Scattering of Particles by Potentials

Transcription

1 Scattering of Partices by Potentias 2 Introduction The three prototypes of spectra of sef-adjoint operators, the discrete spectrum, with or without degeneracy, the continuous spectrum, andthe mixed spectrum, as we as the corresponding wave functions, contain important information about the physica systems that they describe. Yet, from a physicist s point of view, the resuts obtained unti now remain somewhat academic as ong as we do not know how to render the information they contain visibe by means of concrete experiments. For exampe, the static spectrum of the Hamitonian describing the hydrogen atom and the spatia shape of its stationary wave functions, a priori, are not observabe for us, the macroscopic observers, as ong as the atom is not forced to change its state by interaction with externa eectromagnetic fieds or by interaction with scattering beams of eectrons. In other terms, the pure stationary systems that we studied so far, must be subject to nonstationary interactions in set-ups of reaistic experiments which aow for preparation and detection, before we can decide whether or not these systems describe reaity. As this question is of centra importance I insert here a first chapter on the description of scattering processes in what is caed potentia scattering, before turning to the more forma framework of quantum theory. 2.1 Macroscopic and Microscopic Scaes In studying cassica macroscopic systems our experience shows that it is aways possibe to perform observations without disturbing the system: In observing the swinging penduum of an upright cock we measure its maxima eongation, its period, perhaps even the veocity at the moment of passage through the vertica, a stop-watch in our hand, by just ooking at it and without interfering, to any sizabe degree, with the motion of the penduum. Even extremey precise measurements on sateites or on panets by means of radar signas and interferometry are done with practicay no back-reaction on their state of motion. This famiiar, amost obvious fact is paraphrased by the statement that the object, i.e. the isoated physica system F. Scheck, Quantum Physics, DOI: 1.17/ _2, 123 Springer-Verag Berin Heideberg 213

2 124 2 Scattering of Partices by Potentias that one wishes to study, is separated from the observer and his measuring apparatus in the sense that back-reactions of the measuring procedure on the object are negigibe. The perturbation of the system by the measurement either is negigiby sma, or can be corrected for afterwards. In particuar, the system is obviousy not infuenced by the mere fact of being observed. There is a further aspect that we shoud reaize: The ength scaes and the time scaes of macroscopic processes are the typica scaes of our famiiar environment, or do not differ from them much, that is, do not go beyond the ream of what we can imagine based on our day-to-day experience. Exampes are the miiseconds that matter in sportive competitions, or the very precise ength measurements in micro mechanics. Matters change when the object of investigation is a microscopic system such as a moecue, an atom, an atomic nuceus, or a singe eementary partice: 1. Every measurement on a micro system is a more or ess intrusive intervention that can strongy modify the system or even destroy it. As an exampe, think of an atom described by a wave function ψ(t, x). If this atom is bombarded by a reativey coarse beam such as, e.g., an unpoarized ight beam, one intuitivey expects the subte phase correations that are responsibe for the interference phenomena of the wave function, to be partiay or competey destroyed. In fact, the impossibiity of separating measuring device and object beongs to the most difficut aspects of quantum theory. 2. Furthermore, the spatia and tempora scaes of typica quantum processes, in genera, are sma as compared to spatia distances or time intervas of experiments set up to detect them. An exampe may iustrate this: The size of a hydrogen atom is of the order of the Bohr radius (1.8), that is about 1 1 m. This is a very sma quantity as compared to the distance of the hydrogen target from the source emitting the incoming beam that is used to investigate the atom, as we as from the detector designed to detect the scattered beam. A simiar remark appies to the tempora conditions in atoms. Characteristic times of an atom are defined by the transition energies, τ(m n) = 2π c. c E m E n For the exampe of the (2p 1s)-transition in hydrogen this time is τ(2 1) s a time interva which is very short as compared to the time scaes in a typica experiment. The genera concusion from these considerations is that in genera we can ony observe asymptotic states, reaized ong before or ong after the process proper, and at arge spatia distances from it. More concretey this means the foowing: We wish to investigate a quantum system which, when isoated, is a stationary one, by means of a beam of partices. The system is the target, the partices are the projecties which are directed onto the target, or which are seen in a detector after the scattering has taken pace. The interaction process of the beam and the system happens within a time interva Δt around, say, t =. It is ocaized within a voume V of space around the origin x = that is characterized by the radius R. At time t

3 2.1 Macroscopic and Microscopic Scaes 125 the beam is constructed in a controed way at an asymptoticay arge distance from the target. The beam and the target in this configuration define what is caed the in-state (abbreviation for incoming state ). For t + the scattered partices, or more generay the reaction products of the scattering process, are identified in the detector which aso has an asymptoticay arge distance from the target. The scattered partices, together with the target in its fina state, are in what one cas the out-state (abbreviation for outgoing state ). Other situations where we actuay can perform measurements are provided by systems which are stationary by themseves but which are unstabe because of interactions with other systems. For exampe, the 2p-state of hydrogen is unstabe because it decays to the 1s-state by emission of a photon, the typica time for the transition being on the order of 1 9 s. In this exampe the in-state is the atom in its excited 2p-state, the out-state consists of the outgoing photon and the atom in the stabe ground state. In this exampe, too, our information on the (unstabe) quantum system stems from an asymptotic measurement, the decay products being detected a ong time after the decay process happened and very far in space. 1 As a genera concusion we note that experimenta information on quantum mechanica systems is obtained from asymptotic, incoming or outgoing states. We cannot penetrate the interaction region proper and cannot interfere with the typica time scae of the interaction. In quantum mechanics of moecues, atoms, and nucei, the important methods of investigation are: scattering of partices, i.e. eectrons, protons, neutrons, or α partices, on these systems; excitation and decay of their excited states by interaction with the eectromagnetic radiation fied. Scattering processes which can be deat with by means of the notions and methods deveoped in Chap. 1, are the subject of this chapter. The interaction with the radiation fied requires further and more comprehensive preparation and is postponed to ater chapters. Likewise, scattering theory in a physicay more genera, mathematicay more forma framework wi be taken up again ater. 2.2 Scattering on a Centra Potentia We assume that a potentia U(x) is given which describes the interaction of two partices and which is sphericay symmetric and has finite range. Somewhat more formay this means U(x) U(r) with r = x, im [ru(r)] =. (2.1) r 1 Of course, the unstabe state must have been created beforehand and one may ask why the preparation procedure is not taken to be part of the in-state, or under which condition this becomes a necessity. The answer is a quaitative one: The tota decay probabiity of the unstabe state, when mutipied by, yieds the energy uncertainty, or width Γ of the unstabe state. If Γ E α,i.eif the width Γ is sma as compared to the energy E α of the state then this state is quasi-stabe. The process used to prepare it can be separated from the decay process.

