JANUARY, 1940
NUMBER 1V '
¡ Æ THE BELL SYSTEM
T O H N I C A L J O U R N A L
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION
TM V V The Physical Basis of Ferromagnetism— R. M. Bozorth . . 1 Contact Phenomena in Telephone Switching CircuitsI
— A. M . Curtis 40 Effect of the Quadrature Component in Single Sideband
Transmission— H. Nyquist and K. W. Pfleger . . . . 63 Low Temperature Coefficient Quartz Crystals— W. P. Mason 74 A New Standard Volume Indicator and Reference Level
— H. A. Chinn, D. K. Gannett, and R. M . Morris 94 Metallic Materials in the Telephone System
— Earle E. Schumacher and W. C. Ellis 138 Technical Digest—
An Interesting Application of Electron Diffraction—
L. H. Germer and K . H. Storks 152 Abstracts of Technical P a p e r s ...156 . Contributors to this I s s u e ...159
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F. B. Jewett A. F. Dixon S. Bracken
EDITORIAL BOARD H. P. Charlesworth
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W. Wilson
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Copyright, 1940
American Telephone and Telegraph Company
P R I N T E D I N U . S . A .
T he Bell System Technical Journal
Vol. XIX January, 1940 No. 1
The Physical Basis of Ferromagnetism
By R . M . B O Z O R T H
A fte r a n in tr o d u c to r y rev iew of th e g e n e ra l n a tu r e of th e th e o r y of m a g n e tic p h e n o m e n a a n d th e m a g n itu d e s of th e a to m ic forces in v o lv e d , th e r e is a d iscu ssio n of E w in g ’s th e o ry , its re s u lts a n d lim ita tio n s . T h e la te r th e o r y of W eiss is th e n giv en b riefly in o rd e r to fix th e c o n c e p t of th e m o le c u la r field. In o rd e r to e lu c id a te th e n a tu r e of th is field a d ig ressio n is m a d e to d iscu ss th e a to m ic s t r u c tu r e of th e fe rro m a g n e tic e le m e n ts a n d e le m e n ts h a v in g sim ilar s tru c tu r e s . W ith th is as a b asis th e p h y sic a l n a tu r e of th e m o lecu lar field is d iscu ssed a t som e le n g th . I t s re la tio n to th e s tr u c tu r e of d o m a in s, p a rtic u la rly th e n a tu r e of th e b o u n d a rie s b e tw e e n d o m a in s, is b r o u g h t o u t.
F in a lly th e r e is a rev iew of th e g y ro m a g n e tic effect, its signifi
c a n c e fo r m a g n e tic th e o r y , th e p rin c ip a l e x p e rm e n ta l m e th o d for its d e te r m in a tio n , a n d th e n u m e ric a l re s u lts s u p p o rtin g th e id ea t h a t th e s p in of th e e le c tro n a n d n o t its o rb ita l m o m e n t is re s p o n sible fo r fe rro m a g n e tis m .
In t r o d u c t i o n
I
N T H E la s t five or te n y ea rs th e th e o ry of ferrom agnetism has show n in d icatio n s of m a tu rity . F o r th e first tim e a plausible sto ry can be to ld concerning th e u ltim a te m agnetic particle, th e essential n a tu re of th e ato m of a ferrom agnetic substance, th e kind of forces w hich d ete rm in e th e p ro p erties of m agnetic crystals, th e effect of s tra in on m a g n e tic m a te ria ls and th e m a n n er in w hich these various p h en o m en a com bine to d eterm in e th e properties of com m ercial m ate ria ls. I t is tru e t h a t th e sto ry is largely q u alitativ e, an d th a t th e re are still m a n y p o in ts t h a t are u n c e rtain or m issing en tirely , b u t n evertheless it is possible to describe th e m ajo r fe atu res w ith som e confidence.T h e fu n d a m e n ta l m ag n etic p article is th e spinning electro n . O ne m ig h t th in k t h a t th e o rb ita l m otions of th e electrons in th e a to m would also c o n trib u te to ferrom agnetism , owing to th e ir m agnetic
1
I t l t y
V,
2 B E L L S Y S T E M T E C H N I C A L J O U R N A L
m o m en ts, b u t it h a s now been esta b lish e d t h a t w hen th e m a g n e tiz a tio n is a lte re d all t h a t changes is th e d irectio n o r “ s e n s e ” of th e spin of c e rta in of th e e lectro n s in th e a to m s— th e o rb ita l m o tio n s rem ain p ra c tic a lly u n ch an g ed .
T h e electro n s t h a t a re resp o n sib le for th e m a g n e tic p ro p e rtie s of iron, c o b a lt, nickel a n d th e ir alloys lie in a d efin ite “ s h e ll” in th e ato m . A s show n in Fig. 1, th e re a re four shells o r regions, m o re or
less well defined, in to w hich all th e e lec tro n s circ u la tin g a b o u t th e nuclei of th e se a to m s m a y be d iv id e d w h en th e a to m is s e p a ra te d from its neig h b o rin g ato m s, a s it is, for ex am p le, in a gas. Som e of th e se shells a re su b d iv id e d a s show n. W h en th e a to m s com e closer to g e th e r as th e y do in a solid, th e fo u rth or o u te rm o st shell of each becom es d is ru p te d , a n d th e tw o elec tro n s w hich c o m p rised it w a n d e r from a to m to a to m a n d are th e “ fr e e ” ele ctro n s resp o n sib le for
t h e p h y s i c a l b a s i s OF F E R R O M A G N E T I S M 3 electrical conduction. T h e electro n s in th e o u te r p a r t of th e th ird shell are th o se responsible for th e d istin ctiv e kind of m ag n etism found in iron, co b alt an d nickel. Som e of th e se electrons spin in one directio n an d som e in th e opposite, as in d icated , so t h a t th e ir m ag n etic m o m en ts n eu tralize each o th e r p artia lly b u t n o t w holly, and th e excess of those spinning in one directio n over th o se spinning in th e o th e r causes each a to m as a w hole to b eh av e as a sm all p e rm a n e n t m ag n et.
T h e w ell-established k in etic th e o ry of m a tte r tells us t h a t if each ato m w ere to a c t in d ep en d e n tly of its neighbors, th e a to m s w ould be v ib ra tin g an d ro ta tin g so en erg etically t h a t th e y could n o t be aligned even w ith th e stro n g e st field t h a t can be produced in th e la b o rato ry . T o explain th e kind of m ag n e tic p ro p ertie s found in iron, therefore, it is necessary t h a t th e re be som e in te rn a l force cap ab le of m aking th e m agnetic m o m en t of a g roup of neighboring a to m s lie parallel to each oth er— th e sm all ato m ic “ p e rm a n e n t m a g n e ts ” of each g ro u p m u st po in t in th e sam e d irection so as to provide a m ag n e tic m o m en t g reat enough to p e rm it a realig n m en t w hen su b jected to ex tern al fields.
R ecen tly it has been show n b y in d ep e n d en t m eans t h a t th e re is such a force in ju s t those elem en ts w hich are ferrom agnetic, a n d it is from th is force t h a t th e difference b etw een m ag n etic an d non-m agnetic m a te ria ls arises. T h e force is e le c tro sta tic in n a tu re an d is called
“ exchange in te ra c tio n ” by th e a to m ic -stru c tu re experts, th e w ave m echanicians, w ho h av e show n its existence an d calcu lated its order of m ag n itu d e. T h is force m a in tain s sm all groups of a to m ic m ag n ets parallel ag ain st th e forces of th erm al a g itatio n . (W hen th e m aterial is h ea te d so h o t th a t th e disordering actio n of th e a g ita tio n becom es stro n g enough to overpow er th e forces of “ exchange in te r a c tio n ” th e m aterial loses its ferrom agnetism ; in iron th is h ap p en s a t 770° C.)
