Dirk Walecka, “Electron Scattering by Nuclei” - LNS46 Symposium: On the Matter of Particles
WALECKA: I want to go back to the beginning and talk about why we do nuclear physics. Why is your physics interesting? First the nucleus consists is a unique form of matter consisting of many baryons in close proximity. All the forces of nature are present in the nucleus, strong, electromagnetic and weak. Provides a unique microscopic laboratory for testing the nature of the fundamental interactions.
The nucleus manifests remarkable properties as a strongly interacting quantum mechanical many body system. You saw some of these illustrated in the lovely pictures of Steve Koonin yesterday. After all, most of the mass and energy in the visible universe comes from nuclei and nuclear reactions.
We now know that the nucleus is composed of a new underlying set of degrees of freedom, quarks and gluons held together with remarkable new forces described by QCD, quantum chromodynamics. The baryon itself is a complicated many body system. Nuclear physics is crucial to the understanding of the universe, the early universe, the formation of the elements, supernova, neutron stars.
To me nuclear physics is really the study of the structure of matter. In 1989 the field of nuclear physics formulated a long range plan for the next decade. And I've just listed the chapter titles in the section titles in the chapter the Scientific Frontiers to give you a taste of where nuclear physics is and where it's going, nucleons and nuclei, hadrons and nuclei, quarks and nuclei and hadrons, flavor physics, essentially strangeness in nuclei, the nucleus under extreme conditions, the dynamics and thermodynamics of colliding nuclei, confined and deconfined phases of nuclear matter, precision tests of the fundamental interactions and nuclear physics in the universe.
We actually have three levels of understanding of nuclear physics. The first is what I like to call the traditional nonrelativistic many body problem. We use two body potentials, many body Schrodinger equation and then construct currents from the properties of free nuclei. At another level, we can describe the nucleus as an interacting relativistic many body system employing hadronic degrees of freedom, baryons and mesons and quantum field theory. I like to call this QHD, or quantum hadrodynamics. At the deepest level, we know the nucleus is a strongly coupled system of colored quarks and gluons interacting. Our description is based on a Yang-Mill's theory, based on internal color symmetry.
That theory has two absolutely remarkable properties. Asymptotic freedom, at short distances the particles are effectively weakly interacting. And confining, we do not see the underlying degrees of freedom as asymptotic scattering states in the laboratory.
Why electron scattering? Well first the interaction is known. It's given by quantum electrodynamics. The interaction is relatively weak and this means that we can make measurements on the target without greatly disturbing its structure. We interact with the local electromagnetic current density in that target. What we measure is essentially the Fourier transform of the transition matrix element of the local electromagnetic current density and the target multiplied by the molar potential, or the appropriate lepton matrix element. And electron scattering has the great advantage that we can vary the three momentum transfer and energy transfer independently.
Thus for a fixed energy transfer to the target, we can vary the three momentum transfer and map out Fourier transform of the transition densities. And just to give you the classic example, this is the diffraction pattern you see if you scatter electrons elastically from the calcium 40 nucleus. This is cross-section, or intensity, versus momentum transfer, or effectively scattering angle at fixed incident energy. And here is an actual diffraction pattern, maxima and minima. You see these rings of maxima and minima in the laboratory.
I ask you to notice the scale here. The scale goes over 13 decades. By inverting that Fourier transform, we can actually determine the charge density in the calcium 40 nucleus. This is charge density. It's a function of distance from the center in fermis. The little green band indicates the experimental uncertainty in inverting that Fourier transform. The heavy dashed line indicates a description of the charge density in QHD. Dotted and dashed lines are a description in the traditional picture as shown by Steve Koonin yesterday.
By varying the electron kinematics, we can actually separate the interaction with the charge and current density in the target. That interaction is actually very rich, because not only is there Coulomb interaction with the charges in the target, there is a magnetic interaction with the convection current in the target arising from the motion of the charges and also with the curl of the intrinsic magnetization in the target. There's an intrinsic magnetization because after all nucleons are little magnets.
And another advantage of electron scattering is by going to large momentum transfer, we can bring out high multiples. Electrons, thus, provide a precision tool for looking at the nucleus. And what of course the great advantage of electrons is they can be copiously produced in the laboratory. In addition, their interference effect arising from the exchange of, in addition to a photon, of a Z0. And those can be seen by looking at parity violation, for example, looking at the difference in cross-section for right and left handed electrons.
