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Quantum nucleonics is the study for electromagnetic transitions in nuclei that is most analogous to quantum electronics for atoms. It is rich with potential for application. Here particular attention is focused upon the nuclear analogs of spin-metastable helium to illuminate the potential utility of two and four quasiparticle isomeric nuclei.
Any grouping of electrically charged particles can radiate electromagnetic waves. Generally the characteristic size of the distribution of the charges determines the type of photons most efficiently emitted. Antennas emit radio waves, waveguide structures emit microwaves, electrons oscillating against the positive nuclei in atoms emit light and x rays, and protons and neutrons moving in the nuclei emit gamma rays. Once emitted, gamma rays are no different than x rays which often have the same energies. Since the oscillating charges in the nucleus emit their energy as short wavelength electromagnetic waves, this process is not a nuclear reaction. None of the interior particles of the nucleus escape to cause a nuclear reaction and the atom finishes as the stable (non-radioactive) ground state of the same isotope of the same element.
The nucleus is the smallest part of an atom which in turn is the smallest structural unit of physical matter. Thus, quantum mechanics teaches that the motions of the charged particles found within the nucleus will represent the highest velocities of circulation possible in a sample of any material. This is a fundamental precept that means that the very highest density of (non-nuclear) energy storage will be found in the motions of those charges in nuclei because they are confined in the smallest place known. Just as in the case of atoms, in a nucleus the movement of charges can absorb photons of electromagnetic waves, which in this case are x rays, and make a transition to an excited state of higher energy. Because of the high energy densities and great velocities, the charges usually reradiate such energies in times too short to be measured (<10-18 sec.) However, in rare cases selection rules forbid the coupling of the particle motion to the electromagnetic field and then the high energies are stored for tens and even thousands of years in those special nuclei. Such long-lived, high-energy states of excitation are termed isomeric levels and the materials are simply known as isomers. They are analogous to the metastable states of atoms and molecules.
The use of nuclear isomers to produce intense pulsed power for electromagnetic applications such as the gamma-ray laser is a truly interdisciplinary problem [1,2]. Such concepts arise more naturally in the disciplines of atomic and molecular physics and quantum electronics. However, the underlying unity of concepts is perhaps best appreciated by reference to Table 1 which charts a side-by-side comparison of the physics fundamentals governing the familiar and less-familiar domains for induced photon emission. The compelling similarities strongly support the use of the terminology, first suggested by Prof. Lev Rivlin, of Quantum Nucleonics for this clear analog of the field of quantum electronics.
Table 1: Essentials of atomic and nuclear structure at the most basic level of approximation.
|Quantized movement of electrons (-).||Quantized movement of protons (+) and neutron voids in the average positive fluid of the nucleus (relatively -).|
|Electrons move in Coulomb potential.||Particles move in spherical oscillator potential.|
|Basis states are:
|n,l,m> = Gnl(r) Ylm(q,j)
|Basis states are:
|n,l,m> = Fnl(r) Ylm(q,j)
F and G are not so different hypergeometric and exponential functions. BUT
|L < N; 1p, 2d, etc. orbitals are NOT allowed.||L is unrestricted; 1p, 1d, 1f , etc. are allowed.|
|Energy levels are: E = -Ry / n2||Energy levels are: E = (n + L/2 + 3/2) hn
where hn = few MeV.
|The Uncertainty Principle teaches that because the nucleus is very small, velocities (and so magnetic effects) will be large in comparison to atomic analogs, THEREFORE:|
|Spin-orbit coupling is a perturbation.||Spin-orbit coupling is of major importance.|
|Coupling to the radiation field is generally mediated by electric dipole transitions.||Coupling to the radiation field is important for many electric and magnetic moments.|
For the same basic reasons that one has single particle states for atoms in which an electron is promoted to a higher energy level from the ground by some form of external excitation, there are single particle states for the charges in nuclei. Moreover, the perception of building up more complex systems with more charges works about the same in both atoms and nuclei. Figure 1 shows for nuclei, the analog of the hydrogenic manifold into which electrons could be inserted, subject to the exclusion principle to build heavier atoms. The principle conceptual difference is that for nuclei there are two of the ladders of energy levels to fill independently, one for protons and one for neutrons which act as localized concentrations of relatively negative charge since they are holes in the average positive fluid of the nucleus.
