22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Overview of Cross Section Cacuation References P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements of Neutron Interaction Theory (MIT Press, 1971), Chap 4. The interaction of radiation with matter is a cornerstone topic in 22.101. Since a radiation interactions can be described in terms of cross sections, ceary the concept of a cross section and how one can go about cacuating such a quantity are important considerations in our studies. To keep the detais, which are necessary for a deeper understanding, from getting in the way of an overa picture of cross section cacuations, we attach two appendices at the end of this Lecture Note, Appendix A Concepts of Cross Section and Appendix B, Cross Section Cacuation: Method of Phase Shifts. The students regard the discussions there as part of the ecture materia. We begin by referring to the definition of the cross section σ for a genera interaction as discussed in Appendix A. One sees that σ appears as the proportionaity constant between the probabiity for the reaction and the number of target nucei exposed per unit area, (which is why σ has the dimension of an area). We expect that σ is a dynamica quantity which depends on the nature of the interaction forces between the radiation (which can be a partice) and the target nuceus. Since these forces can be very compicated, it may then appear that any cacuation of σ wi be quite compicated as we. It is therefore fortunate that methods for cacuating σ have been deveoped which are within the grasp of students who have had a few ectures on quantum mechanics as is the case of students in 22.101. The method of phase shifts, which is the subject of Appendix B, is a we-known eementary method worthy of our attention.. Here we wi discuss the key steps in this 1
method, going from the introduction of the scattering ampitude f (θ ) to the expression for the anguar differentia cross section σ (θ ). Expressing σ ( θ ) in terms of the Scattering Ampitude f ( θ ) We consider a scattering scenario sketched in Fig.7.1. Fig.7.1. Scattering of an incoming pane wave by a potentia fied V(r), resuting in spherica outgoing wave. The scattered current crossing an eement of surface area dω about the direction Ω is used to define the anguar differentia cross section dσ / d Ω σθ ( ), where the scattering angeθ is the ange between the direction of incidence and direction of scattering. We write the incident pane wave as Ψ = be i(k r ωt ) (7.1) in where the wavenumber k is set by the energy of the incoming effective partice E, and the scattered spherica outgoing wave as i( kr ωt ) Ψ = f ( θ ) b e (7.2) sc r where f ( θ ) is the scattering ampitude. The anguar differentia cross section for scattering through dω about Ω is 2
sc σ (θ ) = J Ω = J in 2 f (θ ) (7.3) where we have used the expression (see (2.24)), J h * = Ψ Ψ Ψ * (7.4) 2 µi Ψ ( ) ( ) Cacuating f ( θ ) from the Schrödinger wave equation The Schrödinger equation to be soved is of the form h 2 2 V() r 2µ + ψ ( r ) = E ψ () r (7.5) where µ = mm /(m + m ) is the reduced mass, and E = µv 2 /2, with v being the reative 1 2 1 2 speed, is positive. To obtain a soution to our particuar scattering set-up, we impose the boundary condition ikr () e ψ r e ikz + f ( θ ) (7.6) k r>>r o r where r o is the range of force, V(r) = 0 for r > r o. In the region beyond the force range the wave equation describes a free partice. This free-partice soution to is what we want to match up with the RHS of (7.6). The most convenient form of the free-partice wave function is an expansion in terms of partia waves, ψ ( r, θ ) = R () r P (cos θ ) (7.7) =0 3
where P (cos θ ) is the Legendre poynomia of order. Inserting (7.7) into (7.5), and setting u ( r) = rr ( r), we obtain d 2 + k 2 2µ V ( r ) ( + 1) u () dr 2 h 2 2 r = 0, (7.8) r Eq.(7.8) describes the wave function everywhere. Its soution ceary depends on the form of V(r). Outside of the interaction region, r > r o, Eq.(7.8) reduces to the radia wave equation for a free partice, d 2 + k 2 ( + 1) 2 2 u () r = 0 (7.9) dr r with genera soution u ( r ) = B rj ( kr ) + C rn ( kr ) (7.10) where B and C are integration constants, and j and n are spherica Besse and Neumann functions respectivey (see Appendix B for their properties). Introduction of the Phase Shift δ We rewrite the genera soution (7.10) as u ( r ) kr >>1 (B / k )sin( kr π /2) (C / k)cos( kr π /2) = ( a / k)sin[ kr (π /2) + δ ] (7.11) 4
