INTRODUCTION TO ATOMIC SPECTRA BY HARVEY ELLIOTT WHITE, PH.D, A ssi sta nt P rofessor of P hy sics, at th e Universit y of Califo rnia McGRAW-HILL BOOK COMPANY, INc. NEW Y ORK AN D LO NDO N 1934 CHAPTER VII PENETRATING AND NONPENETRATING ORBITS IN THE ALKALI METALS We shall now turn our attention to t he formulati on of an atomic Inodel which treats the interacti on of a single valence electron with t he nu cleus when it is screened by an inter venin g core of elect rons, i .e., by inner complet ed subshells of electrons. This must be don e if K - -HG1rt r ee M 0.5 - - - - - PauIJng - ' - ' -Thomas LO 1.5 zo --- -- 25 30a N FlO . 7.1. -Proba bilit y- d en si t y- di st r ibu t ion cu rves for t he ru bidium-a t om co re of 31) electrons. ( A f ter H a rtree.) we are t o calculate from theoreti cal considerations t he energy levels of t he alka li metals. On t he classical picture of t he atom t he nu cleus is surrounded by va rious shells and subshells of electrons in orbits resembling t he various possible states of t he Bohr-Sommerfeld hydrogen atom. The qu an tum-me chani cal picture, on t he other hand, appears in the form of a probabili t y-d ensit y distributi on for t he same electrons. 7.1. The Quantum-mechanical Model of the Alka li Metals.- By mean s of successive approximations to t he so-called self -consistent field, 100 SEC. PENETRATING AND NONPENETRATING ORBITS 7.1] 101 Hartree ? has calculated the ra dial den sit y distributions of each core elect ron for t he different alka li met als. Although a t reatment of the methods by whi ch t hese calc ulations are made is out of place here, the calculations are not difficult but are long and tedious . It will suffice l' Lithium L 25 2 3 4 5 6 701 6 70 1 Sodium M ~ 35 ~ C'l '- 2 ~ <;;t 3 5 4 a• Potassium N 4s 2 3 4 5 6 l ccl Rub idium 0 55 o 2 3 4 5 --r - 6 F IG. 7.2.-Probabilit y- d ensi t y di stribu ti on curves for t he n eu tral alkali a to ms, lithium. sod iu m , potassium, and r u bid iu m . I n each case t he core is shown by one cu rve and t he valence electron by another. to say, however, t hat t he resultant electric field obtained for any atom is su ch that t he solutions of Schr odinger's wave equation for all of t he core electrons in t his field give a distributi on of electrons whi ch reproduces t he field. 1 H ARTREE, D . R. , Proc. Cumbo Phil. S oc., 24, 89, 111, 1928. 102 I N TRODUCTION TO A TOM IC S P EC T RA [ C HA P . VII As a n example of H artree's results, probability-density-distribution curves for the rubidium core are shown in Fig. 7.1. The radial curve for ea ch subshell is sho wn in t he lower half of the figure. The he avy curve represen t s t he sum of all of t he 36 core elect rons . The four loops, or humps, in t he la t t er are take n to represen t t he K , L, M , and N shells, even t ho ugh eve ry electron contributes something to eac h shell. The dot t ed curves sho wn above for com parison purposes have been calcu lated from hydrogen wave fun ctions by Pauling I by using approximation methods and by Thomas? from stat ist ical considerat ions. In com pa ring t hes e curves with t he corresponding hydrogen-like functions shown in Fig. 4.6, it is obse rved t hat each den sity curve, due to a very large nuclear charge, has been drawn in t oward t he nucleus by a considerable amount. Prob abilit y-density-distribution curves for lithium, sodium, potassium, and rubidium are given t oget her in Fig. 7.2. In ea ch case the core charge is sho wn by a shade d curve and the valence elect r on in it s normal state by anothe r curve. Radially it should be not ed that the major she lls in eac h atom lie well insid e t he first Bohr circ ular orbit of hydrogen r = aI, and t hat t he valence electron lies well inside t he corres ponding hydrogen state. In hydrogen , for exa m ple, t he den sity distribution D for a 58 electron is apprec ia bly large as far out as 50al, whereas in rubidium t he nodes and loop s have been pulled in to about one-tenth of t his. The dot s on t he r axis represen t t he extremities of t he classical orbits based on mod el a (see Fi g. 4.8) . In t hese 8 states t he kin eti c energy of t he electron at t he end of t he orbit is zero, and t he t otal ene rgy is all pot ent ial , e2 TV = P. E. = - - = - eV , (7.1) r where r is in centimeters . 