2000 by BlackLight Power, Inc. All rights reserved. DERIVATION OF ELECTRON SCATTERING BY HELIUM

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1 000 by BlackLight Pwer, Inc. All rights reserved. DERIVATION OF ELECTRON SCATTERING BY HELIUM 193 CLASSICAL SCATTERING OF ELECTROMAGNETIC RADIATION Light is an electrmagnetic disturbance which is prpagated by vectr wave equatins which are readily derived frm Maxwell's equatins. The Helmhltz wave equatin results frm Maxwell's equatins. The Helmhltz equatin is linear; thus, superpsitin f slutins is allwed. Huygens' principle is that a pint surce f light will give rise t a spherical wave emanating equally in all directins. Superpsitin f this particular slutin f the Helmhltz equatin permits the cnstructin f a general slutin. An arbitrary wave shape may be cnsidered as a cllectin f pint surces whse strength is given by the amplitude f the wave at that pint. The field, at any pint in space, is simply a sum f spherical waves. Applying Huygens' principle t a disturbance acrss a plane aperture gives the amplitude f the far field as the Furier Transfrm f the aperture distributin, i.e., apart frm cnstant factrs, (x,y) A(, )exp ik ( x + y) f d d (8.1) Here A(, ) describes the amplitude and phase distributin acrss the aperture and (x,y) describes the far field [1] where f is the fcal length. Delta Functin In many diffractin and interference prblems, it prves cnvenient t make use f the Dirac delta functin. This functin is defined by the fllwing prperty: let f ( ) be any functin (satisfying sme very weak cnvergence cnditins which need nt cncern us here) and let ( ' ) be a delta functin centered at the pint ' ; then b f ( ) ( ')d f ( ') (a < '< b); 0 therwise (8.) We nte, therefre, that a ( ' )d 1 (8.3) the Furier transfrm f the delta functin is given by (x) ( ')exp ikx f d (8.4) which by definitin f the delta functin becmes (x) exp ikx ' f (8.5)

2 by BlackLight Pwer, Inc. All rights reserved. The amplitude is cnstant and the phase functin ikx ' f the rigin. depends n The Array Therem A large number f interference prblems invlve the mixing f similar diffractin patterns. That is, they arise in the study f the cmbined diffractin patterns f an array f similar diffracting apertures. This entire class f interference effects can be described by a single equatin, the array therem. This unifying therem is easily develped as fllws: Let ( ) represent the amplitude and phase distributin acrss ne aperture centered in the diffractin plane, and let the ttal diffracting aperture cnsist f a cllectin f these elemental apertures at different lcatins n. We require first a methd f representing such an array. The apprpriate representatin is btained readily by means f the delta functin. Thus, if an elemental aperture is psitined such that its center is at the pint, the n apprpriate distributin functin is ( n ). The cmbining prperty f the delta functin allws us t represent this distributin as fllws: ( n ) ( ) ( n )d (8.6) The integral in Eq. (8.6) is termed a "cnvlutin" integral and plays an imprtant rle in Furier analysis. Thus, if we wish t represent a large number N f such apertures with different lcatins, we culd write the ttal aperture distributin Ψ( ) as a sum, i.e., N Ψ( ) ( n ) (8.7) n1 Or in terms f the delta functin we culd write, cmbining the features f Eqs. (8.6) and (8.7), N Ψ( ) ( ) ( n )d (8.8) n1 Eq. (8.8) may be put in a mre cmpact frm by intrducing the ntatin N A( ) ( n ) (8.9) n 1 thus, Eq. (8.8) becmes Ψ( ) ( )A( )d (8.10) which is physically pleasing in the sense that A( ) characterizes the array itself. That is, A( ) describes the lcatin f the apertures and ( ) describes the distributin acrss a single aperture. We are in a psitin t calculate the far field r Fraunhfer diffractin pattern assciated with the array. We have the therem that the Fraunhfer

3 000 by BlackLight Pwer, Inc. All rights reserved. pattern is the Furier transfrm f the aperture distributin. Thus, the Fraunhfer pattern Ψ (x) f the distributin Ψ( ) is given by Ψ ( x) ( )exp i x d (8.11) f substituting frm Eq. (8.10) gives [ ] Ψ i x ( x) ( )A( )d exp f d (8.1) A very imprtant therem f Furier analysis states that the Furier transfrm f a cnvlutin is the prduct f the individual Furier transfrms [1]. Thus, Eq. (8.1) may be written as Ψ (x) (x) A (x) (8.13) where (x) and A (x) are the Furier transfrms f ( ) and A( ). Eq. (8.13) is the array therem and states that the diffractin pattern f an array f similar apertures is given by the prduct f the elemental pattern (x) and the pattern that wuld be btained by a similar array f pint surces, A (x). Thus, the separatin that first arse in Eq. (8.10) is retained. T analyze the cmplicated patterns that arise in interference prblems f this srt, ne may analyze separately the effects f the array and the effects f the individual apertures. Applicatins f the Array Therem Tw-Beam Interference We use Eq. (8.13) t describe the simplest f interference experiments, Yung's duble-slit experiment in ne dimensin. The individual aperture will be described by Ψ( ) (C < a; 0 > a) rec( a) (8.14) Here C is a cnstant representing the amplitude transmissin f the apertures. This is essentially a ne-dimensinal prblem and the diffractin integral may be written as ( ) ( )exp ik x Ψ x a d C exp ik x f d (8.15) f The integral in Eq. (8.15) is readily evaluated t give kax Ψ (x) Cf sin ikax exp +ikax ikx f exp f ac f kax f a 195 (8.16) The ntatin sinc sin is frequently used and in terms f this functin Ψ (x) may be written as

