BSTJ 40: 5. September 1961: The Calculations of Metallic Hall Constants: Topological Aspects and Applications to Copper. (Klauder, John R.)

8.6 MB·English

The Calculation of Metallic Hall
Topological Aspects
and Applications to Copper

Constants:

By JOHN RL KIATIIER

"he teed enti of ho Tt eastant i wets ype:
igh waguetie ett diene, om pat om a ropond oneptatins of
oe Simovionls Bank nf score govaed eoodimsioat
‘laws io ony oso nat Be ona te $d ge
Shee ofthe Fon aac, the wonder of wncduactng toed contr
tht suscot m atrmy put. The sutra te le etoe
io Be charge ie reo” ete eco i fh plone eae
Diva sshow exnambaton (ake Ho eon Hated an tes ots ee,
pple of the abn Wirusrt sm thde paper kaw bn wk fa
Iie Pipa sel for eo, so serv ourertontoce die raat
to an fn Bema hve tein othe Hat conan

(he we Be srs tay aus a prin yal

praphie arte.
When a eureat earring rovtllie sumple i pared in ¢ unifort mage

hel a eure tae rie generally ceed, Thx phe
omeno, kaw as magertisrastny, ie ie ss for Stnperatt
fexpoviseta erhniqne for -inyng he tad steoctuge of mals Ha
imple midel ie taken to epreeuta rata ey an effective mates

Dresimato, sesstantwlaxatio tine el), cn the experted hare
jn sistance con be theoretically praised Even oth mote teal
ode, i fond that ou expisimensally claersahls Seatore can sil
be protic, aly, the Hall eoustant. The alti espresion or
the Hall onstant fprernted i (41 isan integral aver a feity of
parallel yvuinnsnmed ies, Hor cach plane the ingen ie pe
1 su pats SveTrAC nIATeNR Sor, MeSH 4

portional ve she ar of thet part of che ple wurst by 2 eld
Inurber uf remcntuars la tnction dein’ in Pal ae
Tredition a” Us rvnlanes pexameer tures ott eee to be a
onowtat somplivatnd analyrical tet, al toe pepor sree some
Snevesing certs oa nantes mesial deed yet ance to
fala he Hall stant. pas, the aera precede repel
rettese apoaya! veg oa Tari oto tines
Plane in oder to evante n pectin tpgieal mratiant seated

‘vit hat plane
Stripped of the qurtiular pyelacl problem witch eotivales the
study, the eswoe aft Lopiegial ingniry © che following: Suppoee

ther i 9 vomtainer with may piers oF string of varying length. The
fede ofeach reve of sng a ed tater 0 tac pi oes a
flowed loos Axsnany lope of sng ws desired ane paved et edke,
subject ntly to the roguiemens That sey di ot tie one another
itshould be enmobisize} (hat IoayswithiaLoope ane povmitted (se Pi
1 fra site example). Now, sleet & point on the table ror an a
string iel atl gk fr the washer of lope whieh swround tut point
{tun or examnpe, pint 1 it ig. 1 iesueeounded by axe loop, loop a
‘Theis clely topological questi ave the munsbor of loops which
faveonnd point ie avast yan abitasydieplceraat of eal
hop — displscemert whieh adit does not permit aay loop 0
se thtoagh the pf Savenigation.
WeSKLNe nau Cavers ste 151
1 simple peti ce soni help testy the role af topology in

Ve eal ose bight Hal const. Fig. 2 sha hype

cal energy surfiee wise eahie rsmutiy, compre of “aphid
‘rylinon.” "The inert ofthis saface with ech of family of

tall panes deli the self conta that are perliwat for the
Hal coset vallatiss, Foe deiiesoosy,evurider thew pas,
Sf which ane Mustaced img. 2, peepenionie to the [O01] ans. If
he pla, ihe plane afi Fig 2 eta ara the elinders, then ite the
ara of the varone eylindhral cre sections tbat enntabutes to the
Tal onstantraulation. However or plane the second plane shown
in Fg. 2, the comtslstion to The Hal onan i based oh the plane

ava ona Un spl aud eylinders sane now the vegion? in that
plane surrunded by an ed aie of centoure be uae the yes
UOnher plates anv adalyerd st a ima fasion
va