where we have repaced B and C by two other constants, a and δ, the atter is seen to be a phase shift. Combining (7.11) with (7.7) the partia-wave expansion of the free-partice wave function in the asymptotic region becomes sin[kr (π / 2) + δ (, ) ] ψ r θ kr>>1 a P (cos θ ) (7.12) kr This is the LHS of (7.6). Now we prepare the RHS of (7.6) to have the same form of partia wave expansion by writing f ( θ ) = f P (cos θ ) (7.13) and e ikr cosθ = i (2 +1) j ( kr ) P (cos θ ) kr>>1 i (2 +1) sin( kr π / 2) P (cos θ ) (7.14) kr Inserting both (7.13) and (7.14) into the RHS of (7.6), we match the coefficients of exp(ikr) and exp(-ikr) to obtain 1 i iδ f = ( ) [ a e i (2 +1)] (7.15) 2ik a = i (2 +1) e iδ (7.16) Combing (7.15) and (B.13) we obtain f ( θ ) = (1/ k ) (2 +1) e iδ sin =0 δ P (cos θ ) (7.17) 5
Fina Expressions for σ ( θ ) and σ In view of (7.17), the anguar differentia cross section (7.3) becomes σ ( θ ) = D 2 (2 +1) e iδ sin =0 2 δ P (cos θ ) (7.18) where D = 1/ k is the reduced waveength. Correspondingy, the tota cross section is σ = dω ( ) = 4 D 2 2 σθ π (2 +1)sin δ (7.19) =0 S-wave scattering We have seen that if kr o is appreciaby ess than unity, then ony the = 0 term contributes in (7.18) and (7.19). The differentia and tota cross sections for s-wave scattering are therefore ( ) = D 2 2 σθ sin δ o ( k) (7.20) 4π D 2 2 σ = sin δ ( k ) (7.21) o Notice that s-wave scattering is sphericay symmetric, or σ ( θ ) is independent of the scattering ange. This is true in CMCS, but not in LCS. From (7.15) we see o f o = (e iδ sin δ o )/ k. Since the cross section must be finite at ow energies, as k 0 f o has to remain finite, or δ o ( k ) 0. We can set o im k 0 [e iδ ( k ) sin δ o ( k )] = δ o ( k) = ak (7.22) 6
where the constant a is caed the scattering ength. Thus for ow-energy scattering, the differentia and tota cross sections depend ony on knowing the scattering ength of the target nuceus, σθ ( ) = a 2 (7.23) σ = 4π a 2 (7.24) Physica significance of the sign of the scattering ength Fig. 7.2 shows two sine waves, one is the reference wave sin kr which has not had Fig. 7.2. Comparison of unscattered and scattered waves showing a phase shift δ o in the asymptotic region as a resut of the scattering. any interaction (unscattered) and the other one is the wave sin(kr + δ o ) which has suffered a phase shift by virtue of the scattering. The entire effect of the scattering is seen to be represented by the phase shiftδ o, or equivaenty the scattering ength through (7.22). In the vicinity of the potentia, we take kr o to be sma (this is again the condition of owenergy scattering), so that u o ~ kr ( a), in which case a becomes the distance at which the wave function extrapoates to zero from its vaue and sope at r = r o. There are two ways in which this extrapoation can take pace, depending on the vaue of kr o. As shown in Fig. 7.3, when kr o > π /2, the wave function has reached more than a quarter of its waveength at r = r o. So its sope is downward and the extrapoation gives a distance a 7
which is positive. If on the other hand, kr o < π /2, then the extrapoation gives a distance a which is negative. The significance is that a > 0 means the potentia is such that it can have a bound state, whereas a < 0 means that the potentia can ony give rise to a virtua state. Fig. 7.3. Geometric interpretation of positive and negative scattering engths as the distance of extrapoation of the wave function at the interface between interior and exterior soutions, for potentias which can have a bound state and which can ony virtua state respectivey. 8
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Appendix A: Concepts of Cross Sections It is instructive to review the physica meaning of a cross section σ, which is a measure of the probabiity of a reaction. Imagine a beam of neutrons incident on a thin sampe of thickness x covering an area A on the sampe. See Fig. A.1. The intensity of the beam hitting the area A is I neutrons per second. The incident fux is therefore I/A. Fig. A.1. Schematic of an incident beam striking a thin target with a partice emitted into a cone subtending an ange θ reative to the direction of incidence, the 'scattering' ange. The eement of soid ange dω is a sma piece of the