nuclear physics

299 CHAPTER 11 Nuclear Structure Contents 11.1. Requirements of a nuclear model 300 11.1.1. Some general nuclear properties 300 11.1.2. Quantized energy levels 301 11.1.3. The nuclear potential well 302 11.2. Rotational energy and angular momentum 303 11.2.1. Rotational (mechanical) energy 303 11.2.2. Angular momentum 304 11.2.3. Coupling of spin and orbital angular moments 305 11.2.4. Magnetic moments 307 11.2.5. Precession 309 11.3. The single-particle shell model 311 11.3.1. Quantum number rules 311 11.3.2. Nuclei without nucleon spin-orbit coupling 312 11.3.3. Nuclear level scheme with nucleon spin-orbit coupling 312 11.3.4. The nuclear spin 314 11.4. Deformed nuclei 316 11.4.1. Deformation index 316 11.4.2. Electric multipoles 316 11.4.3. The collective nuclear model 317 11.5. The unified model of deformed nuclei 318 11.6. Interaction between the nuclear spin and the electron structure 320 11.6.1. Hyperfine spectra 320 11.6.2. Atomic beams 322 11.6.3. Nuclear magnetic resonance 323 11.7. Radioactive decay and nuclear structure 324 11.7.1. Gamma-decay 324 11.7.2. Beta-decay 326 11.7.3. Alpha-decay theory 326 11.7.4. Spontaneous fission 330 11.8. Exercises 333 11.9. Literature 333 Throughout the ages and in every civilization, pe ople have developed explanations of observed behaviors. These explanations are based on the principle of causality, i.e. every effect has a cause and the same cause produc es always the same effect. We call these explanations models . Scientists are professional model-builde rs. Observed phenomena are used to develop a model, which then is tested through new experiments. This is familiar to every chemist: although we cannot see the atoms and molecules which we add into a reaction vessel, we Radiochemistry and Nuclear Chemistry 300 certainly have some idea about what is going to happen. It is indeed our enjoyment i n developing models which causes us to experiment in science. We also want to be able to make quanti tative predictions based on our models which we therefore formulate in mathematica l terms. To allow tractable calculations most m odels involves simplifications of the "real world". Of course, since man is fallible some models may turn out to be wrong but as new dat a accumulate, wrong or naive models are replaced by better ones. We have alread y shown how one model for the nuclear structure, the liquid drop model, has helped us to explain a number of nuclear properties, the most important being the shape of the stabil ity valley. But the liquid drop model fails to explain other important properties. In thi s chapter we shall try to arrive at a nuclear model which takes into account the quantu m mechanical properties of the nucleus. 11.1. Requirements of a nuclear model Investigat ion of light emitted by excited atoms (J. Rydberg 1895) led N. Bohr to suggest the quantized model for the atom, which became the foundation for explaining the chemica l properties of the elements and justifying their ordering in the periodic system. From studies of molecular spectra and from theoretic al quantum and wave mechanical calculations, we are able to interpret many of the most intricate details of chemical bonding. In a similar manner, patterns of nuclear stabilit y, results of nuclear reactions and spectroscopy of radiation emitted by nuclei have yielded information which helps us develop a picture o f nuclear structure. But the situation is more complicated for the nucleus than for the atom. In the nu cleus there are two kinds of particles, protons and neutrons, packed close together, and there are two kinds of forc es ! the electrostatic force and the short range strong nuclear force. This more complex situa tion has caused slow progress in developing a satisfactory model, and no single nuclear model has been able to explain all the nuclear phenomena. 11.1.1. Some general nuclear properties Let us begin with a summary of what we know about the nucleus, and see where that leads us. In Chapter 3 we observed that the bindi ng energy per nucleon is almost constant for the stable nuclei (Fig. 3.3) and that the radius is proportional to the cube root of the mass number. We have interpreted this as reflecting f airly uniform distribution of charge and mass throughout the volume of the nucleus. Other experimental evidence supports this interpretation (Fig. 3.4). This information was used to develop the liquid drop model, which successfully explains the valley of stability (Fig. 3.1). This overall view also supports the assumption of a strong, short range nuclear force. A more detailed consideration of Figures 3.1 and 3.3 indicates that certain mass number s seem to be more stable, i.e. nuclei with Z- or N-values of 2, 8, 20, 28, 50, and 82 (see also Table 3.1). There is other evidence for the uniqu eness of those numbers. For example, if either the probability of capturing a neutron (t he neutron capture cross-section) or the energy required to release a neutron is plotted for different elements, it is found that Nuclear Structure 301 maxima occur at these same neutr on numbers, just as maxima occur for the electron ionization energy of the elements He, Ne, Ar, Kr, etc. (i.e. at electron numbers of 2, 8, 16, 32, etc.). The nuclear N- or Z-values of 2, 8, 20, 28, 50, and 82 are called "magic numbers". It seems logical that these magic numbers indicate some kind of regular substructure in the nucleus. Moreover, since the same magic numbers are found for the neutrons and for th e protons, we would further assume that the neutrons and the protons build their substructur e independently of each other, but in the same way. Another fact that must be indicative of the nuclear substructure is the stabilit y for nuclei with even proton or even neutron numbers. Since we know that the individual nucleons h ave spin, we could postulate that nucleons in the nucleus must pair off with opposed spins. 11.1.2. Quantized energy levels The nucleus would thus seem to consis t of independent substructures of neutrons and protons, with each type of nucleon paired off as far as pos sible. Further, the nucleons obviously grouped together in the magic numbers. From the decay of radioactive nuclei we know that the tota l decay energy ( Q-value) of any particular nuclide has a definite value. Moreover, (-emission from any particular nucleus involves discrete, definite values. These facts resemble th e quantized emission of electromagnetic radiation (X-ray, UV, visible light, Radiochemistry and Nuclear Chemistry 302 FIG. 11.2. The Sn neutron and proton shell structure in the potential well. (According to S.116 G. Nilsson.)etc.) from atoms. We may conclude a similar explanation for the nucleus: decay of radioactive nuclei, whether ", $, or (, involves a transition between discrete quantized energy levels. 11.1.3. The nuclear potential well In our development of a model of the nucleus we have so far not considered the nuclea r binding energy. Let us imagine the situation wherein a ne utron of low kinetic energy approaches a nucleus (Fig. 11.1). Since th e neutron is uncharged it is not affected by the Coulomb field of the nucleus and approaches the nucleus with no interactio n, until it is close enough to experience the strong nuclear force F, which is always attractive, i.e. F is a positive quantity. At th e n n point r the neutron experiences the strong attraction to the nucleus and is absorbed. The surface n of the nucleus is assumed to exte nd to r since this distance represents the radius of the nucleuss over which the nuclear force is constant. When the neutron is absorbed, energy is released and emitted in the form of a (-quantum. The energy of the (-ray can be calculated from the known masses of the reactants and product nuclides: E = ! 931.5( M ! M ! M). ( A+1 A n The energy released is the (neutron) binding energy of the nucleus E. The total energy of theB nucleus has thus decreased as is indicated in Figure 11.1(B); it is common to refer to thi s decrease as a potential well. The nucleons can be considered to occupy different levels in such a potential well. The exact shape of the well is uncertain (parabolic, square, etc.) and depends on the mathematical form assumed for the interaction between the incoming particle and th e nucleus. Nuclear Structure 303 Protons exp erience the same strong, short range nuclear force interaction as they contact the nucleus. However, they also experience a long range repulsive interaction due to the Coulomb force between the positive incoming protons and the positive charge of the nucleus. Thi s repulsion prevents the potential well from being as deep for protons as for neutrons. Figur e 11.2 shows the energy levels of the protons and of the neutrons in the nucleus of Sn.116 50 11.2. Rotational energy and angular momentum It is an intriguing fact that nucleons in nuclei, electrons in atoms, as well as large cosmi c objects such as solar systems and even galaxies, are more dominated by rotational than by linear motion, although in our daily life the latter seems to be a more common phenomenon. I n rotation there is a balance between two forces: the centrifugal force of inertia, which tries to move a body away from a center point, and an attractive force (gravitational, electrostatic , etc.), which opposes the separation. In the preceding section we ass umed that the nucleus existed in some kind of a potential well. One may further assume that a nucleon moves around in this well in a way not too differen t from the way the electron moves around the atomic nucleus, i.e. with an oscillation between kinetic and potential energy. With some hypot hesis about the shape of the nuclear potential well we can apply the Schrödinger wave equation to the nucleus. Without being concerned at this point with t he consequences of assuming different shapes we can conclude that the solution of the wave equation allows only certain energy states. These energy states are defined by tw o quantum number s: the principal quantum number n , which is related to the total energy of the system, and the azimuthal (or radial ) quantum number l , which is related to the rotationa l movement of the nucleus. 