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A Determination of the Parameter for Hauerite

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JV IAU .IT ·

~

••1•

Jr••

nt•4

b7

11• (ttf :r

In .-:> rt1

Cal:ltorala ;tut!

tut•

of eotmolosr · aa

4-.

Ca.lif·orrda
(2)

'o.r l ... ln. r dui.:ti cn

'.Ork

Id

.

.. art

II -

A

ew

l.1 r tu.re

r

i· .nee

thod

ot

tio

of

t

t

ion

rite

7

1

18

(3)

Introduction

l a.. CoYalent Bon4a, a.ncl Covalent :\adii

Tb.e elec t.ron-pa.ir bond wa.s o dginally postulated by ·;. ii. I.ewis in 191'.~ ae e..n expl anati on of coval ency. Since

"'he adTent of the quantWtl mechanics, a much better unde

r-st".ndlne; of he n:.i,ture of the bond haa been obtained. Eei tl er and Londonl a.nd others lu1ve treated the hydrogen molecule by c-u·~.ntum mechanics. In the original t reatment,

r::f He1 tler 'J.n(! London, they consider~d simple resonance

of the elect rons between the two nuclei, v;i th one electron ~,.h,,ays on each nu.cleue. Thia gi Tee rise to what might be

termed a pure covalent bond, It ha• been a)Jown by Pauling2• 3

.~nd ot.ht)ra how this treatment 1lJtJ.Y be extended to more

com-pli catcd molccul~:s. In general, e ha.Te bond formation

henever we have equival ence degeneracy1 this is the

Just-ification of t.he electron-pair bond, for indistinguishable states are had if we consider the first electron of the pair on the first nucleus, and the s£cond electron on the

second, or vice verea.

In this simpl P. treatment, however, the possibility ia neglected

ot

both electrons being on one nucleus at the

same time1 thnt is, the possibility of ions r.!xiating is

ip,-nored. To determine the approximate contribution of the ionic character to the total state of the molecule, we can

oet up the wave function having the correct symmetry proper-f,ies, R-S was done by l.feinba.um't for the hydrogen molecule:

(4)

-1-( coval ent t.erm) ( ior~c term)

Herc,

to.

,

~ ar e we.Te f net ~ons bout nuclei a ·~n b, respect iYely, n.nd (1' ( ~~), indicate the electron eow.:d -dered. The paramet(ffs,tA.,/l are Tnried so that :,he resul ~~

-ant energy is a minim:.1.nlf thla has been showuv to give the best. wave function of the type. 'l'!:1en, presumably, the

ratio o\"' giTee, at lea.et approximately, the relat ive con

-tri butiona of ionic a.nd coTal ent char acter to the s•-.ate of

the ! OlE!CUle.

~'he exact · iatinction be~.ween ionic and coualent bonds

i s neoestrn~·ily somewhat a.rbi trary • in as much as ;~he wave

functions corresponding to the normal and all excited e :.ates

of an ion form a complete orthogonal set, so that any waTe

function, PTen one that 1a purely cov .. lent in the sense

ueed f1bove, can be accurately represented as a seri ,a

expan-sion of the ionic functions. Pauling6. ,herofore prefers

to consider in an approx! i ~a.te t:.reatmcnt only the 11aTe tune-tionc corresponding to ions or atoma in their lowest states.

He then 6.efines a normal coTalent bond as one in which ionic

terms

or

the wave functions for each atom occur with the

arune co~fficir'nt. Bonde in l:!l.Olecules of the type H:H,

dis-cussed above, are necessarily norm.al covalen:, from

coneid-era 'ti ons of eymmt~try. In addi t:ion • we mn.y h:._>ve bonds in

molecules of the type A:13 :1hich are of this type, if A and

13 ·.re of the same degree of electrom:.gat ivi t,y. 1'hen terms

+ - - +

corres:µondi ng to the e ~a ,.~~s A B , and A B wi 11 ocr.;ur r:i th

;,he same coeffici1-0nt . ·_'he bond in HI is vory n1'arly purely n

(5)

normal ooTal lnt. That in NaCl • on t e other hand, is

largely 1 on! c.