cone (see aso Fig. A.2). If the nucear density of the sampe is N nucei/cm3, then the no. nucei exposed is NA x (assuming no effects of shadowing, i.e., the nucei do not cover each other with respect to the incoming neutrons). We now write down the probabiity for a coisioninduced reaction as {reaction probabiity} = Θ / I = NA x σ (A.1) A where Θ is the no. reactions occurring per sec. Notice that σ simpy appears in the definition of reaction probabiity as a proportionaity constant, with no further justification. Sometimes this simpe fact is overooked by the students. There are other ways to introduce or motivate the meaning of the cross section; they are essentiay a 9
equivaent when you think about the physica situation of a beam of partices coiding with a target of atoms. Rewriting (A.1) we get σ = {reaction probabiity} / {no. exposed per unit area} = Θ IN x = 1 Θ I N x (A.2) x 0 Moreover, we define Σ=Nσ, which is caed the macroscopic cross section. Then (A.2) becomes Θ Σ x =, (A.3) I or Σ {probabiity per unit path for sma path that a reaction wi occur} (A.4) Both the microscopic cross section σ, which has the dimension of an area (unit of σ is the barn which is 10-24 cm 2 as aready noted above), and its counterpart, the macroscopic cross section Σ, which has the dimension of reciproca ength, are fundamenta to our study of radiation interactions. Notice that this discussion can be appied to any radiation or partice, there is nothing that is specific to neutrons. We can readiy extend the present discussion to an anguar differentia cross section dσ / dω. Now we imagine counting the reactions per second in an anguar cone subtended at ange θ with respect to the direction of incidence (incoming partices), as shown in Fig. A.1. Let dω be the eement of soid ange, which is the sma area through which the unit vector Ω passes through (see Fig. A.2). Thus, dω = sinθ dθ dϕ. 10
Fig. A.2. The unit vector Ω in spherica coordinates, with θ and ϕ being the poar and azimutha anges respectivey (R woud be unity if the vector ends on the sphere). We can write 1 dθ dσ = I d N x (A.5) Ω dω Notice that again dσ / dω appears as a proportionaity constant between the reaction rate per unit soid ange and a product of two simpe factors specifying the interacting system - the incient fux and the number of nucei exposed (or the number of nucei avaiabe for reaction). The normaization condition of the anguar differentia cross section is dω( dσ /d Ω) =σ, which makes it cear why d σ / dω is caed the anguar differentia cross section. There is another differentia cross section which we can introduce. Suppose we consider the incoming partices to have energy E and the partices after reaction to have energy in de' about E'. One can define in a simiar way as above an energy differentia cross section, dσ / de ', which is a measure of the probabiity of an incoming partice with energy E wi have as a resut of the reaction outgoing energy E'. Both dσ / dω and dσ / de' are distribution functions, the former is a distribution in the variabe Ω, the soid ange, whereas the atter is a distribution in E', the energy after scattering. Their dimensions are barns per steradian and barns per unit energy, respectivey. Combining the two extensions above from cross section to differentia cross sections, we can further extend to a doube differentia cross section d 2 σ / dωde ', which is a quantity that has been studied extensivey in therma neutron scattering. This cross section contains the most fundamenta information about the structure and dynamics of the scattering sampe. Whie d 2 σ / dωde ' is a distribution in two variabes, the soid ange and the energy after scattering, it is not a distribution in E, the energy before scattering. 11
