Lecture 12 Krane Enge Cohen Williams

Transcription

1 Lecture 1 Krne Enge Cohen Willims -decy Energetics Systemtics Coulomb effects ngulr mom g-decy Multipole rdn Decy constnts Selection rules Problems 1 Clculte the nd the energy of the emitted α, for α-decy from 4 Po, 08 Po, 4 Cm. Indicte the residul nucleus. Use the uncertinty principle to estimte the minimum speed nd energy of n α- prticle inside hevy nucleus. 3 In semi-clsicl picture n l=0 α-prticle is emitted long line tht psses through the centre of the nucleus.. () How fr from the centre of the nucleus must n l= prticle be emitted? ssume = 6 MeV nd = 30. (b) Wht would be the recoil rottionl energy if ll the recoil went into rottion of the dughter nucleus. 4 For the following γ-ry trnsitions, give ll permitted multipoles, nd indicte which would be the most likely. 9/ - 7/ +, ½ - 7/ -, 1 - +, 5 decy process leds to finl sttes in n even-even nucleus, nd gives only three γ-rys of energies 100, 00 nd 300 kev, which re found to be of E1, E nd E3 multipolrity. Construct two different possible level schemes for tis nucleus nd lbel the sttes with their most likely I π ssignments. 6 nucleus hs the following sequence of sttes beginning with the GS. 3/ +, 7/ +, 5/ +, ½ - nd 3/ -. Drw level scheme showing the intense γ-rys trnsitions nd indicte their multipolrity.

2 Review Lecture 11 Isospin 1 The nucler force is chrge independent, hence the level structure of excited sttes in nuclei tht hve the sme, nd the sme configurtion (but differing N nd Z) will be the sme. The exmple given ws =14. fter llowing for the different coulomb energy e (Z - 1) D E c = 3/ 5, nd the mss difference between proton nd neutron 1/ 3 4per0 (0.78 MeV), the sttes in 14 C nd 14 O (mirror nuclei) line up in energy, together with nlogue sttes in 14 N. The isospin No. of configurtion is the T z vlue (T z = (N-Z)/) of the most neutron-rich form of the configurtion tht cn be mde consistent with Puli principle. The low-lying sttes in 14 C nd 14 O, nd the nlogue sttes in 14 N re ll T=1, with projection T z = -1, +1, nd 0 respectively. The GS nd most lowlying sttes in 14 N hve T=0 (nturlly T z =0) Bet Decy The bet decy process - X e fi X + n + Z Z -1 The energy of b - -prticle emitted from nucleus,z to form nucleus, Z+1 Z X fi b - Z + 1 X + e = {[ m( X ) - Zm Z - e ]- [ m( Z+ 1 X ) - ( Z + 1) m ] -m = [ m( X ) - m( X )] c b - Z Z + 1 β represents the energy shred by the e - nd the ntineutrino b- = Te + E n nd since when one is mx the other is zero ( ) ( E n T e = = - mx ) mx b - e e } c For positron emission, similr clcultion gives tht: + X fi X + e Z b + Z -1 = [ m( X ) - m( X ) - m ] c Z Z -1 The log ft term nd wht it indictes. The concept of forbidden-ness. e The importnce of super-llowed trnsition, when the Nucler wve-function overlp is exct.

3 Lecture 1 We discussed β-decy lst lecture, nd tody I will discuss α nd γ decy. We need to sk wht determines whether decy of ny kind will occur, nd then look little more closely s to wht is the probbility of certin decy. Firstly let us remember we re discussing nuclei which lie outside the collection of 00 or so stble nuclei., nd ultimtely they will rech stbility in order to ttin minimum energy of the nucler system. These nuclei hve either n excess of protons or neutrons, nd in generl they will ttin stbility by β-decy, chnging n p or p n. However for nuclei with > 150 α-decy is the most likely possibility. Ultimtely the requirement for prticle decy is tht the finl system is more stble thn the originl. One cn show by considering the (energy difference between the finl nd initil system), tht for light prticle emission the only possibility is α. Here tble 8.1 For α decy the process is Z x N -4 4 fi Z- x' N-+ Energy considertions give the is defined s = ( m Thisenergyisshred between the ndthedughter nucleus so ( m x x - m - m x' x' - m - m ) c ) c = = T x' + T So if is positive, in theory α-decy cn occur. The emitted α will of course not hve KE (T) equl to, since the dughter nucleus will recoil (T x ). T since m T = m 1+ m» 4 1+ x' << m x'

