Nuclear Chernistry-State
of the Rrt for Teachers
Nuclear Shapes From the Mundane to the Exotic Steven W. Yates Department of Chemistry, University of Kentucky, Lexington, KY 40506-0055
The shape of the atomic nucleus is one of the most fundamental nuclear properties, and along with the mass and volume, i t is a n important characteristic to consider i n describing nuclei. Nuclei are known to be composed of nucleons-protons and neutrons-and to he surrounded by electrons. (The internal structure of the nucleons can be ignored in this discussion.) Because the nucleons are much more massive t h a n t h e electrons, the nucleus contains most of the mass of the atom. Moreover, i t has been known for many decades that nuclei are only a few femtometers (lo-'' m j in diameter, whereas atomic diameters are several orders of magnitude larger, on the order of hundreds of picometers. A useful way to think of the scale of the nucleus is in terms of more familiar dimensions. If a typical atom were the size of a football stadium, with the electrons whizzing around the goal posts, the hot dog stands, etc., then the nucleus would be a small marble on the fifty-yard line, and i t would have a mass greater than 10 million tons. In making this analogy and suggesting that the nucleus is marhle-shaped, we have already expressed an expectation that nuclei are spherical. Surprisingly, most atomic nuclei are not spherical. Although considerable progress has been made in recent years through inventions that "see" objects as small as individual atoms (e.g., the scanning tunneling microscope), the information we obtain about the shapes of nuclei must be of a n indirect nature. The experimental methods used will not be described here, but i t is clear that any probe of the nucleus must have a wavelength comparable to the nuclear dimensions. Nucleon scattering or the observation of the decays of excited nuclear states are among the more common methods used to explore nuclear shapes.
Shapes of Classical Objects
By examining the shapes of classical objects, we can perhaps gain some insight into the shapes of nuclei. About 300 years ago, Sir Isaac Newton suggested t h a t a rotating spheroid would flatten a t the poles. This prediction clearly contradicted the views of the leading astronomers of the day. For example, J e a n Dominique Cassini, t h e wellknown Italian-born French astronomer, contended that the Earth, the best-studied macroscopic spheroid, was elongated a t the poles. We now know, of course, that Newton was correct and that the Earth as well as other planets in our solar system are slightly flattened a t the poles; they are oblate. The extreme case of this effect is exhibited by Saturn (Fig. 1) for which the equatorial diameter is more than 10% larger than the polar diameter. The extent of this flattening of the planets can be quantified by defining a n ellipticity. The ellipticities of the planets in our solar system are shown in the table ( I ) .Remember that nuclei are quanta1 (microscopicj systems and not macroscopic objects, so the oblate character of the planets may not be representative of nuclei.
Oblate Nucleus Quadtupole Moment c 0
Spherical Nucleus
Ouadruoole Moment.
o
Prolate Nucleus 3uadrupole Moment z 0
Fgure 1, Image of Saturn fromthe Voyager 2 mission. The flattening at the poles gives rise to an oblate shape.
Figure 2. Shapes of the nucleus. The deformationis indicated by the quadrupole moment. (Reproducedfrom ref 2 with permission.) Volume 71
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Shape and Stability of Nuclei Deviations from Sphericity Like theelectron configurations ascribed to thenoblegases, wrt:iln numbers of'nuclwns e h b i r unusual stahlhty These "magic nurnheri" of nucleons a n 2,8,20,28,50, $2, and 126, andnuclei with these numbers of protons or neutrons appear to be nearly spherical. As the number of nucleons deviates from thwc ;na& numbers, d c h m e d nuclear shnptti acquire weator iitabihty In fact, in their ground stares most nuclei are prolate, rather than oblate likethe planets. The Nuclear Quadrupole Moment The nuclear quadrupole moment, a property somewhat analogous to the ellipticity, is used to describe the deviations of nuclei from sphericity As illustrated in Figure 2, spherical nuclei have no quadrupole moment. Oblate nuclei display negative quadrupole moments, whereas prolate nuclei have positive quadrupole moments. The magnitude of the quadrupole moment indicates the degree of the deviation from a spherical shape (2).
to nuclei in excited states.) Nuclei with such unusual distortions are called superdeformed (4). Even greater elongations, leading to axis ratios of approximately 3:1 are anticipated; recent data suggests the observation of such an extreme nuclear shape, a "hyperdeformed" nucleus (5).Oblate superdeformed nuclei with axis ratios of 1:2 might also be expected, but none have been observed to date. These exotic nuclear shapes are presented in Figure 4, which also shows two examples of more complex deformed shapes. These pearlike and bananalike nuclei are related and are produced by higher-order, socalled octupole deformations. Although evidence for the former is abundant, searches by nuclear spectroscopists for the latter continue.
Excitations What evidence do we have that nuclei exhibit deformations? If most nuclei are indeed prolate, they should be subject to excitation through rotation, much like a diatomic molecule. Chemists are very familiar with the regularly spaced lines that occur in the spectra obtained from rotational excitations of molecules. As in Figure 3, similar spectra (of y rays, because we are dealing with much more energetic transitions in nuclei) with evenly spaced lines provide evidence that nuclei undergo collective rotations. The energies of these excitations are given by the expression Figure 3. Gamma-ray spectrum observed from the de-excitation of a superdeformed nucleus. i is the moment of inertia of the deformed nucleus, where . and I is the nuclear spin, that is, the total angular momentum of a specific nuclear state. Surprisingly, this expression is identical to the one long used to describe the rotations of molecules. Moreover, from the energy spacings between the regularly spaced y rays, illustrated in Figure 3, we can determine the moment of Inrrtln and the d&ree of d~,formatlonFrom such data, the ratlo ofthe axes ol the wherold that cave n w to the rotations can be deduced.