In Section Ih ve defing the esstonuy sist neuen terme

cof the gonna sistance tar tht series Ue elertie eld
terms om ape exon Lhe aula extn fr Ue Ligh eed
Fl const sete to the einer ef ae maghetie el che Per
savor nthe metal, aad che symimeley of errata. es Setion HL

thace tacts won ealelstion peel thal study the topology of
hninfersering planar contours ane iene im deal, and loge
pnredce is developed, suitable for sly with the eid of a digs
Smpter, whic the relevant ava may be ascertained in tho most
fetal eae, Lv din Te! disoneion is iacuded ofthe signee
Sf open eoniou's, fy Hine coos Hae posers nt det al
ever de on Ueda

‘ethan TY presents sone of the sonmpsraleeslte obtained in the
alvin of the LL eorstt for reper. "Ther rats ace based on
the plisaesoiogce! rel Tor copper thet was propowd ty
fhant! Several Agures awe sions thal epics obits, bot coord and
‘ayn, pomputed for vows chin aries i planes teanevere 20th
Hee ase (2 (OT, a [ML

1 the mis gence stasior, che wletion between st appli eur
rect demsigy Cit al hv lv lacie Bold 2 i deere by
Fes tivity tensor (wren tl eatin

Fo ons we
re ard inh flown ll elon ran fem 1S se son
tion tnavention i ape. In the presnce of w ragnetin fl (He,
the tensor ps loving fiesion of TT), esrieted by Tae Onsager
Irlatons 10 Tsp Hg} gp). To iavasiant expressions eve
fominosly te serio Me sett ens ‘The fet i the
rssetivicy defined by

ts die anmarion inven ae uit vector which en Hh dine
tion ofthe ures je- Ty seen fvasint pasumeter is Ue Hall

+ defined by

aed a

By ~ Hell fey
whore, 1 ssn, HE — deat sua ie the thavesindes, complevey
futisynmeeis tes deed st (or —1) sbenevne (Al) ian even

foe mld} peetntasien of (121 Ohsetvw that he eset. 12), i
menue wats commis i)

seanitve only to the semmitsis part ofthe msn (eon, wil the
all constant, {3} sents nthe wntsgmse pant fp

“The theory developed ay Iii, Aa, ad Reyaay and by Ci
shit ad Precast treats conduction ig magic Geld ie ey
fenesal way, The underlying pietere of ehir thnary is mack el
Ssnolzed st erystmmomentars space, the dil spa of he veins
Xbversdimessonal space st which the meal atice lies. wey Iw
teanpersares the motion of the change nezrens sn etl i ened

fa use energy eras, che Keri sorfans le exrine io is
als confied eon the phe ramon ee penn aa
torte neti of the rye Bel JT, The parodieity nt sbe ate
in physical pace los tn pernisty ub Hee Swral Laie te

tens spe snl Us peril ity ef he wripnwal fallow ie poe
tothe nengy enfecesisbr veaentu e

1a wel rezment cf the era alae fxs ii Ble
lou sere thefulagesl cine erally whic iy th
the nel or yonimily” ann, Heh Keo mite om whic the eet
ronine sis a all caw taut, when sted for any ditzetion af
{he gen Gl Te gi ‘min sithin cobs comars
reaps toe hay the Tee enemy teal wie dees he Bo
Stn I thar sd 6 Be “eletaeclie* Covert, i ho
Treacy interior rio tas engi greer thn fp Hex che oti
eet

Whew the Fern surline esc e bury of che Bon
son, then the peririty sequbemteat cap lead to counacte suse
sterling sanragied the wate scomeecont spare, Bor ssa, the
ple euskal" ig. 2 opt eomarored Lee ara so
Dat of =pteree” eatin 9 separate Bllonsn wore ane naire
Ty “rylidetn” a steh a opeh surfue, ome bile — sna oie
cn Have at infinite eatenson, never eewnng slaw Te prtnular
Plans apt orbits ote ang vcore xe Zor the mode a
Tistrced Fig 2, Neverthe ex ine spe Pera sacs, corti
lrcetios ofthe maguet lt whee dietinn ditt the fail of
Poca phone sa hich th one ie) Hoa tall eave obits be
fel bil in wont. Blae ie elord least in some ite muir of
Briloa anins, ‘The fadoments) property of ll ena ets har
Teviaos the esas of magic Hel eens are Fall emastat
Fy can te theoretically predicted, ler these ceeumatanves, the
‘analytics exprsion,