In 22.106 we wi be concerned with a three types of cross sections, σ, the two differentia cross sections., and the doube differentia cross section for neutrons, whereas the doube differentia cross section is beyond the scope of 22.101. There are many important appications which are based on neutron interactions with nucei in various media. We are interested in both the cross sections and the use of these cross sections in various ways. In diffraction and spectroscopy we use neutrons to probe the structure and dynamics of the sampes being measured. In cancer therapy we use neutrons to preferentiay ki the cancerous ces. Both invove a singe coision event between the neutron and a nuceus, for which a knowedge of the cross section is a that required so ong as the neutron is concerned. In contrast, for reactor and other nucear appications one is interested in the effects of a sequence of coisions or mutipe coisions, in which case knowing ony the cross section is not sufficient. One needs to foow the neutrons as they undergo many coisions in the media of interest. This then requires the study of neutron transport - the distribution of neutrons in configuration space, direction of trave, and energy. In 22.106 we wi treat transport in two ways, theoretica discussion and direct simuation using the Monte Caro method, and the genera purpose code MCNP (Monte Caro Neutron and Photon). 12
22.101 Appied Nucear Physics (Fa 2006) Lecture 7 (10/2/06) Appendix B: Cross Section Cacuation: Method of Phase Shifts References -- P. Roman, Advanced Quantum Theory (Addison-Wesey, Reading, 1965), Chap 3. A. Foderaro, The Eements of Neutron Interaction Theory (MIT Press, 1971), Chap 4. We wi study a method of anayzing potentia scattering; it is caed the method of partia waves or the method of phase shifts. This is the quantum mechanica description of the two-body coision process. In the center-of-mass coordinate system the probem is to describe the motion of an effective partice with mass µ, the reduced mass, moving in a centra potentia V(r), where r is the separation distance between the two coiding partices. We wi sove the Schrödinger wave equation for the spatia distribution of this effective partice, and extract from this soution the information needed to determine the anguar differentia cross section σ ( θ ). For a discussion of the concepts of cross sections, see Appendix A. The Scattering Ampitude f ( θ ) In treating the potentia scattering probem quantum mechanicay the standard approach is to do it in two steps, first to define the cross section σ ( θ ) in terms of the scattering ampitude f ( θ ), and then to cacuate f ( θ ) by soving the Schrödinger equation. For the first step we visuaize the scattering process as an incoming beam impinging on a potentia fied V(r) centered at the origin (CMCS), as shown in Fig. B.1. The incident beam is represented by a traveing pane wave, Ψ = be i(k r ωt ) (B.1) in where b is a coefficient determined by the normaization condition, and the wave vector k = k ẑ is directed aong the z-axis (direction of incidence). The magnitude of k is set by 13
Fig.B.1. Scattering of an incoming pane wave by a potentia fied V(r), resuting in spherica outgoing wave. The scattered current crossing an eement of surface area dω about the direction Ω is used to define the anguar differentia cross section dσ / d Ω σθ ( ), where the scattering angeθ is the ange between the direction of incidence and direction of scattering. 2 2 the energy of the effective partice E = h k /2µ = hω (the reative energy of the coiding partices). For the scattered wave which resuts from the interaction in the region of the potentia V(r), we wi write it in the form of an outgoing spherica wave, i( kr ωt ) Ψ = f ( θ ) b e (B.2) sc r where f ( θ ), which has the dimension of ength, denotes the ampitude of scattering in a direction indicated by the poar ange θ reative to the direction of incidence (see Fig. B.1). It is cear that by representing the scattered wave in the form of (B.2) our intention is to work in spherica coordinates. Once we have expressions for the incident and scattered waves, the corresponding current (or fux) can be obtained from the reation (see (2.24)) J h = Ψ * ( Ψ ) Ψ( Ψ * ) (B.3) 2µi 14
2 The incident current is J = vb, where v = hk / µ is the speed of the effective partice. in For the number of partices per sec scattered through an eement of surface area dω about the direction Ω on a unit sphere, we have 2 J ΩdΩ = v f (θ ) dω (B.4) The anguar differentia cross section for scattering through dω about Ω is therefore (see Appendix A), σ (θ ) = J Ω = J in f (θ ) 2 (B.5) This is the fundamenta expression reating the scattering ampitude to the cross section; it has an anaogue in the anaysis of potentia scattering in cassica mechanics. Method of Partia Waves To cacuate f ( θ ) from the Schrödinger wave equation we note that since this is not a time-dependent probem, we can ook for a separabe soution in space and time, Ψ( r,) t = ψ r τ t, with τ () = exp( is of the form () () t ite / h ). The Schrödinger equation to be soved then h 2 2 V() r 2µ + ψ ( r ) = E ψ () r (B.6) For two-body scattering through a centra potentia, this is the wave equation for an effective partice with mass equa to the reduced mass, µ = mm 1 2 /(m 1 + m 2 ), and energy E equa to the sum of the kinetic energies of the two partices in CMCS, or equivaenty 15
