The nuclear optical model and its optical-scattering analog: Mie scattering

Electromagnetic, atomic, and nuclear scattering are major tools in modern physics research,^1–6 but they are often only briefly covered in undergraduate curricula. This paper reveals important similarities between two models of scattering on very different length scales: the Mie optical-scattering model and the nuclear optical model. The historical development of these models is outlined along with some of their important applications. We show the close analogy between the models and, in particular, that the Mie optical-scattering model, with only a complex index of refraction together with a suitable choice of the ratio of wavelength to object size, λ/D, can also model nuclear scattering and enable extraction of realistic nuclear dimensions. This analogy provides insight into the nuclear optical model and its parameters as well as many other scattering processes over a wide range of dimensions.

Gustav Mie's model was published in 1908 and successfully reproduced the scattering of light from gold particles embedded in a colloidal suspension.¹ Such scattering produces a spectrum of colors depending on the angle of observation and includes as a special case Rayleigh scattering, where the size, i.e., diameter D, of the scattering objects is much smaller than the incident wavelength, typically by a factor of ten or more.^3,4 For example, the Rayleigh scattering of sunlight from air molecules produces the blue overhead sky on Earth. In contrast to Earth, scattered sunlight on the planet Mars is mainly caused by scattering from dust particles in the Martian sky, given that there is little gaseous atmosphere on Mars and, hence, there is little Rayleigh scattering. In this case, Mie scattering with λ ≪ D favors scattering of the red part of sunlight at large angles, so the overhead sky appears red, and sunsets appear blue on Mars (Fig. 1). These planetary examples illustrate the importance of the ratio λ/D in determining important characteristics of optical scattering, which we show is also true for nuclear scattering.^5,6

In Sec. II, we review optical scattering and diffraction in more detail. In Sec. III, we outline the development of the nuclear optical model, and in Sec. IV, we show that there is a close and useful relationship between optical and nuclear scattering processes, over a wide range of dimensions. The latter spans optical wavelengths (nanometers, 10⁻⁹ m) to nuclear dimensions (femtometers, 10⁻¹⁵ m).

In Mie optical-scattering calculations, the complex function f(θ,φ) is determined via an expansion of an incident plane wave and outgoing spherical wave (Eq. (1)) into partial-wave components in spherical coordinates using spherical harmonics. These partial-wave components must satisfy Maxwell's equations for waves incident on, transiting, and then exiting the scattering object. The outgoing scattered wave is constructed from the resulting modified partial waves, and the scattering amplitude f (θ,φ) is determined. Hence, the latter depends on the dimensions, shape, surfaces, and index of refraction of the scattering object and the surrounding medium.⁹ The angle φ is relevant for polarized waves, but often polarization states can be averaged to yield a scattering function independent of φ, i.e., f(θ). The mathematical details are given in several references⁴ including those describing student optical-scattering experiments using Mie-model calculations for data analysis.^10,11

The scattered intensity vs angle, and in particular, the differential scattering cross section away from zero degrees, is given by dσ/dΩ(θ) = f*(θ)f(θ).^5–8 Because f(θ) varies with wavelength, if there is a range of incident wavelengths, some wavelengths may be preferentially scattered at certain angles, as is the special case for Rayleigh scattering of sunlight that results in the blue sky.

Several student- and instructor-friendly computer programs are available to calculate Mie scattering. One program, particularly well-suited for student use, is MiePlot by Philip Laven.⁹ This program includes options to calculate Mie, Debye, Rayleigh, and other scattering models, as well as Fraunhofer diffraction.