There are seTeral considerations which enable one to

dist inguish between coTalent and ionic bonde in many

eub-et ncea. For exe.mple, certain spatial arrangements of

coval ent bon s are t o be ex1>· c ed3, and we might expect

molecules ha.Ting th~se co~igurat ons, .s d ·t .. rmin . lr.r

crystal atru.cture investigations, to be largely covalent.

Pauling and Tiugg1ns7 haTe prepared n table of

cova-lent radii or atems. '~bey ba.ae the1~ radii on obserTed

Taluee f om crystal structure and band spectral data,

with we.Te mech:u1ica.l foundation• '.o obtain the radii of

tetrahedral atoms, they first took ( C)*, (Bl); (Ge}, and

(Sn) ae one half the obaenecl interatomic distances in

the dit1.:"Mttd t)"pe crya tal.e

ot

those e lem~nta. They next

took (s) aa one ho.lf S·S* aa determin·d trom write.

FeS 2, and hauerlte, llDS2~ J'rom the Taluea reported for

s-

s

in these substances, faee next section). the nlue

of

(S)

was taken as

1.06

l.

HaTlng

th1a Talue, the radii

of a number of other a·toma could be determin ~·d from their

r..,spectiTe sulfides, by subtracting this Yalue of

(s)

from the1ab8erTed interatoJDic distances. he best amaooth

curves were then drawn ror1~each row or the periodic table,

tetrahedral radius Tersua atomic number. It 1 as then notice~

that the atomic radius na read from these curves for ~he

hal-ogens

differed considerably from the values given by band

"p\:-c tral data. 1he values read from the curves were about

_. mhe symbol ( C} means the covalent O".radius1 ~nd

s

-s

m~ans sulfur - sulfur distance, etc.

(6)

-3-.02 too small for (I), bile (Cl) and (F) were each

about 0.01 oo large. As i t eemed doubtful that the d1rect1o e of the di c.rep nciea should be different in

Cl d I , tho Yalu of {S) s ass ed to b 1.04

X

,

d

ll ~he calculations re1eatcd. ~his gave perfect u~ree­ ment

tor

(Cl) nd (P), whil ao e discrepancy in (I ,

( · ) was t t.ri but-...,d to the dif1 erence in the oI bi ::als

in-'f'O·lTe\l in the bon e. Pauling and Huggins con.s1d .r .d the

ifference be ween the obeerTed value for (S) and that

required by h 1r table to indoicate that the sulfur radiua

mi .. ht be dep ndant on the natur of the crystal.

he d1Tal~nt manganese radius as estimated by the extr polationi

N

?'t1.

3

9

1)1

Co&(l.3

l)f

.. Fe&(l.23

i),

to

be 1.15

i.

Aleo,

by extrapolation

of

the 1aoelectronic

sequence,

xf

{1.21 i)1

ccf

(1.22

X

·

),

:re

1 (l.23

i}

up er limit. ot 1.24

I

ia placed on (KJt). HoweTer. the obseT~d Talue of YnS in h uerite, aa calculated from

~ald and Friedrich's Talue of the parameter ( hicb le

discussed 1 t~r) ia 2.58

l,

glTing (v:il'} ae 1.55

l,

and from fnTe 2, l t ie found to be 1.59

I.

Paulin and Huggins

could offer no explanation

ot

thia anonaaloueJ.7 large rad-iua.

It s .ema worth while to inTeet igate whether the d1a-crepanc7, 1.06 Ta. 1.04

!

·for (S) is real, aa su«gested by

P uling and Huggins, who made the suggestion thnt it re&ults from a change in the orbitals}, or ia 1ue to error in the report d ve.lue of the parameter. In view of this, and of the anomoloualfU'Dga.neee radius, I have JUt·i1'lveet-!girt1ed

(7)

-4-er,,fa al batuttri te-. In the following eootions there i s

g1:ven a d!acueaion of prev os worii..1\ ... 1th an es im~1tion ~

of rr,orta, a.nd a descri •tion of th~ method used in my

erlte the par ter val

u

Qt 0.4012 •

o

.

