Appendix A. Angular Momentum and Spherical Harmonics
The total angular momentum of an isolated system is conserved; it is a constant of the motion. If we change to a new, perhaps more convenient, coordinate system related to the old by a rotation, the wave function for the system trans- forms in a defmite way depending upon the value of the angular momentum. If the system is made up of two or more parts, each of them may have angular momentum, and we need to know the quantum rules for combining these to a resultant. We cannot do more here than introduce and summarise a few salient points about these matters. More detailed discussions of the quantum theory of angular momentum are available in several treatises (for example, Brink and Satchler, 1968; see also Messiah, 1962). A general word of caution is in order. The literature on this subject presents a wide variety of conventions for notation, normalisation and phase of the various quantities which occur. Sometimes these conventions represent largely arbitrary choices, sometimes they emphasise different aspects of the quantities involved. If the reader is aware of this possibility, and if he adopts a consistent set of formulae, little difficulty should be encountered.
AI ANGULAR MOMENTUM IN QUANTUM THEORY
An isolated system will have a total angular momentum whose square has a discrete value given by j(j + 1)1i 2 , where1i is Planck's constant divided by 21T and where j is an integer or integer-plus-one-half according to whether the
283 284 INTRODUCTION TO NUCLEAR REACTIONS system as a whole is a boson or a fermion.* That is to say, the wave function of the system will be an eigenfunction of 12 , the operator for the square of the angular momentum J2 lj) = j(j + l)lj) (AI) (Henceforth, we shall use natural units so that1i = 1.) Note that we often speak of 'the angular momentum j' when strictly we mean that the square of the angu- lar momentum isj(j + 1). The state lj) cannot be an eigenfunction of the operator J because the various components of the vector J do not commute with each other. Instead their commutation relations can be summarised by the formula
J X J = iJ (A2) Hence a system cannot simultaneously be an eigenfunction of the operators for more than one component of J. Each component of J commutes with J2 , how- ever, so we are free to choose J2 and one component to generate angular momen- tum eigenfunctions. It is conventional to choose the z-component Jz and we label its eigenvalues by m Jz ljm) =mljm) (A3) The lack of commutativity between the components J; can then be pictured by using the vector model, Figure Al. In this model, the vector J has a definite magnitude (j (j + 1)] 1/ 2 and a definite projection m on the z-axis, but its direction is uncertain because of the uncertainty in Jx and Jy. This uncertainty is represented by imagining the vector to be precessing around the z-axis so that its x andy components fluctuate. The prototypical angular momentum operators are those corresponding to the orbital angular momentum L of a particle with momentum p. Explicitly L = r X p, with Px = -i atax, etc. or in polar coordinates L± =Lx ±iLy
= ± e±il/> {aao ± i cot 8 ~} at/> . a L =-r- (A4) z atJ>
*A nucleus is made up of A nucleons, each of which is a fermion and has an intrinsic spin of 111. Their relative orbital motions can have angular momenta which are integral multiples of1r. The rules of angular momentum addition (gee below) then ensure that the total angular momentum or 'nuclear spin' is an integral multiple of11 if A is even or an integer-plus-one- half times11 if A is odd. Therefore nuclei with A even are bosons, while those with A odd are fermions. APPENDIX A 28S
I
' ' z' ' ' ' m ' ' ' ' ' '
Figure Al Vector model of an angular momentum vector /with a magnitude j [iU + 1)] 1•/2 and a z-projection equal tom. It precesses about the z-axis so that its x- andy-components are variable. Its projection m' upon another z' -axis also fluctuates; classically only the range of m' values shown is allowed but quantum indeterminacy allows -j .;;; m' .;;; j and
2 22 {AS) L =- ti~ o aao (sin 8 a:)+ si~ 2 8 a: } The expression AS for L2 is one which occurs when the Schrodinger equation for a particle moving in a central field is separated into radial and angular equa- tions (Messiah, 1962). The angular part of the wave function is an eigenfunction of Lz and L2 ; these eigenfunctions are called spherical harmonics and denoted Yf{' (8, ). They obey the eigenvalue equations
L2 Y:' (8, ) = Q(Q + 1) Y:' (8, ) {A6) Lz Y{' (8, )=mY{{' (8, ) {A7) Spherical harmonics and their properties are discussed in more detail in section A3. Another important set of angular momentum operators are those for spin-i particles. For these we have j =i and J =t d, with the vector o representing the three 2 x 2 matrices of Pauli. With lz diagonal these are 286 INTRODUCTION TO NUCLEAR REACTIONS
{AS) ax= (01 01 )' ay = (0i -i)0 ' az = (10 -10) with {A9) Then the eigenfunctions for an intrinsic spin of 1/2 obey the eigenvalue equations {AlO) Besides the commutation properties embodied in equation A2, the a matrices obey the anti-commutation relations {All) in particular, a~ =a~ =a; =1 , where 1 is the 2 x 2 unit matrix. Together with 1 , the a1 are sufficient for a complete description of a spin-t system, i.e. a system with two possible states (m = ± t). For this reason they can also be used to represent the isospin of an isospin doublet such as the neutron- proton pair. When used to represent isospin, the a, matrices are usually denoted Tt (see Wilkinson, 1969). It is often convenient to use two different coordinate systems which are oriented in space in different directions. [One example is in the description of a nucleus with a permanently deformed (non-spherical) shape. It is helpful to use a set of axes fiXed along the principal axes of the nucleus {the body-fixed axes) as well as a set independently ftxed in space {the space-fzxed axes). If the nucleus is rotating, the body-ftxed axes will be rotating with respect to the space-fiXed ones.] Then a state with angular momentum j which has a definite projection m on one z-axis will have a distribvtion of projections m' with respect to the other z' -axis. This distribution comes from the quantum uncertainty in the x andy components and can also be visualised using the vector model as being due to the precession of J about the z-axis (Figure Al). The rotational transformation which determines this distribution in m' is described in standard texts {Brink and Satchler, 1968; Messiah, 1962).