A first-order method for determining the dimensions of microscopic objects embedded in an optical medium or vacuum is far-field Fraunhofer diffraction, which was first studied in detail in the 1700 s. Recall that Fraunhofer diffraction from a circular aperture with diameter D, which is comparable in size to the incident wavelength λ, yields a far-field diffraction pattern described by sin(θ) = m λ/D, with the first two maxima at m = 1.63 and 2.68, and the first two minima at m = 1.22 and 2.23.^4,12 This is illustrated in Fig. 2 for λ/D = 1/4. Invoking Babinet's principle, this also is the type of pattern expected for a circular, opaque object of the same dimensions.^4,12 Also shown in Fig. 2 is a Mie scattering calculation from a transparent spherical object with λ/D = 1/4, having a real or complex index of refraction.⁹ A complex index of refraction is often needed for many types of optical and E&M scattering.^2,13,14 As expected, the effect of adding an absorptive component to the index of refraction reduces the scattering at large angles, where the optical path for many incident rays through the scattering medium is larger than the path of many of the rays at forward angles.⁹ 

As seen in Fig. 2, analysis using Fraunhofer diffraction to fit the minima and maxima of the Mie scattering would imply a larger object size, D (i.e., smaller λ/D) by 15%–20% for the calculations shown. This also is the case for diffraction analyses of nuclear scattering, as often done in introductory physics classes. However, when the size of the object is very large relative to the incident wavelength, corresponding to a very small value of λ/D, Mie scattering more closely resembles Fraunhofer diffraction.

The nuclear optical model (NOM), which utilizes a complex scattering potential, represents an evolution of several early nuclear scattering models that used only real potentials. This includes Rutherford's model for classical scattering of low-energy ions from nuclei via a long-range Coulomb potential, published in 1911.^4–6 Only many years later (1930s and beyond), when accelerators with higher-energy nuclear beams became available (creating shorter λs) were widespread deviations from Rutherford's classical scattering model observed. Figure 3 shows examples of elastic scattering for a range of projectiles incident on a nickel target foil, all at approximately the same projectile velocity (ca.10 MeV/nucleon). There is a wide variation in the scattering patterns, and several resemble the Fraunhofer pattern shown in Fig. 2, as expected for small values of λ/D.

The quantum-mechanical (QM) scattering of a subatomic particle by an atomic nucleus can also, like Mie optical scattering, be described by Eq. (1) using a partial-wave expansion where k and λ are the incident particle's de Broglie wave number and de Broglie wavelength, λ = 2π/k. The wavenumber, k, for non-relativistic particles (as most cited here) is k = 0.2187 sqrt (A_I E_I) f m⁻¹, where A_I is the incident particle's atomic mass number, and E_I is its kinetic energy in MeV. The plane-wave Born approximation (PWBA) was one the early QM nuclear scattering models treating incident and scattered particles as waves, and it is introduced in many QM and nuclear physics textbooks.^7,16,17 However, calculations using PWBA with only a real nuclear potential usually do not reproduce the observed nuclear scattering data in the energy regions discussed here (Fig. 3), as the model does not include the attenuation and other alterations of the scattered waves due to nuclear reactions.

In the late 1930s, several researchers then proposed a major improvement of the NOM by making the nuclear scattering potential complex with an imaginary, absorptive component U(r) =V(r) + i W(r), together with the Coulomb potential for charged ions.^18–20 In 1954, Roger Woods and David Saxon introduced a realistic NOM potential distribution that followed the Fermi shape of the known charge distributions,^4,21 but with an increased diffuseness.²² This potential is denoted as a Woods–Saxon (WS) potential, but other nuclear potential forms are now included in the NOM.^23,24 Especially important was the addition of a spin–orbit potential to account for nucleon–nucleus scattering of spin-aligned nuclei.

