JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Looking into the Nucleus*

FRANCIS BITTER

Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Received December 22, 1952)

This paper provides a brief elementary account of the insight into nuclear structure which has been ob-tained from investigations of the hyperfine structure of atomic-energy levels.

INTRODUCTION

THE light radiated by atoms contains informationabout nuclear structure. Because of the small-

ness of the nucleus as compared to the atom-thenucleus is smaller by a factor of the order of 10-12 orless-it was at first thought that the nucleus might betreated as a point. The nucleus, consequently, wasfirst treated as a point charge, having a mass, a mag-netic moment, and an angular momentum. The chargeand mass we shall not consider further. They are ac-curately known from other than optical experiments.The magnetic moment of the nucleus is coupled to theelectronic motion by means of the magnetic field pro-duced by the atomic electrons. The coupling energiesfor various mutual orientations of nuclear and electronicangular momentum give rise to the hyperfine structuresplitting of energy levels and spectral lines.

The electron, however, does not revolve around thenucleus as the earth revolves around the sun. Theelectronic wave functions indicate that electronssweep out the entire atomic volume in the course oftheir motions. In particular, they penetrate the nu-cleus, and in some states are much more intimatelycoupled to the nucleus than indicated above.

The standard procedure for investigating nuclearstructure is to bombard the nucleus with mono-energetic particles and to observe the fragments thatfly away from such collisions. Perhaps it is not too far-fetched to regard the atom as a fabulously well-regu-lated van de Graaff generator in which a beautifullycollimated mono-energetic beam of electrons is directedat the nucleus. We can observe the electromagneticquanta which are emitted, and from these draw con-clusions about nuclear structure. These require us topicture the nucleus, not as a point, but as an extendedobject in which neutrons and protons move about in amanner which must be described by nuclear wavefunctions.

It now becomes reasonable to suppose that, if suf-ficiently precise observations of hyperfine structurecan be made, we can measure nuclear radii. The Cou-lomb potential due to the nucleus follows a 1/r functionoutside of the nucleus. Within the nucleus it does notapproach an infinite, but a finite value. The deviationof the electrostatic potential energy from the 1/r formwithin the nucleus will have a slight influence on energy

* This work has been supported in part by the Signal Corps,the Air Materiel Command, and the U. S. Office of Naval Re-search.

levels which becomes important when we considerisotopes. The addition or removal of neutrons from anucleus to form isotopes modifies the proton distribu-tion and therefore the potential within the nucleus.The energy levels and spectral lines radiated by afamily of isotopes will therefore be slightly shiftedwith -respect to each other, and this isotope shift cantell us something about nuclear-charge distribution.

Although the electron distribution in closed shellsand S states is spherically symmetrical, the opticalelectron or electrons are generally in states having apreferred axis. The same is true of nuclei. The chargedistribution of the protons is generally not spherical,the departure from spherical symmetry being measuredby the nuclear electric quadrupole moment Q. There isthen an electrostatic directional coupling between anucleus and atomic electrons which modifies the mag-netic coupling. The spacing of the magnetically pro-duced hyperfine structure components is thereforemodified, and we have a means for studying nuclearelectrical asymmetry.

Finally, we must consider not only the extendedelectrical structure of nuclei but their magnetic struc-ture. If a nucleus cannot be considered a point, itsmagnetic properties cannot be described in terms ofa dipole moment. We shall see further on that themagnetic structure of nuclei is best discussed in termsof quantum numbers, much as we discuss atomicmagnetism in terms of quantum numbers. There seemsto be a great deal of evidence that, to a first approxi-mation at least, we may consider the magnetic prop-erties and the angular momentum of a nucleus to bedue to an odd, unpaired particle. Thus stable nucleiwith an even number of protons and an even numberof neutrons all seem to have no angular momentumand no magnetic moment. The odd proton and/orneutron is then characterized by an orbital and a spinquantum number, and these orbital and spin quantumnumbers form the basis for a discussion of magneticstructure.

This is the formal basis for spectroscopic investiga-tions of nuclear structure. How much has actually beenlearned as the result of such studies? The known nucleiare shown in Fig. 1.1 As to orders of magnitude, thereare something like 100 stable nuclei having an oddparticle, and at least one piece of evidence is availablefor about 80 percent of these nuclei. This is not meant

1 Figures 1 and 2 are taken from the author's Nuclear Physics(Addison Wesley Press, Inc., Cambridge, 1950).

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234 FRANCIS BITTER Vol. 43

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LOOKING INTO THE NUCLEUS

to imply that the required job is done. Complete in-formation is available on only a small part of thesenuclei, and it is probably correct to say that in not asingle instance have we extracted all the available in-formation. The electrons still know a lot more aboutstable nuclei than we do; and when it comes to unstableradioactive nuclei, the field has hardly been touched.Of the thousand or so known radioactive nuclei, wehave some optical information on perhaps a dozen.

We shall now review in greater detail the specificitems about nuclear structure learned from detailedexaminations of atomic energy levels. In the firstplace, the results shown in Fig. 2(a) and (b) were firstobtained predominantly by optical experiments duringthe last few decades. Certain typical differences be-tween particles with an odd proton and an odd neutronare shown. The results may be qualitatively under-stood by noting that the nuclear angular momentumis made up of an orbital contribution L plus or minusthe spin of the odd particle. If this is a proton, weshould expect the magnetic moments to fall in thevicinity of L nuclear magnetons, modified by thecontribution due to the intrinsic moment of the oddproton. If it is a neutron whose contribution we areconsidering, then the orbital contribution vanishes,and there is only the intrinsic moment to be considered.This, by and large, is the behavior shown in Fig. 2.

