag ne f i c ene r gy , and E q . (13 .31 ) de -s c r i bes how t he ene r gy o f t he s e me t a s t ab l e mi n i ma i s r e l a t ed t o t he b(hc)l ine.A cco r d i ng t o t h i s i n t e r p r e t a t i on , t he s t a t e o f min imum f ree energy canbe o b t a i ned by s t ud y i n g t he e x t r ema o f E q . (13 .31 ) w i t h r e s pec t t o a r b i t r a r yva r i a t i ons o f b(hc). W h a t w e h a v e t o d o is t o so l v e t h e e q u a t i o n o b t a i n e dby e qua t i ng t o ze ro t he va r i a t i on 8g con s equ en t t o a s ma l l, a r b i t r a r yva r i a t i on 8b(hc). By co ns id er ing tha t 8fM = --tZoHMFSM (see Eq. (13.19)),Eq. (13.31) giv es

    ~ = 21.~oM~ [b(hc) - Ha - HM f] 8b(hc)p[hc,b(hc)] dhc = 0 (13.33)ow h i ch , g i ven t he a r b i t ra r i nes s o f ~b(hc) an d t he f ac t t ha t p(hc,hu) i s a pos i t iv equan t i t y , l e ads t o t he s o l u t i on

    beq(hc) = S - - Ha + HM F (13.34)The equi l ibr ium s ta te des cr ibe d b y Eq . (13 .34) i s sho w n in F ig . 13 .12 .T he s i mp l i c i t y o f E q. ( 13 .34 ) i s on l y app a r en t , be caus e t he m ean - f i e l d HM Fi s a f unc t i on o f M , w h i ch i n t u r n i s a f unc t i on o f beq(hc) through Eq. (13.10) .

    FIGURE 13.1 2. Equ ilibrium state described b y Eq. (13.34).

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    456 CHA PTER 13 Co llections of Bistable UnitsSe l f - cons i s t en t so l u t i ons a re t o be sough t , w h i ch m ay no t be un i que . Inadd i t i on , w e shou l d t e s t w he t h e r t he equ i l i b r i um desc r i bed by Eq . (13.34)i s s t ab l e o r uns t ab l e . These a spec t s can be d i scussed by cons i de r i ng t hemagnet i za t ion of the equi l ib r ium s t a t e , ob ta ined by inser t ing Eq. (13 .34)into Eq. (13.10),

    f ~ f S a + S M r -[M e q(S a)]Meq(Ha) = 2 M s d h c p ( h c, h u ) d h u ( 1 3 . 3 5 )oand i t s Gibbs f ree energy , ob ta ined f rom Eq. (13 .31) ,g e q ( H a )

    + 2 tz~ . 'fo (hu - H a )p (h c , hu )d hufMF[Meq(/-/d)] Ha 4- HMF[Meq(Ha)] (13.36)

    Let us con s ider the der iva t ive of Eq. (13 .35) w i th respec t t o H a. We h av e[H a = ,u 1 4- aMe----- d H a (13.37)w h e r e

    fo~= 2 M s J o P ( h c , H a + H M F ) d h c (13.38)uBy i n t roduc i ng t he no t a t i on 5

    k M F - - ~ 3 H M F 1 (13.39)LOM Jran d by so lv ing Eq. (13 .37) w i th re spec t t o d M e q / d H a, w e f i nd

    d M e q _ Xeq- (13.40)d H a 1 - - k M F , ) ( e q

    The s t a t es assoc ia t ed w i th Eq . (13 .35) an d Eq. (13 .36) fo l low the l awsof equ i l i b r i um t he rm ody nam i cs , d i scusse d i n Sec ti on 4.2 . In pa r t icu l a r ,b y t a k in g t h e H a der iva t ive of Eq . (13 .36) an d by m ak ing use of Eq. (13.37)thro ug h Eq. (13 .39) , one can ve r i fy tha tM e q = 1 (13.41)~0/0Ha Jr

    Sin gen eral , the q ua nt i ty kMF def in ed b y Eq. (13.39) is no t a cons tant , bu t a fu nct io n ofM, excep t fo r the case wh ere HMF i s p rop or t ion a l to M.

