INTRODUCTIONTO

ATOMIC SPECTRA

BY

HARVEY ELLIOTT WHITE, PH.D,A ssistant P rofessor of Phy sics, at th e

Universit y of Cali fo rnia

McGRAW-HILL BOOK COMPANY, INc.N E W Y ORK AN D LO NDO N

1934

CHAPTER VII

PENETRATING AND NONPENETRATING ORBITS IN THE ALKALIMETALS

We shall now turn our attention to t he formulation of an atomicInodel which treats the interacti on of a single valence electron withthe nu cleus when it is screened by an intervenin g core of elect rons,i .e., by inner complet ed subshells of electrons. This must be don e if

K

--HG1 rt r ee- ----PauIJng- '-'-Thomas

M

0.5 LO 1.5 zo----- -

25 30a

N

FlO . 7.1.-Probabilit y- densi ty- di st r ibu t ion curves for t he rubidium-a tom core of 31) elec­trons. (A f ter H artree.)

we are to calculate from theoreti cal considerations t he energy levelsof t he alkali metals. On t he classical picture of the atom t he nu cleusis surrounded by va rious shells and subshells of electrons in orbitsresembling t he various possible states of the Bohr-Sommerfeld hydrogenatom. The qu an tum-me chani cal picture, on t he other hand, appearsin the form of a probabili ty-d ensit y distribution for t he same electrons.

7.1. The Quantum-mechanical Model of the Alka li Metals.- Bymean s of successive approximations to t he so-called self -consistent field,

100

SEC. 7.1] PENETRATING AND NONPENETRATING ORBITS 101

Hartree? has calculated the ra dial density distributions of each coreelect ron for t he different alka li metals. Although a t reatment of themethods by whi ch t hese calculations are made is out of place here, thecalculations are not difficult but are long and tedious. It will suffice

632o

l'

Lithium

L

25

2 3 4 5 6 701

Sodium

M

~ 35~

C'l'-~ 2 3 4 5 6 701

<;;t

•aPotassium

N

4s

2 3 4 5 6 l ccl

Rub idium

0

55

4 5--r­F IG. 7.2.-Probabili ty- densi ty distribu ti on curves for the neu tral alkali a to ms, lithium.

sodiu m , potassium, and ru bidiu m. In each case t he core is shown by one cu rve and t hevalence electron by another.

to say, however, t hat t he resultant electric field obtained for any atomis su ch that t he solutions of Schr odinger's wave equation for all of t hecore electrons in t his field give a distribution of electrons whi ch reproducesthe field.

1 H ARTREE, D. R. , Proc. Cumbo Phil. S oc., 24, 89, 111, 1928.

102 I N TRODUCTION TO A TOM IC SPEC T RA [C HA P. VII

As an example of Hartree's results, probability-density-distributioncurves for the rubidium core are shown in Fig. 7.1. The radial curvefor each subshell is shown in t he lower half of the figure. The heavycurve represen ts t he sum of all of t he 36 core elect rons . The fourloops, or humps, in the la t t er are taken to represen t t he K , L, M , and Nshells, even t hough eve ry electron contributes something to each shell.The dot ted curves shown above for comparison purposes have beencalculated from hydrogen wave fun ctions by Pauling I by using approxi­mation methods and by Thomas? from stat ist ical considerat ions. Incompa ring thes e curves with t he corresponding hydrogen-like functionsshown in Fig. 4.6, it is observed t hat each density curve, due to a verylarge nuclear charge, has been drawn in t oward t he nucleus by a con­siderable amount.

Prob ability-density-distribution curves for lithium, sodium, potas­sium, and rubidium are given t ogether in Fig. 7.2. In each case thecore charge is shown by a shaded curve and the valence elect ron in it snormal state by another curve. Radially it should be not ed that themajor shells in each atom lie well insid e t he first Bohr circular orbit ofhydrogen r = aI, and t hat t he valence electron lies well inside thecorres ponding hydrogen state. In hydrogen , for example, t he densitydistribution D for a 58 electron is apprec iably large as far out as 50al,whereas in rubidium the nodes and loop s have been pulled in to aboutone-tenth of t his.

