396

17.8.4 Effect of Focusing and C hanneling on the Number of D isplaced Atoms

I f in the c ourse of formation of a cascade a recoil becomes channeled or develops in to a dy namic c ro svdion. the k inetic energy of the rec‹›il is lost to the ca.evade; i.e., its energy is transformed to heat through electronic stopping or su bth resh o I d atomic collisions. "I'he probability of the occurrence of a ‹'ry stat effect is a fund tion of recoil energy. "the notation P ( E ) is u md for either of the pro babilities Pt or P,. , . However, the effect of focused collision sequences on the displacement cascade is quite sma 11 owing to the u pper energy limit E t of 100 eV in the focu sing process. ’l'he basic integral equation governing i'ascade r»rinaticin

c'an be mod ified to accou nt for cry stal ef feels by amending Eq. 1 i . fi 5 to

i ( E I - P ( E ) * [ 1 — P ( E 2E, 2 /' r ( "I’ ) dT ( 17. 11 1 j

The first term on the righ t represen As the lone displa‹'ed atom ( i.e., the PKA itself} wli ie h resu Its if the PKA is

‹'han neled or focused on its fi rst collisio n. The set ond term, which is weigh ted with the probability 1 — P{ E ) , gives the number of displacemen Cs ‹ reated by a PKA that ma kes an ord inary d isplacing fi rst collision. T his equation t'an be solved by thP method used in the pres ion s set tion if the probability P is assumed to be iudependen t of enrrgy . Tak in g the derivative of E q. 17. 11 I sv il h respe‹ I Its Ei then y ields

whi‹'h c an be in tear:ited t‹› gi ’e

CE' ' " ’ — P " ' 1 — 2 P

'I'he integration const an I C can be found by substitu ti ng this so lu tion into Eq. 17.111:

The c'onip lete srilu tion is therefore

pr‹iduced by a single PKA that receives energy E from a collision with the bombarding particle. 1 n this secti on the su pply of energy to the atoms of a metal from fast neu Iron s is con pled with cascade theory to permit calculation of the rate at which va‹'ancies and interstitials are produced in a

.specified neu Iron flu x spec trum. N‹i a‹ ct›un t is taken of the reduction in the n umber of displacements due to recom bi- nation \ v ithin the volume of tht ‹ decade.

Let ‹i„ ( E„ , E ) d E be t,he differential energy-transfer c ross section for the production of PKA s sv ith energies in ( E,dE ) due to neu Irons of energy E„ . Eat h PKA gors on to produce r ( E ) d isplaced atoms. I f the differential neu tron flux is O ( E„ 1, flu rate at wh ie h at‹ims are displac-rd is

x ‹ „ ( E„ , E ) r ( E ) d E d isplaced at‹›ms ( 17.113 j

’l'he energy tr‹insfer parameter. . \ , is given by Eg. 1 7. H, u'1iicli. for th e case of neutrons, can be iv ritten

( 17.114 )

where A is the mass nu mbcr of‘ the lattice atom in atomic mass units. ’1’he u pper limit o n the inner in tegral of Eq. 17.11.1 is the maximum-e nergy P KA that can be produk ed by a neu tron of energy E„ , and the to ever 1 imit on the ou ter integral is the m inim um neutron energy that produ ces a P KA tif energy E,; . Neu trr›ns of energies less than Ei,; /. \ t whi c'h is abo ut 200 eV ft›r the major ‹'on sti tu ents of stai rile ss steel ) create no displacements by elastic c‹illisiOns sv ith the n uclei of lat tice atoms.

