Chapter 10: QUANTUM SCATTERING

Transcription

1 Chapter : QUANTUM SCATTERING Scattering is an extremely important tool to investigate particle structures and the interaction between the target particle and the scattering particle. For example, Rutherford scattering finds that atoms have small and massive nuclei, surrounded by tiny and light electrons. High-energy photons can be scattered by a neutron, revealing its inner structure formed by quarks. Quantum scattering is a very extensive subject, and we will study the basic ideas. We have briefly discussed photon scattering, now our focus is on matter particles, such as electrons and neutrons, within the formal and exact framework, though approximations will be taken as well in practice.. Cross Sections Goal: finding the differential cross section dσ /dω of an incident particle beam with definite momentum scattered by an object described by a (localized) potential. Simplification: A steady particle beam with definite momentum can be described by a planewave, and we can only consider one particle. The results for a current density of a particle beam can be obtained readily. Let s begin with one dimension (D). Aeikx + Be ikx (x ) Ce ikx + De ikx (x +) () D =. (.) () The reflection rate R = B / A. (3) The transition rate T = C / A. Fig..: One-dimensional scattering. Both R and T are a function of k, therefore (4) Theoretically, if we know V(x), we can calculate R(k) and T(k). (5) Experimentally, we can measure R(k) and T(k) to gain knowledge of V(x). Now let s look at the 3D, which shares many features with the D case. However, it is much more Fig..: 3D scattering. complicated due to angle dependence. The differential scattering cross section is 67

2 dσ dω = number of particles scattered in dω/sec number of incident particles/sec/area dω. (.) The following is the derivation of dσ /dω based on definition of Eq. (.). Neglecting the trivial normalization factor, the total wave function is where the incident wave is e ikz = e ikz + ψ sc (r,θ,φ), (.3) and the scattered wave is ψ sc (r,θ,φ). When r, assume rv(r) to guarantee a non-divergent solution of ψ sc : ( + k )ψ sc =, for r. (.4) The general solution for the free-particle equation in spherical coordinates: ψ sc (r,θ,φ) r = [A l (kr)+ B l (kr)]y m l (θ,φ), (.5) l m where Y m l is the spherical harmonics, and the spherical Bessel and spherical Neumann functions, respectively. The asymptotic behaviors of and : (kr) r sin(kr lπ / )/ kr, (.6) (kr) r cos(kr lπ / )/ kr. (.7) To get a purely outgoing wave for ψ sc = e ikr / r, A l and B l must satisfy A l / B l = i, then ψ sc (r,θ,φ) r = eikr kr l m ( i) l ( B l )Y l m (θ,φ) = e ikr r f (θ,φ) (.8) where the scattering amplitude r e ikz + f (θ,φ) eikr r (.9) f (θ,φ) = ( i) l ( B k l )Y m l (θ,φ). (.) l m The probability flux (current density) for a particle with mass m and wave function ψ : j(r,t) = which satisfies the continuity equation ( im ψ* ψ ψ ψ * ), (.) ρ t + j =. (.) 68

3 Here ρ(r,t) = ψ(r,t) is probability density, and Eq. (.) is derived from Schrödinger equation. In polar coordinate systems, The magnitude of the incident flux: The scattered flux: j inc = = ˆr r + ˆθ r θ + ˆφ r sin θ φ. (.3) im e ikz e ikr cosθ e ikz e ikr cosθ ( ) = k m. (.4) j sc = im ψ * * ψ sc sc ψ sc ψ sc ( ). (.5) At the limit of r, the derivatives along ˆθ and ˆφ directions behave as ~ (Q: why?), r while the derivative along ˆr is (θ,φ)eikr r f r Dropping the ~ r terms at the limit of r, we obtain = f (θ,φ)ik eikr r + f (θ,φ)( )eikr r. (.6) j sc = k f (θ,φ) ˆr. (.7) m r The probability (proportional to number of particles) flows into solid angle dω at the rate R(dΩ) = j sc r dω, (.8) The incident flux j inc is proportional to the number of particles/sec/area, thus dσ dω = R(dΩ) j inc dω dσ dω = f (θ,φ) (.9) Notes: () f (θ,φ) is also written as f (k, k ), with k and k the incident and scattered wave vectors. () We have considered elastic scattering, i.e., k = k ; in general, dσ dω = k k f (k, k ). 69