The dots on t he r axis represen t t he extremities of t he classical orbitsbased on model a (see Fi g. 4.8) . In t hese 8 states t he kin eti c energyof the electron at t he end of t he orbit is zero, and the total energy isall pot ent ial ,

e2

TV = P. E. = - - = - eV , (7.1)r

where r is in centimeters . Expressing 11 in volts and r in Angstroms,

(7.3)

(7.2)r

11 = 300e = 300 X 4.77 X 10-10 = 14.31.r X 10-8 r X 10 8

Expressing r in units of al (al = 0.528 A),

27.1 27.1r m RX = V = ionization potential in vo lts'

From the ioniz ation pot en ti als of t he alkali met als given in Table 6.1t he following va lues of the orbital ext remit ies are obtained :

Li Na K Rb Csrma • = 5.0al 5.3a l 6.3al 6.5al 7.0al

1 PAULING, L., Proc. Roy. S oc., A, 114, 181, 1927.2 THOMAS, L. H ., Proc. Camb. Phil. Soc., 23, 542,1927; see also GAUNT, Proc. Camb.

Phil . Soc., 24, 328, 1928; and FERMI, Zeits. f . Phys., 48, 73, 1928.

SEC. 7.2] PENETRATING AND NONPENETRATING ORBITS 103

The above derivation is only a close approximation, for we have assumeda rigid core of unit charge.

7.2. Penetrating and Nonpenetrating Orbits.- T he quantum­mechanical model of the sodium atom is shown in Fig. 7.3. In additionto the shaded curve for the 10 core electrons the three lowest possiblestates for t he one and only va lence electron are also shown . The corre -

Penetrating

j j38 3p

SODIUM

Non-Penetrating

l.3d

3d

FIG . 7.3.- Compari soll of the quantu m-mechanical with t he classical m od el of theneu tral sodium atom . Three of the low est possible states for t he single valence elect ro nare a lso sho wn .

sponding 38, 3p, and 3d classical orbits based on mode l b (see Fig . 4.8),are shown in the lower part of the figure . A comparison of these orbitswith the corresponding hydrogen orbits shows that, due to penetrationinto the core, 38 and 3p are greatly reduced in size radially. The 3dorbit, on the other hand, remains well outside the main part of the coreand is hydrogen-like. Corresponding to t he penetration of the 38 and 3porbits the probability-density-distribution curves (above) have smallloops close to t he nu cleus.

104 I N T RODUCTIO N TO A TOM IC SPECT RA [CHAP. VII

e

Consider now t he classica l picture of a valence electron describinganyone of various types of orbits about the spherically symmetricalsodium-ato m core. In Fig. 7.4 six different orbits ar e shown representingvalence-electron states wit h the same total qu an tum number n (thesame maj or axis) bu t slightly different azimuthal qua ntum nu mber l(different minor axes). With a core-densit y distributi on of chargealways finite bu t approaching zero as r -? 00, all valence-electron orbitswill be more or less penet rat ing.

In Fig. 7.4a t he electro n moves in a path well outside t he majorpart of the core. Sin ce t he field in t his outside region does not deviate

NON-PENETRATINGORBIT

-------~·f

PENETRATINGORBIT

FIG . 7.4 .-Sho wing the chang e in electron orbits with in cr ea sing p enetration.

greatly from a Coulomb field , t he orb it will be a Ke pler ellipse pr ecessingslowly (due to small deviations from a Coulom b field ) about t he atomcenter. In t he rema ining figures increased penetration is sho wn accom­panied by an increase in t he precession at each turn of t he orbit . Ast he electron goes from aphelion (rma» to per ihelion (rmi") ; it leavesbehind it more and more of t he core charge. With t he steady in creasein force field t he electron is dr awn from its original path into a moreand more eccentric path, with t he result t hat at its closest approachto t he nu cleus t he electron has t urned through somew hat mo re t han180 deg. Up on reachin g rm " . again, t here has been an advance, i .e., aprecession, of t he aphelion. The increased force of attraction betweennucleus and electron at penet rati on increases t he binding energy , thekinetic energy, and t he term values bu t decreases the total energy of t heatomic system [see Eq. (2.15)].