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    13.2 ENERG ETIC ASPECTS 457(com pare w i th Eq . 4 .43) . We kn ow f rom Sec t ion 4.2 tha t t he Meq s t a t e wi l lbe sta ble if O2geq~ OH2 < O. The refore , a cc ord ing to Eq. (13 .41), the s tabi l i tyof the s ta te i s cont ro l led by the s ign of Eq. (13 .40) , tha t i s , by the fac t tha tt h e re e x i s t s o r n o t a m a g n e t i z a t i o n i n t e rv a l w h e re k MFXeq > 1. This is theres ul t of two di ffere nt a c t ions . Xeq des cr ib es th e effec t of f ine-scale s t ruc-t u r a l d i so rd e r , w h e re a s kMr_ charac te r i zes l a rge -sca l e mean-f i e ld e f fec t s .A cco rd in g to Eq . (13.38) , Xeq > 0 un de r a l l c i rcum stances . Con verse ly , kM Fm a y t a k e b o t h s ig n s , d e p e n d i n g o n t h e r el a ti v e im p o r t a n c e o f t h e v a r i o u sm e c h a n i sm s c o n t r i b u t i n g t o t h e m e a n - f i e l d e n e rg y . I f kM r_ > O, t hen , de -p e n d i n g o n w h e t h e r t h e re is o r is n o t s o m e m a g n e t i z a t i o n i n t e r v a l w h e r ekMr_Xeq > 1 , t he equ i l ib r ium m agn e t i za t ion cu rve , Eq . (13 .35) , an d theco r re sp on d in g geq energy , Eq . (13.36) , w i l l t ake o ne o f t he tw o fo rmssh o w n i n F i g 1 3 .1 3 . I n t h e i n t e rv a l w h e re k MFXeq > 1 , there are threecoex i s t ing energy b ranches , and Eq . (13 .34) admi t s t h ree so lu t ions , o fw h i c h o n e i s s t a b l e , o n e m e t a s t a b l e , a n d o n e u n s t a b l e . C o n v e r se l y , o n es t ab le so lu t ion on ly ex i s t s i f kM r_Xeq < 1 e v e r y w h e r e .Th e re a r e t w o a sp e c t s t h a t r e m a i n t o b e d i s c u s se d . F i r s t , w e w o u l dl ike to see i f t he re i s some par t i cu la r f i e ld h i s to ry capab le o f genera t ingt h e M eq s t a te , a n d , s e c o n d , w e w o u l d l ik e t o c o m p re h e n d h o w , s t a r t i n gf ro m a g e n e r i c n o n e q u i l i b r i u m s t a t e d e sc r i b e d b y a g i v e n b(hc) l ine, the

    TvQ):E Y /

    " rvQ)t: n

    - ~ u

    j r n si

    /T. m s' i , s

    H a H aFIGURE 13.13. Equilibrium magn etization Meq(Ha) (Eq. 13.35) and correspo ndingGibbs energy geq(Ha) (Eq . 13.36). Left: kMFXeq < 1; Rig ht: kMrXeq > 1 in the intervalindicated b y the dashe d l ine. The symbols s, ms, and u denote a stable, metastable,and unstable state, respectively.

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    458 CH APT ER 13 Collections of Bistable Units

    s y s t e m w i l l a p p r o a c h t h e r m o d y n a m i c e q u i l i b r i u m u n d e r t h e a c t i o n o ft h e r m a l f l u c t u a t i o n s . T h e f i r s t a s p e c t i s c l o s e l y r e l a t e d t o t h e c o n c e p t o fanhysteretic m a g n e t i z a t i o n c u r v e , t h e s e c o n d t o t h e s u b j e c t o f magneticviscosity. T h e y w i l l b e d i s c u s s e d i n S e c t i o n 1 4 . 1 a n d S e c t i o n 1 4.2 .