Figure 1: Energy level diagram for the single particle states available to be filled by protons or neutrons in order to build larger nuclei. From the left is first the ladder for the spherical oscillator approximation, then a refinement to accommodate the finite depth of the oscillator potential, and then the splitting caused by the strong spin-orbit coupling.
The possibility for the nuclear analogs to metastable atoms is already set at this simple level of approximation by the occurrence of levels of high angular momentum embedded in sequences of levels with low angular momentum. The cases of the 1h11/2 and the 1i13/2 levels in Fig. 1 are of particular importance for the storage of electromagnetic energy in the nuclei.
To make the transition from the nuclear to the atomic perspective one addition to the concepts is required. Most nuclei of interest for the storage and controlled release of electromagnetic energy are not quite spherical in shape, often being ellipsoids of revolution. This has the effect of adding a second dimension to energy level diagrams such as seen in Fig. 1. The energies of the levels must be plotted as functions of the eccentricity of the deformation. Conceptually the continuous deformation from spherical to the prolate shape has three basic effects upon the single particle structure as shown in Fig. 2.
As shown in Fig. 2 the stretching of one axis requires a change in the nomenclature because additional quantum numbers are introduced. Shown to the right in the figure is a typical designation of two levels. Only the first number, 9/2 or 7/2 is significant at the level of analysis of interest here. While the example is for neutrons, the analogous situation for proton states is not much different.
Figure 2: Schematic representation of the effect of a passage of a nucleus from an ideal spherical shape to the usual shape of a prolate ellipsoid for an example of neutron single particle states in the mass region above 82 neutrons filling lower states.
Of major significance in Fig. 2, is the splitting of each single particle state into (2J+1)/2 sublevels, according to the absolute value of the magnetic quantum number, m, as shown for the example of the 1i13/2 level in Fig. 2. Each sublevel should have a different dependence upon the degree of deformation from spherical, some gaining energy with deformation and some losing as shown. A few levels have exceptional stability, such as the 9/2 and 7/2 states shown in Fig. 2. These are very important because they form families of ground and isomeric states which span long chains of successive elements in the periodic table.
Once split by |m|, the resulting sublevels have a capacity of only two particles of the same type; one with J "up," and one with J "down." This has a major effect in the "building up" of heavier isotopes of an element by adding more neutrons. Because spin-orbit coupling is so much stronger in nuclei than atoms, the addition of an additional neutron brings not just an addition of spin 1/2, but an addition or subtraction of the value of J describing that particular sublevel because it is bringing orbital L and spin S "locked together." Since there is a strong "pairing interaction" tending to fill lower shells in a way to align the resulting J's of the occupying particles in antiparallel, the net angular momentum for a sublevel filled with its pair is J=0. In Fig. 2 it can be safely assumed that all 82 neutrons lying below the first level shown in the drawing are paired to give J=0 for all contributions below the scale of the figure. Each neutron added to a nucleus forms the ground state of the next isotope by falling into the lowest vacant or singly occupied level. The building of heavier isotopes in this way by filling the next shell is simplified by the self-adjustment of the eccentricity of the nucleus caused by the greater number of particles as summarized in Fig. 3.
Figure 3: Schematic diagram of the filling of single particle (neutron in this example) states to build heavier isotopes from left to right. In the center two columns, the filling of the 7/2 level and addition of a neutron to the next open level has changed the eccentricity of the nucleus causing a sensitive, but unpopulated open level above 9/2 to lose energy and take a place in the hierarchy below the 7/2 level. In that case, as seen in the last column, the population of the filled level drops into the newly depressed level leaving the single "outer" neutron to begin to fill again the 7/2 level.
The significance of the such self-adjustment of the sequence of levels to be filled by additional particles is that there develop long sequences of elements and isotopes which have very similar structure. When those sequences provide for metastable levels there is the opportunity to store and then trigger the release of electromagnetic energy from such exceptionally stable structures.
At the top of each column in Fig. 3 is the value of the total angular momentum of the nucleus. For example, in the first column it is assumed that all levels below the 7/2 are filled, each with two antiparallel neutrons so that the contribution to the total from this "core" of particles is Jcore=0. Of course, the protons are filling a separate sequence of levels and may finish with an "odd" outer proton which will combine with the result of the J for the neutrons. For this example it is assumed that there is an even number of protons and so, no net contribution to the nuclear J from the protons. Then, when the first neutron is added to the 7/2 as shown, it brings J=7/2 to add to the Jcore=0 to give the total J=7/2 seen at the top of the first column.