B u t w h y th e n is n o t ev ery piece of iron a com plete p erm a n e n t m ag n et? F o r som e reason n o t und ersto o d a t presen t, a t o rd in a ry te m p e ra tu res th e e le c tro sta tic forces of exchange in te ra c tio n m a in ta in th e ele m e n ta ry m ag n ets parallel only over a lim ited volum e of th e specim en. T h is volum e is usu ally of th e o rd er of 10-8 or 10-9 cubic cen tim eters a n d co n tain s a million billion a to m s an d is of course in visible. Such a volum e is said to be s a tu ra te d because th e ato m ic m ag n ets a re all po in tin g in th e sam e directio n , a n d has been given th e nam e “ d o m a in .” T h u s a m ag n etic m aterial a t room te m p e ra tu re, before it has been m agnetized b y su b jectin g it to th e influence of a m ag n etic field, is d ivided in to a g re a t m a n y dom ains each of w hich is m agnetized to s a tu ra tio n in som e d irectio n gen erally different from th a t of its neighbors. T h e n e t o r v ec to r sum of th e m ag n etizatio n s is zero, an d ex tern ally th e m a te ria l ap p e a rs to be u n m agnetized b u t in
4 B E L L S Y S T E M T E C H N I C A L J O U R N A L
re a lity th e m a g n e tiz a tio n a t a n y one p o in t is v e ry in ten se. W h en a m a g n e tic field is a p p lied b y bringing n e a r th e m e ta l a p e rm a n e n t m a g n e t o r a coil of w ire c a rry in g a c u rre n t, th e m a g n e tiz a tio n of th e m a te ria l as a w hole is in creased to a definite v alu e. W e believe t h a t w h a t th e n ta k e s p lace is sim p ly a change in th e d ire c tio n of th e m a g n e tiz a tio n s of th e d o m ain s. If we re p re se n t th e m a g n e tiz a tio n of a n y d o m ain b y a v e c to r, th e effect of th e e x te rn a lly a p p lied field will be re p re se n te d b y th e r o ta tio n of th ese v ec to rs— ro ta tio n s n o t acc o m p a n ie d b y a n y chan g es of length.
/ n / V
r1 / ///
7
D IR E C TIO N S OF EASY M A G N E T IZ A T IO N
IR O N N I C K E L
Fig. 2—T he positions of th e atom s and th e directions of easy m agnetization in crystals of iron and of nickel.
R e c e n tly m u ch h as been learn ed a b o u t th e m a g n e tic p ro p e rtie s of m a te ria ls b y a s tu d y of single c ry sta ls. O rd in a ry m e ta ls are com posed of a g re a t m a n y c ry sta ls o ften to o sm all to b e seen easily b y th e n a k e d eye. B u t in th e la s t few y e a rs m e th o d s h a v e b ee n fo u n d for m a k in g large c ry sta ls of a lm o st all th e com m on m e ta ls, c ry sta ls as large as th e m o re fa m ilia r ones of ro c k c a n d y a n d even of q u a rtz . E x p e rim e n ts on such c ry sta ls of iron show t h a t th e y are m u ch m o re easily m ag n etize d in som e d irectio n s th a n in o th e rs.
T h is d ep en d en ce of ease of m a g n e tiz a tio n on d ire c tio n is illu s tra te d in F ig. 2 for iron a n d nickel in re la tio n to th e p o sitio n s of th e a to m s in
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 5 th e cry stals. T h e circles re p re se n t th e positions w hich cen ters of ato m s ta k e up on a n im ag in a ry fram ew o rk or la ttice. B ecause of th e sm allness of ato m ic dim ensions o n ly a sm all fraction of th e a to m s in a cry stal of o rd in a ry size are show n, b u t th e sam e p a tte rn , th e u n it of w hich is o u tlin ed b y solid lines, e x ten d s th ro u g h o u t th e w hole of th e single c ry sta l. T h e arrow s in d ic a te th e d irectio n s of “ e a s ie s t” m a g n etizatio n , w hich are d ifferent for th e tw o m aterials a s m ay be noticed.
In o rd er to give a n o tio n of th e ab so lu te and relativ e sizes of c ry sta ls and dom ain s and ato m s w ith w hich m ag n etic processes are concerned, it m ay be p o in ted o u t t h a t a piece of o rd in a ry iron a cubic ce n tim e te r in volum e m a y co n ta in a b o u t 10,000 single cry stals, an d t h a t each cry sta l co n tain s on th e av erag e 100,000 dom ains each w ith from 1014 to 1016 ato m s.
A lth o u g h th is article is n o t concerned p rim arily w ith th e d etails of th e changes in m a g n e tiz a tio n t h a t occur w hen a m ag n etic field is applied, a b rief d escrip tio n of such changes is desirable. In a cry sta l of iron th e d irectio n s of e asy m ag n e tiz atio n are parallel to th e cubic axes, t h a t is, th e y are th e six d irectio n s parallel to th e edges of th e cube w hich rep resen ts th e s tru c tu re . W hen such a m ag n etic m aterial is u n m ag n etized as a w hole a po rtio n of one of th e cry stals in it m ay be rep resen ted b y th e highly schem atic Fig. 3(a). As show n, each of th e dom ains, rep resen ted b y th e arrow s, circles an d crosses, is m ag n etized in one of th e directions of easy m ag n etizatio n , equal n u m b ers in each of th e six d irections. W h en a w eak field is applied in th e directio n in d ic a te d an d its stre n g th g ra d u ally increased to a high value, th e m ag n etizatio n s of th e dom ains change su d d en ly a n d th e ir d irectio n s ap p ro a c h coincidence w ith th a t of th e m agnetic field. T h is is usu ally accom plished b y th e displacem ents of dom ain boundaries, th ese m oving so t h a t som e dom ains grow a t th e expense of o th ers in w hich th e m ag n etizatio n lies in a directio n fu rth e r from t h a t of th e field. W h en th e field has been increased to such a stre n g th th a t p ra c tic a lly all th e dom ains are o rien ted as show n in (b) a n d th e crystal is really ju s t one large dom ain, a second process com m ences: th e m ag n e tiz atio n changes slow ly in direction u n til finally it is parallel to th e field, a n d th e n changes no m ore. T h e m aterial is th e n said to be s a tu ra te d , as show n in (c).
F ig u re 3 is d raw n to illu stra te th e changes in m ag n etizatio n t h a t occur in a single c ry sta l of iron. Iro n as we o rd in arily see it is com posed of a g re a t m a n y m in u te single c ry stals, b u t th e changes in m ag n etizatio n t h a t occu r in each one of th ese c ry stals are ju s t those w hich hav e been described, th e m a g n etiz atio n of th e w hole p o ly cry sta llin e m a te ria l being th e sum of th e m ag n etizatio n of th e p a rts.
T h e m o st d efin ite e v id en ce of th e existence of d o m a in s is th e B a rk h a u se n effect. T o p ro d u ce an d d e te c t it, a piece of m a g n e tic m a te ria l is w o u n d w ith w ire th e ends of w hich are c o n n e c te d to a v a c u u m tu b e am plifier. W h en th e m a g n e tiz a tio n of th e m a te ria l is ch an g ed , a s e.g. b y m oving a p e rm a n e n t m a g n e t n e a r it, a ru stlin g so u n d o r a series of clicks m a y be h ea rd in phones o r in a loud s p e a k e r 6 B E L L S Y S T E M T E C H N I C A L J O U R N A L
i
Fig. 3— D omains in a single crystal of iron. As th e m agnetic field increases in strength the magnetic mom ents first change suddenly (a to b) by displacem ent of th e boundaries between them , then ro tate sm oothly (b to c).
co n n e c te d to th e o u tp u t e n d of th e am plifier. E v e ry su ch click is ascrib ed to th e su d d e n c h an g e in d ire c tio n of m a g n e tiz a tio n in a single d o m ain , a n d from m e a su re m e n ts of th e sizes of th e clicks w e g e t o u r b e s t e s tim a te of th e sizes of th e d o m ain s. E v e n m o re d ire c t evid en ce of th e existen ce of d o m a in s a n d th e ch an g es t h a t th e y u n d erg o h a s been o b ta in e d re c e n tly b y sp re a d in g colloidal iron oxide o v er th e su rface of a m a g n e tic m a te ria l a n d looking a t it u n d e r a m icroscope.