SEC. i .3] P E N E T RA TING AND NONPENETRA T ING OR BITS 105

FIG. 7.5.- Sch em a t ic represen t a ti onof t he pola rization of t he atom core byan ext ern al electron.

7.3. Nonpenetrating Orbits.- Nonpenet ra t ing orbits are defined ast hose orbits for which the observed energies are very nearl y equa l tot hose of t he corresponding hydrogen-like orbits. Such orbits on eit hert he class ical or qua ntum-mecha nical model do not appreciably penetratet he atom core and ar e t hose states for which the azimuthal quantumnumber l is more nearly equa l to the total quant um number n. The forbits in all of the alkali metals are good examples of nonpenetratingorbits. T erm va lues for t he 4f, 5f, and 6f states given in T abl e 7.1 willillustrate t his.

TABLE i .I.-TERM VALUES OF NONPENETRATING f ORBITS IN THE ALKALI M ETALSCO~IPARED WITH T HOSE OF HYDROGEN

Electron designation . ... . . .. . . 4f 5f 6fT erm design ation . . . . . .. .. . . . . 421" 5 21" 62/<'

H ydrogen 6854 .85 438 i . 11 3046 .60Li 6855 . 5 4381.2 3031 .0Na 6858 .6 4388. 6 303 9 .7K 68i9 .2 4404 .8 305 i . 6Rb 6893 .1 4413 .i 3063 . 9Cs 6935 .2 4435.2 30i 6 . 9

With the exception of caesium the observed va lues are hydrogen-liketo 1 per cent or bet ter.

T erm values plot ted as t hey are in Fig. 5.2 show, in genera l, thatp, d, and f orbits in lithium, d and f orbits in sodium and potassium,and f orbits in~rubidium and caesiumar e nearly hydrogen-like. In t heenha nced spectra of the ionized alka ­line earths t he term va lues are tobe compa red wit h t hose of ionizedhelium, or t hey are to be div ided by 4(i .e., by Z 2) and compared wit h hydr o­gen as in Fig. 6.5. In t hese energylevel diagrams it is observed that p, d,and f orbit s in Be II, d and f orbits inMg II, and f orbits in Ca II, Sr II, andBa II are hydrogen-like and therefore corres pond to nonpenetrating orbits.

Although nearly hydr ogen-like, t he te rm values of nonpenet ratingorbits (see T able 7.1), in genera l, are greater t ha n those of hydrogen. Bornand H eisenberg- attributed t hese small differences to a polariz ationof t he core by the va lence electron (see Fig. 7.5). In t he field of t he

1 BORN. M ., and W. HEISENBERG, Zeits. f. Phys., 23, 388, 1924.

106 I NTRODUCTION TO A TOM IC SPEC T RA [CHAP. VII

valence electron t he atom core is pushed away and the nucleus is pulledt oward t he electron by virtue of t he repulsion and attraction of like andunlike charges, respe cti vely . The effect of t his polarization is to decreaset he total energy of t he system. On an energy diagram t his means alowering of t he level, i.e., an increase in the t erm value. Theoretical valuesof the polarizati on energy calculated for the alka li metals with the aidof t he quantum mechanics are found to account for t he major part oft hese very small deviati ons from hydrogen-like te rms ."

7.4. Penetrating Orbits on the Classical Mode1. - Although no sharpline of demarkation can be drawn between penetrating and nonpene­trating orbits, the former may be defined as t hose orbits for whi ch theterm values are appreciably differen t from t hose of hydrogen. Referringt o Fig. 5.2, t he s orbits of Li, the s and p orbit s of Na and K, and thes, p, and d orbits of Rb and Cs come under this rough classification . Cer­t ainly on t he qu an tum-mechanical model all orbits are penetrating.