4 (3) We only study the elastic scattering in this class. Methods to compute the scattering amplitude f (θ,φ) : () Time-dependent perturbation theory () Time-independent perturbation theory (3) Exact theory. Formal Theory and the Born Series Up to a normalization parameter, the wave function of a free particle is φ k = e ik r, which satisfies ( + k )φ k =. (.) Under full potential V(r) a particle s wave function can be obtained by solving ( + k ) (r) = m V(r)ψ (r). (.) k We write the total wave function in the form: = e ik r + ψ sc, with eigenenergy E k = k m, but momentum is not well defined. Here k is the wave vector of the incident wave ψ inc = e ik r, while we have derived that the scattered wave ψ sc r f (θ,φ) eikr r. We need to solve the full potential Schrodinger equation [Eq. (.)] to obtain f (θ,φ). We use Green s function G (k;r, r ) to find the formally exact solution, which satisfies ( + k )G (k;r, r ) = m δ(r r ). (.) The exact solution to Eq. (.) can be obtained by solving the Lippman-Schwinger equation: (r) = e ik r + G (k;r, r )V( r )ψ ( r ) d r. (.3) k As usual, we can solve it iteratively starting from the zeroth order: Zero-th order (for zero potential V): () = e ik r. First-order Born approximation: ψ () k (r) = e ik r + G (k;r, r )V( r )e ik r d r. (.4) 7

5 You might see the above expression in symbolic form iiterature: The Born Series for exact solution: ψ () = ψ () +G V ψ (). (.5) ψ = ψ () +G V ψ () +G VG V ψ () +. (.6) In this class, we focus on the first-order Born approximation, which requires an explicit form to Green s function G (k;r r ), which can be obtained using contour integral: Then and (r) = e ik r + ψ sc (r). G (k;r, r ) =G (k;r r ) = eik r r r r m π. (.7) (r) = e ik r m e ik r r π r r V( r ) ( r ) d r, (.8) Now we derive the formally exact f (θ,φ) for ψ sc r f (θ,φ) eikr r. When r, r r r( r r / r ), (.9) then r r r + r r r, (.3) k r r kr r r r = kr k r, (.3) where the scattered wave vector k = kˆr for elastic scattering. Thus, e ik r r r r eikr r e i k r, (.3) ψ sc (r) r eikr r m e i k π r V( r )ψ ( r )d 3 r, (.33) k and finally we obtain the exact expression for scattering magnitude: f (θ,φ) = f (k, k ) = m e i k π r V( r )ψ ( r )d 3 r (.34) k 7

6 Note that e i k r V( r )ψ ( r )d 3 r = (π) 3 k k V (.35).3 T-Matrix Define the T-matrix as!t(k, k ) e i k (π) r V(r)ψ 3 k (r)d 3 r, (.36) so that dσ dω = 4π m! "T(k, k ). (.37) The Born series ψ = ψ () +G V ψ () +G VG V ψ () + can be employed to obtain the T-matrix. Under the (st-order) Born approximation:!t () (k, k ) = e i k (π) r V(r)e ik r d 3 r, (.38) 3 which is essentially the Fourier transformation of potential: Let!V(k k ) = e i k r V(r) e ik r d 3 r. (.39) (π) 3 T(k, k ) = T(k k ) = e i k r T(r) e ik r d 3 r, (.4) combining with the definition of T-matrix [Eq. (.36)], we obtain T(r)φ k (r) =V(r) (r). (.4) Obviously, T () (r) =V(r) and then T () (k, k ) = T () (k k ) = V(k k ). The expansion (Born series) of T is similar to that of ψ, as we expect. Recast this problem more formally: H = H +V, and H φ k = ε k φ k (.4) with H = p / m, ε k = k / m, and φ k (r) = e ik r. The full-potential Schrödinger equation: H = E k, (.43) 7

7 and = φ k + φ sc. For elastic scattering, E k = ε k = k. Eq. (.43) m (E k H ) =V (E k H )ψ sc =V, (.44) because (E k H )φ k =. Operating (E k H ) on both sides of Eq. (.44): ψ sc = (E k H ) V, (.45) we obtain the Lippman-Schwinger equation: = φ k +(E k H ) V. (.46) We introduce a Green s function defined as G =G H (E) = lim η (E H + iη). (.47) Here H is an operator, and positive η ensures that the scatted wave ψ sc is outgoing. Thus (r) = φ k (r)+g V(r) (r). (.48) Using Eq. (.4), i.e., T(r)φ k (r) =V(r) (r), we derive the T-matrix equation (Dyson s Eq.): which can be solved iteratively: T(r) =V(r)+V(r)G T(r), (.49) Define Then T =V +VG V +VG VG V + (.5) G =G H (E) = lim η E H + iη = [E H + iη]( G V ) = G G V =G +G VG +G VG VG + (.5) T =V +VGV. (.5) We have derived the Born series for T and ψ using two slightly different approaches, but they essentially the same. One can verify that Alternatively, G (k;r, r ) = r r E H + iη = r k m H + iη r = m e ik r r π r r. (.53) 73