Therefore, thermal neu tron s {mean energy - 0. 1 eV ) are incapable of causing damage to strut tural or c Padding metals by d irect colli sion energy transfer. H owes er, thermal neu trons ‹!an t'au se display ements by becoming absorbed in a nucle u s and producing a radioactive species that decays by eini si‹in of a high -energy gamma ray . The decay- product atom recoil s from th is ei'ent with su fficient energy to disp lace itsei r and per haps a few other lattice a toms. We do not treat this process here, in asmu ch as the .s‹ altering collisions between lattice atoms and energetic neutrons are

1 — P E $* 1 — 2P \ 2E,t /

“ P 1 — 2P

( 17.112 )

far more important in fast reactors than is the damage caused by capture reacti ons involving slow neu trons. Problem 17. 7 at the end of the chapter dea 1s with the recoil

fi quatiun I 7.112 was first obtained by Oen and

Robinson.' Equation 17. 112 reduces to the Kinchin— Pease result ( Eq. 17.68 svh en P - 0. 'the crystal effect ( principally chan neling ) is most important for large P KA energies, whir'h simply reflects the greater n umber of recoils su e ptible to loss from the cascade by th is means. For P - 7 , for example, a 10- keV P KA in iron produces 100 di.sp1aced atoms according to Eq. 17.112. When channeling is neglected, twice this number is generated.

17.9 DISPLACEMENTS AND DAMAGE IN

energy of latti re atoms that become rad ioactive by virtue of neutron capture.

17.9.1 Displacement Cross Section

Equation 17.113 can be written in terms of the displacement cross section:

Rm ' N i d ( E ) p ( Eg ) dE ( 1'7.115 )

w here nd is

A FAST-NEUTRON FLUX

d!* › ,

d

* " v ( E, ,E ) r ( E ) dE ( 17.116 )

Up until this point we have been concerned with the methods of calculating r ( E ) , the nu mber of displaced atoms

The displacement cross section can be comp uted if the nuclear scattering cross section for neutrons w ith the

elemen I comprising the lattice is k nown. As mentioned above, r ( E ) m ust be k now n as well. Graphs giving d as a function of neutron energy can then be constructed for each nuc fide ( isotopes included ) contained in an alloy su ch as steel or zircaloy. This graphical information can then be combined with the neutron-flux spectru m characteristic of the particular location in the reactor in wh ie h irradiation

397

the di fferential angular cross sections for the scattering reactions by use of the first equ a lily in Eq. 17. 2 2:

d ( co s dE

d ( co s 0

occu rs in the same manner prescribed by Eq . 17.115. I n this way the results of experiments conducted in one flux spectrum can be used to estimate material be havior in a

* m in

dE ' rt E ) dE ( 17.11b l

reactor with a different neu tron-fJ ux spectrum.

T he scattering of fast neu trons by the nucleus of a lattice atom can be elastic or inelastic. I n elastic scattering the nu cleu s of the stru ck atom is not excited to a h igher energy state as a result of the collision; k inetic energy is conserved in the scatterin g event. In inelastic scattering the nu cleu s recoils from the collision in an excited state. The ex citation energy , Q, is provided at the ex pense of the k inc tic energies of the scattered neutron and the recoil ing n u cleu s; total energy rather than kinetic energy is conserved in the collision. Inelastic scatterin g become s tmp or tan t when the neu tron energy becomes just a bit larger than the excitation energy , Q. When Eg Q, inelastic scattering is energetically impossible. ’l“he lowest ex cited state of the nucleu s generally has an energy of 1 Me V a bove the gr ound state energy.

I n inelastic scattering, one neutron is ejected from the nucleus fur each neutron absorbed. At higher neutron energies the n ucleu s may be left in such a high ly excited state as a resu It of m omentari ly absor b ing the bombard ing neutron that two neu Iron s are emitted in the decay of the compound nucleus. 'l'h is interaction is the ( n, 2n ) reaction. flecau se the fl ux of fast reactors is low at the th resho ld energies of the ( n, 2n ) reaction, the contribution of th is reaction to damage is sma ller than elastic or inelasti c neu Iron scattering.