To effect a calcula ti on of t he t erm values for penetrating orbits on theclassical t heory, one is led by necessity to simplify somewhat the atom-coremodel given in Fi g. 7.4 . A suit able idealized model was first put forward

by Schrodinger " in whi ch t he core elec-Penetrcrln q Orbit t rons were t hought of as being distrib-

uted uniformly ove r t he sur face of one ormore concentric spheres. This samemodel has been t reated by Wentzel ,"Sommerfeld, " Van Urk," P auling andGoudsmi t ," and others. Sin ce theclass ica l t reatment of penetrating orbitsis so closely ana logous t o the quantum­mechanical t reatment of t he same

FIG. i .G.- Valence elec t ron penetrut- orbits, to be taken up in the nextillr~ an ideal core where the core electronsale distr ibuted uniformly ov er th e sur- sectio n, Schrodinger's simplified modelface of a sphe re . will be considered here in some detail.

Cons ider the very sim plest model in whi ch the core electrons aredistribut ed uniformly ove r the sur face of a sphere of radius p (see Fig. 7.6).Let Z ie represen t t he effect ive nu clear charge insid e the charge shell andZoe t he effective nu clear charge outside the shell. Usually Zo is 1 for thealkali metals, 2 for t he alkaline earths, etc. The potential energy

1 For the qu antum-mechani cs formula giving the polariza tion ene rgy see L.PAULING and S. GOUDSMIT, "Struc ture of Line Spectra ," p . 45, 1930; and also J . H.VAN VLECK and N. G. WHITELAW, Phys. Rev., 44, 551,1933.

2 SCIIRODINGER, E ., Z eits. f . Ph ys., 4, 347, 1921.3 WENTZEL, G. , Zeits. f . Phy.~ . , 19,53, 1923.• SOMMERFELD, A., "Atomhau," 5th German ed. , p. 422, 1931.6 VAN URK, A. T. , Zeits. f. Phys., 13,268, 1923.5 P AULING, L. , and S. GOUDSMIT, "Structure of Line Spectra," p. 40, 1930.

SEC. 7.4] PENETRATING AND NONPENETRATING ORBITS 107

of the valence electron when outside the thin spherical shell of charge(Zi - Zo)e will be

(7.4)

while inside it becomes

Z ie2 Zie2 Zoe2

V i = -- + - - -0 (7.5)r p p

The total energy for an elliptic orbit, in polar coordinates rand 'P, isgiven by Eq. (3.26) as

W = T + V = ~(p2 + p~) + V (7.6)2m T r2 ,

where PT and P<p are the radial and angular momenta, respectively.'From the results obtained in Chap. III it is seen that outside the chargedshell the motion will be that of an electron in a Coulomb field of chargeZoe, and inside the shell the motion will be that of an electron in a Cou­lomb field of charge Zie.

Applying the quantum conditions to the orbital angular momentump<p, which must be constant at all times throughout the motion,

LZ

1Cp<pd'P = kh, P<p = k2:· (7.7)

That part of the electron path which is outside the shell is a segmentof an ellipse determined by the azimuthal quantum number k and theradial quantum number r o, whereas the path inside the shell is a segmentof an ellipsedetermined by the same azimuthal k (p<p = constant) but adifferent radial quantum number, rio Substituting successively thepotential energies of Eqs. (7.4) and (7.5) in the total energy [Eq . (7.6)]and solving for PTI the radial quantum conditions can be written down as

(7.8)

(7.9)

The total quantum numbers to be associated with r o and ri will, as usual,be given by

and n i = k + rio (7.10)

Since the electron does not complete either of the two ellipses inone cycle, the integrals of Eqs. (7.8) and (7.9) are not to be evaluatedover a complete cycle as indicated but over only that part of the ellipseactually traversed. The radial quantum number r for the actual pathtraversed is therefore given by the sum of the two integrals

108 I NTROD UCTION TO ATOM IC SPECT RA [CHAP. VII

i Rodr + r R idr = rhoouts ide J iDBide

(7.11)

The total energy in t he outside region by Eqs. (2.14), (2.15), (2.30), (2.33),and (7.4) is

(7.12)

where al is t he radius of the first Bohr circular orbit.The total energy inside is

W = -T = V = _ Z;e2 + (Zi - Zo)e2.

2 2aln; p

Since the energy inside and outside must be the same,

Z~e2 Z;e2 (Zi - Zo)e2- -= --+ .