A2 ANGULAR MOMENTUM COUPLING AND SYSTEMS COMPOSED OF TWO OR MORE PARTS
Often we have systems made up of two parts, each with angular momentum. These parts may be two different particles, or perhaps the spin and orbital properties of a single particle. Each part has associated with it an angular momentum operator and its z-component. Let these be J 1 , J1z, J2 , and J2z. Then we have two choices for a set of four commuting operators for the com· bined system. One choice is {Al2) APPENDIX A 287
The corresponding eigenfunctions are products of the eigenfunctions for each part and obey the eigenvalue equations Jlli1m1} lj2m2> = it(it + l)lhmt> lhm2> (A13) (Al4) where i = 1 or 2. The other choice of a set of commuting operators is in terms of the total angular momentum of the combined system, J = J 1 + J2 • It is (Al5)
The eigenfunctions of the coupled system we will write as Ij d 2JM}. They obey the eigenvalue equations Jllitj2JM> = j;(j; + l)lid2JM>, i = 1, 2 (A16) 121iti2JM> =J(J + l)ljihJM> (A17) lz lithlM> =Miid2JM> (A18)
For given i1 and j 2 , the values of J are restricted by the triangular condition of vector addition (Al9) and the allowed J ranges between these limits in integer steps. The choice Al2 would be especially useful if the system were isolated and if there was no interaction between the two parts. Then the angular momentum and its orientation for each part would remain constants of the motion, as is expressed by equations Al3, Al4. However, if there is an interaction between the two parts, it is likely that the individual components m; will not remain constant even if the magnitudes it do: the two vectors ji will tend to precess around their resultant J instead of each precessing independently about the z-axis. If the system as a whole is isolated, J2 and lz will remain constants. These two extremes are pictured using the vector model in Figure A2. These two descriptions are not independent but their eigenfunctions are related by a unitary transformation. This may be written explicitly as lid2JM} = ~ ljlml}lhm2} ljlml}lj2m2} = ~ liti2JM} These equations defme the (real and symmetric) Wigner or Qebsch-Gordan coefficient (A22) 288 INTRODUCTION TO NUCLEAR REACTIONS z ------ (D) (b) Figure A2 Vector picture of two angular momentaj1 andj2 • (a) In the coupled (J, M) representation, i 1 and i 2 have a resultant J with projection M. J precesses about the z-axis, while j 1 and j 2 precess about J; m 1 and m 2 are indeterminate. (b) In the uncoupled (j 1 m 1 j 2 m2 ) representation i 1 and i 2 precess independently about the z-axis; their resultant J fluctuates in direction and magnitude For brevity, this is often written (j1hm1 m2 1JM) and we shall follow this con- vention. The coefficient vanishes unless J satisfies the condition A19; further, since lz = JlZ +l 2z, we must have M= m 1 + m2 • The vector model gives a physical meaning to the transformations A20 and A21. The left side of Figure A2 represents a system with definite J and M; the individual j; then precess around J in a correlated fashion and their correspond- ing projections m; fluctuate. Equation A20 expresses the distribution of the m1; ~ Hence no given coefficient may exceed unity. An explicit general expression can be given for the coefficients, as well as simple formulae for special cases (Brink and Satchler, 1968). Numerical tables are also available. Physical arguments can be used in some special cases. For example, suppose we have the system ljd1) llzh.) in which each part has the maximum allowed z-component. Then the resultant M = j 1 + j 2 is also a maximum and the rule A19 shows that only J = j 1 + j 2 is allowed. Consequently, because of the normalisation condition A24, (A25) so that (A26) Similarly, changing the sign of the z-components is equivalent to inverting the direction of the z-axis. This can have no physical significance for an isolated system so we expect a corresponding symmetry relation. We find TABLE AI Qebsch-Gordan coefficients [or j 2 = 0 or J = 0 = li;J limM TABLE A2 Clebsch-Gordan coefficients whenj2 = 1/2 J m2 = 1/2 m2 = -1/2 jl +t [i1 +M+fJ/2 [i1 _M+tr' 2 2jl +1 2jl +1 ,,• -,1 _[i1 -M+fJ 12 [j1 +M+fJ/2 2jl +1 2j, +1 290 INTRODUCTION TO NUCLEAR REACTIONS TABLE A 3 Clebsch-Gordan coefficients when j 2 = 1 I J m2 = 1 m 2 =0 m2 = -1 [Vs +M)(is +M+l)r/2 e~ -M+1)(is +M+l)J'2 [ M -[Vs +M>Us -M+l)J'2 [Us -M>U1 +M+l)]l/2 is 21s Us + 1) UsUs +1)]1/2 2/s Us + 1) [Us - M) (fs - M + 2 [Us -M) Us +M)J/2 [ul +M+ n u~ +M)T/2 11 -1 l)J' 2/s (2/s + 1) Is (2/s + 1) 2/s (2/s + 1) h = 0, 1/2 and 1 from which numerical values may be obtained. They should be used in conjunction with the relations A25 and A27. As an exercise, the reader may also use these tables to verify the relations A23 and A24. Among the various representations of the Clebsch-Gordan coefficient which may be encountered, a popular one is the Wigner 3 - j symbol. This is renorma- lised so as to give it a high degree of symmetry (see Brink and Satchler, 1968) {a b c)= (-)11-b--y (2c + 1)-1/2 (abCi{Jic- 1> (A28) ~0: t3 'Y In most systems of physical interest there are more than two component parts and we are faced with the problem of constructing states with good total angular momentum from the angular momenta of these parts. In general there will be more than one way of doing this. For example if we have three angular momenta h, j 2 and h, there are three ways in which we may couple them to a resultant J by applying twice the coupling relations A20 or A21. One is (A29) while another is (A30) Any of the three possibilities is a valid representation but they are not independ- ent. The corresponding eigenfunctions are related by a linear transformation; for example, I (jlj2)J12, h ;JM) = I: ljl, (hh)J23 ;JM) (jl, V2h)J23 ;JI (jd2)J12 ,h ;J) J23 (A31) The transformation coefficient is real and independent of M; it is used to defme the Racah coefficient W APPENDIX A 291 A3 SPHERICAL HARMONICS Spherical harmonics are solutions of the differential equation A6 [L2 - Q(Q + 1)] Yr (8, rp) =0 (A34) with integral values for Q. The operator L2 is givett explicitly by equation AS. The dependence on rp may be factored out Yf (8, rp) =Sf (8) exp(imrp) (A35) then the eigenvalue equation A7 follows immediately from the explicit form A4 for the operator L 11 • We have man integer with -Q EO; m EO; Q. The explicit defini- tion of the er involves an arbitrary choice of phase; the one most commonly used is that of Condon and Shortley (1951) er (8) =(-)'" [(2Q + 1) (Q- m)!] 