E = µv 2 /2, with v being the reative speed. The reduction of the two-body probem to the effective one-body probem (B.6) is a usefu exercise, which is quite standard. For those in need of a review, a discussion of the reduction in cassica as we as quantum mechanics is given at the end of this Appendix. As is we known, there are two kinds of soutions to (B.6), bound-state soutions for E < 0 and scattering soutions for E > 0. We are concerned with the atter situation. In view of (B.2) and Fig. B.1, it is conventiona to ook for a particuar soution to (B.6), subject to the boundary condition ikr e ψ () e ikz k r + f ( θ ) (B.7) r>>r o r where r o is the range of force, V(r) = 0 for r > r o. The subscript k is a reminder that the 2 2 entire anaysis is carried out at constant k, or at fixed incoming energy E = h k /2µ. It aso means that f ( θ ) depends on E, athough this is commony not indicated expicity. For simpicity of notation, we wi suppress this subscript henceforth. According to (B.7) at distances far away from the region of the scattering potentia, the wave function is a superposition of an incident pane wave and a spherica outgoing scattered wave. In the far-away region, the wave equation is therefore that of a free partice since V(r) = 0. This free-partice soution is what we want to match up with (B.7). The form of the soution that is most convenient for this purpose is the expansion of ψ ( r ) into a set of partia waves. Since we are considering centra potentias, interactions which are sphericay symmetric, or V depends ony on the separation distance (magnitude of r ) of the two coiding partices, the natura coordinate system in which to find the soution is spherica coordinates, r ( r, θ, ϕ ). The azimutha ange ϕ is an ignorabe coordinate in this case, as the wave function depends ony on r and θ. The partia wave expansion is ψ ( r, θ ) = R () r P (cos θ ) (B.8) =0 16
where P (cos θ ) is the Legendre poynomia of order. Each term in the sum is a partia wave of a definite orbita anguar momentum, with being the quantum number. The set of functions { ( P x of the properties of P( x )are: )} is known to be orthogona and compete on the interva (-1, 1). Some 1 dxp ( x ) P ( ' x ) 1 = 2 2 +1 δ ' P (1) = 1, P ( 1) = ( 1) (B.9) P( x ) = 1, P( x ) = x P( x ) = (3 x 2 1)/2 P( x ) = (5 x 3 3 x)/2 0 1 2 3 Inserting (B.8) into (B.6), and making a change of the dependent variabe (to put the 3D probem into 1D form), u ( r) = rr ( r ), we obtain d 2 ( + k 2 2µ V() r +1) u () r = 0, r < r o (B.10) 2 2 2 dr h r This resut is caed the radia wave equation for rather obvious reasons; it is a onedimensiona equation whose soution determines the scattering process in three dimensions, made possibe by the properties of the centra potentia V(r). Uness V(r) has a specia form that admits anaytic soutions, it is often more effective to integrate (B.10) numericay. However, we wi not be concerned with such cacuations since our interest is not to sove the most genera scattering probem. Eq.(B.10) describes the wave function in the interaction region, r < r o, where V(r) = 0, r > r o. The soution to this equation ceary depends on the form of V(r). On the other hand, outside of the interaction region, r > r o, Eq.(B.10) reduces to the radia wave equation for a free partice. Since this equation is universa in that it appies to a scattering probems where the interaction potentia has a finite range r o, it is worthwhie 17
to discuss a particuar form of its soution. Writing Eq.(B.10) for the exterior region this time, we have d 2 2 + k ( + 1) u r (B.11) 2 2 () = 0 dr r which is in the standard form of a second-order differentia equation whose genera soutions are spherica Besse functions. Thus, u ( r ) = B r j ( kr ) + C rn ( kr) (B.12) where B and C are integration constants, to be determined by boundary conditions, and j and n are spherica Besse and Neumann functions respectivey. The atter are tabuated functions; for our purposes it is sufficient to note the foowing properties. j ( x ) = sin x/ x, n ( x) = cos x/ x o o sin x cos x cos x sin x j ( x ) =, n ( x ) = 1 1 2 x x x x x j ( x ) n ( x ) 1 3 5...(2 +1) 13 5...(2 1) x 0 x 0 +1 x (B.13) j ( x ) 1 1 x>>1 sin( x π /2) n ( x ) cos( x>>1 x π /2) x x The Phase Shift δ 18
