Retrospective Theses and Dissertations
1935
Calculation of the ionization potential of lithium
Charles P. Wells
Iowa State College
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Recommended Citation
Wells, Charles P., "Calculation of the ionization potential of lithium " (1935). Retrospective Theses and Dissertations. Paper 12921.
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UMI
CAT^IMTIOH OP fia
loiiimfioa fOTiiifiAi. Of tjifuiim
m
?• iells
A
Suteiitteci to tlae QmAim^m Faealtj
for the Dsgree of
BOCffOE OS' PliltOSOPHS"
mk$m %n
Upproved
Signature was redacted for privacy.
Signature was redacted for privacy.
Signature was redacted for privacy.
•»«a
Xom &tat;e C#3.1®@i
^ ,. 1935
UMI Number: DP11983
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I., IlfROBUCflOI
%m Variatloa Principle*
1&# allowec,
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uMotm
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at«
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= 6yi)r H \)/' dv
to %13# ,ii0i^li»iiig -eondition
W&w^' H r©pr«aetits tb® 'Iteailtonlm
is ttoe cliaracteristic function,
f©ip
Im
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tloiivj'
system tm&
if tMs wfla tsrl®!
wlilob coxitain® pat'ttra^tei'S *sich 'mm- ¥«ri«Ha «ii%ll
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te tto© #»©ri3y,t, felM%'\]/'i»
tls»: fo\m^&%im fer
%l»
.iibo» tfeal; W im mi
lialt
#» app2?o:s^toat#
oa mmy laody pTOt>l#is8 ms laid
If ails©afe#f»g%; ia ^ileb fat. t»lai0<i tti# tli^w Bo% pi^bliWi a«
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»tmmA Mm to mm $©»«#% s«Msg®»%rl« mt®:
Pm 171 and 19#» Julitx® Spring®!'# Berlin, (1931) •
s 2eits« £* Pl^s» 33, 411 (1926).
s Zeits, f» Pbys. 41, 239 (1927),
« Zelts. f. Pbys, 40, 492 (1926).
frnicttonm for fctaa lltMom ateta, tooth -usiiig tli®
-f©miitatloii
of
Mm.0^ later •§«© &
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dl«i»S8i.oii
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ant
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antisfasiet*msiog m {i®te«ciii-
elements sr® fmietlons of tli© ooaraiamfe&s of
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tion »m. -m 'mm mt ^rodnet# of twa&tlom ot slagl# ©leetroB,®*,
•ffa@y also sfeow€
tti# tMi»%-slx tieims-
Into tn©
mergf taitot^als: iite b® 'ipedmced to tw*
Mm
Wm'U »a Mtiifiait* ^f'l^ p»l3l« of #ol¥lng tli©
SslmsedlBger e^iuatios fos? -Wm hydrogen atom laid tii© mowm&% stmt#
of tia© hellws Atoa. to&s imm ©arrled out to a Mgli
of
«-etiid-ss, "but tlis ^b«0fc ^proxS»tioa ^ileli toas li#«& otot»ln#.d for
th.© lowest state of litlaiiira l.s .not nearly go
Hftipgreaves'^ ^ «ittg tli« »tiio€ of ^##lf««oit®ist«ot
•mS. lo^fesrt®0. ttsiuf
tJ» ^^riatlcm priaclpl# witli liyarog»ft*»llls
»*@ fisnotlons «if»® tl» first to publish aay results
!««, fli»-l3P
litli-»
t&« «;^ri»atal % mor®. thai
oao' pmw wntm
yiiiajit^aa'''')l«cjffiiScs,' p^*" l§S»i^lV,
I^ndon C19S0).
FhjB* Sev, 34, 1293 (102?^).
ZclU. f. Phys. 61, 199 (19S0)»
Proc. Camb* Phil* Soe« 25, 75 (1929)*
Phys^ Bov. m, &7B (1930).
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^
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,,#,Ja,8a i^iaa +
, 33?ig •
.1.1,i»
S^s
• 81^3.®' .
z
I=jdvj[i^ ^ i
i-
- ski t •
#•
S ~ ^ Jd'S'J
Iff n r^irJ f i.^iiii.. * ,i..^»i, * Swiaaa -«
J
I oFj.
,. &Px OTs
asp:}|
^ i..i'&:#a,. , .,^,,^^63.
