Chapter 17

Radiation Damage

17.1 INTRODUCTION

The preceding chapters have dealt with some of the observable consequences of fission-fragment irradiation of ceramic fuel materials; succeeding chapters will be con- cerned with changes in the properties of cladding metals resulting from fast- neu tron bombardment. These macro- scopic, obser s'able, and often tech nologicall y crucial results of ex posure of solids to energetic particles are collectively k now n as radia lie ii e//?cfs. The primary, microscopic events that precede the appearance of gross changes in the solid are termed i a dia I in it da i tiagc. This branch of physics attempts to predict the number and configuration of the poin t de fects t vacancies and interstitial a toms} produced by the bombarding particles. Radiatio n-dama ge analyses are not concerned with what the de fects go on to do in the solid—such processes are properly categorized as radiation effects. Radiation damage and radiatio n effects can also be distinguished by their characteristic time scales; the primary e \ ’en ts produced by nuclear irradiation are over in less [han 10 ' I sec after the bombarding par tic le has interacted with the solid. Su bsequen I proces.ses require m uch I on ger times; the diffusion of radiation-produ ced porn t defects to sin ks in the solid can take milliseco nds. The time scale of the nucleation and growth of voids in metals by agglomeration of radiation-produced vacancies iS of the order of mon ths.

The primitive damage-producing processes in volve the interaction of lattice atoms with particles possessing en- ergies far in e xcess of thermal energy ( kT ) . Consv9uentls , the temperature of the solid is of no importance in t he an alj sis or radiation damage. 'l'he processes included under radiatio n effects, however, are concerned with point de- fects, or c lusters thereof, whic h are in thermal equ ilibriu m wit h the host crys tal. T he kinet ics of such processes are therefore hig hl y dependent on solid temperature, w hich invariabl y appears as a Bol tzma nn factor, exp ( —E kT ) , where E is the characteri.stic energ y of a thermodynamic process or a migrator y event.

T he energy transferred to a stationary lattice atom in a collision with a high-energy bombarding particle is of the order of tens to hundreds of k iloelectron volts. This quantity of energy is so much larger than the energy binding the atom in its lattice site that displacement of the struck atom is virtuall y certain . The lattice atom first struck

and displaced by the bombarding particle is called the

;»‘imai x' /z iioc/i-on a In m, or PKA, Because a PKA possesses su bstan tial kinetic energy, it becomes an energetic particle in its oiv n righ t and is capable of creating additional lattice disp lacemen ts. These subsequent generations of displaced lattice atoms are k nown as li ig/ter oi'de r h no cli -o ii s, or rr coil a lo in s. An atom is considered to have been displaced i f it comes to rest su fficien tly far from its original lattice site that it can not return spontaneously. lt must also be outside the recomb ination region of any other vacancy created in the process. The displaced atom ultimately appears in the lattice as an interstitial atom. The emp ty lat tice sites le ft behind by the displaced atoms ( equal in number to the displaced atoms ) ate indistinguishable from ordinary ther- mally produced vacancies. The ensemble of point defects created by a single primary k nock-on atom is k now n as a displace m e itt cascaJe.

The earliest and simplest theory of radiation damage

treated the cascade as a collection of isolated vacan cies and interstitials and gave no consideration to the spatial distribution of the point defects. In the crudest appro xima- tion the number of displaced atoms is computed by approximating the collision partners as hard spheres; the only physical property of the solid needed in th is model is the energy that a lattice atom m ust acquire in a collision in order to be displaced. Many improvements on this simple collision model has'e been made, but the idea of a cascade consisting of isolated point defects has been retained. Hard-sphere scatterin g can be replaced by energy-transfer cross sections based on realistic interatomic potentials. The loss o f energy of a movin g atom by in ter action with the elec trons of the medium, in addition to elastic collisions between atoms, can be added to simple cascade theory. Finally, the sim ple model can be improved by considerin g energy -loss mec hanisms peculiar to the periodicity of the crystalline lattice, the most important of which are focusing and channel ing.

Radiation damage is not restricted to the isolated point de fects produced by the bombardin g particles. Indeed, vacancies and interstitials can be produced so close to each other that clustering of the point defects occurs sponta- neously within the sh ort time required for completion of the primary event. When the distance between successive collisions of a recoil atom and the stationary lattice atoms approaches the interatomic spacing of the crystal structure,

373

37.4

it is clearly in appropriate to infidel the ‹cascade as a collec tion o f iso lated vacancies and in tcrstitials. 1 nstead, a dense cluster of point defects called a disp lu ‹ eiti en I sh› ik e or de file ted non e is formed. because of the proximity of the point de fee As in a displacement spike, the pr‹i bab ility of near-inst an taneous an ni hilation of man y of the vacancies and interstitials produced by tfie h igh -energy c ollisions becom es large. I n fact, the nu in her of point defects that actually surv ive a cascade and are c apable of producing observable radiation effects can be as low as 1’ of the num her predicted by sim ple ‹'ascade theory.

