BSTJ 30: 4. October 1951: On the Reflection of Electrons by Metallic Crystals. (MacColl, L.A.)

8.5 MB·English

On the Reflection of Hlectrons hy Metullic Crystals
Bi by A. MaccOuL

priate ate
eae cera daeriageee
SERS actetrabtcre gt tol tele ede
Berra lesietrt ads ued Keyan ce
Sani

1, Fowswors
I wor whi is presented in this paper was undertaken aa resle
‘of converting kad with Dr.C. J. Davison at various times during
the yours 1858 and 1999, when he was investigating the reflection of oe
trons impinging onthe sf of metallic crystal, The results for a simple
petal vast te general peobla were published in 1539 Thereafter the
‘ork onthe general polen continued intermittent, an it was almost
‘Complten y the ean prt of 1942, when it was brought toa halt by the
net of walle stvitie. Sin then nothing bas been done on the prob
tem, ead the ets alrearyoblaised bave never Boe plished ie ete
owes Herring snd MCE Nichols have fncnded an umipatig ts
ion of some otha tote sgn of the eels in thelr rceot mam
raph on thermionic emission
“Althonh the intervening yeas, by binging new problems in pysis to
the fore, have exe thi wor nose some of theinters which it posesed
‘the Line it being dere, stil sess to Be worth wile to pat te fol
teas upon record. The preset acasion, when his nds and formes cle
Teagues are cleratng Ir. Davieorsreveniet Mella, is en especially
septate oe for this purpose

2, Romarmamon op sine Ponte
Wie conser electrons moving with energy E and impinging on a plane

sce ofa metalic crystal. (Fig, 1.) According to quentam mechani here

iscertan probability A, gnerlly neither Oe 1 chat aa electron il be

felleted Uy the erystly and cra fo move backwards tnvarl the sarce;

find there isthe eourplementary pobsbiiy 1 tbet the eect wl

epg ee: 5, 1 he a ee nh
Me Wen Pont,» 2, p57 8,
(On RILRCHON OF RIRETEONS we ATETALIE cHYaEALS BD

penetrate the crys, and fw vay through the remind of che cert
We call the vecton eect, aid ean tie it alesatvely a he
ratio of the intensity of the redestad electron beam to the iatesiy of the
ineiden bea.

‘We wit a calulate Ras &Suneias uf Boner tobe ale to do this
ectvaly, ie necasry to danliae the setusl physical stvation quite
‘hastily. Gonever, the Weston whic we sl we preserve what
er to be the most important eal of Ue physical sition.) Om Une
other and, once the Wellton bas been act up, te mathematical calcula
tions themirlves wil be eased though without eppresinetins!. Hence,

a eto a

any dscrepuacies between the theorticl cess std the reels of exper
tannt are to be ateributed tothe Saudequncy ofthe model, and not tof
legitimate spe inthe athertion! work

‘Oar iesiztin ofthe phyicletation cia be Gescrbed in the form of
‘be tine following assumptions:

“Azsumpion I The probiem may be ave as one conemning one-ime
sional mation of eletron. Ths we set up rectangular coordinate event
fn spaces and ne asrame thatthe eeyetal veeupin the hallspace = <0,
ul That all of the point functions with which we are caer deer
solely upon the eaodinate

Atvurpton I. Thete existe function ¥(#, suc that a eleetun atthe
point «has potential energy ¥(s); andthe Bebavar of an elocren is gov
frac by the Scudinger weve equation
200 aus weLL erste tremmcaT JOURNAL, oeTUMER 19S

Mave

Cog = 0 o

(Cte San where ia Pinel’ conan lm the mas a an
fet) Th sey deals strmary ws” with ari compl
fated pransors volving, sections im crystals. Discussions of the validity
ff the esumption are be fund inns works um the election theory
of meta

Avent IL, Sessa, te function V2) i ven by she female
Vi) — = Fob Yesinaie— 8, <a
ita, fas 2i
Hv),

wen: the abaolte vale ofthe electronic change and iy Ve and

re culails aoienegative conmants. (A grap of this fuaction Vs) B
‘own in Fig. 2) Aevoring tn this stucyptiog, an eacton in the relon
[> ty obit hla aeons i nen 2 Be in
Sgrecment vith the fst, at leat ie nat too smal Als, acre
the assrplion, Se pletil energy’ af an elector. fn the depths uf the
cyst specie psi veto witha nezaive meca value. This par,
tthe assumption isonet as any aaautption can he whieh ater Ce
Scout fr tne complicate wal poasases fa terre of «potential energy
fuaction, However, wir pelculr eves of a peiede function is based
langle pon mere consieraion of eatemstis! convenience. Filly, we
ne WEELECHIOW OF RERETHONS RW ARTA eRVTA'S SL