11 = Expressin g 11 in volts a nd r in Ang stroms, 300e = 300 X 4.77 X 10- 10 = 14.31. r r X 10-8 r X 10 8 (7.2) Expressin g r in units of a l (al = 0.528 A), 27.1 r m RX =V 27.1 = ionization potential in vo lts' (7.3) From t he ioniz ation pot en ti als of t he alkali met als given in Table 6.1 t he following va lues of t he orbital ext re mit ies are obtained : Li rma • = 5.0al Na 5.3a l K 6.3al Rb 6.5al Cs 7.0al L., Proc. Roy. S oc., A, 114, 181, 1927. L. H ., Proc. Camb. Phil. Soc., 23, 542,1927; see also GA UNT, Proc. Camb. Phil . Soc., 24, 328, 1928; a nd FERMI, Zeits. f . Ph ys., 48, 73, 1928. 1 PAULIN G, 2 THOMAS, SEC. 7.2] P ENETRATING AND NONPENETRATING ORBITS 103 The above derivati on is only a close approximation, for we have assumed a rigid core of un it charge. 7.2. Penetrating and Nonpenetrating Orbits.- T he quantummechanical model of the sodium atom is shown in Fig. 7.3. In addition to t he shaded curve for t he 10 core electrons the three lowest possib le states for t he one and only va lence electron are also shown . The corre - SODIUM Penetra ting j 38 j 3p Non-Penetrating l. 3d 3d FIG . 7.3.- C ompari soll of the quantu m-mechan ical with t he classical m od el of the neu tral sodium atom . Three of t he low est possible states for t he sin gle valen ce elect ro n are 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 orbits with the corresponding hydrogen orbits shows t hat, due t o penetration into the core, 38 and 3p are greatly reduced in size radially. The 3d orbit, on t he other hand, remains well outside the main part of the core and is hydrogen-like. Corresponding to t he penetration of the 38 and 3p orbits t he probability-density-distribution cur ves (above) have small loops close to t he nu cleus. 104 I N T RODUCTIO N TO A TOM IC SPEC T RA [CHAP. VII Consider now t he classica l picture of a valence electron describing anyone of various t ypes of or bits about t he spherically sym met rical sodium-ato m core. In Fig. 7.4 six differen t orbits ar e shown representing valence-electron states wit h t he same total qu an tum number n (t he sa me maj or axis) bu t slightly differen t azim uthal qua ntum nu mber l (different minor axes). With a core-de nsit y distributi on of charge alw ays finite bu t a pproaching zero as r - ? 00, all valence-electro n orbits will be mo re or less penet rat ing. In Fig. 7.4a t he electro n moves in a path well outside t he major part of t he core. Sin ce t he field in t his outside region does not deviate NON-PENETRATING ORBIT e -------~· f PENETRATING ORBIT FIG . 7.4 .-Sho wing the chang e in electron orbits with in cr ea sing p enetration. greatly from a Coulom b field , t he orb it will be a Ke pler ellipse pr ecessing slowly (due to small deviati ons from a Coulom b field ) about t he atom center. In t he re ma ining figures in creased penetration is sho wn accompanied by an in crease in t he precession at eac h t urn of t he orbit . As t he electron goes fr om aphelion (rma» to per ihelion (rmi") ; it leaves behind it more and more of t he core charge. With t he steady in crease in for ce field t he electron is dr aw n fr om its original path into a m ore and more eccentric path, with t he result t hat at its closest a pproach to t he nu cleus t he electron has t urned t h rough somew hat mo re t han 180 deg. Up on re achin g rm " . again, t here has been an advance, i .e., a precession, of t he aphelion. The in creased for ce of attraction between nucleus and electron at penet rati on in creases t he binding energy , t he kinetic energy, and t he term values bu t decre ases t he total energy of t he atomic system [see Eq. (2.15)]. SEC. i .3] 105 P E N E T R A TI N G AND NONPENE TRA T ING OR B ITS 7.3. Nonpenetrating Orbits.- N onpenet ra ting orbits are defined as t hose orbits for which t he observed energies are very nearl y equa l to t hose of t he corresponding hydrogen-like orbits. Su ch orbits on eit her t he class ical or qua ntum-mec ha nical model do not appreciably penetrate t he atom core and ar e t hose states for which t he azimuthal quantum number l is more nearly equa l t o the total qua nt um number n. The f orbits in all of t he alkali metals are good examples of non penetrating orbits. T erm va lues for t he 4f, 5f, and 6f states give n in T abl e 7.1 will illustrate t his. TABLE i .I.