'J'he angular dependence of the elastic-scattering cross

section can be written in a series of Legendre pol \ ’norn ials:

where o, t t E n ) is the total elastic-scattering cross section for a neu tron energy E„ , P is the Uh Legendre polyn omial, and values of a I are the energy -dependen t coefficients of the cross-section expansion . A I the neu tron energies en-

countered in fast reactors. it is su f ficien t to retain onl v the

1 0 and 1 = 1 terms in the series expansion of Eq. 17.11 fi. Since o ' 1 tlHd P , = cos fi , eve can sv ri ie

| 1 * a , ( E„ I cos 0 ] ( 1 7. 1 20 )

where at has been set equal to unity for normaliz ation and a , ( E . i represents the degree ‹if anisotropy ‹i f the el as ti‹' scat tering reaction. I I a i = 0, the di fferen tial ‹'runs se‹'ti‹›iJ for isotropic elastic scattering is recovered.

\ Vhen s‹ attering is elastic, the angle— e nergy tran.sforrna- t iota derii ative is given by Eq. 17. S w ith 4’ a nd E replaced b ’ Rd n respecti vel j .

Neu tron scattering can also be c haracterized as isotropic or a nisotropic. I n inelasti c scattering the incident neutron is first absorbed by the nucleu s, and the scat tered neu tron

d ( cos*V

dE

( 1 7. UI )

is in reality emitted a very short time later from the com ptiund nucleus. Because absorption precedes reemission of the neu tron, the angular distribu tion of the inelastically

.scattered neu trons is to a very good approximation iso-

tropic in the center-of-mans sy stem.

Equation 17.121 i ••n d r both isotropic and an isotropic

elastic scattering.

Since inelastic scattering is isotropic in the center-of mass svstem , o„, I E„ , ( 7 l simplifies to

Below about 0.1 Wle V, elastic neutron scattering is also

isotropic in the cen ter-of-mass system. A I h igher energies,

o„, ( E„ ,0 I - u;, ( E )

4z

( 17. 12 2 )

however, the elastically scattered neu trons have a distin ct forward bias. T his phenomenon is k nown as p -wave scat tering.

To explicit ly accou nt for elastic and inelastic neu tron scattering, we can write Eq. 17.11G as

u here • i . i E n ) is the to tal inel as tic-scattering cross section.

The inelastic-scat tering process can excite the struc k

nu cleus to a num ber of discrete levels having energies Q, above the grou nd state or to a con tinuum of levels at high energies. For sim plicity , we treat here the case in wh ich on ly a sin gle discrete state with ex ci tation en erg y Q is produced.

Because the recoil in g nucleus has absorbed energy in

X u q ,

( Eg .E l r ( E ) d E ( 1 7. 1 17 )

the collision, the el asti c-scattering form ula relating energy transferred to scattering an gle, Eq. 17. 9, is no longer valid.

where u,., tE„,E} ‹ind u„, ( E„,EJ are the dirrerential energy'-transfer cross se‹-tions for elastit and inelastic neu tron scattering, respective 1 \ ', and E„, i q i : t nd E,p „ , are

the limiting recoil energies in the inelastic-scattering

process. Equation 1 7.117 can also be w fi t ten in terms of

Instead. the collision k inemati cs must be based on conser va- tion of total ( rat her than kineti c ) energy , iv hich re.en Its in addition of a term Q to the righ t-hand side of Eq. 17. 4. The analog of Eq. 17.5 for an inelastic collision wherein the stru ck nu cleus retain s an energy Q is

398 F I “.IDA 3IEL'TA L ABS PF. C TLS O F .4 UCLE A R REAC TGR F €'F. L F. L F. hf FNT!S

E- 1 . \ Eq 1 — ' A Q

2 2A E,

— 1

1 + , j Q ‘i

A Eq

C O S 0 ( 1'7 . 1 23 )

Except for resonances, the elastic scattering cross sec- tion, w e t ( E n ) , is more or less constan t with neu tron energy. The inelastic-scattering cross section, . however, sharply

increases with energy above the thresh old ( E n ) p ip . The anisotr Op y factor a, ( Eg } tends to decrease the displace-

w hich reduces to Eq. 17.5 if Q - 0. T he maximum and miniirium recoil energies are obtained by setting cos 0 equal to — 1 and 1, re cspectively:

1 + A Q

2A E„

ment cross section because forward scattering transfers less energy, on the average, than does isotropic mattering. If both inelastic scattering and anisotropic elastic scattering are neglected and the elastic-scattering cross section is assu med to be ener gi' indepen dent, Eq. 17.129 reduces to

m d ( E„ ) — ( 17.130 )

E,p , q — 2 \ E„ 1 1

— 1

( 17.125 )

I n this sim plest of cases, the displacement crO ss section increases linearly w ith neutron energy.

Inasm uch as \ Eg , 2 is the average energy transferred to t he lattice atom by a neutron of ener E p , the coefficien I

. \ Ep 4E d is the average nurn ber of displac'emen ts produced by‘ a neu Iron of ene rgy E„. For 0.5 -Me V ne utro ns in iron

"the th resh old energy for production of the excited state is given by the requiremen t that the term under the square- root .sign be greater than zero, or

( A - 5 S ) , the displacemen t cross sectio n is - 350 times larger than the nu clear scatte rin g cross sect io n. The t‹i fat displacement rate for th is case ca n be obtained by inserting Eq. 17.130 into Eq. 17.115:

where c,q ( E„ } is zero for E p ( E q ) p q .

2’he transformation from scatterin g angle to energy transfer is

R,-

'

‹p ( M.131 )

d ( cos 0 ) 2

t 1 + A $

17. 127 )

where Eg is the average neutron ei erjjy and ‹I› is the tutal

dE „, . \ Ep A Eq

Substituting Eqs. 17. 120, 17.121, 17.122, and 1'7.127 in to

17.11 d y ields

neu trun flux ( wi th energies abu \ 'e E d ‘.1 ) . For the condi- tions

N =0B5x 10°' atoms,em'

u,. \ - 3 bariJs

‹I› ' 10' ’ neuf rons c‘tn ' se‹‘

_ 2E

. \ E

› ( E ) dE

4E d

- 350 displaced atun \ s,’neutrun c‹›llisiun

x "rr a x r ( E I dE

( 17.1 2b )

we find that R d is S x 10' " displa‹'ed atoms rm ’ see ' Or, div id in g by N, the displacetren t ra ie per at‹ilr ( dpa sec ) is 1 II"’ : eac'h atom in the metal is disp laced from a n Ormal

If more than one excited state contri but ecs to the inelastic scattering process, the last term in Eq. 17.128 is replaced by a sum over the excited states. eat h with its particular u „, , Q, E„, , , and Eg „, .

To proceed further, we m ust specifv rt E ) . .A simple result can be obtained by using the Kinchin—Pease expre+ sion for n ( E ) . Su bstitu tin g Eq. 1 7.68 in to Eq. 17.126 and

neglecting d compared to . \ E„ its the first integral restt Its in

( 17.1291

We ha \ ’e assumed fur illu rra£i \ ’e purposes that The maxi m u m P KA energy . \ E„ is less than the i onization limi I given by Eg. 17.4'4.

lat tice site once ever y 12 days.

A lt hough Eq. 1 7.130 ice useful for illti strat in g the order of magnitude of the di.splacement cross section, it is not suf ficiently accurate for predicting mechanical-pr Opert y behavior u nder irradiation . Doran ' " an d Pierc y' " h ave calculated displace me nt cross sections for stainless steel and z ir‹ onitim, respectively , using the Lindhard model fOr r{ E ) l Eqs. 17. 90 to 7. 9 3 ) and available data on the energy dependence uf the el a.stic- and inelastic-scattering cross sect,ions and the anisotropy parameter a , I Ep 1. Figure 17. 17 shows the d isp laceinent cross .section fOr stainless steel. ’I’he jagged appearance of the cur ves is due to resonances in the elastic' scattering c'ross section.