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 7 T h e regular p a tte rn o bserved 1 is sim ilar in n a tu re to th e fam ilia r one o b ta in ed w hen iron filings are sprin k led n ear a p e rm a n e n t m a g n e t;
th e fine colloidal p a rticles are n ecessary in th is case b ecau se th e whole scale is sm all. T h is m ic ro -p a tte rn changes w hen th e ap p lied field changes, an d th e difference is a ttr ib u te d to th e re d istrib u tio n or re o rie n ta tio n of gro u p s of dom ains. T h ese p a tte rn s are o b ta in e d only on m ag n e tic m a te ria ls a n d are found on th e m even w hen th e m a te ria l is unm ag n etized ; such a one is show n in Fig. 4.
Fig. 4— The powder pattern produced by colloidal iron oxide on the surface of a demagnetized silicon-iron crystal, showing the presence of inhomogeneous mag
netic fields. Magnification about 1000.
Ma g n i t u d e s o f Ma g n e t i c Fo r c e s
F erro m ag n etic th e o ry h as been m ad e difficult b y th e fa c t t h a t th e m ag n etic forces b etw een th e electrons in an a to m are sm all com pared to th e e le c tro sta tic forces. T h e la tte r force betw een tw o electrons of charge e (in e.s.u.), a d istan ce a a p a rt, is eq u al to
e 2/ a 2.
T h e m ag n etic force betw een th e sam e electrons dep en d s on th e speed of th e charges as well as on th e ir m ag n itu d es, an d , w hen th e d irection of m otion is p erp en d icu la r to th e line joining them , is eq u al to
w here v/c is th e ra tio of th e speed of each electron to th e speed of light. Since vie is u su ally of th e o rd er of 0.01, th ese m ag n etic forces
1 L. W. M cKeehan and W. C. Elm ore, Phys. Rev., 46, 226-228 (1934). See also the earlier experiments by F. B itter, Phys. Rev., 41, 507-515 (1932). See also the account by Elmore in F . B itter’s Introduction to Ferrom agnetism , McGraw-Hill, New York, 55-66 (1937).
8 B E L L S Y S T E M T E C H N I C A L J O U R N A L
a re a b o u t 10~4 of th e e le c tro s ta tic forces. T h e difference is even g re a te r w hen e le c tro sta tic forces betw een electro n s a n d nuclei, or b etw een nuclei, are c o m p ared w ith m a g n etic forces. T h e m a g n itu d e s of th e se forces for a specific h y p o th e tic a l a rra n g e m e n t are show n in F ig. 5.
E LECTRO N
-e
/ /
/ ELECTROSTATIC FORCES
I IN DYNES
f i ---2 X IO "3 ---
--- 2 X I O '1--- * 0 NU CLEUS
+ 2 6 e
0
\
M A G N E T IC FORCES IN DYNES
— O R BITAL M OTIO NS, 2 X IO - 5 ---- ---S P IN S , 3 X tCT1 4 ---
ELECTRO N
-e
Fig. 5— T he m agnitudes of the forces in a hypothetical iron-like atom , showing th a t electrostatic forces are more powerful th a n m agnetic forces.
C onsider th e m a g n itu d e of m a g n etic forces from a n o th e r p o in t of view . T h e m a g n e tic en erg y of a p e rm a n e n t m a g n e t of m o m e n t ha
in a field of s tre n g th H is
E = - haH ,
w hen ha a n d I I a re p arallel. In a m a g n e tic su b sta n c e w e m a y re g a rd th e a to m ic m a g n e ts as being held p arallel b y a fictitio u s field H i.
W h en th e m a te ria l is h e a te d to th e C u rie te m p e ra tu re , 6, th e e n e rg y of th e rm a l a g ita tio n (p= kd) d e stro y s th e a lig n m e n t of th e ato m ic m a g n e ts b y th e fictitio u s o r “ in te r n a l” field H {. T h e n
kd ~ haHí.
F o r iron, 6 = 1043° K . a n d ha = 2.04 X 10 20 erg /g au ss, th u s th e
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 9 energ y p e r a to m is
kd = 1.4 X 10-13 e r g = 0.09 electro n -v o lt a n d th e in te rn a l field
H i = 7,000,000 oersteds.
A lth o u g h th is field is m u c h stro n g er th a n a n y so far produced in th e la b o ra to ry , th e en erg y involved is sm all com pared to t h a t w hich c o n tro ls chem ical b in ding. F o r exam ple, th e energ y of ionization of th e helium a to m is a b o u t 25 electro n v o lts. A n o th e r w a y of show ing t h a t th e m a g n e tic forces are sm all co m p ared to th e e lec tro sta tic forces holding a to m s to g e th e r, is to co m p are th e C urie te m p e ra tu re w ith th e te m p e ra tu re of v a p o riz a tio n .
T h e calcu latio n of m ag n e tic forces b y th e o ry is th u s ex trem ely difficult, b ecau se th e y are b u t sm all a d d itio n s to th e ele c tro sta tic forces w hich th em selv es c a n n o t u su ally b e c a lcu lated w ith m uch precision.
E w i n g ’s Th e o r y
E w ing 2 w as one of th e first to a tte m p t to explain ferrom agnetic p h en o m en a in te rm s of th e forces betw een ato m s. H is th e o ry will be d escribed briefly here, since m a n y physicists to d a y , w hen th in k in g a b o u t m a g n etic phenom ena, still go b a c k to E w in g ’s ideas of fifty y e a rs ago. H e assu m ed w ith W eb er t h a t ea ch a to m w as a p e rm a n e n t m a g n e t free to tu r n in a n y directio n a b o u t its ce n ter. T h e o rien tatio n s of th e v a rio u s m a g n e ts w ith re sp e c t to th e field an d to each o th e r w ere supposed to b e d u e e n tire ly to th e m u tu a l m ag n etic forces.
T h e / , H cu rv e an d h y steresis loop w ere calc u lated for a lin ear gro u p of su c h m a g n e ts a n d w ere d eterm in ed e x p erim en tally using m odels h av in g as m a n y as 130 m a g n e ts a rran g ed a t th e p o in ts of a plane s q u a re la ttic e .
T h e c alc u latio n s for a lin ear ch ain show t h a t as th e field is g ra d u ally in creased in m a g n itu d e from zero th e re is a t first a slow continuous ro ta tio n of th e m ag n ets, th e n a su d d en change in o rie n ta tio n and finally a fu rth e r co n tin u o u s ro ta tio n u n til th e m a g n ets lie parallel to th e field. T h e I , H cu rv es c a lcu lated for such a g ro u p of m a g n ets resem ble in g eneral form th e a c tu a l curves of iron : th e y show a p erm e
a b ility first increasing th e n decreasing, a n d s a tu ra tio n a n d hysteresis.
A m a g n e tiz a tio n cu rv e an d a h y steresis loop o b ta in e d 3 w ith a m odel of 130 m a g n e ts in sq u a re a rra y , are show n in Fig. 6. E xperi-
2J . A. Ewing summarized in “ M agnetic Induction in Iron and O ther M etals,”
T he Electrician, London, 3d ed. (1900).
3 J . A. Ewing and H . G. Klaassen, P hil. Trans. Roy. Soc., 184A, 985-1039 (1893).
10 B E L L S Y S T E M T E C H N I C A L J O U R N A L
m e n ts w ith th e m odel show ed a v a rie ty of o th e r p h en o m en a in clu d in g ro ta tio n a l h y ste resis loss a n d its red u ctio n to zero in high fields, th e effect of s tra in on m a g n e tiz a tio n , th e existence of h y steresis in th e s tra in vs. m a g n e tiz a tio n d ia g ra m , th e effect of v ib ra tio n an d th e ex isten ce of tim e lag a n d acco m m o d atio n w ith re p e a te d cycling of th e field.