8 G (k;r, r ) = φ (r)φ * ( r ) k k ε k ε k + iη d 3 k = ε k H + iη = ε k H + iη δ(r r ). e i k (r r ) d 3 k (.54).4 Applications and Validity of the First-order Born Approximation With the dramatic progress in computational power, now we can calculate exact differential cross sections for any potential. But analytically, first-order Born approximation is still widely used to reveal the basic physics and estimate the magnitude. In the following, we discuss two examples using the first-order Born approximation. To the first order, T () (r) =V(r) and then!t () (k, k ) =!T () (k k ) = V(k! k ) = e i(k (.55) k ) r V(r)d 3 r. (π) 3 The scattering magnitude f () (k, k ) = 4π m "T () (k, k ), (.56)! and the differential cross section dσ dω = 4π m! "T () (k, k ). (.57) As shown in Fig. 9.3, for elastic scattering, T () (k, k ) =T () (k k ) =T () (q), and here q = k k = k sin(θ / ). (.58) For the special case of spherical potential, V(r) =V(r), T () depends only on polar angle θ : T () (θ) = π π V(r)e iqr cos( θ ) r sin( θ )d θ dφdr 8π 3 = π q rv(r)sin(qr)dr (.59) Example : finite square well, a simple model for nuclei. Fig..3: Scattering with angle. 74 Fig..4: differential cross section.

9 V(r) = V (r a) (r > a) We use Eq. (.59) to find that T () (θ) = V a 3 sin(qa) π (qa) qa (.6) cos(qa), (.6) dσ dω = 4m V a sin(qa) cos(qa)! 4 q 4 qa Question: in Fig. 9.4 there are many minima in differential cross section, why?. (.6) Example : Yukawa potential (Screened Coulomb potential), for massive scalar fields. V(r) = V e µr µr. (.63) Using Eq. (.59) it is straightforward to obtain with (.6) In the following we discuss the physics involved: T () (θ) = V 6π µ q + µ, (.64) q = 4k sin (θ / ) = k ( cosθ). (.65) () Let µ but keep V / µ a constant, Yukawa potential reduces to Coulomb potential. If V / µ = Z Z e, dσ dω = mz Z e 4k 4 ( cosθ) = 6 Z Z e E K sin 4 (θ / ). (.66) It recovers the classical Rutherford scattering! The first-born result is valid in the high-energy limit, for which classical and quantum results typically agree. () The total cross section: σ = mv µ 4π µ (µ + 4k ). (.67) 75

10 For Coulomb potential, it becomes infinity, which means that all incident probability current will be scattered, no matter how large the incident cross section is. Therefore, the Coulomb interaction is an infinite-range interaction, while the Yukawa potential is a finite-range interaction. Validity: we replace = e ik r + ψ sc by just e ik r so that we need to ensure that ψ sc e ik r =. (.68) The scattered wave has the maximum amplitude at scattering center r =, then ψ sc (). Then ψ sc () ψ sc (r) = m e ik r r π r r V( r )e ik r d r, (.69) For spherical potential V(r) =V(r), the validity condition is m e π ikr r V(r)e ik r d 3 r. (.7) m k e ikr sin(kr)v(r)dr. (.7) In the following we consider r within a small scatting region of. () For low energy, kr, e ikr, sin(kr) kr. ψ sc () Using the simplest finite square well model: V(r) = V (r r ) (r > ) we obtain the condition for the first-order Born Approximation: m mv rv(r)dr. (.7) V, m. (.73) () For high energy, k. e ikr sin(kr) = i (e ikr ); since k, the e ikr term oscillates fast in the scattering range, which averages to zero. Therefore, ψ sc () 76

