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Background for the Spheroidal Nuclear Model Proposal

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Background for the Spheroidal Nuclear Model Proposal
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On December 11, 1975, James Rainwater delivered his Nobel Lecture “Background for the Spheroidal Nuclear Model Proposal” in Stockholm, before an audience of academicians and other dignitaries. Slightly more than six months later, on July 1, 1976, he repeated the lecture in Lindau, before an audience mainly consisting of students and young scientists. For a Nobel Laureate coming to a Lindau Meeting the year after having received the Nobel Prize, this is not untypical. For us, it becomes an asset, since it means that many (or even all) of the illustrations shown on the lecture slides are available in the Nobel Lecture. In those days, such lectures tended to become rather technical and sometimes difficult to understand. Not so this time, for several reasons. One reason is that Rainwater for a long time had been teaching students, another is that he basically was an experimental physicist lecturing on a theoretical topic. As an experimentalist, he was not so interested in the theoretical machinery but more in the physical effects. Actually, it is an interesting fact that the Royal Swedish Academy of Sciences included Rainwater among the recipients of the 1975 physics prize. The other two, Aage Bohr and Ben Mottelson, were bona-fide theoreticians and had spent many years developing the spheroidal model of the atomic nucleus, starting with a paper by Bohr published in 1951. But the experimentalist Rainwater had published a paper already in 1950, where he suggested that the model might be useful, and that is certainly the reason that he was included among the three physics laureates. But a detailed understanding of the arguments are locked up in the Nobel archives at the RSAS. According to the Statutes of the Nobel Foundation, a time interval of 50 years must go before historians of science are allowed to look through the material, which means that not before January 2026, the detailed arguments will become known! Anders Bárány
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Transcript: English(auto-generated)
One was related to something that more recent that I had been involved with and
The other was the subject of the Nobel lecture Which refers to things from? 1949 1950 which in a way is somewhat ancient history the decision was that the ancient history should Provide so I'll ask you to be
Understanding that I'm not speaking about things which are quite recent during the period 1949 1950 academic year at Columbia and in fact During my main career as a physicist. I have been involved with experimental
subjects that particular time We had received funding from the Office of Naval Research For a synchro cyclotron something like the Berkeley synchro cyclotron and At that time I was involved mainly with trying to get the radio frequency system drive for this
Operating so that most of my time was spent at the cyclotron Making these attempts the cyclotron at that time the radio frequency system Operated but such that all of our meters read zero and you look inside and you see saw the effect of a fluorescent tube and
our main problems at that time were to put in sweeping grid so that We were able to operate with the proper voltage for acceleration During that period I was sharing an office room 910 in the pupine physics building
With all bore this was To prove of great benefit for subsequent developments that I will mention during the period from about 1948 to about 1962 I was also involved in teaching an advanced
nuclear physics course at Columbia University And of course when you teach a course in some subject you are somewhat forced to Become more expert than you would be and more familiar with topics than you would be if you were not teaching it What I will try to describe now
Is how I understood things as well as I can reconstruct it at that time For my proposal which was made in the paper published in mid 1950 suggesting that one
except the mayor Johnson shell model, but Remove the constraint that the nucleus is spherical and allow the condition that it be distorted and in particular The emphasis was on the fact that the shell model itself contains the mechanism
for the distortion Well, let me take things now in a little bit more orderly fashion the conceptual developments of the Concepts of the nuclear theory in a sense began with Ernest Rutherford's alpha particle experiment
Scattering experiments in 1910 where he showed in fact that the nucleus was very small But 10 to minus 12 centimeters or smaller and that the atom as a whole then Had the electrons around previously there had been models For example where you had electrons and positive charge and some sort of a
intermixed mush of some kind but this immediately led to a picture where in 1913 Niels Bohr was able to exploit the concept for the electron orbits about the nucleus with the
introduction of the Quantization conditions where you have in essence the earliest shell model Picture for electrons about the nucleus. This was of course a great triumph and of course led to his Nobel Prize This was extended by many workers and in particular. There was the Wilson Sommerfeld
quantization rules that generalized the quantization condition to all the degrees of freedom or the integral of P-sub-i d-q-sub-i is an integer times 8 planks constant where
P-sub-i and q-sub-i are generalized momentum and coordinate in terms of understanding Say atoms more completely the chemistry of atoms one needed more conditions and in 1925 there was the concept by Goudsmit and Uhlenbeck
That the electron in fact was not just a simple negative charged object but that it had a span of a half and then when you add to that the proposal by Pauli of the exclusion principle You are then able in principle to build up the concept of the periodic system of the elements and understand in a way