112 pr (cos 8) if m ;;;. 0 41T (Q +m)! (A36) Here Pr (cos 8) is the associated Legendre polynomial (Abramowitz and Stegun, 1970.) The spechl case with m =0 is the Legendre polynomial, P: (cos 8) =P'J. (cos 8). From equation A36 we have that 292 INTRODUCTION TO NUCLEAR REACTIONS (A37) Further, the parity operation (reflection through the origin) replaces (8, 1/J) by (1r -- 8, 1/J + 1r). Since (A38) we see that the spherical harmonics have a definite parity of (-)Q. They are also orthogonal over the unit sphere and, as defined, normalised so that (A39) where liab = 1 if a= b but liab = 0 if a* b. We also have a sum rule l: I Y:Z (8,1jJ)I 2 = 2Q+ 1 (A40) m 47T A few examples of low order are 2 112 12 ~ = (·L)'' ; Y? = (:7T) cos 8; Yf1 = :j{;7Ty sin 8 exp(±ii/J) 5 )1/2 2 1 ( 15)1/2 Y~ = ( - 167T (3 cos 8- 1); Yf = + -87T cos 8 sin 8 exp(±iljJ) 15 )1/2 Yf 2 = -( sin 2 8 exp(±2i P:Z(cos8)~(-Q)m (-2 sin[(Q+t)8+(2m+l) -7TJ Q7TSin-)l/?. 8 4 which holds for Q>> 1, m << Qand 8 not too close to 0 or 7T. The Y;' (8, 1/J) provide us with a complete set offunctions with which we may expand any function of the angles 8 and 1/J, as was done in equation 3.34, 4.19, 4.60 or 4.61 for example. The Rayleigh expansion 3.35 of a plane wave is a special case. When m = 0, a spherical harmonic Y3 reduces essentially to a Legendre poly- nomial PQ (Abramowitz and Stegun, 1970) Ji (8, 1/J) =~] PQ (cos 8) APPENDIX A 293 These Pfl constitute a complete orthogonal set which may be used to expand any periodic function f of 9 which is symmetric about 9 = 0,[(-9) = [(9). Since l 2 Pfl(x)PQ'(x)dx = -- liQQ' (A41) f-1 2Q + 1 which is a special case of equation A39, we may write [(9) = 1: aflPQ (cos 9) Q where afl 21 (2Q 1) 1r f(9)PQ (cos 9) sin 9d9 (A42) = + f0 The results of measurements of an angular distribution or differential cross- sections may be expressed in this form, with the coefficients afl being chosen to optimise the fit of [(9) to the data. Some Legendre polynomials oflow order are P0 (x)=1; P2 (x)=t(3x2 -1); P3 (x)=t(5x3 -3x) P4 (x)=i(35x4 -30x2 +3) For further properties of the spherical harmonics, see, for example, Brink and Satchler (1968), also Messiah (1962) and Morse and Feshbach (1953). A4 EXAMPLE 1: RADIOACTNE DECAY OF A NUCLEUS Consider a nucleus, with spin J and z-component M, at rest; the wave function describing its internal state is t/JJM(T). Suppose it decays by the emission of a particle with zero spin, such as an a-particle, with orbital angular momentum Q. Suppose the daughter nucleus has a spin J' and wave function t/J.t M'(r'). Then the initial wave function breaks up as follows (compare equation A20) (A43) where m = M - M' and IJ' - QI .;;;; J .;;;; (J' + Q). Here Now we do not observe the internal state of the a-particle or the daughter nucleus, but only the direction of emission, so we must integrate over the internal variables of thea-particle and the daughter nucleus. Since f dr' t/I;•M'(r') x t/IJ'M"(r') = f>M 'M", etc., there is no coherence between transitions to differ- ent final states with M' =I=M''. Further, the radial function u11(r) is a constant at a flxed value of r. Consequently, the angular distribution of the radiation is simply proportional to (A45) where m =M - M'. Because the dependence of lQ' on 1/J enters only through the factor eimcf>, we see that WM is independent of 1/J; the distribution is symmetric around the z-axis. In the special case that J' = 0, so that Q =J and m =M, then the Oebsch-Gordan coefficient is unity and WM(8)= I Y_f (8, 1/>)1 2 (A46) Another special case occurs when J = 0, so that Q = J', m = - M' and the Oebsch-Gordan coefficient has the value (-)12 -m (2Q + 1)- 112 • Then Wo(8) = _1_ 1: Yfr (8, 1/>)12 (A47) 2Q+ 1 m I = 47T from equation A40; that is, the ~gular distribution is constant or isotropic. This is a general property of the angular distribution of products from a spin-zero system; it may be shown to be true for J = 1/2 also. Other cases follow by inserting explicit values for the coefficients in equation A45. If the initial nucleus was not prepared in a single substate M but oriented with a distribution of M values with probabilities PM, the angular distribution of the decay radiation becomes (A48) AS EXAMPLE 2: FORMATION OF A COMPOUND NUCLEUS AND STATISTICAL WEIGHTS One way of preparing the radioactive nucleus discussed in the previous section is to form it as a compound nucleus in a nuclear reaction. Co11sider the collision of two nuclei A+ a with spins h and ia, respectively. Following equation A21, their wave functions may be combined to form channel-spin functions (compare section 3.8.1) APPENDIX A 295 (A49) where M 5 = MA + m8 and Ih - ia I.;;;; S.;;;;; (h + i 8 ). Here the Oebsch-Gordan coefficient is the probability amplitude for finding a particular value S of channel spin with z-component Ms when the colliding pair has z-componentsMA and ma. If the incident beam and target are unpolarised, the probability of any given 1 1 MA is (2h + 1r and of any given m8 is (2i8 + 1)- • Consequently the proba- bility of finding a given S and Ms in such a beam is where the sum is constrained to values such that MA + m8 =Ms. Now equation A23 tells us that this sum is just unity so that p - 1 S,Ms - (_2/_A_+_1_)(_2_ia_+_1_) (A51) This is independent of Ms, as would be expected since the two nuclei are not polarised and therefore there is no preferred direction in space. The probability of finding S irrespective of the value of Ms is 2S+ 1 g(S) = ~ Ps M = ----- (A52) 5 Ms ' (2/A+1)(2i8 +1) which is just the statistical weight for channel spin introduced in section 3.8.1. The total angular momentum J of the system is obtained by combining the channel spin S with the relative orbital angular momentum Q (A53) where M = M 5 + m and IS - Q I.;;;; J.;;;;; (S + Q). Then the spin of any compound nucleus which is formed is limited to one of these J values. Now the Oebsch- Gordan coefficient is the probability amplitude for finding a particular J value in a system with channel spinS, Ms and orbital Q, m. Including equation-A49, we see that the probability amplitude for finding a particular value of J in a system of two nuclei with MA and m8 moving with relative angular momentum Q, m is just (A54) In particular, the vector L is always perpendicular to the direction of motion; if we take this direction (the beam direction in an experiment) as z-axis, then m = 0 only andM=M5 • When the incident spins h and i 8 are randomly oriented, the probability of finding the channel spin S and total angular momentum J is 296 INTRODUCTION TO NUCLEAR REACTIONS PsJ= 1 ~ kiAiaMAma1SM> PsJ= 1 ~ I REFERENCES Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical Functions. New York; Dover Publications Brink, D. M. and Satchler, G. R. (1968). Angular Momentum. 