Lecture 6 Povh Krane Enge Williams Properties of 2-nucleon potential

Transcription

1 Lecture 6 Povh Krane Enge Wiiams Properties of -nuceon potentia Meson Theory of Nucear potentia I recommend Eisberg and Resnik notes as distributed Probems, Lecture 6 1 Consider probem 15 in Krane Chap 4 (page 115). This requires a dea of thought. Because both the neutron and the proton have magnetic dipoe moments, there is a difference in the energy when they are aigned or.estimate this difference when their centres are separated by r d. It might be interesting to compare this vaue with the tensor force as potted in Cohen fig 3.7ot Krane, Fig Use energy conservation, in the ight of the uncertainty principe, to estimate the reation between the range of the nucear force and the mass of the π meson. Use this reation to estimate the pion mass if r = fm. 4 Because pions had not been discovered in 1936 when Yukawa proposed the meson theory of the nucear force, it was suggested that the µ meson was responsibe. What woud the range of the nucear force be if this were true? 34

2 Review Lecture 5 Nuceon-nuceon scattering 1 Study of deuteron tes us ony about S=1, =0 components of nucear potentia Need to consider scattering of nuceons to study S=0 and >0 components. Concept of nucear cross section the number scattered in thickness t is N = N o (1 e nσ t ) 3 Doing scattering experiments require acceerators to produce a beam of protons or neutrons of a controed energy, and detectors to measure the number scattered at a particuar ange. 4 Consider a pane wave incident on the target. The effect of the potentia is to produce a phase change. Interference of the incident pane wave and the scattered wave, some distance from the nuceus expains the scattering probabiity. 5 The theory is most easiy done by repacing the incident pane wave with a sum of spherica waves. ikz ikr cosθ = ϕ = e = e = B r) ( θ ) in ( 0 Y =, 0 We imit the scattering theory to =0. You shoud be abe to show that an incident nuceon with <10 MeV KE ab has a cassica AM simiar to a wave with =1. Thus if KE is < 10 MeV or so, we can put B = 1, and Y m! (1/4π) 1/. 4π sin δ σ = 6 The cacuated cross section is k This has a vaue of 5 b, which is ony 0% of the measured vaue. Reason is that we used nucear potentia parameters for from our soution of the deuteron S=1. As the neutron, with spin ½ approaches the proton in the target (with spin s = ½, the probabiity of couping to 1 is 3 times that of couping to 0. Thus the measured cross section is σ = ¾ σ 3 + ¼ σ 1. So 0 b = ¾ (5 b) + ¼ σ 1 ¼ σ 1 = 0 - ~4 = 16 b So that σ 1 ~ 64 b. Confirms our concusion that nucear potentia depends on reative spins of interacting nuceons. 7 The (n,p) scattering cross section at high energy when observed in the CM frame was not forward peaked as expected. About haf of the scattering was at arge anges. This impied conversion of a p!n and n! p during the interaction. 35

3 Lecture 6 In the ast ecture, I indicated that components of the nucear potentia, other than S=1 (which we get from the deuteron BE), must be obtained by N-N scattering. Since in this case we can have the spins aigned to give S=1 or S=0. (hence see if there is a difference in the potentia depending on spin aignment). In addition we can vary the energy of the projecties and hence the AM, and check any dependence on the re AM of the nuceons. We showed that if we kept the energy of the incident proton ow (<a few MeV) the AM quantum number was =0, and the cross section was isotropic. The debrogie waveength was > width of the scattering potentia, and by anaogy with optica scattering, the diffraction pattern was isotropic. We cacuated the cross section for n-p scattering and using the parameters for the S 3 potentia (from the deuteron) we cacuated a vaue of 5 b. We saw that the theoretica vaue is significanty beow the measured vaue at ow energies. The reason for this was that in scattering we are not imited to S=1, we aso have S=0. As we noted before, the nucear potentia for S=0 is smaer than for S=1. (Reca the S 1 state of the deuteron was unbound). Assuming that the arge difference is due to singet scattering we determined the reative strengths of σ 3 and σ 1 as 5 b and ~64 b respectivey. This is a huge difference, and an important indication of the difference in the singet and tripet components of the nucear potentias. This confirmed our observation from the fact that the S 1 excited state was ess bound than the S 3, that the nucear force must be spin dependent. In fact it is not possibe to quantify this difference from these scattering experiments. It was resoved by scattering VERY ow-energy neutrons (0.01 ev) from hydrogen MOLECULES, which come in forms Para- (proton spins S=0) and Ortho- (Proton spins S=1). These neutrons have debrogie waveengths of 0.05 nm (much arger than the separation of the nucei in the H mo. So that the neutron interacts coherenty with BOTH nucei, and scatters coherenty. The resuting interference pattern (scattering cross-section) aows the S=1 and S=0 potentias to be quantified, and the S=0 one is 40% weaker. The reason scattering of nuceons with energies <a few MeV is not much hep in determining the compete nature of the N-N potentia is that their debrogie waveength is bigger than the potentia radius: the Anguar distribution is isotropic. In terms of the optica anaogue, the ange of the first diffraction minimum given by sinθ =λ/d is at an ange greater than π (in the CM system.) As we use higher energy probes, we woud expect that the scattering cross section woud begin to show forward peaking since now the waveength is smaer that the potentia width.. So et us ook at some resuts for n-p scattering when E p is much higher. From our optica anaogue we woud expect the cross section to peak at smaer anges as λ/d decreases. 36