The S-matrix is essentially what we wrote before with an additional term, which for heavy Z's is, again, just the weak lepton matrix element. Now times the Fourier transform of the matrix element of the weak neutral current density in the target. We know this has both a vector and axial vector component. And thus by looking at the product of these two terms and the product of those two turns and interfering them with this, we have parity violation.
If this is now assumed known and measured, we can in fact measure this. And that means we can use electrons to measure the distribution of weak neutral current in the nuclear system. These experiments are terribly difficult. But in fact, in a tour de force experiment completed at Bates doing elastic scattering from carbon-12, the feasibility of such measurements has now been established. And we're ready for the next generation of those measurements.
In addition, Bjorken scaling in the deep inelastic region provides dynamical evidence for quarks, allows a measurement of the quark momentum distribution and allows us to test QCD predictions on approached scaling.
Cross-section for inclusive electron scattering essentially measures two response surfaces, which can be separated by varying the electron kinematics. These response surfaces are functions before momentum transfer squared and nu, the Lorentz invariant which reduces to the energy transfer in the laboratory system.
Let me give you a sketch of what these response surfaces look like, just for your orientation in this and the following talk. Here I've sketched the response surfaces both for a nucleus and for nucleon. This is the four momentum transfer. And this is essentially the energy transfer except I've divided by the four momentum transfer squared and call that variable 1 over x. So here, we see a elastic scattering from the nucleus falling off with the characteristic form factor.
We see excitation of discrete states in the target, giant resonances, quasar elastic scattering, scattering from a single nucleon Doppler broadened by the Fermi motion. Scatter, we see excitation of various excited states of the nucleon. And in the case of nucleon itself, if you look off here at very large K squared but fixed values of 1 over x, in fact, the cross-section, the response surfaces, are independent of K square. That's Bjorken scaling. And that, because of the absence of the form factor, that gives us evidence for the point like dynamic-- in point like scattering from the quarks in the interior.
If we scatter from discrete nuclear states, then in fact we can make a multiple analysis of these two response surfaces. The transfer surface is the sum of the squares of the reduced matrix elements of the transverse electric and transverse magnetic multiples. And the logitudinal response, or Coulomb response, is the reduced matrix element of the Coulomb multiples.
Now in fact in a talk like this, it doesn't make any sense to try and give formula. But I wanted to show this transparency for a particular reason. I thought I would tell you how I got into electron scattering. I was a graduate student here working with Viki. And of course we were brought up on Blatt and Weisskopf, which I devoured. It's a marvelous book. And I mastered the material in chapters 12 and Appendix B. And for those of you who don't have total recall, those are the sections on electromagnetic interactions and multiple analysis.
And I went to Stanford, and as a young faculty member there, I taught nuclear physics. And I went through all those multiple analyzes. And I also did the Weisskopf estimates, which Viki mentioned yesterday. And the Weisskopf estimates are simple estimates for the relative strength of the various electromagnetic multiples in gamma transitions.
And also at that time at Stanford, there was a young faculty member who every time he saw me would corner me and show me this lovely data on the form factor for the excitation of the first quadrupole resonance in nuclei, which had the same form factor across a wide variety of valence structures. That was Henry Kendall. And so too Jerry Friedman was also there at that time. I think Jerry owned the deuteron during that period.
So to try to get some insight into this, I put together the multiple analysis, which I learned from Blatt and Weisskopf. And it's all there. It's all in Blatt and Weisskopf, together with a simple dynamical model of collective excitations. And I was hooked. And I haven't looked back since.
These are the multiples. Now the Weisskopf estimates are based on the long wavelength limit of the multiples. In long wavelength, it's only the minimum multipolarity that contributes to the transition. And the thing that always excited me about electron scattering was that in fact by varying the momentum transfer, you can actually see all of these multiples. And in fact, under right conditions, you can actually see the maximum multiple that can contribute to a scattering.
And let me just give you a couple of examples of that. This is elastic magnetic scattering from niobium 93 done at Bates. This is the momentum transfer, transverse response. At low momentum transfers, you see just the magnetic dipole scattering from magnetic dipole moment. But then you see the M3, M5, M7 and out here you in fact see the M9. Niobium has 9-- is 9 has plus. And the shell model configuration is a G9 halves proton.
Well, what do you learn from that? In fact, what you learn from it is this. Now I've actually sketched this for an F7 halves in vanadium so that I could get it on a 10 fermi square. If you take the vanadium nucleus and line it up along the z-axis, the intrinsic magnetization also points along the z-axis. And this is the surface of half maximum magnetization density just to give you a picture of where the proton actually is in the vanadium nucleus.