Be = (fae f nto
TWH aus vane prsmae niceecan aoeasAe, surm 8

Aetines My hese his Pinch" instant, e > Us the valu a the lee
tone share, sd is Use spl of light, Phe integration over pe ep
fence a sumtin gver the vats momentin plates peace
to the mmzncie cl; ant Reaeate, tho drsetion tar ong
the direeton of the nga Hal, The domain of Sstegeation i (1)
ta ns sie a igh roi ze, il he dincised in Beton
B11. The ont inigza escenario cons ei ita sc tp
‘lave Tru ochit is lettonliee the ea teed in (1) is positive
Shomer proportional to the ares of he inter domain. Oa the other
Tua, ian ont Hs nee, she cana atta i (4 ie apace
‘unbes proportonel tthe ee de ines domain

"kite table seale rlnes ans inteied i suoee to stm
ompirtely geomet frior fw Tall sonstant, which we call
Ter po Pra where the Py repssens sppmine tins va
bee the

ee Jar gran, %
“hie eral ve ly ache eatin of ce wage
hh wees fies ry 2b tbe ser lve tary a
Sather ated es Icy act Fra
vile mated nh ein oes Insts oas @ ar
Ie eminent St inri' eng fo te

4: Compuiatinal Prete

uta Domasn of Ineoration aut Swann of th Caentatin

"Ths formal definition proviousy give fr (4) for 5] tates thas
he invgeatins densa cavers all of ant epaee fies
tenth bat That the aecsnvssey of the fatal rab af Blouin sone
{ali invite) sb be dv out, The perio the energy sare
races tt Ue noeieak resale olin Irom at tegration over
this weed iliieeogin shone en tr the role of a rao-
“nite inegttion rsteited to only enw Brion sone. Tedd, i all

the eee ene lay wat a singe sre (ms a small sphere), thea
Sse iige Hans HOSA 388

te ntti dona send oni Bo seated. to ony one Beilin
Fwever, when the Fern sirare ise, #8 8 omen
Tha allo the cfersst eve is dot Ue when wg se. For
‘snmp: for cet ively of Mh raguee eld, won af te large
ini may lose ony efter exrioas Yatouet gabe of dierent
thus, Ieie the pools Taat teae may es bag omits, whieh
Ticats che semingly snus iaegrsion exer » respond
huge dorain of rnin eae a9 cat thi ch Pe ple, he
wile sf the ages teen, fy Meter Dey one eloon- oF
Tavietika, maybe acess! diy
Tan ndepeaadeterinatinsw ¢he nazare of all te oxbits
say. at of Hiss egetnsn nt le ge enarter were aleady in
Hand ths eal ulsubted By ater over only one Blouin
fumes Kites fora al val ou Hae" aol
fwady spo he tre mete ogo era far whieh oly a pron?
a ofeil yen ale pate cn one Brien zoe 8 this
res sen ne empl tL dana negra on tev the
the point. — Py

factory tet Beni zone weuine a
raat

"Mot pe esa sy ost iv am weiter Pepa for cr io"?
Jo, forbunaaly a 6 tllowingaipunra shes, «fx omens
Finn, inaltoows elton sd desk orbits ou exit om ain
fet nthe sem sla nly HF se oes pe eneoaRe another
‘Seve ecw sity ae petri fll ow Fig, eepezac by
‘ear aval sls mgincs waich repestr foe example, earmece bos
find below he Fevuc nergy, este yuntienine Plas she Fig
Tile eassea acter eu] net easiest he nis
ea thin the exbit tags ani loc, so ap opseile gh, the
tan ep aera iteion toch oot marked & Lhe we me oyst i
ings tw sng sigs ibo aed ae heer Eo
tena thie siane bree awning apple the etciner of Use
Celts in hie ives sn tha je te alles ano willie 20 boundaries
Cite ist Hla ong cates to G, Ana woth sly the "ade
Sve or nse nbal a i hetatter, he “vlev an area” eorex
tena eho the energy eels fn te Fos oun, once it
thecal of the Ll ete fear place san. Ts,
Coubten the-eqnlens of gL" oo the otis sea for eating
fe ese tse ta ae pa. TE tae pote serum hy at
SEMI even) sumer 0! swstanes the ee vith outed) ra aelevant
tiv, all pints whose nergy wadive vy che Hori eum Ly i of
the aare foie sig Tei ¥en eva fea
sina Covad Ont Chora Uterine ty 0 Papal Stoay

‘The wlovant are in big. ie indepesdnt ef whether th Saetion
pints have evar: gov ter ur low thar he Fasrierengy, Jaded, Fig
2 srophically lhusnutes the flowing oropere: vere poiut in the
elect aren He cot ben edd muvee felted orbit. Ber esezape,
point I thin shel para gH al i evel pene op,
Sud 0 en Zor pein? weld Puivbonmeny, ths chevseeriation ofthe
Pits che selevanit ea 2 wart under au detranation of the
unto. ae that 9 fark ease hvu the govt ana sly
Th iveriance property ie epoca uv nareze, bee ndepesdeot ef
sve dela ef hr wb shapes