11 m k we obtain the condition for the first-order Born Approximation: V(r)dr (.74) mv k V k m. (.75) Conclusions: (i) If the Born approximation is good at low energy, it is also good for high energy; (ii) The Born approximation is normally good for high-energy limit, but it might fail miserably at the low-energy limit..5 Partial Wave and Phase Shift; Optical Theorem Motivations: () At low energy the first-born approximation breaks down except for very weak scattering potential; () For the scattered spherical-like wave (especially at low energy), it is of great interest to study the angular momentum dependency on scattering amplitude. The incident planewave can be decomposed of angular-momentum states e ikz = e ikr cosθ = (l +)i l (kr)p l (cosθ), (.76) where is the spherical Bessel function, and P l the Legendre polynomials, l= P l (cosθ) = 4π l + / Y l (θ). (.77) For spherical potential V(r) =V(r), f (θ,φ) = f (θ), and it also depends on k : f (θ,k) = (l +)a l (k)p l (cosθ), (.78) l= where a l (k) is the l-th partial wave amplitude, which is the measure of the scattering strength for the state with the (orbital) angular momentum l. For the low-energy case, l max kr (Q: why?), so you don t need to compute many a l (k). Now the task is to find a l (k) for a given potential V(r). Let s begin with a free particle, Then, (kr) r sin(kr lπ / ), (.79) kr 77

12 e ikz r (l +)i l ikr ei(kr lπ/) e i(kr lπ/) P(cosθ) l l= = (l +) eikr ik r e i(kr lπ) r P l (cosθ). l= Here there are one incoming and one outgoing wave for each value of l, and their probability current densities cancel out, i.e., no net probability flux flows into or comes out of the origin. When potential V(r) is turned on, the wave function is not e ikz, instead, (.8) (r) = (l +)i l R l (kr)p l (cosθ), (.8) l= where the radial wave function R l (kr) should reduce to that of a free-particle at r for each value of l: A l sin kr lπ / + δ l (k) R l (kr) r kr = A l ikr eiδ l e i(kr lπ/) e iδ l e i(kr lπ/), (.8) where δ l (k) is the phase shift. To keep the incoming wave unchanged, we set A l = e iδ l. Then (r) r (l +) e iδ e ikr l ik r e i(kr lπ) r P l (cosθ) l= = e ikz + (l +) e iδ l ik P(cosθ) e ikr l l= r Comparing with the expression for f (θ,k), i.e., Eq. (.78), we obtain (.83) The scattering amplitude a l (k) = e iδ l (k) ik = k eiδ l sin(δ l ). (.84) The total cross section σ = f (θ) = (l +)e iδ l sin(δ k l )P l (cosθ). (.85) l= dσ dω dω = f (θ) dω = 4π using the orthogonality of Legendre polynomials P l (x): (l +)sin (δ k l ), l= 78

13 On the other hand, P l (cos) = P l () =, then Thus we have proved the famous optical theorem: π P l (cosθ)p (cosθ) l sin(θ)dθ = δ l l +. (.86) l Im[f ()] = (l +)sin (δ k l ) (.87) σ = 4π k An alternative proof is based on Lippman-Schwinger equation: Operating k V from the left side on Eq. (.89): l= Im[f ()]. (.88) = k +G V. (.89) k V = V G V ψ = ψ k k V V G V. (.9) Here the Green function (operator) is G = lim η (E Ĥ iη) + δ(e E ) = lim η E Ĥ iη d E = iπδ(e Ĥ ). (.9) Since G has purely imaginary part, we look at the imaginary parts on both sides of Eq. (.9): Im( k V ) = π V δ(e Ĥ )V ψ = π k T δ(e Ĥ )T k. (.9) k We have used Eq. (.4), i.e., T =V k. Using dσ dω = 4π m T(k, k ), we can derive! k T δ(e Ĥ )T k = k T δ(e Ĥ ) k k d 3 k T k = k T δ(e E ) k k T k d 3 k = T(k, k ) δ(e E )d 3 k =! 4π m dσ dω δ! k m! k k m d k dω =! k dσ 4π m! k / m dω dω = k! (π) 4 m σ. (.93) 79