4 by BlackLight Pwer, Inc. All rights reserved. Ψ (x) ac sinc kax f (8.17) Thus the result that the elemental distributin in the Fraunhfer plane is Eq. (8.17). The array in this case is simply tw delta functins; thus, A( ) ( b) + ( + b) (8.18) The array pattern is, therefre, A (x) [ ( b) + ( + b) ] exp i x f d (8.19) Eq. (8.19) is readily evaluated by using the cmbining prperty f the delta functin, thus, A (x) exp ibx ibx f + exp bx f cs f (8.0) Finally, the diffractin pattern f the array f tw slits is Ψ (x) 4aC sinc ax bx f cs f (8.1) The intensity is I(x) 16a C sinc ax bx f cs (8.) f Frm Eq. (8.), it is clear that the resulting pattern has the appearance f csine fringes f perid f / b with an envelpe sinc ( ax / f ). The distributin pattern bserved with diffracting electrns is equivalent t that fr diffracting light. Nte that Eq. (8.15) represents a plane wave. In the case f the Davisn-Germer experiment, the intensity is given by Eq. (8.13) as the prduct f the elemental pattern crrespnding t a plane wave f wavelength h / p and the array pattern f the nickel crystal. CLASSICAL WAVE THEORY OF ELECTRON SCATTERING The fllwing mathematical develpment f scattering is adapted frm Bnham [] with the exceptin that the Mills mdel is a Furier ptics derivatin fr an exact elemental pattern, a plane wave, and an exact array pattern, an rbitsphere. In cntrast, Bnham derives similar scattering equatins fr an incident plane wave via an averaged prbability density functin descriptin f the electrn, the Brn mdel. In scattering experiments in which Fraunhfer diffractin is the mst imprtant mde fr scattering, measurements are made in mmentum r reciprcal space. The data is then transfrmed in terms f real space, where the structure f the scatterer is expressed in terms f distances frm its center f mass. There are, frtunately, well knwn mathematical techniques fr making this transfrmatin. If we are given a mdel f the scattering system, we can, in general, uniquely calculate the results t be expected in reciprcal space fr scattering frm the

5 000 by BlackLight Pwer, Inc. All rights reserved. mdel. Unfrtunately, the cnverse--deducing the nature f the scatterer uniquely by transfrming the experimental results btained in reciprcal space--is nt always pssible. But, as we will see, certain pssibilities can be eliminated because they vilate fundamental physical laws such as Special Relativity. In classical ptics, a diffractin pattern results whenever light is scattered by a slit system whse dimensins are small cmpared t the wavelength f light. In rder t develp a mathematical mdel fr diffractin scattering, let us represent the amplitude f an incident plane wave traveling frm left t right as e i(k r t ), where the abslute magnitude f the wave vectr k is k. The quantity λ is the wavelength f the incident radiatin and hk is the mmentum p. The vectr r represents the psitin in real space at which the amplitude is evaluated, and and t are the frequency and time, respectively. A plane wave traveling in the i (k r + t ) ppsite directin is e where the sign f k r changes, but nt the sign f t. That is, we may reflect a wave frm a mirrr and reverse its directin, but we cannt change the sign f the time since that wuld indicate a return t the past. The intensity f a classical wave is the square magnitude f the amplitude, and thus the intensity f a plane wave is cnstant in space and time. If a plane wave is reflected back n itself by a perfectly reflecting mirrr, then the resultant amplitude is e i(k r t ) + e i(k r + t ) e i t csk r, and the intensity is I 4cs k re i t e i t which is independent f time and given as 4cs k r which clearly exhibits maxima and minima dictated by the wavelength f the radiatin and the psitin in space at which intensity is measured. In an experiment, we measure the intensity f scattered particles, which is related t plane waves in a simple fashin. T see this, cnsider a cllimated plane-wave surce, whse width is small cmpared t the scattering angle regin where the scattering is t be investigated, incident upn a diffractin grating. If we integrate the incident intensity ver a time interval t, we btain a number prprtinal t the energy cntent f the incident wave. We may safely assume in mst cases that the scattering pwer f the diffractin image des nt change with time, s that a cnstant fractin f the incident radiatin and hence cnstant energy will be transferred int the scattered wave. We further assume that the effect f the diffractin grating n the incident radiatin ccurs nly in a regin very clse t the grating in cmparisn t its distance frm the detectin pint. Fr elastic scattering (n energy transfer t the grating), nce the scattered prtin f the wave has left the field f influence f the scatterer, all parts f the scattered amplitude at the same radial distance frm the scatterer must travel at the velcity f the incident wave. Fr simplicity, we neglect resnance effects, which can 197

6 by BlackLight Pwer, Inc. All rights reserved. intrduce significant time delays in the scattering prcess even if the waves are scattered elastically. The effects f resnance states n the scattering at high energies is usually negligible and hence will nt be discussed here. In the case f inelastic scattering, in which waves are scattered with varius velcities, we can fcus ur attentin successively n parts f the utging scattered radiatin which have velcities falling within a certain narrw band, and the fllwing argument will hld fr each such velcity segment. The result f the integratin f a cnstantvelcity segment f the scattered intensity ver the vlume element, R+ R r dr sin d dφ, (8.3) R 0 0 is prprtinal t the energy cntent in that prtin f the scattered wave, and the result must be independent f R. This restrictin, which is a direct cnsequence f cnservatin f energy, then demands that the utging scattered waves have in plar crdinates the frm ( ) (8.4) Ψ sc (R,, ) eikr R f, where the term 1/R is a dilutin effect t guarantee energy n an everspreading wave. Ψ sc nly describes the scattered amplitude after the scattered wave has left the field f influence f the scatterer and is thus an asympttic frm. The functin f (, ) is called the scattered amplitude and depends n the nature f the scatterer. The classical thery tells us that the scattered intensity is prprtinal t the square magnitude f the scattered amplitude; s, the intensity will be directly prprtinal t f (, ). R Let us next cnsider the expressin fr the scattering f a plane wave by a number f disturbances in sme fixed arrangement in space. Cnsider the scatterers cmprising a nucleus and electrns; this wuld crrespnd t a plane wave scattered by an atm. We shall chse the center f mass f the scatterer as ur rigin and shall fr the mst part cnsider dilute-gas electrn scattering in the kev energy range, where the electrn wavelength λ lies in the range 0.03 Å < < 0.1 Å. The scattering experimental cnditins are such that t a high degree f apprximatin, at least 0.1% r better, we can cnsider the scattering as a single electrn scattered by a single atm. Nte als that n labratry t center-f-mass crdinate system transfrmatin is required because the rati f the electrn mass t the mass f the target will be n the rder f 10 3 r smaller. Let us cnsider an ensemble f scattering centers as shwn in

7 Figure by BlackLight Pwer, Inc. All rights reserved. 199 Figure 8.1. An ensemble f scattering centers. We may write the ttal scattered amplitude in the first apprximatin as

