Nuclear and Particle Physics - Lecture 22 Alpha decay

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Darren Morris
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1 1 Intoduction Nuclea and Paticle Physics - Lectue 22 Alpha decay We have looked at gamma decays (due to the EM foce) and beta decays (due to the weak foce) and now will look at alpha decays, which ae due to the stong/nuclea foce. In contast to the pevious decays which do not change A, alpha decays happen by emission of some of the nucleons fom the nucleus. Specifically, fo alpha decay, an alpha paticle, 4 2He, is ejected so geneically A ZX Z 2 A 4 Y + 4 2He fo some X and Y. Y clealy has a diffeent numbe of nucleons to X. Compae how alpha and beta decays move the nuclei aound in Z,N plane N + Z Note that gamma decays cannot change Z, N o A. 2 Fission and fusion Alpha decay is in fact only one specific case of a whole ange of pocesses which involve emission of nucleons. These ange fom single poton o neuton emission up to splitting the nucleus into two oughly equal pats. Paticulaly in the latte case, these ae called fission decays. Which nuclei would we expect to fission? The binding enegy pe nucleon cuve shows that the maximum occus aound 56 26Fe and dops off to eithe side. Hence, both small A and lage A nuclei ae less stongly bound pe nucleon than medium A. This means thee will be some enegy elease if two small nuclei ae combined into a lage one, a pocess called fusion which will be discussed late in the couse. Similaly, thee will be an enegy elease if a lage nucleus is split into two smalle pats; this is fission and what concens us hee. We ae specifically looking at alpha decays in this lectue as this is the most common fom of fission. Why is alpha decay the most common? The binding enegy pe nucleon cuve does not fall vey fast; it dops by only 10% ove the whole A ange above 56 26Fe. The slope is oughly 0.01 MeV/A. Hence, the enegy elease is quite small unless the poducts ae paticulaly stongly bound. This is the case fo 4 2 He which is doubly magic and has B E = 28.3 MeV, which although it has an A well below 56 26Fe, is still 7 MeV pe nucleon. Even fo alpha decay, it is only enegetically possible fo nuclei with A > 150 and needs significantly highe values of A than 1

2 this fo a easonable enegy elease. Hence, alpha decay is seen mainly in heavy nuclei with lage A > 200. Howeve, such nuclei afte alpha decay will leave a daughte nucleus with A 4 which will also nomally be above 150 and so will itself be able to alpha decay. Hence a sequence of alpha decays is often seen and can be many decays long. They continue until they each a nucleus which is stable (o semi-stable). One example is the decay sequence stating fom U which ends at Pb. (Note that the Pb nucleus has Z at a magic numbe, hence being quite stable.) N Pb Bi Po Rn Ra Th Pa U Z Note thee ae seveal beta decay steps hee too. Why? Alpha decays educe Z and N equally, specifically educing them each by two pe decay. Howeve, the heavy, lage A stating nucleus will have N > Z as that is what is needed to be nea the beta-stability cuve. Hence, a pue alpha decay sequence would leave a lowe A nucleus with a highe and highe faction of neutons, wheeas the beta-stability cuve equies the faction of neutons to become lowe as A is educed. Hence, the beta decays bing the intemediate nuclei back close to the betastability cuve. Note, thee will always be too many neutons, not too few, so positon emission o electon captue ae basically not seen in these sequences. This is why, although adioactivity was discoveed in the 19 th centuy, antimatte (specifically the positon) was not found until One final point; beta decay does not change A but alpha decay changes it by 4. Hence, all heavy nuclei with the same A/4 emainde (A modulo 4, o witing A = 4n + m then the value of m) will end up at the same nucleus. Hence, thee ae effectively only fou such decay sequences. 3 Alpha decay ates One obvious question is why do we see any of these sequences at all? This is a stong foce decay and they have had aound 5 billion yeas to decay. The answe is that some of the lifetimes fo 2

3 these decays ae billions of yeas, despite being due to the stong foce. The ange of lifetimes fo alpha decay is found to vay by ove 25 odes of magnitude and this is effectively totally detemined by the Z value and the amount of enegy elease, Q. This extemely stong exponential dependence on Z/ Q can be undestood by consideing the alpha decay pocess in moe detail. To be emitted, the alpha paticle has to fom outside the nucleus Howeve, we can take a simple model whee we conside it to have an independent existance within the nucleus befoe the decay. What potential would it then feel? Thee ae two foces, the nuclea and the EM foce. The nuclea potential fo the alpha will be effectively the same as fo the nucleons, i.e. something like the Saxon-Woods shape which we discussed peviously. V() N Thee is also a lage Coulomb epulsion due to the positive chages on both the nucleus and 3

4 alpha so the EM contibution to the potential looks like V() N The total is then V() N which shows the alpha paticle has a lage potential baie to ovecome in its decay. If the enegy elease Q is lage, then its enegy will be above the maximum of the potential and it can decay vey fast. Howeve, the maximum of the potential will be V max 2(Z 2)e2 4πɛ 0 n 2Ze 2 Z = 2.4 4πɛ 0 0 A1/3 A 1/3 MeV Fo the nuclei which alpha decay, Z/A 1/3 15 and so this baie is seveal 10 s of MeV, much lage than most obseved alpha decay Q values. The usual case is that the enegy elease means it is well below this maximum so it can only exist and subsequently get though by QM tunnelling. As you will know, any QM tunnelling pocess dops exponentially as the baie width o height inceases. Fo a squae baie V d the amplitude goes as e k, whee k = 2m(V 0 E)/ h. This means the pobability of tansmission though the whole baie is popotional to e 2kd. Hence, a lage d o small E give a vey small pobability. 4

5 The above is fo a constant V 0. Howeve, the Coloumb baie is a function of and so in this case the pobability fo an alpha getting though is popotional to whee the Gamow facto G is 2m G = k() d = h e 2 k() d = e 2G 2mQ 2Ze V () Q d = 2 h 4πɛ 0 Q 1 d The integal is ove the ange of adii fo which the alpha enegy is below the baie, i.e. fom the nucleus adius out to the adius whee the alpha enegy is geate than the potential. Fo small Q, this latte adius is much bigge than the nuclea adius. This integal is not tivial (but can be found in most text books) but in the appoximation of the uppe limit being much bigge than the lowe, the integal is This means the Gamow facto is 2Ze 2 4πɛ 0 Q 2Ze2 π 1 d 4πɛ 0 Q 2 = Ze2 4ɛ 0 Q G = 2mQ Ze 2 h 4ɛ 0 Q = e2 2m Z = 2.0 Z MeV 1/2 4ɛ 0 h Q Q The ate pe nucleus R (equal to the invese of the lifetime) is then expected to be R = 1 τ = ae 2G so log R = log a (2 log e)g = log a 1.7 Z Q The plot shows a slope of 1.7 which agees well with most of the measued values. Hence, tunneling is what causes the huge vaiation in lifetimes; fo Q values which only diffe by 1 MeV, a diffeence of 10 5 is seen. 5

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