    13 .3 H Y S T E R E S I S P R O P E R T I E ST h e m e t h o d s d i s c u s s e d i n t h e p r e v i o u s s e c t i o n s s h o w h o w t h e P r e i s a c ha p p r o a c h c a n b e a p p l i e d t o th e d e s c r i p t i o n o f s e v e r a l a s p e c t s o f h y s t e r e s is ,b u t t h e y s a y n o t h i n g a b o u t t h e f a c t t h a t a g i v e n s y s t e m m a y o r m a y n o tb e d e s c r i b e d i n t h a t w a y . D r a s t i c s i m p l i f i c a t i o n s h a v e b e e n i n t r o d u c e d i nt h e d e s c r i p t i o n o f t h e m e t a s t a b l e s t r u c t u r e o f t h e s y s t e m f r e e e n e r g y ,a n d i t i s n a t u r a l t o c o n s i d e r h o w t h e s e s i m p l i f i c a t i o n s c a n b e p h y s i c a l l yj u s ti fi e d . A p l a u s i b l e a p p r o a c h m i g h t s e e m t o b e t h e i d e n t i fi c a t i o n o f r e a lp h y s i c a l u n i t s w i t h p r o p e r t i e s s i m i l a r t o t h o s e o f a P r e i s a c h u n i t . I n t h i sr e s p e c t , p a r t i c l e a s s e m b l i e s c o u l d b e i n p r i n c i p l e g o o d c a n d i d a t e s , w i t ho n e p a r t i c le a s s o c i a t e d w i t h o n e P r e i s a c h u n i t. H o w e v e r , o n e s o o n r e a l iz e st h a t a d i r e c t j u s ti f ic a t i o n o f t h e P r e i s a c h d e s c r i p t i o n b y s u c h a c o n s t r u c t i v em e t h o d c a n n o t g o v e r y f ar, d u e t o t h e d i ff ic u l ti e s in h e r e n t i n t h e t r e a t m e n to f p a r t i c le i n t e r a c ti o n s . I n c o n t i n u o u s b u l k s y s t e m s , P r e i s a c h u n i t s s h o u l da r i s e f r o m s o m e d e c o m p o s i t i o n o f t h e m a g n e t i z a t i o n p r o c e s s i n t o B a r k -h a u s e n j u m p s , b u t n o o n e k n o w s h o w t h i s d e c o m p o s i t i o n s h o u l d b ec a r r ie d o u t o n g e n e r a l g r o u n d s . T h e p o i n t i s t h a t p r o v i n g o r d i s p r o v i n gt h i s o r a n y o t h e r p h e n o m e n o l o g i c a l a p p r o a c h t o h y s t e r e s i s e v e n t u a l l yc a l l s f o r d e t a i l e d t r e a t m e n t s b a s e d o n t h e n o n e q u i l i b r i u m s t a t i s t i c a l m e -c h a n i c s o f m e t a s t a b l e s y s t e m s , a n a r d u o u s a n d d e l i ca t e t a s k in it se lf .

    H o w e v e r , t h e r e is a n o t h e r s t r a t e g y t h a t m a y b e f o ll o w e d . T h e P r e i s a c hd e s c r i p t i o n o f h y s t e r e s i s , d u e t o i t s i n t e r n a l m a t h e m a t i c a l s t r u c t u r e , p r e -d i c ts m a g n e t i z a t i o n c u r v e s c h a r a c t e r i z e d b y c e r t a i n q u a l i t a t iv e p r o p e r t i e s ,i n d e p e n d e n t o f t h e i r d e t a i le d s h a p e . T h e s e p r o p e r ti e s a r e k n o w n a s th ewiping-out (o r deletion, o r return-point-memory) p r o p e r t y a n d t h e congruencyp r o p e r t y . T h e f a c t t h a t t h e s e p r o p e r t i e s a r e o b e y e d b y a g i v e n s y s t e m i st h e f i n g e r p r i n t o f t h e P r e i sa c h n a t u r e o f th a t s y s t e m . T h e s e p r o p e r t i e s c a nb e e x p e r i m e n t a l l y t e s t e d . If t h e t e s t g i v e s a p o s i t i v e r e s p o n s e , t h e n w ek n o w f o r s u r e t h a t i t i s p o s s i b le t o d e s c r i b e t h a t s y s t e m b y t h e P r e i s a c ha p p r o a c h . T h e d e c o m p o s i t i o n o f t h e m i c r o m a g n e t i c f re e e n e r g y i n to e l e-m e n t a r y b i s ta b l e c o n t r ib u t i o n s i s p o s si b le , e v e n i f w e d o n o t k n o w h o wt o c a r r y i t o u t i n d e t a i l . I n t h i s s e c t i o n , w e d i s c u s s t h e m a i n r e s u l t s t h a tc a n b e o b t a i n e d a l o n g t h i s l i n e .