More detailed investigations have been concernedwith the interpretation of the observed nuclear spinsas well as their magnetic moments. It has been shownby a variety of authors2 that the single particle modelcould account for the main features of all the data.It is assumed that the odd particle moves about in asteep-sided potential energy well which is attributedto all the other nucleons. The energy levels and wavefunctions for such a system are readily calculable. Ifstrong spin-orbit coupling is assumed, and the energylevels are successively filled, it is possible to under-stand the bulk of the known facts about nuclear spinsand magnetic moments. Only in a few cases are thefacts flatly in contradiction with the above predictions,indicating that a "one-particle" model is really onlyan approximation.

Atomic-shell structure is understood on the basis ofthe hydrogen energy levels, plus the Pauli exclusionprinciple, plus a certain amount of manipulation ofthe data to take the inner structures of heavy atomsinto account. Similarly, one would expect that thefilling up of the energy levels of a single particle in asteep-sided well, plus the exclusion principle, pluscertain additional assumptions about the crossing-overof sublevels in heavy nuclei, might lead to a nuclear-shell structure. This is indeed the case. The number ofparticles in a nuclear shell is commonly called a "magicnumber." The magic numbers are 2, 8, 20, 28, 50, 82,126. These nuclear configurations are particularlystable, and a variety of experimental facts support the

2 A clear exposition of the basic ideas is given by Maria Goep-pert Mayer, Phys. Rev. 78, 16 (1950).

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magic number theory. An important and revealingsummary of data is shown in Fig. 3. Here quadrupolemoments are plotted as a function of the number ofodd nucleons. Without attempting to account for thesedata in detail, we may notice here the appearance ofmore or less marked periodicity, and the tendency forzero quadrupole moments at the magic numbers, witha negative slope of the curve at these crossover points.

The remaining area to be covered concerns the veryprecise measurements of nuclear magnetic momentsand hyperfine-structure constants which have beenmade possible by atomic beam and nuclear magneticresonance' methods. Experiments making use of lightare limited by the sharpness of the radiated spectrallines, if not by the resolving power of the instrumentsused. The limit of the sharpness of these lines is givenby the uncertainty principle and the mean-life of theatom in the states involved in the radiation.

E At. (1)

For excited states At is of the order of 10-8 sec, andthis determines the sharpness of the radiated line. Formore precise measurements longer times At are greatlyto be desired.

To obtain these longer mean-lives it is necessary towork in the ground state. Spontaneous transitions be-tween the magnetic sublevels of the ground state maybe completely neglected, and the mean-life of an atomin a given state is determined by the conditions of theexperiment. The magnetic resonance method consistsof placing the atom in a magnetic field so that themagnetic sublevels are split, subjecting the atom to aradio frequency magnetic field, and observing theresonance absorption frequencies corresponding to theenergy differences between the sublevels, divided byPlanck's constant. At resonance, the component of themagnetic moment of an atom in the direction of thefield is altered. In an atomic beam experiment, this re-orientation is detected by noting the altered trajectoryof the particle in an inhomogeneous field through whichthe beam is passed. In this experiment the At of Eq. (1)is the time that the atom spends in the rf field. Withatomic velocities of the order of 104 cm/sec, and dis-tances of the order of centimeters, times of the orderof 10-3 sec are available, and consequently much more

235April 1953

FRANCIS BITTER Vol. 43

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FIG. 3. [Reproduced from "Nuclear quadrupole moments and nuclear shell structure," Townes, Foley, and Low,Phys. Rev. 76, 1415 (1949)]. The plotted oints are quadrupole moments divided by the square of the nuclear radius(1.5X1O-13 A 3)2 . Moments of odd proton and odd proton, odd neutron nuclei (except Li6 and C3 8) are plotted ascircles against number of protons. Moments of odd neutron nuclei are plotted as crosses against number of neutrons.Arrows indicate closing of major nucleon shells. The solid curve represents regions where quadrupole moment be-havior sems established. The dashed curve represents more doubtful regions.

precise measurements can be made with such a tech-nique.

In a nuclear magnetic resonance experiment, a solid,liquid, or gaseous sample of diamagnetic atoms in aconstant field is subjected also to an rf field, and theresonance condition is detected by noting the changein the electrical impedance of the rf coil. The criticaltime At in these experiments is the magnetic relaxationtime, or the time that a particle in thermal equilibriumwith its neighbors spends in a given magnetic sublevelof the ground state. In paramagnetic substances or sub-stances in which the atoms have magnetic moments ofthe order of Bohr magnetons, the relaxation time is soshort that hyperfine structure measurements can bemade only crudely, if at all. In diamagnetic substances,however, in which the only permanent magnetic mo-ments are of the order of nuclear magnetons, relaxationtimes much longer than 13 sec are common, and very

sharp resonance absorption curves can consequentlybe observed.

By combining the very accurate observations ofnuclear magnetic moments made by magnetic reso-nance methods with very accurate measurements ofhyperfine structure separations made by atomic beammethods, the interesting conclusion to which we re-ferred above is reached; namely, that in some instancesthe magnetic field due to atomic electrons is sufficientlyinhomogeneous over the nuclear volume, and the dis-tribution of magnetization of nuclear matter is suffi-ciently inhomogeneous, to require us to compute inter-action energies not by a product of the nuclear momentand the field at the center of the atom, but by an inte-gral over the nuclear volume. Although relatively fewmeasurements of this accuracy have so far been made,they have provided significant information concerningnuclear structure, particularly in the case of hydrogen.

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