There is an important subtlety in the addition of particles and the computation of the resulting angular momentum. Referring back to Fig. 2, it can be appreciated that the first quantum number associated with the name of the state, the 7/2 in the case of the 7/2 is not a vector of angular momentum, but a scalar because it represents one of the possible projections of the total angular momentum of the parent 2f7/2 state (in this example the maximum 7/2 projection,) on the direction corresponding to the figure axis of the prolate nucleus. In this example, that projection can be +7/2 or -7/2, both having the same energy, but no other possibility. Such projections of angular momenta onto the long axis of the nucleus are termed K quantum numbers and are quite significant in the descriptions of nuclear structure.
When a second neutron is added, as shown in the second column of Fig. 3, the strong pairing interaction forces it to enter the state antiparallel so that it contributes K=-7/2 to the total. Since these are scalars, the resulting total J for the state can only be J=0 as shown at the top of the second column. Without a change of shape of the nucleus, the third neutron would enter the 9/2 state as shown in the third column giving a total J (and total K) of J=9/2 (and K=9/2). However, the rearrangement of the levels with changed eccentricity creates the situation shown in the fourth column and results in J=7/2, K=7/2, to use the full designation. At this point K=J, but that will later change for excited states.
Within the same framework of states just used to illustrate the building of different elements by adding protons, and the building of heavier isotopes by adding neutrons to the single particle manifold of states; the creation of excited states can be understood to result from the promotion of a particle from its normal "ground state" to some higher open level, just as in the case of atoms. The energy needed to promote the neutron or proton can be imparted by collisions as is the usual case in nuclear physics, or by the absorption of a photon from an electromagnetic field as is more customary in atomic physics. To dispense with two other exotic analogies, it should be noted that if the promoted particle is given enough energy it can enter a free state, as in the case of the ionization of an atom. For nuclei the analog is termed an evaporation of the particle. The other exotic corresponds to the double excitation of an atom and can result from the promotion of neutrons or protons from filled shells in the core of the nucleus to various open levels of higher energy. Generally we are not concerned with such processes.
As shown in Fig. 4, in the case of atoms, the ground state of a helium atom can be conveniently conceived as being two electrons filling the 1s level with spins antiparallel. At this level of approximation the helium triplet metastable can be understood to be the result of promoting one of the electrons from the 1s to the 2s state and then "flipping" its spin so that the two are parallel. The resulting state has J=1 and a long lifetime against spontaneous radiation because of the difficulty of coupling a spin-flip transition to the radiation field.
Figure 4: Schematic comparison of the formation of metastable atoms and isomeric nuclei.
In the lower half of Fig. 4 is shown the analogous situation to the formation of metastable helium for the excitation of protons in the interesting region of the periodic table near hafnium and tantalum. The ground state is a persistent 7/2 for protons shown with that (and lower levels) filled for the case of Hf. By promoting one of the protons to the first open state 9/2 and "flipping" it, one possible analog of metastable He is formed as shown. Since there are two unpaired particles contributing to the total J such a combination is termed a 2-quasiparticle state of excitation (the prefix being explained later.) However, in the nuclear case because of the strong spin-orbit coupling, flipping the spin of the proton also flips the orbit with its angular momentum. Such a "flipped" state is termed a "time-reversed orbital" by the nuclear community to emphasize the fact that both components of J for the particle turn over while locked together. As can be seen in Fig. 4 the result is a total J = 7/2+9/2 = 8 and K = 8 as well. This is a very large total J to try to radiate away spontaneously. In an archetypical example, 178Hf, this K = 8 state has a half life of about 4 seconds; quite long for nuclear excited states. The energy of excitation is equal to the energy of a broken pair, 1 - 1.5 MeV, a value which depends very little on the state of the particle or even whether it is a proton or neutron.