W5t«
or"®
., 4 l3.ftlia!r:rfe^ .lisa , I±w. + i ,£a.lm8.^y.af,. Utmi i&gi
•fr 1 ,y:a",tKa,g^j:Sa^ ijbm i^.iii,i *1
2 ., ,
BTs
2
, I
^%m^m
L* » fd'^'3'l^
J
(.ri
JTr
•t'g
Wia
r©
•»
or lie-
5?s» J
or 3,3
^Smn
«5s*a#
,
.} \1/
\J/
Tis« T«aa* ,
=: i JfivJ >)^s,8«
llfch ttmm vm3.«#S|t ,*# ©btaia as tbe.imw mlnimm problem.
k®{H »-«..«..i..Al«.i...l...a.|.l.,mililt.il
an - k(I. •!.»)„
„...„ W
„ W|
.m...„W.->
f s aitt«
or •
- {I - I')®
4(ii - fan (ti »iH'i
" *•
This will yield a get of equatlorss is %. tsj^, 0, d as
2
dCJu • L*)
YTZW'
"*•
1
dCil - M*)
1
aC8 » M*l « A
mr
- TTW
InvGlv@d»
ticialar dlfficttltj.
Ttm integrals 1,
flii'- work of iBtegFatfing L
BhoTten&d becaus# , ef fcli©
to T% mfl Ts*
II eatiss' tm par**
of
tm soias^mt
fttnetloat
y#sp&et
Tlnis %%• %•» &milj mmn tiiiti-
3r*
J
t'a
aft#.
1
Jia^
jlliuia »
fbe itttegrstion of 11*;^, M, i* is moti. longer and coataina
several difficulti#® «l5leli are diacass##
I.* la sysKkBtri© mltli j»esp«et to
as abo'®#
T.tie integral
and y®, -mA fflay b#' ahoytened
Thusi
f^-^T jilasaJICa&i s LivJ
J' •
r%
• J • •
.r^
and
Jtej
d^J
y»i3
S*sa
The mluea of IIh, U .«»€ M% !»»;>- H» in tetM of th© pai'ametera
ar© glt®n ia tbe a«xt ##eiion#
^^13® integrals
Ac
j c3rs
-/o
d3P»
ai»© of tl» fom
arts
-'lil-rj
- ti-rjl
I
J?
„y"^
rsf, ,.rift.
' N|*T5^"^Pa.a) ® (' "x^s"»
)
ife©F» all t l m ^ ^ m ' t t m r*s m&y talc# ©a ©if values.»
In
of
i^onential
©o%.ial» ftjp. p
ji., tl*
©f th© ©xpon*
y l»av# %ii» 'valw® o: » 1 #r
4- x)^ § m %._g
y - a or |(a • 1), eorr»apouding to Integrals o^r
"^XBS "^3131•
exponent t never exceeds 3,
first lnt»gX'atioa %m
yor t « S,
or
fftien t » 1,. th©
easily fo\ai€ to h&vm the
lnt©gi*al ims tli# ifialme
or 2, til© first Integral trnmrms elliptic^
it Is 6Xm.jB posslfel© t«
tMa difficulty i# ©^.I'co -e.
^ i®)*:-*
-S*
If % « 0'
liitli on% •mm&mption
the «r<ler of liit#^aticm so feiiat
Tlzla cm.# exception «ill. M. di#cia®a©a
later#
file second ^aad tMrfi itttegrati^as oaa toe e.srrie€ ••out with*
out diffieulty,
til® subftitmtlois ©f tto© liiaits squires tiiat
q and s Iwtli l3® <?#!»
Ihen this la tfe#
Wm reaainltig integ-'
rals are of the fomi
f«ri firg fdrg r| r^ ri
•'O
'-'o
®3ad can be integrated toy tt:® »ll-feioTO f^wtila infc©
It -either t or « i» #¥©0.,^ the integpsl ©wr r^s srmst be separates; into t«> parts#
Fop '©xsn^l©,!,. If f is even,
til® integrisl ofer T%^ hemmm
[dTa. farg
•^o
•'o
-'o
* j'^to'a. Jdf® Jtr® rl'
p. 51 (18G7).
[dras 2*1® r-^ r@ r« e°"'"
•'In-tii
^
-J<ii»^J<p'g|cir3 rr^ r| e*''
^
-gg-'
thm- t±m%
.^d.
of integral#
mmn&
lao^
mm %im tpsc© whmm rs.>»®
wimmm' Tx'^m-
%• part® mi.