4 Expected n-p scattering cross section Ep = 5 MeV Ep = 100 MeV The cassica reason for this expectation is that the impuse given to the nuceon as it gets into the range of the nucear potentia is sma if it is moving quicky (the SPEED bus-effect). 90 Scattering ange (CM) 180 fina momentum Momentum transferred p Scattering ange Initia momentum p K p/p ~ F t/p ~ F(r/v)/mv r = width of potentia -V o r ~ V 0 /mv (F = -dv/dr) ~ V 0 /K If K > V 0 ange of scattering is sma When they did n-p scattering at 90 MeV they got the foowing: (Note we are in the n-p CM ref. Frame.) Not what was expected! There is no peaking at forward anges. There is an equa number of neutrons (or protons) scattered through arge anges as there are through sma. The expected forward-peaked cross section woud have invoved an interaction as shown in the top diagram of the figure beow. The expanation of what was observed is shown in the ower diagram. Scattering of neutrons by protons at 90 MeV. Resuts potted in the CM reference frame. 37

5 The interpretation of the data was that in about haf of the scatterings (whist in the region of the N-N potentia) a p changed to a n, and vice verca. This exchange effect is one of the terms in the N-N potentia (caed the exchange term). The NN potentia might be written: + P V Scattering occurs via Ψ * VΨ f i so + P VΨ = Ψ i i = Ψ + PΨ i i repacing i with one can write this as Ψ + PΨ since interchanging a P with a N is equivaent to spatia exchange, it is aso equivaent to the parity operation. So PΨ = ( 1) Ψ so that VΨ = Ψ + PΨ [1 + ( 1) ] = Thus if is odd V is near zero! i.e. Nucear potentia depends strongy on. It woud be unfair of me to suggest that we have reay covered N-N scattering. Most of the information about the nature of the N-N force was obtained from more compicated scattering than =0. I do not intend to cover that. You wi find it in Povh Chap16.1, Enge Chap. 3 and Krane Chap. 4. At higher vaues of the interference pattern becomes more compex, and the anaysis more difficut (and more subjective). The foowing is a ist of the components of the N-N potentia and what they depend on V ( r ) = V ( r ) 0 + V + (V + V ss ( r )s T LS 1.s ( r )( 3( ( r )( s 1 s / h 1 + s.x )( s ).L / h.x ) / r s 1 s ) / h

6 1 The centra potentia (a function of r ony) attractive short-range centra potentia with a repusive core. Modeed as a square we for nuceons with and parae spins c = 0.4 fm (core width) V o = 73 MeV width b = fm Spin dependence Depth significanty ess for the p and n aigned anti-parae. In genera the potentia depends on whether the two nuceons are in a reative singet (S=0) or tripet (S=1) state. We observed this from facts. The S=0 state of the deuteron is unbound, The n-p scattering cross section is very different depending on whether the nuceons coupe to S=1 or S=0. 3 Tensor potentia We have aready noted that this is necessary to expain the QM of the deuteron. We stated that it must depend on S.r. 4.s dependence (spin-orbit) As we as the so-caed static components to the nucear potentia, scattering experiments show that it aso depends on dynamic variabes such as the reative nuceon momentum (in practice the AM).The potentia is deeper if and s are parae. The simpest form of this is a term such as V SO (r).s N-N potentia depends on: Couping of spins Couping between and s s and r (tensor component) From n-p scattering From deuteron For L>0, and study of repusive core need higher energy scattering. The figure above shows the forms of the various components of the nucear potentia In essence one of the questions we need to ask, and to resove at east quaitativey, is how is this force promugated? A cue to the answer to this question can be got from the (n,p) scattering at higher energies data that I mentioned above. 39