The neutrons in 178Hf are in a similar situation with the ground 7/2 being filled, as shown in Fig. 3, and with the first open level being 9/2. Another possible analog of metastable He can be formed by promoting one of the 7/2 neutrons into the time-reversed 9/2 state. The result is again a state with J = K= 8 and an energy of one "pairing interaction." So there are two possible metastable states of 178Hf with the same J, same K, and about the same energy of excitation. In this case quantum mechanics is relentless and the two possibilities "interfere," so that the state which actually occurs in nature is a superposition of about equal parts of the proton metastable and the neutron metastable arrangement, with J = K = 8 and the energy of a pairing interaction. In this case it is a "quasiparticle" which has been promoted and flipped, being part proton and part neutron, and hence the seemingly peculiar terminology is actually appropriate. Admittedly, this is an unusual case resulting from the accidental coincidence of proton and neutron states with the same J, K, and energies and it usually does not arise. However, it is the same type of mixing of otherwise pure states which is believed to be essential to the controlled release of the excitation energy stored in other metastables.
The 4 sec. state of 178Hf is interesting as an example of many analogs and extensions of more familiar atomic structure, but it is not a very good metastable for any use because of the short lifetime against spontaneous radiation. In our 10 years of study of this problem the best metastable that has been found [2,3] is the 4-quasiparticle isomer of 178Hf obtained by promoting into time reversed orbitals both a proton and a neutron as described above. In this case the "quasi-" is unnecessary since there is no accidental degeneracy and the result is a state with J = K = 16 and an energy of two pairing interactions. Because of the 16 units of angular momentum to be flipped, this state has a half life against spontaneous radiation of 31 years. It stores 2.445 MeV per nuclei.
To affect the lifetime of such an isomer as the 4-quasiparticle (4-qp) state of 178Hf there must be more states of excitation to mix when desired into the single particle states described so far. These can arise from the rotation of the nuclei as a whole (or nearly so) about an axis perpendicular to its long axis. This rotation is completely analogous to that of a diatomic molecule, giving excitation energies reasonably approximated by the expression for a rigid rotor, E = EJ,K + BI(I+1), where the first term is the energy accruing from the single-particle configurations and the I are vector totals of the rotational quanta about the axis for this type of rotation and the J of the single particle states. The result for nuclear structure is that each -qp state forms the origin of a "rotational band" built upon the state. Since the I and the basic J's from the -qp states do add vectorially, the resulting I (including the rotor quanta,) increase even though there are no changes in the promotion of the constituent particles. However, since the I and K are not parallel, the growth of I arising from a more energetic rotor does not affect the projection quanta, K upon the long axis of the nucleus. Each rotational band of excited states is customarily distinguished by the value of K of the parent quasiparticle state. An example of the rotational band built upon the ground state of 170Hf is shown in Fig. 5, together with a graph of energy of excitation as a function of the value of I.
Figure 5: Graph of the excitation energies of the "rotational band" built upon the ground state of 170Hf, plotted on the left as a ladder of successive energies and on the right as function of the parameter I, where I is the quantum number in the simple rigid rotor approximation.
Such a band for which the I values contain the largest contributions from rotor quanta is termed the "yrast" band, and is usually the ground state band, GSB. If the rigid rotor model were absolutely accurate, a graph such as shown in Fig. 5 would be a linear function of I(I+1), just as in the analogous case of the energy levels of a diatomic molecule. However, as in that comparable situation the moment of inertia is not absolutely constant so the rotational constant B has some dependence upon the amount of rotation and higher-order terms are necessary. What is most distinctive in the nuclear case is the relatively small range of variances of the rotational constants with respect to changes of mass or element, when compared to the ranges found for diatomic molecules. In such a comparison, nuclei appear to be all about the same size for mass changes of the order of 10%, so the rotational bands do not vary greatly with element or isotope. Then, it is reasonable to expect that in the interesting region of Hf and Ta there should be open energy levels with J = 8 and 16 found at energy levels around 1.0 and 2.8 MeV, respectively, that are built upon the ground single particle state with K = 0. (The example is constructed assuming an even number of each kind of particle, and in the case of an odd number, the odd rotational levels appear and the K is different by the contribution of the single unpaired particle. However, the general features do not change.) Cascading transitions between such rotational levels occur very rapidly because of the strength of electromagnetic moments of higher order than are common for atoms and molecules.
The energy levels seen in Fig. 6 are organized into two pairs of single particle states, one pair on either side of the GSB in the center column. The left member of each of the pairs is the sequence of states for protons and the right member is for neutrons. In the right pair the excitation energies are assumed to be additive, so that the total energy from promoting one neutron and one proton together can be compared with the corresponding energy of the levels of the ground state band.