004, the s-~

1

.

Ib. Early erk

Tbe- O;'l';f1Jta.l e .naetlilr• of pyrite wa• fir t worked

out Qr- ·•

x4

·J:.

~

9

,

by tb:e uee of hie. apcurtr9Itleter, a.e

one. of hia early etructure detenninat1one. and tb.e first

i n•olYiilg a par6JMter. He. attributed t.o tb.e crystal a

synilll:etry wh!elil •OUld p~ce 1 t. i n th• &pace. ttoup T: .

'!.'he Fe ate> a lie op a taoe-eentered cubic ~a~tice. ~f

we pass planee, parallel to Ure C\lbe face••· tltru ea.ch

Fe atom., we obtain eigllt uall cubes

trom

the

unit cube.

;:e now draw 41agonale ()f .these eight cubes 1~ such a wa;y

that none inter•ect . Each d.iagonal will lmYe an Fe atom

at one end . ., and an em11ty corner at he other, A S atom

is p.laced. on each diagonal a.t a distance ua0.J3 from the

Fe a.tom at tbe end of the diagonal.

ao

ie the length of

the edge or the unit cubeJ; a.0../3 is thurr the length of the

diagonal .of the unit <..Ube• and ·u le a pare.meter to be

i. '" illustrated determined experiniental,l.y. Thtt structure is ::f'~ ,~~ , . .,. ' 4. in Flg. 1. For clearness-• the cube hsa been rep e ~1 ..,I:)

as d.1'11.ded into halves. The let tering indicates how the

(8)
(9)

-5-e to Joined. Bragg belieTed from his spe

c-tral lntenaity inteneity meaeurm.ente that u wae between

o.at5 an4 o.405.

Shortly after the work ot Bragg, .Evntld an.d }'r ied.r1cb9

retlxamin d. p;yri ta by mea.ne of Laue photographs• .vhi ch

had been first taken a.bout a year preTiouely by Friedrich

and Knipping, at the suggestion of Laue. In thi n me" od,

continuous, or "whi te" X radiation is 2fll!.~<.. thru ~he cryst•

al, in contrast to the method of 2ragg, . n which mGnoahro•

m tic radiation is reflected from a crystal ta.ce.

As a result of heir work, ~hich I sa.11 discuss

mere fully in the next sect.ion, Ewald and Friedrich c e

to the conclusi ... n that Bragg's nlue for the par@meter in

l'Yri

te is slightly too high. Thay a,ssi ned to u n value

bet.ween 0.3875 and 0.,3885. In addition, they exlllmined

ho.ueri te., whi ch is isom rphoue with pyrite., and assigned

to 1 t a parameter va ue of 0.4000 ~ J.0005. A.a mention.eel

before, 1lt:f investigation incUcatea thia to be in slight

error

·

.

Parker and '.'!hi

t

ehous~

iS

redetermined the . parameter

for pyrite, obtaining a value of 0.3861 but their Talue

is probably not a.s accurate as tl at of Ewald · nd }11

ried-rich, for i t wa: obtained by Fourier analysis of the

(10)

~he net p a ed in preparing Laue photographs 1a

shown di agramatically tn Fig. •

et T of the X-ray tube i s coll

s.41a.t1on from l'.he ta

r-ted by t o pin holes,

t ia then pasaea thru thin section of the crystal,

d t e eoa': t~, ·ed and undetl<,cted ra,ya strike the hoto•

raph c plate, F. A small unield, 3, is usually l aced

bef ore ~he pla e to int ercept th undeflect ed

ray,

~o pre•

Tent the formation or an overly intdnse central image.