E w in g ’s g eneral m e th o d m a y b e illu s tra te d b y c alc u la tin g th e m a g n e tiz a tio n cu rv e a n d h y ste resis loop for an infinite line of p arallel
Fig. 6—A m agnetization curve and hysteresis loops of a Ewing model of 130 pivoted magnets in square array.
e q u ally spaced m a g n ets (Fig. 7a). I t is done m o st sim ply b y co n sid erin g first th e m a g n e tic p o te n tia l en erg y 4 of a m a g n e t of m o m e n t Ha and length I, in th e field of a sim ilar m a g n e t:
w *= - 7 T
-
V ?P‘W - ‘y r P>(>)
--- • (*) H e re r is th e d ista n c e b etw een th e c e n te rs of th e m a g n e ts an d th e P ( 6 ) ’s are L egendre fu n c tio n s of th e angle, 6, b etw een th e d irec tio n of th e m o m e n t of th e m a g n e t a n d th e line jo in in g th e m a g n e t cen ters.Ps(9) = (1 + 2 cos 26)/4,
P i(d) = (9 + 20 cos 26 + 35 cos Ad)/64,
Pe(9) — (50 -f- 105 cos 26 + 126 cos 4 9 + 231 cos 60)/512.
T h e p o te n tia l en erg y p e r m a g n e t, W i, for a n infinite s tra ig h t row of m a g n e ts can easily be o b ta in e d b y sum m in g W for all pairs.
W i = - — [1.20 P 2(0) + l.O 4P4(0)(i/r)2
+ l.O lP e(0)(Z/r)4 + • • • ] • (2) 4 G. M ahajani, Phil. Trans. Roy. Soc., 228A, 63-114 (1929).
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 11 T h e b eh av io r of th e line w hen su b je c te d to a field H m a y be found b y a d d in g to W i th e energy te rm — Hha cos (0O — 6), w here 60 is th e angle b etw een th e line of c e n te rs an d th e d ire ctio n of th e field, an d
FIELD STRENGTH
Fig. 7—A m agnetization curve and hysteresis loop for an infinite line of equally spaced magnets originally “ demagnetized.”
finding th e v a lu e of 6 w hich m akes th is to ta l energy a m inim um for given valu es of do a n d I I :
~ [ W i - I Iha cos (0„ - ^ )] = 0.
T h is gives
n = (d/dd)W i
H Asin (00 — 6 ) '
12 B E L L S Y S T E M T E C H N I C A L J O U R N A L
T h e co m p o n e n t of m a g n e tiz a tio n parallel to H is I = /« c o s (0O - 0),
w here I , is th e s a tu ra tio n m ag n e tiz a tio n . B y s ta r tin g w ith h alf of th e line of m a g n e ts p o in tin g in a d irectio n o p p o site to t h a t of th e o th e r half, th e in itia l m a g n e tiz a tio n is zero an d an u n m a g n e tiz e d or d e m a g n etized m a te ria l is sim u la te d . T h u s a m a g n e tiz a tio n cu rv e an d a h y steresis loop of th is assem blage are o b ta in e d b y p lo ttin g H a g a in s t I.
S uch a p lo t is show n in F ig. 7(b), w ith th e scale of H d e te rm in e d b y th e m a g n itu d e s of h a a n d r. T h e cu rv es are o bviously sim ilar to th o se for real m a te ria ls.
Li m i t a t i o n s o f Ew i n g’s Th e o r y
So far, th is c a lc u latio n is e q u iv a le n t to w h a t E w ing d id o v er four d ecad es ago. B u t now w e know th e c ry sta l s tru c tu r e of iron a n d in p a rtic u la r th e d ista n c e s b etw een th e ato m s. W e also know th e m a g n e tic m o m e n t of eac h iron a to m a n d know , th erefo re, th e v a lu e of Ma/r3 w hich d e te rm in e s th e scale of H . U sing th e a p p ro p ria te v a lu e s ha = 2.0 X 10-20 erg /g a u ss a n d r = 2.5 X 10-8 cm , th e coercive force H c for l/r = 0.1 is found to be 4600 oersteds. T h is is affected som e
w h a t b y th e ra tio l/r, b u t in a n y case H c is found to be of th is o rd e r of m a g n itu d e unless l/r is v e ry close to u n ity . T h is m a g n itu d e of H c is g re a te r b y a fa c to r of 106 th a n th e low est v a lu e o b ta in e d ex p erim en ta lly , 0.01. S im ilarly th e in itia l p e rm e a b ility , h o, acco rd in g to th e m odel is a b o u t u n ity w hile ob serv ed v alu es for iro n ra n g e fro m 250 to 20,000. A d ju s tm e n t of l/r to h ig h er valu es d ecreases h o-
T h is ca lc u la tio n of th e m ag n e tiz a tio n cu rv e a n d h y steresis loop are b a se d on a v e ry m u c h idealized m odel, an d it is difficult to e s tim a te th e e rro r to w hich i t m a y le ad . O ne fa c to r t h a t h a s b een co m p letely n eg lected is th e flu c tu a tio n in en erg y . A m u ch b e tte r a p p ro x im a tio n w ould b e to c a lc u la te th e m a g n e tic p o te n tia l e n erg y of a g ro u p of m a g n e ts a rra n g e d in sp ace in th e sam e w ay t h a t th e iron (or nickel) a to m s are a rra n g e d in a c ry sta l. T h is h as been done b y M a h a ja n i 4 w ho show ed t h a t a p p lic a tio n of E q . (1) w ith / = 0 (b u t su m m ed to a c c o u n t for th e effects of all m a g n e ts in th e s tru c tu re ) lead s to th e re su lt t h a t th e m a g n e tic p o te n tia l of th e sp ace a rra y is in d e p e n d e n t of 0, in o th e r w ords one o rie n ta tio n of th e dipoles is a s sta b le a s a n y o th e r a n d th e m a g n e tiz a tio n cu rv e w ould go to s a tu ra tio n in in fin itesi
m a l fields no m a tte r in w h a t d ire c tio n H m ig h t be a p p lied . I f I is finite, th e sta b le p ositions of th e m a g n e ts are p arallel to th e b o d y - d iag o n als of th e cu b e w hich is th e u n it of th e c ry s ta l s tru c tu re , a n d
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 13 th is becom es therefore th e d irectio n of easy m ag n etiza tio n , a situ a tio n w hich is co rrec t for nickel b u t decidedly n o t so for iron. T h e b e st correspondence b etw een th e ac tio n of th e m odel a n d of iron itself is o b ta in e d if th e m odel is m ad e b y placing a sm all circu lar c u rre n t of electricity , in ste ad of a m a g n e t w ith finite len g th , a t each la ttic e p o in t of th e space a rra y . In th e la tte r case we can explain th e d irectio n of easy m ag n e tiz a tio n in iron an d th e v a ria tio n of m ag n etic en erg y w ith d irectio n in th e cry stal.
In considering E w in g ’s m odel it is a p p ro p ria te to e stim a te th e energy of th e rm a l a g ita tio n an d to co m pare it w ith th e m ag n etic p o te n tia l energ y as c a lc u la ted from th e m odel. S u b stitu tin g in E q . (2) th e sam e v alu es of ha an d r as w ere used above, we o b ta in 10-16 erg per a to m for th e m a g n e tic p o te n tia l energy in zero field. T h is is to be co m p ared w ith th e ro ta tio n a l energy of a single m olecule a t room te m p e ra tu re , 2 X 10-14 erg p er a to m as given b y th e kin etic th e o ry . T h u s th e en erg y of th e rm a l a g ita tio n is 200 tim es as g re a t as th e calcu la te d m a g n e tic energy. E v en a t liquid a ir te m p e ra tu re s th e th e rm a l a g ita tio n w ould p re v e n t th e ato m ic m a g n e ts from form ing sta b le co n figurations. W ith o u t som e ad d itio n a l force th e m odel E w ing used w ould b eh a v e as a p a ra m a g n e tic ra th e r th a n a ferrom agnetic solid.