2aln~ 2aln; p

(7.13)

(7.14)

(7.15)r = r: + rio

Penetrating Orbit

Consider now t he special case shown in Fig. 7.7, in which the t wopar tial Kepler ellipses are almost complete.' If t he outside orb it were

a complete Kepler ellipse the elect ronwould never penetrate t he shell,whereas if t he inner orbit were com­plete t he elect ron would always remaininside. As t he outer ellipse is madeless and less penetrating, t he twoellipses become more and more com­plete and t he integrals of Eq. (7.11)approach t hose of Eqs. (7.8) and (7.9).Expressing t his in terms of t he radialquantum nu mbers,

FIG . 7.7.- Special case wher e in ne rand outer ellipses are almost com plet e.Co re charge distr ibuted unifor ml y overt he su r face of a sp here.

In a similar fashion the periheliondist an ce of the outer orbit ao(1 - Eo)

approaches t he aphelion of t he inner ellipse ai(1 + Ei) , both approac hingat t he same t ime t he radius of the spherical shell p. We write, t herefore,

ao(1 - Eo) = ai(1 + Ei) = p, Ei = ..!!. - 1.ai

(7.16)

In terms of the qu an tum numbers [Eqs. (3.23) , (7.10), and (7.15)]

n - no = ni - k = fJ.. (7.17)

This difference n - no is the so-called quantum defect and no is the Ryd­berg denominator nell . The semimajor axis and eccentricity of the

lSee VAN URK, A. T ., Zeits.f. Phys., 13, 268.1923 .

..-

SEC. 7.4) PENETRATING AND NONPENETRATING ORBITS 109

inner ellipse ar e by Eqs. (3.32) and (3.22), respecti vely,

Substituting a i in Eq. (7.16) and squa ring, there results

(PZ i )2 k2

E~ = - - 1 = 1 - - ,'n;al n~

from which

(7.18)

(7.19)

(7.20)

Replacing k2 and k by the corresponding quantum-mechanical valuesl(l + 1) and 1+ t, respectively [see Eq. (4.52)), and substituting inEq. (7.17), the quantum defect becomes

(7.21)

This expresses the experimental resul t, well known before the quantummechanics, t hat for a given atom the quantum defect J.L is a fun ctionof the azimuthal quantum number and is independent of the to talquantum number (see Table 7.2).

T ABL E 7 .2.-ExPERIMENTAL V AL UE S OF T HE QUANTUM DEF E CT n - n o = p. FO R

P E N ET RATI N G ORBITS IN L I T HIUM AND SODIUM

Element Term El ect ron n = 2 n = 3 n = 4 n = 5 n = 6 n = 7

Li 2S s, l = 0 0 .41 0 .40 0 .35 0 .35 0 .35 0 .35~

Na 2S s, l = 0 . ... 1.37 1.36 1.35 1.35 1.352p p, l = 1 .. . . 0 .88 0 .87 0 .86 0 .86 0 .86

In order to calculate qu an tum defect s from Eq. (7.21), it is necessaryto determine values for the electron-nuclear distance p. Successfulattempts t o calculate suitable values of p from quantum-mechanicaldensi ty-distribution curves similar to t hose shown in Figs. 7.1 and 7.2have been made by Pauling. I In t he case of Na, Pauling obtains t heaverage value of the electron-nuclear distance PK for the two K electrons

as 0.132al, and for the eight L electrons PL

= 0.77al. Assuming that

the va lence elect ron penetrates only t he outermost shell of eight electrons,the effecti ve nucl ear cha rge Z ie will be ge. Sub stituting Z i and P

L

in Eq. (7.21), t he values of J.L = 1.36 and J.L = 0.85 are obtained for the 8

and p orbits of sodium, in very good agreement with the observed valuesgiven in Table 7.2. For lithium with Z i = 3 and P

K= O.53aI t he value

1 P AULIN G, L., Proc. Roy. Soc., A, 114, 181 , 1927.

110 I N1'RODUCT ION TO ATOM IC SPECTRA [CHAP. VII

(7.22)

IJ. = 0.39 is computed for the s orbits, also in good agreement withexperiment .