-TERM VALUES OF NONPENETRATIN G f ORBITS IN THE ALKALI M ETALS CO ~IPARED WITH T HOSE OF HYDROGEN Electron de signation . ... . . .. . . T erm d esign ation . . . . . .. .. . . . . 4f 4 21" 5f 5 21" 6f 6 2/<' H ydrogen Li Na K Rb Cs 6854 . 85 6855 . 5 6858 .6 68i9 .2 6893 .1 6935 .2 438 i . 11 4381.2 4388. 6 4404 .8 44 13 .i 4435.2 3046 . 60 3031 .0 303 9 .7 305 i . 6 3063 . 9 30i 6 . 9 With t he exception of caesium t he observed va lues are hydrogen-like to 1 per cent or bet ter. T erm values plot ted as t hey are in F ig. 5.2 sho w, in genera l, that p, d, and f orbits in lithium, d and f orbits in sodium and potassium, and f or bits in ~rubidium and caesium ar e nearly hydrogen-like. In t he enha nced spectra of t he ionized alka line earths t he term va lues are to be com pa red wit h t hose of ionized heliu m, or t hey are to be div ided by 4 (i .e., by Z 2) and com pared wit h hydr ogen as in F ig. 6.5. In t hese energy level diagrams it is observed t hat p , d, FIG. 7.5.- S ch em a t ic r epresen t a ti on t he pola rization of t h e atom co re by and f orbit s in Be II, d and f orbits in of an ext er n al electron. Mg II, and f orbits in Ca II, Sr II, and Ba II are hydrogen-like and t herefore corres pond to nonpenetrating orbits. Althou gh nearly hydr ogen-like, t he te rm values of nonpenet rating orbits (see T able 7.1), in genera l, are greater t ha n t hose of hydrogen. Born and H eisenberg- attributed t hese small differences to a polariz ati on of t he core by t he va lence electron (see Fig. 7.5). In t he field of t he 1 BORN. M ., and W. HEISENBERG, Z eits. f. Phys., 23, 388, 1924. 106 I NTROD UCTIO N TO A TOM IC S PEC T RA [CHAP . VII valence electron t he atom core is pu shed away and t he nucleus is pulled t oward t he electron by virtue of t he repulsion and attraction of like and unlike charges, respe cti vely . The effect of t his pol arizati on is to decrease t he total energy of t he system. On an energy di agram t his means a lowering of t he level, i.e., an in crease in t he t erm value. Theoretical values of t he polarizati on ene rgy calculated for t he alka li met als with t he aid of t he quantum mechanics a re found t o account for t he m ajor part of t hese ver y small deviati ons from hydrogen-like te rms ." 7.4. Penetrating Orbits on the Cla ssical Mode1. - Alt hough no sh ar p lin e of d em arkati on can be drawn between pen etrating and nonpenetrating orbits, t he former may be defined as t hose orbits for whi ch the term values a re a pprecia bly differen t from t hose of hydrogen. Referring t o Fig. 5.2, t he s orbit s of Li, the s a nd p orbit s of Na a nd K, and the s, p, and d orbit s of Rb and Cs come under this rough cla ssification . 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 the classical t he ory, one is led by ne cessity to sim plify some what the atom-core mod el given in Fi g. 7.4 . A suit able idealized m odel was first put forward by Schrodinger " in whi ch t he core elecPenetrcrln q Orbit t rons were t ho ught of as bein g distribut ed uniformly ove r t he sur fac e of one or more conce ntric sphe res. This same mo del has been t reated by W en tzel ," Somm erfeld, " Van Urk," P auling and Goudsmi t ," and others. Sin ce the class ica l t reatme nt of pen etrating orbits is so closely a na logous t o t he quantummechanical t reatme nt of t he same FIG. i .G.- Va lence elec t ron penetrut- orbits, to be take n up in t he next illr~ an ideal core where the co re electrons ale distr ibuted uniformly ov er th e sur- sectio n, Schrodinger's sim plified model face of a sphe re . will be consid er ed her e in some detail. Cons ider t he ve ry sim plest mod el in whi ch the core electrons are distribut ed uniformly ove r t he 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 t he charge shell and Zoe t he effective nu clear cha rge outsid e the she ll. Usu ally Zo is 1 for the alkali metals, 2 for t he alkaline earths, etc. The potential energy 1 For t he qu antum-mechani cs formula giving t he polariza tion ene rgy see L. PAULING and S. GOUDSMIT, "Struc t ure of Lin e Spectra ," p . 