It ia easily eho"11 that if we b ·Te a series of ident

-io l p rallel pl es b.ru pointe

ot

the rystal l r.ttlce,

separated by a istance d from one a.not~er, then a r q

incident to the planes t an angle 19 ··ill be reflected

by th series of lanes it th' wa•e length of' the r93 and

the ,ngle of incidence satisfy the relation

n X • 2d ain '9

(1}

.here n is A-117 int eger. 7he angle l»f retleotion

ta,

as

usual; e ual to ·,:.ha angle of incid~nc~. In a cubic crystal.

the diat 'l.llce bc'~ween such reflecting :r lanes of indicea (bkl) 1 that ie, intersecting t.he crystallographic 3.Xea at

dlsta.ncee proportional t o l/h, l/k, 1/1, is easily shown

to be

( 2)

The condition for reflection from such a plane is thus that

n~• 2 &C)5lll1 s

.Jif

+

iC".

11 ( 3)

l3ut

the radiation usP.d in preparing Laue photographs ie
(11)

It 1

•••

con

t

a

i

n

.

.1 ion

o

r

l waYe len«it~e

re

ter

th h cu •oft proYi.d.&d by the n~.» m condition,

eV

°'

h1J

r

eV ~

~

er·

i he charge on the el ctron, e.nU the po te

n-ti ppl ied to the X-ray tube. '.1.'hu.a, proTided

( 4)

t

r..

s · t ! pl , os { hkl ) wi l. alwn o g ve ..,, reflecti on.

1r .,e t ke so e :oii. t of the ag: ce latti ce <.l.S or1t~in,

... ! ~n tl e lrecU.0.1 in a~ace of a lire :•'s,.i :,"lg thru the

orif}in

"y

be ' ... '>resented by the coor l :1at e

x

,

y,

z,

o

r

o e other int t u uhlch t.1e :.inc p, a .sa. l f' :x,y,z,

a r t1 mi.l re la ti ona ~P t o one :i.nother • ,he line p ssee tlu~ a point u,T,w, w· re u,v,w, a.re al l into.g~re.

'h•n, 111 gen ral, t iere are a l arge number of plane pa

aa-ing tbru thi• line and intercepting o ... h r lattice po nt e.

11 these plan• ro ·a a

Pt

end the l ine 1a i. 1rmed

!!Bl

&!!.

and d .sig.nated by the in icea [1.IYWQ .•

l

(hkl)

a.nd

(

h

'k

'

l')

are two of

t

h

e

pla.uee

i

n

t

h

e

200 , the inltio~s of the zone ~e y be ahown to be

gi•en by

u. kl

' -

lk

'

T a lh' - hl t

w . hlt' - kh'.

In a similar manner , . s t"Vo str 1ght l ines de+. r1,in :- a.

plan•~ the indices of jle.ne paosing thru the zone axes

[un] a.nd [u.'T'w' ] may be shown to be

(12)

-a-h • f t ' • WY'

1t • wu' - uw'

l • UT' TLl' ,

Now, for e111ery pl.a.ne ln a aonf61, t'ier ·· /$ a l n.rge

llWll~.r ( 1a1"1nite, for an infinite lal..tice ) of identic<.i.l

pa.r-allel planes a" H ata.noe<· g1Ten by (2). Then, , ... ray

paaeing into the la• .. ice 11·111 lw· reflected by ea.ch

o

r

-!:he

pan.o•

ot the zone, and thi.s mult11iJ i ci ty of r, f lecti i:us

will form a eircular t ne

or

ra.ya, with t.h~ aone a.xis ae

th.e axie of the cone, a.n<l the undeflected ray will lie OD

thft cone. i'lgure 3 represent a auch a zone axis, ; . ·1·he

pl

t••

?1 -i ll cu.t the ge:1er~ited cone, and as ia 1ell

known, tl1e int er cept of the cone on the plates wi 11 be wi

ellip0e1 thus, all the reflections o! a given zone lie on an ellipse paaeing tbru the central image, l .