11.2.1. Rotational (mechanical) energy If a mass m circles in an orbit of radius r at a constant angular velocity T (rad s ), the-1 tangential velocity at this radius is v = Tr (11.1) r The kinetic en ergy in linear motion is E = 2mv, and we therefore obtain for the (kinetic) kin 2 rotational energy E = 2mTr (11.2a) rot2 2 The angu lar velocity is often expressed as the frequency of rotation, L = T/2B s. Equation r-1 (11.2a) can also be written E = 2IT (11.2b) rot rot 2 where Radiochemistry and Nuclear Chemistry 304 FIG. 11.3. A particle spinning in (a) field-free space and (b) in an external field along the z-axis.I = 3mr (11.3) rot i i2 is the rot ational moment of inertia . We consider as the rotating body a system of i particles of masses m, each individual particle at a distance r from the axis of rotation; then (11.2b) i s i i valid for any rotating body. In nuclear science we primarily have to consider two kinds o f rotation: the intrinsic rotation (or spin) of a body around its own axis (e.g. the rotation of the earth ev ery 24 h), and the orbital rotation of an object around a central point (e.g. rotation of the earth around the sun every 365 days). Equation (11.2b) is valid for both cases, but (11.2a) only for the orbital rotation of a particle of small dimensions compared to the orbital radius, in which c ase I = mr. For a spherical homogeneous spinning body (e.g. the earth's intrinsic rot 2 rotation) of external radius r, (11.2b) must be used, where I = 2mr/5. ex rot ex2 11.2.2. Angular momentum Like linear motion, rotation is associated with a momentum, called the angular momentum . For the orbital rotation the orbital angular momentum (p) isl p = mvr (11.4a) l r while for spin the spin angular momentum ( p) iss p = TI (11.4b) s rot Angular momentum is a vector quantity , which means that it has always a certain orientation in space, depending on the direction of rotation. For the rotation indicated in Nuclear Structure 305 Figure 11.3(a), the vector can only point upwards. It would not help to turn the picture upside down because the coupling between rotational direction and its vector remains the same, a s students south of the equator will agree. Quan tum mechanics prescribes that the spin angular momentum of electrons, protons, an d neutrons must have the magnitude p = SS[s(s+1)] (11.5) s2 here s is the spin quantum number . For a single particle (electron or nucleon) the spin s is alway s 1/2. In addition to spin, the three atomic particles can have orbital movements. Again quantum mechanics prescribes that the magnitude of the orbital angular momentum of thes e particles p = SS[l(l+1)] (11.6) l2 We shall refer to l as the orbital (angular momentum ) quantum number . Only certain values are permitted for l, related to the main quantum number n: for electrons: 0 # l < n!1 for nucleons: 0 # l For nucleons but not for electrons l may (and often does) exceed n. 11.2.3. Coupling of spin and orbital angular moments A rotating charge gives rise to a magnetic moment F. The rotating electron and proton cans therefore be considered as tiny magnets. Because of the internal charge distribution of th e neutron it also acts as a small magnet. In the absence of any external magnetic field thes e magnets p oint in any direction in space (Fig. 11.3(a)), but in the presence of an external field they are o riented in certain directions determined by quantum mechanical rules. This i s indicated by the angle 2 in Figure 11.3(b), when we have the spinning particle in the center of a coord inate system. The quantum mechanical rule is that the only values allowed for th e projections of spin angular momentum p(z) on the field axes are:s p(z) = SSm (11.7) s s For composite systems, like an electron in an atom or a nucleon in a nucleus, the ( magnetic ) spin quantum number m may have two values, +1/2 or !1/2, because the spin vector has two s possible orientations (up or down) with regard to the orbital angular momentum. The orbital movement of an electron in an atom, or of a proton in the nucleus, gives rise to anoth er magnetic moment ( F) which also interacts with external fields. Again quantu m l mechanics prescribes how the orbital plane may be oriented in relation to such a field (Fig . 11.4( a)). The orbital angular momentum vector pP can assume only such directions that it s l projection on the field axes, p(z), has the valuesl Radiochemistry and Nuclear Chemistry 306 p(z) = SSm (11.8) l l m is ref erred to as the magnetic orbital quantum numbers; m can have all integer value s l l between !l and + l. For a single-particle, nucleon or electron, the orbita l and spin angular moments add vectorially to form a resultant vector (Fig. 11.4(b)), pP = pP + pP (11.9) j l s pP will orient itself towards an external field so that only the projectionsj p(z) =SSm (11.10) j j are obtained on t he field axes, m is the total magnetic angular momentum quantum number ; it j can have all integer values between !j and + j. The magnitude of p isj p = SS[j(j+1)] (11.11) j2 where j = l ± s (11.12) Here j is the total (resultant ) quantum number of the particle . Nuclear Structure 307 11.2.4. Magnetic moments a. Single particles . Dirac showed in 1928 that the spin magnetic moment of the electron is: F(electron) = SSe/2m = B (11.13) s e e where B= 9.273 × 10 JT (joule per tesla). This value is referred to as one Bohr e!24 !1 magneton (B.m.). For the proton one would expect the spin magnetic moment F to bes B@m/m = B (11.14) eep n B (n for nucleon) has the value 5.051 × 10 JT and is referred to as one nuclear magneton n!27 !1 (n.m.). However, measurements show that F(proton) = 14.1 × 10 JT = 2.793 n.m. s!27 !1 The reason for the higher value is found in the uneven charge distribution within the proton. Recall that the neutron also has an uneven charge dist ribution. This gives rise to a neutron spin magnetic moment F(neutron) = !1.913 n.m. s The magnetic moment F, caused by a charge q in circular orbit , can be calculated fro m l classical physics: the current caused in the orbit, qv, times the area encircled, Br. Therefore r2 F= qvr/2 (11.15) l r Dividing (11.15) by p according to (11.4a), givesl ( = F/p = q/2m (11.16a) ll This ratio ( is called the gyromagnetic ratio . If ( and p are known, F may be calculated . l l Equation (11.16a) is valid for electrons ( e, m), but for protons only the left part. Because ofe the quantization of p(z) (11.8) the component of the orbital magnetic moment in the fiel d l direction (Fig. 11.4(a)) is also quantized: F(electron) = mB (11.17) l le Such a simple approach i s not possible for a nucleon in a nucleus, because no nucleon moves completely independent of the other nucleons, nor is the orbital path always circular. b. Atoms and nuclei . In an atom with many electrons, the spin and angular moments of the electrons couple vectorially and separately to form resultant quantum numbers (using Radiochemistry and Nuclear Chemistry 308 Line Property Electron Proton Neutron _______________________________________________________________________________________ 1 Mass(u) 0.000 5486 1.007 276 1.008 665 2 Charge ( e units) !1 +1 !1 3 Spin quantum number s 1/2 1/2 1/2 4 Spin dipole moment µ=1 B.m. µ=2.793 n.m. µ=!1.913 n.m.e p n .))))))))0)))))- 5 Orbital quantum number 0#l#n!1 0#l 6 Permitted orbital field projections !SSl…+SSl !SSl…+SSl 7 Total angular quantum number ( j=l±s ) j=l±s 8 Total angular momentum ( p=SS/(j(j+1)) ) p=SS/(j(j+1))j j 9 Particle symbolism nl nli j _______________________________________________________________________________________ Notes 2. The electron unit charge: e = 1.602 × 10 C.!19 3. The spin angular momentum has the magnitude p=SS/(s(s+1)), with the permitted projections on an external6 field axis: ± sSS (for e, p, and n = 2SS). 7. l and s couple only in the one-electron system; see §11.2.4. 