How chemistry works? This was in a rather crude period Whether immediately thereafter quantum mechanics evolved with the Schrodinger equation Heisenberg Dirac
Bringing it into full bloom. It was immediately applied to practically everything that one can think of and For English-speaking people. This is seen by reference to the text or the Treatise of 1935 by Condon and shortly the theory of atomic spectra where you see in fact that the
Theory is a rather advanced stage now for the electron orbits and shells about the nucleus one notices that one knows to a high order of accuracy what the force law is namely it's the Coulomb force and
The Treatment for the hydrogen atom could be done before quantum electrodynamic complications essentially with complete accuracy When you got to two electron system, it became something that you had to do approximately
and in fact for the helium atom, you know that you do this with Variational methods and you can get results quite closely but you have to take an enormous number of terms a variational approach if you try to do lithium or potassium or
Iron or something like that things tend to get somewhat out of hand for a reasonably exact calculation and you have to use somewhat more hand-waving arguments and Plausibility Arguments as to what would take place in particular where you would average over the effect of the other electrons
on a given electron and the treatment of their interactions in the case of the nucleus Attempts were similarly made but in the 20s the only particles that were known The electron the proton and when you tried to make a nucleus with protons and electrons
this led to great complications and The progress was essentially zero when the Neutron was discovered by Chadwick in 1932 This was a breakthrough and immediately
One had proposals that you should in fact consider the new nucleus It's made up of neutrons and protons and Except that you didn't really know what the force law was, you know that it was strong Nuclei were reasonably stable, but you could produce
Reactions or you had radioactivity artificial radioactivity and natural radioactivity But the exact force law equivalent to the Coulomb force was not really known during the 30s 1930s The subject There was an initial attempt to
develop the concept of a nuclear shell theory in the this you would consider say a spherical box kind of problem and you would say put neutrons and protons in this box and If you do this and Treat the neutrons and protons as obeying the Dirac equation and the Pauli principle
Then the first shell would be the lowest shell the first s state for the neutrons and protons and You can put two neutrons and two protons to fill it up and you get helium-4 which is known to be unusually stable
And in fact the binding energy of the last neutron and proton for helium-4 is somewhat over 20 million volts If you try to add After you have helium-4 you try to add an additional neutron or proton You have to put it in the next shell and for a spherical box this would be the
first P state L equal 1 state and this shell that you're filling there with neutrons and protons extends from between helium-4 and oxygen 16 But to try to put the first one in it turns out that it won't even stick if you try to add a neutron or proton to helium-4 you
See the ground state of the system mainly as a scattering resonance that are a little over 1 million volts When you add two or more you start getting balanced systems like lithium-6 lithium-7 beryllium-7 and so on the next shell though
For a shell model a simple shell model particles in a spherical box would be closed in oxygen 16 Which is an unusually stable nucleus The next shell that one would obviously arrive at would be the one or the second s state and the first d state Come in and this is filled at calcium 40 and calcium 40 is unique
in that it's the Heaviest stable nucleus that has equal numbers of neutrons and protons So in a sense you do have something like a closed shell picture there but beyond that attempts to
Match to what you would picture a shell model would give for regions of extra stability or very frustrating because the Actual numbers were not the right numbers Now my own learning of the subject was quite a bit through
Professor betas reviews of modern physics articles in 1936 1937 one with bacher one by himself one with Livingston and in particular for the shell model
There was a paper in 1937 by Feynberg and Phillips which Attempted to Use a Hartree-Fock approach for nuclei In the region of the filling of the first pea shell beyond helium-4 towards oxygen
This was something that You obtained then the relative energies of ground states the Predicted magnetic moments and various other things the excited states The agreement was Not really very satisfying and there was great frustration for the shell model
Another feature which intended to complicate the situation Was that in the early 30s when one considered reaction theory for example a neutron or proton? incident on the nucleus how would you predict where resonances occur and
So on and the picture that was used then was something like what we call the optical model now Namely where you consider the nucleus as a region of potential some average potential and you have resonances which are very far apart in energy and and
size of the nucleus that you're involved with this was fine as long as you didn't have any experimental checks on it, but a very few years later Slowed neutrons were invented and patented by professor Fermi immediately huge numbers of experiments were done with slow neutrons or
Neutrons above the thermal energy and it was found in fact that things like gold and indium and so on Had large numbers of resonances and these resonances were of Wids less than a volt and
Depending on the element might be on the average 10 volts or 100 volts apart And this was something that was completely not able to be understood on the basis of the picture Where you considered the average potential and this led Niels Bohr and?