Swerve thas our Pf continues, a shyla! consi. one faite
ts sh be,
‘We es oi late te mathemati ples ee the felling
EG} being defn hy (2), we ae bv aaa seuton 9) a 2) aati
ying the Gallon eoion
(a) Te tho region » > 1 the fonction ps) exp (21/4) represents an
incident beam of electrons moving toward the st, abd a reflected beam
of letrons moving tower the igh
(@) Te the regen + < the average cleewon How, fit is ot zero, dee-
tod toward the lt.
(6) The function ple) and its derivative #4) ace everywhere continuous
Hoving btained sich a solution Ye), we ace then to empue the rac of
the itesity of the reflect elect beam tothe intensity ofthe indent
boa Tn particule, we ate tn study the dependence a this ratio upon the
quantice B, Vand V
"Tae pape [LAM, 1930] sendy referze to dealt ith the epoca casein
sabi Pi = 0, be the casein which Vs) i assamed ta to constzat io the
Fegion a, Consequesty, we are now concerzed shed ita she eases
lnwhich F,> 0

5. Gustwatstns Conernsase nie Crear

Inthe regan = 2 the wave expat (1 aes he arm

fr e[er ipo

re gener oto of thi aati fo the Soren

40)

Wie) + MBA,

ry rans 2) als) ae ta pi
Tar ations which we hoe a that he Sth fle) ee (28)

ster A sal Baa

ad als} exp (“20H E/H) sepresent beams of deri, of uit teri
‘moving. ta the lft ad ight expecta.
Th the region 2 <a the wave equation lakes the far

= RUE + Ve — Pesin ate — aol = ©

e

‘We are concerned with a elation ofthis equation ofthe form

vial = CH

sehore Cie eanesnt and Pfs i particular tion aach phat the fare
902m us sxereu cenUEAL JOUDNE, OCTOMEH 195
ion yale exp (20384) serene a staze Sa which the aseage fw of
lectus in She capa ile vauies 0: ie recto tard the et
"Phe actual fous of the fustins Yale), dole, Hof) wil he sous
resol
"Nom the coatnnity ofthe foneione ye sy") gives us the system
of equations

Anon) | Bout) ~ Chsked

Avie + BOs = Chen),
roo which we can clelate the ratio B/A in ver of the ved, Ya
(ur seguied election eaficient Fi | 8/4 (nnd ea we ata the fre
‘mula

sea Bo)

vica ~ BS 0 ©
Say BE

Fie) FS tea

1 xasshoxn‘n LAM, 1939] thot *h functions}
by the fenmulae
8) = HG, le) = a8

vl Yale) are own

wyhere
eo kat, n= inva,

ul the sano WiC, Wa4l—0 denote the seul fonctions excusing

in the thony ofthe corfiwes Pypergoametsle Garston. The salar x

tyes us el the iafermationearcening Yl) al fel “hak we eal gue

Hans ie erder ta elewate me have affect, aly 22 ena suitable

otion fa) epaon (8), el the Uo esata $l)

4. Tum Soren oF

esT6¥ 8)
1 oader to Saltate the uee of known resale, i 8 convenient t write

He 3 By
we@-geu-f, Bass Byam,
‘hen equation (3) takes dae fra
0 et meme
TE + ht mamta = 0 io)

This we ofthe canaica forme of Mothie's<iferentalenustin, for

c "Moen a” (Caper XD, Catee
fs RRPECLON uF KEMUTLONS mY AerDATEEG cRYAEAI BIH

which en extensive

ony ete We hell call afew of Li hie fasts
rough tn chit Chea

‘Vases the constants and <tsty somo ose of esta spl oe
tions she genet] Solutio fel (0) bf the fora

oo

“+ Kereta,

shete es vermont determined by and, (2) i w fonction wach is
evil with the pecid wend che Ke are ewan a gestion

Tu erlin range of elas of che ie aan se ae other
anges ite pe iery. When rel we ea vy ake it fo be
bosses an thes, vont Ye) amy Be Bode nthe range
‘ens ehuvee fo} tne Ee fanston eh). Wea a are iapinary
Recon tae = to ies [joc the, Im onder hae ale hal presen
State ir which the Pow w elven fe to the lef a he cyst, rat
oo les lo be he Farin ef.

‘When pi purinsyary we yew non-vanishing Sooo eeetons to the
left inthe etal Consegnersly, te inten oft lati Tarn
Dees da de intesity ft incident am, eat = Ihe rn er
emust lave © 1. Ou thecier hind when is sea there sro aveeage
slzczon tow in Lie eryalal. Coroenty der thi oat the ten
Sis ofthe uit ea reeclel beam mist oe gual, 20 thet & =
“Phone eaneidetions pone to t36 pata fess aly
online der hich pean page gy.