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    13.3 HYSTERESIS PROPERTIES 45913 . 3 . 1 W i p i n g - ou t p r op e r t y an d c on g r u e n c y p r op e r t yW i p i n g - o u t p r o p e r t y . W e h a v e s e e n t h a t t h e b(hc) l i n e d e s c r i b i n g t h e s t a t eo f a c o ll e ct io n o f P r e is a c h u n i t s i s d e t e r m i n e d b y t h e s e q u e n c e o f r e v e r s a lp o i n t s, t h a t is , o f l oc a l m a x i m a a n d m i n i m a r e a c h e d b y t h e a p p l i e d f i e ldi n i ts p a s t h i s t o r y ( F ig . 13 .5 ). A s a m a t t e r o f f a c t, t h e s e q u e n c e o f f i e l dr e v e r s a l p o i n t s g i v e s a r e d u n d a n t d e s c r i p t i o n . I t i s p o s s i b l e t o i d e n t i f y as u b s e q u e n c e c o n t a in i n g t h e m i n i m u m c o m p l e t e s e t o f i n f o r m a t io n n e c e s-s a r y t o d e f i n e t h e s t a t e o f t h e s y s t e m . T h i s s u b s e t i s o b t a i n e d b y e x t r a c t i n gf r o m t h e i n it ia l s e q u e n c e t h e s u b s e q u e n c e o f m o n o t o n i c a l l y d e c r e a s i n gf i e l d m a x i m a , H M 1 > H M 2 > . . . , a n d o f m o n o t o n i c a l l y i n c r e a s i n g f ie ldm i n i m a , H m l < Hm 2 < . . . , a l t e r n a t i v e l y r e a c h e d i n t im e , t h a t i s, a t ti m e st in1 < tM 1 < tin2 < tM2 < . . . (F ig . 13. 14). Th e o the r ex t re m a h av e n oi n f l u e n c e o n t h e f i n a l s t a te o f t h e s y s t e m . H o w t h is c a n o c c u r is i l l u s t ra t e db y F i g . 1 3 . 1 5 , w h i c h s h o w s t h e p a r t i c u l a r c a s e w h e r e t h e f i e l d i s f i r s ti n c r e a s e d u p t o t h e v a l u e H 1, t h e n d e c r e a s e d d o w n t o H 2, a n d f i n a ll yi n c r e a s e d a g a i n . W h e n H r e a c h e s a g a i n t h e v a l u e H 1, t h e b (h c ) l i n e a c q u i r e sa g a i n t h e s a m e s h a p e i t h a d w h e n H = H 1 f o r th e f i r st ti m e . T h e e f f e cto f t h e i n t e r m e d i a t e f i e l d o s c i ll a ti o n is c o m p l e t e l y e r a s e d ( w i p e d o u t ) a n dt h e s ta t e o f t h e s y s t e m h a s n o l o n g e r a n y m e m o r y o f t h a t in t e r m e d i a t eh is to r y. N o t e t h a t t h e m o m e n t w h e r e H c r o ss e s t h e l ev e l H 1 is th e m o m e n ta t w h i c h w e k n o w f o r s u r e t h a t , w h a t e v e r t h e s u b s e q u e n t f i e l d h i s t o r y ,t h e r e w i l l b e a l a t e r fi e ld m a x i m u m H M > / - / 1 , w h i c h e l i m i n a t e s H 1 f r o mt h e s e t o f m o n o t o n i c a l l y d e c r e a s i n g f i e ld m a x i m a .

    H M 1

    ' " f ~ " ~ . . . H M 2~ . tim e

    FIGURE13.14. Ext ract ion of the reversa l -poin t subsequen ce ident i fy ing the s ta teof a Preisach system.