45, 1930 ; a nd 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. , Z eits. f . Ph y.~ . , 19,53, 1923. • SOMMERFELD, A., "Atomhau," 5t h German ed. , p. 422, 1931. 6 VAN URK, A. T. , Z eits. f. Ph ys., 13,268, 1923. 5 P AULING, L. , a nd S. GOUDSMIT, "Struct ure 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 Vi = -r + -Zipe 2 Z oe 2 - 0 p (7.5) The total energy for an elliptic orbit, in polar coordinates rand 'P, is given by Eq. (3.26) as W +V = T = 2m ~(p2 T + p~) r2 + V , (7.6) 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 charged shell the motion will be that of an electron in a Coulomb field of charge Zoe, 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 momentum p<p, which must be constant at all times throughout the motion, L Z p<pd'P = kh, 1C P<p = k 2:· (7.7) That part of the electron path which is outside the shell is a segment of an ellipse determined by the azimuthal quantum number k and the radial quantum number r o , whereas the path inside the shell is a segment of an ellipsedetermined by the same azimuthal k (p<p = constant) but a different radial quantum number, rio Substituting successively the potential 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 (7.10) n i = k + rio Since the electron does not complete either of the two ellipses in one cycle, the integrals of Eqs. (7.8) and (7.9) are not to be evaluated over a complete cycle as indicated but over only that part of the ellipse actually traversed. The radial quantum number r for the actual path traversed is therefore given by the sum of the two integrals 108 I NTROD UCTION TO ATOM IC S PECT RA i outs ide Rodr + r J iDBide [CHAP . R idr = rho VII (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 t he first Bohr circula r orbit. The total energy inside is W = -T = V = _ Z;e2 + (Zi - Zo)e2. 2 2aln; p Sin ce t he energy inside and outside must be t he same, Z~e2 Z;e2 (Zi - Zo)e2 - -= - -+ . 2aln~ 2aln; p (7.13) (7.14) Consider now t he special case shown in Fi g. 7.7, in which t he t wo par ti al K epler ellipses are almost complete.' If t he outside orb it were a complete K epler ellipse t he elect ron Penetrating Orbit would never penetrate t he shell, whereas if t he inn er orbit were complete t he elect ron would always remain inside. As t he outer ellipse is mad e less and less penet rating, t he t wo ellipses become more and more complete and t he integrals of E q . (7.11) approach t hose of Eqs. (7.8) and (7.9). Expressing t his in terms of t he rad ial qua ntum nu mbers, FIG . 7.7.- S pecial case wher e in ne r and outer ellipses are a lmost co m p let e. Co re charge distr ibuted unifor ml y over t he su r fa ce of a sp here. r = r: + rio (7.15) In a similar fashion t he perihelion dist an ce of t he outer orbit ao(1 - Eo) approaches t he aphelion of t he inner ellipse ai(1 + Ei) , both approac hing at t he same t ime t he radius of t he 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 semima jor axis and eccentricity of t he lSee VAN URK, A. T ., Zeits.f. Phys., 13, 268.1923 . ..- SEC. 7.4) 109 PENETRATING AND NONPENETRATING ORBITS inner ellipse ar e by Eqs. (3.32) and (3.22), respe cti vely, (7.18) Substituting a i in Eq. (7.16) and squa ring, t here results i )2= Z - 1 ' n;al E~ = ( -P k 2 (7.19) 1 - - , n~ from which (7.20) Replacing k2 and k by the corresponding quantum-mechanical values l(l + 1) and 1 + t, respecti vely [see Eq. (4.52)), and substituting in Eq. (7.17), t he quantum defect becomes (7.21) This expresses the experimental resul t, well known before the quantum mechanics, t hat for a given atom t he quantum defect J.L is a fun ction of the azimuthal quantum number and is independent of the to tal quantum number (see Table 7.2). T ABL E 7 .2.-ExPER IMENTA L V AL UE S OF T HE QUANTUM DEF E CT n - n o = p. FO R P E N E T R ATI N G ORBIT S I N L I T HIUM AND SODIUM El ement T erm El ect ron n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 Li 2S s, l = 0 0 .4 1 0 .40 0 .35 0 .35 0 .35 0 .35 Na 2S 2p s, l = 0 p, l = 1 .... 