8 by BlackLight Pwer, Inc. All rights reserved. a sum f amplitudes, each f which is prduced by scattering frm ne f the single scattering centers. In this view, we generally neglect multiple scattering, the rescattering f prtins f the primary scattered amplitudes whenever they cme in cntact with ther centers, except in the case f elastic scattering in the heavier atms. Clearly a whle hierarchy f multiple-scattering prcesses may result. The incident wave may experience a primary scattering frm ne center, a prtin f the scattered amplitude may rescatter frm a secnd center, and part f this amplitude may in turn be scattered by a third center (which can even be the first center), and s n. An incident plane wave will bviusly travel a distance alng the incident directin befre scattering frm a particular center, depending n the instantaneus lcatin f that center. T keep prper accunt f the exact amplitude r phase f the incident wave at the instant it scatters frm a particular center, we select ur rigin, as mentined previusly, t lie at the center f mass. The phase f the scattered wave depends n the ttal distance traveled frm the center f mass t the detectr. We can nw write the scattered amplitude as N exp[ ik( z l + R r l )] Ψ ttal f l 1 R r l (, ) (8.5) l where z l + R r l is the distance traveled frm a plane perpendicular t ( ) the incident directin and passing thrugh the center f mass and f l, is the scattered amplitude characteristic f the l th scattering center. It shuld be clear at this pint that the term exp [ ik R r l ] f R r l, l 1 f a plane wave in the scattered directin with the dilutin factr R r l ( ) is made up t accunt fr energy cnservatin and with allwances made thrugh f l, ( ) fr any special influence that the scatterer may have n the scattering because f the detailed structure f the scatterer. The additinal term e ikz 1 enters whenever tw r mre scattering centers are encuntered and accunts fr the fact that the instantaneus lcatin f ur scattering centers may nt cincide with planes f equal amplitude f the incident plane wave. That is, in a tw-center case, the first particle may scatter a plane wave f amplitude +1 while at the same time a secnd scatterer may encunter an amplitude f -1. The amplitudes f the incident plane wave which the varius particles encunter depend n their separatin frm each ther alng the z-axis and n the wavelength f the incident radiatin. By adding t the phase, the prjectins f the varius r l vectrs nt the incident directin, referred t the same

9 000 by BlackLight Pwer, Inc. All rights reserved. rigin, this prblem is autmatically crrected. As lng as ur cmpsite scatterer is n the rder f atmic dimensins, the magnitude f R will be enrmusly larger than either z l r r l. This allws us t expand R r l in a binmial expansin thrugh first-rder terms as R R R r l. In the denminatr, the first-rder crrectin term R can be neglected but nt in the phase. T see this, suppse that R is X 10 6 and R R r l is /. Clearly / wuld seem negligible cmpared t X 10 6, but lk what a difference the value f a sin r cs functin has if / is retained r mitted frm the sum f the tw terms. The prduct kz l may be rewritten as k i r l, where the subscript i n k dentes the fact that k i is a vectr parallel t the incident directin with magnitude k. Similarly, since R is a unit R vectr whse sense is essentially in the directin f the scattered electrn, we may write k R R r l as k s r l where k s is a wave vectr in the scattering directin. The phase f Eq. (8.5) nw cntains the term (k i k s ) r l, where k i k s must be prprtinal t the mmentum change f the incident particle n scattering, since hk i is the initial mmentum and hk s is the final mmentum f the scattered electrn. This vectr difference is labeled by the symbl s. The asympttic ttal amplitude is nw expressible as Ψ ttal eikr R N e is r l l 1 f l, 01 ( ) (8.6) Classical Wave Thery Applied t Scattering frm Atms and Mlecules. Let us first apply Eq. (8.6) t scattering frm atms. We will cnsider the theretical side f high-energy electrn scattering and x-ray scattering frm gaseus targets as well. In the x-ray case, the intensity fr an x-ray scattered by an electrn is fund experimentally t be a cnstant, usually dented by I cl, which varies inversely as the square f the mass f the scatterer where I cl is the Thmpsn x-ray scattering cnstant. This means that x-rays are virtually unscattered by the nucleus, since the rati f electrn t nuclear scattering will be greater than m 4 p 1 X 10 m e 9X10 8 ~10 6, where m p is the prtn rest mass and m e is the electrn rest mass. The ttal amplitude fr x-ray scattering by an

10 0 000 by BlackLight Pwer, Inc. All rights reserved. atm can then be written as xr Ψ ttal N I cl e i cl e ikr e is r l (8.7) l 1 where cl is a phase factr intrduced because f a pssibility that the x- ray scattered amplitude may be cmplex. The intensity can be written as xr I ttal N N I cl (N + e is. r l k ) (8.8) l k l k where r lk r l r k is an interelectrn distance. Bth expressins, Eqs. (8.7) and (8.8), crrespnd t a fixed arrangement f electrns in space. Fr electrns, the intensity f scattering by anther charged particle prceeds accrding t the Rutherfrd experimental law I I e Z s 4, where Z is the charge f the scatterer and I e is a characteristic cnstant. Nte that bth I cl and I e include the 1 dilutin factr and depend n R the incident x-ray r electrn beam flux I and n the number N f target particles per cubic centimeter in the path f the incident beam as the prduct I N. We may take f l, Z ( ) I e s exp [ i ( Z) ],where Z ( ) is again an unknwn phase shift intrduced because f the pssibility that the amplitude may be cmplex. In the x-ray case fr scattering by an atm, the intensity is independent f the phase cl, and we need nt investigate it further. In electrn scattering, this term is different fr electrns and nuclei since they cntain charges f ppsite sign and usually different magnitude. The amplitude fr this case is ed e ikr Ψ ttal I e s Ze i N ( ( Z )+is r n ) i 1 + e ( ( )+is r i ) i1 (8.9) which fr an atm simplifies further, since the nuclear psitin vectr r n is zer because the nucleus lies at the center f mass. The term ( Z) is the nuclear phase and ( 1) is the phase fr scattering by an individual electrn. The ntatin 1 signifies a unit negative charge n each electrn as ppsed t +Z n the nucleus, where Z is the atmic number. The intensity with r n 0 becmes ed I ttal I e {Z N N N + Z cs[ (Z) ( 1) s r i ] + N + e is r i j } (8.30) s 4 i 1 i j i j Nte that the last tw terms n the right in Eq. (8.30) are identical t thse in Eq. (8.8). Accrding t Huygens' principle, the functin N e is r i i1 f Eq. (8.9) represents the sum ver each spherical wave surce arising frm the

