928 A. W. JOHNSON AND J. B. GERAR+0

With the values of ~ and ~, given in Ref. 5 at 15Torr, p is evaluated to be equal to 5. 1 x 10 ~ cmssec-' and is presented in Tabl. e I under "D."Again, because p is evaluated at low pressures,it should be nearly equal to g.

Using the entire range of n and o, (Ref. 5) whichwas measured at gas pressures between 15 and 55Torr, we evaluated p with Eq. (6) and found ityielded values from 4. 5 ~ 10 9 to 5. 1 x 10 9 cmsec '. In this pressure range, the conditionschanged from the case at l.ow pressure, where thetotal metastable density was nearly equal. to theatomic-metastable density, to the case at highpressure, where the molecular metastables com-pl.etel.y dominated after a time into the afterglow.The fact that P changed by only a small amountover this pressure range is a self-consistent justi-

fication that all metastaWbe ionizing rates are nearly equal.

V. DISCUSSION

In the above calculations, various severe, al-though reasonable, assumptions were made in eachcase. A major justification for the derived valueof p, is that the values of p, determined by differ-ent techniques vary about the mean value by only20%. All of the derived values of P, evaluated fromthe different techniques are between the maximumand minimum values given under "Range" in TableI. A more accurate evaluation of P, requires amore accurate value for the oscillator strength ofthe 4650-A band of He~. When this oscillatorstrength is more accurately known, the techniquedescribedhere canalsobe used to evaluate p&2 and p2,

~Work supported by the U. S. Atomic Energy Commission.'A. V, Phelps and J. P. Molnar, Phys. Rev. 89, 1202 (1953).'P. A. Miller, J. T. Verdeyen, and B. E. Cherrington, Phys.

Rev. A 4, 692 (1971).'A. Wayne Johnson and J. B. Gerardo, Phys, Rev. Lett,

28, 1096 (1972).'A. Wayne Johnson and J. B. Gerardo, Phys. Rev. Lett.

27, 835 (1971).'A. Wayne Johnson and J. B. Gerardo, Phys. Rev. A

5, 1410 (1972).G. J. Schulz and R. E. Fox, Phys. Rev. 106, 1179 (1957).J. D. Jobe, dissertation (University of Oklahoma, 1968)

(unpublished).

PHYSICAL REVIEW A VOLUME 7, NUMBER 3 MARCH 1973

Photoionization of Atomic Nitrogen and Atomic Oxygen

P. S. Ganas~University of Florida, Gainesvitle, Florida 32601

(Received 10 November 1972)

We utilize an atomic independent-particle model to calculate photoionization cross sections for atomicnitrogen and atomic oxygen as a function of the wavelength of the incident radiation, from thresholddown to 10 A. Comparison is made with available experimental data and the results of Hartree —Fockcalculations.

I. INTRODUCTION

Atomic photoionization cross sections are useful.in probing the details of atomic structure. A knowl-edge of such cross sections is helpful in the under-standing of gaseous discharges, ionospheric l.ayers,and stel1ar atmospheres. Photoionization is knownto play an essential role in the propagation ofsparks, lightning, and other types of discharges.It is important for the atomic theorist to be able tomake accurate g priori calculations of atomic pho-toionization cross sections. When accurate exper-imental cross sections exist, the calculations serveas a test of atomic theories and methods of calcu-lation.

