FILLED-EPOXY MIRRORS 1111

Reactor radiation has little effect on physical propertiesfor doses up to the equivalent of about 107 r. It is ex-pected that all of the above parameters and propertieswill be important in future applications of epoxy-replicamirrors where mirrors may be rotating, mounted inunconventional manners, and subject to temperatureextremes or radiation fields. It is further expected thatthe general use of epoxy mirrors will increase becauseof the cost advantage, the extreme flexibility in designand fabrication, and the weight savings which can berealized.

VIII. ACKNOWLEDGMENTS

The authors wish to thank James Walker, LelandEminhizer, Clair Smith, and Carl Volz, Jr. for assistancein carrying out the experimental work. Cooperation ofthe Staff of the Pennsylvania State University NuclearReactor Facility is also greatly appreciated. In addition,the authors acknowledge the helpful comments ofmembers of the Singer-Bridgeport Military ProductsDivision of the Singer Manufacturing Company, wherethe replica process is being developed under the tradename Replikote.

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 50, NUMBER 11 NOVEMBER, 1960

Free-Free Continuum of Oxygen*R. G. BREENE, JR.,t AND MARIA C. NARDONE

Aerosciences Laboratory, Missile and Space Vehicle Department, General Electric Company, Philadelphia, Pennsylvania(Received July 14, 1960)

Oxygen is treated in an example calculation of the free-free absorption cross section. Wave functions forthe free electron in the field of an oxygen atom under the assumption of no electron exchange or corepolarization are calculated on the IBM 704. These wave functions are applied to the calculation of thefree-free absorption corresponding to a particular continuum frequency and initial state energy. In thiscalculation the dipole acceleration form of the matrix element is computed. The matrix element calculationis carried out on the LGP-30. These partial cross sections are smeared over a Maxwell-Boltzmann distribu-tion of initial electronic energies in order to obtain the free-free cross section. The results are in approximateagreement with the familiar classical Kramers result for an effective Z of 0.35, although the frequencydependence is noticeably different. We carry out the calculation for the temperatures 8000, 6000, and400 0 'K and find there to be no temperature dependence in the cross section, at least in this range. Thecalculation is also carried out under the assumption of a mean electron velocity at 8000'K. The magnitudedoes not differ much although the wavelength dependence is somewhat less.

I. INTRODUCTION

W E have discussed the application of our freeVV electron wave functions to one of the familiarradiation continua, namely, the bound-free continuum,in our example calculation, that of carbon. We nowturn our attention to an even more popular type ofcontinuous radiation, the free-free or bremsstrahlungwhich is encountered when the electron makes atransition between two of its free states in the presenceof a force center. For our consideration here the forcecenter will be an atom, and in our example calculationwhich is to follow the atom is oxygen.

It has been common to treat this problem by anapplication of the Kramers theory2 which is doubtlessquite adequate when the perturbing center is an ion.In the case of neutral atoms it has been supposed thatthe "nuclear charge" appearing in the Kramers theoryis usually some fraction of unity, and this fraction is

* Based on work performed under the auspices of the U. S. AirForce Ballistic Missile Division.

t Consultant. Correspondence address: 48 Maple Drive, Center-ville 59, Ohio.

'R. G. Breene, Jr., J. Planetary Space Sci. 2, 10 (1959).2 H. A. Kramers, Phil. Mag. 46, 836 (1923).

determined by an adjustment of the continuum tofit experiment. There is probably little doubt that thefree electron does feel as if it were being affected bysome fraction of one nuclear charge and behavesaccordingly. Nevertheless we have no idea as to whatthis fraction may be without recourse to experiment.The treatment we propose, although admittedly con-taining inaccuracies, will give us some idea as to thisfraction without such an appeal.

A treatment similar to ours has been carried out byHammerling and Kivel.A They applied their freeelectron wave functions to the computation of crosssections at a sufficient number of wavelengths so thatan analytic expression for the free-free continuumcould be fitted to the results. This expression was thenused in their subsequent work.4 Although their calcula-tion was tied in with experiment through the cutoff inthe Buckingham expression for the polarizationpotential which they used, this could have been

3 P. Hammerling and B. Kivel, AVCO Research Laboratory,Research Note 49, June, 1957.

4 J. C. Keck, J. C. Camm, B. Kivel, and T. Wentink, Jr. Ann.Phys. 7, 1 (1959).

November 1960

R. G. BREENE, JR., AND M. C. NARDONE

avoided to the degree of our approximation here byignoring the polarization potential.

The treatment which is to follow is similar to the

earlier treatment of hydrogen by Chandrasekhar andBreen.' We first utilize our free electron wave functionprogram to determine the wave functions among whichour free-free transitions are to take place. From thesethe cross section for the transition corresponding to agiven absorbed radiant frequency may then be deter-mined. Now here, of course, a given frequency may beabsorbed by an infinite number of free states. There-fore, before we may find the cross section for a givenradiant frequency we must smear the individual crosssections over a Maxwell-Boltzmann distribution of thelower free electron states. It is apparent that thismust be done for a particular temperature, so, in ourillustrative example, we work at 80000 K. With theobtention of these latter cross sections the free-freecontinuum is effectively determined.