to and others to propose that the nucleus in fact acted like a drop of a liquid say a Drop of water where you have a tightly bound system, and if a particle comes in from the outside it doesn't really just go through, but it shares its energy with the other particles he and
born Calcar evolved the liquid drop model to explain nuclear reaction theory this indicated that in fact you would consider the Residences that you see with the neutrons to be Situations where you have an ability to excite the full degrees of freedom of the system
so that it isn't just the excitation of the incoming the rest instances of the incoming particle, but Residences for the complete nucleus Now during the period of the 40s. I was involved under professor dunning at Columbia in
Evolving slow Neutron Spectroscopy where we had our small cyclotron we would pulse it detector some distance, and we were able to Study the interaction Cross-sections as a function of energy we were of course also very familiar with the Bohr wheeler paper on fission which
Indicated that the nucleus doesn't have to be spherical if it's going to fish in it can't Fish in one at the same time stay spherical during that period when we were investigating huge numbers of residences and
field we were quite familiar with the fission concept when I began in 1948 to teach the nuclear physics course I was quite interested in the particle in the box picture and the
Concept of the shell model and gradually At least by early 1949 I became convinced that the shell model should have a high degree of validity Even before Mayer and Jensen came out with what we now accept is the correct model If I can have the first slide
And the lights down This Indicates another feature which it involved by white soccer The figure actually is from the poor models and text it shows the Binding per nucleon as a function of atomic weight for stable nuclei here we go from
0 to 250 covering the range of the stable and Man-made elements and the binding per particle This is seven and a half million volts per particle eight million volts eight and a half nine the picture that it evolved with the liquid drop picture was that as in the liquid drop that set in fact a
Given nucleon is surrounded by other nucleons And you have a nearly constant binding due to the fact that it's immersed in the nuclear fluid however for a small nucleus
The nuclei at the surface will have less binding and you have a decrease in binding Which is proportional the area this makes it such that the lighter nuclei have less binding the dots indicate this is a large scale for the lighter nuclei in this region here and
You can see the shell model effects, which are always superimposed even in atomic physics Superimposed on something that would represent average trends well the rise at the beginning indicates the contribution of the surface term, which is proportional to the
surface area in addition It was early agreed that probably neutrons and protons were in basically the same Particle in different charge states and in fact the neutron can decay into a proton
and the positron and a neutrino or If it's favorable in the nucleus of the one of the protons can change I'm sorry the neutron to electron and an anti neutrino and a proton and the proton
Could change if it were energetically favorable So the condition of stability then is the condition that You have the same number of neutrons plus protons, but you have the condition of greatest ability It's a compromised balance always where different effects tend to make you want to go different ways
if you consider the Statistics of the problem of filling a box you can put two neutrons in each space state two protons in each space state the lowest kinetic energy I would tend to think of it in terms of kinetic energy in terms of just a spherical box with say infinite walls
the Kinetic energy you fill successive levels and if you want to change for example Protons and the neutrons you have to take some of the Protons from the upper fills proton states and move them into the unfilled neutron states
so that you Increase the energy by an amount where the amount that you the average distance you move them up is proportional the Number that you've moved and the number that you move well The net effect is that you get a quadratic term and that quadratic term Tends to favor equal numbers of neutrons and protons
There's an additional effect Which says of course that the proton has an electric charge the neutron doesn't So you have the protons repelling each other z times z minus 1 Over the radius kind of term and this favors that you would only have neutrons
Well, whenever they're one term wants to go one way and another term wants to go the other way you arrive at the minimum energy condition of balance and when you go into this region you start Getting a larger and larger excess of neutrons over protons
For the stability condition and also a decrease in binding because of the decrease due to the Coulomb term the net effect at the minimum point in addition to these terms there is a pairing term where particularly for the ground state you prefer to have
Neutrons balancing each other and spin for net spin zero You can think of it in terms of the two particles per space-state but there are other ways of thinking of it which result in somewhat the same effect this has the effect that the
Nuclei that have even numbers of neutrons even numbers of protons are unusually stable And in fact most of the atoms in the universe are of that type except for hydrogen the heavier element if you have an odd number of
Nucleons, that is the atomic number odd then it's equally likely to be stable for even number of neutrons odd number of protons or the other way around and this leads for a particular I'd say even number of nucleons into two parabolas for the
energy of the system as a function of the difference between the neutron and proton number Well one of the points here is at the top it indicates neutron number and you see 20 28 50 82
126 The proton number which is smaller for the stable nuclei 20 28 50 82 and this point here, which is unusually bound is for the double closed shell Nucleus led 208 which has 82 protons 126 neutrons
Well as I say the old shell model was not able to arrive at things beyond the closed shell 20 but in 1949 in one of the issues of the physical review there were three papers in the same issue
proposing three different models which would explain things so speak in terms of the Different magic numbers the one that held up was the paper by Maria Mayer At the same time there was work going on in Germany Jensen which I wasn't familiar with but
Between the two of them they developed the subject in this picture one says well, how can we make the shell model function and in the case of Maria Mayer It was the result of a suggestion by Fermi. Is there any evidence for spin orbit coupling?