1.37 0 .88 1.36 0 .87 1.35 0 .86 1.35 0 .86 1.35 0 .86 ~ .. . . In order to calculate qu an tum defect s from Eq. (7.21), it is necessary to determine values for t he electron-nuclear distance p . Successful attempts t o calculate suitable values of p from quantum-mechanical densi ty-distributi on curves similar to t hose shown in Figs. 7.1 and 7.2 ha ve been made by P auling. I In t he case of Na, P auling obt ain s t he average value of t he electron-nuclear distance PK for the two K electrons as 0.132al, and for the eight L electrons PL = 0.77al. Assuming that t he va lence elect ron penetrates only t he out ermost shell of eight electrons, t he effecti ve nucl ear cha rge Z ie will be ge. Sub stituting Z i and PL 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 values given in T able 7.2. F or lithium with Z i = 3 and PK = O.53aI t he value 1 P AULIN G, L., Proc. Roy. Soc., A, 114, 181 , 1927. 110 I N1'R OD UCT IO N TO ATOM IC S PECTRA [C HAP. VII IJ. = 0.39 is computed for t he s orbits, also in good agreeme nt with exp eriment . 7.5. Quantum-mechanical Model for Penetrating Orbits.-The solut ions of t he Schr odin ger wave equation for hyd rogen-like atoms and for t he normal state of t he helium-like and lithium-like atoms h ave been carried out to a high degree of accuracy. The calculations for lithiumlike atoms are of par ti cular interest in t hat t hey appear to give t he qua ntum-mec ha nical ana logue of t he inner and outer orbit segments, so well kn own 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 the lithium atom in its normal state 28. P erh aps t he most recent and accurate calculations are t hose made by Wilson. ' In pr evious determin ations single-elect ron wave fun cti ons of t he hydrogen type were used inst ead of a wave function for t he at om as a whole. The rather brief treatment t o be given here is t hat of Wilson based up on a principle previously introduced by Slater. > Slat er has shown t hat a wave fun ction representing an atom cont aining many electrons may be pr operly constructed by expressing it in t he form of a det ermin an t t he elements of which are built up out of t he Schrodinger wave equation. In t he case of lithium, for example, t he wave fun ction is writ ten Yt 1 -'!: I -'!:2-'!:3 = V(3 A I A 2 A 3 , (7.22) B I B2 B3 where t he A's and A's are I s hydrogen-like wave fun ctions for t he two K elect rons and t he B 's are 2s wave fu ncti ons for t he L elect ron. From Eq. (4.55) t he solutions of t he wave equation for I s and 2s electrons are (7.23) (7.24) where N I = - (Z3/7l')1, N 2 = (Z3/87l') 1, Z is t he atomic number, and aI, the radius of the first Bohr circular orbit, has been chosen as unit of length. Now the 2s func tion [Eq. (7.24)] for t he valence electron is orthogonal to t he I s fun ctions [Eq . (7.23)] for t he K elect rons. This would no longer be so, if different valu es for Z should be subst ituted in Eq s. (7.23) and (7.24) . F or t his case Slater" has shown t hat t he t wo fun ctions may be or thogonalized by adding a fraction of one of the I \VILSO N, E . B., J our. Chern. P hys., 1, 210, 1933. pap er. 2 S LATE R, 3 S L AT ER, J . C., P hys. Rev., 34, 1293, 1928. J . C., P hys. tu«, 42, 33, 1932. For othe r referenc es see this SEC. 7.5] PENETRATING A ND N ON P ENE T RA T I NG ORBITS 111 Is functions to the 2s function. Now a determinant possesses t he property that when any row is multiplied by any fa ctor and added t o any other row t he dete rminan t remains unch anged in value. Wil son's normalized wave functi on for t he lithium atom therefore takes t he form (7.25) where the A's and B's no w represen t t he sim ple hydrogen-like wave functions with modified v alues for Z. When determinant s involvin g hydrogen-like fun cti ons with Z pu t equal to t he at omic number are used to calculat e t he en ergy, t he result 30 Hydrogen 25 L Lithium2s ---- - --- -- --2 3 4 - r_ 5 6 70 , FIG. 7.8.