It we construct the perpendicular P to all the planee

of t te zone, ~.hey will eTidently al l lie in a plane, and

thi• plane will, of course, intersect the plate in a

straight l ine. "'hue, 1! w~ make 11 p1.ot from the Laue

photograph t o conTe1·t •rom the o·oserTed posit ions of

re-f'lect 1 ons f'rom plOJ1e to that of the Litercept with the

perpendicular to a. plane, the ellipses are conTer '..ed to str .ight linee -- a gr30.t ':lid in ai:,signing indices to

t:~e reflections, and interpreting the photogrnph. ''his ia

.. ermed a. "'gnomonic projection•. ?o ma.lee this plot we

not.tee that, if !X ie the cryatn.l to plate distance, 16

the distance froru ref'leet ion to centrr-i.l image, a.n··' lp

(13)

IX la u.sually fixed at 5 cm. In pl otting the gz~Jmoni~

when the i (c:. i i;es d.nd e i :~

e.p~:.roxima.te YfaL1.e fo1· a.01 foI· \1e ,;""u t ake Lhe smal la:;. t.

valu.•

or

._.!.lr._"'Q - f ound. ·~nd. 9\lb.&t.1 ta-::.e 1 t 1::1

rela-,J

11"

+-·F~•

1'

:ii:>~ (4), and thon lhe am.a.ll~st v~lu.e o:f a

0 for vhi cb

th~ relation hol..!s ,}r ovide• a. lo,.e1· ll::d t for "0• I! a

lar.ge ~n.unb.:'r o.f ~ ('fl~ctions nc.s ~rcr~ent, it is often p

oa-enough that tLe nwnber of m. le cul es 1 n tb1.1 u.ni ~ cell oaa

b0 (&1 Tan. ·~hen, fron a kaowledge of the dene1 ty, tbo

o-..-lcal compoai tion, the atomic weights of the constituent••

·;.:nd. Avogadro's n~:ber, it 1• JGSei't>le to aaeign a lllOTe

ac<:urate "f·a.J.ue to a.0 • For .greatest accl.U'acy, however,

the size of the unit is obtained from oeci11ation

photo-graphs or spectro1uet.ric measurment.a. When the size of"

the un.it has been determine~, it iA possible to aasign

n 1' 'Yalues to every rafle<'',ion by su:>atitution in (;;).

If the Tal.ue of 11~ for a reflect icn liea beh1een the

short waTe CU't-off :~or the radiation, ·· nd twice _;his Talue,

i t is GT1f,enf; t.nat n must be uaity1 t.hat is, the reilection

is 1"irat order . For reflections of' greater n,.., i t lo

im-poe..=.ibl.ft to aaaign the order of the reflection.

(14)

Al.the eo• 1n.fo1--.tion c<1tacernlug the i:Jtructare ot

a cn·y, ta.l l • to be had t r.om t d.e synmic try of !·n.Ue phot

o-graphu, to get cor:ipl\:±te inf orms.tio •• if", is nee.eeaa.ry to

(hx + ky

X; •Y; • z; Ll

.. "' ~ - }~ + )fl.,, +

.

J

,

.i

]ao :;a J

J

"1t-

+

l?"

+

t

l ue t r0:11 7.h.: o,~igi1 1:.;

AN

t .kYa + l?Y .

.J hL+ IC+

f

7'or refleeti on, i~he phase <tiff tll'~""ce for iden ti ceJ.l planes

{hkl) mu.at be 21rn. The distance bet\veem ident.!.~a.l planes

WG.8 giTea (eq. (2)) as

'!tli.a:n the phase ~1ff e:rence: ~ of a '¥f~Ye r.efl: cted f l'Onl the

pJ.anea {bkl} Llentic:al -.,,1th that thn.&

x;,

y~,a3, s.:itd a

wn.Yc re.fleeted b.i ..Plrui0a 1deat..i cal wi ·'.:h tht\t tbru th,. oJ:igin i.s g1 ven by

It ill shewn in the theory of waYEl motion tbat the

tuupll tu4e F

ot

the resultant of a number of waves ot the

same frequency, and of u1pli t udea

r., , "'-, .

,

.

~

and ~eea

.

"'8

t

ip'f.