The photoionization cross sections of atomic ni-trogen and atomic oxygen for incident radiationfrom threshold (=900 A) down to 10 A are calcula-

ted. These particular cross sections are impor-tant in the interpretation of solar-terrestrial ef-fects. They have been calculated by Dalgarno andParkinson, ' Bates and Seaton, McGuire, Thomasand Hel. liwell, Kahler, and Henry, using theHartree-Fock approach. In contrast to this ap-proach, the present calculations are based uponthe independent-particle model (IPM) of the atom.In comparison to Hartree-Fock-Slater calcul. ationsand to experiment, a simple two-parameter IPMpotential has been found to provide a good repre-sentation of atoms and electron-atom interac-tions. v ' The potential for an electron in a neu-tral atom may be written (in atomic units)7

where d and H are adjustable parameters. This

PHOTOIONIZATION OF ATOMIC NITROGEN AND ATOMIC OxyGEN

-O. I—

7s6s5s

4p

momentum transfer, 5'is the energy loss, and

df/dWis the electron-impact ionization GOS. Forthe photoionization problem,

W= he/X= T.&I,

4s v

C9ELLLJz:-02LU

-0.3—

3p

EXP=- l.069RyPp3

O =-IX)70Ry

NITROGEN

FIG. 1. Excited-state energies of atomic nitrogen.The lines denote average of experimental levels. Theposition of the ground state is indicated by numericalvalues. The symbol Q denotes theoretical IPM energylevels based upon the parameters d=1.1922, H=2. 9593.

where X is the wavelength of the incident radiation,I is the ionization potential of the atom, and T isthe kinetic energy of the photoelectron.

An el.ectron-impact ionization process is consid-ered in which an electron in the ~olo subshell. isejected into the continuum with a momentum k.I.et the initial and final momenta of the projectileelectron be koand k, respectively; then K=k —kis the momentum transfer. By using the first Bornapproximation and assuming the Russell-SaundersI S-coupling scheme for the initial and final states,it can be shown that'0 in atomic units

where g,'., is the radial matrix element0

(d l(l+ 1)

q~s- ~ii—&(")+~ i)&.i(~)=O (2)

potential. is inserted into the radial. Schrodingerequation, .

Zi i, =f,"

pi +'&)jr («) p.,&, (~)«and the coefficient B is given by

B=NF .

(6)

to obtain the energy eigenvalues E„„and wavefunctions P„,(x)/x, using a subprogram of the Her-man-Skillman computer code. The potential param-eters d and H are determined by requiring theenergy eigenvalues to be in good agreement withthe experimental single-particle energy levels.The latter are obtained from the tables of Moore"by averaging over multiplets, as described in Ref.8. The experimental averages for atomic nitrogenare displayed in Fig. 1 together with the computedenergies corresponding to the parameters d=1.1922 and H=2. 9593. One sees that the agree-ment between the theoretical and experimental l.ev-el.s is very precise. The electron-impact excita-tion and ionization of atomic oxygen has been dis-cussed in Ref. 10 using the Born approximation andthe IPM. The parameters d= 0. 8164, and H= 2. 224obtained in Ref. 10 will be used in the present cal-culations on atomic oxygen.

In Eq. (6), P, (k'v)/x is the radial component ofthe Eth partial wave of the continuum wave function,P„, (r)/x is the bound-state radial wave functionnolqfor the nolo orbital, and j,(«) is a spherical Besselfunction. The continuum wave function is obtainedby solving the radial Schrodinger equation with thesame analytic IPM potential as the initial. state.Then the bound and continuum wave functions areorthogonal to each other. In Eq. (5), the array inround brackets is a 3j symbol. In Eq. (7), N is thenumber of electrons in the active subshell, and Eis the coefficient of fractional parentage'3 for con-structing the initial state(nolo) "SOLO from the corestate (nolo)" 'S,L, and an n lo electron. We notethat B varies according to the core. For oxygen,which has a (1s~2s 32p') 'P ground state, the possi-ble O' states are 48, ~P, 3D for which the valuesof 8 are 3, 1, 3, respectively. These core statesarise in the following processes:

II. THEORY OP P) + h~ -0'('.S) + e for X &910 A, (Ba)

e = (0. BOV x 10 '~) lim cm~z. 0dS' (3)

Here 0 isthe photoionizationcross section, It. is the

The calculations are based upon the close rela-tionship between the photoionization cross sectionof an atom and the generalized oscillator strength(GOS) for the electron-impact ionization of an atom.From the discussion given in Secs. II and IV of thereview article by Stewart, '3 one has the relation

0( P)+ h&-0'( D)+ e" for A. &732 A,

OPP)+ h&-0'( P)+ e for X &665A.