In a real m a te ria l, how ever, it is now well established t h a t th e re are v e ry pow erful forces, n o t co n te m p la te d w hen E w ing m ade his m odel an d proposed his th e o ry , w hich m a in ta in parallel th e dipole m o m en ts of neighboring ato m s. T h ese are th e e lectro static forces of exchange (see p. 24) w hich H eisenberg suggested are pow erful enough to align th e e le m e n ta ry m a g n ets a g a in st th e disordering forces of th e rm a l a g ita tio n , forces m u ch larg er th a n th o se of m ag n etic origin. T h e o ry acco u n ts only for th e o rd er of m a g n itu d e of th ese forces. O u r b e s t e stim a te of th e c o rresponding energy of m ag n e tizatio n is o b tain ed b y assum ing t h a t it is eq u al to th e en erg y of th e rm al a g ita tio n a t th e C urie point,
\k d . F o r iron (6 = 1043 °K ) th is gives 7 X 10“ 14 erg per ato m . Th e We i s s Th e o r y
In o rd er to u n d e rsta n d how ato m ic forces give rise to ferrom agnetism i t is d esirab le to review briefly W eiss’s th e o ry 6 of ferrom agnetism , w hich in tro d u ces a so-called “ m olecular fie ld ’’ th a t p resen tly will be identified w ith th e n a tu re of th ese forces. T h is th e o ry is a n extension of L a n g e v in ’s th e o ry of a p a ram ag n e tic gas. T h e original L angevin th e o ry c u lm in a te d in a fo rm u la re la tin g th e m ag n etizatio n , I , to th e field-strength, I I, a n d th e te m p e ra tu re , T ; th is is th e h yperbolic co-
6 P. Weiss, Jour, de physique (4) 6, 661-690 (1907). P. Weiss and G. Foex, “ Le M agnetism e,” Colin, Paris (1926).
14 B E L L S Y S T E M T E C H N I C A L J O U R N A L ta n g e n t law,
I _ . . ¡J-aH k T h ~ C n k T haH '
In d e riv in g th is th e a ssu m p tio n s a re m ad e t h a t th e e le m e n ta ry m a g n e ts, each of m o m e n t /xa, a re su b je c t to th e rm a l a g ita tio n a n d m o m e n ta rily m a y h a v e a n y o rie n ta tio n w ith re sp e c t to th e d ire c tio n of th e field, a n d t h a t th e y a re to o fa r a p a r t to influence each o th e r. Q u a n tu m th e o ry a lte rs th e second of th o se a ssu m p tio n s b y s ta tin g t h a t in such a n en sem b le of e le m e n ta ry m a g n e ts (atom s) th e re will be o n ly a lim ite d n u m b e r of possible o rie n ta tio n s, in th e sim p lest case o n ly tw o , one p arallel a n d th e o th e r a n tip a ra lle l to th e d irectio n of th e field. In th is case th e e q u a tio n co rresp o n d in g to L a n g e v in ’s is
7. = tanhT # - (3)
T h e se tw o th e o re tic a l relatio n s are p lo tte d for v a ria b le H a n d c o n s ta n t T (room te m p e ra tu re ) in F ig. 8, th e c o n s ta n ts b ein g th o se for
Fig. 8— W ith no helpful m utual action between atom s, enorm ous fields would be necessary to satu rate a m agnetic m aterial.
iron (/o = 1740, ha = 2.04 X 10-20 erg /g au ss). I t is o b v io u s t h a t w ith th e h ig h e st fields so fa r a tta in e d in th e la b o ra to ry (a b o u t 300,000 o ersted s) th e m a g n e tiz a tio n w ould a tta in o n ly a sm all fra c tio n of its final v a lu e Io if th is law w ere obeyed, a n d in th is ra n g e I w ould be sen sib ly p ro p o rtio n a l to th e fie ld -s tre n g th :
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 15 w here C is a c o n sta n t. T h is relation, know n as C u rie ’s Law , is obeyed b y som e paramagnetic th o u g h n o t b y ferro m ag n etic su b sta n c e s. I t is usually w ritte n w ith I / H d e n o te d b y th e sym bol
x>
rep resen tin g su sc e p tib ility :C X = J . ’
M a n y m ore p a ram ag n etic su b stan c es obey th e sim ilar “ C urie-W eiss Law ” :
I = (4)
t - e K J
W eiss p ointed o u t th e significance of 6 in th is e q u a tio n : it m eans t h a t th e m a te ria l b eh av es m ag n etically as if th e re w ere an ad d itio n a l field, N I , aiding th e tru e field I I . T h is equivalence is show n m a th e m a tic ally b y p u ttin g 6 = N C in E q . (4) w ith th e re su lt
C ( I I + N I ) T
T h e q u a n tity rep resen ted b y N I is called th e "molecular fie ld ” an d t h a t b y N th e “ molecular field constant.” I t is in te rp re te d b y su p p o s
ing t h a t th e elem en tary m a g n e t does hav e an influence on its neighbors, c o n tra ry to th e assu m p tio n s of th e sim ple L angevin th eo ry .
T h e significance of th e m olecular field for ferrom agnetism is now a p p a re n t if we replace th e H b y H + N I in th e m ore general E q . (3) and exam ine th e resu ltin g e q u a tio n :
. f . t a n h
S A S + S H .
(5)1 o Kl
T h is eq u a tio n is perh ap s th e m o st im p o rta n t in th e th e o ry of ferro
m agnetism . I t in d icates t h a t even in zero field th e re is still a m ag n e tiz atio n of considerable m ag n itu d e, provided th e te m p e ra tu re is n o t too high. P u ttin g I I = 0 and
E q . (5) reduces to
6 — haN I o/k,
T, = T je ' (6)
T h is p u rp o rts to specify th e m a g n etiza tio n a t zero applied field b y a fu n ctio n t h a t is th e sam e for all m aterials, w hen th e m a g n etizatio n is expressed as a fractio n of its value a t ab so lu te zero and th e te m p e ra tu re as a fractio n of th e C urie te m p e ra tu re on th e ab so lu te scale. T h is m a g n etiza tio n vs. te m p e ra tu re relation, p lo tte d as th e solid line of Fig.
16 B E L L S Y S T E M T E C H N I C A L J O U R N A L
9, m ean s t h a t at all temperatures below 0 the intensity o f magnetization has a definite value even when no field is applied.
H ow is it th e n t h a t a piece of iron can a p p a re n tly b e u n m ag n etized a t room te m p e ra tu re ? T h e answ er, given b y W eiss, is t h a t below th e C u rie p o in t all p a r ts of th e iron are m ag n etized to s a tu ra tio n b u t t h a t d ifferen t p a rts a re m ag n etized in d ifferen t d ire c tio n s so t h a t th e o v erall
Ldt t 0 .4 II
J?|J?
0.3
0
0 0.1 0 .2 0 .3 0 .4 0.5 0.6 0.7 0 .8 0 9 10
J_e
Fig. 9— Dependence on the tem perature of the saturation m agnetization of iron, cobalt and nickel, as com pared w ith theory.
effect is zero. T h is is th e c o n c e p t of th e d o m ain , a lre a d y d iscu ssed . A cco rd in g to th is c o n cep tio n th e I of E q . (5) is t h a t of a d o m a in an d is d e te rm in e d e x p e rim e n ta lly b y m e a su rin g th e m a g n e tiz a tio n of a specim en w hen all d o m a in s a re parallel, i.e., a t (tech n ical) s a tu r a tio n ( / = I s). E q . (6) should th e n b e w ritte n
h P h
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 17 I t is a problem of th e o retical physics to d e te rm in e th e n a tu r e of th e m olecular field. Before discussing w h a t progress h as been m a d e in d o in g th is it will be necessary to review som e of o u r know ledge of th e s tr u c tu r e of th e a to m s w ith w hich we are co ncerned.