11 000 by BlackLight Pwer, Inc. All rights reserved. scattering f an incident plane wave frm each pint f the electrn functin where the wavelength f the incident plane wave is given by the de Brglie equatin h / p. The sum is replaced by the integral ver and f the single pint element aperture distributin functin. The single pint element aperture distributin functin, a(,,z), fr the scattering f an incident plane wave by the an atm is given by the cnvlutin f a plane wave functin with the electrn rbitsphere m functin. The cnvlutin is a(,,z) (z) [ (r r )]Y l, ( ) where a(,,z) is given in cylindrical crdinates, π(z), the xy-plane wave is given in Cartesian crdinates with the prpagatin directin alng the m z-axis, and the rbitsphere functin, [ (r r )]Y l (, ), is given in spherical crdinates. Using cylindrical crdinates, N e is. r i i 1 a(,, z)e i [s cs( Φ )+ wz] d d dz (8.31) 0 0 The general Furier transfrm integral is given in reference [3]. Fr an aperture distributin with circular symmetry, the Furier transfrm f the aperture array distributin functin, A(z), is [3]: N e is. r i i 1 a(,z)j (s )e iwz d dz (8.3) 0 03 A(z)e iwz dz (8.33) 0 F(s) (8.34) The same derivatin applies fr the tw-pint term N e is r i j f Eq. (8.30). The sum is replaced by the integral ver and f the single pint element autcrrelatin functin, r(,, z), f the single pint element aperture distributin functin. Fr circular symmetry [3], r(,, z) a(,,z) a(,,z) (8.35) and N N e is r i j i j i j a(,z)j (s )e iwz d dz (8.36) 0 R(z)e iwz dz (8.37) 0 And R(z) A(z) A(z) (8.38) Fr clsed shell atms in single states such as rare gases, Y(, ), the spherical harmnic angular functin f the electrn functin is a cnstant, and nly tw expressins are pssible frm all rders f averaging ver all pssible rientatins in space. Fr the x-ray case the

12 by BlackLight Pwer, Inc. All rights reserved. scattered intensities are and I xr 1 I cl [ A(z)e iwz dz] I cl F(s) (8.39) 0 I xr I cl [N + R(z)e iwz dz] (8.40) 0 while fr electrns, the scattered intensities are I ed 1 I e {Z + Zcs[ (Z) ( 1)]F(s) + F(s) } (8.41) s 4 and I ed I e s 4 {Z + Z cs[ (Z) ( 1)]F(s) + N + R(z)e iwz dz} (8.4) where the subscript 1 dentes an amplitude derivatin and an intensity derivatin. The aperture functin f the nucleus is a delta functin f magnitude Z, the nuclear charge. The Furier Transfrm is a cnstant f magnitude Z as appears in Eqs. (8.41) and (8.4). Nte that the Furier cnvlutin therem prves the equivalence f Eq. (8.39) and Eq. (8.40) and the equivalence f Eq. (8.41) and Eq. (8.4). The aperture array distributin functin, A(z), Eq. (8.33), crrespnds t the electrn radial distributin functin f Bnham, and the aperture array autcrrelatin functin R(z), Eq. (8.37), crrespnds t the electrn pair crrelatin functin f Bnham []. ELECTRON SCATTERING EQUATION FOR THE HELIUM ATOM BASED ON THE ORBITSPHERE MODEL The clsed frm slutin f all tw electrn atms is given in the Tw Electrn Atm Sectin. In the helium grund state, bth electrns rbitspheres are at a radius r a The helium atm cmprises a central nucleus f charge tw which is at the center f an infinitely thin spherical shell cmprising tw bund electrns f charge minus tw. Thus, the helium atm is neutrally charged, and the electric field f the atm is zer fr r > 0.567a. The Rutherfrd scattering equatin fr islated charged particles des nt apply. The apprpriate scattering equatin fr helium in the grund state can be derived as a Furier ptics prblem as given in the Classical Scattering f Electrmagnetic Radiatin Sectin. The aperture distributin functin, a(,,z), fr the scattering f an incident plane wave by the He atm is given by the cnvlutin f the plane wave functin with the tw electrn rbitsphere Dirac delta 0

13 000 by BlackLight Pwer, Inc. All rights reserved. 05 functin f radius 0.567a and charge/mass density f 4 (0.567a ). Fr radial units in terms f a a(,,z) (z) [ (r 0.567a )] (8.43) 4 (0.567a ) where a(,,z) is given in cylindrical crdinates, (z), the xy-plane wave is given in Cartesian crdinates with the prpagatin directin alng the z-axis, and the He atm rbitsphere functin, [ (r 0.567a )], is given in spherical crdinates. 4 (0.567a ) a(,,z) Fr circular symmetry [3], F( s) 4 (0.567a ) (0.567a ) z (r (0.567a ) z ) (8.44) 4 (0.567a ) (0.567a ) z ( (0.567a ) z )J (s )e iwz d dz (8.45) 0 z 4 F(s) (z 4 (0.567a ) 0 z )J (s z z ))e iwz dz ; z a 0 (8.46) z Substitute z z cs F(s) 4 z sin 3 J 4 z (sz sin )e iz 0w cs d (8.47) 0 Substitutin f the recurrence relatinship, J (x) J ( x) 1 J x x int Eq. (8.47), and, using the general integral f Apelblat [4] (sin ) +1 J (bsin )e ia cs d a + b 0 with a z w and b z s gives: F(s) (z w) + (z s) 1 ( ) ; x sz 0 sin (8.48) 1 b a + b 1 J +1/ (a + b ) z s (z w) + (z s) J ((z 3/ [ w) + (z s) ) 1/ z ] s (z w) + (z s) J 5/ [((z w) + (z s) ) 1/ ] (8.49) (8.50) The magnitude f the single pint element autcrrelatin functin, r(,,z), is given by the cnvlutin f the magnitude f the single pint element aperture distributin functin, a(,,z), with its self.

14 by BlackLight Pwer, Inc. All rights reserved. r(,,z) a(,,z) a(,,z) (8.51) The Furier cnvlutin therem permits Eq. (8.51) t be determined by Furier transfrmatin. r(,,z) e iw z ( 0.567a ) z e iw z dz 0 dw (8.5) 0 r(,,z) e i sin( w z) J 0.567a w 1( ) w dw + C (8.53) 0 where C is an integratin cnstant fr which R( ) equals zer at r 1.134a r(,,z) 1 4z 3 1+ z E z + 1 z K z 4z z 0 4z z + C 0 < z z ; z 0.567a (8.54) Eq. (8.54) was derived frm a similar transfrm by Bateman [5]. The electrn elastic scattering intensity is given by a cnstant times the square f the amplitude given by Eq. (8.50). 1 (z I ed w) + (z s) 1 I e z s (z w) + (z s) J ((z 3/ [ w) + (z s) ) 1/ z s ] (z w) + (z s) J 5/ [((z w) + (z s) ) 1/ ] (8.55) s 4 sin ; w 0 (units f Å 1 ) (8.56) Results The magnitude f the single pint element aperture distributin functin, a(,,z), cnvlved with the functin (z 0.567a ) is shwn graphically in Figure 8. in units f a. The functin was nrmalized t.