4 126 2 Scattering of Partices by Potentias Sphericay symmetric potentias such as U(r) = U Θ(r r) or U(r) = g e r/r, r the first of which vanishes outside of the fixed radius r, whie the second decreases exponentiay, fufi this condition. The Couomb potentia U C (r) = e 1e 2 r does not. The aim is to investigate the scattering of a partice of mass m on the potentia and to cacuate the corresponding differentia cross section. (Note that in the case of the two-body system m is the reduced mass.) The differentia cross section is an observabe, i.e. a cassica quantity. Its definition is the same as in cassica mechanics (cf. [Scheck (21)], Sect. 1.27): It is the ratio of the number dn of partices which are scattered, in the unit of time, into scattering anges between θ and θ + dθ, and of the number n of incoming partices per unit of area and unit of time. In other terms, one determines the number of partices that were actuay scattered, and normaizes to the incoming fux. In contrast to the cassica situation these numbers are obtained from the current density (1.54) (with A ) describing the fow of probabiity, via Born s interpretation, not from cassica trajectories (which no onger exist). The correct way of proceeding woud be to construct a state coming in aong the 3-direction as a wave packet at t = which custers around the average momentum p = pê 3, then to cacuate the time evoution of this wave packet by means of the Schrödinger equation, and eventuay anayze the outgoing fux at t +. As this is tedious and cumbersome, one resorts to an intuitive method which is simper and, yet, eads to the correct resuts. One considers the scattering process as a stationary situation. The incoming beam is taken to be a stationary pane wave, whie the scattered state is represented by an outgoing spherica wave. At asymptotic distances from the scattering center the wave fied then has the form r : ψ Somm (x) e ikx3 + f (θ) eikr r, k = 1 p, (2.2) whose first term is the incoming beam with momentum p = pê 3 = kê 3 whie the second is the outgoing spherica wave. This ansatz is caed Sommerfed s radiation condition The roe of the (in genera) compex ampitude f (θ) is carified by cacuating the current densities of the incoming and outgoing parts. It is usefu, here and beow, to denote the skew-symmetric derivative of (1.54) by a symbo of its own, f (x) g(x) := f (x) g(x) [ f (x)]g(x) (2.3) where f and g are compex functions which are at east C 1 (once continuousy differentiabe). For the incoming wave one finds j in = 2mi e ikx3 e ikx 3 = k m ê3 = vê 3, with vê 3 the veocity of the incoming partice.

5 2.2 Scattering on a Centra Potentia 127 Using spherica poar coordinates for which ( = r, 1 r θ, 1 r sin θ ), φ and the foowing expressions for the gradient of ψ = f (θ)e ikr /r ( ψ) r = ψ ( r = 1 r 2 + ik ) f (θ)e ikr, r ( ψ) θ = 1 ψ r θ = 1 f (θ) r 2 θ eikr, ( ψ) φ =, the outgoing current density is easiy cacuated, j out = k f (θ) 2 m r 2 ê r + 1 2mi r 3 f (θ) f (θ)ê θ. The first term, upon mutipication by the area eement r 2 dω of a sphere with radius r and center at the origin, yieds a probabiity current in the radia direction proportiona to f (θ) 2. The second term, in contrast, yieds a current decreasing ike 1/r which must be negected asymptoticay. The fux of partices across the cone with soid ange dω in an outgoing radia direction becomes (for arge vaues of r) j out ê r r 2 dω = k f (θ) 2 m r 2 r 2 dω (r ). Normaizing to the incoming fux, the differentia cross section is found to be dσ e = j out ê r r 2 dω = f (θ) 2 dω. j in This resut carifies the physica interpretation of the ampitude f (θ): This ampitude determines the differentia cross section dσ e dω = f (θ) 2. (2.4) The function f (θ) is caed scattering ampitude. It is a probabiity ampitude, in the spirit of Born s interpretation. The square of its moduus is the differentia cross section and, hence, is a cassica observabe. As we wi see soon it describes eastic scattering. The tota eastic cross section is given by the integra taken over the compete soid ange σ e = dω f (θ) 2 = 2π π sin θ dθ f (θ) 2. (2.5) Before continuing with a discussion of methods that aow to cacuate the scattering ampitude and the cross sections we add a few compements to and comments on these resuts.

6 128 2 Scattering of Partices by Potentias Remarks 1. The resut obtained for the asymptotic form of j out shows that it was justified to ca the two terms in (1.135), Sect , outgoing and incoming spherica waves, respectivey. 2. In the two-body probem with centra force, the variabe r is the moduus of the reative coordinate, m is the reduced mass, m m 1m 2, and θ θ m 1 + m 2 is the scattering ange in the center-of-mass system. Thus, the ampitude f (θ) is the scattering ampitude in the center-of-mass system. 3. The potentia U(r) must be rea if the Hamitonian is to be sef-adjoint. If this is so, then there is ony eastic scattering. No matter how it is scattered, the partice must be found somewhere in the fina state, or, in the spirit of quantum mechanics, the probabiity to find the partice somewhere in space must be conserved. Therefore, the expression (2.4) describes the differentia cross section for eastic scattering, the expression (2.5) gives the integrated eastic cross section. However, there are processes where the fina state is not the same as the initia state. An eectron which is scattered on an atom, may oose energy and may eave behind the atom in an excited state. A photon used as a projectie may be scattered ineasticay on the atom, or may even be absorbed competey. In those cases one says that the fina state beongs to another channe than the initia state. Besides the rea potentia responsibe for eastic scattering, the Hamitonian must aso contain interaction terms which aow to cross from the initia channe to other, ineastic channes. Loosey speaking, in such a situation the tota probabiity is distributed, after the scattering, over the various fina state channes. In Sect. 2.6 we wi deveop a buk method for describing such a situation without knowing the detais of the reaction dynamics. 4. Even though the ansatz (2.2) is intuitivey compeing, in a strict sense the ensuing cacuation is not correct. The condition (2.2) assumes a stationary wave function where both the incoming pane wave and the outgoing spherica wave are present at a times. In particuar, when one cacuates the current density (1.54) there shoud be terms arising from the interference between the incoming and outgoing parts. Instead, in our cacuation of the current densities we proceeded as if at t = there was ony the pane wave, and at t =+ there wi be ony the scattered spherica wave. Athough this derivation rests on intuition and, stricty speaking, is not correct, it yieds the right resut. This is so because the pane wave is but an ideaization and shoud be repaced by a suitaby composed wave packet. We skip this painstaking cacuation at this point and just report the essentia resut: One foows the evoution of a wave packet which at t described a ocaized partice with average momentum p = pê 3. For arge positive times and at asymptotic distances there appears an outgoing spherica wave with the shape assumed above. There are indeed interference terms between the initia state and the fina state, but, as