8. Permitted projections on an external field axis: SSm, where !j#m#j.j j 9. n is the principal quantum number, l the azimutha l (orbital, radial) quantum number, i is the number of electrons in the particular n,l state, and j is the total angular momentum quantum number. For l=0, 1, 2, etc., the symbols s, p, d, f, g, etc., are used.TABLE 11.1. Summary of the pro perties of the atomic constituents (independent movements in a central potential field) . Electron-electron Nucleon-nucleon Line Property interaction interaction ________________________________________________________________________________________ 10 Spin-spin ( s-s) coupling, q.n. S=Gs (strong) :For each nucleon j=l±s i 11 Total spin magnetic moment µ=2(S(S+1)) B.m. =and I=Gj s i2 12 Orbit-orbit ( l-l) coupling, q.n. L=Gl (strong) ;For ee nuclei: groundstate l=0 i 13 Total orbital magnetic moment µ=2(L(L+1)) B.m. =For eo, oe nuclei: groundstate l=j L2 14 Spin-orbit coupling, q.n. J=S+L (weak) <For oo nuclei: groundstate varies 15 Resulting angular momentum p=SS(J(J+1)) B.m. p=SS(I(I+1)) n.m. J I2 2 16 Resulting magnetic moment µ=g(J(J+1)) B.m. µ=g(I(I+1)) n.m. J J II2 2 17 Transition selection rules )L±1, )J=0 or ±1 )=integer, )>0 I j 18 Multipole moment Electric dipole Electric or magnetic multipole ________________________________________________________________________________________ Electron-nucleon interactions ________________________________________________________________________________________ 19 Grand atomic angular mom., q.n. F = J + I = S + L + I 20 Grand atomic angular mom. p=SS(F(F+1)) ; field projections 0,…, SSF F2 21 External field: none Hyperfine spectrum (hfs): J levels split due to nuclear spin I 22 External field: weak ( #10 T) Hfs F-levels split into 2 F+1 levels (Zeeman effect)!2 23 External field: average ( -10 T) Electron-nucleon q.n. decouple, producing (2 J+1)(2 I+1) levels 24 External field: very strong Electron spin-orbit decouple, producing separate S- and L-levelsTABLE 11.2. Summary of atomic and nuclear p roperties associated with particle interactions (q.n. = quantum number). Nuclear Structure 309 conventional symbolism) S = 3sP (11.18a) i L = 3lP (11.18b) i which couple to form the total angular momentum (or internal) quantum number of the atom JP = LP + SP (11.18c) (see Table 11.2). Equations (11.18) ar e referred to as Russell-Saunders coupling . For the atom as a whole, the magnetic moment is F(atom) = g e p /2 m = g B m (11.19) j j e j e j where g is the Landé factor and + J # m # !J. The Landé factor accounts for the effect o f j j mutual screening of the electrons. The nuclear m agnetic moment depends on the spin and angular moments of the neutrons and protons. For the nucleus it is given by F(nucleus) = g e p /2 m (11.20) I I p where g is the nuclear g-factor and p, the magnitude of the nuclear spin angular moment . I I Because this moment c an have only the projections SS m on the axes of a magnetic field whereI m is the nuclear magnetic angular momentum quantum number ( !I # m # I), we may write I I this equation F(nucleus) = g (e SS /2 m)m = g B m (11.21) I pI I n I I is the total nuclear spin . We shall see in the next section how I can be determined. In Table 11.1 w e have summarized the most important properties of the atomic constituents, and in Tabl e 11.2 their modes of interaction. In Table 11.3 some spin and magnetic moments are given for stable and radioactive nuclei. 11.2.5. Precession Before going into the details of nuclear structure, there is one more property of the nucleon which must be considered. Both types of angular momenta, p and p, as well as thei r s l correspo nding magnetic moments, are vector quantities. Quantum mechanics forbids pP (and consequently F) to be exactly parallel wi th an external field. At the same time the external field tries to pull the vector so that the plane of rot ation becomes perpendicular to the field lines. The potential magnetic energy is E = BP@FP = B@F cos 2 (11.22) magn Radiochemistry and Nuclear Chemistry 310 Stable nuclides Radioactive nuclides ________________________________ ________________________________________________ X I (±) µ Q$ X I (±) µ Q$ tA A I I 2 ________________________________ ________________________________________________ H 1/2 + +2.793 H 1/2 + +2.979 12.33 y1 3 H 1 + +0.857 +0.003 C 0 + 5730 y2 14 B 3 + +1.801 +0.085 Na 4 + +1.690 14.959 h10 24 B 3/2 !+2.689 +0.041 P 1 + !0.252 14.262 d11 32 C 0 + Cl 2 + +1.285 !0.018 3.01×10 y12 36 5 C 1/2 !+0.702 Ca 7/2 !!1.327 +0.046 163.8 d13 45 N 1 + +0.404 +0.019 Fe 3/2 ! 2.73 y14 55 O 0 + Co 5 + +3.799 +0.44 5.271 y16 60 O 5/2 + !1.894 !0.026 Cu 1 + !0.217 12.70 h17 64 F 1/2 + +2.629 Zr 5/2 + 64.02 d19 95 Na 3/2 + +2.218 +0.101 I 7/2 + +2.742 !0.40 8.0207 d23 131 P 1/2 + +1.132 Cs 7/2 + +2.841 +0.051 30.0 y31 137 S 3/2 + +0.644 !0.076 La 3 !+0.730 +0.094 1.678 d33 140 K 3/2 + +0.391 +0.049 Au 2 !+0.593 +0.68 2.6952 d39 198 Co 7/2 !+4.627 +0.404 Th 0 + 1.405×10 y (")59 232 10 Sr 9/2 + !1.094 +0.335 U 7/2 !!0.38 +4.55 7.038×10 y (")87 235 8 Pr 5/2 + +4.275 !0.059 U 0 + 13.9 4.468×10 y (")141 238 9 Au 3/2 + +0.146 +0.547 Pu 1/2 + +0.203 2.411×10 y (")197 239 4TABLE 11.3. Spin I, parity (+ even, ! odd), nuclear magnetic moment (µ n.m.) and quadrupole moment Q $ (10 m)!28 2 for some nuclides XA (see Figs. 11.3 and 11.4) where B is the magnetic field strength. However, because of th e inertia of the particle this does not occur until the particle has moved somewhat along its orbit, with the consequence that the vector starts to rotate around the field axes as shown in Figure 11.4(a). The angular momentum vector therefore precesses around the field axes, like a gyroscope. We may rewrite (11.16a) as ( = FP/pP (11.16b) rot This form makes it more obvious why ( is referred to as the gyromagnetic ratio. Equatio n (11.16b) is vali d both for angular momentum and spin, but of course the value is different for electrons and protons. The angular velocity T of the precession is found to be( T = FBP/pP (11.23) ( rot By replacing angular velocity with frequency ( T = 2BL) L = (@B /2Bp (11.24a) ( rot or L = (@B/2B (11.24b) ( Nuclear Structure 311 Number of states Accumulated l State Possible quantum values (including spin) nucleon number ________________________________________________________________________________________ 0 s 0 1 x 2 = 2 2 1 p !1,0,+1 3 x 2 = 6 8 2 d !2,!1,0,+1,+2 5 x 2 = 10 18 3 f !3,!2,!1,0,+1,+2,+3 7 x 2 = 14 32 4 g !4,!3,!2,!1,0,+1,+2,+3,+4 9 x 2 = 18 50 l - !l, …, + l 2(2 l+1) !TABLE 11.4. Energy levels derived on basis of permitted values for the azimuthal quantum number l and spin quantum number swhere L is the Larmor precession frequency.( 11.3. The single-particle shell model 11.3.1. Quantum number rules Let us assume that a nucleon moves around freely in the nuclear potential well, which i s spherically symmetric, and that the energy of the nucleon varies between potential and kinetic like a harmonic os cillator, i.e. the potential walls (see Figs. 11.1 and 11.2) are parabolic. For these conditions the solution of the Schrödinger equation yields: E(nucleon)= SS(2U/mr)[2(n!1)+l] (11.25) o22 where U is the potential at radius r = 0 and m is the nucleon mass. We have defined n and l o previousl y. The square root, which has the dimension s , is sometimes referred to as th e!1 oscillator frequency ( T in Table 11.6). The follow ing rules are valid for nucleons in the nuclear potential well: (a) l can have all positive integer values beginning with 0, independent of n; (b) the energy of the l state increases with increasing n as given by (11.25); (c) the nu cleons enter the level with the lowest total energy according to (11.25 ) independent of whether n or l is the larger; (d) there are independent sets of levels for protons and for neutrons; (e) the Pauli principle is valid, i.e. the system cannot contain two particles with al l quantum numbers being the same; (f) the spin quantum numbers must be taken into account (not included in (11.25)). Radiochemistry and Nuclear Chemistry 312 Levels Number of nucleons Accumulated nucleons _______________________________________________________________________ 1s 2 2 1p 6 8 1d and 2s 10 + 2 20 1f and 2p 14 + 6 40 1g, 2d, and 3s 18 + 10 + 2 70 1h, 2f, and 3p 22 + 14 + 6 112 … … …TABLE 11.5. Energy levels according to equation (11.25) Following these rules, the nuc leons vary greatly in energy and orbital motion. However these rules do not ex clude the existence of two nucleons with the same energy (so-called degenerate states), provided the quantum numbers differ. 