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    460 CH APT ER 13 Co l lections of Bis table Un i ts

    FIGURE 13 .15 . W ip ing-ou t m echan i sm . Wh en H reaches aga in the pas t reversa lpoint H1, t he e f fec t o f the in te rmed ia te f ie ld osc i ll a tion be tw een H 1 and H 2 i se rased .Congruency property. C o n s i d e r t w o d i f f e r e n t b(hc) s t a t e l i n e s a s s o c i a t e dw i t h t h e s a m e f i el d H 1. I f w e b e g i n c y c l i n g t h e f ie l d b e t w e e n H 1 a n da n o t h e r f i el d H 2, t h e s a m e t r i a n g u l a r r e g i o n o f th e P r e i s a c h p l a n e w i l le v e n t u a l l y b e s w e p t b a c k a n d f o r th b y t h e b(hc) l i n e , i n d e p e n d e n t l y o ft h e i n i t i a l s t a t e ( F ig . 1 3 .1 6 ). A c c o r d i n g l y , t h e s y s t e m w i l l d e s c r i b e t w om i n o r l o o p s g e o m e t r i c a l l y c o n g r u e n t ( F i g . 1 3 . 1 7 ) . T h e d i f f e r e n c e i n t h ei n i t i a l s t a t e s w i l l m e r e l y r e s u l t i n a v e r t i c a l s h i f t o f t h e m i n o r l o o p s i nt h e Ha-M p l a n e . T h is i d e n t i t y of m i n o r l o o p s e x p r e s s e s t h e c o n g r u e n c yp r o p e r t y .

    T h e i m p o r t a n c e o f t h e s e p r o p e r t i e s l ie s i n t h e f o l l o w i n g t h e o r e m , d u eto M a y e r g o y z . 6

    FIGURE 13 .16 . P re isach p lane represen ta t ion o f m inor loops w i th iden t ica l peakfields but d i fferent pas t h is tor ies .6The proo f can be fou nd in the references given in the bibliograhical notes.

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    13.3 HYST ERESIS PROPERTIES 461

    1 .0

    0.5

    0.0

    -0 .5

    J-1- 0_ 2 ' _~ ' 0 ' ~ ' 2

    H / H o

    FIGURE 13.1 7. Mino r loops for the s am e field histories as Fig. 13.16, calculatedfrom the factorized -Loren tzian Preisach distrib ution (Eq . 13.12), cr -- 1.Applicability of Preisach model. A s y s t e m w i t h h y s t e r e si s c a n b e d e s c r ib e db y t h e P r e is a c h m o d e l i f and on ly/ f i t o b e y s t h e wiping-out p r o p e r t y a n d t hecongruency p r o p e r t y . T h e s e p r o p e r t i e s r e p r e s e n t n e c e s s a r y a n d s u f f i c i e n tc o n d i t i o n s f o r t h e a p p l i c a b i l i t y o f t h e P r e i s a c h d e s c r i p t i o n .

    T h i s t h e o r e m c l a ri fi e s th e l i m i t s o f a p p l i c a b i l i t y o f t h e P r e i s a c h m o d e l .I n a d d it i o n , i t i n d ic a t es w h i c h m e a s u r e m e n t s o f m a c r o s c o p ic m a g n e t i z a -t i o n c u r v e s s h o u l d b e c a r r i e d o u t i n o r d e r t o d e c i d e w h e t h e r a g i v e ns y s t e m c a n b e d e s c r i b e d b y t h e P r e is a c h a p p r o a c h o r no t.

    1 3.3 .2 R e t u r n - p o i n t m e m o r yT h e w i p i n g - o u t p r o p e r t y c a n b e r e c a s t i n a d i f fe r e n t f o r m t h a t i s c o n c e p t u -a l l y i m p o r t a n t . L e t u s r e c o n s i d e r F i g . 1 3 . 1 5 . W e s a i d t h a t , a t t h e p o i n tw h e r e t h e f ie l d a c q u ir e s a g a i n t h e v a l u e H = H 1, t h e m e m o r y o f t h ep r e c e d i n g f i e l d o s c i l l a t i o n i s l o s t . W e c a n l o o k a t t h e s a m e b e h a v i o r f r o ma d i f fe r e n t v i e w p o i n t . W h e n t h e f i el d r e t u r n s b a c k t o H = H 1, t h e s y s t e mr e t u r n s b a c k t o t h e e x a c t s a m e s t a t e i t o c c u p i e d a t t h e t i m e w h e n t h ep o i n t H = H 1 w a s r e a c h e d f o r t h e f ir s t t im e . I n o t h e r w o r d s , t h e s y s t e mi s a b l e t o s t o r e m e m o r y o f i t s p a s t s t a t e s a n d i t i s a b l e t o r e t u r n b a c kt o t h o s e e x a c t s t a t e s u n d e r a p p r o p r i a t e c o n d i t i o n s . T h i s i s t h e c o n c e p te x p r e s s e d b y t h e t e r m return-point memory.