Figure 17. lb show's the di f feren tial neu tron-flu x spectra in two fast reactors and one t hermal reactor. ’1’he average neutron energy in the all-metal Experimental Breeder Reactor 1 I ( E OR -II ) core is 0. 85 i \ 4e V. 1 n the mixed o x ide Fast 'I'est Reac't Or ( F'I’R l core, the average neu tron energy

D ispla‹ emt nt c russ st 'tion fu r sta i mess steel.

i nor Ref. I S. )

Fig. 1 7.1 8 Go mparison of ne utro n -flu x spectra r« th » reacto rs ( FV R, Fast Test Reactor; Erin -i I, E qriniental Breeder Reactor II; ETR, Engineering Test Reactorl . The FTR and EB R-II are fast reacto rs; t h‹• ET R is a t hermal reactor. Th rissio n-ne ut ro n -energy spectru m is show n f‹ir comparison. ( After \ V . N. Me El ro y and R. E . IN ahl, Jr., ASTM Special Technical P ublicatioii No . 184, p. 37 5, American So ciet y for Test ing and materials, 197 0. )

is 0.45 MeV. The fission neutron spectrum ( average energy = 1 Me V ) is sh ow n fo r comparison. To compute the displacement rate in stainless stee 1, we m u hip ly the cu n'e of F ig. 17.17 by one of the spo‹ tra in F ig. 17. lS and in leg rate t he produ ct according to Eq. 17. 115.

17.9.2 Damage Functions

The ultimate objective of calc mating R d is to permit prediction of the extent of a particular mec hanical-property change in a fast reactor from the results of experiments

conducted in irradiation facilities that have considerably different neu tron-flu x spectra. Ty pical mechanical- property changes induced by fast-neu tron irradiation are the yield strength , the ductile-to-brittle transition temperature, and swelling. I t is by no means generally true that the change in any of these properties is proportional to the number of

displaced atoms produced by an irradiation or k nown duration. Although the exten t of vo id formation in metals appears to depend primarily on the number of vacancy— interstitial pairs created by irradiation, mechanical proper- ties such as y ield strength are determined by the clusters of vacancies and interstitial loops that remain a fter the nascent cascade has annealed and the isolated vacancies and interstitial.s have disappeared at the various sinks in the solid. The proper theoretical approach in the latter case is to compute the produc tion of stable point-defect clusters resu lti rro ni radiation, not the total number of displaced atom s. 4’his can be accomplished by replacing r ( E } in Eq. 17.11b w ith the number of clusters that are produced b›' :i PKA of energy E , w hich may be estimated from com pu ter simulations of radiation damage. The restil ting

rate of cluster formation, R . ...‹. ., should be a better measure of the damage ( i.e., the y ield-strength change ) than is the rate Of formation ‹if total displaced atoms, R„ .

Ca l‹ ii lati ons of lh is sort have been performed by R usscher and Dahl . 2 '

’l'hese completely tliet›retical attempts to predi‹ t some iniv rose opic property of radiation damage t e.g., rate of formation of di.splaced atoms or rate of formation of clu hers ) .me n‹it su ffi‹'ient to correlate macroscopic prop- erty changes in reactors of di f ferent flux spectra primarily because other c on sequences of irradiation besides the number uf displac em ents or clusters affect the macrosc‹›pi‹ property in question. 3’hus, although void formation certainly de pends ‹›n th e rate of producti on of vacancies rind interstitia I atom s b y radiation, it is also a fun‹ tie o of the' quantity ‹if hel iti in gas generated by ( n,o 1 rea‹'tions in I he metal bt•‹-au se hel rum appears to be ne‹'essary to stab ilize em bry‹i v‹i ids. t'a lc'ulation of the disp la‹'eirient rate R,; , rio matter lit›iv a‹ ‹ urate, provides no inf onnation on the hel iu m-producti on rate.