Tire a around deez she mele f eure earning
fs giver pnt prion othe pane P, ~ soca mel be "wea

1 pition gery uch lager han thot lying ft the Seat Bellin
ons, nbite which suru 9 poitt ca ea" oe Sead i they aw ce
nals enecmered the izvestaaciek: anwewes, 2b6 wari ha a
Finite rogon et investigation fo ema prion

Tn ta election of shes, srocedure to ind che anaes of costes
oecsing & parbiessat point sips 2: vertsaly a desiuble festure

{ine son ip eae fst He nae xd ae ie
tee reoseny eve absent ae He balls irk
tail on wonstone Cin BB) atl, acting them, serial the

ng eae fe agg eequace of eloely avd pate ll ing in
Slev'uht Cue. Iau cca sve) ate of rasings ae Ber eaeogy
ie Inu, he osiginal pine ive fie outside Ie teva: ira. Lie
Tnuesive prncedtse is, urforsusately, ro: always come. Fig. 3 ie
Iietes a sft of omen fare ple sl anes there possole pace
ll could 9 ne termine the rape oa Ws the sreedng chore
oth pete nel 2 evriaine thie sate, el a Tene
shal tog, as, mw rire, evi rie By
hve ah, Eoseven, path sant ae dint wita this tora
ig, Tn thie essai tae path ed aside a sad-eapy” at ly
ra cova im eersesed “hia, ae wintertime ence
tha she iia! pont sil ne He sabia the nlevant are, Chany @
tune saphisleeted pronrcare of invewigatin ie hauine hat eid
The poniility of ning in a andtrap hy aking ata i
‘Sevecal aiamuate scr ot invetgation came to mind whisk bave
ue desiree propery of skirting around "sund-eps." In eae of chee
an original tect paint i selected, and weenie of pnss Ising in some
prowl simu lw wn lle for He erasing of Hee ssa

Fenton TE euslaar'sninnlicl the enptone He fwe, ie
Surat ans wayne war

aya turning ith “eee poset ial «tn de lL Th
"eile i aH sont 2 Sloss bf th Kees
‘is supp orton hae "be pals ave the comtour as sean ae 2
fen prnvel, weve ersigg the ere Hou mil ee

nae nai the tgiel ot ant. tnple suey reves vert
lve techie eons. Aether shen 0 lightly eeoter samples
rhe pach lle (vente sso ign teigh
lve ef ively se Tol suranioues Av the pale path Lecce
fhe eovtsae ara vunivan 149 oped tseeios wares for #0
‘oui, This sone ikerise noe took eee, ahd eh se
Sonos Among Ue eat itn to anslyae eae iy ee
apd ie the Ieoer © if eet pb Flee hk” aa To
inertia he fall is nmunds CF ers, iv gener ae,

bration of Lie avigeal pot ga the Gnetae of ane a wants
iter olive the pigs al ss tate ys ean

ence
‘Te meth of jestebge ie ae teato ia Pig% allhon gl al ae
solute, ns bw sa agers eo of the arte sarong mF
Ted Who aca fe esemsnceres i ts pneaut seam, that cc
Vrs sda cup ill eprint sree at hi re ree
{GK in RMR TaN THERA IOEUNAL, HHO set

sof heaton rand Stee SRS

‘wan ovigaully enguged, While the eutee contour ot being exanined,
the mimiber of posts on tke contaar ane emgsre chute slong i ne
whee exiension of the evigina, lina of sesigatin, Whetr this
rer f cea of ever detemmton whaster st mtr does or ies
ot uous Ue orginal toa pul” Afi the evar ieee,
the nih apt te arte of the pe cad Hen pane tally
tui aig the rgial ditch seahing fe @ new santa to ty
Tt ante ifn in staid in emilee the anne ft
Fig. scbemacrally iLustrseoe eh steps in ommplce eaidy of ze
‘ent by the phosae:raeiigne: the gest Si tea sont, the ease
hing he santo, i He ashing on betving in te engin
reetion to concizue thr topcegical tavseigation. ‘This investgssion
proved i the ane emp. in tte topo study of che vezi
‘Feplnes a ealevlatng she geonsrtssa pur ofthe Mal eoustant by (3
A Aovighi orwell ioteypasier wal iwagesl er, we tt

Indore te nauengeine anon in gach ya tle saleline 6