'the ‹ ascade is initiated by a primary k Hoc k-un atom. The• cascade therefore vu n i sts of man y interactions be- tween m cv ing and statio nar y atoms of the same kind . ’I’1ic primary knock -on atom, on the ‹it her hair d, is produ ce d by a bombarding particle arising directly front some nu clear event, pri Hcipall y the fission prot ess. I ii ( erms of daiage- produ cing capabilities, the most im pt›r tan t nuclear partit les are the fission fragmen Is ( in fu el materials ) and fast neu trons ( in the cladding and stru‹:tural In aterials j. Other energetic en batornic partic les, such as electrons, pro toti s, alpha particles, and gamrn a rays, ‹-an also initiate disp lace- ieit cascades. However, these particles are e•ither far less damaging than fission fragmen Is ( e and ) ur are produced in such small quanti ties in reactor fuel elements tha I their cr›ntribution to the total damage is negligible tp and o ) . Only fast neutrons and fission f ragmen Is are c onsidered as born bard ing partic les in this cha pier, and on ly the theo- retical me a tmen I of radiati‹in damage in m‹;liatomi‹ solids w ill be reviewed. f“or practical purposes of estimating damage in reactor ir aterials, the calculations for elemental sol ids are usually simply applied w ith ou t m odificatit›n It› m ul tielemen I sq stems, such as the fuel I U,Pu ) O or the alloy stainless steel.

'rci calculate the displacement rate, we must know the total flu x and the energy spectrum of the bombarding particles. For fast neu tron s the differential flux, Qt Eg ) , is obtained from reactor-physics calculations. 'l'he equivalent quan tity for fission fragments can be obtained from the fission density , F, and a reasonable assumption concerning the energy loss of the fragments in the fuel. 1 f the energy spectra ir of the flu x of bombarding particle s and the energy -transfer cross section for co li isions between these particles and atoms of the lattice are k now n, the number of primary knock -on atoms in a differential energy range c-an be compu ted. T he final step is to use this source spectra m of the primary k n ock -on atoms to determine the total nu mber of recoils, or displaced atoms, usin g cascade theory. Such a com pu tatio n provides the best available estilrate of the damage inflicted on a solid by irradiation for those properties which depend primarily on the presence of isolated point defects ( e ¿. irradiation creep and void growt h ) . O n the other hand, when su ch forints of damage as irradiation hardening or embrittlemeiit are of interest, the size and nu mbcr density of displacement spikes are more important than the concentration of isolated vacancies and in terstitials that have escaped from the spike. I n this instance, analytic cascade theories that predict only the number of d isp laced atoms, no matter how sophisti cated from the point of view of atomic collisions, are not germans. The c'haracteristics of thy cl usters of defe‹ ts

created by a F KA can best be ascertained by cont pu ter sim ulation of the r adiation-damage process.

To predict either the n um ber of disp lac'ed at orns by an a naly tical isolated poin I-defect c decade minded or to com- pute the configu ratiun of a displacement spike by a com pu ter experiin en I requ ires that the in teratomic poten- tial between atoms of the solid be k now n. A great deal of in formation on atoiN ie interaction potentials has been obtained by anal ysis of the equilibrium properties of a solid ( Chap. 4 ) . \ Tnf‹›rtunateIy , these potential functions repre- sent the i n[eracti on at separatiun distances of the urder uf a lattice c'onstan I, whereas the pu ( ential at much smaller operations is rele \ ’an I in radiation-damage c alc•ulations, whi‹'li ii vul \ ’c‘ mutli higher partit ie energic's. ñt›r vc'r \ ’ h igh energies Thu toll iding atoms approach each usher su closet}’ that the bare init lei interac't in a tuan ner pres ribed by a Coulom b potential. I n the encrg y range c harac teriz ing most of the collisions respt›nsible for cascade production, how ever, the n uclear ‹ harges are partially reened by the at untie- electron.s, and no r oinplett•l y sati sf actor y iti ter- atoiiiic potential describes the in terat ti on. The screened Con lomb po ten tial ( sometime.s t alled the i3o hr ptitential ) , the in verse power law potential. and the Born—hlayer pO ten tial are frequent 1 y used. l3ecause ‹if the cumpu National difficulties involved in dealing w ith potential functions that lrad to nonisotro pie' sc at terin g in the center-of-mass sq stein, these potentials are uften used unly t‹› compute the rad ius of the eqn ivalent hard sphere ‹'harat'teri zing the coll ision, but they coll isio ri dy nannies are determine d from the hard-sphere in odel ( which gives isotropic .'«-uttering iri center-of-mass coord inates ) . I n radiation-damagv calcula- tions only the repulsive p orti‹in of t he in teratomic potential fu nction is needed. The attractis'e forces between lat title at‹ims, which are impor tan I in the equil ibriu m proper ties t›f the solid, play no part in the esents asso ciated with radiation damage.