7 This was unexpected, and particuary interesting reative to the microscopic source of the nuceon potentia. If we ook at the reaction from the CM frame where the p and n approach with equa mom and energy we can see the reason. What do we detect neutrons! Where shoud they appear? At forward anges since KE is greater than V o, the higher KE the smaer the transverse impuse. Yet there are as many neutrons scattered through arge anges as sma. How come? In about haf the cases the n-p scattering ange is sma as expected since KE is much greater than V o. For the rest the neutron has been transformed into a proton and vice verca! So that when viewed in the CM system, the exchanged partice scattering ange appears to be arge. This observation eads on to the ast section of the ectures deaing with the N-N potentia. I want to discuss the meson exchange theory of this potentia. π-meson theory of nucear potentia We now ook at the source of the binding potentia between the nuceons. The so-caed one pion exchange potentia (OPEP). It was proposed way back in 1935, by Yukawa, that the nuceons were surrounded by a fied of massive partices. These virtua partices, now caed π-mesons or pions, were emitted from a nuceon, and existed for a short time. At the end of their ifetime they were re-absorbed back into the nuceon. However if within their ifetime another nuceon came within the range of this fied, the pion coud be absorbed by this nuceon. The momentum woud thus be transferred from one nuceon to the other, and transfer of impuse is akin to a force. This exchange of mesons was postuated as the source of the nuceon attractive potentia. The picture is a direct anaogy of the source of the eectromagnetic fied. Each charge is pictured as surrounded with a fied of virtua photons, and it is the exchange of these photons that produces the EM potentia. Because the photons are massess, they trave at the speed of ight, and can have an infinte range. The pions on the other hand have a mass of about 140 MeV/c, and its range is consequenty finite. More on this in a moment. There are severa questions that shoud be raised: 1. How come the existence of these virtua partices does not vioate energy conservation?. How does exchange of partices ead to an attractive potentia? 3. Does the mode fit the bi? 40

8 1 The mass of the pions The pion comes in 3 kinds: π +, π - and π o. The masses are π + and π MeV/c π o 135 MeVc This sma difference has some impications that we wi ook at ater. You shoud reca that we said the nuceon was a partice with an isospin of ½, with two projections t z = + ½ being a neutron, and t z = - ½ being a proton. In the same way the pion is a partice that has an isospin t = 1, and thus can have 3 projections: t z = +1 is a π -, t z = 0 is a π o, and t z = -1 is a π +. It is an isospin tripet. Now how come these pions can be created without vioating conservation of energy? In cassica mechanics this is vorboten, however in quantum mechanics vioations are aowed within the Heizenberg uncertainty imits. E. t ~ h in this case the energy vioation is 140 MeV/c, so we see that the pion can exist for a time t ~ h / E ~ (ev-s)/ (ev) ~ sec not very ong, however if we assume that the pion has a veocity cose to that of ight (an over estimate) it can trave a distance ~ x m. ~ m, 1.4 fm The approximate range of the nuceon potentia. Interestingy, the π meson athough predicted in 1937 was not discovered unti 1947 in cosmic ray studies. The pion is in fact unstabe and decays to a mu meson. A rea meson can be made from the coud of virtua mesons surrounding the nuceon by providing sufficient energy to pay back the energy bank. This is usuay done by bombarding a nuceon with another nuceon with at east 140 MeV of kinetic energy. Why an attractive potentia? First et s ook at what we might expect to expain the N-N potentia. In terms of the 90 MeV n-p scattering data we discussed before, where the AD was symmetric about 90 degrees. We said that this was expained in terms of a p " n. The mechanisms possibe are: n " p + π - and p " n + π + That is a neutron emits a virtua π - into its fied and becomes a p, or conversey a p emits a virtua π + into its fied and becomes a n. The interacting p or n absorbs the pion and changes into the opposite nuceon. Interestingy in that scattering experiment, the p " n and n " p exchange accounted for about haf of the events. The non-charge exchange is accounted for by the creation of a virtua π 0 meson. i.e. p " p + π o and n " n +π o. p-p and n-n interactions can ony come about by exchange of a neutra pion. Note that these possibiities dictate that a proton can have in its meson fied either a π + or a π o, but never a π -. A neutron on the other hand can have in its meson fied either a π - or a π o, but never a π +. Thus the charge distributions of these two nuceons wi be markedy different. 41

