Nuclear and Particle Physics - Lecture 18 The nuclear force

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1 1 Itroductio Nuclear ad Particle Physics - Lecture 18 The uclear force Protos ad eutros ca actually bid together through the strog force strogly eough to form boud states. These ca alteratively be thought of as boud states of 6, 9, quarks. These boud states form the uclei of atoms ad the sum of the roto charges is what holds the outer atomic electros i their orbits. 2 Nuclei A ucleus is described by the umber of rotos, Z, ad the umber of eutros, N. The total umber of rotos ad eutros (collectively called ucleos) is deoted by the atomic mass umber A which is by defiitio Z + N. Some omeclature is that uclei with the same atomic umber Z but differig values of N are called isotoes of each other. Because they have the same uclear charge, they ted to be very similar chemically, although the uclear roerties themselves are very differet. Less ofte used terms are that uclei with the same N but differig Z are called isotoes ad uclei with the same A but differig Z (ad hece N) are called isobars. N Isotoes (costat Z) A Isotoes (costat N) Isobars (costat A) Z You might thik that if the baryos (ad mesos) are colourless, i.e. have o et colour charge, the there would be o strog force betwee them. This is ot actually true; the same argumet i terms of EM charge would argue that there would be o force betwee eutral atoms. This is clearly wrog; if true all matter would be a ideal gas ad o liquids or solids could form. 3 Nucleo forces I the case of atoms, although they have o charge, there are residual multiole electric fields which ca iteract although these otetials fall off a lot quicker tha 1/r. These forces are ormally refered to a Va de Waals forces ad are ofte arametrised by a otetial of the form a/r 12 b/r 6. There is a similar effect for baryos; there is a residual Va de Waals-like strog force betwee them which, although much weaker tha the full strog force, is still able 1

2 to overcome the roto EM reulsio ad bid ucleos together ito uclei. We will call this force the uclear force. Like the Va de Waals force, it falls off quickly with r ad so is short rage. Also like the Va de Waals force, it becomes large ad reulsive at very short distaces. Oe imortat roerty of the uclear force is that it is aroximately ideedet of the ucleos ivolved. Both the roto ad eutro ad made of three quarks with very small masses comared to the mass of the QCD field surroudig them, which results i the roto ad eutro masses beig very close; m = MeV/c 2 ad m = MeV/c 2, which oly differ by 0.1%. All the quarks have the same strog charge ad are i equivalet wavefuctios, so the residual force should be the same for both. Clearly, there is a reulsive EM force betwee rotos which is abset for eutros, but the uclear force is effectively the same for both tyes of ucleo. There is aother hysical icture of the same residual uclear force i terms of boso exchage, which the makes it more like our other forces. The obvious questio is; what bosos are exchaged? They caot be gluos as that would leave et colour charge o the radiatig ad absorbig uclei. I fact, the uclear force ca be well described by meso exchage, i.e. radiatio ad absortio of ios, rhos, etc. 0 π 0, ρ... + π +, ρ... Sice for ay force, the rage h/mc, the for all but the shortest distaces, io exchage domiates as the ios are so much lighter tha the other mesos. Sice m π 140 MeV/c 2, the the rage is h/m π c = 1.4 fm. This agai shows the uclear force is very short rage ad ow also sets a scale for this rage. This icture is most clearly eeded for the case of roto-eutro elastic scatterig. As might be exected, the scatterig is maily eaked forward, meaig small agles are most likely; this is similar to eµ scatterig. However, there is also a large eak at very large agles ear 180 ; this is easy to uderstad i terms of the above charged io exchage diagram, where the kiematics are very similar to eutral io exchage but the roto ad eutro aear to swa directio. Both ictures of the uclear force are valid i certai ways ad both hel to uderstad asects of its characteristics. We will use these cocets to try ad uderstad how the ucleos bid together ito uclei. First, we shall look at a few simle uclei ad the at the behaviour of larger uclei with may ucleos. For the latter, we shall try to quatitatively justify the bidig eergy B E, for which m Nucleus = Zm + Nm B E c 2 It is obvious that for a ucleus to be stable, the B E > 0. 4 Simle uclei The simlest ucleus besides a sigle ucleo is the deutero, with Z = N = 1, i.e. oe roto ad oe eutro. This is writte as 2 1H as it is a isotoe of hydroge (ad whe made ito water, 2