11.3.2. Nuclei without nucleon spin-orbit coupling If we c alculate the sequence of energy levels on the assumption that n is constant, we obtain the pattern in Table 11.4. This is the sequence of electronic states in atoms but it does not agree with the observed magic numbers 2, 8, 20, 28, 50, etc., for nuclei. Applying the rules of the previous secti on a new set of nucleon numbers is obtained: 2, 8, 20, 40, 70, etc. (Table 11.5, and left column Table 11.6). This level scheme allows a large amount of degenera cy. For example, for n = 1 and l = 3 (1f-state) we find from (11.25) tha t [2(n!1)+l] = 3, which val ue is obtained also for n = 2 and l = 1 (2p-state). Since the f-state can have 14 and the p-state 6 nucleons, the degenerate level of both states can contain 2 0 nucleons. However, the numbers still do not correspond to the experimental magic numbers. A further refinement is possible if we assume tha t the nuclear potential well has straight walls, i.e. the potential energy U(r) is !U at r < r while it is infinite at r $ r, where r is the o n n n nuclear radius. T his assumption, when introduced (see Fig. 11.2 and figure in Table 11.6) in the Schrödinger-equation, leads to a splitting of the degenerate levels, so that the lowest energy is obtained for the state with lowest main quantum number n. For our example of the 1f and 2p-states the 1f orbitals are lowe r in energy than the 2p. This refinement yields the middle row of levels in Table 11.6 but still does not lead to the correct magic numbers. 11.3.3. Nuclear level scheme with nucleon spin-orbit coupling In multielectron atoms, the Russell-Saunders coupling is present in light atoms. However, in the heaviest atoms of ma ny electrons and in highly charged nuclei, the j-j (spin-orbit) coupling better describes the systems. Haxel, Jensen, Suess, and Goeppert-Mayer in 1949 suggested that the nucleons always experience a strong spin-orbit coupling according to (11.12) Nuclear Structure 313 Radiochemistry and Nuclear Chemistry 314 FIG. 11.5. Part of the decay scheme for A=47; the data are experimental values.j = l ± s (11.12) and that the total spin of the nucleus I is the sum of the nucleon spins I = 3j (11.26) In this way, all I levels are split into two levels with quantum values l + 1/2 and l ! 1/2, of which the former has the lowest energy value (the opposite of the electron case). This yields the row o f levels on the right in Table 11.6. Because of the energy splitting the new level s group together so they fit exactly with the experimental magic numbers. As an example, consider the level design ation 1i . This has the following interpretation: the11/2 principal quantum number is 1; i indicates that the orbital quantum number l is 6; the angular momentum quantum number j is 11/2 ( j = l ! 1/2). The number of permitted nucleons in each level is 2 j + 1, thus 12 for j = 11/2. The mag ic numbers correspond to sets of energy levels of similar energy just as in the atom the K, L, M, etc., shells represent orbitals of similar energy. The N-electronic shell contains the 4s, 3d, 4p set s of orbitals while the 4th nuclear "shell" contains the 1f , 2p , 2p , and 5/2 3/2 1/2 1g sets of orbitals. Remember that there are separate sets of orbitals for protons and fo r 9/2 neutrons. 11.3.4. The nuclear spin The nuclear spin I is obtained from (11.12) and (11.26). Since j is always a half-integer , nuclides wi th odd number of nucleons (odd A) must have odd spin values (odd I), while those with even A must have even I. The nucleons always pair, so that even numbers of proton s produce no net spin. The same is true for even numbers of neutrons. For nuclei of eve n numbers for both N and Z, the total nuclear spin I is always equal to zero. Some ground state nuclear spins are given in Table 11.3. Nuclear Structure 315 For odd A the nuclear spin is wholly determined by th e single unpaired nucleon (single particle model). Let us take the nucleus C as an example. It contains 6 protons and 7 neutrons. From13 6 Table 11.6 we conclude that there are 2 protons in the 1s level and 4 protons in the 1p1/2 3/2 level. A simi lar result is obtained for the first 6 neutrons but the 7th neutron must enter th e 1p level. The value of I for C is thus 1/2. Another example is V, which has 23 protons 1/2 2313 51 and 28 neutrons. Since N = 28, the neutrons do not contribute to I. From Table 11.6 we see that we can accommodate 20 protons in the orbitals 1s , 1p, 1d ,2s. The next 3 protons must2 6 10 2 go into the 1 f level (1f ), where, however, 2 of them are paired. Therefore, the singl e 7/2 7/23 unpaired proton has j = 7/2, leading us to predict I = 7/2, which also is the measured nuclear spin value. Whe n a nucleus is excited, either through interaction with other particles or in a deca y process, for nuclei with N or Z values near magic numbers the paired nucleons seem not to be perturbed by the excitation (if it is not too large). As a result, we can associate the excitation with any unpaired nu cleons. Let us choose the decay of Ca to Sc (see Fig. 11.5). Ca has47 47 47 27 neutrons and in the ground state the last 7 neutrons must occupy the 1f level. The ground7/2 state of (the unstable) Sc has 21 protons, the unpaired proton can only be accommodated in47 the 1f level. Thus both of the ground states have I = 7/2. The next higher energy states for 7/2 Sc involve the levels 1f and 2p . In the Figure it is seen that these levels are observe d47 5/2 3/2 although their order is reversed from that in Table 11.6. This reflects some limitation of the single-particle model, and is explained in §11.5. For odd-odd nuclei the nuclear spin is given by I(odd-odd) = j + j = (l ± 1/2) + ( l ± 1/2) (11.27a) p n p n According to the rules formulated by Brennan and Bernstein (useful in the range 20 < A < 120 for ground states and low-lying longlived isomeric states), if the odd particles are both particles (or both holes) in their respective unfilled subshells then if j + j + l + l is even, then I = *j ! j* (11.27b) p n p n p n if j + j + l + l is odd, then I = *j ± j* p n p n p n but if we have both particles and holes, then I = j + j ! 1 p n The use of these rules can be illustrated by the case of Cu which has its odd proton in the 1f64 29 5/2 orbital and its odd neutron in the 2p orbital. Thus for the proton, j = 5/2, l = 3; for th e 3/2 neutron, j = 3/2, l = 1. Because j + j + l + l is even, we use I = *5/2 ! 3/2* = 1, which p n p n is the observed spin value. The lightest nuclei a re exceptions to this rule since they often exhibit LS coupling according to (11.18). B is an example; it has 5 protons and 5 neutrons, the fifth10 nucleon being in the 1p state. Thus I = j + j = 3, which is observed. 3/2 1 2 We mentioned in §11.2.4 that theoretical calculations of the nuclear magnetic moment , (11.21), are usually not satisfactory. The value of F(nucleus) can have a number of value s depending on m, with a maximum value of I. I Radiochemistry and Nuclear Chemistry 316 Parity is the behavior of wave functions when all coordinate signs are reversed. Even parity — no effect, else od d1 parity. The parity of the nucleus follows the rule s: (i) when both particles are in states either of even1 parity or of odd parity, they combine to a system of even parity. (ii) one particle in a state of even parity and one in a state of odd parity combine to a system of odd parity. 11.4. Deformed nuclei 11.4.1. Deformation index Both the liquid -drop model and the single-particle model assume that the mass and charge of the nucleus are spherically sym metric. This is true only for nuclei close to the magic numbers; other nuclei have distorted shapes. The most common assumption about the distortion of the nuclide shape is that it is ellipsoidal, i.e. a cross-s ection of the nucleus is an ellipse. Figure 11.6 shows the oblate (flying-saucer-l ike) and prolate (egg-shaped) ellipsoidally distorted nuclei; the prolate shape is the more common. Deviation from the spherical shape is given by $ = 2( a!c)/(a+c) (11.28) where a and c are the elliptical axes as shown on in Fig ure 11.6c. For prolate shape $ > 0, and for oblate shape $ < 0. The maximum deformation observed is about $ = ±0.6. The deformation is related to the nuclear shell structure. Nuclei with magic numbers ar e spherical and have sharp bou ndary surfaces (they are "hard"). As the values of N and Z depart from the magic numbers the nucleus increases its deformation. 11.4.2. Electric multipoles In a sph erical nucleus we assume the charge distribution to be spherical and the nucleus acts as a monopole. In the deformed nuclei, the nuc lear charge has a non-spherical distribution. The potential at a point x6,y6,z6 (Fig. 11.6) will be found to vary depending on the charge distribution and mode of rotation of the nucleus. The nuclear charge may be distributed to form a dipole, a quadrupole, etc. Nuclei are therefore divided into different classes depending on thei r electrical moments: monopoles, dipoles, quadrupoles, octupoles etc. It has been found that nuclei with spin I = 0 have no multipole moment. According to theory, nuclei with I = 1/2 can have a dipole m oment, but this has not yet been shown experimentally. Nuclei with I = 1 have quadrupole moments; they are fairly common. The quadrupole momen t Q$ can be calculated for sph eroidal (i.e. deformation not too far from a sphere) nuclei in terms of the electron charge, e, by Q$ = (2/5) Z (a ! c) (11.29a)2 2 Nuclear Structure 317 FIG. 11.6. Three nuclidic shapes: (a) spherical nucleus, (b) oblate (extended at the equator), and (c) prolate (extended at the poles). Q$ is usually referred to as the internal quadrupole moment, i.e. the expected value for a rotation around the z-axis. However, quantum mechanics makes this impossible and gives for th e maximum observable quadrupole moment Q$ = Q$(I!1/2)/( I+1) (11.29b) obs Thus Q$= 0 for I # 1/2. Q$ is usually given in area (m ). Most commonly 10 m is used obs obs2 !282 as unit and referred to as one barn, Q$ is > 0 for the more common prolate shape, and <obs 0 for oblate. Some measured values are given in Table 11.3. The rotation of nuclei with electric multipoles gives rise to formation of magnetic multipoles. Nuclei can therefore also be divided according to the magnetic moments in the same way a s according to their electrical moments. 