4 However lone does not determine whether or not we will observe n α decy. Since the decying nucleus hs lrge Z, the coulomb brrier cts to suppress decy. The probbility of the α getting out depends very criticlly on the energy of the α. This ws found out by Geiger nd Nuttll in 1911 (the Geiger Nuttll lw) Note the Verticl scle is logrithmic, nd rnges over some 5 orders of mgnitude. Nuclei with the sme Z re joined. We re now in position to estimte the rte of α-prticle emission. V B = 1. (Z-)e 4πε o V Coulomb pot. ½(b-) ½(B-) b r b r KEα KEα -V o -V o We model the system s n α nd dughter nucleus, with rdius. Within the rdius r< the α is free to move, but it cnnot escpe becuse of the coulomb brrier. Between nd b, clssiclly the α cnnot exist, since V>KE. Clssiclly he prticle will bounce bck nd forth in the region r=0 to. However quntum mechniclly there is chnce of lekge, or tunnelling through the brrier, nd one cn clculte the probbility P of escpe. The decy constnt λ = fp per second. Where f is the frequency of presenttion t the boundry. s n exmple, for 38U the lph tkes ~10 38 tries before it succeeds (~10 9 yers.)

5 The M problem is simplified by using squre pot brrier. The brrier height vries from (B-) bove the prticles energy t r=, to zero t r=b. Lets tke n verge height ½(B-). The width of the true coulomb brrier is b-. Lets tke the squre one s ½(b-). (NOTE b is the vlue of r when the lph is free of the coulomb potentil. It is equivlent to the stopping distce from r=0 for in incoming lph of energy. So conservtion of energy gives KE = PE 1 = 4pe 1 b = 4pe o o ( Z - ) e b ( Z - ) e P of α being outside nucleus is prop to (WF outside) /(WF inside) = prob fter normlizing t boundry P» e where 1 k = h -k ( b- ) m 1 ( B - ) Typicl vlues For Z=90 ~7.5 fm, B~ 34 MeV. With =6MeV, b~ 4 fm. This gives P ~ x 10-5 if V (potentil) ~ 35 MeV, nd ~ 6 MeV, then f = 5 * 10 1 Hz. So tht remembering tht λ = fp per second, λ = 10-3 per second t 1/ =700 s. You cn see the criticlity of the dependence on E α (~) since if we chose = 5 insted of 6, P 10-30, nd t 1/ 10 8 s. This is roughly consistent with the observtions.

6 The Effect of WF For emission not only does the need to be quite high, but the overlp of the initil nd finl WF must be lrge. q = G y * f. y * i 1g9/ consider 1 Po 08 Pb(GS) + Wht does y i look like? 1h9/ 08 Pb y I ~ y ( 08 Pb + n + p) these p nd n in essence form n prticle since they re in reltive l=0 stte. q G = y f. y * i q G = y (08Pb). y * (08Pb)( h9 / ) ( g9 / ) y ( ) = 1. y ( )( h9 / ) ( g9 / ) In fct these WF re to ll intents nd purposes the sme. So the decy should proceed strongly if is OK.

7 Effect of M l 4 Cm 38 Pu gin the WF overlp is essentilly 1 between the two GSs. However only 74% goes to GS. The rest goes to excited sttes. Note tht Pu is rottionl nucleus. The first 5 sttes ll hve the sme intrinsic WF. So overlp is sme. The difference is tht 4 Cm strts with M = 0 nd to get to 1 st excited stte the α must crry wy units of M. This mens it must overcome centrifugl brrier of l ( l + 1) h mv (hence problem 3) Electromgnetic decys Two mrks of person literte in nucler physics: 1. they sy nu-cler, not nucu-lr they know the difference between x-rys nd γ-rys Wht re g-rys? Very-high-energy EM rdition E=hf hc/λ From nucleus (cf x-rys from tom) s we sid erlier. if nucleus is in n excited (not GS) stte, it will, if energeticlly possible decy by emission of n α prticle, or by β emission.