I • • • f is gi ¥9Jl by

J!

r.,.

exp 1y,+

&,

exp 1 ~ .. • • •

Tlu1$ .• putting in the ~xPreseion for the phase found above,

ttt• aapli tli4e • f ~Jit:~i'ei\dtant we.Te reflected b;y all planes

(15)

•ire 'the coordi tee of the

J

th e.tom

in he unit cel l• and fa; the sca.t·ter ing power of the

j~ g.tom, called the "atooic scat tering factor .. of t.he

ntom. As8Uluitlg the a.tome t o be spherically symmetric i .

t c atomi• scat :,erlng factor& t· e functions of the nature

of the a t c:Hl:l, ~nd of sin-e /').. '?h y have een tabulated

for all t oms for v~rioue valuPa of oin l'/r. • F is te~

e{>. the ttstructurt~ fnotor". The intensi ty of the refleo•

tion ia p1·oportionnl to :Tl!*. n calculating intensi ties,

it ia usually more eonvenicn(. to con'f'ert the exponent ial

to tbe tr1gine.mecric fornu

eXJ) i21111(hx; + ~ + lz,-) • COB l21r'Dthlt,- + ~-+ 183·)

+ 1 ein 2rn{ hx;+ lcy;; + 1•3)

";'he actual am:pli t ud of the reflected 'beam depends in a

complicated way on a number of other fa.etora, each aa9

?.nd:A, but 11" comparisons are only made between

ref'lec-ions for which all theae other factors are constant,

it 1 s suf~i cl ent to use only the F Yaluee lu compar1 son.·

; Id A Di scussion ot the Parameter Determination of Ewal d Rnd Fri rich

on hauerite

Ewald an Friedrich in their work"',_r-e:f'erred to a.boTe,

carried thru the method essentially a.s deseri bed above,

-tho they didn't attempt to fix the size of the uni cell.·

In their determination, they found that the reflection

(

&&

I}

does not appear. As the struct•re factor for this

..

reflection dis ppears for u •

o.4.

they took this as the
(16)

-Tal.ae of u. The7 conalclered th tr reeulte t• be accur•

ate with1a 0.0005. HoweYer·, u the etnaeture factor

tor

(1

2

1)

increases bu.t elowJ.7 on ei ther aide

ot

ite

point of disappearance• and eapecia.Jly a.a the intensity

of l"efleotion i s propor·~ional to th square of the st

ruc-ture fact.or, hLi ie riot a. very sensi t-1 ve method for

de-termining t.he value of u. !he refl ec·uion probably would

not be ob&erT•d even if u differed as much as o.0015

froa t.be Talue 0.4000, for the inle:nui ty of {5.21) woulcl

at ill be only 'ctween one two-hundredth and one t

hree-h .. 1ndredth that. of ( 25i}, a reflec'.:.ion of medium int. en-ai ty.

(17)
(18)

:Part I I A .Jew Parameter Determination for Hauerite

In this part ie described a new determination of

the parameter for hnueri te which I huvc ca.rri{'d oJt by

the use of the Laue method.

7he size of the unit of structure of hauerite has

'7

been Dea.sured 11 nu.mber of t imes. Hl.'tling ~rnd liugr;ina

0

give a

0 :a 6 .097 t J. 005 A, aa determined from o~ici l lu -tion :photor,rA.phs. I have uaE·d th.is value of a0 in the

present determln~tion.

A J.aue photograph ~··as ~>re.x.1. · ed with the incidr-nt

beam ne~rly norm:\l to .( 111). Tl1e ;:ihotogr...,_pb so obta

in-ed gt ve a la.rge m1 1bf'r of well shaped spots. Figure 4

l s a print made from the photograph. Upon ma.king a

gno-monic projection, assigning indices, tnd calculating n

for the reflf'Ct.ions, no spots with n 'X less than 0.24

1,

the short wave cut-off of the radiation used, were found.

There is, therefore, no indication that the unit is la

rg-er than previously reported.

If he.ueri te hne the pyrite structure, its r-;pace group

1 8 .,, l,._ • " This rec;uires fi rst order refl ections wi th

incU-cea (Okl), with k odd, to be H.bsent*. Seve::--al such r

e-fl eetions were f o!md, hut a.11 but one of these weJ. e very

failnt , and all were shaped differently from the ot:i1er

o::iots on the photograph. 'l'he one fairly strong spot was

(o5~), ','lhich was at nA= 0.38

R.