3 makes it heavy water ). If the force betwee ucleos is ideedet of ucleo tye, the we might exect there to be ad uclei as well. However, either of these combiatios is stable, i.e. they do ot have ositive bidig eergy. This is because, i these cases, the two ucleos are idetical ad the Pauli exclusio ricile comes ito lay. I fact, the deutero has si 1 which is formed from the two si 1/2 ucleos i a S = 1 state. This lowest state has L = 0 as is usually the case. The ext highest state is S = 0, agai with L = 0, so these states are slit by a hyerfie iteractio, just like for the mesos; however, i the meso case, the S = 0 was lower. But, as we have see several times, the S = 1 state formed from two si 1/2 articles is symmetric uder iterchage of the fermios. There is o colour art of the wavefuctio ow to hel out, so this state caot be occuied by idetical articles. Hece, although the force betwee the ucleos is the same, the ad states are forbidde ad so do t exist. For the higher S = 0 state, where the fermios are atisymmetric, all three combiatios of, ad are foud (where the state is the first excited state of the deutero) but this state has egative bidig eergy ad is ot stable. B E Deutero S= MeV 0 Z S=0 The deutero itself actually has quite a low bidig eergy E B = 2.23 MeV, i.e. 1.1 MeV er ucleo, which, as we will see, is quite small comared with the average value of aroud 8 MeV er ucleo for most other uclei. Similarly, goig u a ucleo, it turs out there are two boud states of three ucleos; tritium, 3 1 H, ad helium-3, 3 2He, which are states of two of oe ucleo ad oe of the other with equivalet wavefuctios. (Tritium i fact beta decays to helium-3.) There is o boud state of or as Pauli exclusio forbids these combiatios i the tritium ad helium-3 wavefuctio. There is agai a wavefuctio with a higher eergy which allows these, as well as excited tritium ad helium-3 states but, as for the deutero, these are uboud. For four ucleos, helium-4, 4 2He, also called the alha article, is extremely stable with B E = 28.3 MeV, i.e. 7.1 MeV er ucleo. It is si 0 ad has the two rotos ad two ucleos all acked close together i a low eergy level. There are o other boud uclei for A = 4. We ca check the uclear force is ideedet of the ucleos ivolved most easily by comarig uclei where all the rotos ad eutros are iterchaged, as there are the o differeces due to the Pauli exclusio ricile. Such uclei are called mirror uclei ; tritium ad helium-3 are oe examle. Aother examle, secifically the excited states of 23 11Na (which has Z = 11 ad N = 12) ad 23 12Mg (which has Z = 12 ad N = 11), is show below. 3

4 Hece, the levels are see to be very similar, with the differeces exlaiable i terms of the EM force o the rotos due to their charge. Hece, what we fid is that the strog force betwee rotos ad eutros seems to be equal but that the Pauli exclusio ricile is very imortat for determiig which states are allowed ad ideed teds to favour states with aroximately equal umbers of rotos ad eutros, i.e. Z N. We will ow look at the roerties of uclei with may ucleos, i.e. ucleos i bulk. 5 The liquid dro model We have see that the uclear force has a very short rage 1 fm, which is oly the size of a ucleo radius. It also saturates, i.e. becomes very large ad reulsive for short distaces. This meas that i a ucleus with may ucleos, they will ot all crowd together at the origi but will sread out to occuy a fiite volume each, liked acked sheres. I additio, the force each the imoses o the others is so short rage that it is egligible for all but the earest eighbours. This is hysically similar to the saturatio i a water dro, where the Va de Waals forces betwee the water molecules effective make oly the earest eighbour iteractio sigificat. 4

5 Because of this aalogy, the resultig uclear model is ofte called the liquid dro model. It gives a good descritio of the bidig eergies of the groud states of uclei with may ucleos, where may i ractise meas A 20. There are two immediate cosequeces of this model. Firstly, the volume of the ucleus clearly would be exected to icrease liearly with the umber of ucleos, so V A = N + Z. Sice V = 4πr 3 /3, the uclear radius r should go as r A 1/3. This is a good aroximatio ad is usually exressed as r = r 0 A 1/3, where r 0 = 1.2 fm, roughly the radius of a ucleo. The secod cosequece is that bidig eergy for each ucleo would be exected to be costat. This is because each ucleo oly bids to its earest eighbours, so the cotributio to the bidig eergy from each is a fixed value. Hece, we would exect B E A, which is agai foud to be roughly true; as stated above, the bidig eergy for heavy uclei is foud to be roughly 8 MeV er ucleo. However, there is a obvious roblem with this aroximatio ad that is the ucleos o the surface of the ucleus. These will ot have as may earest eighbour bods as the oes well withi the uclear volume as there are o earest eighbours outside the ucleus. This is exactly the same effect which gives rise to surface tesio i a water dro, so agai the aalogy holds. We would therefore exect that there will be a correctio to the bidig eergy roortioal to the shere surface area 4πr 2. Sice r A 1/3, the our exressio for the bidig eergy becomes B E = a v A a s A 2/3 where a v (the volume term costat) ad a s (the surface term costat) are arameters to be determied from data. 5