At o m i c St r u c t u r e o f Fe r r o m a g n e t i c M a t e r i a l s
T h e s tru c tu re of an iso lated iron a to m h a s a lre a d y b een show n in F ig. 1. T h e tw e n ty -six elec tro n s a re d iv id ed in to four principal
“ sh ells,” each shell a m ore o r less well defined region in w hich th e electro n s m ove in th e ir “ o rb its .” T h e first (in n erm o st) shell c o n tain s tw o electro n s, th e n e x t shell e ig h t, th e n e x t sixteen, a n d th e la st tw o.
A s th e periodic sy ste m of th e ele m en ts is b u ilt u p from th e lig h test elem ent, hyd ro g en , th e fo rm a tio n of th e in n erm o st shell begins first, an d w hen co m p leted th e n u m b e rs of electro n s in th e first four shells are tw o, eig h t, eig h teen , a n d th irty -tw o , b u t th e m ax im u m n u m b e r in each shell is n o t alw ays re ac h ed before th e n e x t shell begins to be form ed. F o r exam ple, w h en fo rm atio n of th e fo u rth shell begins, th e th ird shell c o n ta in s o n ly e ig h t electrons in ste a d of e ig h te e n ; it is th e su b se q u e n t b u ild in g u p of th is th ird shell t h a t is in tim a te ly connected w ith ferro m ag n etism . In th is shell som e electrons w ill be sp in n in g in one d irectio n a n d o th e rs in th e opposite, an d th e se tw o senses of th e spins m a y be c o n v e n ie n tly referred to a s positive a n d negative. T h e n u m b ers on th e circles show how m a n y electrons w ith + a n d — spins are p re se n t in e a c h shell in iron a n d it will be noticed t h a t all ex cep t th e th ird shell c o n ta in as m a n y electro n s spinning in one directio n as in th e opposite. T h e m a g n e tic m o m e n ts of th e electrons in each of these shells m u tu a lly c o m p e n sa te one a n o th e r so th a t th e shell is m ag n etic
ally n e u tra l a n d does n o t h a v e a p erm a n en t m agnetic m om ent. In th e th ird shell, how ever, th e re a re five electrons w ith a positive spin an d one w ith a n e g ativ e so t h a t four electron spins are u n b alan ced or u n co m p en sa ted a n d th e re is a re su lta n t polarization of th e a to m as a w hole. T h e existence of a p erm an e n t m agnetic m o m en t for each ato m ob v io u sly satisfies one of th e requirem ents for ferrom agnetism .
In th e free a to m th e o rb ita l m otions of th e electrons also co n trib u te to th e m a g n e tic m om ent. W hen th e iron ato m becom es p a r t of m e ta llic iron th e electron o rb its becom e too firmly fixed in th e solid s tru c tu re to be influenced appreciably b y a m ag n etic field. T h e co rresp o n d in g m om ents do n o t change w hen th e in te n sity of m a g n e tiz a tio n c h a n g e s — th is is shown by th e gyrom agnetic experim ents discussed l a t e r a n c j ; t is s u p p o s e d th a t th e o rb ita l m om ents of th e electrons in
n e u t r a l i z e one a n o th er, v a rio u s ato m s n euuaiu-
18 B E L L S Y S T E M T E C H N I C A L J O U R N A L
In th e solid s tr u c tu r e n eighboring a to m s influence th e m o tio n an d d is trib u tio n of electrons, p a rtic u la rly in th e th ird p a r t of th e th ird shell (3d shell) a n d th e first p a r t of th e fo u rth shell (45 shell). In F ig. 10 th e difference b etw een a free a to m an d one t h a t is p a r t of a m e ta l is illu stra te d . E a c h of th e te n places for electro n s in th e 3d shell is re p re se n te d b y an a re a w h ich is sh ad ed if t h a t p lace is occu p ied . T h e d is trib u tio n corresponds in (a) to a n isolated a to m of nickel, in (b) to a nickel a to m in a m e ta l; in th e la tte r s itu a tio n th e re is on the average 0.6 electro n p er a to m in th e 45 shell (these electro n s are loosely b o u n d a n d are th e free electrons responsible for electric co n d u c tio n ) a n d a v a c a n c y o r hole of 0.6 electro n p e r a to m in th e 3d-shell.6 In th e 45 shell th e n u m b e r of electro n s w ith + an d w ith — sp in a re a lm o st e x a c tly eq u al, b u t in th e 3d shell all of th e sp aces for + spin a re filled.
T h e difference b etw een th e n u m b ers of + an d — spins is e q u a l to th e n e t m a g n e tic m o m e n t p er a to m . E x p e rim e n ta lly th e difference in th e n u m b e r of + spins an d — spins in an a to m is d e te rm in e d fro m th e s a tu ra tio n in te n s ity of m a g n e tiz a tio n a t a b so lu te zero. W h e n th is difference is one th e a to m h a s a m o m en t of one B ohr m a g n eto n ,
mb = 9.2 X 10-21 erg/gauss
co n seq u en tly th e n u m b e r of B o h r m a g n e to n s ca n b e c a lc u la te d from th e a to m ic w eig h t, A , a n d th e d e n sity , d :
B ohr m a g n e to n s/a to m = /3 — —IoA• Msd
In Fig. 10 (/) th e d iag ram for nickel is re p eated , th is tim e w ith th e to p s of th e unfilled positions on th e sam e level to b rin g o u t a n an alo g y w ith th e filling of vessels w ith w a te r. D ia g ra m s for m an g an ese, iron, c o b alt, nickel a n d c o p p e r are show n in p a rts (c) to (g). I n ea c h case th e 18 electro n s in closed shells are n o t show n. In iron th e s itu a tio n is so m ew h at d ifferen t from t h a t in nickel, n e ith e r th e 3 d + n o r th e 3d — shell is filled. T h is follows from th e re la tiv e c o n sta n c y of th e n u m b e r of electro n s in 45, from th e excess of holes in 3 d + o v er th o se in 3d — (/3 = 2.2), an d from th e to ta l n u m b er, 26, of e x tra -n u c le a r electrons.
T h e d istrib u tio n in sp ac e of electro n s belonging to th e 3d a n d 45 shells is know n a p p ro x im a te ly 7 a n d is d e p icted in Fig. 11. In (a) th e o rd in a te show s th e n u m b e r of electro n s th e re a re a t v a rio u s d ista n c e s from th e nucleus. T h e 3d shell is th u s seen to be a ra th e r d en se rin g
6 E . C. Stoner, Phil. Mag., 15, 1018-1034 (1933); N. F . M o tt, Proc. Phys. Soc.
47, 571-588 (1935); L. Pauling, Phys. Rev., 54, 899-904 (1938).
7 Calculations were based on th e equation given by J. C. Slater, Phys. Rev 36
57-64 (1930). ’ ’
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 19
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20 B E L L S Y S T E M T E C H N I C A L J O U R N A L
of electro n s, a s c o n tra s te d w ith th e 4s shell w hich ex te n d s fa rth e r from th e nucleus, so fa r t h a t in th e solid th e shells of n eig h b o rin g a to m s o v e r
lap c o n sid erab ly . In (b) th e n u m b e r of e lectro n s h a v in g e n e rg y b e
tw een E a n d E + d E is p lo tte d a g a in st th e e n e rg y E ; th is re p re s e n ta tio n is sim ilar to t h a t of Fig. 10 b u t now th e sq u a re s a n d rectan g les are rep laced b y th e m ore a p p ro p ria te cu rv ed surfaces. If (b) is tu rn e d 90 re la tiv e to (a) th e tw o p airs of cu rv es b e a r som e resem b lan ce to each o th e r. T h is is so because th e e n erg y of b in d in g is g en erally less a t g re a te r d ista n c e s from th e nucleus. T h e 3d-\- level is re p resen ted as low er in energ y th a n th e 3 d — since one of th ese b a n d s is p referred .