The cross section is also calculated for temperaturesof 6000 and 4000'K. The result indicates an all butnegligible variation of the cross section with temper-ature. We recall that the Kramers result is proportionalto T-1. Finally we replace the Boltzmann smearing bythe assumption that all electrons have the meanvelocity (irkT/2m) . The magnitude of the crosssection in the range 5000 to 30 000 A is not greatlyaffected although the wavelength dependence changessomewhat.

II. CROSS SECTIONS FOR INDIVIDUALTRANSITION

We have detailed elsewhere our method of obtainingwave functions for a free electron in the field of anatom.6 In this connection we remark our neglect of (1)polarization and (2) exchange. It would seem that thisneglect would have a somewhat lesser effect on thefree-free than on the bound-free matrix elements. Thisis because our interest here is in the combination ofpairs of free electron wave functions referred to thesame interaction base. At any rate the procedure willbe precisely the same when our free electron wavefunctions incorporating polarization and exchange areapplied to this calculation.

We begin by supposing the electron to exist in theCoulomb field of the oxygen atom where the field isgiven by

Z r L3 p(0)J2V= -~~dr. (1)r reJ

The wave function in the integrand of Eq. (1) is theeighth-order determinant corresponding to the 3 P

ground state of oxygen and which we obtained from

I S. Chandrasekhar and F. H. Breen, Astrophys. J. 104, 430(1947).

G R. G. Breene, Jr., and M. C. Nardone, Phys. Rev. 115, 93(1959).

our small variation program.7 The evaluation of Eq.(1) permits us to set up the Schrddinger equation forthe free electron. Having done this we suppose oursolution to this equation expressible in partial waves

X, (r)le= E-Pi (6). (2)

r

It has been demonstrated several times and againby us6 that the Born approximation for Xl(r) is quitesatisfactory for 1l 1. Therefore, after substitution ofEq. (2) in the Schrbdinger equation, only the numericalsolution to the s-wave equation need be sought. Wehave programmed the solution to this equation on theIBM 704 and s-wave functions have been obtainedtherefrom as required. The normalization of thesefunctions is obtained by fitting the machine solutionto (1/k) sin(kr+5o) at eight atomic units of lengthfrom the nucleus. Here k is the electronic linear mo-mentum in atomic units. These, then, are the wavefunctions which are to be used in our free-free matrixelement calculation.

In obtaining any free-free cross section for a transitionarising from a particular lower state the electric dipolematrix element for the transition in question mustfirst be determined. In obtaining such a matrix elementbetween two wave functions of the form Eq. (2) it isapparent that every partial wave I may only combinewith a partial wave 1/i1. Thus, for example, an s wavein a lower state of momentum ko will combine with ap wave in an upper state of momentum ki. We choosethe dipole acceleration form of the dipole momentoperator. This is equivalent to the derivative of Eq.(1) and, approximately, it goes to zero only a few unitsfrom the nucleus.

From Gaunt8 we may obtain the following expressionfor the cross section for the absorption by a freeelectron in a state of energy ko2/2 a quantum of energyAk 2 /2[= (k1

2 /2)- (k/2)]:

2.5441 X 10-41 a (ko; zAk2

) = .41X 0i (k 12

1I (1,k 2 1 IIl- , k12) 2

ko2k1 (Ak2 )3 1=1

+ k,2| (I -1, k'o21 | |,ki 2) 1 2} )CM5, (3)

where the matrix elements are to be given in atomicunits.

Now we really need consider only the first term inthe sum of Eq. (3) as Chandrasekhar and Breen5 haveillustrated and as we may point out qualitatively. Byabout two units of length from the nucleus the deriva-tive of the free electron potential has fallen to zero.Therefore the contribution to the matrix element mustnecessarily arise from values of the integrand lyingwithin this radius. For p waves and higher we will be

7 R. G. Breene, Jr., Phys. Rev. 111, 1111 (1958).8 J. A. Gaunt, Phil. Trans. Roy. Soc. (London) A229, 163

(1930).

Vol. 501112

FREE-FREE CONTINUUM OF OXYGEN

dealing with products of the form Jp+Jp+a. If we recallthe form of these functions near the origin it will beclear that their products will contribute little to theintegral within the required range of the potentialderivative. Therefore, all terms in the sum of Eq. (3)save the first may be neglected. Thus, the desiredsum is

7rkoklr dV2 =_ J J (kor)-Xo(kir) (r)tdr

2 fJ dr

6-

4

0.05 0.10 015: 0.20 Q.25 0.30 0.35 0AO405 0.4Q50 0.55

FIG. 2. The Maxwell-Boltzmann distribution for T=800 0 'K.-7rk 02 ki dV 2

2__ J xo(kor)-J1(k1r)(r)2dr . (4)

In Eq. (4) the Xo(kr) are the s-wave functions whichwe have obtained from our free electron calculation.The computation of Eq. (4) has been programmed forthe LGP-30, and all matrix elements to which werefer have been evaluated thereon. The relationshipbetween Ak2 and X is given by

X =911.3/zAk 2 A. (5)

It is most convenient to obtain the matrix element asa function of wavelength and arising from a givenlower state. As an example 1i

2/k02k 1(Ak 2)8 is plotted in

Fig. 1 against wavelength. We illustrate only the oneko, but it is to be understood that we have evaluatedthese as required. To determine the ko's that must betreated we must first decide on the electronic distribu-tion which will govern the initial population. A simpleMaxwell-Boltzmann distribution is assumed so that

5xI03

to3

5XlO 2

1j

to

IZ

V4.