- Qu a n t um-m ech anical model of t he lithium atom . is no t so good as migh t be expected . Much bet t er agreement is obtained by considering t he Z's as paramet er s and adj usting t he m, by variation me thods, until a minimum energy is obtaine d . Wil son found t he lowest energy value with t he wave functions and their parameters as follows: A i = 1/;18 = N 1e - B, + bA i = N2 { z_ 2 0"1 rie - (Z -U2)" 2 - (7.26) ( Z -v 1 )r" - (Z -u, )" e 2 + be - (Z - Ul)"} , (7.27) wher e 0"1 = 0.31, 0"2 = 1.67, 0"3 = 0, and N 1 and N 2 are no rmalizing fa ctors. The prob ability-density-distribution curve obt ained by plotting 47rr 21/;2 against t he electron-nucl ea r dist ance r is show n by t he. heavy curv e in F ig. 7.8 . The dot t ed curve for t he hydrogen 2s state does no t righ tly belon g her e but is sho wn as a com parison with t he 2s state of lithium. The latter curve is ob t ained by plot ting 41l"r21/;~ •• The pulling in of t he inner loop of t he Li 2s curve over t he inner loop of t he hydrogen 2s curve is du e to t he lack of scree ning by t he I s electrons of t he core and is to be compared wit h t he deeper pen etration 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, is screened from the nucleus by the two core electrons. This average screening is well represented by the screening constant 0"2 = 1.67. The screening of each Is electron from t he nucleus by the other Is electron should lie between 0 and 1. The value 0"1 = 0.31 is in good agreement with this. When the 2s electron is inside the core (i. e., the smaller loop), the screening by the outer Is electrons is practically negligible. The value 0"3 = 0 is in good agreement with this. This same analogy between the quantum-mechanical model and the orbital model should extend to all elements. The accuracy with which Eq. (7.25) represents the normal states of lithium and singly ionized beryllium (see Figs. 5.2 and 6.5) is shown by the following values: Li I, 28 Bc II, 28 Spe ctroscopic Calculated T = 43484 cm - 1 T = 146880 43089 cm- 1 145984 Calculations of the quantum defect for the alkali metals from a purely theoretical standpoint have also been made by Hartree.! Hartree, employing his self-consistent field theory (see Sec. 7.1), has determined the quantum defects and energy levels for a number of states in several of the alkalis. After determining a probability-density-distribution curve for the core of the atom, as in Fig. 7.1, the energy of the valence electron moving in this field can be calculated. In rubidium, for example, he obtained the following values of p,: TABLE 7.3.-0BSERVED AND CALCULATED VALUES OF THE QUANTUM DEFECT Jl. FOR RUBIDIUM (Af ter Hartr ee) Electron t erm 58 68 78 5p 6p 4d 5 2S 6 2S 7 2S 5 2P 6 2P 4 2D Jl. (obs. ) 3 .195 3.153 3 .146 2 .71 2 .68 0.233 Jl. (calc. ) 3.008 2.987 2.983 2.54 2 .51 0 .028 Dill. 0 .187 0 .166 0.163 0.17 0 .17 0 .205 Since the quantum defect is a measure of the penetration of the valence electron into the atom core, the orbital model as well as the quantum-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. The first 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; and beneath them are drawn the hydrogen probability-density-distribution curves of the same quantum numbers. On the orbital model the perihelion distances are very nearly the same. On the quantum-mechanical model the lengths of the first cod I~~_ _- I I I I I I I t j 3d 4d 5d 6d ld 1 o 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 sa me l on eithe r t he cla ssical or t he q uantu m-mechanic al model. loop (indicated by t he first nodal points), with the exception of 3d, are nearly the same. Figure 7.9 brings ou t be tter than any other, perhaps, the close analogies t hat may be drawn between the orbits of the early quantum theory and t he probability-density-distribution curves of t he newer quantum mechanics. This is one of the reasons why the terms orbit, penetrating orbit, nonpenetrating orbit, etc., are st ill used in discussing quantum-mechanical processes. Problems 1. Assu m ing t hat t he 8 and p or bit s in potassium penetrate only the shell of eight M electrons, com pute t he value of PM fr om Eq. (7.21). Compare these values of P with. the d ensity-distribution curve in Fig. 7.2. If the values of t he qu antum defect are no c known from Prob. 2, Chap. V, t hey are readily calcula ted from the t erm values dir ectly. 2. Compute, on the class ica l t heory, m od el a, the m aximum elect ron-n uclea r distances attaine d by t he valen ce elect ron of sodium in t he first 10 8 orbits. Compare these values with t he corres pondin g 8 orbits of hydrogen by plotting a graph. Plot Tmax aga inst n for hydrogen a nd n eff for sodiu m.

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