The interaction between a mt›ving atom and the lattice atoms is almost universall y treated as a sequence of two-body elastic coll isions. 'the binary -coll ision assumption is quite satisf actory at high inte'ra‹‘ti‹in energies becau se the approach distances givin g sii bstan tial energy transfer arv very m uc'h .smaller I man the distances betivee n lattice atoms; thus the collisions ‹-an be considered to occur between isolated pairs of at oms. A t energies approaching the thresh old energy f‹ir displacemen I, however, the cross section for atom—at tim interaction is large, and the incomin g atom can interact w ith more than one atom at the same time.

3’he co llisi‹in f›etween a recoil and a lattice atau is often assu m vd to be el as tic, whit h mea us that k inetic ener ’ is conserved in the event. Inelastic it}’ can arise frum exci La£iun ur ionization of the orbital elecfruns uf the at‹›ms invulved in the r-‹›llisiun. Indeed, iilteractiun of moving atuins ur iuns w itlJ the elec ( rons ‹›f the sulid c‹›nstitufe the majur energy -l‹›ss prune a £ high c•nergie . 4’ransfer c›f energ y from the moving ation to electr‹ans does not lead tO displacement, only to heat; the low electr c›n mans mean that the} ‹ arr}' little’ mumen turn e \ ’en £li‹›ugh they may be q uite energetic. Co nsequentlj , it is impor tant to be able to estimate the degree to wh ic'h the energy of a recoil atom is partiti‹›ned bet ween elevtrt›nic ex ‹ italian and

lb..l .I/.1 C• /: 375

ela.stic at om -‹itom collision s. O my the energy transferred in the lat ter p rracess is assail a ble for cau sing displa ce nents. Energy is transferred to thr electrons in .small in creme nts .so closel y spaced that I he process can be regarded as a r'onti n ur›tLs 1 ass of en r'rg \ b \ I li e muvi ng ‹itr›m. '1’hc' at r›ni con tin ties to travel i n a str‹ii gh 1.1 i in' flu I sl owe down as il‘ it is e re pas.si n g th rough a i is‹ one ni‹ di u m. 'I'hr ‹it o m— atom i n t v r‹ic I i ‹i us, o n th e o I li er It ‹tn d, ‹›c-t' ur at u'i del j s}iac't'd int‹ ri als, transfer a .signific'ant p‹›rt i ‹›n ‹›f tht' init i:it k i nr'ti‹ t'ilt'rg \ ‹› r I lir‘ mc \ i i y ‹ttr›m i n an t‘ se »ti ally i nñfal1[aMLcMn

inc i i rig atr›rn can lie a‹'r'u rat ct j se par‹ited i n t o twri p ‹irt s: t 1.1 ri is‹'rt'tt' ct ‹is tic at‹i rn- at‹›ni encou n te re ivli icli b‹i I h r‹ ‹1 uc- ‹' th e tuiertij c›f th ‹' inc'i‹1en I atom and pri›‹lute lat t i‹ t' ri i s pl at'e nu' n Is ‹in ri i 2.1 a e r›n t i It u r›iis |i ruc'e s,s r› 1’ t'lt't'I ri› n ie

ti is pl ‹it‘e nu'n ts.

Not all the energy transrerred to a stationary lattice atom b} a recr›il atom by process 1 is used to displace the former. A substantial portion of the ini tial energy of the PKA is degraded to heat by atom—atom collisions that do not del iver the requisi ie disp lacemen I energy to thr struck a tom. I n this event the struck at orri simp ly rat ties about in it.s lat title st ie, ul timatel y degrading the energy i I rece i \ 'ed i n the e‹il 1 ision to heat.