ers ot t%B
litt@:gi»atioii &wmr i»g. latist
long and t©€io«s fo^ iitc^ pow­
r®#
0a® dlXfieiilty
^£©14
3P»
tegr-al _1«
tm tm
In £«•-.•
©f tti®- t«iaa
Aft#r integration. ©w»r
|[<iri fdrs ("is fdxxg f^ga
"^o
•'o
-"o
Jln-rii
'iri-rji
^-<^c>-.-vr»)-r^
I^a
« + |sji?3t jdrg Jil&g
• rs^iCSrg®
ipg®)
+ Isjii Jipg J^n
f3.Fgrj5(33C»i« • I?|8«H^0® + ra®)
+ |Sjd]:.»t, Jlr® Jute's
W^^Wsimm^ *
4-
[jjy- f
^ Jo
Jr,
f itFs
• yg®)
-(03?#* :+
•* 3?a®l ©
Jo
fM® lest @#t &t iiit#grala ims or. «i>psi?©nt diffio-ulty, b«-'
cause of tlie ©xponost at
After integrsti#ii
parts over
Ts, «® find rni lateral, ©t thm fom
f&^3,
-Jo
Jr,
T% e"''
a as ^(a + 3. J and p •« « •
0^.©3? wm mppmmrs t-«s to# divergent.#
-li.
r®
At fl»":®t glsnee tlie'- integral
flii.s..^, liowswjr*. a»:^
l.f the order of iat«^,&tioii ts. chmigmA^
ral Im
,
«vold®d
Ttm m^&nltlng f»te.g*
•O
f
^ i*rt#
-
£
8
^
-
„
_
4 -«f|
9%%
-f-L
m -.-jgs
•^e
w% gimm
a
*
^ -
'ija-ei^ %«» fe#
a
mmw
tw,. «iii«li appis** t#
^
m% «»% o>-.((> . fe tl» lest
'fl^f vm
'
OO
"
^!
iwm •
tim m,itim)M§ I# «%•
«« ti^|.|jiii ^ eiiom
t0 0:OBV9^f« Sl^ hmm tl»
^ «li lor, g
A m^mA .tiffleuitf
Sjj t,B@
©f tt»
1•
'*11^ #»!» .to 1*#
»i^
' t»»t3i® stiifto wltli itl» lisfc#g»»
tl<» of tt» l*§t t«Ks
i»8sifc#*»g -Aisii |£i»^ 5^^**
S» fsator-1® • t® ««»t«s t^ss well, m r*®»
tbtslaeata mimmm
A
^ *» €l»Ps«e%» fe\® a«s»
tt», Satw5^«a# i>w®r Wm ^
oiii|.l&r %»
twi
m fiad aft» »f««fmS.
e s #f list«gr«i»
•6
r -«<j- -e«'i
..a..
J
Lei%
j-a
flisscs^' of
'% imf^s
f
06
»r, -«••»•
-«''«•
-«t.
B«a 88 - 6_.o
J
Ora.
55®
Serf#®* })p« 4S4*42i5,
fli# Hyst h.m
f&xm
b#®®. lnt©gpa"fe#fl*.
tfe# second i» ot t!j©
oe
I:
. -ax
-b*
A#,
B©
wiiich may hm aho-m
^
_cX
+ Ce
b# .iatagjra'bl-©
Ait -fr Sb
C© «•
fiiaii^atw©®^,, p^»ovi«l®€
4- C « §,j|
smA imvm tti© m2.«» !«•: logs • m logte ••*•
iog^e*
It is easily s#0ii tlm$ tli@ ^©©fflelenfcs an^ #^oiiea%» la
atw integral amtlsfy tbis:»0 eondlfeioim «at tlist tli« valui® of the
integral is
2p 1©SP • P
P
logl«
1 thlyd •diffieml.ty Is that io®atl-on<t<i
uiian feli© fia^st
lategratioB is ©Hipti.e mA Int&rGtmngMg 0S»ii©f of Integfatloii
does not SMpr&ve %}m #itwtioa»
as the cjoeffioleat Uff
exact integral 1#
flils typ#' of lategyal ocetij*#:
i» the potential tea
IflvJ £a-sS-aa.
of L**.
• •
fh®
Both trasJ?##. wii©s included wltli
the Jlacotoiaa.. Imv© #v«n ©:xpoii«iits aat
has a 2#^ «3Cpon.#iit:,
Tims intagration® owr m;y of tb® tiir@« will b& ®,llipfclc#.
. la order to aj|>i*©xlsat« thia tern w BmpmS.
i#8 of Ii®g©ndJre Polimoaial#®.*
»i [
P
in- « s«r*
fhus
) + h? X (/*)
and Co., Loudon Cl908)«
a ^tiittak©r^ Cotarse la f3oci«3m Aimlysis,,
Ifeiwrsltj? Press, Lonrlon
p* 296-.-S30,^ Csal33Pi«lg©
-25mimm 'h «
^®
la cr*
that the series he. eoiwefgeiit h »
< 1*,- so fefeafe w» a«#t
s© two series on© -rrtt.-". V:^t% ^<1 lii® etfe©?' witli ri<f»s^
SubstifeiitloB ®f tiii« ia t3^ sb©t# lmMm.^m.1 glv©#.
l^p 1^ J
4-Jx,.vJ
fojp"
|V (jj) 4. iPj^ (jj) +• ii:%.gfjj> *«*] foy f-tCTs*
Sow til#
Pljj) s! 1,
1
Polynomials ar«
Pa(jj)
« eos0,
•wli©y« cos# » ^'*"'
Ps(jJ) a?