11.4.3. The collective nuclear model In 1953 A. Bohr and Mottelsen suggested that the nucleus be regarded as a highly compressed liquid, undergoing quantized rotations and vibrations. Four discrete collective motions can be visualized. In Figu re 11.6 we can imagine that the nucleus rotates around the y-axis as well as around the z-axis. In addition it may oscillate between prolate to oblate forms (so-calle d irrotation) as well as vibrate , for example, along the x-axis. Each mode of such collectiv e nuclear movement has its own quantized energy. In addition, the movements may be coupled (cf. coupling of vibration and rotation in a molecule). The model allows the calculatio n of rotational and vibrational levels as shown in Figure 11.7. If a U nucleus is excited ab ove its ground state through interaction with a high energy heavy238 ion (coulomb ex citation), we have to distinguish between three types of excitation: (a) nuclear excitation, in which the quantum number j is changed to raise the nucleus to a higher energy level (according to Table 11.6); (b) vibra tional excitation, in which case j is unchanged, but the nucleus is raised to a higher vibrati onal level, characterized by a particular vibrational quantum numb er (indicated in the figure); (c) rotational excitation, also characterized by a particula r rotational quantum number. The Figure shows th at the rotational levels are more closely spaced and thus transitions between Radiochemistry and Nuclear Chemistry 318 rotational levels involve lower energies than de-excitation from excited nuclear or vibrational states. In case of even-even nuclei, the rotational energy levels can be often calculated from th e simple expression E = (SS/2I)n(n+1) (11.30) rot rot rr2 where I is the moment of inertia and n the rotational quantum number; this equation i s rot r identical to (2.29). The validity of this equation depends on whether the different modes o f motion can be treated independently or not, which they can for strongly deformed nuclei like U.238 11.5. The unified model of deformed nuclei The collective model gives a good description for even-even nuclei but cannot account fo r some of the discrepancy between observed spins and the spin values expected from th e single-particle shell model. The latter was developed on the assumption of a nucleon moving freely in a symmetrical potential well, a situation which is valid only for nuclei near close d shells . The angular momentum of an odd- A deformed nucleus is due both to the rotationa l angular momentum of the deformed core and to the angular momentum of the odd nucleon. Consequently the energy levels for such a nucleus are different from those of the symmetric shell model. Nuclear Structure 319 This situation wa s taken into account by S. G. Nilsson, who calculated energy levels for odd nuclei as a function of the nuclear deformation $. Figure 11.8 shows how the energies of the Nilsson levels vary with the deformation $ of the potential well. Each shell model level o f angular momentum j splits into j + 1/2 levels (called Nilsson levels or states ). Each level may contain up to two nucleons and form the ground state of a rotational band. In addition th e undefo rmed levels ( $ = 0) appear in somewhat different order than for the symmetric shel l model (Table 11.6). This leads to a reversal in o rder for some of the levels, e.g. 1f and 2p5/2 3/2 (this explai ns the observed level order for Sc, Fig. 11.5). The Nilsson levels are quit e47 different in all characteristics from the shell model states, and their prediction of energies , angular momenta, quantum numbers, and other properties agrees better with experimental data for deformed nuclei than those of any other model. Radiochemistry and Nuclear Chemistry 320 As an example we may choose Na, which has a quadrupole moment of 0.101 barn .23 11 Assuming the nuclear radius to be 1.1 A = (a+c)/2 (fm) we can use (11.28) and (11.29a)1/3 to calculate a deformation index $ = 0.12. (The value 1.1 for the constant r in (3.7) give s o better agree ment in nuclei where the inertia of the nucleus is involved.) From Figure 11.8 the 11th proton must enter the 3/2 level rather than the 1d level as the symmetric shell mode l 5/2 indicates in Table 11.6. The experimental spin of 3/2 confirms the Nilsson prediction . Similarly, for t he deformed F and Ne, we expect from Table 11.8 the odd nucleon to give19 19 9 10 spin of 1/2 and not 5/2, as wou ld be obtained from Table 11.6. Again, experiment agrees with the prediction of 1/2. 11.6. Interaction between the nuclear spin and the electron structure We have already seen how the s pin and orbital angular momentum of the electrons and of the nucleu s produce magnetic fields that interact with each other. The field produced by th e electrons is much la rger than that of the nucleus, and consequently the nuclear spin is oriented in relation to the field produced by the electron shell. By contrast the effects of the nuclear spin on the ele ctron structure is so small that it usually neglected. Nuclear physics has provided us with instruments of such extreme sophistication and resolution that there are many ways o f measuring with great accuracy the inter action between the nucleus and the electrons. The result has been new research tools of utmost importance, most prominent being the nuclear magnetic resonance (nmr) techniques. The separate disciplines of chemistry, atomic physics, nuclea r physics, and solid state physics approach each other closely in such techniques, and a n understanding of the theory and experimen tal methods requires knowledge of all these subjects. In this sec tion only a few important aspects of the interaction between nuclear spin an d electronic structure are reviewed. The methods described are usually not considered to fal l within the framework of nuclear chemistry, but in all scientific fields it is important to be able to reach the border and look at developments and techniques used on the other side. Suc h information is often the seed to further scientific development. 11.6.1. Hyperfine spectra In §11.3.4 we mentioned that the electron s in the atomic shell have Russell-Saunders coupling (11.18), JP = LP + SP, where JP is ref erred to as the internal quantum number . The magnetic field created by the electrons int eracts with that caused by the nuclear spin to yield the grand atomic angular momentum vector FP = JP + IP (11.31a) The magnitude of this momentum is p = SS[F(F+1)] , (11.31b) F2 Nuclear Structure 321 The letters S and P stand for PL=0 (El=0) and 1, respectively. The superscript 2 refers to the number of possible S-1 j values; thus 2 S+1=2, i.e. S=Es=2. The subscript gives the PJ-value. iFIG. 11.9. The development of hyperfine lines in an optical spectrum of sodium due to th e nuclear spin I, which enters through FP=JP+IP. but only projections p(z)=0,... SSF are permitted on an external (to the atom) field axis. This F leads to a large number of possible energy levels, although they are limited by certain selection rules. The nuclear spin can orient itself in relation to JP in 2I+1 directions if I # J 2J+1 directions if J # I Consider Na as an example (Fig. 11.9); it has a nuclear spin I = 3/2. The yellow sodium23 1 line of 589.6 nm is caused by de-excitation of its electronically excited p state to the groun d state 2s .1/2 The difference between the two p-states is very small, only 0.0022 eV (or about 0.6 nm), and can only be observed with high resolution ( fine spectra ). To each of these three levels th e nuclear spin I has to be added, yielding the quantum number F according to (11.31). It is easy to determine that the level number rule holds, e.g . for I = J we must have 2 I + 1 = 4 possible levels. These levels can be observed in optical spectrometers only at extremely high resolution (hyperfine spectru m, hfs). The energy separation between the hfs lines depends on the nuclear magnetic dipole F, the spin value I, and the strength of the magnetic field produced at th e I nucleus, as discussed in §11.2.4 (see also §11.6.3). The energy separation is very small, on the order of 10 eV, co rresponding to a wavelength difference of about 1/1000 of a nm. Even-5 if there is great