There are several excellent books deal in g with the subject or radiation damage in a comprehensi \ ’e and detailed manner.’ ' 1 n this chapter only th ose aspects or the theory pertinent tu the perr rmance or n \ tcle‹i rum elements are con sidered. Details of some derivatio us have been omitted w hen I her ‹‘dn be f ound in one of the boo ks dev'oted solely to the field of radiati‹›n damage.

17.2 BINARY ELASTIC-COLLISION DYNAMICS

17. 2.1 Scattering Angles and Energy Transfer

Many useful aspects or b inary collision s can be ob tained iv ithou t k noivledge of the interatomic potential by ap plica- tion of the laws of momentum and energy co nserration. Oniy nonrelativistic elastic collisions are considered. The masses or the interacting particles are denoted by M ; and M; . Particle 1 ( the projectile ) approaches stationary par ti- cle 2 ( the target ) iv ith speed v , . Figure 17.1 ( a ) shows the speeds and directions of the particles berore and after the collision in the laboratory rrame of reference, ivh ich is at rest w ith respect to the observer. "the a naly sis is simplified by transrorming the coordinates from the laboratory system to one that m oves wit h the vel ocity o r the center or mass of the two-particle system. The speed of the cen ter of

Fig. 17. 1 Hi nary coll isio n bet wet n a projt•et il e t› f mass M , and a targ‹‘t part it I‹• r mans M . ( a ) Iab‹›ratr›ry frame or

«•r‹• t‘. ( b ) Center or-mans c‹›crdinatt‘s. ( c ) Vect‹›r dia- gramrelating the vrlocit ies in the I wo c‹›ordinat‹' s}'stems.

the center-of-mass sy.stem are related to those in the laboratory system by

( 17.2a )

u 2 — v„ ( 17.2b )

The directi on of u is opposite to that or , , . The scattering angle in the center-of-mass system is fi .

When the collision is iewed in the center-or-mass system , the recoil in g particles appear to move away rrom eac h other in opposite directi ons. Mo mentu m conservation along the axes of approach and departure yield

Mt u * \ t; u t, • * M; u; t ( 17. 3 ) and conservation of k ineti c energy requires that

mass is given by

( 1’7.1 )

1 M t u M,

2 " 2

M, “ 2 M z u z I 17.4 )

Since the cen ter-of-mass velocity is unchanged by the collision, the event ap pears in the new coord in ate system as sh oiv n in F ig. 17.1 ( b ) . The initial speeds of the particles in

Equations 17.3 and 17. 4 are sari ried only ir

° i i = u i o ' "i o *‹ n u, t - u t ' •'. ..

( 17. 5a )

( 17. 5b )

376

’I'he particle velocities in the laboratory system af ter the collision are determined by vectorially adding the center of-mass vel oci ty to u , t a nd u, t , or

v t = u; t + v,. ,p ( 1 7. Sb )

’the magnitu des of v , t and v j t can be obtained f roin the vector diagrams show n in f'“ig. 17. 1 ( c ) . Application of the law of cosines to the louder diagram y ields

2›’t,m ( 1 — cos d ) ( 17.7 )

where ut t was expressed by Eq. 17. b b in o rder to arrive at the second equality in the above equation. We can eliminate v, from Eq. 17. 7 by using Eq. 17. 1, which produces

Combining these three equ ati cHis and rearranging yiel de

tan ( 17.11a

S imil arly , the law uf sines for the vector diagra m for the recoil particle v ields

wli ich, when com bined w ith kq s. 17. a b and 17. 7, resu Its in the relation

11 7. 11b j

17.2.2 Some Properties o1‘ the Head -On Collision

’The preceding analysis is valid for any nonrelativistic elastic collision for any cen ter- of-mass scat tering angle fi provided the collision partners in the ini tial and final states

Noting that ' E } o

M } * u 2 is the k inetic energy of the

are sufficiently far apart that the interaction energy

projectile and E ; t = S'I, v \ t 2 is the k inetic energy of the recoil part icle, we can w rite the above form ula

E 2 E , ( 1 — cos fi )

'to sim plif y th is notation for subsequent use, we replace E t by E and denote E d by T, the energy transferred to the struck particle by the collision. T he group containing the mass num bers is given a special sy mbol:

between them is negligible com pared to their k inetic energies. During the collision event, however, the separation distance is small, and I he con version of k inetic energy to potential energy is import an t. I n particular, for a head -on collision ( fi s ) , the k inetic energy ( exclusive of the k inetic energy of the center of mass ) becomes zero at the poin I where the particles I urn around and begin to retrace their paths. During a head-on collision, m omentu m conservation can be expressed by

( 17.8 )

and the energy-transfer equ ation becomes

T 1

2

where v t and v; are the laboratory sVstem speeds of the two particles at some point during the collision. The

( 17.9 ) relative speed of the two particles is defined by*

The maxim um possible energy transferred from the moving particle to the stationary one occurs in a head-on collision , for which P - and

( 17. 10 )

I f the particles are identical, A = 1, and any energy between

g = v , — v; t 17.1 2 )

Rearrangement of the above two formulas permits v a nd vt to be expressed as functions of v, „, and g:

( 17.13a )

0 and E can be transferred in the collision.