SUPERDEFORMED a:b=3:1
Exotic Shapes There are many known examples of prolate nuclei with axis ratios of 2:l (see Fig. 4); these can be taken a s our first examples of exotic nuclear shapes (3).(However, this extreme deformation, like most of those below, is applicable Ellipticities of the Planets Planet
Ellipticity 0 0 -0.0034 -0.0059 -0.0637 -0.1 02 -0.024 -0.0266
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto ? Ellimicity isdefined as (R,- R,)/R,, where R ~ a n dRa atethe polar and equa-
torial radii (I).
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Figure 4. Examples of exotic nuclear shapes. Nuclear radii along the symmetry axis and a principal axis perpendicular to the symmetry axis are denoted by a and b.
Different Distributions within the Nucleus The Neutron Halo Throughout our discussion, we have assumed that the protons and neutrons in the nucleus behave similarly. But is this necessarily the case? Recent evidence suggests that in a number of light nuclei with a large excess of neutrons the neutrons act as a "halo" around a relatively inert nuclear core
composed of both neutrons and protons (6).This concept is illustrated in Figure 5, a representation of the "Li nucleus. Although the nucleus of 'Li, the heaviest stable isotope of lithium, contains three protons and four neutrons, "Li has eight neutrons in its nucleus. The plot in the top portion of Figure 5 illustrates that the density of protons in "Li falls off much more rapidly than the density of neutrons. Therefore, it may be appropriate to envision this nucleus as composed of a core of neutrons and protons surrounded by the aforementioned neutron halo. Independent Behavior of Nucleons
___
The independent behavior of neutrons and protons in nuclei is not a recent observation; it was first identified as the giant electric dipole resonance. In this mode, illustrated in Figure 6 for a spherical nucleus, the distributions of neutrons and protons are viewed as interpenetrating fluids in oscillatory motion. Such nuclear oscillations have been confirmed in many nuclei, both spherical and deformed. Additional torsional modes are available in deformed nuclei. In Figure 7, the possible angular oscillations of deformed neutron and proton distributions are illustrated. Althoughevidence of the entire nucleus under~oinr .. .. scissorslike motions upper figure, has nalr I n w ~,tlt:~intd, n1.m) nucle~have propemles succesrinr?,that this t\we of motion 15 vo?i~hlefor the outermo~nucle& around'; core of neutrons and protons (lower figure) (7).We have recently measured the lifetimes of these oscillations in our labratories a t the University of Kentucky; they exist for only a few tens of femtoseconds.
Radmr [fm]
I - proton
,1 @ ,
Figure 5. Artistic conception of the " ~ i halo nucleus with two neutrons outside a neutron-proton core. The plot indicates the proton and neutron radial densities of " ~ i .
Summary From the above examples, i t is clear that a number of unusual nuclear shapes are known and have been characterized. I t seems inevitable, however, that as our knowledge of these quanta1 systems increases even more exotic nuclear shapes will emerge.
Acknowledgment I would like to acknowledge R. F. Casten, D. P. DiPrete, W. D. Ehmann, E. A. Henry, and B. M. Sherrill for providing figures used in this paper. I also wish to thank W. D. Ehmann, J. D. Garrett, J. D. Robertson, and M. Villani for their helpful comments.
Proton Distribution
Giant Scissors Mode Excitation
Low-lying Scissors Mode Excitation
Neutron Distribution
0
Z
Figure 6 . Time evolution of the giant electric dipole resonance. The soherical neutron and oroton distributions are envisioned as oscillatiiginterpenetrating fluids
neutron distribution proton distribution F g-re 7 Sc ssors nooe exc tallons n nLc e n tne g an1 sc ssors mooe the en! re prolon and ne-lron o strloJl ons "noergo a 5 ~ : s sors ne osc at on wnereas lne lon- y ng sc %ors mooe nvo ves only the outermost nucleons Volume 71
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4. %in.P. J.:Nyak6,B.M.:Nelson.A.H.:Simpaon.J.:Bentley.M.A.;CranmerGordon. H. W.; Forsyih. P D.; Hows. D.: Mokhfar.A. R.; Morriaon,J. D.: ShsrpeySehafer. J . F:Sleften.G.Phvr Rea kt(. 1986.57.811-814. 1. Garrett, J. D. In The R c p o m e d N u d e i Unde? Erlmme Condilionr; Bmglia. R. A . 5. Gaiind~Uribarti,A.:~~d~~w~.H. R.; Bsll.d.C.:Drake.T.E;Janzen,V.P;Kuehner. Bertsch. G . F.. Eds.: Plenum: New York. 1988: pp 1 5 8 . J. A.: Mullins. S. M.: Persaon. L.: Pr#uost. D.; Radford, D. C.:Waddington. J. C.: 2. Ehmann. W D.;Vance. D. E.R~diiihhhiif'y'~'~'~dNz~clmrM~!hod~~fAn~l.~si~: Wiley: Wsrd.D.: Wyss. R. Phyr Rev. Lpli 1993.71.231-234. New York. 1991: p58. 6. Tanihata, I. Nucl Phys 1991.A522.27Sc-292c. R. F:Ra&.A.; 7. RichtecA InCol,Bmwra~nipt?ririNi~clmrSlruchnP/zy~iii;C8rten, 3. The IroSyin Lahorntoy RmmICI1 Opp011unili~swith RndimcIioe Nuclmr B e a m . Moshinsky, M.: Plttel, S.. Eds: World Scient36c:Singapore. 1988. pp 127-164. R e p o t No. LALP91-51. Los Alamos National Laboratory. 1991.
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