9 Note that (p,p) scattering is forward peaked at high energy as we expect, whi (p,n) has forward and backward scattering. You shoud carify this in your own mind in terms of what type of meson is exchanged in the interaction. High energy eectron scattering, simiar to that which we mentioned in ecture 1 confirms that the charge distribution of protons and neutrons is very different The proton charge distribution is everywhere positive, way out to the range of the nucear potentia. For the neutron it is positive at sma r, where the p from n " p + π - woud be, it is positive, but where the π - woud be at arger r it is negative. You no doubt have reaised why the p and n have intrinsic magnetic dipoe moments that are arger than for the eectron, and in particuar why the neutron has one at a! Exchange eading to attractive potentia Now the expanation as to how the exchange of a meson eads to an attractive force is not simpe to understand. In any cassica iustration, such as that used in 1 st year texts of scatters exchanging piows, eads to a repusive force (I have actuay thought of exchanging boomerangs, but this aso eads to repusion). There is no cassica exampe. The exchange force is a purey quantum mechanica phenomenon. The exampe I have used, and is used by Eisberg, is the H + ion. In this situation we have a stabe combination of two +ve protons, which must repe each other, remaining stabe by virtue of the exchange of the singe eectron between the two protons. Above is the eectrostatic potentia of the eectron associated with the two protons in the ion. The protons are separated by ~1.1 angstrom (10-10 m). You can see that this is the PE diagram for two protons. Note the energy scae ev is the BE of a singe p-e system. Aso shown is the WF of the eectron. Note again that it ooks ike a H s-state WF for R arge. The important thing to note is that the vaue of ψ shows that the singe eectron spends most of its time BETWEEN the two protons. Now we want to know if the p-p system wi form a stabe system with the energy of the system being a minimum for a particuar separation of the protons. This is iustrated in the next figure. 4

10 Here the energy of the tota system (e + p) is potted as a function of Proton separation. When the p s are we separated, the eectron sees itsef as beonging to one of them. Hence it has a potentia energy of ev. As the separation decreases the energy of the eectron decreases. It begins to fee the attraction of the other proton. Remember it spends most of its time between the protons. The imit of this attractive potentia on the eectron woud be when R=0, and we have Z= and the eectron woud bind as for the He + with BE x 13.6 ev = ev. On the other hand the repusion between the protons can increase essentiay to infinity, so it dominates, and we get the PE diagram shown. potentia energy ev ev ev tota energy Proton energy 1.0 A Eectron energy The H + ion Potentia energy as a function of proton separation Separation of protons (A) Note that there is a minimum, and hence a stabe separation for the ion. This comes about ony because of the motion of the singe eectron which is exchanging between the protons. This is an extremey good anaogy for the OPEP situation. 3 Does it do the job? We it certainy gives a potentia that has the basic shape to fit the experiments. I don t want to spend a ot of time on it, but you shoud know that the meson fied that surrounds the nuceon is anaogous to the photon fied around a charge. The eectrostatic potentia is obtained from the reativistic energy of a photon: E = p c or -p +E /c = 0 repacing p and E with the appropriate operators: E= ih gives ( h d dt h c 1 d V V = 0 c dt Lapace' s equation e V = 4π ε r, p= ih d dt )V = 0 43

11 For the meson with mass m the energy equation becomes E = p c + m o c 4 equation which eads to an 1 d ψ moc ϕ = ψ c dt h Kein Gordonequation soutionis e ψ = g r µ r mc whereµ = h Ψ(r) The form of the Yukawa potentia This is as far as I want to go with discussion of the N-N potentia. We are now going to ook at more compicated nucei. A that a we can do for these is to provide modes. These modes have assumptions, and are aways ony approximations. 44