7.5. Quantum-mechanical Model for Penetrating Orbits.-The solu­t ions of t he Schrodinger wave equation for hydrogen-like atoms andfor the normal state of the helium-like and lithium-like atoms have beencarried out to a high degree of accuracy. The calculat ions for lithium­like atoms are of par ti cular interest in that t hey appear to give thequantum-mec hanical ana logue of t he inner and outer orbit segments, sowell known on t he classical t heory and t reated in t he last section.

Numerous attempts have been made t o calculate t he energy of thelithium atom in its normal state 28. Perh aps t he most recent andaccurate calculations are t hose made by Wilson. ' In previous deter­minations single-elect ron wave fun cti ons of the hydrogen type wereused inst ead of a wave function for the atom as a whole. The ratherbrief treatment to be given here is t hat of Wilson based up on a principlepreviously introduced by Slater. >

Slater has shown t hat a wave fun ction representing an atom con­t aining many electrons may be properly constructed by expressing itin the form of a determinan t t he elements of which are built up outof the Schrodinger wave equation. In the case of lithium, for example,t he wave fun ction is writ ten

1 -'!: I -'!:2-'!:3Yt = V(3 A I A 2 A 3 ,

BI B2 B3

where t he A's and A's are I s hydrogen-like wave fun ctions for thetwo K elect rons and t he B's are 2s wave functions for t he L elect ron.

From Eq. (4.55) the solutions of the wave equation for I s and 2selectrons are

(7.23)

(7.24)

where N I = - (Z3/7l' )1, N 2 = (Z3/87l') 1, Z is the atomic number, and aI,

the radius of the first Bohr circular orbit, has been chosen as unit oflength. Now the 2s func tion [Eq. (7.24)] for t he valence electron isorthogonal to the Is fun ctions [Eq . (7.23)] for t he K elect rons. Thiswould no longer be so, if different values for Z should be subst itutedin Eq s. (7.23) and (7.24) . F or this case Slater" has shown that thet wo fun ctions may be orthogonalized by adding a fraction of one of the

I \VILSON, E . B., J our. Chern. Phys., 1, 210, 1933. For other references see thispap er.

2 S LATER, J . C., Phys. Rev., 34, 1293, 1928.3 S LAT ER, J . C., Phys. tu«, 42, 33, 1932.

SEC. 7.5] PENETRATING A ND N ON PENE T RA T I NG ORBITS 111

Is functions to the 2s function. Now a determinant possesses the prop­erty that when any row is multiplied by any fa ctor and added t o anyother row t he dete rminan t remains unchanged in value. Wilson'snormalized wave functi on for t he lithium atom therefore takes t he form

(7.25)

where the A's and B's now represen t t he simple hydrogen-like wavefunctions with modified values for Z.

When determinant s involvin g hydrogen-like fun ctions with Z pu tequal to t he atomic number are used to calculate t he energy, the result

30

Hydrogen 25L Lithium2s---- - --- -- ---

2 3 4 - r_ 5 6 70 ,

FIG. 7.8.- Qu ant um-m ech anical model of the lithium atom .

is no t so good as might be expected . Much bet ter agreement is obtainedby considering the Z's as parameters and adj usting them, by variationmethods, until a minimum energy is obtained. Wilson found t he lowestenergy value with the wave functions and their parameters as follows:

A i = 1/; 18 = N 1e - ( Z-v1)r" (7 .26)

{z _ 0"1 - (Z -U2)" - (Z -u,)" - (Z -Ul)"}

B , + bA i = N 2 2 rie 2 - e 2 + be , (7.27)

where 0"1 = 0.31, 0"2 = 1.67, 0"3 = 0, and N 1 and N 2 are normalizingfa ctors. The probability-density-distribution curve obt ained by plotting47rr21/;2 against the electron-nuclear dist ance r is show n by t he. heavycurv e in F ig. 7.8 . The dot ted curve for t he hydrogen 2s state doesno t righ tly belon g here but is shown as a comparison with t he 2s state oflithium. The latter curve is obt ained by plot ting 41l"r21/;~••

The pulling in of t he inner loop of t he Li 2s curve over t he innerloop of the hydrogen 2s curve is du e to t he lack of scree ning by t he I selectrons of the core and is to be compared wit h the deeper penetration