(Bb)

(Bc)

The values of B to be used for the various wave-length ranges are listed in Table I. For nitrogen,which has a (1s 32s ~2ps) 4S ground state, the possi-ble N" states are '9, 3P, 'D for which the valuesof B are 0, 3, 0, respectively. It follows that the'S and 'D states do not contribute to the photoion-ization cross section.

S30 P. S. GANAS

TABLE I. Values of coefficient B in Eq. (5). TABLE II. Computed IPM photoionization cross sectionsfor atomic nitrogen.

AtomWavelength

range (A)

Contributingcores 0 (10 3 Cm2}

A, &732732 &A. &665

X& 665

A, & 850

4S

S 2D

48 2D 2P

3P

433

where S... is the radial dipole matrix element0

s... = f"I, , (k ~)~p„, (~)d~0 0 0 p

Finally, it may be noted that the limit of the GOSas K-0 may be evaluated explicitly:

I

oo =o

o'+"' '" o o o~) I "oI'

101520253035405060708090

100150200

0.00140.00570.0370.0390.0640.1060.1550.3090.5350.8511.351.862.487.2

14.9

III. RESULTS AND DISCUSSION

The computed photoionization cross sections foratomic nitrogen and atomic oxygen, correspondingto the ejection of a 2P electron, are displayed inFigs. 2 and 3, respectively, for wavelengths down

to 200 A. For wavelengths from 200 A down to10 A, the computed cross sections are tabulated inTables II and III. One sees that the cross sectionsdiminish rapidly below 200 A. The computationsare based upon Eq. (5) with K2 set equal to iO 4, asmall but finite value. It was convenient to useEq. (5) rather than Eq. (9) because of the existingcomputer code for generating the GOS. FromFigs. 2 and 3, one sees that the over-all agree-ment between theory and experiment is quite rea-sonable.

In Fig. 2 the computed cross sections for atomicnitrogen are compared to the experimental data ofEhler and Weissler, ' Samson and Cairns, and

1,6—

Comes and Elzer. ' The data of Samson and

Cairns, ~5 between 209 and 53'7 A, pertain to mo-lecular nitrogen; their values have been halvedhere. The assumption that the photoionizationcross section of N may be taken as one-half thatof N2 appears to be justified for the shorter wave-lengths. '4'~ In the range 200-650 A, the theoret-ical cross sections are centrally located betweenthe three sets of experimental data. Above 650Athe discrepancy between theory and experiment be-comes more pronounced. It may be that the modelis not accurate enough near threshold. This maybe due to the neglect of electron-exchange effects,and the distortion of the wave functions of the pas-sive electrons by the transition.

In Fig. 3, thecomputedcross sectionsfor atomicoxygen are compared to the experimental data ofCairns and Samson, ' Samson and Cairns, "and

TABLE III. Computed IPM photoionization cross sectionsfor atomic oxygen.

0. (10 '~ cm )

1.2-o+W+ ++

I-0.8 — +IJJCO

o0.4(ZO

0. . . I. . . . . . , . I. . . , . , I. . . . , . . . I. . . , . . . , I, . . . . , . . I

900 800 700 600 500 . 400 300 200WAVELENGTH (A)

FIG. 2. Photoionization cross sections for atomicnitrogen. The solid curve represents the theoretical IPMresults. The experimental data points denoted by thesymbols i, 0, + are taken from Hefs. 14, 15, 16, re-spectively.