Figure Shows. wel cron eager, sided sighs ta vil wu seme
purpses Heer oul Gare taken to be metangaar cantata oi
fa ple 2! the plone i eve into desi of ne ina
Urabe y= mtr of ens TY the pout 2) he Ch inti
Le ofthe la! raging above i el i te oi ei the atesion
ff ene ofthe wl! reine pre rsgirry, Uf, @) Bis escty
fn the boundary of ane of the region, we awe a mriertat marr comp
ed stustion whic me a aot ed ta cen ben) “Pai gram er
ables a> exile cetsrmins, fo ey Exel valucsof Fyoad Vy th ena
alee of Bis wth wubave Wm 1, We atal eel these exes aloe
OEE the Sifraton bends

‘Now our rfl 3s boon reduced to thas of cating fa veles of
wek a net Ha litearton nds. Tn sting tie pase ofthe subject
feel fou the coun of the ret enltleins, witha: any exaafiae
on f ways in which the work might have bos dove nn ished

11 isan, 2, Siete en GX Warn one‘ 4, Chale XT
S04 tue gua svar mmeusies. jovTat, acroREs 1951

(Of the mary’ metliw's which heve been devil for Gnding ston of
Mails’ dfecntisl equa, the ore which i concepruly simplest is
hat cue to Kou. hs usthod ean be des a ellos

Unde ste tert.
oreo [aoe
sh Mathewson 4 ges vein ira ets
gah theo ts — oy

We sck 2 goltion of this equstion im the barat a pose occ in the
pareese ay

re

arc oe ale “nd thatthe Fer tismse 2 ox, mnt ally he
ern uations

ba go

4 dont Bewets =

Slee bey
2

| Dans —
s+ Baap bt = 0,
De WNERCHION OF ROREMRONS HF WEA KavTARE 885

‘vical in rer chat we Sl es the solution whick me rnin
oust reduce to iy when Ty = 02 Al fa} rush be pes wil the

onuiins, gether with he tions (0), i deer
ede)» socesaively and wiiquely. We tually aban the

<a “en” a tava=a]

a

o0--Elg ana aay ta alee,
Tale =ar@— VG — 09) 7 4 are T ame @
beam ral
WH Ora ane ta) 7 1 REPAID

‘ema
vo S eee ara = wT

4is— 144 e"————_ 6—396 410)
a WER wea Sama
(15 + 18% + S66)

BOT wre | Fos
ag ane ]
THae FOG + + a9
te functions nf) and) ave ben compet lo; but, bus
ecosinty tine tectiegies

T snc tnt Been gf) pac egetion

Aheste seg is omeel renee a

Te principe wee havea he inormaion a ned eae tae
fect cain A. Burlserrre, ear eeperence showed that fee cxite

‘acy U rompine By ths metho, proved taat the vale of oss wot
He too near su edge ofa eration bane However, ean tho proved
10 be unstable Gr calling values of Vor values of Ebr the mehr
hood afadifsction band edge, aid we were fro Loser ance mete
‘e-ohtsin Hose values, Aer sme tentative work with other metho, we

fice star rhage the ees a
te Heer smn fe

mith rae in
se Hane = WC
NMG oe mete Sveti HHL FOCMRAL, OCTOBEE 1951

settled ypon a metho dus ta Whitaker, aod ths wasted tw complete the
‘lesions

Whitaker's metkoé is dees Une ia Whittaker end Watson's

Afsisen Avalysn” 44 Kaiton, p42. The mevied > dewelope mare
fully ia papers by Ine. We sal conne ourselves hee to ome sammary
indications ofthe notre of he wal.

“Tie method las to eprserations of the soon of Matha’ equa
tion by formolse eich der instactare depending en le yatt of che
{26} plne in which we ae working. We eho give the ferme eitabie
Jor ose in the eigorhaed ofthe peat @ = T, f=; the formulae ar ase
in otver pars of she plane ace given 5 Tne,

Giver He wales! and ye st determi a number o by menos of
‘oimple qin

f yco0 20 + % (2 + cos 4e) — B cos 22

Bim 1 oose 20 + 8 (m2 + cos de) ~ F cos 2
+ EL (2— weante) +
+h nos)e

‘Then we sek soto ol eation (8) i che form,
4 = °F, tenes salle | De a= tay sin (eb De ~ ey

here the 3's andthe #5 ace corstents, Wes aubatitate the expreston for
{Pte the diflorntal equim, ard determine veluce ofthe rrwtarts by
Tncposing the coratiom thatthe sesulting relation sll be > etiy in
After sone rfher inset algbuaie masipletons we faalyezeve atthe

4 st ait
ao ein 26 — ai to — ME inte
20 hed

or gigas ante +
ow mevivcnow or RexemROS a were cxvseats $97

= OW)

ostet

a
oa

4

‘The ealesaled tenue whic ave exited kere cule us to ealulate the
selation ple toa certtin eccurey, and this accurscy proved to besuficient
for our purposes

“Although this method is very corplccednnaltcally, it was found to
1s ute eavenient for purposes 0 mere ovtn,

5. Tue Reroecriow Conmaeor vie Laue Watts ov 2
is work is concern ally ith Ce welseson coefficient fr tall
values of BGetly up 920 lect volte). Monever it interesting that
tre can obisin a ple approximate formule for or indsaiely lage
‘hs of i ce ietervalebeoween the erection bands

"Fur this purpase ns go back c run's method de write the dependent
vera im equntion (5) inthe form g = — fe} a. We find thal ee
Aependent variable a ste the equation

Bo aitya = oF + 200825 ~ 0,

ond we seek colstion ofthis equation i he fom

aman tO

"he funetons() are corel nel we Gly ative, la entely
stright forward way al the renal

eereres o