2nd edn. Oxford; Oxford University Press Condon, E. U. and Shortley, G. H. (1951). The Theory of Atomic Spectra. Cambridge; Cambridge University Press Messiah, A.M. (1962). Quantum Mechanics I and II. Amsterdam; North-Holland Morse, P.M. and Feshbach, H. (1953). Methods of Theoretical Physics. New York; McGraw-Hill Wilkinson, D. H. (1969), ed. Isospin in Nuclear Physics. Amsterdam; North- Holland Appendix B. Transformations between LAB and CM Coordinate Systems As was discussed in sections 2.2 and 3.1, it is particularly convenient to use a moving coordinate frame in which the centre of mass of two colliding nuclei is at rest. This is called the CMS or centre-of-mass system. A coordinate frame at rest in the laboratory is called the LAB system. We shall only consider non- relativistic kinematics here. (Marmier and Sheldon (1969) give the relativistic case.) Suppose the target particle A, with mass mA, is at rest in the LAB and the projectile a, with mass m 8 , is incident with velocity v8 • The CMS is moving in the LAB with a velocity (B1) hence the projectile has a velocity in the CMS of , mA Va =va -VcM = Va (B2) mA +ma where we use primes to denote quantities measured relative to the CMS. The target has a velocity in the CMS of vA. = - VcM. The total momentum of the pair is zero in the CMS so that Thus their speeds are in the ratio , Va mA ---, (B3) VA ma 297 298 INTRODUCTION TO NUCLEAR REACTIONS The bombarding energy E = t mav~ becomes transformed into _t v2 E -2ma a =tmaV~2 +t mAv~ +t(ma +mA)VJM =t JlaV~ + t MVtM (B4) =EQ +EcM where Ita is the reduced mass of the pair IJ.Ol = (BS) and M is the total mass, M =m 8 + mA. We recognise EcM as the kinetic energy associated with the motion of the centre of mass, while £ 01 is the kinetic energy of relative motion in the CMS; also (B6) ma+mA After a collision, the centre-of-mass motion, hence EcM and VcM, are un- changed. The energy of relative motion E01 will be unchanged if it is an elastic collision, although the directions of motion of the two particles will change. If the Q-value of a non-elastic reaction A(a, b)B is Q01 fJ, the energy of relative motion after the reaction will be Efl=£01 +Q01 fl (B7) The reduced mass in the exit channel will be mbms IJ.(J=--- (B8) mb+ms In the special case of inelastic scattering, mb = m8 and ms = mA so that IJ.(1 = 1J.01 • Since the total momentum in the CMS must remain zero, the two residual particles separate in opposite directions with equal but opposite momenta. Hence their speeds in the CMS are related by (B9) Bl ELASTIC SCATTERING After an elastic collision, the speeds of the two particles in the CMS are un- changed (see equations B3, B9}. This is not true in the LAB because some momentum has been transferred to the previously stationary target. Figure Bl APPENDIX B 299 illustrates the velocity relations after collision. From the sine rule for triangles we have sin (8cM - 8L) VcM -, = x,say (B10) sin 8L Va I I I I I I I I I I I I I I I I I I I Figure Bl Velocity relationships in the LAB and CM systems for the elastic scattering of two particles (We follow the popular convention of using subscripts CM and L to denote angles measured in the CMS and LAB, respectively.) Equations B1 and B2 give rna X=- (Bll) mA Equation B 10 relates the CMS and LAB angles of scattering of the projectile a. This relation is shown graphically in Figure B2 for several values of x. When x.;;;;; 1, 8L increases monotonically from 0 to 11 as 8cM increases from 0 to 11. For x > 1, two values of 8cM contribute to a given value of 8L and 8L has a maximum value which is smaller than 'IT. This can be understood physically; x > 1 means the projectile is heavier than the target and even a head-on collision will leave the projectile still moving forward. In the CMS this would appear as backward scattering. The corresponding angles of recoil of the struck particle A (see Figure BJ) are related by (B12) 300 INTRODUCTION TO NUCLEAR REACTIONS 0 90 180 Bc.m. (dtQ) Figure B2 Relationship between scattering angles in the LAB and CM systems. For elastic scattering, xis the ratio of the masses of the two particles, x = mafmA. For non-elastic scattering, x is given by equation B20 because v~ = VcM. Further' acM = 1T - 8cM' so that Cl:L = t (1T- 8cM) {B13) Another useful relation is obtained by equating components of the momenta perpendicular to and parallel with the beam v~ sin 8cM = Va sin 8L v~ cos 8cM + VcM = Va cos 8L These yield sin 8cM tan 8 L=----- (Bl4) x+cos8cM or x +cos 8cM COS 8L = ----::------:-:- (BlS) (1 + x 2 + 2x cos 8cM)1/ 2 The definition of a cross-section implies that the same number of particles are scattered into the element dilL of solid angle in the direction (8 L, tPL) as are scattered into dilcM in the corresponding direction (8cM, tPcM)· Thus the APPENDIX B 301 differential cross-sections are related by aL(lh)dnL = :rcM(8cM)dncM (B16) Since the transformation between LAB and CMS is symmetric in azimuth about the beam direction, we have tPL = 1/JcM, = 1/J say. Hence we need -=--= (B17) aL dncM d(cos 8cM) From equation B15 we soon find d(cos 8L) 1 +xcos8CM =------(B18) d(cos 8cM) (1 +x2 + 2x cos 8cM)312 It is also convenient to have this relation expressed in terms of the LAB angle; it can be shown that d(cos 8L) (B19) d(cos 8cM) B2 NON-ELASTIC COLLISIONS We shall not derive these results here but leave that as an exercise for the reader. The relations BlO and B14-B19 remain valid if the expression for xis generalis- ed. For the reaction A(a, b)B the expression to use is 2 X = V~M = [ mamb E01. Jl/ (B20) Vb mAmB E01. + Q01.p We note that x is still the ratio of the speed of the centre of mass to the speed of the outgoing particle in the CMS (compare with equation BIO). The relation B12 no longer holds because in general we do not have v8 = VcM. B3 SPECIAL CASES When x = 1, as for the elastic scattering of two particles of equal mass, equation B10 gives 8cM = 28L so that 8L cannot exceed ~1T (see Figure B2). The CMS and LAB cross-sections are then related l:Sy aL(8L) 4 8 ---'---'-- = COS L