Well, if you take the neutron and proton in orbits They have their orbital angular momentum and their spin angular momentum and certainly in atomic physics. The couplings are somewhat important for the coupling between These but in nuclear physics, it turns out that you get rather large effects. And if I have the next slide
Well, what I'm showing here is the plot Again from the Bohr model some text of the position of the various single particle states as a function of atomic weight using a
Potential which includes spin orbit terms terms of The type that we've considered this shows the lowest s state how it goes As a function of the size of the nucleus zero the bottom of the well is somewhere down here
The P shell is now split into a p3 halves and a p1 half The D is split into d5 halves t3 halves and s2 half and you notice here that the state of highest orbital angular momentum lies lower and also the state or the spin or an orbit couple to give the largest value is lower and
This was a thing that higher up or the splitting skip got larger Gave a splitting of things where the high Total angular momentum state went down to the next shell and gave you the proper numbers for the 50 For the 82 and the 126 with things like sub shells at 28
the thing I wanted to point out here though was that the average binding Per particle or the or the valence nucleons tend to have an energy up here zero enter binding is there and
somewhere around 8 million volts and we note that the nuclear forces are short-range the nucleus is small and The net effect is that the kinetic energy and the potential energy Inside the nucleus not the average expectation value which includes
Contributions from where you're outside the range of the force are huge compared to the net binding so it seemed to me that if you were Considering what would happen if you tried to make the wave functions very different from just those that minimize kinetic energy in a box that
If you put very much of the states that had the same general symmetry Properties that you would make the energy go up so very high that in terms of the long-range Properties of the wave function they must look very close to what you would expect
From the shell model picture. This was something that I'd convinced myself of Before the mayor Jensen papers came out and I was rather appalled at all nuclear physics Conferences that I went to until about 1955 that some very respected theorists would get up and say The shell model seems to work. There's no basis for understanding why it should work and so on
fortunately this effect is stopped You go to the next slide, please We have to have the lights can you focus it better this
indicates an attempt to use an alpha particle picture for rillium 8 carbon 12 oxygen 16 neon 20 magnesium 24 silicon 28 sulfur 32 This is an alternate way of looking at some of the night light nuclei
Which are multiples of the alpha particle and see if it makes sense to? Consider them such structures in the case of brilliant 8 which would be 2 you would think of it as a dumbbell carbon 12 is a triangle and so on and Then you ask for how many bonds you would have and the average binding per bond the case of brilliant 8
It isn't stable it flies apart But for all of the others you get a surprisingly constant value in 2.4 something million volts per bond And in fact Linus Pauling for the last few years has been trying to do nuclear physics in this fashion
where he considers nuclei is made up of alpha particles and tritons which form the shells placing them the way the chemists do and Structures like you would have for molecules and there may be a certain validity to this next slide please
This if you take the alpha particle model. This is from Steven Muskowski's article in the hanbuch der physique of flug in the 1950s this says the Theoretical value for ground state first excited second excited and at that time these were the observed values you see
in each case that the agreement is certainly remarkably good compared to the 1937 paper of Feynberg and Phillips where they attempted to do the real show model next slide please
In at that time in 1949 in the fall semester when the Maria Meyer paper had come out we had a seminar at Columbia where all bore and I divided the time I Reviewed the evidence for the mayor at that time. I didn't know about the Jensen work
which immediately clarified much of nuclear physics that is you understood systematic relations for beta decay You understood where you had isomeric states And just an enormous amount of information that had been
You had individual evidence from individual nuclei suddenly you had an ordering of things over a large range The magnetic moments were particularly of interest If you take their picture in the earlier stages literally and look at the odd
Atomic mass nuclei you would say that all of the even nucleons pair off to give you a spherical System of zero angular momentum so that the angular momentum of the nucleus as a whole is just out of the last nuclei and If you take this literally you get the what are known as the Schmidt
limits which For L plus a half or L minus a half for odd neutron or proton The actual values are not on these limits, but they're in between But the remarkable thing was that all of those that should go with the upper limits Seem to be above all of those that should go with the lower limits. So although they weren't