Becau se of the ina bil ity of displacement calculations to cope with the comp lex ity of most macroscopic radiation effects. a semiempirical method, k now n as the da mage ( u nc I icon me lhod, has evolved. 2 2 In this method the rate of displaced-atom production appearing on the left-hand side c›f Eq. 17. 115 is replaced by the change in a partic'ular macroscopic property in a time t of irradiation, and the di.sp1acement cross section on the righ t is eliminated in favor of a function G ( E q ) , w hich is to be determined. The damage fun‹ tion ror the particular mechanical property is I, ( E„ ) . Th u s, Eq. 17. 115 is replaced by

In th is eqn a tion, AP; represents the change in the proper ty labeled by the index i during an irradiation of time t in a neutro n flu x +. The spectru m of the riux in the irradiation facility is O ( Eg ) . The equation has been multiplied and divided by the total neutron fiu x

l- u ( E, ) dE, ( 1’7.133 )

so the ratio d ( E q I J fi ( Eg ) dEp is a normalized flux sped tr mm. T lie produ‹ t 'let is the total neutron fluence.

'I he term G ( E ) is the damage fu nction for property i for neutrons of energ Eg. The conditions under which the

400

property Pi is measured after irradiation and the conditions ( exclusive of the neu tron flu x I during irradiation must be carefully speci fied. 'the damage fu ncti on depends on these iionneutronic conditions. For instance, if P, is the yield streng th of a part icu lar metal, the temperature at wh ich the irradiatio n and the .su bseqtient mechanical test are carried on t must be k now n. "the derived damage function may chan ge if either of these tw o auxiliary conditions is at tered.

The technique for obtaining Gi ( Eg ) is to measure AP; in as many dif feren t ( bu t k now n ) neu tron-flux spectra as possible. O ne then at temp ts to deduce a single function G,l Eg I from the data obtained in eac h irradiation by using equations of the form give n by Eq. 17. 13 2. 'l'his process is called damage-function II nfold ing. Deduction of G;l E" I from a set of mea.sured IP, value.s i n different neu tron-fl u x spectra is ana Ing ous to the determination of the flti x sper'trum of a reactor by activation of foils of a nuns ber of neutron absorbers of different energy dependent capture cross section.s. The damage function is determined by iterative solti tion of the set of equati on s given by Eq. 17.1 3 2; a first guess of G ( E„ I is inserted into the set tif integrals, and the ‹-al ‹'u lated property changes IP, are compared w ith t he measured val ues. 'The function G, ( E„ ) is then adjusted, and th e calculation is repeated un til the measured property chan ges are reproduced as closel \ ' as possible by the in tegrals on the right of Eq. 17. 1 3 2.

1 n th is process both the n timber of iteration s required and even the accuracy of the damage fu nctit›n ultimatel

‹obtained depend on the ‹ivailabilit ' of a gotid first gues.s of t he damage fu n ction. "F he best initial estimate of G, ( E" is the di splat'ern ent t ro s ser‘tion o, I Ep ) o n the as sum pt itnl I flat t lie damage I i .e.. the r!hange in the rnec han ical p rt›perty in questit›n 1 sly ou ld be rough ly’ proptirtional to the n i mber of displaced a turns.

Figure 1‘7. 1é ( aj sho \ vs the damage function for UJe

yield strength and swell ing of .stainless steel determined bv the met hod described above. "rhe u nit.s of the dama ge function s are those of the pr‹ i p •• y I › '' eld strength in k ilo New to rms per sq care meter ( kN ' m° 1, swelling in percen t ( " ) l divided b the total neu tron fluence ( u nits of neutrons/ cm ) . Eiach damage function was determined from tests