112 INTRODUCTION TO ATOMIC SPECTRA [CHAP. VII

and speeding up of the electron in the inner part of the classical orbit.The 2s electron, since it is most of the time well outside the core, isscreened from the nucleus by the two core electrons. This averagescreening is well represented by the screening constant 0"2 = 1.67. Thescreening of each Is electron from the nucleus by the other Is electronshould lie between 0 and 1. The value 0"1 = 0.31 is in good agreementwith this. When the 2s electron is inside the core (i. e., the smallerloop), the screening by the outer Is electrons is practically negligible.The value 0"3 = 0 is in good agreement with this. This same analogybetween the quantum-mechanical model and the orbital model shouldextend to all elements.

The accuracy with which Eq. (7.25) represents the normal states oflithium and singly ionized beryllium (see Figs. 5.2 and 6.5) is shown bythe following values:

Li I, 28Bc II, 28

Spe ctroscopic

T = 43484 cm- 1

T = 146880

Calculated

43089 cm-1

145984

Calculations of the quantum defect for the alkali metals from apurely theoretical standpoint have also been made by Hartree.! Har­tree, employing his self-consistent field theory (see Sec. 7.1), has determinedthe quantum defects and energy levels for a number of states in severalof the alkalis. After determining a probability-density-distributioncurve for the core of the atom, as in Fig. 7.1, the energy of the valenceelectron moving in this field can be calculated. In rubidium, for example,he obtained the following values of p,:

TABLE7.3.-0BSERVED AND CALCULATED VALUES OFTHE QUANTUM DEFECTJl. FOR RUBIDIUM

(Af ter Hartr ee)

Electron term Jl. (obs. ) Jl. (calc. ) Dill.

58 52S 3 .195 3.008 0 .18768 62S 3.153 2.987 0 .16678 72S 3 .146 2.983 0.1635p 52P 2 .71 2.54 0.176p 62P 2 .68 2 .51 0 .174d 42D 0.233 0 .028 0 .205

Since the quantum defect is a measure of the penetration of thevalence electron into the atom core, the orbital model as well as thequantum-mechanical model would be expected to show that the pen~

1 HARTREE, D. R., Proc. Camb. Phil. Soc., 24, 89,111,1928.

SEC. 7.5] PENETRATING A ND NONPENETRATING ORBITS 113

t ration is not greatly different for all states of the same series. Thefirst seven of a series of d states, I = 2, for example, are shown in Fig.7.9. The orbits given above are drawn according to model a; andbeneath them are drawn the hydrogen probability-density-distributioncurves of the same quantum numbers.

On the orbital model the perihelion distances are very nearly thesame. On the quantum-mechanical model the lengths of the first

cod

I~~__-II

I

II

I

tI j 3d 4d 5d 6d ld

1o 10 20 30 40 50 60 70 80 90 .01

F IG. 7.9.- Seri es of d orbits illustra ti ng nearly eq ual penetration for all orbits with t he samel on eithe r t he classical or the quantu m-mechanical model.

loop (indicated by t he first nodal points), with the exception of 3d, arenearly the same. Figure 7.9 brings out better than any other, perhaps,the close analogies t hat may be drawn between the orbits of the earlyquantum theory and t he probability-density-distribution curves of t henewer quantum mechanics. This is one of the reasons why the termsorbit, penetrating orbit, nonpenetrating orbit, etc., are st ill used in discussingquantum-mechanical processes.

Problems

1. Assuming t hat t he 8 and p or bit s in potassium penetrate only the shell of eightM electrons, compute t he value of PM from Eq. (7.21). Compare these values of P with.the density-distribution curve in Fig. 7.2. If the values of t he quantum defect are no cknown from Prob. 2, Chap. V, t hey are readily calcula ted from the t erm valuesdirectly.

2. Compute, on the class ica l t heory, mod el a, the maximum elect ron-nucleardistances attaine d by t he valen ce elect ron of sodium in t he first 10 8 orbits. Comparethese values with t he corresponding 8 orbits of hydrogen by plotting a graph. PlotTmax aga inst n for hydrogen and n eff for sodium.