7 2.2 Scattering on a Centra Potentia 129 these terms osciate very rapidy, integration over the spectrum of momenta renders them negigiby sma. Except for the forward direction, that is for scattering where p = p, the ansatz (2.2), together with the interpretation given above, is correct. 5. It is instructive to compare the quantum mechanica description with the theory of eastic scattering in cassica mechanics. The definition of the differentia cross section, of course, is the same (number of partices per unit of time scattered into the soid ange dω, normaized to the incoming fux). The physica processes behind it are not the same. The cassica partice that comes in with momentum p = pê 3 and impact parameter b, moves on a we-defined trajectory. It suffices to foow this orbit from t = to t =+ to find out, with certainty, where the partice has gone. In quantum mechanics we associate a wave packet to the partice which, for exampe, contains ony momenta in the 3-direction and which, at t =, is centered at a vaue p = pê 3. Vaues for its 3-coordinate can be imited within what the uncertainty reation aows for. As the partice has no momentum components in the 1- and the 2-directions, or, in other terms, since p 1 and p 2 have the sharp vaues, the position of the partice in the pane perpendicuar to the 3-axis is competey undetermined. At t + quantum mechanics yieds a probabiity for detecting the partice in a detector which is positioned at a scattering ange θ with respect to the incoming beam. It is impossibe to predict where any individua partice wi be scattered. The probabiity dσ e /dω which is defined by the compex scattering ampitude f (θ) wi be confirmed ony if one aows very many partices to scatter under identica experimenta conditions. 2.3 Partia Wave Anaysis Ceary, the scattering ampitude must be a function of the energy E = 2 k 2 /(2m) of the incoming beam, or, what amounts to the same, a function of the wave number k. Whenever this dependence matters we shoud write, more precisey, f (k,θ)instead of f (θ). The cross section has the physica dimension [area]. Hence, the scattering ampitude has dimension [ength]. In the physicay aowed region θ [,π] or z cos θ [ 1, +1], and for fixed k, the ampitude f (k,θ) is a nonsinguar, in genera square integrabe function of θ. 2 Therefore, it may be expanded in terms of spherica harmonics Y m (θ, φ). However, as it depends on θ, by the spherica symmetry of the potentia, and does not depend on φ, this expansion contains ony spherica 2 Assuming the restriction (2.1) the ampitude f is square integrabe, indeed. In the case of the Couomb potentia the ampitude is singuar in the forward direction θ =, its behaviour being ike 1/ sin θ, and, hence, is no onger square integrabe. Nevertheess, the scattering ampitude can sti be expanded in terms of Legendre poynomias. However, the series (2.6) is no onger convergent in the forward direction, and the expression (2.7) for the integrated cross section diverges.

8 13 2 Scattering of Partices by Potentias harmonics with m =, Y, which are proportiona to Legendre poynomias, Y = 4π P (z = cos θ). As a consequence one can aways choose an expansion in terms of Legendre poynomias, f (k,θ)= 1 (2 + 1)a (k)p (z). (2.6) k = The factor 1/k is introduced in order to keep track of the physica dimension of the scattering ampitude, the factor (2 + 1) is a matter of convention and wi prove to be usefu. The compex quantities a (k) which are defined by (2.6) are caed partia wave ampitudes. They are fuctions of the energy ony (or, equivaenty, of the wave number). The spherica harmonics are orthogona and are normaized to 1. Therefore, the integrated cross section (2.5) is σ e (k) = 4π (2 + 1)(2 k 2 + 1) a (k)a (k ) dω Y Y, which, by the orthogonaity of spherica harmonics, simpifies to σ e (k) = 4π k 2 (2 + 1) a (k) 2. (2.7) = Like the formuae (2.4) and (2.5) these expressions are competey genera, and make no use of the underying dynamics, i.e. in the case being studied here, of the Schrödinger equation with a centra potentia U(r). We now show that the ampitudes a (k) are obtained by soving the radia equation (1.127) for a partia waves. The arguments presented in Sect and the comparison to the anaogous cassica situation show that this is not ony an exact method for studying eastic scattering but that it is aso particuary usefu from a physica point of view. By assumption the potentia has finite range, i.e. it fufis the condition (2.1). In the corresponding cassica situation a partice with a arge vaue of anguar momentum c stays further away from the origin r = than a partice with a smaer vaue of c, and the action of the potentia on it is correspondingy weaker. Very much ike in cassica mechanics the quantum effective potentia U eff (r) = 2 ( + 1) 2mr 2 + U(r), for arge vaues of, is dominated by the centrifuga term. Thus, one expects the ampitudes a to decrease rapidy with increasing, so that the series (2.6) and (2.7) converge rapidy. If the -th partia wave is taken to be R (r)y m = u (r) Y m r