So in fact by measuring all of those multiples, you can in fact determine the detailed microscopic distribution of the current and magnetization of that last valence particle in the nucleus. You can actually see that the vanadium nucleus is a little current loop, is a little magnet and a little current loop. And you can actually measure the distribution of that current loop. And of course since the neutrons also have intrinsic magnetization, you can measure neutron distributions this way also.
Here's an inelastic cross-section again done at Bates. 60 degrees q of 2 inverse fermis, there is one level that dominates the spectrum. And from its form factor, that is the area under that curve, it's identified as an M6 transition. Well in fact, an M6 is the maximum multipolarity you can get when you promote a particle from the 1D5 half shell to the 1F7 half shell. The maximum multipolarity you can get 6 to 6 minus magnetic transition. And by going to high enough momentum transfer, you can make that dominate the cross-section.
Let's move on to another part of the spectrum. This is the quasielastic peak. The data is in fact from HEPL. And when I checked the reference, it was Moniz et all. If they can put me in an experimental session, they can put you on an experimental paper.
The curve is a calculation by Rosenfelder, and it's based on a relativistic mean field theory and local density approximation using a density from the relativistic Hartree calculation. So in fact, there are no parameters in that. It's a fully relativistic calculation, conserved current and the description of that data is all that you could ask. So you think you really understand what's going on in the quasielastic region.
Well what would be even simpler would be the Coulomb response, because after all that just comes from the charges. The Coulomb response is significantly less than this. And to separate it, you have to do make a Rosenbluth plot. People have worked very hard to do that over the past several years. And when you do it, you find this.
If you take the area under the Coulomb response, which is the Coulomb sum rule, then this is in fact what you see in calcium 40. These are the experimental points. And this is a curve calculated exactly the same way as the theoretical curve on the previous picture. So in fact, the experimental Coulomb response falls below our best theoretical estimates by something like a factor of 2.
Now, you can say, OK, maybe that's correlations. Because if you remember the Coulomb sum rule properly normalized, the area under the Coulomb response is 1 plus the Fourier transform of the two body proton proton density. In fact, this is familiar in condensed matter physics. With neutron scattering, you measure the Fourier transform of the two body density of the time.
Well I did some calculations. This is 1 minus the inelastic sum rule. And this is what you get if you just had a Fermi gas. This is for nuclear matter. These are poly correlations, just with a non interacting Fermi gas. And then I took the Fourier transform of the two body density that Vicky and Luis Carlos Gomez and I had used in our discussions of nuclear matter. If you have just a hard core, you get that.
If you have a more realistic [? Muskovski-Scott ?] potential, you get something like this. Weill in the region of highest energy on the previous picture, you might get something like a 10% reduction. It's very difficult to see how you can get anything like a factor of two reduction. I would say the fact that the experimental value of the Coulomb sum rule, which should be the simplest thing that we have in nuclear physics, the fact that we're-- our theoretical understanding of that is off almost by a factor of two is really one of the major outstanding problems in nuclear physics.
Well people have talked about this correlation function for years. It is after all the basis of our calculation of all the properties of nuclear matter. What does the two body density look like at short distances in the nucleus? In fact, it's never been measured. That is it's never been measured until recently. It was measured at Bates.
It was measured for a simple system, which is helium 3. But one nice thing about helium 3 is that you have a calibration check. Because you can also do the experiment from tritium, which has only one proton. And there is no proton proton correlation in one proton system.
So in fact here the-- this is the Fourier transform of the two body proton density in tritium. And here's what it is in helium 3. And for the first time, we have some knowledge of what the two body distribution is in short distances in nuclei coming from electron scale.
Let me go on to another part of the spectrum. This is the excitation of the nucleon. This is part of the original Slack, MIT collaboration experiment. It's beautiful data. It's even more impressive if you realize that the elastic peak has been suppressed. The elastic peak doesn't occur on this curve.
So in fact, this is the excitation of the delta. And then these are higher-- higher resonance groups. Now I can look at the form factor, or area, under that curve as a function of momentum transfer. That's what it looks like. In fact, this is from-- these points are from Marty [? Bridenbuck's ?] thesis here at MIT.
There is one more point down here that's hard to see. But there is in fact a point here at K squared equals zero coming from photo production. The theoretical curve is a calculation with essentially within the framework of QHD. It's actually-- we did it. But it's actually a synthesis of work done by a lot of people, [? Fubini, ?] [? Nambouin, ?] [? Van ?] [? Togen, ?] [? Zagery, ?] [? Denery, ?] [? Adler. ?]