acM(8cM) Consequently, even if the angular distribution is isotropic in the CMS (a eM = constant, as for the scattering oflow-energy neutrons from protons) the angular 302 INTRODUCTION TO NUCLEAR REACTIONS distribution in the LAB is proportional to cos 8L. Further, equation B13 shows that for elastic scattering that is, the scattered and recoil particles move at right angles in the LAB. When x << 1, we may expand in powers of x. For example BcM R: 8L + x sin 8L and if 8 is also small Also aL(OL) R: 1 + 2x cos 8L acM(OcM) and acM (BcM) R: 1 - 2x cos BcM aL(8L) REFERENCE Marmier, P. and Sheldon, E. (1969). Physics ofNuclei and Particles. New York; Academic Press Appendix C. Some Useful Data The physical constants were obtained from E. R. Cohen (1976), Atomic Data and Nuclear Data Tables, Vol. 18, 587. Note that m =metre, g =gramme, s =second, J =Joule= 107 erg, 1r = 3.14159265, e = 2.71828183. Cl PREFIXES tera (T) = 1012 deci (d) = to-t nano (n) = 10-9 giga (G) = 109 centi (c) =10- 2 pico (p) = 10-12 mega (M) = 106 mil1i (m) = to-3 femto (f) = t0-15 kilo (k) = 103 micro (lJ.) = 10-6 atto (a) = w-11! C2 PHYSICAL CONSTANTS Speed of light c = 2.99792458 x 1011 m s-1 ~ 3.00 x 1023 fm s-1 Elementary charge e = 4.803242 x to- 10 esu = 1.602t89 X 10- 19 C e2 = 1.4400 MeVfm Planck's constant h = 6.626t76 X 10-34 J S =4.13570 x 10-21 MeVs 'fi =h/21r = 0.65822 x t0- 21 MeV s 1i2 = 41.802 u MeV fm 2 Fine structure constant a= e2 /ftc= 7.29735 x 10-3 = 1/137.036 Avogadro constant N A = 6.022 x 1023 mol- 1 Boltzmann constant k8 = 0.8617 x 10-4 eV K- 1 Electron volt eV= 1.602t89 x 10- 19 J 303 304 INTRODUCTION TO NUCLEAR REACTIONS C3 REST MASSES atomic mass unit u = 1.660566 x 10-24 g ={1/12) mass ofneutral 12C atom uc2 =931.502 MeV electron me =0.54858 X 10-3 U mec2 =0.51100 MeV muon m,., = 0.1134 u m,.,c2 = 105.7 MeV pion m1r± = 0.1499 u m1r±c2 = 139.6 MeV m1ro = 0.1449 u m"0 c2 = 135.0 MeV proton mp = 1.007276 u 2 m p c = 938.280 MeV neutron m0 = 1.008665 u m0 c2 = 939.573 MeV deuteron md =2.013553 u mdc2 = 1875.628 MeV binding energy = 2.225 MeV a-particle m01 = 4.001506 u m01 c2 = 3727.409 MeV binding energy= 28.30 MeV C4 RELATED QUANTITIES Compton wavelength: electron -fi/mec = 386.16 fm proton fi/mpc= 0.2103 fm Non-relativistic wave number for mass m with energy E k = 2rr/'A =0.2187 [m(u)E(MeV)] 112 fm-1 Non-relativistic speed for mass m with energy E ftk v = - = 1.389 x 1022 [E(MeV)/m(u)] 112 fm s- 1 m Wave number for photon of energy E k = 2rr/'A= 5.068 x 10-3 (£(MeV)] fm- 1 Sommerfeld (Coulomb) parameter for two particles with charges Z 1 e and Z 2 e, reduced mass m, CM energy E and velocity v Z Z e2 1 2 1 n = = 0.1575 Z 1Z2 [m(u)/E(MeV)] / 2 -ltv APPENDIXC 305 cs THE ELEMENTS listed are the elements with their chemical symbols and their atomic numbers Z. Also given is the mass number A of the most abundant naturally occurring isotope. When there is no stable isotope, the A for the isotope with the longest known lifetime is given in parentheses. Note that some elements have several stable isotopes; the largest number occur for tin, Sn, which has 10. Element Symbol z A Element Symbol z A hydrogen H 1 krypton Kr 36 84 helium He 2 4 rubidium Rb 37 85 lithium li 3 7 strontium Sr 38 88 beryllium Be 4 9 yttrium y 39 89 boron B 5 11 zirconium 'h: 40 90 carbon c 6 12 niobium Nb 41 93 nitrogen N 7 14 molybdenum Mo 42 98 oxygen 0 8 16 technicium Tc 43 (97) fluorine F 9 19 ruthenium Ru 44 102 neon Ne 10 20 rhodium Rh 45 103 sodium Na 11 23 palladium Pd 46 106 magnesium Mg 12 24 silver Ag 47 107 aluminium Al 13 27 cadmium Cd 48 114 silicon Si 14 28 indium In 49 115 phosphorus p 15 31 tin Sn so 120 sulphur s 16 32 antimony Sb 51 121 chlorine Cl 17 35 tellurium Te 52 130 argon Ar 18 40 iodine I 53 127 potassium K 19 39 xenon Xe 54 132 calcium Ca 20 40 caesium Cs 55 133 scandium Sc 21 45 barium Ba 56 138 titanium Ti 22 48 lanthanum I.a 57 139 vanadium v 23 51 cerium Ce 58 140 chromium Cr 24 52 praseodymium Pr 59 141 manganese Mn 25 55 neodymium Nd 60 142 iron Fe 26 56 promethium Pm 61 (145) cobalt Co 27 59 samarium Sm 62 152 nickel Ni 28 58 europium Eu 63 153 copper Cu 29 63 gadolinium Gd 64 158 zinc Zn 30 64 terbium Tb 65 159 gallium Ga 31 69 dysprosium Dy 66 164 germanium Ge 32 74 holmium Ho 67 165 arsenic As 33 75 erbium Er 68 166 selenium Se 34 80 thulium Tm 69 169 bromine Br 35 79 ytterbium Yb 70 174 306 INTRODUCTION TO NUCLEAR REACTIONS Element Symbol z A lutetium Lu 71 175 hafnium Hf 72 180 tantalum Ta 73 181 tungsten w 74 184 rhenium Re 75 187 osmium Os 76 192 iridium Ir 77 193 platinum Pt 78 195 gold Au 79 197 mercury Hg 80 202 thallium Tl 81 205 lead Pb 82 208 bismuth Bi 83 209 polonium Po 84 {210) astatine At 85 {210) radon Rn 86 {222) francium Fr 87 {223) radium Ra 88 {226) actinium Ac 89 {227) thorium Th 90 {232) protactinium Pa 91 {231) uranium u 92 {238) neptunium Np 93 {237) plutonium Pu 94 (244) americium Am 95 {243) curium Cm 96 {247) berkelium Bk 97 {247) californium Cf 98 {251) einsteinium Es 99 {254) fermium Fm 100 {253) mendelevium Md 101 nobelium No 102 {255) lawrencium Lw 103 rutherfordium Rf 104 {261) hahnium Ha 105 Appendix D. Penetration of Potential Barriers and the Fusion of Very Light Nuclei The most tightly bound nuclei are those near the middle of the periodic table (A- 50 to 100, say). Consequently, a sufficiently heavy nucleus may release energy by splitting (fissioning) into two lighter ones (Preston and Bhaduri, 1975). This may occur either spontaneously or after the capture of a neutron, and is the source of energy in a nuclear reactor (as well as the fission or 'atomic' bomb). On the other hand, two lighter nuclei may release energy by combining (fusing) to form a heavier one. Many such fusion reactions have been studied experimentally using beams of heavy ions from accelerators to bombard target nuclei (sections 2.18.12 and 4.12), but only the fusion of two of the lightest nuclei is likely to provide a practical source of energy. The reason is that the repulsive Coulomb force between the two nuclei is proportional to the product of their charges, Z 1Z 2 e2 , and this Coulomb barrier (section 2.18.8 and Figure 2.35) must be overcome before the attractive nuclear forces can initiate the reaction. This fusion process powers the sun and other stars, as well as the fusion, or 'hydrogen', bomb. It is the focus of attention in attempts to produce a controlled thermonuclear reactor. It also provides an interesting example of a practical application