Figure 6: Schematic representation of the way in which the accidental coincidence between energies and J values might provide for quantum mechanical mixing of states with different K-values of the projections. Such "K-mixing states" could contain "parts" of both the large-K isomer and the low-K ground structure, superposed so that they could be radiatively connected to either type of band with electromagnetic transition moments of low order.
As emphasized in Fig. 6 there is an accidental coincidence in both energy and total I around I = 8 and I = 16. If the bases sets of states shown in Fig. 6 were perfectly orthogonal, there would be no chance for an electromagnetic transition to occur between the 4-qp isomer at the top of the right pair of single particle states and the corresponding member of the GSB. There is a selection rule for electromagnetic transitions which greatly hinders those for which the multipolarity, M of the moment mediating the transition is less than the change in K needed. Since transitions requiring a high multipolarities are very improbable, a transition for which K = 16 would be highly forbidden. However, if the states in the different columns were not perfectly "pure," then the accidental resonances could be expected to mix components from each pure state into the levels actually occurring in nature, just as happens in mixing the components of the actual J = K = 8 level in 178Hf. It is the expectation that such mixing actually occurs  which has motivated proposals to use x-ray photons to excite the 4-qp isomer in 178Hf to a proximate state of mixed parentage with respect to K in order to trigger the release of the stored energy into a cascade of electromagnetic transitions between members of the GSB populated from the decay of the K-mixing level.
As reviewed previously  and elsewhere in these Proceedings, this plan for the exploitation of K-mixing levels for the triggering of rapid isomer decay had been proven already in previous experiments on other isomers, such as 180Ta. That isomer 180Tam, was one of the most initially unattractive vehicles for experiment because it required a change of 8 quanta of angular momentum between isomer and ground state. However, because a macroscopic sample was readily available in mg quantity, 180Tam became the first isomeric nuclide to be excited by the absorption of x-rays to a K-mixing level from which cascade to the ground state subsequently occurred.
The 180Ta nuclide is the rarest stable isotope found in nature at 0.012% abundance and is the only naturally occurring isomer. The 1+ ground state has a halflife of 8.1 hours while the 9- isomer, 180Tam stores 75.3 keV with a half life in excess of 1.2 x 1015 years. In an experiment conducted in 1987, 1.2 mg of 180Tam was exposed to bremsstrahlung from a 6 MeV linac and a large fluorescence yield was obtained. This was the first time a (g,g') reaction had been excited from an isomeric target and was the first evidence of the existence of giant pumping resonances as transitions to K-mixing levels were called at those times. Simply the observation of fluorescence from a milligram sized target proved that an unexpected reaction channel had opened since grams of material had been usually required in this type of (g,g') fluorescence experiment. Analyses [5,6] of the data indicated that the partial width for the dumping of 180Tam was nearly 0.5 eV.
Transition energies to the K-mixing level were determined for the 180Tam(g,g')180Ta reaction in a series of irradiations  made at the S-DALINAC facility in Darmstadt using fourteen different endpoints in the range from 2.0 to 6.0 MeV. Fitting of the data to accepted models of resonance absorption by adjusting trial values of the integrated cross section, (sG) determined values for the dumping of 180Tam isomeric populations into freely-radiating states. Integrated cross sections  were 1.2 x 10-25 cm2 keV and 3.5 x 10-25 cm2 keV for K-mixing levels at 2.8 ± 0.1 MeV and 3.6 ± 0.1 MeV, respectively.
The integrated cross sections for 180Ta were of unexpected magnitude, exceeding anything previously reported for transfer through analogous atomic or molecular process for "optical pumping" of lasers by two orders-of-magnitude. In fact, they are 10,000 times larger than the values usually measured for nuclei.[8-12] They serve as a convenient basis for an estimation of the parameters which might reasonably be expected for the triggering of the 178Hf 4-quasiparticle isomer. The Breit-Wigner expression for the resonant absorption of photons by nuclei , scales as the inverse square of the transition energy. Systematic studies  have suggested that the excitation energy of the K-mixing level in the 178Hf nuclide could differ little from the excitation energy of the 16+ isomeric state, itself. The difference in energies which would have to be supplied by the absorption of an x-ray has been reported  to be less than 300 keV, a value 10x smaller than the corresponding transition energy for 180Ta. Since that was an upper limit, it is interesting as well, to consider the possibility of a transition energy in 178Hf which is 100x smaller as well. Then assuming other factors were more slowely varying between the Ta and Hf examples, the two hypotheses would result in cross sections larger in Hf by factors of at least 102, and perhaps 104, respectively.