9 2.3 Partia Wave Anaysis 131 the radia function u (r) obeys the differentia equation ( ) 2m u (r) 2 U eff(r) k 2 u (r) =, (2.8) (see aso Sect ). At the origin, r =, the radia function must be reguar. This means that we must choose the soution which behaves ike R r,oru r +1, respectivey, in the neighbourhood of the origin. For r the effective potentia becomes negigibe as compared to k 2 so that (2.8) simpifies to the approximate form r : u (r) + k2 u (r). Therefore, the asymptotic behaviour of the partia wave must be given by ( r : u (r) sin kr π ) 2 + δ (k). (2.9) The phase δ (k) which is defined by this equation is caed the scattering phase in the partia wave with orbita anguar momentum. The foowing argument shows that the asymptotic form (2.9) is very natura: On the one hand, the function u (r) is rea (or can be chosen so) if the potentia U(r) is rea. Under the same assumption the scattering phase must be rea. On the other hand, the asymptotics of the force-free soution is known from the asymptotics (1.133)of spherica Besse functions (which are reguar at r = ), u () (r) = (kr) j (kr) sin ( kr π 2 If the potentia U(r) is identicay zero a scattering phases are equa to zero. Therefore, the scattering phases measure to which extent the asymptotic, osciatory behaviour of the radia function u (r) is shifted reative to the force-free soution (r). There remains the probem of expressing the scattering ampitude, or, equivaenty, the ampitudes a (k) in terms of the scattering phases. In soving this probem, the idea is to write the unknown scattering soution ψ(x) of the Schrödinger equation in terms of a series in partia waves, u () ψ(x) = c R (r)y (θ) = 1 r = ). c u (r)y (θ) (2.1) and to choose this expansion such that the incoming spherica wave α in e ikr /r which is contained in it for r, coincides with the incoming spherica wave contained in the condition (2.2). We begin with the atter: Expanding the pane wave in terms of spherica harmonics (cf. (1.136)), e ikx3 = i 4π(2 + 1) j (kr)y, = =

10 132 2 Scattering of Partices by Potentias and making use of the asymptotics (1.133) of the spherica Besse functions, j (kr) 1 (kr kr sin π ) = 1 2 2ikr (eikr e iπ/2 e ikr e iπ/2 ), the piece proportiona to e ikr /r can be read off. The outgoing spherica wave, on the other hand, in (2.2), in addition to the term proportiona to the scattering ampitude f (θ), aso contains a piece of the pane wave that can be read off from the same expression. This is to say that the Sommerfed condition (2.2) is rewritten in terms of spherica waves as foows: ψ Somm (x) e ikr 2ikr ( ) i 4π(2 + 1) e iπ/2 Y = ( + eikr i 4π(2 + 1) e iπ/2 Y + 2ikf(θ). 2ikr = This is to be compared to the representation (2.1) for the scattering soution for r which becomes, upon inserting (2.9) once more, ( ) ( ψ(x) e ikr ) c e iδ e iπ/2 Y + eikr c e iδ e iπ/2 Y. 2ir 2ir = = The incoming spherica waves in ψ Somm and in ψ are equa provided the coefficients c are chosen to be c = i k 4π(2 + 1) e iδ. For the rest of the cacuation one just has to compare these formuae. Inserting the resut for c into the outgoing part of (2.1), one finds f (θ) = 1 e 2iδ 1 4π(2 + 1)Y k 2i = = 1 e 2iδ 1 (2 + 1)P (cos θ), k 2i = where use is made of the reation Y (θ) = 4π P (cos θ). Comparison with the genera expansion (2.6) yieds the foowing exact expression for the ampitudes a (k) as functions of the scattering phases, a (k) = e2iδ 1 2i = e iδ (k) sin δ (k). (2.11) )

11 2.3 Partia Wave Anaysis 133 Appications and Remarks 1. It was indeed usefu to define the ampitudes a by extracting an expicit factor (2+1) in (2.6). The so-defined ampitudes then have modui which are smaer than or equa to 1. The second equation in (2.11) is correct because the phase δ is rea The resut (2.11) for the partia wave ampitudes which foows from the Schrödinger equation, has a remarkabe property: The imaginary part of a is positive-semidefinite Im a (k) = sin 2 δ. Cacuating the eastic scattering ampitude (2.6) in the forward direction θ =, where P (z = 1) = 1, its imaginary part is seen to be Im f () = 1 (2 + 1) Im a (k) = 1 (2 + 1)sin 2 δ (k). k k = = In turn, the integrated eastic cross section (2.7) is found to be σ e (k) = 4π k 2 (2 + 1)sin 2 δ (k). = In the case of a rea potentia there is eastic scattering ony, the integrated cross section (2.7) is identica with the tota cross section. Comparison of the resuts just obtained yieds an important reation between the imaginary part of the eastic scattering ampitude in the forward direction and the tota cross section, viz. σ tot = 4π k Im f (). (2.12) This reationship is caed the optica theorem. 4 Loosey speaking it is a consequence of the conservation of (Born) probabiity. 3. As we wi see in Sect. 2.6 and in Chap. 8, the optica theorem aso hods in more genera situations. When two partices, A and B, are scattered on one another, besides eastic scattering A + B A + B, there wi in genera be ineastic processes as we, in which one of them, or both, are eft in excited states, A + B A + B, A + B A + B, or where further partices are created, A + B A + B + C +. As a short-hand aowed fina states are denoted by n. The optica theorem then reates the imaginary part of the 3 This remark is important because the same anaysis can be appied to the case of a compex, absorptive potentia. In this case the scattering phases are compex functions. The first part of the formua (2.11) sti appies, the second part does not. 4 Indeed, this term was coined in cassica optics. It was known, in a different context, before quantum mechanics was deveoped.

12 134 2 Scattering of Partices by Potentias eastic forward scattering ampitude at a given vaue of the energy, to the tota cross section at this energy, σ tot = n σ(a + B n). The more genera form of the optica theorem (2.12) is σ tot (k) = 4π k Im f e(k,θ = ), the quantity k being the moduus of the momentum k in the center-of-mass system How to Cacuate Scattering Phases The asymptotic condition (2.9) can be interpreted in sti a different way that, by the same token, gives a hint at possibiities of obtaining the scattering phases. As the potentia has finite range the interva of definition of the radia variabe r spits into an inner domain where the (true) potentia U(r) is different from zero, and an outer domain where either it vanishes or becomes negigiby sma, and where ony the centrifuga potentia is active. Thus, in the outer domain every soution u (r) is a inear combination of two fundamenta soutions of the force-free case. For instance, these may be a spherica Besse function j (kr) and a spherica Neumann function n (kr), so that u (k, r) = (kr) [ j (kr)α (k) + n (kr)β (k) ], (for r such that U(r) ). One now takes this formua to arge vaues of r, and compares the asymptotic behaviour (2.9) with the asymptotics of the spherica Besse and Neumann functions (1.133) and (1.14), respectivey. Using the addition formua for Sines ( ) sin kr π 2 + δ (k) ( = sin kr π ) cos δ (k) + cos 2 then yieds an equation for the scattering phase which is ( kr π ) sin δ (k) 2 tan δ (k) = β. (2.13) α This shows that the differentia equation for the radia function must be soved ony in the inner domain of the variabe r: One determines the soution reguar at r =, e.g. by numerica integration on a computer, and foows this soution to the boundary of the outer domain. At this point one writes it as a inear combination of j and n, and reads off the coefficients α and β whose ratio (2.13) yieds the scattering phase in the interva [,π/2].