of the theory of the tunnelling through potential barriers that is allowed by quantum mechanics (section 3.6). Deuterium, D, is an attractive fuel to burn because it is readily available from the 'heavy water', D2 0, that is to be found in ordinary water, H2 0. Two deuterium nuclei (deuterons) also have the lowest Coulomb barrier to be over- come, with Z 1 =Z 2 = 1. The two reactions of interest are d + d-+ t+ p + 4.03 MeV, d +d-+ 3He + n + 3.27 MeV. 307 308 INTRODUCTION TO NUCLEAR REACTIONS (The capture reaction d + d ~ 4 He + 'Y releases 23.9 MeV of energy, but the branching ratio into this channel is extremely small.) The relative velocity between two deuterons needed to surmount their mutual Coulomb barrier may be obtained by heating a plasma of deuterium to a very high temperature (hence the term thermonuclear reaction). The barrier height is of the order of hundreds of ke V. Since Boltzmann's constant is kB ~ 10-4 eV K-1 (Appendix C), it takes a temperature 8 of over a billion (109 ) degrees for the average kinetic energy of thermal motion(- kBO) to surmount this barrier. Actually, a lower temperature is sufficient (Rolfs and Trautvetter, 1978) because (i) the deuterons may tunnel through the potential barrier (section 3.6) even when their energy is below its top, and (ii) the dis- tribution of velocities in the plasma will provide a considerable proportion of the deutrons with kinetic energies greater than the mean (which would be ikBO for a Maxwell distribution). The appropriate conditions for fusion to occur exist in the hot centres of stars, but their achievement on earth, and the containment of the hot plasma, remains a formidable challenge to engineers that is still being addressed. The quantum mechanical tunnelling through the Coulomb barrier is possible even when the deuterium is not at a high temperature, so that the deuterons have very little thermal energy. This would lead to cold fusion. However, its probability is very small (otherwise there would be very little deuterium left on earth!). For example, consider a gas of deuterons at room temperature (0 ~ 300 K). If the distribution of velocities is Maxwellian, the mean kinetic energy of relative motion of two colliding deuterons is ikBO = 0.039 eV (Clayton, 1968). Then we may use the low energy limit (equation 3.79) for the barrier transmission factor, T. This gives the probability for two deuterons with this mean energy of penetrating their mutual barrier to be T(E = 0.039 eV) ~ w-2176 , an extremely small number! However, this is very misleading because there is a distribution of velocities in the gas which we must average over in order to find the overall probability of fusion. The high energy tail of this distribution is very important because the barrier transmission factor, T(E), increases so rapidly with energy, E. What is required in order to evaluate the fusion rate (Clayton, 1968; Rolfs and Trautvetter, 1978) is the average (a(E)v), where vis the relative velocity of the pair of deuterons, and then, from equation 3.78, their cross-section for fusion at energy E can be written in the form a(E) = E-1ST(E), with the 'astrophysicalS factor' essentially constant (Clayton, 1968). Also, from equation 3.79 we have Qn T(E) = -bE-112 , with b constant. The probability, P(E), of finding a pair with relative energy, E, is proportional to E 112 exp( -E/kBO) if the distribution is Maxwellian (Clayton, 1968). Then the product a(E) v P(E) is proportional to exp( -E/kBO - b/E112 ), which is peaked atE= E 0 = (!bkBOi13 with a full width at half-maximum of M/E0 = 4(3E0 /kBOr 112 . Usually E 0 ~ kBO, so that the peak is sharp. In our case, b ~ 990 (eV) 112 and, for example, E 0 = 3.05 kBO at a temperature of 108 K. APPENDIX D 309 However, at a terrestrial room temperature (J = 300 K, the peak moves out much further on the tail of the distribution. Then E0 R:: 211 kaO and AE/Eo R:: 0.16. The barrier penetration factor at this energy E =E 0 is now T(E0 ) R:: 10-183 , an enormous increase over the value for the mean energy of tk8 0. (Of course, this advantage is partially off-set by the much smaller probability, P(E0 ), of finding two deuterons in the gas with this relative energy.) Nonetheless, the fusion probability remains negligibly small. The probability would be enhanced if some external environment could be found that would 'squeeze' the deuterons closer together and thus assist in overcoming the barrier. For example, the electrons in a deuterium molecule bind the two deuterons and partially shield the Coulomb repulsion between them. It has been estimated (Van Siclen and Jones, 1986) that this may greatly increase the barrier penetration factor' perhaps to greater than w-80' but still it is small, ensuring that a gas of deuterium remains quite stable under normal conditions. A further gain is possible by replacing an electron in the deuterium molecule by a muon (Massey et al., 197 4 ). The greater mass of the muon (207 times that of the electron -see Appendix C) results in its occupying an orbit with a much smaller radius than that of the electron, thus binding the two deuterons closer together by a factor of about 200. The barrier penetration factor is again estimated to be increased dramatically, perhaps to better than 10-4 (Van Siden and Jones, 1986) This is now sufficiently large for spontaneous fusion to have been observed in muonic deuterium molecules (Massey et al., 1974). Because the muon plays only a transitory role, it has been called muon-catalysed fusion. Difficulties in utilising this process include (i) the muons have to be produced independently, and (ii) they only live a short time (2.2 x w-6 s) before decay- ing into an electron and two neutrinos. REFERENCES Clayton, D. D. (1968). Principles ofStellar Evolution and Nucleosynthesis, Chapt 4. New York; McGraw-Hill Massey H. S. W., Burhop, E. H. S. and Gilbody, H. B. (1974). Electronic and Ionic Impact Phenomena, Vol. 5. Oxford; Oxford University Press Preston, M A. and Bhaduri, R. K. (1975). Structure of the Nucleus. Reading, Mass.; Addison-Wesley Rolfs, C. and Trautvetter, H. P. (1978).Ann. Rev. Nucl. Part. Sci. Vol. 28, 115 Van Siclen, C. DeW. and Jones, S. E. (1986). J. Phys. Vol. 12, 213 Solutions to Exercises 1.1 (i) 2.4 x 1018 MeV 2.39 x 10- 23 fm (ii) 1.2 x 1014 MeV 2.39 x 10- 19 fm (iii) 10.1 MeV 9.02 fm 4.02 x 10- 12 MeV 1.01 x 10- 12 MeV 1.2 (i) 70 kW 0.35 g weight (ii) 70 kW 0.17 g weight (iii) 140 kW 0.17 g