15 000 by BlackLight Pwer, Inc. All rights reserved. Figure. 8.. The magnitude f the single pint element aperture distributin functin, a(,,z), cnvlved with the functin (z 0.567a ) in units f a. 07 The magnitude f the single pint element autcrrelatin functin, r(,, z), cnvlved with the functin (z 1.134a ) is shwn graphically in Figure 8.3 in units f a. The functin was nrmalized t and the cnstant f was added t meet the bundary cnditin fr the cnvlutin integral. Figure The magnitude f the single pint element autcrrelatin functin, r(,, z), cnvlved with the functin (z 1.134a ) is shwn graphically in units f a. The experimental results f Brmberg [6], the extraplated experimental data f Hughes [6], the small angle data f Geiger [7] and the semiexperimental results f Lassettre [6] fr the elastic differential crss

16 by BlackLight Pwer, Inc. All rights reserved. sectin fr the elastic scattering f electrns by helium atms is shwn graphically in Figure 8.4. The elastic differential crss sectin as a functin f angle numerically calculated by Khare [6] using the first Brn apprximatin and first-rder exchange apprximatin als appear in Figure 8.4. Figure The experimental results f Brmberg [6], the extraplated experimental data f Hughes [6], the small angle data f Geiger [7] and the semiexperimental results f Lassettre [6] fr the elastic differential crss sectin fr the elastic scattering f electrns by helium atms and the elastic differential crss sectin as a functin f angle numerically calculated by Khare [6] using the first Brn apprximatin and first-rder exchange apprximatin.

17 000 by BlackLight Pwer, Inc. All rights reserved. 09

18 by BlackLight Pwer, Inc. All rights reserved. These results which are based n a quantum mechanical mdel are cmpared with experimentatin [6,7]. The clsed frm functin (Eqs. (8.55) and (8.56)) fr the elastic differential crss sectin fr the elastic scattering f electrns by helium atms is shwn graphically in Figure 8.5. The scattering amplitude functin, F(s) (Eq. (8.50), is shwn as an insert. Figure The clsed frm functin (Eqs. (8.55) and (8.56)) fr the elastic differential crss sectin fr the elastic scattering f electrns by helium atms. The scattering amplitude functin, F(s) (Eq. (8.50), is shwn as an insert.

19 000 by BlackLight Pwer, Inc. All rights reserved. 11

20 1 000 by BlackLight Pwer, Inc. All rights reserved. DISCUSSION The magnitude f the single pint element autcrrelatin functin, r(,, z), cnvlved with the functin (z 0.567a ) (Figure 8.3) and the electrn pair crrelatin functin, P(r), f Bnham [8] are similar. Accrding t Bnham [8], the electrn radial distributin functin, D(r), calculated frm prperly crrelated CI wave functins fr He is similar in shape t the P(r) functin but its maximum ccurs at a value f r almst exactly half f that fr P(r). Thus, the functin D(r) is similar t the magnitude f the single pint element aperture distributin functin, a(r,,z), (Figure 8.). D(r) and P(r) lead t a mst prbable structure fr the He atm in which the electrns and the nucleus are cllinear with the nucleus lying between the tw electrns []. This is an average picture cmpared t the Mills mdel. Hwever, it is apparent frm Figure 8.4 that the quantum mechanical calculatins fail cmpletely at predicting the experimental results at small scattering angles; whereas, Eq. (8.55) predicts the crrect scattering intensity as a functin f angle. In the far field, the slutin f the Schrödinger equatin fr the amplitude f the scattered plane wave incident n a three dimensinal static ptential field U(r) is identical t Eq. (8.5) nly if ne assumes a cntinuus distributin f scattering pints and replaces the sum ver l in Eq. (8.5) with an integral ver the scattering pwer f l f pint l replaced by the instantaneus value f the ptential at the same pint. This result is the basis f the failure f Schrödinger's interpretatin that Ψ(x) is the amplitude f the electrn in sme sense which was superseded by the Brn interpretatin that Ψ(x) represents a prbability functin f a pint electrn. The Brn interpretatin can nly be valid if the speed f the electrn is equal t infinity. (The electrn must be in all psitins weighted by the prbability density functin during the time f the scattering event). The crrect aperture functin fr the Brn interpretatin is a Dirac delta functin, (r), having a Furier transfrm f a cnstant divided by s which is equivalent t the case f the pint nucleus (Rutherfrd Equatin). The Brn interpretatin must be rejected because the electrn velcity can nt exceed c withut vilating Special Relativity. Slutins t the Schrödinger equatin invlve the set f Laguerre functins, spherical Bessel functins, and Newmann functins. Frm the infinite set f slutins t real prblems, a linear cmbinatin f functins and the amplitude and phases f these functins are sught which gives results that are cnsistent with scattering experiments. The Schrödinger equatin is a statistical mdel representing an apprximatin t the actual nature f the bund electrn. Statistical

21 000 by BlackLight Pwer, Inc. All rights reserved. mdels are gd at predicting averages as exemplified by the reasnable agreement between the calculated and experimental scattering results at large angles. Hwever, in the limit f zer scattering angle, the results calculated via the Schrödinger equatin are nt in agreement with experimentatin. In the limit, the "blurred" representatin can nt be averaged, and nly the exact descriptin f the electrn will yield scattering predictins which are cnsistent with the experimental results. Als, a cntradictin arises in the quantum mechanical scattering calculatin. Fr hydrgen electrn rbitals, the n rbital is equivalent t an inized electrn. Accrding t the quantum mechanical scattering mdel, the incident inized electrn is a plane wave. Hwever, substitutin f n int the slutin f the Schrödinger equatin yields a radial functin that has an infinite number f ndes and exists ver all space. The hydrgen-like radial functins have n l 1 ndes between r 0 and r. The results f the Davisn-Germer experiment cnfirm that the inized electrn is a plane wave. In cntrast, fr the present rbitsphere mdel, as n ges t infinity the electrn is a plane wave with wavelength h / p as shwn in the Electrn in Free Space Sectin. Althugh there are parallels in the mathematical derivatins wherein the Schwartz inequality is invked, the physics f the Heisenberg Uncertainty Principle is quite distinct frm the physics f the rise-time/band-width relatinship f classical mechanics [10] as given in the Resnant Line Shape and Lamb Shift Sectin. The Heisenberg Uncertainty Principle is derived frm the prbability mdel f the electrn by applying the Schwartz inequality [9]; whereas, the risetime/band-width relatinship f classical mechanics is an energy cnservatin statement accrding t Parseval's Therem. The Brn mdel f the electrn vilates Special Relativity. The failure f the Brn and Schrödinger mdel f the electrn t prvide a cnsistent representatin f the states f the electrn frm a bund state t an inized state t a scattered state als represents a failure f the dependent Heisenberg Uncertainty Principle. In cntrast, the present exact rbitsphere mdel prvides a cntinuus representatin f all states f the electrn and is cnsistent with the scattering experiments f helium. References 1. Reynlds, G. O., DeVelis, J. B., Parrent, G. B., Thmpsn, B.J., The New Physical Optics Ntebk, SPIE Optical Engineering Press, (1990).. Bnham, R. A., Fink, M., High Energy Electrn Scattering, ACS Mngraph, Van Nstrand Reinhld Cmpany, New Yrk, (1974). 13