101520253035405060708090

100150200

0.00740.0170.0390.0860.1350.2090.3270.624l. 061.802.453.424.39

12.123.7

PHOTOIONIZATION OF ATOMIC NITROGEN AND ATOMIC OXYGEN

I

O

O+o 0.5LLIV)

0KC3

FIG. 3. Photoionization cross sec-tions for atomic oxygen. The solidcurve represents the theoretical IPMresults. The experimental data pointsdenoted by the symbols +, 0, + aretaken from Hefs. 17, 15, 18, , respec-tively.

0.2— 0+hz = 0 +e+ I. . . I. . . I. . . I. . . I. . . I

900 800 700 600 500 . 400 500 200WAVEI ENGTH {A )

Comes, Speier, and El.zer. ' As in the case ofatomic nitrogen, the values of Samson and Cairns"have been halved. The agreement between theoryand experiment is quite reasonable over the wholerange 200-900 A. The discontinuities in the photo-ionization curve at 732 and 665A are due to thepossibility of ionization to the ~D and 3P states of0', as discussed in Sec. II.

As noted above, the main deficiency in our re-sults occurs near the photoionization threshold ofatomic nitrogen, where the calculated values aresomewhat higher than experiment. The resultsobtained by Bates and Seaton' in this region arequite close to experiment; however, this is due totheir use of more refined wave functions, whichincorporate the effects of distortion as mell as ex-change. The values in Tables II and III are some-what smaller than those given by McGuires; thisis due to the inclusion by McGuire of inner-sub-

shel. l. photoionization. The computational simplic-ity of the IPM is an advantage over Hartree-Pockmethods. An essential feature of the IPM is theflexibility achieved by having adjustable parame-ters in the potential. The Hartree-Pock method,on the other hand, has no corresponding facility.By tuning up the parameters of the potential the ex-perimental single-particle energies can be charac-terized very accurately. The IPM is perhaps mostadvantageous in situations requiring the calculationof a large number of cross sections with good accu-racy.

ACKNOWLEDGMENTS

It is a pleasure to thank Alex E. S. Green forreading the manuscript. The author appreciatesthe marm hospitality of Professor Green and hisassociates during the author s visit to the Univer-sity of Florida.

~Work supported by the National Science Foundation.tPresent address: Physics Department, California State Uni-

versity, Los Angeles, Calif. 90032.'A. Dalgarno and D. Parkinson, J. Atmos. Terr. Phys.

18, 335 (1960).D. R. Bates and M. J. Seaton, Mon. Not. R. Astron. Soc.

109, 698 (1949).'E. J. McGuire, Phys. Rev. 175, 20 (1968).G. M. Thomas and T. M. Helliwell, J. Quant. Spectrosc.

Radiat. Transfer 10, 423 (1970).'H. Kahler, J. Quant. Spectrosc. Radiat. Transfer 11, 1521

(1971).R. J. W. Henry, Planet. Space Sci. 15, 1747 (1967).

'A. E. S. Green, D. L. Sellin, and A. S. Zachor, Phys. Rev.184, 1 (1969).

'P. S. Ganas and A. E. S. Green, Phys. Rev. A 4, 182

(1971).'T. Sawada. J. E. Purcell, and A. E. S. Green, Phys. Rev. A

4, 193 (1971).' P. A. Kazaks, P. S. Ganas, and A. E. S. Green, Phys,

Rev. A 6, 2169 (1972)."C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.)

Circ. No. 467 (U. S. GPO, %ashington, D. C., 1958), Vol.I.

' A. L. Stewart, in Advances in Atomic and MolecularPhysics, edited by D. R. Bates and I. Estermann (Academic,New York, 1967), Vol. 3.

"B.W. Shore and D. H. Menzel, Principles of AtomicSpectra (Wiley, New York, 196&).

"A. W. Ehler and G. L. Weissler, J. Opt. Soc. Am.45, 1035 (1955).

"J. A. R. Samson and R. B. Cairns, J. Opt. Soc. Am.