14 Thus we have proved the optical theorem: Im f ( θ = ) = 4π m! ( π)! k (π) 4 m σ = k 4π σ (.94) Another interesting result is e iδ l sin(δ l ) = mk and one can use this to determine phase shift δ l. (kr)v(r)r l (kr)r dr, (.95).6 Determination of Phase Shifts Let s begin with a brief review of radial wave functions for central potential d χ l m dr +V (r)χ = Eχ eff l l, (.96) l(l +) where χ l (r) = rr l (r), and the effective potential V eff (r) =V(r)+. For free particles, mr R l (r) = c (kr)+c (kr). (.97) Asymptotically, when kr, (l ) (kr) (l = ) ; (kr). Thus R l (kr) = c (kr). (.98) We want to determine phase shifts δ l for a scattering potential V(r) that vanishes for r >. Outside the scattering center, it is a free particle, but R l (r) = c (kr)+c (kr), since the origin is excluded. The wave function after scattering: (r) = (l +)i l R l (kr)p l (cosθ). (.99) We use the asymptotic behaviors of, and to determine c and c : l= (r) r (l +) e iδ e ikr l ik r e i(kr lπ) r P l (cosθ), (.) l= 8

15 sin(kr lπ / ) (kr) r, kr cos(kr lπ / ) (kr) r, kr (.) we obtain c = e iδ l cos(δ l ) and c = e iδ l sin(δ l ), so that R l (r) = e iδ l cos(δ l ) (kr) sin(δ l ) (kr), for r > r. (.) We can match the radial wave function R l (r) at r = to find the phase shift δ l. Example : The Hard Sphere V(r) =, r, r > r. Boundary conditionr l ( ) = tan(δ l ) = (k ) (k ) j δ l = tan l (k ) (k ). (.3) () Low energy limit k, the s-wave ( l = ) scattering dominates. sin(kr δ = tan )/ kr cos(k )/ k = tan tan(k ) (.4) δ = k. (This is actually exact!) f (θ) = k eiδ sin(δ )P (cosθ), (.5) Fig..5: Phase shifts for attractive and repulsive scattering potentials. and P (x) = dσ dω = f (θ) = k sin (δ l ) = tan ( k ) k + tan ( k ). (.6) At the low-energy limit, dσ dω r, and we obtain a surprising result: k σ = 4π k sin (δ ) 4πr kr, (.7) 8

16 the total cross section is four times the classical results. For other l-values, Then (x) x x l (l +)!!, (l )!! (x) x. x (l+) (.8) tan(δ l ) kr (k ) l+ δ l kr (k )l+. (l +)!!(l )!! (.9) At low energies, the scattering into the high angular momentum states is negligible. () High-energy limit k, (k ) sin(kr lπ / ) kr, k (k ) cos(kr lπ / ) kr. k (.) Then tan(δ l ) = tan(k lπ / ) δ l kr kr +lπ /, (.) and we obtain sin (δ l ) = sin ( k +lπ / ) sin (δ l )+ sin (δ l+ ) =, (.) k σ = 4π k (l +)sin (δ l ) l= The total cross section at high-energy limit is twice the classical results. Example : The finite Square Well V(r) = V, r, r > r To simplify our discussions, let s consider only the s-wave scattering. For r >, = 4π k (kr ) = πr. (.3) V and > V < (repulsive) (attactive). For r <, R (r) = e iδ cos(δ )j (kr) sin(δ )n (kr) = sin(kr + δ ) eiδ. (.4) kr 8

17 χ (r) = rr (r) = A sin(κr), (.5) where E V = κ m. Note: when E <V, κ = i k with k real. Then sin(κr) = sinh( k r). Boundary conditions: χ (r) and its derivative χ (r) are continuous at r =, then e iδ k sin(k + δ ) = A sin(κ ) e iδ cos(k + δ ) = A κ cos(κ ) (.6) tan(k + δ ) = k κ tan(κ ) tan(δ ) = k κ cot(κ )tan(k ) κ cot(κ )+ k tan(k ). (.7) Thus for attractive potential, V <, κ > k, δ > ; for repulsive potential, V >, κ < k, δ <, as illustrated in Fig Ramsauer-Townsend Effect Let s focus on the attractive case. For a given k (E is fixed), when V increases, δ also increases; one can adjust the values of V and so that δ = π /, resulting in the maximum cross section for the s-wave scattering: σ = 4π k. If V increases still, eventually one can make δ = π, so that σ =, i.e., the s-wave partial cross section vanishes. Therefore, in the low-energy limit ( k ), strong attractive potential could lead to perfect transmission without scattering. This is known as the Ramsauer-Townsend effect, which was first observed for electron scattering by rare gas atoms such as Ar, Kr and Xe in 93. Fig..6: Electron scattering from rare gases. Rev. Mod. Phys. 5, 57 (933). To understand the Ramsauer-Townsend effect, we need to use Levinson s theorem: δ l (k = ) = N l π (.8) 83