conducted in several different reac tors w ith different flux spectra. 4’he dashed lines in the graphs are the displacement cross section of F ig. 17.17 extended to lower energies than in Fig. 17.17. The increase of uq ( E„ } a nd Ci, ( E„ } at neutron energies below 1 0* â4 eV is due to damage produced by recoil atoms activated by ( n.j ) reaction s iv ith slow neutrons ( the cruss section s for cap ture reaction s are proportional £o the inverse of the neu tron speed ) . A lt hough the damage fun ction is appreciable at very low neu tron energies. the property change IP, is n ot greatly affected by this low -energy tail of G; ( E„ ) because the flux spectru m of fast reactors co mains relatively few low-energy netl trons ( Fig. 17. 18 ) . T he insensitivity of damage to low -energy neu trons is reflected by the bread th of the error band for Eg 10 ñIeV in F ig. 17. 19 ( a ) .

’f'he y ield -strength damage fu nction is very t lose to t he

displacement cr oss section used as the input first guess of Gi ( Eg J. This accord implies that w hatever features of the displacement cascade are responsible for an increase in the strength of irradiated steel are at least proportional to the nu mber of displaced atoms. The damage function dedu ced

Fig. 17.19 Llamage l'tinctiuns for two radiation effects in 304 stainless stec•l. ( a ) Yi c'l c1 strrua gth for irradiation and test

£c‘mpc‘ratures ‹›f 4 HO“ ñ. | Frc›m R. f.. him mo»s et al. , \ ’«c/. 7’t’‹'// izu/., 1 6 : 11 ( 10 ? 2 l. | I h ñwellin g at t EU“ [ Frum

R. L. Simnio li.s ct at . , '/’i’aiix. I tile r. .Vt«-f. .5o‹ ., 15: 2 4 9 ( 197 2 j.

from the initial que ss G, ( Eg j t onstant is sh ow n as the dotted curve in Fig. 17.19 ( a ) . ’I’his curve is vastly dif feren t from the damage fu ncti on obtained w ith the aid of an in put displacemen t function, for w hich the initial guess is G, ( E„ ) o„ I Eg ) . T he dotted curve is incorrect and reflects the stringent requirement of a good first gue ss if the iterative method is to co diverge to the correct damage fu nction.

Figure 17.191 b} shows the da mage functi on obtained for stainles+stee1 swelling due to void formatio n. T he damage function for this property c hange is similar to, but not identical to, that for the y ield stren gth.

17.9.3 Damage Production by Ion Bombardment

'the e stent t› f radiation d amage produced by ex posure of a structural metal to a fast-ne utrun flux depends on the duration of i rrad iatio n . The damage in creases w ith the fast-ne ut ron flue nce , w hich is the prod uct of the fast- neutro n flux , '1 , and the i rrad iation time, t. The economics of nu clear pow er requi res that the fuel of commercial fast breeder reactors remain in service for a fluence in excess of 10° neutrons /cm 2 ( i.e. , for a year at a flux approaching 10' 6 neutrons cm 2 sec 1 ) . Accurate assessment of the durability of structural metals for use in L6IFBR cores requires that the radiation effects produced at these

RA DIA TI ON DANiA G E

fluences either be measured directly in an irradiation facility where the expected fluences can be obtained or be extrapolated from tests at muc h lower fluences by recourse to an appropriate theoretical model. Acceptable theoretical models are often not available for particular radiation effects, and son nd fuel-element design can be achieved only by testing to the expected service fluences. This situation applies to swelling of the cladding due to void formation;

no accurate theory is available for prediction of void

E L E CT RONIC

40J

production, and extrapolatio n of low-fluence swelling data is risky because the pheno me non is not linear with fluence. Even if adeq nate irradiatio n facilities w ith a fast- neutro n flu x of 10' neutro ns cm ' sec ' were availab le, 3-y ear-d urat ion tests would be required to at tain the design fluences of a n LMFB R core. I n a test facility with a flux of

ION

E NE R GY

STOPPING ATOMIC STOPPING

I I

PE N E T R AT ION

10 ' neutrons c m 2 sec*' , 30 years wo uld be necessary. There is therefore a great incentive to dev ise irradiation tests that can simulate fast -neutron damage at fl uences of

10 2 neutro ns cm 2 in a reasonable amount o f time ( say of

the order of days ) .