(B.12) as Using the asymptotic expressions for j and n we rewrite the genera soution u ( r ) kr >>1 (B / k )sin( kr π /2) (C / k)cos( kr π /2) = ( a / k)sin[ kr (π /2) + δ ] (B.14) The second step in (B.14) deserves specia attention. Notice that we have repaced the two integration constant B and C by two other constants, a and δ, the atter being introduced as a phase shift. The significance of the phase shift wi be apparent as we proceed further in discussing how one can cacuate the anguar differentia cross section through (B.5). In Fig. B.2 beow we give a simpe physica expanation of how the sign of the phase shift depends on whether the interaction is attractive (positive phase shift) or repusive (negative phase shift). Combining (B.14) with (B.8) we have the partia-wave expansion of the wave function in the asymptotic region, sin[kr (π / 2) + δ ψ ( r, θ ) ] kr>>1 a P (cos θ ) kr (B.15) This is the eft-hand side of (B.7). Our intent is to match this description of the wave function with the right-hand side of (B.7), aso expanded in partia waves, thus reating the scattering ampitude to the phase shift. Both terms on the right-hand side of (B.7) are seen to depend on the scattering angeθ. Even though the scattering ampitude is sti unknown, we nevertheess can go ahead and expand it in terms of partia waves, f ( θ ) = f P (cos θ ) (B.16) where the coefficients f are the quantities to be determined in the present cacuation. The other term in (B.7) is the incident pane wave. It can be written as 19
e ikr cosθ = i (2 +1) j ( kr ) P (cos θ ) kr>>1 i (2 +1) sin( kr π / 2) P (cos θ ) kr (B.17) Inserting both (B.16) and (B.17) into the right-hand side of (B.7), we see that terms on both sides are proportiona to either exp(ikr) or exp(-ikr). If (B.7) is to hod in genera, the coefficients of each exponentia have to be equa. This gives 1 i iδ f = ( ) [ a e i (2 +1)] (B.18) 2ik a = i (2 +1) e iδ (B.19) Eq.(B.18) is the desired reation between the -th component of the scattering ampitude and the -th order phase shift. Combining it with (B.16), we have the scattering ampitude expressed as a sum of partia-wave components iδ f ( θ ) = (1/ k ) (2 +1) e sin δ P (cos θ ) =0 (B.20) This expression, more than any other, shows why the present method of cacuating the cross section is caed the method of partia waves. Now the anguar differentia cross section, (B.5), becomes σ ( θ ) = D 2 (2 +1) e iδ sin =0 δ P (cos θ ) 2 (B.21) where D = 1/ k is the reduced waveength. Correspondingy, the tota cross section is 20
σ = dω ( ) = 4 D 2 2 σθ π (2 +1)sin δ (B.22) =0 Eqs.(B.21) and (B.22) are very we known resuts in the quantum theory of potentia scattering. They are quite genera in that there are no restrictions on the incident energy. Since we are mosty interested in cacuating neutron cross sections in the ow-energy regime (kr o << 1), it is ony necessary to take the eading term in the partia-wave sum. The = 0 term in the partia-wave expansion is caed the s-wave. One can make a simpe semicassica argument to show that at a given incident energy E = h 2 k 2 /2µ, ony those partia waves with < kr o make significant contributions to the scattering. If it happens that furthermore kr o << 1, then ony the = 0 term matters. In this argument one considers an incoming partice incident on a potentia at an impact parameter b. The anguar momentum in this interaction is h = pb, where p = hk is the inear momentum of the partice. Now one argues that there is appreciabe interaction ony when b < r o, the range of interaction; in other words, ony the vaues satisfying b = /k < r o wi have significant contriubution to the scattering. The condition for a partia wave to contribute is therefore < kr o S-wave scattering We have seen that if kr o is appreciaby ess than unity, then ony the = 0 term contributes in (B.21) and (B.22). What does this mean for neutron scattering at energies around k B T ~ 0.025 ev? Suppose we take a typica vaue for r o at ~ 2 x 10-12 cm, then we find that for therma neutrons kr o ~ 10-5. So one is safey under the condition of owenergy scattering, kr o << 1, in which case ony the s-wave contribution to the cross section needs to be considered. The differentia and tota scattering cross sections become ( ) = D 2 2 σθ sin δ o ( k) (B.23) 21