It's a hadronic calculation. And the interesting thing, which one really has to contemplate, is that the scale here-- this is 4 GEV squared, which is 100 inverse fermin squared-- and the interesting thing is a hadronic description can work down to that very short distance scale for this process.
Well let me say a little bit about why we do coincident electron scattering. So far I've talked about singles experiments. Why do you want to get coincidence experiments, which is in fact what's going to be made possible by the pulse strecher ring here at Bates and by CEBAF. So now not only do you have the electron, but you can measure what's coming out of the targeting coincidence. And the electron variable set up a nice coordinate system here, the momentum transfer, the normal to the scattering plane and the third perpendicular vector.
And theta and phi are the usual polar and azimuthal angles. What do you measure in a coincidence experiment? Well you measure the same thing, lepton Mueller potential, Fourier transform, transition matrix element of the current. But now the final state consists of the asymptotically of the-- it is a Heisenberg state. It's an eigen state of the full Hamiltonian. Asymptotically, it's the target in whatever state it's in and the outgoing particle. And the only thing is you have to use incoming wave boundary conditions.
Well what you learn from this? Well first thing you can do is you realize that if you go to a discrete state of the target, J pi, then in fact, you're doing-- you can do an angular correlation experiment of the type that Martin Deutsch talked about yesterday. But in this case, you're doing an angular correlation experiment with the virtual photon, whose energy and momentum you can vary independently, and the outgoing particle. That distribution is characteristic of this angular momentum.
Furthermore, you no longer have simply the sum of squares of amplitudes, you have interference terms. And that allows you to determine small amplitudes through interference. Again, you have a dial to turn, the three momentum transfer. And by turning that dial, you can bring out transitions of different multipolarities. In the case of e,e prime p, it's particularly simple and elegant. If you knock a proton up and you measure all the energies, then you can measure the energy of where it came from, or the binding energy of the residual hole.
If you measure all the momenta, the three momentum transfer and the momentum of the outgoing particle, then effectively what you measure is the Fourier transform of that bounced A wave function with respect to the momentum difference. Furthermore, if we have a coincidence capability, we can then do double and even triple scattering experiments on the outgoing particle to measure polarizations.
Furthermore, if we have enough energy and kinematic flexibility, we can access new reactions. For example e,e prime K plus, by associated production, you leave a hyper or in quark language, you leave a strange quark in the middle of the nucleus. Furthermore, by doing double coincidence experiments, we can get another handle on the core nucleon, nucleon correlations in nuclei. And finally, we can study the hadronization of quarks.
The struck quark emerges as a hadron, and we can study how that process takes place in nuclei. I want to give you just a couple of examples of each of these. This is-- I show this because it's close to my heart. This was the original coincidence experiment done on the superconducting accelerator at HEPL. And let me tell you what the configuration is. You arrange the energy transfer so you excite carbon to the giant dipole resonance. And now you look at the angular distribution of photons of the protons going to the ground state of boron 11 with respect to that momentum transfer.
And you see a characteristic dipole pattern. Now, if you say four points do not a dipole pattern make, you a right. On the other hand, at Mainz, they have now done that same experiment and this is the quality of the angular correlation data you now have from the new generation of CW electron accelerators.
Here's an example of the e,e prime p process that I talked about. This is from Nikhef. This is e,e prime p on lead- 208 going to thallium 207. This is the ground state of 3s a half whole, and here is the characteristic fall off of the Fourier transform of a 3s a half wave function. Here's the first excited hold state at 2d 3 halves. Here there's a characteristic rise and fall of the Fourier transform of a 2 d3 halves wave function.
And down here, you see nothing until you increase the momentum transfer. And by golly here, all of a sudden you see an h 11 halves coming up. You can arrange it so you see these high spin levels. And in fact, you can take the nucleus apart, level by level, shell by shell, wave function by wave function.
This is a experiment that is ongoing at Bates. It's actually another tour de force. It's a coincidence experiment. It's actually a triple coincidence experiment. It's a polarization transfer between a polarized electron and a polarized neutron. And by measuring this polarization transfer, you measure an interference term that's directly proportional to the electric form factor of the neutron. Of course the electric form factor of the neutron, it tells you about the charge distribution in the neutron. OK?
It's a triple coincidence experiment because you have to measure to detect-- the scattered electron. You've got to know there's a neutron there. And then you have to measure a scattering asymmetry to determine the neutron polarization. Now this is the actual signal. OK? And this background is accidental coincidences.