* "I:
litli th##»
eos^e -
%lm-
terns are all iat©^abl© .altixougli soiaeirhat ioag aai eoa^lieated*
¥i© gl^® li®r®
s«t*-iip nmrnBsmpf £m tlm first lnt#g»».
tion^^ wMeh is aiiall.&r to th&% of Wm rpaaSnlug intSf^Patioiaa#^
Sine©fciiere
ai*© two part.# for eacii tem <i&Tmspo&S.ln.g %a
•and T%>rm0 •«€ mtnm. S"ts em&. W'm ^nt&r mltii. mmn
integral.
0
1
Jo
h&
firs
Jo
iSfs
j"ip» J^i^n
late six p«rt% m iell&mtr03?a^CSl^3L® ^
r»%3.CSrg'®
• rs®l
rs«'Hll?g'® * 'W%^1
* i?a«|
the
i.
+
®Jo
K,
r&s •
J
* rnm^} Car# •* w%^) 4""^'''''''^'*''-
fi.
|«r» j"^'8 TtTm^iSTg^ • rg®l lm%^ + i?@»)
^
^ ij&s J^« jis
rx^raCSrg^
r®® '! CSi»g® •
Mm it e«a to® -^o« that tli« %©%•! lat#gral Jd'rJ ^Sigfaa
is SMil©^ tlt« ^y #f til# corresporxling mm^ 1;#iiss In L* a»^,.
-tfe# wml-mB of i»* «p© in g©s«»l l.®ss ttssa tlios© of i,
«o tli&t
t«Mis la %im. series' will ifi^e an ©xc©l3.6»t approx^
Isafcloii im tti« Integral,#
Tbls *iil be .dis:cusa#ci Xe^teT Aeo
¥mlm@s of ir>e giai^aiister «» gl^#a.
*91"
111*
CALCiirji^ifioHS
!.• %'al.ii©s of
fbe th©oi^t-ieal laetfcod of det®!*"
mialag tli©
by diffsreatiatt'esii hm hmm des^ibed
til#
is tli# ©xpaient lal #a%sFS ia
siiela ft :e<a!ipliea.fee€ aaam-F tliat it la not p^aotieal %o mt'tempt
I«.st©ad, wm obtain tbB
of ^ lay
and
tli.#s detefsilii© tl» Qtlfiers % cliff©rentlatioB»
In
%&• sel»c% the tmmt vmlm of jis^.. »® use tt» a«cii
siiaplcy trial .fuxictloa
tow tbe oatef ®X€etx»oii*
lltli «. IStfci# eitlcalatim It is fournd
tiiat til# t»st mla© .of a tm t.M.s f-asetioa is. m » ,.t6.» • fliie
is til® amm fmetlai m that
bf Sl«te.r mA results in a
of til© .«@r^ Qmlf ,88 pe,y eeiit in #,rfor»
f'cai©tioii is tlmt u.s#d by
ftm- nesct feial
and
i
•t)!?),*
i
Using «: « -•2i wm fiii4 h .« -..g*.
By trial It is foima.
mltie of a Is the hm% f'or this
tb© ©iierfcg'
tills .
«a# c^vmw a w.mlm of
p«i* «iit itt ©ri'oi'#
••
fii® final fi33ict Oil tt0e«i is
ss
[l -fr. h3*a •*' eiPj.e + dCrx-f^)®]-
Puttlng .« « .2o, »©• find valme© of M » fl*, I^.
feerss &i b, .© sad d to to©
S * IP in
M • 1* » ®(S*r3.3064 4- 996l»9a6b • 20^8*898® • tSgliSSM
+ S0,m.908b^ • S303*&112e^ •
+ 20,S64*845bc + 17,840»l9e4 -I- S6t802.98M),,'
L - L» « 3C(634.3433 -t- 17,458*17b + 4175*044ie 4- 4l3g»497d
+ 144,010^97b^ + 9557.2S5<}'* ^ 76,644*71d^
+ 61j204*87bc. + g9»B2fU74c4
f8»079»28M).,
N - K* s It(163^607 + 4643*485b + 1252.3077© 4- 98^778^
-f-
* 3BS9*61©® -• I8,BS0.'78d®
+ 10,643..865bc + 7639»9844M + I6,712*54M)t
file cQTTeQt ¥alm®s of th^ thrm r©aaiiii.iig paf>«aet'©FS em
tli#« 'De fomd froa the ©q.-nation
V^e need to solve tbe equations
2
a{L ^ L*) - 1
dCl. • iSM
1. dCl • K*)
^
irrt' -Ta—
rrit
— - mrft -^5^— =» 0, etc.