13 2.3 Partia Wave Anaysis 135 Here are some exampes of potentias for which the reader may wish to determine the scattering phases: 1. The sphericay symmetric potentia we U(r) = U Θ(r r) ; (2.14) 2. The eectrostatic potentia U(r) = 4πQ 1 r dr ϱ(r )r 2 + dr ϱ(r )r, (2.15) r which is obtained from the charge distribution 1 ϱ(r) = N (2.16) 1 + exp[(r c)/z] with normaization factor [ N = 3 ( πz ) 2 ( z ) ] 1 3 ( ) n 4πc c c n 3 e nc/z. n=1 The distribution (2.16) is often used in the description of nucear charge densities. The parameter c characterizes the radia extension whie the parameter z characterizes the surface region. Its specific functiona form is aso known from statistica mechanics which is the reason why it is usuay caed Fermi distribution. Figure 2.1 shows an exampe appicabe to reaistic nucei where z is very sma as compared to c. r,16,14,12,1,8,6,4, r Fig. 2.1 Iustration of the mode (2.16) for a normaized charge distribution. In the case shown here, c z. The exampe shows the distribution with c = 5fm,z =.5 fm. The parameter c is the distance from the origin to the radius where the function ϱ(r) has dropped to haf its vaue at r =. The two points where it assumes 9 and 1 % of its vaue in r =, respectivey, are situated at r 9 = c 2z n 3 and at r 1 = c + 2z n 3, respectivey. They are separated by the approximate distance t 4z n z

14 136 2 Scattering of Partices by Potentias 3. The Yukawa potentia, that we mentioned in the introduction to Sect. 2.2, U Y (r) = g e r/r. (2.17) r We wi show ater that this potentia describes the interaction of two partices which can exchange a scaar partice of mass M = /(r c) (c denotes the speed of ight). The name is due to H. Yukawa who had postuated that the strong interactions of nuceons were due to the exchange of partices with spin, the π-mesons, ong before these partices were actuay discovered. The ength r = c/(mc 2 ) is interpreted as the range of the potentia. It is the Compton wave ength of partices of mass M. These remarks emphasize that the exampe (2.17) may have a deeper significance for physics than the pure mode potentias (2.14), (2.15) and (2.16). Consider now two potentias of finite range, U (1) and U (2), and compare their scattering phases. For equa vaues of the energy the corresponding radia functions obey the radia differentia equations [ ] u ( j) 2m (r) 2 U ( j) eff (r) k2 u ( j) (r) =, j = 1, 2, where the effective potentias differ ony by the true, dynamica potentias U (2) (1) eff (r) U eff (r) = U (2) (r) U (1) (r). Both radia functions are assumed to be reguar at r =. As we are factoring 1/r, this means that u ( j) () = ( j = 1, 2). Compute then the foowing derivative, making use of the differentia equations for u (1) and u (2), d ( (1) u dr u(2) u (2) ) u(1) = u (1) u(2) u (2) u(1) = 2m ( U (2) 2 U (1)) u (1) Taking the integra over the interva [, r], one has u (1) (r)u(2) r = 2m 2 (r) u (2) (r)u(1) (r) u(2). ( ) dr U (2) (r ) U (1) (r ) u (1) (r )u (2) (r ). In the imit of r going to infinity, and using the asymptotic form (2.9) as we as its derivative on the eft-hand side of this equation, one obtains an integra representation for the difference of scattering phases: k sin(δ (1) δ (2) ) = 2m 2 dr ( U (2) (r) U (1) (r) ) u (1) (r)u(2) (r). (2.18)

15 2.3 Partia Wave Anaysis 137 This formua is usefu for testing the sensitivity of high, intermediate, and ow partia waves to the potentia, by etting U (1) and U (2) differ but itte. Instead of the scattering phases themseves one cacuates the change in any given partia wave as a function of the change in the potentia. Aternativey one may consider a situation where U (2) vanishes identicay, i.e. where the corresponding radia functions are proportiona to spherica Besse functions, u (2) (r) = (kr) j (kr). With δ (1) δ and δ (2) = the integra representation reduces to sin(δ ) = 2m 2 rdr U(r)u (r) j (kr). (2.19) The Yukawa potentia (2.17) may serve as an iustration of this formua. For the sake of simpicity we assume this potentia to be weak enough so that the corresponding radia function can be approximated by the force-free soution, 5 u (Y ) (r) (kr) j (kr). With this assumption one has sin δ (Y ) 2mk 2 = 2mkg 2 = mgπ 2 k = mg 2 k Q r 2 dru Y (r) j 2 (kr) rdr e r/r j 2 (kr) dϱ e ϱ/(kr ) J 2 +1/2 (ϱ) ( 1 + ) 1 2(kr ) 2. In this derivation the variabe ϱ = kr is introduced, and the spherica Besse function is written in the standard form of Besse functions found in monographs on specia functions, viz. ( ) π 1/2 j (ϱ) = J +1/2(ϱ). 2ϱ 5 This approximation is nothing but the first Born approximation that is studied in more detai in Sect

