Outline. Radioactive decay. Total decay constants. Introduction. Total decay constants. Total decay constants

Transcription

1 Radioactiv dcay Chaptr 6 F.. ttix, Introduction to Radiological Physics and Radiation Dosimtry Outlin Dcay constants Man lif and half lif Parnt-daughtr rlationships; rmoval of daughtr products Summary Introduction Particls insid a nuclus ar in constant motion Natural radioactivity: a particl can scap from a nuclus if it acquirs nough nrgy Most lightr atoms with Z<8 (lad) hav at last on stabl isotop ll atoms with Z > 8 ar radioactiv and disintgrat until a stabl isotop is formd rtificial radioactivity: nuclus can b mad unstabl upon bombardmnt with nutrons, high nrgy protons, tc. Total dcay constants Considr a larg numbr N of idntical radioactiv atoms Th rat of chang in N at any tim dn N W dfin as th total radioactiv dcay (or transformation) constant, it has th dimnsions rciprocal tim (usually s - ) Total dcay constants Th product of t (for a tim intrval t<</) is th probability that an individual atom will dcay during that tim intrval Th xpctation valu of th total numbr of atoms in th group that disintgrat pr unit of tim (t<</) is calld th activity of th group, N Total dcay constants Intgrating th rat of chang in numbr of atoms w find N t N Th ratio of activitis at tim t to that at t = N N t

2 Partial dcay constants If a nuclus has mor than on possibl mod of disintgration (i.., to diffrnt daughtr products), th total dcay constant can b writtn as th sum of th partial dcay constants i : Th total activity is B N N N B Partial dcay constants Th partial activity of th group of N nucli with rspct to th ith mod of disintgration can b writtn t N N i i Each partial activity i N dcays at th rat dtrmind by th total dcay constant sinc th stock of nucli (N) availabl at tim t for ach typ of disintgration is th sam for all typs, and its dpltion is th rsult of thir combind activity Th fractions i N/N ar constant Units of activity Th old unit of activity was th Curi (Ci), originally dfind as th numbr of disintgrations pr scond occurring in a mass of g of 6 Ra Whn th activity of 6 Ra was masurd mor accuratly th Curi was st qual to 3.7 s - Mor rcntly it was dcidd by an intrnational standards body to stablish a nw spcial unit for activity, th bcqurl (Bq), qual to s - Ci 3.7 Bq Man lif and half lif Th xpctation valu of th tim ndd for an initial population of N radioactiv nucli to dcay to / of thir original numbr is calld th man lif =/ N.3679 N ln rprsnts th avrag liftim of an individual nuclus is also th tim that would b ndd for all th nucli to disintgrat if th initial activity of th group, N, wr maintaind constant instad of dcrasing xponntially Man lif and half lif Th half-lif / is th xpctation valu of th tim rquird for on-half of th initial numbr of nucli to disintgrat, and hnc for th activity to dcras by half: N N /.5 / Man lif and half lif Exponntial dcay charactrizd in trms of man lif and half lif

3 Radioactiv parnt-daughtr rlationships Considr an initially pur larg population (N ) of parnt nucli, which start disintgrating with a dcay constant at tim t = Th numbr of parnt nucli rmaining at tim t is N = (N ) - t Simultanously th daughtr will disintgrat with a dcay constant of ( nd gnration doing th dcaying ) Th rat of rmoval of th N daughtr nucli which xist at tim t will - N Radioactiv parnt-daughtr rlationships Thus th nt rat of accumulation of th daughtr nucli at tim t is dn N N t N N Th activity of th daughtr product at any tim t, assuming N = at t =, is t t N N Radioactiv parnt-daughtr rlationships Th ratio of daughtr to parnt activitis vs. tim: N N t If is composd of partial dcay constants, B, and so on, rsulting from disintgrations of, B, typs, thn th ratio for a particular typ is N N t Equilibria in parnt-daughtr activitis Th activity of a daughtr rsulting from an initially pur population of parnt nucli will b zro both at t = and W can find th tim t m whn N rachs a maximum giving N t m t m d t m ln / Equilibria in parnt-daughtr activitis This maximum occurs at th sam tim that th activitis of th parnt and daughtr ar qual, but only if th parnt has only on daughtr ( = ) Th spcific rlationship of th daughtr s activity to that of th parnt dpnds upon th rlativ magnituds of th total dcay constants of parnt ( ) and daughtr ( ) Daughtr longr-livd than parnt, < For a singl daughtr product th ratio of activitis N t N This ratio incrass continuously with t for all tims Sinc N = (N ) - t on can construct th activity curvs vs. tim for th rprsntativ cas of mtastabl tllurium-3 dcaying to its only daughtr iodin-3; and thn to xnon-3: - - 3m 3 3 5T 53I 54X 3h 93h / / 3