Strictly on it. There was a rather clean division for the ones that should go with the upper side did in fact Tend to be above those that should go with the lower limits And this is one of the great triumphs. However There was accumulating evidence on the electrical quadrupole moments which
represents the distortion of the spherical nucleus away from a spherical shape towards a football or a cigar shape or a pancake shape Which indicated that? the nuclei particularly in the rare earth regions were very very non spherical and
in late 1949 Professor Charles Townes gave a colloquium at Columbia University Describing discussing a paper that he and William Lowe and Henry Foley had prepared Which reviewed the evidence and in fact?
His talk was essentially based around this figure and what is plotted here is the quadrupole moment in units of the square of a radius which Was taken then namely 1.5 a to the 1 third times 7 of the minus 13 centimeters or Fermi's
that is now considered a rather large value and The thing that was emphasized was in the region of the closed shells you did seem to be going through zero quadrupole moment as you should and Qualitatively it looked proper but here you have this huge peak
Where in particular say with Lutetium 176 you get a value that if you try to do with ordinary shell model methods Is between 30 and 40 times anything that you can come up with so he left it as That this was something that wasn't understood
while he was talking I Since I had been freshly thinking about the mayor shell model and the other pictures It occurred to me that the shell model itself if you remove the requirement that the nucleus be spherical in fact contained the mechanism For producing this distortion have the lights, please
the Picture that I Considered at first during the his colloquium was that just a particles in a spherical box If you have particles in a spherical box in particular a closed shell of high angular momentum
And then you add a nucleon of high angular momentum to start a new shell this the angular momentum of that nucleon and the nucleus will be the same because the Core balances off to essentially zero contribution so what you're picturing now is something of
A circular orbit along the equator of high angular momentum and then you consider the quantization rule for that that the integral of momentum around the loop be a Some integer times Planck's constant and You find that the energy Has a term which goes inversely as the square of the radius of the equator
now From the theory of fission and other treatment of the nuclei you knew that nuclear matter tended to keep the volume fixed as you distorted it so you say well suppose we let the
Spherical nucleus to start in a way where the volume stays fixed But the equator bulges and if you do this then you can see that the for each 1% increase in the radius At the equator you're getting a 2% decrease in 1 over r squared and This then is a term which is linear in the distortion the restoring term was mainly the surface term
balanced by a Coulomb term that once it in fact to distort and if you put the two together when you have a quadratic term and linear term you get a displaced parabola and the
Magnitude of the predicted effect seemed to be at least as big as the observed effect if anything that somewhat on the too large side But it was in the correct direction but then the question is what do you do in the middle of the rare earth region and the picture there was that suppose you consider the axis of symmetry of the
ellipsoid Rotation as being the Z direction and XY the plane of the equator Then you consider the average value of the kinetic energy for the X component the Y component in the Z component and
if you stretch the X and Y one way and the Z the in a compensating way the Kinetic energy terms if you take as trial wave functions wave functions that you distort the same way Would change by the square of the amount of the distortion and for a closed shell
Since you have all orientations this averages to zero so that there's no net linear term for a closed shell But suppose you take a closed shell and start and take out one nucleon What you have left is a hole and this hole if it corresponds to an equatorial orbit Once the is the absence of a term that wants the nucleus to go disc shape so it tries to go cigar shape
Then you take out two of them, and if you take them out with their Z components as large as possible plus or minus As you start emptying if you do it in a way that doesn't satisfying the momentum for the system as a whole
you Increase the linear term until you get to about a half filled shell And then beyond that you're taking out things you would want to go the other way and it would come back down so Qualitatively at least or semi quantitatively it seemed to give the right answer This essentially was the picture that I had
During professor towns talk it seemed like rather obvious thing that one would ask students on some qualifying exam As a simple problem and that I thought everybody would immediately jump on it For some reason they didn't and I'm grateful and my presence here is due to the fact that it wasn't obvious to everybody else
I apparently were frozen into different views. It was something that I discussed quite a bit with Boa Boa and Professor Lam's Suggestion others decided to get it in a more formal mathematical shape so that the paper that I