22 by BlackLight Pwer, Inc. All rights reserved. 3. Bracewell, R. N., The Furier Transfrm and Its Applicatins, McGraw-Hill Bk Cmpany, New Yrk, (1978), pp Apelblat, A., Table f Definite and Infinite Integrals, Elsevier Scientific Publishing Cmpany, Amsterdam, (1983). 5. Bateman, H., Tables f Integral Transfrms, Vl. I, McGraw-Hill Bk Cmpany, New Yrk, (1954). 6. Brmberg, P. J., "Abslute differential crss sectins f elastically scattered electrns. I. He, N, and CO at 500 ev", The Jurnal f Chemical Physics, Vl. 50, N. 9, (1969), pp Geiger, J., "Elastische und unelastische streuung vn elektrnen an gasen", Zeitschrift fur Physik, Vl. 175, (1963), pp Peixt,E. M., Bunge, C. F., Bnham, R. A., "Elastic and inelastic scattering by He and Ne atms in their grund states", Physical Review, Vl. 181, (1969), pp McQuarrie, D. A., Quantum Chemistry, University Science Bks, Mill Valley, CA, (1983), p Siebert, W. McC., Circuits, Signals, and Systems, The MIT Press, Cambridge, Massachusetts, (1986), pp

23 000 by BlackLight Pwer, Inc. All rights reserved. EXCITED STATES OF HELIUM 15 In the grund state f the helium atm, bth electrns are at r a as given in the Tw Electrn Atm Sectin. When a phtn is absrbed by the grund state helium atm, ne electrn can mve t a radius at r > r 1, the radius f electrn ne. The phtn will generate an effective charge, Z P eff, within the first rbitsphere t keep electrn 1 at the allwed radius 0.567a. We can determine Z P eff f the "trapped phtn" electric field by requiring that the frce balance equatin is satisfied with the superpsitin f the electric fields f the nucleus and the "trapped phtn". Frm Eqs. (1.56) and (1.153), the frce balance equatin fr electrn ne, at r a is m e v h Z r 1 m e 0.567a 3 T eff e a (9.1) 0 Z T eff (9.) where Z T eff is the effective charge f the central field f the "trapped phtn" plus the nucleus. The electric field f the nucleus fr r < 0.567a is E nucleus +e 4 r (9.3) Frm Eq. (.15), the equatin f the electric field f the "trapped phtn" fr r < 0.567a, is ( ) l [ ] E Z e 0.567a m r phtnn,l,m P eff ( l +) Y 4 r 0 (, ) + Y l (, )Re 1+ e i n t 1 [ ] n 0 fr m 0 (9.4) The ttal central field fr r < 0.567a is given by the sum f the electric field f the nucleus and the electric field f the "trapped phtn". E ttal E nucleus + E phtn (9.5) Frm Eqs. (9.-9.5) and the frce balance bundary cnditin at r a, the electric field f the "trapped phtn" fr r < 0.567a, is E r phtnn,l,m ( ) l [ ]e 0.567a r 1 l + ( ) Y 0 0 m (, ) + Y l (, ) Re 1+ e i n [ [ t ] n 0 fr m 0 (9.6) Substitutin f Eqs. (9.3) and (9.6) int Eq. (9.5) gives fr r < 0.567a, ( ) l [ ] E e r ttal 4 r e 0.567a0 1 0 m 1 4 r ( l +) Y 0 (, ) + Y l (, ) Re 1+ e i n [ t ] n 0 fr m 0 (9.7) Recall frm Eq. (.17) f the Excited States f the One Electrn Atm

24 by BlackLight Pwer, Inc. All rights reserved. (Quantizatin) Sectin that the slutin f Z T eff f the bundary value prblem f "trapped phtns" which excite mdes in the rbitsphere resnatr cavity is Z'. In this case, Z' Z 1 1 where Z is the nuclear n charge. Thus, fr 0.567a < r < r where r is the radius f electrn, E e 1 r ttal 4 r n Y 0 m 0 (, ) + Y l (, ) Re 1+ e i n [ [ t ] n 0 fr m 0 (9.8) The frce balance equatin fr electrn is the same as the equatin derived t determine the inizatin energies f tw electrn atms as given in the Tw Electrn Atm Sectin (Eq. (7.18)): m e v h r m e r 1 e 3 n 4 r h 3 s(s + 1) (9.9) n m e r with the exceptin that the magnetic frce is multiplied by ne-half because electrn 1 is held at a fixed radius. With s 1, s(s +1) r n 4 a n,3,4,... (9.10) The energy stred in the electric field, E ele, is given by Eqs. (1.175) and (1.176) e E ele 1 n 8 r (9.11) where r is given by Eq. (9.10). The energy stred in the magnetic fields f tw unpaired electrns initially paired at radius r, E mag, is given by Eq. (7.30) E mag e h m e r 3 (9.1) where r is given by Eq. (9.10). E magwrk is the integral f the magnetic frce (the secnd term n the right side f Eq. (9.9) multiplied by tw because with inizatin, the secnd term f Eq.(9.9) crrespnds t the center f mass frce balance f electrn tw and electrn ne wherein the electrn mass replaced by the reduced mass 1 m e ): r h E magwrk s(s +1) dr (9.13) nm e r 3 where r is given by Eq. (9.10).