16 138 2 Scattering of Partices by Potentias The ast step makes use of a known definite integra which is found, e.g. in [Gradshteyn and Ryzhik (1965)], Eq Here, Q is a Legendre function of second kind whose properties are we known. 6 These functions are known to decrease rapidy both for increasing and for increasing vaues of the argument 1, cf., e.g., [Abramowitz and Stegun (1965)], Fig Potentias with Infinite Range: Couomb Potentia The Couomb potentia vioates the condition (2.1). This means that its infuence is sti fet when the partice moves at very arge distances from the scattering center. This is seen very ceary in the asymptotics of the partia waves, derived in Sect , which in addition to the constant phase σ, contain an r-dependent, ogarithmic phase γ n(2kr) mutipied by the factor (1.156), γ = ZZ e 2 m 2. (2.2) k The scattering soutions of the Couomb potentia certainy do not obey the radiation condition (2.2). Both the outgoing spherica wave and the pane wave are modified due to the ong range of the potentia, and the formuae of partia wave anaysis derived above, cannot be appied directy. A simiar statement hods for any sphericay symmetric potentia which, though different from the (1/r)-behaviour in the inner domain, approaches the Couomb potentia for arge vaues of r. An exampe is provided by the eectrostatic potentia corresponding to the charge distribution (2.16) in Exampe 2 of Sect In order to sove this new probem we proceed in two steps: In the first step we show that the condition (2.2) is modified to r : ψ e i{kx3 +γ n[2krsin 2 (θ/2)]} ei[kr γ n(2kr)] + f C (θ), (2.21) r with r-dependent, ogarithmic phases both in the incoming wave and in the outgoing spherica wave, and cacuate the scattering ampitude f C (θ) for the pure Couomb potentia. In the second step we study sphericay symmetric potentias which deviate from the (1/r)-form in the inner region but decrease ike 1/r in the outer region, in other words, which approach the Couomb potentia for r. In this case it is sufficient to cacuate the shift of the scattering phases reative to their vaues in the pure Couomb potentia, and not reative to the force-free case. Step 1: Athough we aready know the scattering phases for the Couomb potentia from Sect it is instructive to cacuate the scattering ampitude directy, using 6 The function Q (z) and the Legendre poynomia P (z) form a fundamenta system of soutions of the differentia equation (1.112)withλ = (+1) and m =. In contrast to P (z) the function Q (z) is singuar in z = 1, the singuarity being a branch point. For a vaues z > 1 of the argument Q (z) is a one-vaued function.

17 2.3 Partia Wave Anaysis 139 a somewhat different method. Indeed, the (nonreativistic) Schrödinger equation can be soved exacty in a way adapted to the specific scattering situation at hand. 7 With k 2 = 2mE/ 2, U(r) = ZZ e 2 /r, and with the definition (2.2) for γ, the stationary Schrödinger equation (1.16) reads ( + k 2 2γ k ) ψ(x) =. (2.22) r This is soved using paraboic coordinates r x 3, η = ξ = and by means of the ansatz r + x 3, ψ(x) = c ψ e ikx3 f (r x 3 ) = c ψ e ik(η2 ξ 2 )/2 f (ξ 2 ), in which c ψ is a compex number sti to be determined. As before, the direction of the incoming, asymptotic momentum is taken to be the 3-direction. Since no other, perpendicuar direction is singed out in the in-state and since the potentia is sphericay symmetric, the scattering ampitude does not depend on φ. Surprisingy, the differentia equation (2.22) separates in these coordinates, too. The variabe ξ 2 = r x 3 is denoted by u, first and second derivatives with respect to that variabe are written f and f, respectivey. Then, for i = 1, 2: ψ x i = e ikx3 f (u) r x i = e ikx3 f (u) xi r, 2 ψ (x i ) 2 [ = e ikx3 f (xi ) 2 ( 1 r 2 + f r (xi ) 2 )] r 3. The derivatives with respect to x 3 give [ ( ψ x x 3 = eikx3 ikf + f 3 )] (u) r 1 2 ψ, (x 3 ) 2 [ ( x = e ikx3 k 2 f + 2ikf 3 ) ( x r 1 + f 3 ) 2 ( 1 r 1 + f r (x3 ) 2 ) ] r 3. Inserting these formuae in (2.22) one obtains ) (u d2 d + (1 iku) du2 du γ k f(u) =. This differentia equation is again of Fuchsian type and appears to be very cose to Kummer s equation (1.145). The identification becomes perfect if one repaces the variabe u by the variabe v := iku. Indeed, the differentia equation then becomes v d2 f(v) dv 2 + (1 v) d f(v) dv φ + iγ f(v) =. 7 This no onger hods true when the reativistic form of the wave equation is used.

18 14 2 Scattering of Partices by Potentias The soution which is reguar at r = is f (v) = c ψ 1 F 1 ( iγ ; 1 ; v) = c ψ 1 F 1 [ iγ ; 1 ; ik(r x 3 )]. The asymptotics of the confuent hypergeometric function is obtained from the formua (1.147), 1 1F 1 Γ(1 + iγ) eπγ [ik(r x 3 )] iγ 1 + Γ( iγ) eik(r x3) [ik(r x 3 )] iγ +1. Setting i iγ = e πγ/2 and choosing the coefficient c ψ as foows c ψ = Γ(1 + iγ)e πγ/2, the soution ψ takes the asymptotic form postuated above ψ e i{kx3 +γ n[k(r x 3 )]} Γ(1 + iγ) γ Γ(1 iγ) k(r x 3 ) ei{kr γ n[k(r x3 )]}. Inserting r x 3 = r(1 cos θ) = 2rsin 2 (θ/2) shows that this is the asymptotic decomposition, (2.21), into an incoming, but deformed pane wave, and an outgoing, deformed spherica wave. The compex Γ-function is written in terms of moduus and phase, Γ(1 ± iγ)= Γ(1 + iγ) e ±iσ C, so that the scattering ampitude for the pure Couomb potentia is seen to be γ f C (θ) = 2ksin 2 (θ/2) ei{2σ C γ n[sin 2 (θ/2)]}. (2.23) This ampitude contains a phase factor that depends on the scattering ange, and which is characteristic for the ong range of the Couomb potentia. This phase factor drops out of the differentia cross section (2.4) for which one obtains dσ e dω = γ 2 ( 1 ZZ 4k 2 sin 4 (θ/2) = e 2 ) 2 1 4E sin 4 (θ/2), (2.24) where the definition (2.2) and E = 2 k 2 /(2m) were used. The resut (2.24) is caed the Rutherford cross section. Note that it agrees with the corresponding expression of cassica mechanics (cf. [Scheck (21)], Sect. 1.27). This important formua was essentia in anayzing the scattering experiments of α partices on nucei, performed by Rutherford, Geiger, and Marsden from 196 on. These experiments proved that nucei are practicay point-ike as compared to typica radii of atoms. Step 2: Consider now a sphericay symmetric charge distribution which is no onger concentrated in a point but is ocaized in the sense that it ies inside a sphere with a given, finite radius R. One cacuates the eectrostatic potentia by means of the formua (2.15) and notes that for vaues r > R it coincides, either competey or to a very good approximation, with the pure Couomb potentia but deviates from it for vaues r < R. It shoud be cear immediatey, in the ight of the genera discussion of

