JV IAU .IT ·
~
••1•
Jr••
nt•4b7
11• (ttf :rIn .-:> rt1
Cal:ltorala ;tut!
tut•
of eotmolosr · aa4-.
Ca.lif·orrda'o.r l ... ln. r dui.:ti cn
'.Ork
Id
.
.. art
II -
A
ewl.1 r tu.re
r
i· .neethod
ot
tio
of
t
t
ion
rite7
1
18
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 thesame 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:
-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 thearune 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
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 nexttook (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 nlueof
(S)
was taken as1.06
l.
HaTlng
th1a Talue, the radiiof 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.
-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
,
dll ~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.3l)f
.. Fe&(l.23i),
tobe 1.15
i.
Aleo,
by extrapolationof
the 1aoelectronicsequence,
xf
{1.21 i)1ccf
(1.22X
·
),
:re
1 (l.23i}
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 1.55l,
and from fnTe 2, l t ie found to be 1.59I.
Paulin and Hugginscould 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 byP 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
-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.eone. 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 ofthe 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
-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 valuebet.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 . parameterfor 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
~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,
asusual; 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 ieIt 1
•••
con
t
a
i
n
.
.1 iono
r
l waYe len«it~ere
terth h cu •oft proYi.d.&d by the n~.» m condition,
eV
°'
h1Jr
eV ~
~
er·
•
i he charge on the el ctron, e.nU the po ten-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 ex
,
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 oft
h
e
pla.ueei
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
-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
-!:hepan.o•
ot the zone, and thi.s mult11iJ i ci ty of r, f lecti i:uswill form a eircular t ne
or
ra.ya, with t.h~ aone a.xis aeth.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 1ellknown, 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
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::1rela-,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.
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 JJ
"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 awn.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 thesame frequency, and of u1pli t udea
r., , "'-, .
•
,
.
~•
and ~eea.
"'8
tip'f.
I • • • f is gi ¥9Jl byJ! •
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
•ire 'the coordi tee of the
J
th e.tomin 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-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 aideot
itepoint 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.
: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-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 /2,0,1/210,1/
·
,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 accur-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 thesame
form.-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 thepar-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 ratioof 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 ...
-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 adefinite 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. 'fhelast comparison, (651) • (732) giTea a Talue
ot
u from0.4006 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 Talueof the pa.ra.me.ter bas been taken aa
u • 0 • .012
!
0.(1004The 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.JUi,
giTing a. bondradius for 3 of l.043 :t 0.005
X,
in good agreement withthe 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 !.Inwi 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.
-17-Swmary
In
'8-rt
I,
aft.er a iacusaion of covalency and co-valent radii, a brief reyiew of the work of Bragg onJJyri ~.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, theparrun-ater hall been fixed at
u • 0.4012 • 0.0004,
giYing a coTal ent sulfur ra41ue of 1.043
R.
-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 onthe preTioue photperaph waa due either to imperfectiona
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).