25 000 by BlackLight Pwer, Inc. All rights reserved. h s(s +1) E magwrk ; s 1 (9.14) 4nm e a s(s +1) 0 n 4 The magnetic transitin energy, E HF, can be calculated frm the spin/spin cupling energy and the magnetic energy stred in the surface currents prduced by the "trapped resnant phtn". The spin/spin cupling energy arises frm the interactin f the magnetic mment assciated with the spin f ne electrn with the magnetic field generated by the current prduced by the spin mtin f the ther electrn. The spin/spin cupling energy in the excited state between the inner rbitsphere and the uter rbitsphere is given by Eq. (1.136) where B, the magnetic mment f the uter rbitsphere is given by Eq. (1.137). The magnetic flux, B, f the inner rbitsphere at the psitin f the uter is B eh 3 (9.15) m e r Substitutin f Eq. (9.15) and (1.137) int Eq. (1.136) gives h E g e 4m 3 (9.16) e r Phtns bey Maxwell s Equatins. At the tw dimensinal surface f the rbitsphere cntaining a "trapped phtn", the relatinship between the phtn s electric field and its tw dimensinal chargedensity at the rbitsphere is ( ) n E 1 E 0 17 (9.17) Thus, the phtn s electric field acts as surface charge. Accrding t Eq. (9.17), the "phtn standing wave" in the helium rbitsphere resnatr cavity gives rise t a tw dimensinal surface charge at the rbitsphere tw dimensinal surface at r 1 +, infinitesimally greater than the radius f the inner rbitsphere, and r, infinitesimally less than the radius f the uter rbitsphere. Fr an electrn in a central field, the magnitude f the field strength f each excited state crrespnding t a transitin frm the state with n 1 and radius r a t the state with n n and radius r is 1 e (Eq. (.17)) as given in the Excited States f the One n Electrn Atm (Quantizatin) Sectin. The energy crrespnding t the surface charge which arises frm the "trapped phtn standing wave" is given by the energy stred in the magnetic fields f the crrespnding currents. The surface charge is given by Eq. (9.17) fr a central field strength equal in magnitude t 1 e. This surface charge pssesses the n

26 by BlackLight Pwer, Inc. All rights reserved. same angular velcity as each rbitsphere; thus, it is a current with a crrespnding stred magnetic energy. The energy crrespnding t the surface currents, E sc, is the sum f E mag, internal and E mag,external fr a charge f 1 e substituted int Eqs. (1.15) and (1.17) fr bth electrns initially n at r 1. E sc e h 3 n m e h e r 1 3 n m 3 e r 1 (9.18) E sc, the magnetic surface current energy crrespnding t absrbing a phtn such the secnd electrn with principal quantum number n is inized is given by multiplying Eq. (9.18) by the prjectin f the electric fields-the rati f the magnitude f the electric field fr 0.567a < r < r, 1, given by Eq. (9.8) t the magnitude f the electric field n change, 1 1 n. 1 E SC n 1 1 e h 1 n m 3 e h e r 1 ( n 1) n m 3 (9.19) e r 1 n The energy crrespnding t the a singlet t triplet transitin-the hyperfine structure energy, E HF, is given by the sum f Eq. (9.16) and Eq. (9.19) E HF g e h 4m 3 e r + 1 e h (n 1) n m 3 e r 1 (9.0) where r a and r is given by Eq. (9.10). The inizatin energy f helium is given by Eq. (7.8). The inizatin energy f triplet states with 0 is given as the sum f E magwrk, (Eq. (9.14)), the energy t remve the secnd electrn fllwing the absrptin f an inizing phtn which flips the electrns such that they are antiparallel, and the energies terms f Eq. (7.8) where E ele is given by Eq. (9.11) and E mag is given by Eq. (9.1). The inizatin energy f singlet states with 0 is given by the sum f E ele (Eq. (9.11)) and E HF (Eq. (9.0)), the energy f the transitin f the electrns frm antiparallel t parallel such that they repel each ther; s, the secnd electrn is inized. The inizatin energy f triplet states with 0 is given by E ele (Eq. (9.11)) minus the magnetic energy crrespnding t the spin and rbital angular energies which fllw frm Eq. (1.95): E HF (Eq. (9.0)) is multiplied by the magnitude f the maximum spin prjectin which fllws frm Eqs. (1.74) and (1.95) and the magnitude

27 000 by BlackLight Pwer, Inc. All rights reserved. f the maximum rbital prjectin given by Eq. (1.95). The inizatin energy f singlet states with 0 is given by E ele (Eq. (9.11)) minus the magnetic wrk energy crrespnding t the rbital angular energy which fllws frm Eq. (1.95): E magwrk (Eq. (9.14)) is multiplied by the magnitude f the maximum rbital prjectin given by Eq. (1.95). The energy f the excited states f helium are given by the sum f the cmpnent electric and magnetic energies and their interactins as fllws: 19 Excited States with 1s 1s 1 ( ns) 1 Zer (9.1) Triplet States E E ele + E magwrk + E mag (9.) Singlet States E E ele + E HF (9.3) Excited States with Zer Triplet States E E ele s( s +1)+ l ( l +1 ) l + l +1 E ; s 1 HF (9.4) Singlet States E E ele l( l +1) l + l +1 E magwrk (9.5) Table 9.1 gives the rbital factr as a functin f.

28 0 000 by BlackLight Pwer, Inc. All rights reserved. Table 9.1. Orbital factr l( l +1) as a functin f. l + l +1 rbitsphere designatin l( l +1) l + l +1 0 s 0 1 p d f 4 Table 9. gives the radius f electrn and the energy terms as a functin f n. Table 9.. The radius f electrn and the energy terms as a functin f n. n 1 n n 3 n 4 r n (a ) E ele E mag wrk E mag E HF The magnetic splitting energies due t spin angular mmentum and rbital angular mmentum are given by Eq. (.40). In the case that m >1 ( > 1), the magnetic splitting is nnzer, and a spin/rbital cupling energy arises frm the surface currents crrespnding t the "trapped resnant phtn" which gives rise t the net rbital angular mmentum.

29 000 by BlackLight Pwer, Inc. All rights reserved. The spin/rbital cupling energy arises frm the interactin f the magnetic mment assciated with the rbital angular mmentum f ne electrn with the magnetic field generated by the current prduced by the rbital mtin f the ther electrn. Frm Eq. (.40) and Eq. (9.18), E s/, the spin/rbital cupling energy is E s/ ( l 1) e h ( n 1) n m 3 (9.6) e r 1 Spin/rbital cupling increases the inizatin energy f singlet states ( mm s < 0 ) and decreases the energy f triplet states (mm s > 0 ) [1]. Table 9.3 gives the spin/rbital cupling energy terms as a functin f n and (Eq. (9.6)). 1 Table The spin/rbital cupling energy terms as a functin f n and n r n (a ) E s/ Term Designatin 3D D F An electrdynamic spin/rbital cupling energy arises frm the interactin f the magnetic mment assciated with the spin f each electrn with the magnetic field generated by the current prduced by the electrn's wn rbital mtin (Eq. (.84)). The energies f the varius states f helium with spin-rbital cupling crrectins appear in Table 9.4.