19 2.3 Partia Wave Anaysis 141 Sect. 2.3, that high partia waves are insensitive to these deviations. The information on the precise shape of the charge distribution is contained in the ow and intermediate partia waves. This suggests to design the partia wave anaysis for these cases such that it is not the force-free case but the Couomb potentia which is taken as the reference potentia. This means that the phase shift anaysis shoud be designed such as to yied the difference δ = δ U(r) δ C between the true phase and the Couomb phase. 2.4 Born Series and Born Approximation We emphasize again that the expansion in terms of partia waves is an exact method to cacuate the cross section for sphericay symmetric potentias which, in addition, has the advantage of making optima use of the information about the range of the potentia. In the case of potentias which are not sphericay symmetric but may be expanded in terms of spherica harmonics the cross section can aso be computed by expanding the scattering ampitude in partia waves. However, this method becomes technicay compex and cumbersome, and ooses much of the simpicity and transparency it has for sphericay symmetric potentias. The Born series that we describe in this section, does not have this disadvantage. It yieds an exact, though forma, soution of the scattering probem by means of the technique of Green functions, and can equay we be appied to potentias with or without spherica symmetry. Its most stringent disadvantage is the fact that beyond first order it is not very practicabe and becomes cumbersome. The first iteration, or first Born approximation, in turn, is easy to cacuate and aows for simpe and convincing physica interpretation, but it vioates the optica theorem. The starting point is again the stationary Schrödinger equation (1.16) in the form ( + k 2 )ψ(x) = 2m U(x)ψ(x), (2.25) 2 where k 2 = 2mE/ 2. If one deas with a two-body probem the parameter m is the reduced mass; if one studies scattering of a singe partice on a fixed externa potentia then m is just the mass of that partice. 8 The differentia equation (2.25)is soved by means of Green functions, i.e. of functions (more precisey: distributions) G(x, x ), which obey the differentia equation ( + k 2 )G(x, x ) = δ(x x ). 8 The atter case can aso be viewed as the imit of the former in which the mass of the heavier partner is very arge as compared to the one of the ighter partner.

20 142 2 Scattering of Partices by Potentias The we-known reation ( + k 2 ) e±ik z = 4πδ(z) z shows that the genera soution can be given in the form G(x, x ) = 1 1 [ ae ik x x 4π x x + (1 a)e ik x x ]. Formay, the differentia equation (2.25) then has the soution ψ k (x) = e ik x + 2m d 3 x G(x, x )U(x )ψ k (x ). (2.26) 2 It is forma because the differentia equation (2.25) is repaced by an integra equation which contains the unknown wave functions both on the eft-hand side and in the integrand of the right-hand side. Nevertheess, it has two essentia advantages: The constant a in the Green function can be chosen such that the scattering soution fufis the right boundary condition, which in our case is the Sommerfed radiation condition (2.2). Furthermore, if the strength of the potentia is sma in some sense, this integra equation can be used as the starting basis for an iterative soution, i.e. an expansion of the scattering function around the force-free soution (the pane wave). The correct asymptotics (2.2) is reached with the choice a = 1. This is seen as foows: Define r := x, r := x, and assume the potentia U(x ) to be ocaized. As r goes to infinity one has r r : x x = r 2 + r 2 2x x r 1 r x x. The scattering function takes the asymptotic form r : ψ k (x) e ik x 2m e ikr 4π 2 d 3 x e ik x U(x )ψ k (x ). r The reader wi have noticed that we defined kx/r =: k in this expression. Indeed, the momentum of the scattered partice is k. It moves in the direction of x/r and, because the scattering is eastic, one has k = k. By the same token, this resut yieds a genera formua for the scattering ampitude, viz. f(θ, φ) = 2m 4π 2 d 3 x e ik x U(x)ψ k (x). (2.27) (As this equation no onger contains the point of reference x, the integration variabe x was renamed x.) This equation is an interesting resut. If the potentia has a stricty finite range we need to know the exact scattering function ony in the domain where U(x) is sizeaby different from zero. 9 9 An approximation which makes use of this fact and which is particuary usefu for scattering at high energies, is provided by the eikona expansion. The reader wi find an extended description of this method in, e.g., [Scheck (212)], Chap. 5, and iustrated by expicit exampes.

21 2.4 Born Series and Born Approximation 143 If one knew the exact scattering soution this formua woud yied the exact scattering ampitude. Athough this ambitious goa cannot be reached, the formua serves as a basis for approximation methods which are reevant for various kinematic conditions. One of these is the Born series which is obtained from an iterating soution of the integra equation (2.26). The idea is simpe: one imagines the potentia as a perturbation of the force-free soution ψ () k = e ik x such that in a decomposition of the fu wave function ψ k (x) = ψ (n) k (x) n= the n-th term is obtained from the (n 1)-st by means of the integra equation (2.26), i.e. ψ (n) 1 2m k (x) = 4π 2 d 3 x eik x x x x U(x )ψ (n 1) k (x ), n 1 (2.28) Even without touching the (difficut) question of its convergence, one reaizes at once that this provides a method of representing the scattering ampitude as a series whose structure is very different from the expansion in terms of partia waves. Whie in the atter one expands in increasing vaues of, the former is an expansion in the strength of the potentia First Born Approximation It is primariy the first and simpest approximation which matters for practica appications. It consists in truncating the series (2.28) atn = 1. In this case the fu scattering function in the integrand of the right-hand side of (2.27) is repaced by = exp(ik x) so that one obtains f (1) (θ, φ) = 2m 4π 2 d 3 x e ik x U(x)e ik x. Introducing the momentum transfer q := k k with k = k = k, ˆq = (θ, φ), the first Born approximation for the scattering ampitude reads f (1) (q) = 2m 4π 2 d 3 x e iq x U(x). (2.29) ψ () k This formua tes us that in first Born approximation the scattering ampitude is the Fourier transform of the potentia with respect to the variabe q. The formua (2.29) simpifies further if the potentia has spherica symmetry, U(x) U(r). One inserts the expansion (1.136)ofexp(iq x) in terms of spherica