1950 physical review Meantime in discussions with all bore thing that he was interested in had to do With other aspects for example at the seminar where we both spoke The thing that he had discussed was paper that he had been working on
with Weiss cough Victor Weiss cough which had to do with the distribution of magnetic moment inside the nucleus and He was interested in for example The fact that since you would have to have the rest of the nucleus help and balancing the angular momentum that is you don't have something which for the
Partially closed shell is the definite angular momentum state you have to have the rest of the nucleus help make it come out, right? and this in fact you then let you understand why the Magnetic moments are not on the schmitt limits, but are somewhat in between
Also At about this time. It was becoming evident. I think Gertrude Goldhaber others Brookhaven had noted the systematic Behavior of the low-lying excited states in the rare earth region where you seem to have something which?
Well eventually were known to be rotational states the question of rotation of the nucleus had always been something intriguing In the early days when you tried to consider rotational states of the nucleus as a whole taking the rigid body moment you got Rotational states much too close together to agree with experiment
But the thing that all bore said well suppose you have the thing distorted now Into a cigar shape as you have a bump here and bump here this bump can move around It can vibrate Go from this way to this way and back and forth and you can I can also move around you can have
rotation and so on so He became quite interested in the general considerations and While he was at Columbia Prepared a paper which considered how one treats angular momentum and nuclei And this appeared in the January 1951 physical review
Quantization of angular momentum and heavy nuclei the subsequent developments when models and joined him and they prepared they Exploited the field His history now, and I won't discuss it and mainly in the intervening time
Since I'm mainly an experimental physicist with this somewhat accidental Opportunity to contribute to the theory Where the theorists seem to have been? Frozen by some consideration that I still don't understand very thoroughly
That this was not the right way to look at it. I As I say mainly been an admiring observer of the subsequent developments and I have the next slide please with the lights off Well, one of the things that all bore pointed out to me was that if you had a nucleus which was a spheroid
Shape and had some intrinsic quadrupole moment with respect to its symmetry axis then In terms of the time average value that you could get in ordinary experiments The maximum value would be reduced quite a bit
For example, if the angular momentum is zero, you can't see a time average quadrupole moment If it's a half you can't the angular momentum has to be unity or larger And in fact, there is a reduction term that the observed quadrupole moment is smaller by a factor which is something like I times 2 I minus 1
Over 2 I plus 1 2 I plus 3 which is a rather small value Until you get up to rather large angular momentum So in terms of the previous picture that professor towns had shown all of those Quadrupole moments were the measured quadrupole moments and if you interpreted them in terms of the intrinsic
nuclear distortions You would have to put in this correction in reverse fashion and they would be much larger Also, he had used a very large value 1.5 Faramis times a to the 1 third for the nuclear radius and One of the contributions that I was able to make later with Val Fitch with the muonic atom
looking at the transfer from the P state to the S state of the Negative muon about lead was in fact that one should consider for a uniformly charged fear That the effective nuclear size term was 1.2
so that would have made the Numbers in the ordinance larger. Well, this is from a paper by professor towns and I believe volume 39 of Flug's Hahnbuchter physique probably prepared about
1937 the volume itself came out about 1939 and it's a plot of the experimental quadrupole moments the Quantity here is the number of nucleons and In case they're odd. It's the number of odd nucleons
Here is the Intrinsic quadrupole moment that is what you would get when you undo this factor Over R squared are now using the factor 1.2 a to the 1 third Fermi Here I
He includes also Results from the Coulomb excitation during the 1950s it was established that if you have a charged particle alpha particle Proton make a near miss collision on a nucleus
you can excite from the ground state to the second excited state and or higher states and from the Probability of this happening in the cross-section you get a unique value a unique determination of the intrinsic quadrupole moment so a Very large body of information
Was able to be obtained for both odd a and even a nuclei Even a having spin zero in their ground state as to what the intrinsic quadrupole moment was and This is the figure that he had at that time Quadrupole moment in units of R squared I might point out that if you use a
Spherical shell model way of doing it all of the values would be between about here and here so The values tend to be quite large Compared to what you'd expect if you use a spherical base, so obviously the nucleus isn't spherical. It's quite distorted and
here we have two regions which are somewhat mixed up in terms of the atomic number but Represent the two main regions where you have very large angular momentum for the individual particles namely the rare earth region