30 000 by BlackLight Pwer, Inc. All rights reserved. Table 9.4. Calculated and experimental energies f excited states f helium with spin-rbital cupling crrectins. Cnfiguratin Term Energy Energy Designatin (Calculated) (Experimental) 1s 1 S s 1 s 1 3 S s 1 s 1 1 S s 1 p 1 3 P s 1 p 1 1 P s 1 3s 1 3 S s 1 3s 1 1 S s 1 3p 1 3 P s 1 3p 1 1 P s 1 3d 1 3 D s 1 3d 1 1 D s 1 4s 1 3 S s 1 4s 1 1 S s 1 4p 1 3 P s 1 4p 1 1 P s 1 4d 1 3 D s 1 4d 1 1 D s 1 4f 1 3 F s 1 4f 1 1 F

31 000 by BlackLight Pwer, Inc. All rights reserved. References 1. Karplus, M., Prter, R. N., Atms & Mlecules: An Intrductin fr Students f Physical Chemistry, Benjamin/Cummings Publishing Cmpany, Menl Park, CA, (1970), p

32 4 000 by BlackLight Pwer, Inc. All rights reserved. THE THREE ELECTRON ATOM THE LITHIUM ATOM Fr Li +, there are tw spin-paired electrns in an rbitsphere with 3 1 r 1 r a 4 (10.1) 6 as given by Eq. (7.19) where r n is the radius f electrn n which has velcity v n. The next electrn is added t a new rbitsphere because f the repulsive diamagnetic frce between the tw spin-paired electrns and the spin-unpaired electrn. This repulsive magnetic frce arises frm the phenmenn f diamagnetism invlving the magnetic field f the uter spin-unpaired electrn and the tw spin-paired electrns f the inner shell. The diamagnetic frce n the uter electrn is determined belw. The central frce n each electrn f the inner shell due t the magnetic flux B f the uter electrn fllws frm Purcell [1] F m v v e n (10.) r where v r eb (10.3) m e The velcity v n is given by the bundary cnditin fr n radiatin as fllws: v 1 h (10.4) m e r 1 where r 1 is the radius f the first rbitsphere; therefre, the frce n each f the inner electrns is given as fllws: F heb (10.5) m e r 1 The change in magnetic mment, m, f each electrn f the inner shell due t the magnetic flux B f the uter electrn is [1] m e r 1 B (10.6) 4m e The diamagnetic frce n the uter electrn due t the tw inner shell electrns is in the ppsite directin f the frce given by Eq. (10.5), and this diamagnetic frce n the uter electrn is prprtinal t the sum f the changes in magnetic mments f the tw inner electrns due t the magnetic flux B f the uter electrn. Because changes in the magnetic mments are invlved, the determinatin f the diamagnetic frce n the uter electrn is simplified by cnsidering the tw inner

33 000 by BlackLight Pwer, Inc. All rights reserved. electrns as a single entity f twice the mass. The ttal change in magnetic mments f the inner shell electrns due t the field f the uter electrn is then given by Eq. (10.6) where m e is replaced by m e. It is then apparent that the frce given by Eq. (10.5) is prprtinal t the flux B f the uter electrn; whereas, the ttal f the change in magnetic mments f the inner shell electrns given by Eq. (10.6) applied t the cmbinatin f the inner electrns is prprtinal t ne eighth f the flux, B. Thus, the frce n the uter electrn due t the reactin f the inner shell t the flux f the uter electrn is given as fllws: F diamagnetic h eb (10.7) 8r 1 m e where r 1 is the radial distance f the first rbitsphere frm the nucleus. The magnetic flux, B, is supplied by the cnstant field inside the rbitsphere f the uter electrn at radius r 3 and is given by the prduct f times Eq. (1.10) Ḃ eh 3 (10.8) m e r 3 The result f substitutin f Eq. (10.8) int Eq. (10.7) is F diamagnetic e h m e r 3 (10.9) 4m e r 1 r 3 The term in brackets can be expressed in terms f the fine structure cnstant,. Frm Eqs. ( ) e v (10.10) m e r 3 c It is demnstrated in the Tw Electrn Atm Sectin that the relativistic crrectin t Eq. (10.9) is 1 times the reciprcal f Eq. (10.10). Z Z fr electrn three is ne; thus, ne is substituted fr the term in brackets in Eq. (10.9). The frce must be crrected fr the vectr prjectin f the velcity nt the z-axis. As given in the Spin Angular Mmentum f the Orbitsphere with l 0 Sectin, the applicatin f a z directed magnetic field f electrn three given by Eq. (1.10) t the tw inner rbitspheres gives rise t a diamagnetic field and a prjectin f the angular mmentum f electrn three nt an axis which precesses abut the z- 3 axis f h. The prjectin f the frce between electrn three and 4 electrn ne and tw is equivalent t that f the angular mmentum nt the axis which precesses abut the z-axis, and is s( s +1) 3 4 times that f a pint mass. Thus, Eq. (10.9) becmes 5

34 6 000 by BlackLight Pwer, Inc. All rights reserved. h F diamagnetic s(s +1) (10.11) 4m e r 3 r 1 THE RADIUS OF THE OUTER ELECTRON OF THE LITHIUM ATOM The radius fr the uter electrn is calculated by equating the utward centrifugal frce t the sum f the electric and diamagnetic frces as fllws: m e v 3 e r 3 4 r h s(s +1) (10.1) 3 4m e r 3 r 1 With v 3 h 1 (Eq. (1.56), r m e r 1 a 3 fr r (Eq. (7.19)), and s 1, we slve r 3 a 3/ / 4 6 (10.13) r a THE IONIZATION ENERGY OF LITHIUM Frm Eq. (1.176), the energy stred in the electric field is e 8 r ev (10.14) 3 The magnetic field f the uter electrn changes the angular velcities f the inner electrns. Hwever, the magnetic field f the uter electrn prvides a central Lrentzian frce which exactly balances the change in centripetal frce because f the change in angular velcity [1]. Thus, the electric energy f the inner rbitsphere is unchanged upn inizatin. The magnetic field f the uter electrn, hwever, als changes the magnetic mment, m, f each f the inner rbitsphere electrns. Frm Eq. (10.6), the change in magnetic mment, m, (per electrn) is m e r 1 B (10.15) 4m e where B is the magnetic flux f the uter electrn given by the prduct f times Eq. (1.10).