(

.i

o!),

which hi of the

- 0

same form as (034~, and woul1 fall at nJ\= 0.39 A, was

nbsent. Therefore, if the reflect ion

(0

54)

is real, the
(19)

-14-c:ryst 1 ould not be cubic. 'hie le hi hly 1 probable.

In en r l, only intensities of r fl et ions of ne

ly the same n i\ , d the same int rplanar distance were

co 11par d. In this ay, th ca.l cul ·.tions wer '·Juell ai

plified, for t.h ratio of the int nsi tjes of two such

refl ctions is just the square of th· ratio of their

t :;·uctur factor •

The to1i c pooitions i

..

group T., , e:

' i

o

,o,

_

o,

l/

"

,1/2,0,

i /2,0,1/21

0,1/

·

,1/

2

.

Si u,u,uJ 1/2 +u,l/2 -u,u1 u,l/2 +u,1/2 -ut 1/2

-u,u,

1/

2

+u.

u,U,UJ

1/2 -u, 1/2 +u, UJ U11/2-u,l/2+us 1/ 2 +u,u 1/2 -u •

... he structure factor for the refl ct1on (hkl;; 1th h,k,

ndll all odd is t.henii for first order reflections,

F • 4 ~ + 8 £8 COB 21rhu cos 21rb cos 2'1Tlu,.

and for h,k eTen, l odd1 or h,k odd, ..1 even,

F •

$

~ coa 21rbu .aAn 21rlcu ein 21rlu.

The

fc

Taluee used are those given by Pauling and 3herman11 •

It ie to be noticed that in comparing the intensities of

t o reflections of the same int rpl annr dietanc • and where

F f'or both reflections is iTen by the second equation

above, the sc< ~t .ring f actors do not enter into the

expres-sion for the 1· la ti ve intensi ti f?S of the two reflections.

The use

•r

such comparisons makes possible a more a

ccur-ate parameter d . ermination, for uncertaint ies in f.. Ta

l-ues •ill then not effect the results.

*

Because of the symmetry of the group. planes invol-Ti ng cyclic interchange of the same indices e of the

same

form.
(20)
(21)

-15-he po.rAJn t r was quickly n:J.rrowed down to the

re-gion na r o.4 by the comparisons ( 42' ) - ( 24' ) , ·. here )

ia any odd i nteger , and (1311)-- (3 •), where • is any 2 <)

even integer. 'i'ue ra\.i os ~ ( 42' )/ .''~( 241)) , and

1'-.

-

\

13 ")/F2( 3li) are plotted in Figure S against the

par-a.meter. u. All intensities a.r symmetrical about u

=

.5,

so the curves have only been lotted from O.to 0.5. ,the

approximate observed val es were,

1(421)/ 1(241) 3 12 1(427)/1(247) = 8

I ( 423 )A{243) • 15

1(134)/1(31 ) ~ 18 1(136// 1(316) D 9

It will be o~;..rved that the only places where both

cur-res agree with the observed values are in the regions

u

=

o.095 to 0.105, and u • 0.396 to 0.406. The ratio

of 1(535) to 1(614), which ie obaerved to be a."ut

io.

definitely showe that u J11Uet lie in the nelghl>orhoGd of

0.1, for the calculated ratio, F2(535)/.V2(614) is 1.45 for u

=

0.1, and 11.6 for u

=

0. 4.

A fairly l a..rge ntimaer O·f comparisons were used in

determining u more closely. Those used in the f inal

determination a.re listed below:

Inde~

n

"

1obs. R*

832 0.372 0.30 1.6

823 o.362

o.1

e

962 0.3'76 0.15 1.4

676 0.365 0.10

481 0. 308 0.35 1.3

732 o.341 0.40

1.0

651 o.354 0. 48

*Hat!os corrected for difference in nl\. '.'he pproxima.te correction for difference in n"' w },s

ade by plot t in ... a curve of lobs. vc. n ') for 1.he var ...