Before you get to lead 208 and in the region beyond Mass about 220 or 230 or you get the still higher States coming in and you can see that there are values on this scale that correspond to about 25 there's a tendency for the quadrupole moments to be cigar shape and
This is probably due to the fact that the coulomb energy repulsion Gives the lower energy if it's cigar shape than if it's disk shape that is the protons Get further away from each other. Well this Is essentially the same picture but
With the more hugely more detail that had been evolved over about an eight-year period since the first one This is a plot from the Bohr models and book of the distortion parameter Delta these are all things which are crudely speaking the
difference the fractional difference between the major and minor axis of the ellipsoid and you see In the rare earth region, this is atomic number When you get the lead 208, of course, you'd essentially spherical But in going into the rare earth region here erbium, ytterbium, hafnium, tungsten
Disposium gadolinium samarium, you're getting distortion parameters, which are about one-third Then in the region of thorium uranium plutonium and so on you also get large distortions so that There are strong experimental evidence that one should in fact when thinking of a shell model
Think in terms not of a spherical shell model in these regions, but in one terms of one which is distorted there also been generalizations Namely you can have octopole moments and higher order moments. And in fact these show up
as observed quantities now and then my paper in 1950 I pointed out that this was not a complete theory, but a suggestion or a recipe as to what the theorists should carry out and One of the things that I expected to see rather more quickly was something that did the detailed
treatment of the energy levels versus distortion this was finally done by models and a Nelson in a more proper fashion and I Led to what we now know of as the nelson diagrams, and if I can have the next line
this is a nelson diagram in the region of 82 to 126 where for example here we have the h9 halves and The h9 halves depending on its z component of the angular momentum breaks up into the various
Parts which have different slopes so that you have these things that as you have the distortion parameter Each time one gets a slightly different definition of the distortion parameter, but they're for small values they're always essentially the fractional difference between the major and minor axis and You can see that the state energies particularly in the ones that are
mainly Non equatorial favor the cigar shape distortion next slide, please Well, this is from a review article in the Flug handbook their physique I believe volume 40 may be by Steven muskowski an article on nuclear models
where he applies the general concept with harmonic oscillator potential and And Considers the region or you're putting in this case 20 24 30 36 particles in the box of a given kind
What we see here is something where as you go into distortion You get this quadratic increase in this case But then you reach a point where a higher state suddenly crosses and is the lower state So we have a another thing here and here still
Another one crosses and you follow it and you get a thing that looks rather complicated like this in the case of 24 The stable condition is not zero distortion but over here But as you distort still more instead of following up parabola here a higher state comes in with bigger slope
You follow it and another another and so on this indicates That you can have a rather complicated situation next slide, please well in the 1966 this led to the picture that explained some of the odd effects That had been observed for subthreshold fission
Where it was observed for example that if you look at the Levels of the fissionable material that the partial width due to capture Would show just a random
variation along for the levels the partial width of the state for neutrons Would show a random variation, but the partial width for fission would be very very small except in regions where you would see Intermediate structure peaking over a number of levels and then the space in between and then another one
this was explained in terms of the picture indicated here and Was proposed by Strutinski in 9th paper in nuclear physics in 1967 and the picture that one has here which
Is a somewhat smoothed out version of the thing that Minkowski had but and with two Is that the ground state is a distorted state over here and that your excited states are? Systems as you go up one knows that the density of states increases exponentially
As you go up from the bottom here, so you have an exponentially increasing number of states but then you have this barrier top and a Second well, it's up a ways and you would have a number of states here but for the system as a whole these states must fix in with the similar states there and
For the complete fission process you have to have it go over eventually go out the picture then is that those states in the first well Which are close to the energy of the states in the second? Well be the ones that have the strong fission cross-sections where the
strength function for fission is concentrated according to the levels in the second well I Believe that's the last line ours are one more Last slide okay, that's they'll Lights place I'm finished now