Bo mbardment of metals by energetic heavy ions has proven to be a useful tool for compressing t he time scale of irradiation tests by many orders of magnitude. Reasonable currents of H*, C’, and metal-ion beams o f energies from 1 to 10 Me V can be obtained from accelerators. Because the range of heavy ions in solids is quite small ( t y pically 10 km ) , all the initial energy o f t he io n can be dissipated in a small volume of the specimen. Since thP nu mber of displaced atoms in an irrad iat ion e xperiment is a reasonable measu re of t he e xtent of radiation damage, w e calculate the

R p

I b )

DEPTH

rate at w hich a beam of energetic heavy ions causes lattice displacements and compare this figure with that attainable in fast-neu tron irradiatio ns.

Figure 17. 20 shows some features of ion sto pping in solids. In Fig. 17. 20 ( a ) a beam of ions enters a solid target wit h energy E,g . The ions slow dow n in the solid and come to rest at a depth given by the projected range. Figure

17. 20 ( b ) shows the energy-loss characteristics of the ions while traversing the sol id. Because the incident energies are in the million electron volt range, electronic excitation is the principal energy-loss mechanism over most of the range. Figure 17. 20 ( c ) shows schematic plots of the electronic and atomic stopping powers as fu nc tions of ion energy. The electronic stopping power is based on Eq. 17. 5 2, a nd the

Fig. 17. 20 Pat hs and e ne rgy losses of io ns penetrating solids.

unit c ross-sectio nal area and th ick ness d x w hich transfer energy in ( E,dE ) to the atoms in this element is N I ‹i ( E„E ) d E d x. Or, the nu mber of collisions per unit volu me per unit t ime w hich tra nsfer energy in ( E,d E ) at depth x is NI ( Ei ,E ) d E. Now the number o f d ispl aced atoms for each collision that produces a PKA of energy E is r ( E ) . Therefore, the rate of production of displaced atoms at depth x is

. \ Ei

R, ( x ) = N I E g ' o ( E, , E ) r ( E ) d E

atomic stopping power is obta ined by inserting the appro- priate cross section for energy transfer from the ion to the

d ispl aced ato

cm’ -sec

“ ( 17.135 )

lattice atoms into Eq. 17. 29. The ion energy at depth x can be obtained by integrating the electronic stopping- power formula of Eq. 17. 52:

1 2

Ei ( x ) - ( E, j" 2 kx ( 17 .134 )

where E, is given in terms r x by Eq . 17.1 34 a nd . \ is given by Eq . 17.8. Nlult iplication of the abo ve equatio n b y the irradiation t ime t and division by t he lattice atom density N gives t he number of displacements per latt ice ato m in irradiatio n o f fl uence It:

The nu mber of atomic collisions between t he ions and the lattice atoms at depth x ca n be calculated from t he

dpa disp lacemen ts

atom

o ( E„E ) r ( E ) d E ( 17.136 )

following considerations. Let o ( Ei,E ) d E be the di f ferent ia1 cross sect io n fo r transferring em rgy in the range ( E,d E ) to latt ice atoms b y an ion of energy Ei . The pro bab ility o f a collisio n bet ween an ion and a lat tice atom in d x which transfers energy in the range ( E,d E J is N ( E, ,E ) d E d x ( see Eq. 17.19 ) . Since I ions mcm' pa ss depth x per second , the n umber of collisions pe r second in t he volu me element of

Uivisio n o f Eq . 1‘7.136 by the fl uence \ 'ields

dp a at de pth x = J u ( E„E ) r ( F? j d E ( 17 .137 ) ( ions mcm' )

A si mple ill ust rat ive integ rat ion of the right -hand side of Eq . 17.1 37 ca n be obta in ed if t he cross sect io n r ( E, ,E ) is