uncertainty in the energy determination of the levels, simply counting th e number of hyperfine lines for a certain electronic JP-level gives the value of the nuclear spin, because !I # m # I (number of m values = 2 I + 1 ). I I At very high resolutions it may be possible to determine F from hfs, and from this t o I calculate I. The hyperfine splitting and shifting of optical lines have yielded importan t information not only about nuclear spins and magnetic moments, but also about the electri c charge distribution and radius of the nucleus. Radiochemistry and Nuclear Chemistry 322 11.6.2. Atomic beams The hyperfine spectra are obtained from light sources in the absence of any external (to the atom) magnetic field. If the source is placed between the poles of a magnet, whose strength B is progr essively increased, the following sequence of changes takes place (see Table 11.2 , electron-nucleon interaction). Suppose B is increased slowly to 10 T. Each hyperfine level F is found to split into 2 F +!2 1 levels, since the quantum mechanic al rule of permitted projections on an external field vector comes into oper ation. These permitted projections can vary from zero to a maximum value of SSF; for F = 3 seven new lines are obtained. Further increases in B to 10 T leads to a decoupling of F into its components J and I (Fig. 11.10). For each projection of J on the field vector there are 2 I + 1 lines ( !I...0...+ I), giving altogether (2 J + 1)(2 I + 1) lines. Th e splitting of the spectral lines in a weak magnetic field is called the Zeeman effect . The decoupling of the angular momentum of the atom into its electronic and nuclea r components is used in th e atomic beam apparatus to determine nuclear magnetic moments and spin values. A beam of atoms, produced in an oven, is allowed to enter a tube along which a series of magnets (usually three) have been placed. The magnetic field splits the atomic beam into several comp onent beams, each containing only atoms which have the same values for all the quantum numbers. Between the m agnets there is a small coil connected to a high frequency oscillator, which produces a weak oscillating magnetic field. When this oscillator is tuned to an energy hL, which exactly matches the energy difference between two quantum states of the atom, energy may be absorbed producing a transition from one of the states to another . Subsequently the atoms are deflected differently by the magnetic field than they were before the energy absorption. By a combination of homogeneous and heterogeneous magnetic fields the atomic beam apparatus allows only those atoms which have absorbed the energy quantum hL to reach the detector. From the Nuclear Structure 323 properties of the magnetic fields and the geometric dimensions and frequency of the instrument, the magnetic moment of the atom in different quantum states can be determined. This allows calculation of the magnetic mom ent and spin of the nucleus. This technique is of interest to the nuclear chemist because spin values can be obtained for short-lived nuclei in submicroscopic amounts. In contrast the hfs technique requires macroscopic amounts of atoms. 11.6.3. Nuclear magnetic resonance When an atom is placed in an external magnetic field (strength B) so that J and I decouple (cf. (11.31)), the nuclear magnetic moment vector F must precess around the field direction, withI the components in the direction of the field restricted to F = g B m (11.32) I I n I In the external field the states with different m have slightly different energies. The potentialI magnetic energy of the nucleus is E = !FP BP = !g B m B (11.33) magn I I n I The energy spacing between two adjacent levels is: )E = g B B (11.34) I n because )m = ±1. For example, consider the case of the nucleus F for which g = 5.256 I I19 in a field of 1 T, )E = 5.256 × 5.0505 × 10 × 1 = 2.653 × 10 J. The frequency of!27 !26 electromagnetic radiation corresponding to this energy is 4.0 × 10 s or 40 MHz. This lies7 !1 in the short wave length region, 8 = 7.5 m. This frequency is also the same as that of th e Larmor precession, as given by (11.24b). These relatio ns can be used to calculate the nuclear magnetic moment, if I is known, or vice versa. Figure 11.11 shows the results of an experiment in which a sample has been placed in a variable magnetic field containing two coils, one connected to a radio transmitter operating at 5 MHz and the other to an amplifier. The sample is a glass tube (B, O, Na, Al, Si atoms) containing a piec e of copper alloy (Cu, Al) in water (H, D, O). By varying the magnetic field a number of reson ances are observed. For example, for Na one has a resonance at 0.443 T.23 With (11.34) w e can calculate g = 1.48, and, consequently, the magnetic moment of Na is I23 2.218 (Table 11.3). Further, for Na, m = 3/2 and I = 3/2.23 I The magnetic field experienced by the nucleus is not e xactly equal to the external field because of the shielding effect of the electron shell, even if I and J are decoupled. Although thi s shielding of the nucleus is very small, about 10 B, it can still easily be detected with modern!5 equipment. The shielding effect depends on the electronic structure. The structural information that can be provided by this method is very detailed, and a new and deeper insight in chemical bonding an d molecular structure is provided. The nmr technique has therefore become a central tool for the investigation of chemical structures Radiochemistry and Nuclear Chemistry 324 FIG. 11.11. Nmr spectrum for a glass vessel containing water and some copper. The magnetic field is given in tesla (T). in solids, liquids, and the gaseous state. T omography by nmr is also a valuable tool for medical imaging. 11.7. Radioactive decay and nuclear structure In the pr eceding section we have described three methods of determining nuclear spin ! one optical and two magnetic. The nuclear spin plays a central role in forming the nuclear energy states. It is therefore to be expected that it als o should be of importance in nuclear reactions and in radioactive decay. Let us consider some rules for the lifetimes of unstable nuclei, for their permitted mode s of decay, and for the role of nuclear spin. Knowing these rules, it is, fo r example, possible from a decay sc heme to predict the spin states of levels which have not been measured. 11.7.1. Gamma-decay Photons are emitted in the transitio n of a nucleus from a higher energy state (level) to a lower E = hL = E ! E (11.35) ( f i where f and i refer to the final and initial states. Because the photon has spin 1, de-excitation through (-emission is always accompanied by a spin change )I $ 1. This leads to a change in the charge distribution of the nucleus and hence also to a change in its magnetic properties . Depending on the type of change occurring through the (-emission, the radiation is classified as electric or magnetic (§11.4) according to the scheme in the left part of Table 11.7. Although parity need not be conserved in (-decay, the parity change Nuclear Structure 325 Average lifetime ( J) in seconds Spin for (-energy level Type of Name of change Parity____________________________________________________ radiation transition )I=)l change 1MeV 0.2 MeV 0.05MeV_______________________________________________________________________________________________________________________ E1 Electric dipole 1 Yes 4×10 5×10 3×10!16 !14 !12 M1 Magnetic dipole 1 No 3×10 4×10 2×10!14 !12 !10 E2 Electric quadrupole 2 No 2×10 6×10 6×10!11 !8 !5 M2 Magnetic quadrupole 2 Yes 2×10 5×10 5×10!9 !6 !3 E3 Electric octupole 3 Yes 2×10 2 70 h!4 M3 Magnetic octupole 3 No 2×10 180 200 d!2TABLE 11.7. Classification of radiation emitted in (-decay and lifetime calculations of the excited states for given (- energies (From Blatt and Weisskopf.) associated with the different types of (-transitions is listed in that table. Based on the single-particle model, Blatt and Weisskopf have calculated probable lifetimes for excited states assuming a model nucleus with a radius of 6 fm. For 2 -multipole transitions ofL electric (E) or magnetic (M) type they derived the following equations 8=4.4×10 {(L+1)/[ L((2L+1)!!) ]{3/(L+3)} (E/197) r (11.36a) EL (21 2 2 (2L+1) 2L 8=1.9×10 {(L+1)/[ L((2L+1)!!) ]{3/(L+3)} (E/197) r (11.36b) ML (21 2 2 (2L+1) 2L!2 where L is the ang ular momentum carried away by the photon (= )l), 8 is the decay constant in s , E( is the (-ray energy in MeV and r is the nuclear radius in fm.!1 As seen, for decay involving electric multipole transitions the average lifetime J is proportional to r, and for decay i nvolving magnetic multipoles, to r ; in the decay the!2)l !2()l!1) nuclear spin quantum number s does not change. Calculated values are included in Table 11.7. The decay of Na (I = 1+) to Na (I = 4+) provides a useful example. The transition24m 24 involves )l = 3, no (i.e. there is no parity change), so it is designated as M3. The (-energy is 0.473 MeV and the nuclear radius of Na is about 3.7 fm. In order to compare this