As usually done, (g,g ') experiments use bremsstrahlung sources which emit x-ray continua. As a consequence it is not just the cross section which determines the yield, but rather the integrated cross section, the product of cross section and the effective width in energy of the reaction branch leading from isomer to K-mixing level and then to a component of a band built on something not decaying back to the initial isomeric state. If that width were determined by the lifetime of the K-mixing state for nonradiative disassembly into a low-K component mixed into it, the width could be assumed to be comparable for 180Ta and 178Hf. In that case very large values for the integrated cross section, (sG) could be reasonably extrapolated from the measurements on 180Ta to expectations for 178Hf as follows:
(sG) = 10-23 cm2 keV for a transition energy of 280 keV,
(sG) = 10-21 cm2 keV for a transition energy of 28 keV.
While surprisingly large, these values would be consistent with scaling studies of transition energies to K-mixing levels in this region of masses , and would be in general agreement with the Breit-Wigner absorption model usually appropriate . Such values should make an actual demonstration reasonably straightforward with an isomeric target and x-ray source of modest scale and strongly encourage such an experimental effort.
Link to Summary of Ta-180 Experimental Results
 "The Coherent and Incoherent Pumping of a Gamma Ray Laser with Intense Optical Radiation," C. B. Collins, F. W. Lee, D. M. Shemwell, B. D. DePaola, S. Olariu and I. I. Popescu, J. Appl. Phys. 53, 4645 (1982).
 "Progress in the Pumping of a Gamma-Ray Laser," C.B. Collins and J.J. Carroll, Hyperfine Interactions, 107, 3 (1997).
 "Evidence for K-Mixing in 178Hf ," C.B. Collins, J.J. Carroll, Yu.Ts. Oganessian and S.A. Karamian, Hyperfine Interactions, 107, 141 (1997).
 National Nuclear Data Center Online Evaluated Nuclear Structure Data File (ENSDF), Brookhaven National Laboratory, 1994.
 "Depopulation of the Isomeric State 180Tam by the Reaction 180Tam(g,g')180Ta," C. B. Collins, C. D. Eberhard, J. W. Glesener and J. A. Anderson, Phys. Rev. C 37, 2267 (1988).
 "Accelerated Decay of 180Tam and 176Lu in Stellar Interiors through (g,g') Reactions," J. J. Carroll, J. A. Anderson, J. W. Glesener, C. D. Eberhard and C. B. Collins, Astrophys. J. 344, 454 (1989).
 "Resonant Excitation of the Reaction 180Tam(g,g')180Ta," C. B. Collins, J. J. Carroll, T. W. Sinor, M. J. Byrd, D. G. Richmond, K. N. Taylor, M. Huber, N. Huxel, P. von Neumann-Cosel, A. Richter, C. Spieler and W. Ziegler, Phys. Rev. C 42, 1813 (1990).
 "Resonance Fluorescence in Nuclei," F. R. Metzger, in Prog. Nucl. Phys. 7, ed. O. Frisch (Pergamon, New York, 1959).
 "Calibration of Pulsed Bremsstrahlung Spectra with Photonuclear Reactions of 77Se and 79Br," J. A. Anderson and C. B. Collins, Rev. Sci. Instrum. 58, 2157 (1987).
 "Activation of 111Cdm by Single Pulses of Intense Bremsstrahlung," J. A. Anderson, M. J. Byrd and C. B. Collins, Phys. Rev. C 38, 2833 (1988).
 "Calibration of Pulsed X-ray Spectra," J. A. Anderson and C. B. Collins, Rev. Sci. Instrum. 59, 414 (1988).
 "Activation of 115Inm by Single Pulses of Intense Bremsstrahlung," C. B. Collins, J. A. Anderson, Y. Paiss, C. D. Eberhard, R. J. Peterson and W. L. Hodge, Phys. Rev. C 38, 1852 (1988).