More Details on Scattering and Structure

The material in Sections 6.3–6.10 demonstrates that much information concerning subatomic structure can be obtained from scattering experiments. Even a glance at a differential cross section, without detailed computation, can reveal gross features.

As an example, the information contained in Figs. 6.3, 6.5, 6.11, and 6.13 is repro-duced schematically in Fig. 6.24. It highlights one difference between heavy nuclei and nucleons: Typical heavy nuclei have well-defined surfaces; as in optics, interfer-ence effects then produce diffraction minima and maxima in the differential cross section. Nucleons, in contrast, do not have such surfaces; their density decreases smoothly, and they do not show prominent diffraction effects.

The Scattering Amplitude In the present section, we shall treat scattering in somewhat more detail than we have done before. A glance at any current book on scattering⁽⁵⁹⁾ will show that the material presented here constitutes only a minute

58D.F. Geesaman, K. Saito, and A.W. Thomas, Annu. Rev. Nucl. Part. Sci. 45, 337 (1995).

59M. L. Goldberger and K. M. Watson, Collision Theory, Wiley, New York, 1964; R. G. Newton, Scattering Theory of Waves and Particles, McGraw-Hill, New York, 1966; L. S. Rodberg and R. M.

6.11. More Details on Scattering and Structure 173

Figure 6.23: Ratios of the nucleon structure functions deduced from F2(Cu)/F2(d) and F2(Fe)/F2(d). [From J. Ashman et. al., European Muon Collaboration, Phys. Lett. 202B, 603 (1988).] Later data looks similar.

fraction of what is actually used in research. Even so, it should provide some insight into the connection between scattering and structure.

We begin the discussion with a simple case, nonrelativistic scattering by a fixed potential, V (x), and we approximate the incoming particle by a plane wave moving along the z axis, ψ = exp(ikz).

The solution to the scattering problem is a solution of the time-independent Schr¨odinger equation,

−
²

2m∇²ψ + V ψ = Eψ or (6.74)

(∇²+ k²)ψ =2m

² V ψ, where the wave number k is related to the energy E by

k = p

= 1

√2mE. (6.75)

Far away from the scattering center, the scattered wave will be spherical, and it will originate at the scattering center, which is assumed to be at the origin of the coordinate system. The total asymptotic wave function, shown in Fig. 6.25, consequently will be of the form

ψ = e^ikz+ ψs, ψs= f (θ, ϕ)e^ikr

r . (6.76)

Thaler, Introduction to the Quantum Theory of Scattering, Academic Press, New York, 1967; W.O.

Amrein, J.M. Jauch, K.B. Sinha, Scattering theory in quantum mechanics : physical principles and mathematical methods, Reading, Mass. : W. A. Benjamin, Advanced Book Program, 1977.

Figure 6.24: Cross section and charge distribution: The appearance of diffraction minima in the cross section for heavy nuclei implies the existence of a well-defined nuclear surface. Nucleons, in contrast, possess a charge density that decreases smoothly.

Figure 6.25: The asymptotic wave function consists of an incoming plane wave and an outgoing spherical wave.

6.11. More Details on Scattering and Structure 175

The scattering amplitude f describes the angular dependence of the outgoing spher-ical wave; its determination is the goal of the scattering experiment.

The connection between differential cross section and scattering amplitude is given by Eq. (6.2). To verify the relation, we note that for the present case of one scattering center (N = 1), Eqs. (2.12) and (2.13) give for the differential cross section

dσ

dΩ= (dN /dΩ) Fin .

The outgoing flux, the number of particles crossing a unit area a at distance r per unit time, is connected to dN /dΩ by

Fout=dN

Since the flux is given by the probability density current, the computation of dσ/dΩ is now easy. For the incident wave, ψ = exp(ikz), we find

Fin=

2mi|ψ^∗∇ψ − ψ∇ψ^∗| =
k m.

In all directions except forward (0^◦), the scattered wave is given by the second term in Eq. (6.76) so that

Fout=
k

mr²|f(θ, φ)|².

With Eq. (6.77), the relation (6.2) between scattering amplitude and cross section is verified.⁽⁶⁰⁾

In the forward direction, the interference between the incident and the scattered wave can no longer be neglected. It is necessary for the conservation of flux: The scattered particles deplete the incident beam, and the scattering in the forward direction and the total cross section must be related. The relation is called the optical theorem: The total cross section and the imaginary part of the forward scattering amplitude are connected by⁽⁶¹⁾

σtot= 4π

k Imf (0^◦). (6.78)

60The derivation given here is superficial. A careful treatment can be found in K. Gottfried, Quantum Mechanics, Benjamin, Reading, Mass., 1966, Subsection 12.2.

61For derivations of the optical theorem, see Park, p. 376; Merzbacher, p. 532; and Messiah, p. 867.

The Scattering Integral Equation • To find the general solution of the Schr¨odinger equation, Eq. (6.74), we recall that it can be written as the sum of a special solution and of the appropriate solution of the corresponding homogeneous equation, where V = 0. To find a special solution of Eq. (6.74), it is convenient to consider the term (2m/
²)V ψ on the right-hand side as the given inhomogeneity, even though it contains the unknown wave function ψ. As a first step, then, we solve the scattering problem for a point source for which the inhomogeneity becomes a three-dimensional Dirac delta function and Eq. (6.74) takes on the form

(∇²+ k²)G(r, r
) = δ(r − r
). (6.79) The solution of this equation that corresponds to an outgoing wave is

G(r, r
) =−1 4π

e^ik|r−r^|

|r − r
| . (6.80)

To verify that this Green’s function indeed satisfies Eq. (6.79), we set, for simplicity, r
= 0,|r| = r, and use the relations⁽⁶²⁾

The second step in this identity follows from the fact that



d³rδ(r)f (r) and



d³rδ(r) exp(ikr)f (r)

give the same result, f (0), for any continuous function f . The solution of Eq. (6.55) for a potential V (r) is found by assuming that the inhomogeneity (2m/
²)V (r)ψ(r)

62For a derivation of Eq. (6.81) see, for instance, Jackson, Section 1.7.