8 We now wnt to look t the competition tht γ-ry decy offers to α-decy. It turns out tht the decy probbility for α-decy or ny prticle emission is much higher (hlf-life much shorter) thn for γ-decy, so will lwys hppen if energeticlly possible. Otherwise γ-decy. Wht cn we lern from the study of g-decy spectr? You will recll tht in tomic physics it ws the study of the spectr (visible minly but lter IR nd UV) tht led to the unfolding nd confirmtion of the shell model of the tom. The spectrum of H is the supreme exmple. However in more complicted toms it ws necessry to identify the spectr with specific electron configurtions. This involved mesuring the energy (wvelength), the decy probbility (intensity) nd the relted hlf-life of ech excited stte. The γ-rys result from decy of prticulr excited nucler stte to lower one. The resulting spectrum, s succession of γ-rys leding to the GS nd stbility, cn identify the energies of the excited sttes in the nucleus. The energies of these γ-rys cn be determined using scintilltion detectors nd solid stte detectors s most of you used in the prt 3 lb. The reltive intensities of the γ-rys in the spectrum will tell us the decy probbilities of the decy (or hlf-life) of prticulr stte to set of finl sttes. This is reltble to the width of the stte vi the Heizenberg uncertinty reltionship E t~h. Hving collected dt on the nture of the sttes it is then, s in the cse of tomic physics, possible to try nd find out the configurtion of the stte. In the cse of nuclei ner mgic numbers, this is in prt relised vi the shell model, nd in deformed nuclei rottionl sttes cn be identified; however in generl it is very difficult. Clssicl Picture EM rdition results from the ccelertion of chrges or the vrition of currents. The simplest cse is of the oscilltion of n electric chrge such s in rdio ntenn. This is clled electric dipole rdition, E1, nd it hs simple cosθ dependence.

9 The field set up by chnging current through simple mgnetic dipole hs similr behviour, nd is termed M1 rdition. More complicted chrge (nd current) distributions led to higher order rdition modes, with more complicted distribution ptterns. For exmple consider qudrupole chrge distribution. This would led to E, E3, E1 rdition. In generl the rdition is specified s Elor Ml. Where specifies the order of the multipole. NOTE tht photon LWYS crries M. nd it is the mount of M tht must be crries wy (or bsorbed) tht strongly influences the probbility of its emission or bsorbtion. The power rdited by prticulr chrge distribution is given by: ( l + 1) c ω l+ P( E l) = ( ) lm εl[(l + 1)!!] c ( +1)!! = 1 x 3 x 5 x 7. is the moment of the electric chrge distribution, e.g. dipole, qudrupole etc. NB the strong dependence of P on frequency. If we now consider clssicl chrge distribution, the size of nucleus, due to Z protons confined in rdius R we cn mke some estimte of. 1 ~ Zed where d is the mp of vibrtion ~ Ze(3z -r ) 6.5ZeR ( R/R) < ZeR so to zero pproximtion < ZeR l

10 Thus we could write P ( l + 1) c l ( E l) = ( ) ( R) Z e ε l[(l + 1)!!] ω c ω c so tht the rdited power is pproximtely proportionl to ω R l R ( ) or ( ) c λ so interestingly l For medium size nucleus R~10-15 m R/λ = 5 x 10-3 E/E1 power x 10-5 for tom λ ~ 4000 ngst R~0.5 ngst. R/λ = 10-4 E/E1 power 10-8 So lthough in generl the higher the multipole the less power rdited, for toms E rdition is 10 8 times less likely thn E1 (cf 10 5 for nuclei). So in tomic spectr, ll the observble trnsitions give rise to E1 rdition. untiztion The expression for P(E- ) is the power rdited. Nturlly in the clssicl sitution this diminishes with time. Tht is the old problem of the Bohr orbits without quntiztion. The orbits collpsed. In the quntum sitution the energy is emitted s photon of energy hf or h ω, when the nucleus chnges stte from ψ i to ψ f. The trnsition probbility is λ(e- ) nd is P(E- )/ h ω (energy/sec/energy). ( l + 1) ω l+ 1 λ ( E l) = ( ) lm hεl[(l + 1)!!] c For quntum system m is replced by the overlp of the initil nd finl stte chrges s defined by the wvefunctions, nd is known s the multipole mtrix element.

11 ψ er Y ψ dτ lm Z k = 1 * f l k lm i Ech photon of order, emitted in this trnsition, must crry with it n M of h l( l + 1 ) with z projection of mh. Tht is the vector difference between the M of the initil stte I i nd the finl stte I f must be h l( l +1) with the pproprite z component chnge m j. The consequence of this is tht the llowed vlues of re given by The trnsition probbility for emission of g-ry between sttes is l(e- ) (energy/sec/energy) is: I i - I f I i + I f Note tht we hve not yet clculted the probbility of emission of prticulr -photon. l ( l + 1) hel[(l+ 1)!!] multipole Z mtrix element. * l y er Y y dt w c l+ 1 ( E -l) = ( ) lm lm k =1 f k lm i For given energy (w) 1 st order clc. by Weisskopf gives reltive trnsition probbility t 1 MeV s l=1 l~1 l= l~10-4 l=3 l~10-8 l=4 l~10-14 For given l (l=1) 1 st order clcn. Gives the reltive trnsition prob.,s function of energy s: 100 kev l~1 1 MeV l~ MeV l~10 6 Clssifiction of g-ry trnsitions For electric multipole trnsitions p i p f = (-1). For mgnetic multipole trnsitions p i p f = (-1) +1. Note tht trnsition 0 0 cnnot occur, since the photon must crry wy M. Type symbol M chnge Prity chnge Electric dipole E1 1 Yes Mgnetic dipole M1 1 No Electric qudrupole E No Mgnetic qudrupole M Yes h l( l +1) Ech photon of l, must crry with h it n M of with z projection of m. Depending on the M crried off by the photon there re requirements regrding the prity of the sttes involved. For electric multipole trnsitions π i π f = (-1). For mgnetic multipole trnsitions π i π f = (-1) +1. Mgnetic trnsitions re ~10 - times lesslikely thn electric ones of the sme multipolrity (l)