energy24 with those given in Table 11.7, a radius correction from the assumed 6 fm to the observed 3.7 fm must be made; accordingly E = E(r/6) for M-type radiation. This gives a corr obs i!2()l!1) hypothetica l energy of 0.473(3.7/6) = 3 MeV. According to Table 11.7 the lifetime should!4 be > 0.02 s; the observed value is 0.035 s. Th e agreement between experiment and calculation is sometimes no better than a factor of 100. The Blatt-Weisskopf relationship between energy and lifetime is only applicable for excited nucle onic states, not for rotational states. In the decay of these states the rotational quantu m numbe r always changes by two units and the lifetime of the states is proportional to EQ$. (!5!2 The lifetimes are so short that no rotationally excited isomers have been observed as yet. Radiochemistry and Nuclear Chemistry 326 11.7.2. Beta-decay Beta-decay theory is quite complicate d and involves the weak nuclear interaction force, which is less understood than the strong inter action. The theory for $-decay derived by Fermi in 1934 leads to the expression 8 = G*M*f (11.37a)2 for the decay constant, 8. G is a constant, *M* is the nuclear matrix element describing th e change in the wave function during the $-transformation (i.e. of a proton into a neutron fo r positron emission, or the reverse for negatron emission), f is a function of E and Z. *M* max depends on the wave functions before and after the transformation and gives the "order" o f decay. Since t = ln 2/ 8 2 ln 2 = G*M*ft (11.37b)2 2 we see that the product ft should be constant for a decay related to a certain *M*. The ft value 2 (omitting index 2) is often referred to as the comparative $ half-life, and nomograms for its calculation are given in nuclear data tables and decay s chemes. The lower the ft value the higher is the probability for decay, and the shorter is the half-life. Gamow and Teller have give n selection rules for $-decay which are useful for estimating decay energy, half-life, or spin in a certain decay process, if two of these properties are known. These rules are summarized in Table 11.8. For exam ple, the decay of Na occurs 99% through $-emission (with an E = 1.4 MeV)24 max to an excited state of Mg (Fig. 4.7). The log ft value of the transition is 11.1. The groun d24 state of Mg is 0+; the excited state has positive parity. Thus the selection rules indicate an24 allowed transition for which the only spin changes permitted are 0 and ±1. The ground state of Na is 4+, and the observed excited sta te (at the 4.12 MeV level) is 4+, in agreement with24 the rule. 11.7.3. Alpha-decay theory If we calculate the Q-value for the "-decay reaction (4.11) from the mass formulae (4.12) we find that Q > 0 for all nuclei with A > 150, which means that we would expect all elements heavier than the rare earths to be unstable with respect to "-decay. However, the accuracy of eqn. (3.8) decreases when we move away from the region of $ stability. Hence, we should not be surprised that nuclides, with a high Z/A ratio, far from stability have been found to decay by "-emission down to A = 106. For nuclei nearer $-stability, decay by emission of "-particles is observed for some isotopes of rare earths and heavier elements, but it occurs frequently only for A $ 210, i.e. nuclides heavier than Bi.209 In §4.17 we mentioned the discovery by Geiger and Nuttall that the lower the "-energy the longer was the half-life of the "-decay; doubling the decay energy may reduce the half-life by a factor of 10 . Alpha-decay has been observed with energies from slightly greater than 1.820 MeV (e.g. Nd, E = 1.83 MeV, t = 2.1 × 10 y) to about 10 MeV (e.g. Ns, E =144 15 262 " 2 " 10.38 MeV, t = 4.7 ms). The isotopes of the actinide elements typically2 Nuclear Structure 327 Transition type )I Parity change Log ft ___________________________________________________________ Super allowed 0 No 3 Allowed 0,±1 No 4-6(a) First forbidden 0,±1,±2 Yes 6-9 ond forbidden ±2,±3 No 10-13(b) ___________________________________________________________ Not 0 6 0, Also 0 6 0.(a) (b)TABLE 11.8. Gamow !Teller selection rules for $-decay have "-energies between 4 and 10 MeV. The observed stability against "-decay for most nuclei in the range 102 # A # 210 having a positive Q can be explained by assuming the "-particle exists as a (preformed) entity inside " the nucleus but with insufficient kinetic energy to overcome the "internal Coulomb barrier" . This barrier is assumed to be of the same type, although of somewhat different shape, as the external Coulomb barrier, which is discussed in some detail in §12.4. Ass ume that the average kinetic energy is at the level marked E in Figure 11.12. If th e " particles in the nucleus have a Boltzmann energy distribution, an "-particle could in principle form in the nucleus and perhaps acquire sufficient kin etic energy through collisions to overcome the barrie r (E). It would then be emitted with an energy E, which for an element lik e cb cb uranium is 26 MeV. However, the observed "-energy is only 4.2 MeV. This contradiction was explained by Gamow, and independently by Gurney and Condon, in 1928, by using a quantum mechanical model, which retained the feature of the "one-bod y model" with a preformed "-particle inside the nuclear potential wall of even-even nuclei. The time independent solution to the Schrödinger wave equation for an "-particle inside the nuclear potential well is a wave function which has a small, but non-zero, value even outside the poten tial well. The probability, p, to find the "-particle outside the potential well is th e square of the wave function and is found to be Rx p = exp[ !{4B(2µ) /h}I(U(r)!Q)dr] (11.38)2 2 " R where index " refer to the "-particle, µ is the reduced mass of "-particle and residual nucleus, µ = MM/(M+ M), R and R are inner and outer integration limits respectively (where " 1 " 1 x the potential energy of the bar rier is equal to the energy of the emitted particle, cf. Fig 11.12), r is the distance from the center of the nucleus, U(r) is the potential energy of the "-particle, and Q is the total "-decay energy. Ind ex 1 refer to the nucleus remaining after emission of the " "-particle. The decay constant can be r egarded as the product of p and the frequency, f, by which the "- particle hits the barrier from inside. If we assume that the deBroglie wavelength, h/µv, for an "-particle of velocity v inside the nucleus is approximately equal to the nuclear radius, R, we obtain Radiochemistry and Nuclear Chemistry 328 It is imp ortant to recognize the difference between the classical CGS system where the force F between point charges1 in vacuum is F = zze/r and the SI system where F = zze/(4B,,r). 1 2 1 2 2 2 2 2 0 FIG. 11.12. Alpha-penetration through the potential wall. h/µv . R (11.39) The frequency f can then be estimated from v if we assume that the "-particle bounces back and forth inside the potential well with constant velocity. f = v/2R . h/2µR (11.40)2 Combining (11.40) with (11.38) we obtain the following expression for the decay constant Rx 8 . [h/(2µR)] exp{ ![4B(2µ) /h]I(U(r)!Q)dr} (11.41)2 2 2 " R The same relation also holds ! with appropr iate substitutions ! for any charged particle trying to enter the nucleus from outside the potential barrier (cf. §12.4), where, however, only one impact is possible, i.e. f = 1. For some simple mathematical forms of the potential energy U(r) it is possible to fin d analytical solutions to the int egral in (11.41). The simplest form of U(r), a square well nuclear potential according to Fi gure 11.12, yields the following expression after integration and some algebra1 8.[h/(2µR)] exp{ ![(2µ) eZZ/(,,hQ)][arccos( u)!u(1!u)]} (11.42)2 22 2 22 1 " 0 " where u = (E/E) = [4 B,,QR/(ZZe)] (11.43) "cb 0 " 1 " 2 2 2 The radius of the decaying nucleus can be estimated from Nuclear Structure 329 FIG. 11.13. The variation of "-half-life with mass number. Lines connect isotopes.R = rA+ r . 1.30 A+ 1.20 (fm) (11.44) 0 1 " 1a a where A is the mass number of the nucleus after emission of the "-particle and r is the 1 " effective radius of the "-particle. Equation (11.42) can not only be used to compute deca y constants, but also to estimate the nuclear radius for even-even "-emitters from measured half- lives and decay energies. The calculated decay constant is very sensitive to Q: a 1 Me V " increase in Q increa ses 8 (and decreases t) by a factor of about 10 . It is also very sensitive " 25 to the nuclear radius: a 10% increase in R (or the Coulomb radius r; see Fig. 12.4) whic h c means a corresponding decrease of t he Coulomb barrier height, increases 8 by a factor of 150. Decay const ants and partial " half-lives computed from (11.42) are normally within a factor 4 of the measured values for even -even nuclei. The half-lives of even-odd, odd-even, and odd- odd nuclides are often longer than predicted by equations like (11.42), even after Radiochemistry and Nuclear Chemistry 330 inclusion of more elaborate nuclear potentials and angular momentum effects in the theory. The ratio between t he observed and predicted half-lives is called hindrance factor and ranges from one to ~3000. Assuming the Coulomb barrier to be much larger than Q, i.e. u small, designating th e " resulting exponential term in (11.42) by e and taking the logarithm of the resultin g!2G expression we obtain log 8 = constant ! 