10

I I I I I I

200 250 300

the probability that we shall find an electron in a stateof energy ko2/2 is

f(kc 2)= 197.80 koexp(-31.320kc 2); =5040/T, (6)

which is precisely the expression used earlier byChandrasekhar and Breen. In the remainder of theseconsiderations we shall occupy ourselves only with atemperature of 800 0'K. Figure 2 is a plot of Eq. (6)for such a temperature. Now an inspection of thisfigure indicates that the choice of ko=0.04, 0.15, 0.25,0.35, 0.45 should cover the energy region of preponder-ant electron population. The choice appears obviousenough, but, even if it were incorrect, we would berather forcibly reminded of this as the calculationcontinued.

For each of these five lower state energies, then, wemust compute sufficient matrix elements-Eq. (4)-tocover the spectral region of our interest. This, ofcourse, leads to a Fig. 1 for each of the five lower states.

III. CROSS SECTIONS CORRESPONDINGTO CONTINUUM FREQUENCIES

We have now obtained free-free cross sections eachof which corresponds to a particular initial electronicenergy and a particular energy absorption. For aparticular continuum frequency we now wish to weightthe cross sections for the same continuum frequencyand the various initial state energies by Eq. (6). Inthis manner we will obtain the desired free-free crosssection

cT(AkZ2) = a (ko2 Ak2)f(ko2)d(k2)

5.0322X10- 4 0 rp A2- I ko

(Ak2)3 f

Xexp(-31.32 k0ko2 )d(ko 2 ). (7)

In Fig. 3 we give an example of the numericalintegration of Eq. (7). Here we have plotted

5.0322X 1 0&t2ko exp(-31.32otko 2) (2ko)FIG. 1. The matrix element curve for

transitions arising from ko=0.25.(8)

(Ak 2)3 kIk

50 100 150

X Xloo

November 1960 1113

20 ?

11 . G. BREENE, J 1., AND MI. C. NARDONE

110

U

100l

X (A) x /00

FIG. 4. The free-free absorption cross section in the presenceof neutral oxygen. The solid curve is the result of smearing thecross sections over a Maxwell-Boltzmann distribution of lowerlevels. The dashed curve is obtained by supposing all electronspossess the mean velocity corresponding to a temperatureof 8000'K.

that the cross section will be 9X 10-" units larger at4000K than at 8000K. On the other hand it is ap-parent that if we accept our value of the cross sectionat this wavelength, the Kramers equation would predicta difference of about 10OX 10-39 units between thecross sections at 4000 K and 8000 K. It appearscorrect to say, then, that there is no temperaturedependence in the cross section in this region.

Finally we suppose all electrons to be moving withthe mean velocity (7rkT/2m),. The cross-sectioncalculation was then carried out with the inferredneglect of smearing. The result is plotted in Eq. (4).We see that, although the resulting cross section doesnot differ too greatly in magnitude, the wavelengthdependence is somewhat smaller.

400

300

0.40 045 0.50

FIG. 3. The individual cross sections smeared over a Maxwell-Boltzmann distribution of lower levels for 10 000 A.

against ko. We see that our choice of ko values isapparently a satisfactory one. Graphical integration ofthe plot yielded the desired cross section.

This procedure was followed for the wavelengths5000, 8000, 10 000, 15 000, 20 000, 25 000, 30 000 A.The results yield the curve of Fig. 4.

We recall the familiar classical treatment of Kramersleading to the cross section

3.69X 1010 = Z2 . (9)

(T)Yvi

If in Eq. (9) we assume Z=0.406, we obtain ourvalue of the cross section at 10 000 A. In like mannera value of Z=0.347 at 15 000 A and 0.318 at 20 000 Aresults in equality between Eq. (9) and our results.

We obtain about twice the Hammerling and Kivelresult,3 although our frequency dependence is approxi-mately the same. Even so the Kivel et al.4 experimentalresults are such as to allow agreement with our largervalue.

IV. TEMPERATURE VARIATION AND A MEANVELOCITY OF ELECTRONS

The calculation which we have described wasrepeated for temperatures of 6000 and 4000K. Thecross section is very nearly the same; certainly it doesnot contain the T` dependence indicated by Eq. (9).For example, at a wavelength of 25 000 A we predict

-i39

2.5

2.0

I.5'ci

.4e

0 Q0 QI0 0.15 o20 0.25 0.30 0.35

I -

I I I I I I I I I

1114 Vol. 50