P. 8. GA NAB

55, 1035 (1965)."F. J. Comes and A. Elzer, Z. Naturforsch. A 23, 133

(1968)."R. 8. Cairns and J. A. R. Samson, Phys. Rev. 139,

A1403 (1965),"F. J. Comes, F. Speier, and A. Elzer, Z. Naturforsch. A

23, 125 (1968).

I'HYSICAL REVIEW A VOLUME 7, NUMBER 3 MARCH 1973

Energy Dependence of the Total Cross Section for He-HeCollisions from 0.30 to 3.00 kev ~

%. J. Savola, Jr. , F. J. Eriksen, and E. PollackPhysics Department, Univexsi ty of Connecticut, Stoves, Connecticut 06268

(Received 21 September 1972)

The tota]. cross section for scattering of He by He is measured at 100-eV intervals in anenergy range from 0.30 to 3.00 keU. Measurements are made using detector angular reso-lutions of 0.056 and 0.260'. The cross section is found to decrease monotonically from

o212.5 to 5.0 2 and from 7. G to 2.6 A, respectively, as the energy increases. Small-angledifferential measurements are also made out to scattering angles of 1.25' at incident ener-gies of 1.80 and 3.00 keV. At 3.00 keU a He' signal is found at schttering angles beyond0.8'. The ratio of the ion to the total signal increases to about 10% at 1.25'. The total-cross-section data are used to infer a potential.

I. INTRODUCTION

Collisions between beams of fast neutral atomsand thermal target atoms are generally studiedbecause they can provide useful information onshort-range interactions which occur in the collid-ing system. Studies of such collisions have beensuccessful in yielding values for interatomic po™tentials which could then serve as a check on ap-proximations employed in theoretically calculatinginteratomic potentials. Experimental results havealso provided data which are useful i.n predictinghigh-temperature kinetic properties of gases.

Although it is possible to investigate short-rangeinteractions by making differential cross-sectionmeasurements, such measurements usually havebeen. made for ion-atom systems with very littledifferential work having been done on atom-atomcollisions. Experimental studies concerned withatom-atom collisions have generally involved mea-surements of the energy dependence of the totalcross section for scattering a fast neutral beambeyond a detector of fixed acceptance angle. Thisis the technique employed in the present study. Inpioneering investigations, Amdur and co-workers~'have measured such total cross sections for nu-merous neutral systems (see Ref. 2 for a discus-sion of the neutral experiments and additionalreferences to the literature). Similar measure-ments have recently been made by Kamnev,Belyaev, and Leonas. ' In addition, several ofthe important problems relating to the interpreta-tion of experimental data (including quantum-scat-tering corrections and inelastic-scattering effects)

have been discussed in a review article by Masonand Vanderslice, '

The present experiment measures the energydependence of the total cross section for the He-Hecollision in a range from 0. 30 to 3. 00 keV (in 100-eV intervals) at two different detector angularresolutions. One of the detector angular resolu-tions selected is sufficiently high so that quantum-scattering effects may arise at the lower energies.In addition, differential measurements are madeto ascertain the contribution of small-angle inelas-tic scattering to the measured total cross sections.All measurements are made with fast He beamsformed from mass-analyzed He'. Hole-hole geom-etries rather than combinations of holes and slitsare used for collimation of the incident and scat-tered beams.

II. APPARATUS

Figure 1 shows a schematic diagram of theapparatus. The operation of the electron-impact-ion source (A), extractor (8), lenses (C), and

magnet (D) has been previously described. ' Aneutral beam is produced in the charge-exchangeregion (E) by resonant charge-exchange of the mass-analyzed He' with He at a pressure of about 30 p, .The charge-exchange region is 0. 8 in. in diameter,and holes at either end define a charge-exchangepath about 1.2 in. long. Emerging from the holeat (F) is a composite beam of He ions and He neu-trals. An electric field between two plates (6) de-flects the ions into a Faraday cup monitor (H),while the neutrals pass through a collimating hole(I) and form the incident beam. The holes at (F) and