18 where N l is the number of bound states in the potential with angular momentum l. N l is finite for a potential of short range. Previously we have omitted N l π in δ l when we discuss if the phase shift is positive and negative. The phase shift is usually determined by the expressioike tan(δ l ) = F(k). We have encountered the case that F(k = ) = ; since tan(n π) =, how do we determine N? The unique answer is obtained by graphing the phase shift as a function of k. For any finite attractive scattering potential, we demand that this graph has two characteristics: () The phase shift vanishes at very large values of k, and () The phase shift is a continuous function of k. For the attractive square well potential V(r) = V, r, r > r with V <. Set k = mv, then κ = k + k. (Note: k = me ) We have derived tan(δ ) = k κ cot(κ )tan(k ) κ cot(κ )+ k tan(k ) previously, see Eq. (.7). As plotted in Fig. 9.7, in case (), there is no bound state, N =. But in case Fig..7: The s-wave phase shift for an attractive square (), there are one bound state and two well for three values of : (), (), and (3), as plotted in solid, long-dash and dashed lines. bound states in case (3). In case (3), the phase shift passes through π ; however, the Ramsauer-Townsend effect occurs only when k <. Higher value of k (i.e. higher V ) is required. For example,k ~.34 for the experiment of electron scattering by rare gas atoms. At the low-energy limit: 84

19 dσ dω = r tan(k r ) k, (.9) so that the differential cross section can be either smaller or larger (sometimes much larger) than..8 Resonance Scattering In atomic, nuclear and particle physics, the scattering for a given partial wave often has a pronounced peak, which is called the resonance scattering. This is because the phase shift δ l passes through π / or (n +/ )π, then σ l = 4π / k, the maximum value of σ l for a given k. The following is an example to show that resonance condition is equivalent to the boundstate condition. Therefore if the potential is attractive enough to form a bound state, then there is a strong possibility that there sill also be resonance scattering. Let s consider the attractive square well potential V(r) = V, r with V <., r > r This time we look at the bound states for l. The radial wave function R l (r) = A j (κr), r r l l (.) B l h () l (ikr), r > r where κ = m(e V ) /!, k = me /!, and the first type spherical Hankel function h () (x) = (x)+ i (x). (.) Under the low-energy and the strong-attractioimits, i.e., k l and κ l (where κ = mv /! ), respectively, the asymptotic behaviors are sin(x lπ / ) (x) x, x (l )!! h () l (x) x i. x l+ (.) Employing the boundary conditions at r =, i.e., continuity of radial wavefunction and its firstorder derivative, one finds that the condition for a bound state to occur is κ cot(κ lπ ) = l. (.3) 85

20 Since cot(κ lπ ) (for the strong-attractioimit), we obtain κ lπ / = (n +/ )π. (.4) The scattering states have E >, and the general expression for the phase shift δ l is: At a resonance, δ l = π / and cot(δ l ) = cot(δ l ) = k (k ) (κ ) κ (k ) (κ ) k (k ) (κ ) κ (k ) (κ ). (.5) k (k ) (κ ) κ (k ) (κ ) =. (.6) Using the asymptotic properties of and under the limits of k l and κ l : we obtain sin(x lπ / ) (x) x, x (l )!! (x) x, x l+ (.7) (l + )sin(κ lπ / )+ κ cos(κ lπ / ) =, (.8) then again due to κ l, cos(κ lπ / ) κ lπ / = (n +/ )π, (.9) which is exactly the same condition for a bound state. From Eq. (.84) one can show that the partial wave amplitude a l (k) = k cot(δ l ) ik. (.3) We consider the neighborhood of resonance, k = k R (or E = E R ), and δ l (k = k R ) = π /, by expanding δ l (E) as follows Here the width Γ is defined by which leads to cot[δ l (E)] cot[δ l (E = E R )] Γ (E E ) = R Γ (E E ). (.3) R d cot[δ l (E)] de E=E R = Γ, (.3) 86

21 and Γ / a l (k) = k (E E R )+ iγ, (.33) σ l = 4π k (l +)(Γ / ) (E E R ) +(Γ / ). (.34) The peak cannot be infinitely sharp because σ l (k) 4(l +)π k. (.35) Fig..8: scattering. versus k in a resonance 87