(22)

-16-ioue reflections of the aaae form falling at different

Taluea ot n'),.. The aqa..ree of the structure factors are

plotted rroa u • o.398 to u • 0.404 in Figurea 6, 7, and

8. Ee.ch ot the comps.riaons (832) - (823) and (962)

-(676) rlgorouely lim.it.e u to Ta.lues a.boTe 0.4000. Also,

altho (481) ia at greater intcrplanF..r distance than (732),

it le of gree.ter intensity (when corrected for difference

in n "), and tl~erefore 1 ta F 2 Ta.lue must be greater i·han

t.hat of (732). As seen from Fig.

s,

this establishes a

definite upper limit for u at 0.4026. From the compar

i-sons (832) - (823}, 1t is seen that u proba.bl.y lies

be-tween 0.4008 and 0.4016. The comparison (962) - (676)

indicates a Ta.lue

or

u between 0.4008 and. 0.4020. 'fhe

last comparison, (651) • (732) giTea a Talue

ot

u from

0.4006 to 0.4016. Thi• ia probabl)r the moat useful

com-pnrison for the exact determin&tion of the p&rarneter.

From a consideration

or

&11 these compar1eona, the Talue

of the pa.ra.me.ter bas been taken aa

u • 0 • .012

!

0.(1004

The limit• g1Ten are the probable error. Ueiat this

Tal-ue for u, and the value :for a0 giTen before, the

s

-

s

distance is found 't-0 be 2.086

±

0.JU

i,

giTing a. bond

radius for 3 of l.043 :t 0.005

X,

in good agreement with

the vnlue taken by Pauli:ng and Huggins. Thia small change

in t.hP. para.meter, hovtever, leaves the Mn - S distance

practically unchanged nt about 2.59

R,

thus leaving !.In

wi th the auomo.lous radiue of about 1.55

i.

Thie problem was suggee ted by, and carried out und(~r

the direction of, Prof. Linus Pauling, to whom I nm

in-debted f'or much invaluable aid.

(23)

-17-Swmary

In

'8-rt

I,

aft.er a iacusaion of covalency and co-valent radii, a brief reyiew of the work of Bragg on

JJyri ~.e, ruid Ewald ,ind li .. riedricb on pyri i..e rLnd hl:\U ::rite,

with a.n eYalua tion of iccuracy, is given. 'l'he Laue

method for crystal .analysis is desci·i bed, $Uld the t.ueory

outlined.

art II is dt>votod to the description of a new

par-ameter de:ermin&tion for huuerite carried out by the

~rri ter. As a reeul t

ot

this determination, the

parrun-ater hall been fixed at

u • 0.4012 • 0.0004,

giYing a coTal ent sulfur ra41ue of 1.043

R.

(24)

-18-Addenda

Since the preTioue wi.ia writ te~ another Laue

photograph hae been prepared from a new crystal

or

bauerite. On thia photograph, no reflections of

the type (Okl), with k odd, were preaent. It ie thua

quite certain that the presence

or

auch reflections on

the preTioue photperaph waa due either to imperfectiona

(25)

Liter ture Referoncea

1.

w

.

Reitler and F. London,

z.

Ph;raik,

il•

455 (1927).

2. Linue Pauling, Proc. Xat. Acad. Soi., ~ 351 (1928).

3. Li nus Paul 1 ng, •The Nature ot the Chem cal Bond, "

J. Am. Ch••· Soc.~· 1367 (1931) et eeq.

4-. S,Slclne7 We1abaum, J. Chem. Phy•.

!•

5931 $1933).

5. For exaaple,

c.

E~t, Phys. Rey. ~. 878 (1930).

6. Linue Pauling, J. All. Chem. Soc.

Ai•

3670 (1932).

7. Limas Pauling and

:w:.

L. Huggin•,

z.

Krl11t.

II•

205

. (19341. '!'

s.

w

.

L. Braig, Proc. Ro7. Soc. (LondoJJ), A!2,1-t.llt1U914)

9. P.P.EwaJ.d

and

w

.JTle4r1ch,

A.ml.

Pl0'•1k• &t•

·

ll83

. ' . ' (1914).

10. Parker and Whitehouse, Phil. Kag.

li•

'39 (1932).

References

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