    T h e r e a s o n f o r h a v i n g r e t u r n - p o i n t m e m o r y i s q u i t e g e n e r a l , a n d i se v e n t u a l l y d u e t o t h e p o s s ib i l i ty o f i n t r o d u c i n g a n ordering relation a m o n g

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    462 CH APT ER 13 Collections of Bistable Unitst h e s t a t e s o f t h e s y s t e m , w h i c h i s p r e s e r v e d d u r i n g t h e e v o l u t i o n o f t h es y s te m . I n m o r e d e ta il , re t u r n - p o i n t m e m o r y e m e r g e s i n all s y s te m s s a ti s-f y i n g t h e t h r e e r e q u i r e m e n t s l i s t e d n e x t .Partial ordering. L e t u s c o n s i d e r a s y s t e m , w h o s e s t a t e s w e i d e n t i f y b yc a p i t a l l e t te r s (A , B , . . . ). P a r t i a l o r d e r i n g h o l d s if a n o r d e r i n g r e l a t i o n ,i n t h e u s u a l m a t h e m a t i c a l s e n s e , e x i s t s , w h i c h p e r m i t s o n e t o d e c i d ew h e t h e r t w o g i v e n s t a t e s a r e o r d e r e d o r n o t . T h e o r d e r i n g i s p a r t i a l , i nt h e s e n s e t h a t , g i v e n t w o g e n e r i c s ta t e s, t h e y a r e n o t n e c e s s a r i l y o r d e r e d .I f t h e y a r e o r d e r e d , t h e n o r d e r i n g w i l l b e i n d i c a t e d b y t h e n o t a t i o nA>__B.Rate-independent evolution. W e i n d i c a t e b y A ( 0 ) t h e i n i t i a l s t a t e o f t h e s y s t e ma n d b y A(t) t h e t i m e - d e p e n d e n t s t a t e e v o l v i n g u n d e r t h e a c t i o n o f t h ee x t e r n a l f i e l d HA(t). R a t e - i n d e p e n d e n c e h o l d s i f A(t) d e p e n d s o n l y o n t h ef ie ld HA(t) a t th e s a m e t i m e a n d o n t h e s e q u e n c e o f p a s t r e v e r s a l f ie l d s,a n d n o t o n t h e r a t e a t w h i c h t h e f i e l d v a r i e s i n t i m e .Order-preserving evolution. The d y n a m i c s o f t h e s y s t e m a r e o r d e r - p r e se r v -i n g if , g i v e n t w o i n i t i a ll y o r d e r e d s t a t e s A (0 ) ~ B (0 ), w h e n w e a p p l y t ot h e m t w o o r d e r e d f i e l d h i s t o r i e s , t h a t i s , s u c h t h a t HA(t) >-- HB(t) a t a l ls u b s e q u e n t t i m e s , t h e n A(t) >- B(t) a t a l l t i m e s a s w e l l .

    A s y s t e m s a ti s fy i n g th e s e t h re e p r o p e r t i e s d i s p l a y s r e t u r n - p o i n t m e m -o ry . I n f a c t, le t u s c o n s i d e r F i g. 1 3.1 8, w h e r e a s e q u e n c e o f s t a t e s g e n e r a t e db y a c e r t a in f i e ld h i s t o r y i s r e p r e s e n t e d . N o t e t h a t a r e p r e s e n t a t i o n o f t h i sk i n d , w i t h t h e f i e l d a s i n d e p e n d e n t c o o r d i n a t e , i s p o s s i b l e j u s t b e c a u s et h e e v o l u t i o n is r