2G (11.45) which is very similar to the empirical Geiger-Nuttall law. Figure 11.13 sho ws the systematic change of "-decay half-life with nuclear charge and mass for heavy nuclei, lines connect d ata for constant Z. Odd-even effects and influence from magic numbers are visi ble. Such diagrams have historically played an important role in the synthesis and identification of isotopes of the heaviest elements. The "-decay theory was the first successful (quantum mechanical) explanation of radioactive decay, and as such played a major role in further development of nuclear theories and models. Although its simplicity causes it to fail for nonspherical nuclei as well as those near close d shells, such effects can be taken into account in more advanced Nilsson-type calculations. 11.7.4. Spontaneous fission In §4.4 we found that fission of heavy nuclei like U is exoergic and in §4.6 that it is a236 common decay mode for the heaviest nuclei. From the semiempirical mass equation (3.8) it can be found that f ission of nuclei with A $ 100 have positive Q-values. Why is the decay b y spontaneous fission only observed for nuclei with A $ 230? The breakup of a large even-even nucleus into two positively charged fragments of roughly equal mass and charge can be treated in a way similar to that of "-decay. Assuming separation into two spherical fragments, A Z and A Z, in point contact the Coulomb energy can b e 1 1 2 2 calculated to be E = 0.96 ZZ/(A+ A) (MeV) (11.46) cb 1 2 1 2a a where r = 1.5 fm has been assumed. In a real case it is necessary to consider that the newly0 formed fra gments have non-spherical shapes and the value obtained from (11.46) is thus very approximate but sufficient for our discussio n. The spontaneous fission of a nucleus is obviously hindered by a Coulomb barrier, the fission barrier , and the process should be treated as a barrier penetration problem for Q < E. When Q $ E, breakup of the nucleus will occu r cb cb within a few nuclear vibrations, ~ 10 s.!22 The critical condition Q = E for fission of an even-even nucleus into two equal fragmentscb with an unchanged charge to mass ratio can be estimated by equating E from (11.46) with thecb Q-value computed from (3.8). Neglecting the pairing term this results in (Z/A) = 37.89 (11.47a) 2 crit Nuclear Structure 331 FIG. 11.14. Fission barrier height as function of Z/A.2 FIG. 11.15. Qualitative features of the fission barrier for Pu (From Britt). Radiochemistry and Nuclear Chemistry 332 Becaus e asymmetric fission is more common than symmetric and the emerging fissio n fragm ents have non-spherical form, the numerical value derived above is not very accurate . However, the concep t of a critical value of Z/A is important. A more sophisticated treatment 2 results in the equation (Z/A) = 50.883 [1 - 1.7826 ( N - Z)/A] (11.47b) 2 22 crit We can then define a fissionability parameter , x, x = (Z/A)/(Z/A) (11.48) 2 2 crit as a measure of how prone to fission a nucleus is. Figure 11.14 shows fission barrier height as function of Z/A. Calculations by Myers an d 2 Swiatecki using a refined liquid drop model predict that the barrier height should pass through a maximum around Z/A ~ 16. The penetrability of the barrier increases roughly exponentially 2 with its height. Hence, low values of Z/A (but above the value corresponding to the maximum 2 barrier height) implies extremely long half-lives. As an example, U has Z/A = 35.56, E238 2 cb = 5.8 and a partial half-life for spontaneous fissio n, t, of ~ 10 y. By comparison we can 2,SF16 estimate that a nucleus with A = 100 and Z = 44 ( Ru), Z/A = 19.36, has a fission barrier100 2 tens of MeV high and a practically infinite half-life with regard to spontaneous fission. So far we have neglected the effects of pairing, nuclear shell structure and nuclea r deformati on on the fission process. Odd-even, even-odd and odd-odd nuclei exhibit larg e hindrance factors, HF, for spontaneous fission somewhat similar to the phenomenon observed in "-decay. The presence of one or two odd nucleons lead s to spin and parity values which must be conserved during the defor mations leading to fission and thus constrains the possible shapes and energy levels. This contributes to the occurrence of hindrance factors. Hence, o-e, e-o and o-o nuclei have normally m uch longer spontaneous fission half-lives than their neighboring e-e nuclei. The ground state energy, E, of a nucleus can be regarded as a sum of the liquid drop model energy (including deformation) , E , the pairing correction, *, and the shell correction, *, LDM P S to that energy. E = E + * + * (11.49) LDM P S The use of (11.49), with deformation dependent shell corrections, leads to fission barriers with two maxima, see Figure 11.15. Th e occurrence of a secondary minimum is consistent with the observat ion of spontaneous fission isomers. Shell and pairing effects also give rise to lon g spontaneous fission and "-decay half-lives for nuclides around magic N or Z numbers. Detailed calculations based on advanced th eories of nuclear structure lead to the prediction of an area of spheroidal nuclei with in creased nuclear stability (in their ground state) around Z = 114 and N = 184, the so called superheavy elements . However, attempts to synthesize such nuclei in their ground state by heavy ion reactions have been fruitless. Nuclear Structure 333 11.8. Exercises 11.1. The qua ntum numbers s = 1/2 and l = 2 are assigned to a particle. (a) If spin and orbital movements ar e independent, how many space orientations (and thus measured spectral lines if no degeneration of energy states occur) are possible in an external field of such a strength that both movements are affected? (b) How many lines would b e observed if spin and orbital movements are coupled? 11.2. In the hydrogen atom the K-electron radius is assumed to be 0.529 × 10 m (the Bohr radius). (a) Calculate!10 the orbital velocity of the electron assuming its mass to be m. (b) How much larger is its real mass because of th e e velocity? Does this affect the calculations in (a)? 11.3. A beam of protons pass through a homogeneous magnetic field of 0.5 T. In the beam there is a small hig h frequency coil which can act on the main field so that the proton spin flips into the opposite direction. At which frequency would this occur? 11.4. Calculate the nuclear Landé factor for B.11 11.5. In Table 1 1.5 the first degenerate levels have been given. Using the same assumptions, what states will b e contained in the next level and how many nucleons will it contain? 11.6. How deep is the nuclear well for Sn if the binding energy of the last nucleon is 9 MeV?116 11.7. Calculate the spins and nuclear g factors for (a) Ca, (b) Co, and (c) Pr, using data in Table 11.3.45 60 141 11.8. The observed quadrupole moment of Co is 0.40 barn. (a) What is the deformation value $? (b) What spin value59 is expected from the Nilsson diagram? 11.9. Which neutron and proton states account for the spin value I of N?14 11.10. A (-line at 0.146 MeV is assigned to a +4 6 +0 rotational level change in Pu. (a) What should the energy238 of the +2 and +6 rotational levels be? Compare with the measured values of 0.044 and 0.304 MeV. (b) If Pu is238 considered to be a homogenous sphere, what wil l its apparent radius be? Compare with that obtained using relation (3.7). 11.11. A Pu compound is placed in a test tube in a 40 MHz nmr machine. At what field strength does resonance239 occur with the nuclear spin? Is the measurement possible? Relevant data appear in Table 11.3. 11.12. Using the Gamow theory the probability for tunneling of an "-particle in the decay of U is 1:10 , and the238 38 "-particle hits the walls about 10 times per second. What average lifetime can be predicted for U from thi s21 238 information? 11.13. Calculate the half-life for "-decay of Sm assuming that Q is 2.314 MeV. Compare the result with th e147 " measured half-life and compute the hindrance factor. 11.9. Literature I. PERLMAN , A. GHIORSO , and G. T. SEABORG , Systematics of alpha-radioactivity, Phys. Rev. 77 (1950) 26. I. PERLMAN and J. O. RASMUSSEN , Alpha radioactivity, Handbuch der Physik 42, Springer-Verlag, 1957. W. J. MOORE , Physical Chemistry , 3rd edn., Prentice-Hall, I. 1962. W. D. MYERS and W. J. SWIATECKI , Nuclear Masses and Deformations, Nuclear Physics 81 (1966) 1. E. K. HYDE, Nuclear models, Chemistry 40 (1967) 12. H. C. BRITT , in N. M. EDELSTEIN (Ed.), Actinides in Perspective , Pergamon, 1982, p. 245. K. S. KRANE , Introductory Nuclear Physics , Wiley, 1988. G. T. SEABORG and W. D. LOVELAND , The Elements Beyond Uranium , Wiley-Interscience, 1990. D. N. POENARU (Ed.), Handbook of Decay Modes , CRC Press, 1993.