And this is done with a duty factor-- a machine with the duty factor of 1%. Now, I ask you to think of this as a sea mount and this is an ocean. And by increasing the duty factor, you can leave the signal and you can lower the level of the ocean. In fact, if at 100% duty factor, you can lower the ocean by at least two orders of magnitude. And what was a little peak there, now becomes a huge mountain. That's in fact the most dramatic evidence I can show for the advantage of having a CW machine.
How am I doing on time, Bill?
BILL: Five minutes?
BILL: Four minutes.
WALECKA: Four minutes. Why CEBAF? Let me just say a few words about CEBAF to close. There is no single characteristic that makes CEBAF unique. Rather it's a combination of characteristics. It's the kinematic range, 4GEV, the intensity, 200 mircro- amps to 3 n stations. It's the duty factor 100%, and it's the beam quality, the low immittance and the energy resolution in the beam. And my version of CEBAF scientifical is the study of the structure of the nuclear many body system, including the baryon itself, it's quark substructure and the nature of the strong and electroweak interactions governing the behavior of this fundamental form of matter.
We have an approved physics program at CEBAF, and I've just given you my version of the general areas of the approved and conditionally approved proposals, structure a few nucleon systems, single nucleon behavior in nuclei, static properties of the nuclei, strangeness and nuclei, correlations in nuclei, dynamic internal properties of the nucleon, mesons in nuclei, approach to deep inelastic scattering and weak neutral currents in nuclei. We have equipment designed, approved and under construction at CEBAF. We have three experimental halls.
We're going to have complimentary sets of equipment in the three halls. One hall will have a pair of high resolution 4 GEV spectrometers. One hall will have a large acceptance detector. Try and get everything that comes out in coincidence. And the first hall that will come on the air in less than two years from now will have a high momentum spectrometer and a short orbit spectrometer to look at decaying particles. And I'd like to finish showing you just a couple examples of the kinds of data that we expect to get at CEBAF.
These are from proposals. And this is a set of projected errors that includes the design of the equipment efficiency statistics and so on. Here's an experiment which will measure the trans-- the polarization of the outgoing proton in e,e prime P reaction from oxygen 16. And we will have enough resolution to separate those two peaks. OK? And this is what you see if you eject that proton from a p, a half shell.
And there's the expected size of the CEBAF error bars. Here are two theoretical curves. That's with a relativistic [? drak ?] optical potential. And that's with the non-relativistic optical potential in the final state.
Here is the analog of that neutron experiment I talked to about, only done on hydrogen. So it's the polarization transfer experiment on hydrogen, which gets you GEP, charge distribution in the proton. That's the world's data. And that give you an indication of the projected size of the CEBAF error bars and kinematic range on GEP.
Here is the ratio of e2 to m1 transition matrix element for the delta. It's interesting. Bob Leery's going to talk about that a little bit because you can get things like the deformation of the delta and the nucleon. That's the world's data. I asked you to look at the scale. This is now a tenth, and here is the projected CEBAF.
So this is now 0.5, and there's a set of projected CEBAF error bars for that particular experiment. And this is kind of interesting. This is the simulation of the class detector, this 4 pi detector. This is a simulation of the meson spectra that you'll see pi, 0, eta, omega and eta prime.
So in conclusion, I would just like to make two comments about the relation of MIT to CEBAF. The first comment is that the success and quality of the experimental program at CEBAF, the most important-- the single most important factor to me in the quality and success of the experimental program at CEBAF is the caliber of the young people that have been produced by this laboratory. And in fact, the majority of those people have been produced by the chairman of this session. The second is that when I went to CEBAF and set up a program advisory committee, I wanted to get the very best people I could.
I put four people on from MIT. I could have put more. Ernie Muniz, [? Dan Kowalski, ?] Bill [? Bertolozi, ?] Dale [? Donnelley, ?] in fact, Claude Williamson was there. He was chairman of the users group at that time. I got a lot of flack for that. Why do you want to put on so many people from one institution?
But we've got an approved equipment plan. We're building the right equipment and we've got an approved scientific program. And it's absolutely first rate program. So it's a pleasure to come back to MIT. It's interesting. When I come back to MIT and walk to the place, I always feel like I'm coming home.
And I think there are two reasons for that. One is that I have a lot of very close friends here. But the second reason is that this is really where I became a professional physicist, which has really been a deep source of satisfaction to me through my whole life. And to me, MIT has been and is and will continue to be the heart of that profession.