4 Daughtr longr-livd than parnt, < Qualitativ rlationship of activity vs. tim for T-3m as parnt and I-3 as daughtr Daughtr shortr-livd than parnt, > For t >> t m th valu of th daughtr/parnt activity ratio bcoms a constant, assuming N = at t = : N N Th xistnc of such a constant ratio of activitis is calld transint quilibrium, in which th daughtr activity dcrass at th sam rat as that of th parnt Daughtr shortr-livd than parnt, > If th dcay schm is branching to mor than on daughtr ( = + B + ) N N For th spcial cas of transint quilibrium whr th activity of th th daughtr is qual to its parnt s scular quilibrium condition Daughtr shortr-livd than parnt, > To stimat how clos is th daughtr to approaching a transint quilibrium with its parnt w valuat th ratio of activitis at a tim t = nt m to that of th quilibrium tim t : N N N N ntm t n ln / Daughtr shortr-livd than parnt, > Exampl of transint quilibrium: 99 Mo to 99m Tc Two branchs: 86% dcays to 99m Tc, 4% to othr xcitd stats of 99 Tc Only daughtr much shortrlivd than parnt, >> For long tims (t >> ) th ratio of activitis N N th activity of th daughtr vry closly approximats that of th parnt Such a spcial cas of transint quilibrium, whr th daughtr and parnt activitis ar practically qual, is calld scular quilibrium (typically, with a long-livd parnt lasting through th ags ) 4

5 Only daughtr much shortrlivd than parnt, >> n xampl of this is th rlationship of 6 Ra parnt, dcaying to Rn daughtr, thn to 8 Po: 6 88 Ra, / 6y d Th ratio of activitis 86 Rn N / 3.84d -.85d 8 84 Po.7 6 N Only daughtr much shortrlivd than parnt, >> It can b shown that in a cas >> all th progny atoms will vntually b narly in scular quilibrium with a rlativly long-livd ancstor Rmoval of daughtr products For diagnostic or thraputic applications of short-livd radioisotops, it is usful to rmov th daughtr product from its rlativly long-livd parnt, which continus producing mor daughtr atoms for latr rmoval and us Th gratst yild pr milking will b obtaind at tim t m sinc th prvious milking, assuming complt rmoval of th daughtr product ach tim Waiting much longr than t m rsults in loss of activity du to disintgrations of both parnt and daughtr Rmoval of daughtr products ssuming that th initial daughtr activity is zro at tim t =, th daughtr s activity at any latr tim t is obtaind from t N N This quation tlls us how much of daughtr activity xists at tim t as a rsult of th parnt-sourc disintgrations, rgardlss of whthr or how oftn th daughtr has bn sparatd from its sourc Stabl nucli may b transformd into radioactiv spcis by bombardmnt with suitabl particls, or photons of sufficintly high nrgy Thrmal nutrons ar particularly ffctiv for this purpos, as thy ar lctrically nutral, hnc not rplld from th nuclus by Coulomb forcs, and ar radily capturd by many kinds of nucli Tabls of isotops list typical ractions which giv ris to spcific radionuclids Lt N t b th numbr of targt atoms prsnt in th sampl to b activatd: N m Nt whr N = vogadro s constant (atoms/mol) = gram-atomic wight (g/mol), and m = mass (g) of targt atoms only in th sampl 5