The scattering angle in the laboratory system, Q t , and

the direction of the struck atom after the collision, o, , can

M ,

+

( 17.13b )

be rel a ted to the scattering angle in the center-of-mass system with the aid of the vector diagram of Fig. 17.1 ( c ) . Apply ing the law of sines to the triangle representing the scattered projectile y iel ds

sin ( r — fi ) sin fi t

where u t t is gii'en by Eq. 17. b ( a ) , from which v, can be eliminated by Eq. 17. 1, giving

u,; =

Apply ing the law of cosines to the same triangle yields

,2 - u 1 f 2 — 2v, u i r cos ( a — 8 )

The total k inetic energy of the two particles is given by

' KE

2 ' '' + 2

or, when expressed in terms of v, and g, by

KE- l 1

2

* I n the general e lasti c collision treated in Sec. 1 7. 2. 1, the initia 1 a nd fina 1 r elative speeds g and gr are represented by the distances separating the two particles in the diagra m

of Fig. 1 7.1 ( b ) be r ore and a rter the col list on. The values of eo a nd gt have the same magnitude ; the collision si mpl y rotates the r elative velo cit y vector by an angle fi .

where

t 17.14 )

377

Equation 17.18 defines the total c oll isi o ii c ross sec lion between the incident and target species w hen the energy of the former is E. The total cross section is a measure of the probability of occurrence of any type of colli sio n between

is the reduced mass of the system. Thus, the total k inetic energy can be divided into two parts, one due to the motion of the system as a whole described by v,„, and the other arising from the relatii'e k inetic energy of the two particles. The latter is

the two particles. Cross sections of more restricted ty pes of interactions can be similarly de fined. For example, we may requ ire that the collision transfer energy between T and T + dT to the target particle during the collision and define the d if lcrc ii I ial energy -trOris/cr cm ss sect ion by:

E, 1

2

2 ( 1’7.15 )

N o ( E,T } d'f d x Probability of a collision in the distance

dx which transfers energy in the ran ge ( T,dT ) to the target particle ( 17. 19 )

During the colli sion the kinetic energy of the center of mass is unchanged, bu t the relative k inetic energy decreases as the potential energy becomes significant, which occurs at close separation distan ces. Conservation of total energy at any poin t in the collision requires that

E,+4' ( x ) -E„, ( 17. 1fi )

w here V ( x j is the potential energy of interaction at a head-on separation distance x and E, is the relatis'e kinetic energy in t he initial state, which is taken to be at in finite separation. An important special case of Eq. 17.16 occurs at the distance of closest approach, x , where the relative k inetic ener gy is zero. If the collision partners are of the same mass, Al '2, and i f the target atom is initially at rest, g — v , , Eq. 17. 16 then redu ces to

V ( xp ) = ( 17. 17

iv here E - \ 1 , v* ' 2 is the kinetic energy of the projectile. ’l'his fcirmu la z ill be used to dedu ce rq uivalent hard-sphere radii as a fun ction of energy for parti cu lar types of in teratomic potential functions.

17.3 BASIC CONCEPTS

The term in ol ogy pertinent to the collision and energy- loss processes involving large nu mbers of energetic atoms in a solid is reviewed in this section.

17.3.1 Cross Section

The primitive idea of a cross section is sh own in F ig. 17. 2. Consider a Angle projectile ( or bombarding particle j passing through a medium consisting of N target particles per unit volume. Target species are assumed to be distributed randomly. We wish to formulate the probability that the projectile collides with a target particle while tra versing a path length d x in the med ium. If the incident parti cle strikes the front face of the dx -th ick slice within any one of the projected area.s, , characterizing the target particles, a collision occurs. The volume element in the drawing co ntains N dx particles whose p i ected areas occupy a fraction uN dx of the front face of the volume elemen t. The chance of an interaction is therefore:

N ‹ ( Fi ) dx Probability of the collision of an incident particle with a target parti cle in d x ( 17. 1S )

The differential and total cross sections are related by

( 17 20 )

where ’fp is the max imum energy transferable in a collision. For elastic cold ision s, Tp is given by Eq. 1 7. 10.