weight 1.3 143.2 em s-1 = 5.16 km hr-1 10.5 em 1.4 813 MeV 885 kg weight 2.24 X 1027 X g 1.5 See equation 2.19 Vc(O) = t Vc(R) Vc(O) = 25.3 MeV Vc(7)= 16.9 MeV --- 6 Z 1Z 2 e2 Vc(r ,;;;; R) = 5 = 20.2 MeV R 1.6 Potentials: r = 2 fm: 0.72 MeV 5.85 X 10-37 MeV 6.71 MeV r = 1 fm: 1.44 MeV 1.17 X 10-36 MeV 27.4 MeV Forces: r= 2 fm: 0.36 MeV fm- 1 2.9 X 10-37 MeV fm-1 8.15 MeV fm- 1 r = 1 fm: 1.44 MeV fm- 1 1.2 X 10-36 MeV fm-1 47.0 MeV fm- 1 1.7 1.93e 6.45 X 10-24 g 7.27 MeV See Appendix C 1.8 6.07 X 10-14 erg= 3.79 X 10-2 eV 1.35 X 103 m s-1 5.625 X 101° K 1.9 6.05 x 1033 dyn cm- 2 ~ 3 x 1021 x K (steel) 310 SOLUTIONS TO EXERCISES 311 1.10 Q = t Ze(a2 - b2 ) (i) a/b = 1.35 (ii) a/b = 1.373 1.11 m = 2m0 if K = m 0 c2 4MaMA 2 (8Ma£)1/2 2.1 EA = E cos (}A VA= COS (}A (Ma +MA)2 (Ma +MA) Mn""' 1.16Mp En""' 5.7 MeV 2.2 See Appendix B and Figure B2 2.3 Ep = cos 2 Op MeV aL(Ep) = 4 acME~/ 2 aL(Op) = 4acMcos Op 2.4 4.029 MeV 9.40 MeV 107.30 MeV 2.5 38.18 MeV 27.57 MeV Qe 6 o)= 11 or 12 Q(p)::: 4 2.6 359mb 2.7 Ve::: 21.47 MeV VN =- 1.97 MeV Fe= 1.95 MeV fm- 1 FN:::- 3.95 MeV fm- 1 Fe + FN = 0 at r = 11.387 fm 78° 1 2.8 -- (r4 } q4 120 3 oo q2P F(q) = --[sin (qR)- qRcos (qR)] = ~ (-l)P (r2P} (qR)3 p=O (2p + 1)! (r')= _3_ Rn 3 +n Zeros when tan x = x (x = qR): x 0 ""'4.5, 7.75, 10.9 ... (}min :::::; 19°, 31°, 48°; R "'-" 3.6, 3.8, 3.4 fm 2.9 z-axis parallel to k; I perpendicular to k 2.10 [((J):::k- 1 ~ (2Q+l)exp(i52)sin5 2 PQ(cos0) if 11Q:::exp(2i5 2) Q da = k- 2 {sin2 5 0 +6cos(5 0 -5I)sinli0 sin 5 1 cos(} +9sin2 51 cos2 (J dQ 47T [ . 2 • 2 ] a= - Sin 50 + 3 Sin 5 1 k2 2.11 42 MeV 2.12 2.04% 3.72 X 10-3 K 538 kG 2.13 dasym = 4daunsym (spin-0) dasym = daunsym (spin-t) 312 INTRODUCTION TO NUCLEAR REACTIONS s 2.14 2 2.16 1(8) = F0 (8) = sin2 () 1(8) = t = isotropic 3.3 'IJ!=xi(r)l/II(TA)+x2(r)l/12(TA) [V2 - UII (r) + k~] XI (r) = UI2 (r)x2 (r) [V2 - U2 z(r)+k~]X2(r)= U2I(r)xi(r) [V2 - Ueff + k~] XI (r) = 0 1 if Ueff = UII + Lt UI2 U2I e--+0 V 2 - U22 + k~ + ie Lt = 2 3.4 8=tan-I ~tanKR J-kR if K= [~7 (E-1I/ ;;;:.o amplitude for r < R _ [ (Ko KR)2 ]-I/2 h K _ [ 2mVJI/2 -=------_ 1 + - cos w ere 0 - - -- amplitude for r > R k 112 2 4rr . 2 2 tanK0 R ao = - sm 8 -+ 4rrR [ - 1~ ask-+ 0 k 2 KoR a0 = 0 if tanK0 R = K 0 R See section 3.6.1 3.5 mV, R 2 mJI,oR 0 << 1 if kR << 1; ~-- << 1 if kR >> 1 ~2 1i2k 3.6 ( da) =(2m Vo) 2(-1 -)2 if q = 2k sin tO "dn BA rdi 2 o? + q2 Let a-+ 0, keeping (V0 /a) =- ZIZ2 e2 3.7 0.59 fm SOLUTIONS TO EXERCISES 313 3.9 f(O) = k-1 l; {2!2 + 1) CQP!l(cos 0) 2 Oet = 47Tk- 2 l; (2!2 + 1)1 C2 12 Q Oabs = 41Tk- 2 l; (2!2 + 1) [ImCQ- I qzl2] Q Otot = 47Tk- 2 l; (2!2 + 1) ImCQ Q 3.10 f(O) = g(cos 0) + h(cos 0) d·k x k' where cos (J = k·k'/k2 Polarisation vector is parallel to k x k' f(O)=a+b(n•dt)(n·~)i'cn·(dt +d2)+dn•(I(J, -d2) + e(ll•d1) (ll•d2) + g(m·d1) (m·d2) where a, b, ... are functions of k·k' and ll, m, n are orthogonal unit vectors: ll = (k + k')/2k cos~() m = (k- k')/2k sin~() n = (k x k')/k2 sin (J d =0 for identical particles 4.1 (i) If(Jmin = 35°, 34.6°, 34.15°, 33.75° then R = 6.46, 6.51, 6.57, 6.60 fm CorrectedR= 7.26, 7.30, 7.36, 7.38 fm r0 = 1.45, 1.44, 1.44, 1.41 fm (ii)R =6.62 fm correctedR = 7.69 fm Q= 17.8 4.3 1 R [J.I3 -(p.2 -1)3/2] J.l 4.5 U(rp) = 10 [p(rp) + i V2 p(rp) (r2 ) + ...] 4.6 u,(r) =J p(r') v,(r- r') dr' i =0, 1 Vo =-} (Vpn + Vpp) V1 =-} (Vpn - Vpp) Vpn/Vpp =3 Up =- 55.3 MeV Un =- 44.7 MeV 4.8 W:::::: 53+ 25 (N- Z)/A MeV wex:(£+ so)-1' 2 4.9 4.62 fm 0. 79 fm 0.068 fm 4.10 At r = 7.7 fm (see Exercise 4.1) U= (2.23 + 0.52i), (2.29 + 0.411), (2.33 + 0.34i) MeV 4.11 A = (-)n-m (k/K) with (Ka + £\) = m1r and m integer A = (-)n-m with (Ka + £\) =(m + ~ )7r and m integer 2m ] 1/2 -1 ~ [ 4.12 cS 0 = -kR +tan [k-K tan KRJ where K = 112 (E + V) tanK0 R ] 2 OeJ ~ 47TR 2 [ - 1 KoR 314 INTRODUCTION TO NUCLEAR REACTIONS 4.13 v~ 51 MeV A ~83, 292 Vp ~57 (A= 70), 6p (A= 230) 4.14 Tpd = D(K)F(Q) where D(K) =I l/l~(s)Vpn(s) exp(-iK·s) ds, K=tkd -kp F(Q) =I 1/>(rn) exp(-iQ·rn) drn, Q =.kd - (Ms/MA)kp C =[cxfj(a+~)Jl/2 27r{P- a)2 8:e32d) 1/2(a: a )3/2 D(K) =Do ~2/(K2 + ~2), Do=- ( ... .- .- 2 F(Q)= ;-L NL [47r(2L + 1)] 1/ 2 2R 2 [h(QR)h5}>'(iKnR) Q +Kn - hil) (iKnR)j~ (QR)] where J;_ = (dfLfdr)lr=R 4.15 L = 3.0, 1.9, 1.1, 0 R = 8.36, 7.63, 7.34, (?); R ~ 2.0 x A 113 if h(x) maximum at x = 4.5, 3.35, 2.1, 0 for L = 3, 2, 1, 0 4.17 rn = 0.0409 r'Y 4.19 See Blatt and Weisskopf(1952), page 426; Lynn (1968); Lane and Thomas (1958) 4.21 tan 8 1 =sin 204 (:: -cos 29 2) where 9 1 =laboratory angle of projectile of mass m 1 and 92 =laboratory angle of recoiling target of mass m2 4.22 j 2 = (i + 1)2 commutes with Joi Absorption 80, 120-121, 143-145, Classical scattering theory 132-138 152, 187-189 deflection function 134, 135 attenuation of a beam by 30, 31 glory 138 reflection by 120, 121 orbiting 136 See also Strong absorption rainbow 137 Adiabatic approximation 167-170 semi-classical 139-143 Angular correlation 53, 54 Clebsch-Gordan coefficient 287-290 Angular distribution 28, 37, 74-77, Compound nucleus 64-66, 74-77 87,282,293-294 resonances 69-71, 223-234 See also Cross-section, differential statistical theory see Statistical Angular momentum 56, 57, 64, 74, model 75, 81, 104, 105, 126-129, Coulomb barrier 77, 78, 86, 113, 283-291 121, 122 conservation of 7, 27, 127,243 Coulomb excitation 49, 50, 172-174, transfer 202, 205 271 vector model 284, 285, 288 interference with nuclear excitation Antisymmetry 7, 59 172-174 Asymmetry (scattering) 51-53 Coulomb scattering 34-50, 113 amplitude 114 Bessel functions 56, 105-107, 162, classical 34-38 166,170,171 Mott formula 39 Binding energy 9 of electrons 44-49 Blair phase rule 171 orbits 35, 36, 43, 44 Born approximation 100, 101,204- phase shifts 114 206,213 Coupled equations 97-99, 219-221 distorted-wave Born approximation Cross-section 28-30 101, 102,206-210 absorption or reaction 29, 115- Bosons 7 118, 121, 128, 152,260,261, Branching ratio 73, 74 277 Breit-Wigner formula 70 differential 28, 37, 74-76, 95, 96, 115 integrated 115-118 Capture reaction 25 limits 238 Centre-of-mass system 22-24, 91-93, Mott 39 297-302 near threshold 124-126 Centrifugal barrier 59, 105 non-elastic 259-261 Channel 26, 129 Rutherford 37,38 channel spin 126 total 29, 32, 118-120, 177-181, Charge distribution 9, 10,44-47 258 315 316 INDEX Deep-inelastic scattering see Heavy ions Form factor 44, 49, 148 Deflection function 134, 135 Fragmentation 265 Deformed nuclei 11, 14, 19, 167, Fusion 4, 64, 81, 83,307 170,264,265 