(a) (b)
F 'f : 11—T he filling of electron positions in iron, and some elem ents near it in the periodic table. Electron positions for closed shells, containing 18 electrons are not shown.
T h e a re a enclosed b y each 3d cu rv e co rresp o n d s to 5 electro n s w hile t h a t enclosed b y th e 4s co rresp o n d s to 2.
T h e line “ F e ” in Fig. 11(6) re p re se n ts th e lim it to w hich th e 3d a n d 4s shells are filled in iro n ; n e ith e r 3 d + n o r 3 d — is c o m p letely full. T h e low est en erg y levels a re filled first, a n d th e p ic tu re is d ra w n so t h a t th e an alo g y w ith th e filling of co n n ected vessels w ith w a te r is a p p a re n t.
In c o b a lt a n d nickel th e e x tra one a n d tw o electro n s co m p le te ly fill 3d + b u t n o t 3 d - , as in d ic a te d b y th e line “ N i ” for nickel. S ince th e ran g e of e n e rg y in th e 3d “ b a n d s ” is m u c h g re a te r th a n in th e 4s b a n d s th e a d d itio n a l electro n s do n o t a lte r g re a tly th e n u m b e r in 4s, a n d from th e s a tu ra tio n in te n s ity of nickel we e s tim a te th is n u m b e r as 0.6. In co p p e r th e ad d itio n a l ele ctro n is sufficient to fill b o th 3d shells w ith one e lectro n to sp are, a n d th is electro n m u s t go in to th e
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 21 45 shell w hich th e n becom es h alf full as show n b y th e line “ C u ” as well as b y (g) of Fig. 10. T h e d iag ram does n o t show changes in th e re lativ e levels of th e 3d + an d 3 d — b an d s t h a t occur in going from one elem ent to a n o th e r; w hen b o th 3d b a n d s are filled, as in copper, th ese levels a re th e sam e. T h e n u m b ers of electrons an d “ h o le s ” in m etals n e a r iron in th e periodic ta b le are given in T a b le I. A m ore
TABLE I
Nu m b e r o f El e c t r o n s a n d Va c a n c i e s ( Ho l e s) i n Va r i o u s Sh e l l s i n Me t a l At o m s Ne a r Ir o n i n t h e Pe r i o d i c Ta b l e
Elem ent
N um ber of electrons in following shells
T otal
Holes in
Excess holes in 3 d — over 3 d +
3 d + 3d — 4 s + 45 — 3 d + 3 d -
Cr 2.7 2.7 0.3 0.3 6 2.3 2.3 0
Mn 3.2 3.2 0.3 0.3 7 1.8 1.8 0
Fe 4.8 2.6 0.3 0.3 8 0.2 2.4 2.22
Co 5 3.3 0.35 0.35 9 0 1.7 1.70
Ni 5 4.4 0.3 0.3 10 0 0.6 0.61
Cu 5 5 0.5 0.5 11 0 0 0
a c c u ra te d e te rm in a tio n of th e form of th e 3d a n d 4s b an d s for copper is given in Fig. 12, due to S la te r.8
A n especially sim ple an d in te re stin g illu stra tio n of th e atom -m odel d escribed is afforded b y th e alloys of nickel an d copper. T h e s u b s titu tio n of one copper for one nickel a to m in th e la ttic e is e q u iv alen t to a d d in g one electron to th e alloy. T h is electron seeks th e place of low est energy in th e alloy and finds it in th e 3d-shell of a nickel ato m ra th e r th a n in th e copper ato m to w hich it originally belonged. T h is lowers th e m ag n etic s a tu ra tio n of th e alloy b y one B ohr u n it, since th e a d d e d electron in th e 3 d — b an d ju s t n eu tralizes th e m o m en t of one in th e 3 d + b an d . A ddition of m ore copper to nickel decreases th e av erag e m o m e n t u ntil th e e m p ty spaces in th e 3 d — b an d are ju s t full;
th is occurs w hen 60 per c e n t of th e ato m s are copper, and th e n th e m ag n etic s a tu ra tio n a t 0° K will be ju s t zero. T h is is th e ex p lan atio n of th e ex p erim en tal re su lts 9 show n in Fig. 13. T h e re are show n also th e s a tu ra tio n m o m en ts for o th e r alloys of nickel; it is ev id e n t th a t zinc w ith tw o 45 electrons fills up th e 3d b an d tw ice as fa st as copper, alu m in u m th re e tim es as fast, silicon an d tin four tim es an d an tim o n y five, in good accord w ith th eo ry . In each of th ese cases th e ad d ed
8 J. C. Slater, Phys. Rev., 49, 537-545 (1936).
9 V. M arian, A n n . de Physique (11), 7, 459-527 (1937). Some of the d ata for the other alloys shown in Fig. 12 are taken from C. Sadron, A nn. de Physique, 17, 371-452 (1932). T he interpretation of these results is due to E . C. Stoner, ref. 6.
22 B E L L S Y S T E M T E C H N I C A L J O U R N A L
a to m s h a v e filled u p 3d b an d s, losing th e ir m o re loosely b o u n d 4s electro n s w hen th e re a re a v a ila b le places of low er en erg y . T h e d a ta for p a lla d iu m in d ic a te t h a t th is elem en t h as th e sam e n u m b e r of o u te r
NUMBER OF Q U AN TU M STATES PER U N IT EN ER G Y
Fig. 12— E nergy levels in the 3d and 4s shells in copper, according to S later.
Sim ilar levels are believed to exist in nickel and cobalt w ith th e levels filled to “ 10”
and “ 9 ” respectively.
electro n s as n ic k e l; th is m ig h t be ex p ected since p a lla d iu m lies d ire c tly below nickel in th e periodic ta b le . W h en th e sim ilar b u t h e av ier p la tin u m is ad d e d to nickel, th e d ecrease in av e ra g e a to m ic m o m e n t
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 23 indicates t h a t som e of th e o u te r electrons of p la tin u m go in to th e 3d b an d of nickel, b u t t h a t th e y do n o t fill th is level as ra p id ly as th e o u te r electrons of cop p er do w hen th is elem en t is ad d ed .
E lectro n shells t h a t a re co m p letely filled b eh av e m ore like h a rd elastic spheres th a n th o se w hich a re only p a rtia lly filled. In solid co p p er w ith
Fig. 13—The saturation m agnetization of nickel decreases upon the addition of other elements having 1 ,2 ,3 , . . . electrons in th e outerm ost shell.
a com plete 3d shell a n d a 4s shell ju s t begun, th e 4s electrons “ o v erlap ” th o se of neighboring a to m s so m u ch t h a t th e ir connection w ith a n y one a to m is lo st; th e 3d shells on th e o th e r h an d hav e v e ry little overlap w ith neighboring ato m s. In th e ferro m ag n etic m e ta ls th e 3d shells a re in co m p lete a n d th e overlap is g re a te r th a n in co p p er; th is affects th e in te ra c tio n responsible for th e W eiss m olecular field, now to be
24 B E L L S Y S T E M T E C H N I C A L J O U R N A L
discussed. B u t c o p p e r w ould n o t be ferro m ag n etic ev en if th e in te r
a c tio n w ere large, b ecau se th e com pleted shell m ean s t h a t th e s a tu r a tio n m ag n e tiz a tio n is z e ro ; in re a lity co p p er is d ia m a g n e tic .