TAR0£T PART L.FS

n or CT L F

P> R T I C L F. 0 F s

U L! I T A R F. A

Fig. 17. 2 The collision cross sect io n.

The dif’f’ereii!ial aiigiilai’ cross s‹•c'Ii!oii describes the pr‹ibab il ity of an in teraction that resul ts in de flection of the in ciden t parti cle by an an gle 0 in the ce nter-of- mass system:

N ( E,0 J df2 dx = Probability of a collision in dx

which scatters the in ciden t partic ie into a center-of-mass angle in the

range [0 ,dR j ( 17. 21 )

where dfl is an element of solid angle abou t the scattering d irection 0. I nasmu ch as scattering is az.iinu t hally sy m- metri c ( i.e. , equall v probable in any direction in the plane perpendi‹ ular 1.o the x-d irection in F ig. 17. 2 ) , the sol id- a ngle increment is

df2 = 2r d ( cos 0 )

f“or elastic scattering, Eq. 17. S provides a unique relation between ’1’ and 0 ; thus the angular and energy-transfer differential cross sections are connected by

2s r ( E ,0 ) d ( cos fi ) - o ( E,T ) dT

378

O £

Id ( cos )

I d1'

( 17. 22 )

in a flux spectru m for interactions that transfer energy in a particular range. Thus

F ( E,T ) d E dT - Collisions per unit volume per unit

time between target particles and inciden t particles in the range ( E,d E ) s'hich result in energy transfer to the

The ratio of ii ‹i irferential eleme nts d I cos 0 ) and d'l’ is ub tainecl fro ni Ed. 17.9. The second equalit y in Eg. 1 7. 2 2 c‹›n tarns fh is franst‘orinati‹›n. The notation 6 ( T ) means that f is ex presseri as a ftinct i‹in of T using the same equation. Eq uati‹i n 1 7. 2 2 permits the d ifferential energy-transfer cr‹›ss se‹'tion to be determined i1’ the d if feren tial angular

‹'russ secti‹›n is k no \ v n. \ \ 'hull scatterlny is is‹›tr‹›pir in the

17. 3. 2 Nlean free Path

'l'lie mcc ii free ¿ c /i is the at erage distance travelled by an i ncident particle 1aetz'een collisions. Equat ion 17.1 S sho ws I hat th e nu in her of collisions per u nit path length is N ‹i{ E 1. "the reciprocal ‹› I th is cJ Sian t ity is I he a verage path lt•n gt li per ‹'oll is inn, ur

ñlr an l'ree pat li ‹if a particle

target particle in the range ( T,dT )

= N U ( E ) c ( E,T ) d E dT ( 17.28 )

Equat ion 17.28 can also be regarded as a source term expressing the volumetric rate of production of the recoils in the energy range ( T,dT ) :

N $1,* O ( E ) at E,T ) d EJ dT - N mb« ‹›r recoil atoms

produced per unit ›’olu me per unit time with energies in the range ( T,dT )

17.3. 6 Stopping Power and Range

The s to pp ing po nicer is the energy lost by a moving particle per unit of length travelled in the medium. Equation 17.19 gives the probabil itj‘ of a collision in path length dx which results in energy loss between T and

o 1“ energy’ E

17. 3.3 Current anal Flux

( 17.231

'I + dT. The ai'erage energy loss in d x is obtained by multiplying Eq. 17.19 by the energy transl“er T and integrating o \ 't•r all possible values of T:

r in

The c'ii i i ‹ n I desc ribes the rate of transport uf particle,s i f I hey are all travelf ing in one d irecti‹i n, and the //ii- is the analog‹ius measure f‹ir particles that are tn‹›i ing in many different ‹directions.