22 144 2 Scattering of Partices by Potentias harmonics and notes that by integrating over dω x ony the term with = survives. This foows from the fact that Y = 1/ 4π is a constant and that dω x Y m ( ˆx) = 4πδ δ m. Thus, one obtains the expression f (1) (θ) = 2m 2 r 2 dru(r) j (qr) (2.3) where q q =2k sin(θ/2), and where j (u) = sin u u is the spherica Besse function with = (see Sect ). The functiona dependence of the scattering ampitude coud be written as f (q), or, even more precisey, f (q 2 ) because the resut (2.3) shows that it depends ony on the moduus of q and is invariant under the exchange q q. As an aternative, one may express q by the scattering ange θ and write the scattering ampitude in the form f (1) m (θ) = 2 k sin(θ/2) We iustrate this resut by the foowing exampe. rdr U(r) sin[2kr sin(θ/2)]. Exampe 2.1 Let us return to the Yukawa potentia (2.17), U Y (r) = g e μr, where μ = 1. r r The foowing integra is obtained in an eementary way dre μr α sin(αr) = μ 2 + α 2. With α = 2k sin(θ/2) the formua (2.3) yieds f (1) 1 Y (θ) = 2mg 2 4k 2 sin 2 (θ/2) + μ 2. (2.31) Two properties of the resut (2.31) shoud be noticed 1. In the imit μ (though not aowed), and with g = ZZ e 2 and 2 k 2 = 2mE, the ampitude becomes f (1) C (θ) e 2 1 = ZZ 4E sin 2 (θ/2). This is the scattering ampitude for the pure Couomb potentia, except for the phase factor in (2.23). Its absoute square gives the correct expression (2.24) of the differentia cross section.

23 2.4 Born Series and Born Approximation There is a we-known expansion of 1/(z t) in terms of Legendre poynomias and Legendre functions of the second kind, (see [Gradshteyn and Ryzhik (1965)], Eq ) 1 z t = (2 + 1)Q (z)p (t). = Write the ampitude (2.31) as f (1) mg 1 Y (θ) = 2 k 2 2sin 2 (θ/2) + μ 2 /(2k 2 ) = mg 1 2 k μ 2 /(2k 2 ) cos θ, set 1 + μ 2 /(2k 2 ) = z and cos θ = t. This yieds f (1) mg ( ) Y (θ) = 2 k 2 Q 1 + μ2 2k 2 P (cos θ). = One now compares this with the genera expansion in terms of partia waves (2.6) and notices that the coefficients of this series are a = e iδ sin δ sin δ = mg ) 2 k Q (1 + μ2 2k 2 and that they agree with the exampe (2.19) in Sect Note that here and in that exampe, the scattering phases δ are sma. Remark The resuts (2.27) and (2.3) show that the scattering ampitude in first Born approximation is rea, i.e. Im f (1) =. This is in contradiction with the optica theorem. The first Born approximation does not respect the conservation of probabiity Form Factors in Eastic Scattering The first Born approximation eads in a natura way to a new notion, caed form factor, which is important for the anaysis of scattering experiments. This section gives its definition and iustrates it by a few exampes. The question and the idea are the foowing: Suppose a ocaized distribution ϱ(x) of eementary scattering centers is given whose interaction with the projectie is known. If we know the scattering ampitude for the eementary process, i.e. for the scattering of the projectie off a singe, isoated eementary scattering center, can we cacuate the scattering ampitude off the distribution ϱ(x)? The answer to this question is simpe if the Born approximation is sufficienty accurate for the cacuation of the scattering ampitudes. In this case, the ampitude

24 146 2 Scattering of Partices by Potentias for the distribution is equa to the product of the eementary scattering ampitude and a function which depends ony on the density ϱ(x) and on the momentum transfer q. We show this by means of an exampe: Suppose the projectie is scattered by a number A of partices whose distribution in space is described by the density ϱ(x) = Aϱ(x). This is to say that d 3 x ϱ(x) = A and d 3 x ϱ(x) = 1. It is customary to normaize the density ϱ(x) to 1, i.e. to take out an expicit factor A. Let the eementary interaction be described by the Yukawa potentia (2.17). The potentia created by a partices, with their density ϱ(x), then is, using μ = 1/r U(x) = ga d 3 x e μ x x x x ϱ(x ). (2.32) It obeys the differentia equation ( x μ 2 )U(x) = 4πAϱ(x). (2.33) In first Born approximation the scattering ampitude F (1) describing scattering on the distribution ϱ(x) is given by the formua (2.29), upon insertion of the potentia (2.32). The exponentia function is repaced by the identity e iq x = 1 q 2 + μ 2 ( x μ 2 )e iq x. The differentia operator ( x μ 2 ) is shifted to U(x), by partiay integrating twice, and the differentia equation (2.33) repaces the potentia by the density. One obtains the expression F (1) (q) = Af (1) Y (θ) F(q) (2.34) for the scattering ampitude. The first factor is the eementary ampitude (2.31), the second factor is defined by F(q) = d 3 xe iq x ϱ(x). (2.35) This factor which depends on the density ϱ(x) and on the momentum transfer ony, is caed form factor. Its physica interpretation is carified by its properties the most important of which are summarized here: Properties of the Form Factor: 1. If it were possibe to measure the form factor for a vaues of the momentum transfer then the density woud be obtained by inverse Fourier transform, ϱ(x) = 1 (2π) 3 d 3 qe iq x F(q) The density ϱ(x) describes a composite target such as, e.g., an atomic nuceus composed of A nuceons. If the interaction of the projectie with the individua partice in the target is known (in our exampe this is a nuceon), the form factor measures the spatia distribution of the target partices.