A m ore d e ta ile d discussion of th e ato m ic s tr u c tu re of m etals, p a r tic u la rly of th e b a n d p ic tu re of th e ferro m ag n etic m e ta ls, is given in a re c e n t artic le in th is jo u rn a l b y W . S h ockley.10
In t e r p r e t a t i o n o f t h e Mo l e c u l a r Fi e l d
I t w as show n b y H eisen b erg 11 t h a t th e m o lecu lar field can be ex
plained in te rm s of th e q u a n tu m m echanical forces of exch an g e actin g b etw een electro n s in neig h b o rin g ato m s. Im a g in e tw o a to m s som e d ista n c e a p a rt, each a to m h a v in g a m a g n e tic m o m e n t of one B ohr m a g n eto n d u e to th e sp in m o m e n t of one electro n . A force of in te r a c tio n h as been show n to ex ist b etw een th em , in a d d itio n to th e b e tte r- know n e le c tro sta tic an d (m u ch w eaker) m a g n e tic forces. I t is know n th a t, as one w ould ex pect, such forces are negligible w hen th e a to m s are tw o or th re e tim es as fa r a p a r t a s th e y are in c ry sta ls. I t is su p p o sed also, on th e basis of c a lcu latio n s b y B e th e ,12 t h a t a s tw o a to m s are b ro u g h t n e a r to each o th e r from a d ista n c e th e se forces cau se th e elec
tro n spins in th e tw o a to m s to becom e p arallel (p o sitiv e in te r a c tio n ) . As th e a to m s a re b ro u g h t n e a re r to g e th e r th e sp in -m o m e n ts a re held parallel m ore firm ly u n til a t a c e rta in d ista n c e th e force d im in ish es an d th e n becom es zero, a n d w ith still closer a p p ro a c h th e s p in s se t th em selv es a n tip a ra lle l w ith re la tiv e ly stro n g forces (n e g a tiv e in te r a c tio n ). In th e c u rv e of Fig. 14 th e energies c o rresp o n d in g to th e se forces a re show n as a fu n c tio n of th e d ista n c e s b etw een a to m s.
B e th e ’s cu rv e w as d raw n orig in ally for a to m s w ith d efin ite shell ra d ii a n d v a ry in g in tern u clei d ista n ces. I t m a y eq u ally well be used for a series of elem en ts if w e ta k e a c c o u n t of th e d iffe ren t ra d ii of th e shell in w hich th e m a g n e tic m o m e n t resides. T h e c rite rio n of o v erla p p in g or in te ra c tio n for th e m e ta ls of th e iron g ro u p is th e ra d iu s, R , of th e a to m (half th e in te rn u c le a r d ista n c e in th e c ry sta l) d iv id ed b y th e rad iu s, r, of th e 3d shell. In Fig. 14 th is ra tio R / r h as been used as ab scissa a n d th e elem en ts iron, c o b a lt a n d nickel h a v e been given a p p ro p ria te p o sitio n s on th e cu rv e. T h e re c e n tly d isco v ered ferro m ag n e tism of g adolinium 13 is a p p a re n tly a sso ciate d w ith a larg e R / r a n d sm all in te ra c tio n , a s co m p are d to nickel. I t is p laced on th e cu rv e accordingly. S la te r 7 has show n t h a t th e ra tio R / r is larg er in th e
10 W. Shockley, Bell System Technical Journal, 18, 645-723 (1939).
11 W. Heisenberg, Z . f . Physik, 49, 619-636 (1928).
12 H . B ethe, Handbuch der Physik, 24, p t. 2, 595-598 (1933).
13 G. U rbain, P. Weiss, and F . Trom be, Compt. Rend., 200, 2132-2134 (1935).
T H E P H Y S I C A L B A S I S OF F E R R O M A G N E T I S M 25
R_ _ ATOMIC SEPARATION T — DIAM ETER OF 3 d ORBIT
Fig. 14—B ethe’s curve relating the energy of magnetization to the distance be
tween atom -centers, with a fixed diam eter of the unfilled inner shell th a t has the magnetic moment.
ferro m ag n etic elem ents th a n in o th e r elem ents h av in g in com plete inner shells, a n d t h a t th e p o in t a t w hich th e cu rv e crosses from th e n o n ferro m ag n etic to th e ferro m ag n etic region is n e a r R / r = 1.5. V alues of 2R, 2r an d R /r , as calcu lated b y S la te r for som e of th e elem ents w ith in co m p lete in n e r shells, a re given in T a b le II.
TA BLE II
I n t e r n u c l e a r D i s t a n c e s ( 2R ) a n d D i a m e t e r s ( 2 r ) o f I n c o m p l e t e I n n e r S h e l l s o f S o m e A t o m s , i n A n g s t r o m s
Atom 2 R
Inner Shell 2 r
Ratio R/r
Incomplete Inner
Shell
Curie T em perature
6, °K.
Mn 2.52 1.71 1.47 3d
Fe 2.50 1.58 1.63 3d 1040
Co 2.51 1.38 1.82 3d 1400
Ni 2.50 1.27 1.97 3d 630
Cu-M n 2.58 1.44 1.79 3d 600
Mo 2.72 2.94 0.92 U
Ru 2.64 2.33 1.13 4 d
Rh 2.70 2.11 1.28 U
Pd 2.73 1.93 1.41 4 d
Gd* 3.35 1.08 3.1 4/ 290
W 2.73 3.44 0.79 5 d
Os 2.71 2.72 1.02 5 d
Ir 2.70 2.47 1.09 5 d
Pt 2.77 2.25 1.23 5d
* Calculated using Slater’s formula.
T h e e n erg y of in te ra c tio n , J — th e p ositive o rd in a te of F ig. 14— can be e s tim a te d fro m th e v a lu e of th e C urie te m p e ra tu re , 6, in a m a n n e r su g g ested b y S to n e r.14
L e t 2 / be th e difference in th e energy of interaction betw een tw o atom s when th e ir m om ents are respectively parallel and antiparallel. T he to ta l energy of these tw o atom s is therefore
2 E = 2E0 ± J
where E 0 is th e energy of an isolated atom . T he negative sign applies when th e spins are parallel, th e positive when they are antiparallel. Im agine a crystal in which each atom of m om ent ya is surrounded a t equal distances b y z other atom s of which x have th eir spins parallel and y antiparallel. T hen tu rn in g one atom from th e parallel to antiparallel position produces a change of (y — x) in th e num ber of parallel pairs and (x — y) in th e num ber of antiparallel pairs and, therefore, requires an energy
« = 2 J (x - y). (5)
Since in each atom th e m om ent m ust be parallel or antiparallel to th e field, the m agnetization of the m aterial as a whole will depend on the average value of x — y:
I/Io = (x - y)/z. (6)
According to B oltzm ann’s equation an atom will have th e following probabilities of being parallel and antiparallel
P p = 1/[1 + exp ( - e/kT)2
P a = exp ( — e/kT)/[_ 1 + exp ( — t!kT)~\.
Since all atom s behave in th e same way on the average x and y m ust be zP p and zPa. Hence we have
26 B E L L S Y S T E M T E C H N I C A L J O U R N A L
H Io = (x — y)/z = P p — P a ~ ta n h (e/2k T ) or using (5) and (6)
To
= tanh (fifo)-
Com paring th is w ith th e modified Weiss equation, E q. (4), 1 _ u » ¿ N I _ I / I 0
Io k T Tie
we have J in term s of the molecular field constant or th e Curie tem perature:
J = haN IoIz — kd/z.
F o r iron, z = 8, J = kd/8 = 1.8 X 10~14 erg or 0.01 electron volt.
T his derivation indicates th a t J is proportional to d, and th a t the constant of proportionality depends on th e num ber of nearest neighbors. T he num ber of neighbors has not been tak en into account in the following discussion of Fig. 14.
T h e in te ra c tio n c u rv e is s u b s ta n tia te d in a q u a lita tiv e m a n n e r b y th e ob serv ed v a ria tio n of th e C u rie p o in ts of th e iron-nickel a llo y s.15 14 E . C. Stoner, P hil. Mag., 10, 27-48 (1930). S toner’s original work appears to have been in error by a factor of tw o; th e modified tre a tm e n t given here is due to W. Shockley and follows closely the m ethod employed in dealing w ith order and disorder in alloys (see e.g. E qs. 1.11, 1.12, 2.2 and 2.16 in th e article by F . C. Nix and W. Shockley, Rev. Mod. Phys. 10, 1-71 (1938)).
16Sum m arized by J. S. M arsh, A lloys of Iron and Nickel, v. 1, pp. 45 and 142, M cGraw-Hill, New Y ork (1938).