for tM paran^ters
c and d*
fornid
e « -•IS mid
b » ••29^
Yaliie of If « -l#65967*
Kie values #f lii© paralters
d «
fh© experixnentml iralti# is given at
1 a -1*663 a^ci tli« per cent error Is
TrM enero' of
Tbe 'B-alu© li® glws is
milts
<«*1«.6177*
fbeB® glv© a
,M per cent*
Ion lias ts©e» <s€«aptit©d %ll#rtt«@ *
0
•7*2798 in tmlfc# of # /a©.# ©r la G-m-
Subtracting, tlils v»lnB twmi Gm> valu® for tii«
groiaid atst-© ©f ttm at®®, *© fl^ tl» ionlsatlon potential to
tm
,a419.#
ttm exi)@rlae»%al '9'alii© Is glvea
lofes and Becsk*
S3
man
T
s
at
P
h
# ©441#
y s .'6 6 , 2 ' G 9
B©v* S, 8& (192S>»
"•
—
~
'
1?
mimiim
®-i» Scliro«iin0er partial
t©. mn #quim3L#»fe
.©itrntloii t&
is tti# ealcml'tt# pf tarlati.oag miA
feriis problem is Vmn dlrectlj- aolv^ii
th# us# t>i trial mmm
functions,
A proper aiiti^sfiiaeti^le ««-.«,
teractions of th# «l#0%foiis, i« set up
and noBaall»lns
eontalnlng tlm in­
fli® energy irit#gFai
m?9 foaad lii t#3ra«- of pa^ajneters wliieli
©nter In th© wav® function.
Thes© paranefeers aiP# varied i.mtil
t:5© valti© of the ®»#rgy * Is a lalntiuja,*
A laatiiod foi» flu^iag tli« i>eat ^'aiiies of th# puraja^tef® ^
iBoaiiLS of simple fcrisl
wmvm tmrntlons mtA. hf aireet differentia-
tlon is desex'lbeft*
Ivalmtion ©f, aa elliptic int@|rral is gtv&n In teims of
Legeadi'e Polyiia-alal® .ana Metiiods for integj^stiiig. s»¥®ral
1
^
t^onblesoia# ©xposwitis.! tnteG^'als are gi'rem*
!fi3» valtt# ©f til© es©rg^ Is foimcl wttMa
experlniental iralu#*
percent of th©
ffels mlii# co^iM fe#.
Isy
of a
®or# ccaaplleatei «»¥© foaetion &nA «E€cl.ltioiMl p«i»«teters»
ilotet
As t'ais wqwM was nearing eoia^>l0tl«ij| an a. stract
«ippeay©d lis tiie Pi'^slcal l©fl#* Cfol* 47, May
19SS, p«. VQQI stating ttat E, 1.
C:ooiMg« i»v« ccsriputed a val-a« ef
,aiad A*- S* .
fl
eopTOspoadSng
etm9r to the aeespted ©sperlseiital
i
Bine© jio details
gi^®B ia tli# sb-
.©feraet, imWkmw eoaparison of t;li« worlc hm »ot
b##»
*31:.
!• 1
r®:8st0as f#r 1#
pamiaefe^fS*
1 mnd
I* la t^mm-
tl*
fl# tdv® oa fch© following pac;e3 tti® «lg,®lraie #x»
pTOs,sions for tb# iut^gi'al# in terns of th® pm^mmteits &, t)* c,.
and
In ordi®' to sij^liff %li# »itltig., m «## tb#
lag notations
' a « im •# 11/2,
y « a 4- t.
f «• a 4p. s# e • .3,,
-* ->2"»
y«it + -g
1*%
®
«.s
la«
^
a® .
-f* -iii^Wi^'
.M M^J/'
M^TtiiiiHiiHijBr
^ 192b® ,. 4Qb»
+ _iy_ + __J_
a'
4. ^QtM? ^ 1.40bc ^
a
23Bc« ^
aIT
•
Lia =» sfl + eob' 4. 222f
Sla"
*
ggf
3 I as • irp:»
^ -iyS ^
12
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p®7
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• fc
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