6.11. More Details on Scattering and Structure 177 where G(r, r
) is the Green’s function for a delta function potential, Eq. (6.80). The appropriate solution of the homogeneous Schr¨odinger equation describes a particle that impinges on the target along the z axis; the general solution is therefore

ψ(r) = e^ikz+2m

²



d³r
G(r, r
)V (r
)ψ(r
). (6.86) The original Schr¨odinger differential equation for the wave function ψ has been transformed into an integral equation, called the scattering integral equation. For many problems, it is more convenient to start from such an integral equation rather than from the differential equation.

In scattering experiments, the incident beam is prepared far outside the scatter-ing potential, and the scattered particles are also analyzed and detected far away.

The detailed form of the wave function inside the scattering region is consequently not investigated, and what is needed is the asymptotic form of the scattered wave, ψs(x). With ˆr = r/r and k = kˆr, as indicated in Fig. 6.26, |r − r
| becomes and the Green’s function takes on the asymptotic value

G(r, r
) ∼

Inserting G(r, r
) into Eq. (6.85) and comparing with Eq. (6.76) yields the expression for the scattering amplitude,

f (θ, ϕ) = −m 2π
²



d³r
eⁱk^·r^V (r
)ψ(r
).• (6.89)

The First Born Approximation The first Born approximation corresponds to the case of a weak interaction. If the interaction were negligible, the scattering amplitude would vanish and ψ(r
) would be given by exp(ikz
)≡ exp(ik0· r
). As a first approximation, this value of the wave function is inserted in Eq. (6.89), with the result

f (θ, ϕ) = −m 2π
²



d³r
V (r
) exp(iq · r
/
), (6.90)

Figure 6.26: Vectors involved in the description of scattering.

where q =
(ko− k) is the momentum that the scattered particle imparts to the scattering center, as already defined in Eq. (6.3). Equation (6.90) is called the first Born approximation; we quoted this expression in Eq. (6.5) without proof. The scattering of high-energy electrons by nucleons and light nuclei and weak processes can be described adequately by the Born approximation. In Section 6.2, we used it to derive the Rutherford cross section. Next we shall turn to an approximation that is valid under certain conditions even if the force is strong.

Diffraction Scattering—Fraunhofer Approximation When the wavelength of the incident particle is short compared to the size of the interaction region, a semiclassical approach can be used, even if the force is strong. Such an approxima-tion is justified because the average trajectory followed by the particle approaches the classical one. The approximation used for elastic scattering is well known from optics, namely Fraunhofer diffraction. In the scattering of electromagnetic waves, optical or microwaves, the appearance of diffraction patterns has been known for a long time, and their description is well understood.⁽⁶³⁾ A characteristic example, diffraction from a black disk, is shown in Fig. 6.27. Black means that any photon hitting the disk is absorbed. Optical diffraction displays a number of characteristic features of which we stress three:

1. A large forward peak, called diffraction peak.

2. The appearance of minima and maxima, with the first minimum approxi-mately at an angle

θmin≈ λ 2R0

, (6.91)

where R0 is the radius of the disk.

63E. Hecht. Optics, 4th. Ed., Addison-Wesley, Reading, MA 2002; Jackson, Chapter 10.

6.11. More Details on Scattering and Structure 179

Figure 6.27: Optical diffraction pattern produced by a black disk.

3. At very small wavelengths (corresponding to the energy going to infinity) the total cross section for the scattering of light by the disk tends to a constant value,

σ−→ const. for E −→ ∞. (6.92)

A detailed examination of the diffraction pattern for a number of wavelengths per-mits conclusions to be drawn concerning the shape of the scattering object. Diffrac-tion scattering occurs not only in optics but also in subatomic physics, where it is a useful tool for structure investigations. Diffraction phenomena appear because the wavelength of the incident particles can be chosen to be smaller than the dimension of the target particle. The Fraunhofer approximation applies because the incident and the outgoing wave can be taken to be plane waves. To illustrate Fraunhofer diffraction we will present some examples in nuclear and particle physics. Consider first nuclei. Figure 6.28 shows the differential cross section for elastic scattering of 42 MeV alpha particles from ²⁴Mg.⁽⁶⁴⁾ A sharp forward peak and pronounced diffraction minima and maxima stand out clearly. A simple model that considers the nucleus as a dark disk reproduces the position of the minima and maxima well, but with increasing scattering angle, the observed maxima are increasingly smaller than the predicted ones.

The reason for the disagreement is that nuclei are not exactly ‘black disks’.

First, Figure 6.5 indicates that they have a skin of considerable thickness rather than sharp edges, and, further, nuclei are not always spherical but may have a permanent deformation, as will be discussed in Section 18.1. Finally, nuclei are partially transparent for low- and medium-energy hadrons. The simple theory can

64I. M. Naqib and J. S. Blair, Phys. Rev. 165, 1250 (1968); S. Fernbach, R. Serber, and T. B.

Taylor, Phys. Rev. 75, 1352 (1949).

65E. Gadioli and P. E. Hodgson, Rep. Prog. Phys. 49, 951 (1986); P. E. Hodgson, Growth Points in Nuclear Physics, Vol. 1, Pergamon, Elmsford, NY, 1984.