12 The process of emitting photon of quntized energy is not too difficult to imgine when we consider collective oscilltions of the nucleus. The emission of phonon of energy from n λ=1 dipole oscilltion reduces the energy of the nucleus by tht phonon. Similrly for qudrupole oscilltions. similr wy to the tomic shell model. In single prticle model picture, we cn imgine the initil nd finl sttes to involve proton in one orbit jumping to nother with the relese of energy, in Indeed it is only possible to estimte the trnsition probbilities under some model ssumption. fir order-of-mgnitude estimte is mde on the ssumption tht the initil wve function of the proton hs M nd the finl stte is n s-stte ( =0). On this bsis lm e 4π l 3R l + 3 nd the grphs shown indicte vlues of λ for different electric multipole trnsitions for rnge of nuclei See Fig enge 9.13, 14for mgnetic multipole trnsitions. In this cse the expression is identicl except smller by fctor of R.

13 Wht do we observer from these theoreticl estimtes? First note tht T 1/ = ln /λ so tht high trnsition rte short life time the width of the stte (how well its energy is defined) is inversely proportionl to λ. (Heisenberg) The trnsition rtes vry drmticlly with γ-ry energy. E.g. 1 orders of mgnitude for λ=5 trnsitions between 100 kev nd 10 MeV, for E1 it is bout 6 orders. t low energies the effect of is drmtic. 6 orders of mgnitude between E1 nd E t 100 kev. Only orders t 10 MeV. the mgnetic trnsitions show the sme trends, but the rtes re lower by bout orders of mgnitude. How relistic re these numbers? Rel sttes live longer In view of the pproximtions mde it would be surprising if the predictions were correct. In generl (not for E) the clculted trnsition rtes re significntly lrger thn the mesured ones. ( rel sttes live longer thn predicted) This is not surprising since it hs been ssumed tht protons only re involved in the trnsition. In hlf the cses it is the neutron tht moves orbit. The clcultions lso ssume pure simple singleprticle wvefunctions. This is seldom likely to be the cse. E trnsitions for deformed nuclei decy more rpidly. Going ginst the trend E trnsitions for deformed nuclei hve lrger trnsition rtes thn estimted. This is likely to be the cse since the qudrupole moments for these nuclei is much lrger thn ssumed.

14 Internl Conversion These is n interesting effect on these trnsition-rte grphs t low γ- energies for hevy nuclei. Note tht the trnsition rte increses mrkedly bout 100 kev. So tht, prticulrly for hevy nuclei the rte increses by severl orders of mgnitude for low energies. This is the result of internl conversion. Wht is it? In this instnce the energy difference between the initil nd finl sttes of the nucleus is trnsferred to n orbitl (most likely k-shell) electron, nd this is emitted with kinetic energy of E -BE. Thus one sees emission of electrons with very discrete shrp energies tht cn be relted directly to energy differences of sttes within the nucleus. How does this come bout? The whole process of emission of γ-ry is the result of the interction of chrge distribution with the ll-pervding EM field. So tht the Proton in the higher stte drops bck to lower stte within this field (provided by the electric moment of the nucleus), nd the energy difference is emitted s γ-ry. However this EM field lso pervdes the tomic electrons even though they re much further wy (4 or so orders of mgnitude). So it is possible tht the energy difference could be given to n tomic nucleus. This is more likely if the orbitl electrons re close to the nucleus s they re for hevy toms (high Z nd r tom is prop 1/Z). It is lso more likely if the γ-decy hs smll decy constnt (the stte hs long life), since the EM interction probbility goes up. Thus when γ-ry trnsition involves lrge M trnsfer, the electron conversion rte increses. In fct in some instnces (e.g. 0 0 trnsition) when γ-decy is forbidden electron conversion is the only decy mechnism. The picture below shows the energy spectrum of electrons from 1 Pb. The smooth curves re due to β prticles, nd the shrp resonnces re internl-conversion electrons. (Enge 9.19)