4π D 2 2 σ = sin δ ( k ) (B.24) o It is important to notice that s-wave scattering is sphericay symmetric in that σ ( θ ) is manifesty independent of the scattering ange (this comes from the property P o (x) = 1). One shoud aso keep in mind that whie this is so in CMCS, it is not true in LCS. In both (B.23) and (B.24) we have indicated that s-wave phase shift δ o depends on the incoming o energy E. From (B.18) we see that f o = (e iδ sin δ o )/ k. Since the cross section must be finite at ow energies, as k 0 f o has to remain finite, or δ o ( k ) 0. Thus we can set im k 0 [e iδ o ( k ) sin δ o ( k )] = δ o ( k) = ak (B.25) where the constant a is caed the scattering ength. Thus for ow-energy scattering, the differentia and tota cross sections depend ony on knowing the scattering ength of the target nuceus, σθ ( ) = a 2 (B.26) σ = 4π a 2 (B.27) We wi see in the next ecture on neutron-proton scattering that the arge scattering cross section of hydrogen arises because the scattering ength depends on the reative orientation of the neturon and proton spins. Reduction of two-body coision to an effective one-body probem We concude this Appendix with a suppementa discussion on how the probem of two-body coision through a centra force is reduced, in both cassica and quantum mechanics, to the probem of scattering of an effective one partice by a potentia fied V(r) [Meyerhof, pp. 21]. By centra force we mean the interaction potentia is ony a function of the separation distance between the coiding partices. We wi first go 22
through the argument in cassica mechanics. The equation describing the motion of partice 1 moving under the infuence of partice 2 is the Newton's equation of motion, m && r = F 1 1 12 (B.28) where r 1 is the position of partice 1 and F 12 is the force on partice 1 exerted by partice 2. Simiary, the motion of motion for partice 2 is m && r = F = F 2 2 21 12 (B.29) where we have noted that the force exerted on partice 2 by partice 1 is exacty the opposite of F 12. Now we transform from aboratory coordinate system to the center-ofmass coordinate system by defining the center-of-mass and reative positions, mr + m r 1 1 2 r c = 2, r = r 1 r 2 (B.30) m + m 1 2 Soving for r 1 and r 2 we have m r 1 = r c + 2 m r, r 2 = r c 1 r (B.31) m + m m + m 1 2 1 2 We can add and subtract (B.28) and (B.29) to obtain equations of motion for r and r. c One finds (m 1 + m 2 )&& r c = 0 (B.32) µ r&& = F 12 = dv ( r)/ d r (B.33) with µ = mm /(m + m ) being the reduced mass. Thus the center-of-mass moves in a 1 2 1 2 straight-ine trajectory ike a free partice, whie the reative position satisfies the equation 23
of an effective partice with mass µ moving under the force generated by the potentia V(r). Eq.(B.33) is the desired resut of our reduction. It is manifesty the one-body probem of an effective partice scattered by a potentia fied. Far from the interaction fied the partice has the kinetic energy E = µ ( r& ) 2 /2 The quantum mechanica anaogue of this reduction proceeds from the Schrödinger equation for the system of two partices, h 2 2 h 2 2 + V ( r 1 r ) 2 Ψ(r 1, r 2 ) = (E 1 + E 2 )Ψ(r 1,r 2 ) (B.34) 1 2 2m 1 2m 2 Transforming the Lapacian operator 2 from operating on ( r, r ) to operating on ( r, r ), 1 2 c we find 2 2 h 2 h c 2 +V() r Ψ(r c, r) = (E + E)Ψ(r c c, r) (B.35) 2(m + m 2µ 1 2 Since the Hamitonian is now a sum of two parts, each invoving either the center-ofmass position or the reative position, the probem is separabe. Anticipating this, we have aso divided the tota energy, previousy the sum of the kinetic energies of the two partices, into a sum of center-of-mass and reative energies. Therefore we can write the wave function as a product, Ψ(, ) probems, r r = ψ (r ) ψ ( r c c c ) so that (B.35) reduces to two separate h 2 2 ψ ( r ) c c c = E c ψ c ( r c ) (B.36) 2(m + m ) 1 2 h 2 2 + V() r ψ ( r ) = E ψ () r (B.37) 2µ 24
It is cear that (B.36) and (B.37) are the quantum mechanica anaogues of (B.32) and (B.33). The probem of interest is to sove either (B.33) or (B.37). As we have been discussing in this Appendix we are concerned with the soution of (B.37). 25