'l'he tutal current is defined by

I — N u niber of particles cr‹›ssing a plane of unit area perpendicular to the particle

direction per second t 7.24 )

\ \ ’hen the particles are nut mod'iny in a single d irecticn, the fl ux is clefined in lerms u f fhe unit sphere :

'1 - N u mber of parl icles cr‹issing a sphere ‹if unit

Id E ) - N I T o ( E,T ) dT d x

- Average energy loss of a particle o 1’ energy E in mo \ ’ing a distance dx

Liividing this equation by dx and omit ting the as'eraging symbol on d E gives the stopping pc›wer:

dE = N i u ( E,f ) dT ( 17.29 )

dx

4’he minim u m energy transferred, T„ , need not be zero. ’I'he stopping crciss section is defined as

projec te d area per sec on d ( 17 .25 )

The fl ux can be restricted to th‹›se particles cvi thin the

1 dE ’"

N dx ./

T ut E, T ) dT ( 1’i.29a )

energy ril nge from E to E + d E:

It E I d E — Number ‹i f particles with energies in the range t E.d E l crossing the u nit sphere

per wcond ( 17.26 )

where 0 ( E i is the d i f ferential energy Lux, ‹›r simply the di fferential fl ux ‹›r the energy flu x and is related to the total flux by

17.3.4 Collision Density

If the current of incident particles entering the volu me element in Fig. 17. 2 is I and if each particle has a probability given b5 Eq. 17.18 of interacting, the number of c‹illisi‹ins per unit v‹ilu inc per second is N In. This expression can be generalized th describe the cti1lisi‹›n rate

4’h e t'aftgc IS a measure o 1’ the path Ie ng th in the so lid traversed fry a particle I"rum the p‹Jin I o I its hirth in or entry into the solid to the point at which it no longer possesses k i netic energy. Tz'o ranges can be defined: one easy to calculate and the other easy to measure. Figure 17.3 shows a ty pical history of a part icle that makes a number of collisions before it is stopped. The arr‹›ws indicate the path length between successive c‹il1isio ns. 'l'hev are ap pro xi-

Fig. 17.3 Pat h of a t ypical part icle slo wing dow n in a solid showing t he mean and projected rangi+.

mately equal t‹i I.he mean rree path. The I.o t.al rank r is defined as the mean valur ‹› f thr « ‹› r ih‹• linrar sebmt nts

17.4.1 Potential Functions

379

hetwevn ‹ ‹›llisir›ns hetss em birth and stoppinti ‹› 1’ the

R t„ , - Al, )

'I’h‹• total range is relatt•d t‹› tlu' st‹›pping p:›we'r hy

/” dE t 1.7..30 )

'J'he t‹it‹al range can be computed i f the dependence ‹if the stopping p‹›wer on energy is known. Acc‹›rding t‹i Eq. 17.29 tlir tlifferential enertiy-transfer cr‹›ss section is neeclecl f‹›r this calt'ulati‹in.

Tlir pruje‹ ted ra ngt', lt„, is the ‹-‹inip‹inent ‹›f the t‹›tal range at onb the init ial diret'ti‹›n ‹›f the part icle. l'‹›r an interatomic potential I hat varic's as f he in verse sq uam ref the st•parati‹›n clistance, the two ra li gt's are related hy : °

ltecaust' ink single potential fiincti‹›n applies urer tlir

entire raii ge ‹if’ st*pitralit›n distuncen between at‹›ins or ions, it is ust•fitl t‹i c‹›nsicler the limiting ‹-ases uf very -high-energy ct›llisi‹nis 1 siiiall distances ‹›f u ppr‹›‹ i cli ) and near thermal energies ti.e., tens of c•lec'tron volts ) svlierr the elertronir cl ‹›uds of the tw‹› species just beJ,'in t‹› oi'er1up. 'l“here are

two principal contributions to the repulsive potential between tsvo atoms which eorresp‹itid t‹› these extremes:

{ 1 ) the elt ‹'tr‹›static repulsion between the p‹›sitirely charged nuclei and t2 ) the increase in energy rrq uire‹1 to maitz taiM tlte ele‹'tr‹›ns ‹›f nearhy at‹›ms iu tile same r‹‘¿i‹›M

‹›f space witli‹›ut vi‹›lai,ing the Pauli ext lu.sion prin‹ iple. Since n‹i two ele‹'tr‹inn can occupy the same p‹›sition. overlapping ‹›f electr‹›ns fro m two rite ms must lie ac‹ ‹›in- panied by prom‹›tir›n of s‹› mt• of the electrons to higher, unoccu pied lev'els ‹›f the at‹i mic structure. The encrby’ required f‹›r this pr‹›c'ess increases as the atO ms ‹ippr‹i.icli L‘acI1 ‹›tl \ er l›ecause a laryPr nil mher ‹› r tht‘ url› ital tIectr‹› its