6 If is th particl flux dnsity (s - cm - ) at th sampl, and is th intraction cross sction (cm /atom) for th activation procss, thn th initial rat of production (s - ) of activatd atoms is dnact N t Th initial rat of production of activity of th radioactiv sourc bing cratd (Bq s - ) is givn by dn act N t hr is th total radioactiv dcay constant of th nw spcis If w may assum that is constant and that N t is not apprciably dpltd as a rsult of th activation procss, thn th rats of production givn by ths quations ar also constant s th population of activ atoms incrass, thy dcay at th rat N act (s - ) Thus th nt accumulation rat can b xprssd as dn act Nt Nact ftr an irradiation tim t >> =/, th rat of dcay quals th rat of production, raching th quilibrium activity lvl N act N ssuming N act = at t =, th activity in Bq at any tim t aftr th start of irradiation, can b xprssd as t N t act Nact Nt If no dcay occurs during th irradiation priod t (which will b approximatly corrct if t << ) Nact Nt t t Growth of a radionuclid of dcay constant du to a constant rat of nuclar intraction Th xposur-rat constant of a radioactiv nuclid mitting photons is th quotint of l (dx/) by, whr (dx/) is th xposur rat du to photons of nrgy gratr than, at a distanc l from a point sourc of this nuclid having an activity : l dx Units ar R m Ci - h - or R cm mci - h - It includs contributions of charactristic x-rays and intrnal brmsstrahlung Th xposur-rat constant was dfind by th ICRU to rplac th arlir spcific gamma-ray constant, which only accounts for th xposur rat du to -rays is gratr than by % or lss, with xcpt for Ra-6 (%) and I-5 (in which cas is only about 3% of bcaus K-fluorscnc x-rays following lctron captur constitut most of th photons mittd) 6

7 W would lik to calculat spcific -ray constant at a givn point sourc; th xposur-rat constant may b calculatd in th sam way by taking account of th additional x-ray photons (if any) mittd pr disintgration t a location l mtrs (in vacuo) from a -ray point sourc having an activity Ci, th flux dnsity of photons of th singl nrgy E i is givn by 9 ki E i 3.7 ki l l whr k i is th numbr of photons of nrgy E i mittd pr disintgration Flux dnsity can b convrtd to nrgy flux dnsity, xprssd in units of J/s m (xprssing E i in MV): 4 k E 4.77 i i E i (J/s m ) l nd rlatd to th xposur rat by rcalling n X d E,air W air d dr da K K / c air W air c air For photons of nrgy E i th xposur rat is givn by dx n,air d,air and th total xposur rat for all of th -ray nrgis E i prsnt is n dx n i 33.97,air n Substituting th xprssion for th nrgy flux dnsity, w obtain n dx 5 n.389 C/kg s ki l i E, air i This can b convrtd into R/h, rmmbring that R =.58-4 C/kg and 36 s = h: dx 93.8 l n n ki i, air R/h Th spcific -ray constant for this sourc is dfind as th xposur rat from all -rays pr curi of activity, normalizd to a distanc of m by mans of an invrssquar-law corrction: dx l 93.8 n n ki i E i, air R m /Ci h whr E i is xprssd in MV and n / in m /kg If ( n /),air is givn instad in units of cm /g, th constant in this quation is rducd to 9.38 pplying this to an xampl, 6 Co, w not first that ach disintgration is accompanid by th mission of two photons,.7 and.33 MV Thus th valu of k i is unity at both nrgis Using th mass nrgy absorption cofficint valus for air at ths nrgis ar, w find 93.8( ).9 R m /Ci h which is clos to th valu givn in th tabl 7

8 Th xposur rat (R/hr) at a distanc l mtrs from a point sourc of curis is givn by dx l whr is givn for th sourc in R m /Ci h, and attnuation and scattring by th surrounding mdium ar assumd to b ngligibl Summary Dcay constants Man lif and half lif Parnt-daughtr rlationships; rmoval of daughtr products 8