Density-dependent interactions 183, Giant resonance 78, 79, 262-265 193 Glauber method 142, 143 Density of nuclei, distribution 9, 10 Glory scattering 138 Density of nuclei, transition 49, 148 Green function 104 Deuteron stripping 67, 74, 212-214 and deuteron break-up 221 Hamiltonian 96 Butler theory 208, 214, 280 Hard sphere scattering 112, 231 Differential cross-section see Cross- Hauser-Feshbach theory 245, 270, section 271 Diffraction 8 0, 144-14 5 Heavy ions 75, 81-83,267-273 Direct reactions 65-69, 74-76, deep inelastic or strongly damped 167-175,201-218 collisions 81, 268, 269 as energy-averages 256-257 quasi-elastic collisions 268, 271- Born approximation 204-206, 273 213 shock waves 2 71 distorted-wave Born approximation Hyperons 3, 27 206-210 inelastic sea ttering 21 0-212 Identical nuclei 59, 60, 87 knock-out 215-218, 267 Impulse approximation 146-148, multistep processes 218-223 216,266 semi-classical model 202-204 distorted-wave impulse approxi- stripping and pick-up 67, 212-215 mation 216 strong coupling 218, 219 multiple scattering series 146 with heavy ions 271-273 Independence hypothesis 65, 66, 74, Distorted-wave Born approximation 242-244 101, 102, 206-210 Inelastic scattering 25 adiabatic approximation 167 Effective interaction 150 and collective model 210-212, Eikonal approximation 141-143, 220, 221 179,265 and strong absorption 167-172 Elastic scattering 25,41 with Coulomb excitation 172-174 and strong absorption 154-166 Integral equation for scattering 99, compound-elastic 231, 258, 260 100, 102-104 potential or shape-elastic 231, 260 See also Schrodinger equation Electron scattering 9, 44-49 Inverse reactions 60, 61, 87,110, Energy resolution 63, 266, 267 131 Evaporation from compound nucleus Isobaric analogue resonance 239-242 65, 72, 76,78,248-249 Isospin 7, 239-242, 286 Excitation function 63, 68, 69 Knock-out reactions 215-218, 267 Fermions 7 Fission 4, 12, 81,268,270,307 Legendre polynomial see Spherical Fluctuations (statistical) 79, 80, harmonics 252-257 Level density 248-250 and compound elastic scattering Lippman-Schwinger 103 260,261 correlation in energy 253, 254, 256 Magnetisation in nuclei 47-49 distribution of cross-sections Mean free path 31, 80 252-255 Mesons 3, 22, 27, 267 effect of direct reactions 256, 257 Molecules (nuclear) 83 INDEX 317 Neutron scattering 31-33, 69, 72 Quark-gluon plasma 82, 271 optical model at low energies Quasi-free scattering 67 262-265 Q-value 24-27 strength function 261-265, 279 total cross-sections 31-33, 177-181 Racah coefficient 290, 291 Nucleon-nucleon force 42, 43, 57, Radius of a nucleus 9, 10, 30-33, 58, 86, 87, 183 40,41,45-47,198-201 effective interaction 183 charge radius 40, 45, 198, 199 Nucleon-nucleon scattering 42 matter radius 199, 201 potential radius 190, 191, 197, 200,201 Optical model 80, 17 5-198 strong absorption radius 41, 42, at low energies 257-265 154, 157, 158, 196, 199-201, Optical potential 99, 175-177 277 ambiguities 192, 194-197, 279 Rainbow scattering 137, 138 Rearrangement collision 25 empirical potentials for alpha Reciprocity 61, 110 particles 195-197, 279 Reduced mass 93 empirical potentials for neutrons Relative momentum 92, 93 194 empirical potentials for protons Relativity 16, 19, 82,266,271,297 191-194 Reorientation 219 energy dependence. 183, 184 Resolution (energy) 63 folding model for composite nuclei Resonance 69-71, 224, 226-242 186, 187 Breit-Wigner 70, 227, 229 folding model for nucleons 181- isobaric analogue 239-242 183, 185,278 partial width 228, 229, 231,233, for heavy ions 197, 198 234 imaginary potentials 187-189, time delay 235-237 278 wave function 234, 235 mean square radius 193 width 70, 71,227,231 phenomenological form 190, 191 See also Giant resonance spin-orbit coupling 185, 186 R-matrix 131, 132 symmetry potential 194, 278 Rotational states 13, 14, 167 volume integral 193 Rutherford scattering 34-38 Optical theorem 118, 119, 129 orbits 43, 44 Orbiting 136 Scattering amplitude 94, 95, 100, 108, 151 Born approximation 100, 101 Parity 7, 11, 243 distorted-wave Born approximation Partial waves 54-59, 104-107 101, 102 partial wave amplitudes 56, for particles with spin 151 112-114 impulse approximation 146-148 Partition 26 multiple scattering series 146 Peripheral reactions 67 partial-wave expansion 56, 112-114 Phase shift 56, 86, 109-112, 150 See also Strong absorption Pick-up reactions see Direct reactions Scattering matrix 108-110, 129, 130 Plane waves 90-92, 106, 107 poles of 239 Polarisation 50-53, 151, 186 symmetry 110, 130 tensor 50, 51 unitarity 109, 110, 130 vector 51 Schrodinger equation 96-100 Potential barriers 59, 77, 78, 105, coupled equations form 97-99 113, 120-124 for potential scattering 96, 104, Pre-equilibrium reactions 69, 250, 251 105 318 INDEX Schrodinger equation (cont.) and the compound nucleus 244, integral form 99-104 245 radial equation 1OS Blair phase rule 171 Selectivity of reactions 6 2 Coulomb effects 158-161, 164, Semi-classical theories 13 2, 13 9-143, 165, 172-174 145 Fraunhofer diffraction 161-165, effects of absorption 143-145 17Q-172 eikonal approximation or Glauber Fresnel diffraction 165, 166 method 141-143 quarter-point recipe 166 WKB approximation 140, 141 sharp cut-off model 154 SheH model 5, 12-14 smooth cut-off models 15 5-15 7 Single-particle states 12-14 Strong coupling see Direct reactions Sommerfeld parameter 43 Strongly damped coHisions see Heavy Spallation 265 ions Spectrum (energy) 71, 72, 78 Super-heavy elements 83 Spherical harmonics 55, 105-107, Surface diffuseness 10, 45, 157, 158 115,166,170,205, 21Q-211, 213,285,291-294 Threshold 27 Spin 10, 11, 126-129, 151, 185, cross-section near 124-126 285, 286 Time reversal 60, 61 channel spin 126 Transfer reactions see Direct reactions spin-orbit coupling 185, 186, 282 Transition density 49, 148 statistical weight for 61, 127, 128, Transmission coefficient 121-124, 131, 237, 294-296 244,245 Statistical model 242-250 and strong absorption 244, 245 Unitarity 109, 110 angular distributions 246, 247 evaporation model 248, 249 Vibrational states 13, 14, 170 Hauser-Feshbach theory 245 temperature 249 Wavelength 21, 22 Statistical weights see Spin Wick's inequality 118 Strength function 79, 261, 279, 280 Width of resonance 70, 71,227,231 Stripping reactions see Direct reactions partial width 228, 229, 231, 233, Strong absorption 58, 80, 143-145, 234 153-175 Wigner n-j symbols 290, 291 and inelastic scattering 167-172 WBK approximation 140, 141