( 1731 ) At sep«rati‹›ns s‹›in‹' \ vhat smaller than the eqriil ihrium sp. \ ‹'ing cf th+ at‹›ms in thP crystal lat tics, whi‹'IJ is ‹›l tin c›rcltr ‹›f a lat,tic'c' r‹›itslai I, tile n lc‘Icar i”tp \ iIsi‹›n in xmalI

u'lu re SI • and ' \ 4 , arr tht' masses ‹›f the target and prr›je‹ tile species, respeeti vely. A ltli‹›ugh H„ is al ways less than Rt„ „ the difference hetsvec'ii the tw‹i ranges is reclucecl as the ai'erage energy transferret4 per c‹›llision becomes smaller ( i.e., for hI • , 81 t 1 ) .

d'hr' t ‹›nccpLs of stc›pping power and range are mo.st useful when many' small-ene'rgy-transfer collisions ‹›ccur during particle slowing rl‹›wn. 1 n this case the energy l‹›ss process is nearl y continu‹›us, and the deflection per c‹›llision is alst› small. ’l'he intera‹'tion ‹›f at,‹›mir particles z'ith the electrons c›f a solid is an t'x ample of th is type ‹›f slowing down. ’I'he ma ximu m energy transferrecl to an electron by a parti‹'le ‹if maxs 6i is a fraction, 4 m,. DII 0.002 tb1, ‹›f the kinet ie energ y of the mo vine atom, ancl, according to Eq. 1.7. 1 la, the deflection angle per coll is ion is m,. h1 - 1.0 radiant.

17.4 POTENTIAL FUNCTIONS AND ENERGY-TRANSFER CROSS SECTIONS

The manner in which the potential energy of a tiv'o-particle system x'aries with the clistaiirr separating the two centers determines both thr equilibrium proper- ties of an assembly ‹›f atoms and the way that energetic particles interact with a lattice of stationary atoms. The relation between the interatomic potential function and the equilibrium properties of the solid is discussed in Chap. 4. The potential function appears in radiation-damage theory via the differential energy-transfer ‹:ross section, v ( E,T ) , which determines the energy loss rates, the collision density, the mean free path, and other properties of the slowing-down process. The differential energy-transfer cross section is uniquely determined by the potential function, although the connection between V ( r ) and v ( E,T ) is rather complex. Only a few simple potential functions can be converted to analytical expressions for the differential cross section.

because tht' p‹›sitivt• iiiic-lear charges are nearl y romple•tely shielded by the intervening electrons ] Fig. 17.'fi ( c ) ] . In this rcgi‹›n the potential energy of interaction is aclequatrly represented by the l4‹›rn — Prayer p‹›tential:

V ( z ) - A expr ) ( 17.'{2 )

Alth‹›ugh the c‹›nstants A and p in this ftirmula t aiiiu›t he determined fr‹›m the‹›ry, they can be ol›tuined from t lit t‘quilibrium pr‹›perties ‹›f II \ e s‹›lid ( thap. 4 ) .

As the reparatirm distance hetwe‹'n £hc‘ £ \ vr› atc›lns decreases, tlit' cl‹ised -shell repulsi‹›ti ‹lc's‹ rilird hv Eq. 1.7. 32 increase's but, since' th ere are fe we r i*lt•t'trt›ns Incl wr en the tivi› nuclei t‹› sli iel d th t' p‹›sitive t'harges l'ro m ea ‹'li ‹›tht'r, so

‹1‹›es the electrostatic repulsion c‹intributi‹› n t‹› the poten- tial energy. \ \ "hen the interat'ti‹in ent'rgy is st› l‹irpt• th ‹it I he two ii ucle i are se parateti l›y d istaii‹'t's sniallt'r th an thy radius of the in ner vlet'tr‹›n shells ( I ht' K shells l , the prinri pal c‹›ntrib uti‹›n t‹› I h t' t‹›tal p‹›te ntial energy of the system is dur u› thP eect‹stxtir hnre between the two

positively chargt•d nuc'lei | Fig. 1.7 1 a i I ln this li init the interacti‹›n is satisfact‹arily clescribed by the Coult›inh

potential:

( 1‘7.ñ3 )

where Z, and Z, are the atomir numbers of the two atoms or ions and e is the clec'tronic charge ( e' - 1.4.4 e V-.t ) .

The intermediate region where both Coulonibic repul- sion and closed -shell repulsion are ‹›f comparable magni- tudes is the most difficult to describe accti rately. Unfortu- nately, these separation distances are just those most likely to occur in radiation-damage situ atio us. This regio n, which is depicted in Fig. 17.4 ( b ) , is often represented by the screened Co ulomb pot-•ntial, wh ich re fleets the d imin u tion of the pure Coulomb repulsion between the nu clei due to the electrostatic screening of the positive charges b y the intervening inner-shell electrons. This potential is given by