Further developments of the NOM include the determination of “global” NOM parameters^24,25 and the development of a reformulated NOM to allow extraction of nucleon distributions from NOM potentials.^26,27 As with Mie scattering, the NOM has wide-spread applications in modern physics research including the analysis of nuclear reaction data using distorted-wave Born approximation (DWBA),^5,6,17,27 the calculation of total nuclear reaction cross sections,^5,6 and the calculation of single-particle nuclear levels including predictions for super-heavy “magic nuclei” A > 300 (e.g., Z = 114 and 126, N = 148, 184, and 210).^27,29

The special case of heavy-ion scattering (Fig. 3) is discussed in the supplementary material.²⁷ The early development of the NOM and its justification are reviewed in Refs. 18–20. Modern developments are reviewed in Ref. 28. Other related materials, suitable for including in a course in QM, nuclear physics, or modern physics, can be found in the supplementary material for this paper.²⁷ 

Predating the nuclear optical model was the liquid-drop nuclear model (LDM) of Weizsacker, Gamow, Bohr, and Wheeler, which is used to describe many properties of nuclei,^4–6 including nuclear binding energies, LDM excitations, nuclear fission, and fusion–fission processes.³⁰ Given the success of the LDM, it is not surprising that an optical model of nuclear scattering analogous to the Mie optical-scattering model of light from a liquid drop has proven to be very successful.

Changes in the calculated NOM elastic scattering cross sections with variations in certain NOM potential parameters (strengths, radii, diffuseness, and hence D and λ/D) parallel many of the features of Mie optical-scattering calculations when specific Mie-model parameters are varied. Some are expected, for example, changing the NOM potential radii parameters shifts the calculated diffraction-like minima and maxima in the NOM cross sections, as also observed in Mie scattering when the parameter D or n_R is varied (Fig. 4).

Likewise changing the NOM real and complex nuclear-potential strengths, V(r) and W(r), parallel closely the effects seen when the real and imaginary parts of the index of refraction are changed in the Mie-model scattering calculations (Fig. 4). The scattering object appears, as expected, to become more opaque for increasing values of n_I and W(r). Hence, there is less impact on the scattered intensity when the scattering object becomes highly absorbing (Fig. 4). However, this also depends on the magnitude of n_R and V(r), i.e., refraction. Since the parameters used in Mie scattering appear to have close NOM analogs, we can then expect (and will also demonstrate) that nucleon–nucleus scattering can be simulated with a suitable choice of Mie-model scattering parameters. This can provide insight into these and similar scattering processes. We show that, perhaps unexpectedly, Mie scattering can simulate proton-nucleus and alpha-nucleus charged-particle scattering, and even scattering from exotic “halo” nuclei, N ≫ Z.

The close similarity among the Mie optical-scattering calculations, nuclear elastic scattering data (Fig. 5), and related NOM calculations for neutron-nucleus elastic scattering is shown in Fig. 6. (The NOM parameters are from global NOM sets and included in the supplementary material.²⁷) As demonstrated, it is possible to simulate the angular distributions of such scattering quite well by varying a few simple analog Mie scattering model parameters (Sec. IV A). Most important, one needs to adjust the object size, D, i.e., the ratio λ/D, or the parameters determining D, to reproduce the angles corresponding to minima and maxima in the scattered intensity. For example, the n+Pb data at E_n = 14 MeV (Fig. 5) corresponds to an incident neutron with de Broglie wavelength λ = 7.6 fm, and the NOM calculation shown corresponds to a NOM real nuclear potential radius, R = 6.9 fm, and imaginary (absorptive) potential radius, R = 7.5 fm.^24,27 The data implies λ/D = 0.45, and using this with the neutron de Broglie wavelength indicates a nuclear-scattering object with R = 8.4 fm. This is reasonably close to the NOM absorptive-potential radius for lead (R = 7.5 fm).^24,27 An analysis using Fraunhofer diffraction would require a smaller value of λ/D, implying a much larger scattering object (R = ca. 9.5 fm). Likewise, Fig. 6 shows good agreement between the n+Ni NOM calculation with En = 50 MeV (λ = 4 fm) and Mie scattering with λ/D = 0.38. This λ/D implies R = 5.3 fm for Ni (A ca. 58, Z = 28) compared to the real and imaginary n+Ni NOM potential radii of 4.6 and 4.9 fm.

A priori, owing to the long-range repulsive Coulomb potential present, we might not expect Mie optical-scattering calculations to reproduce very well proton-nucleus scattering or other charged-particle elastic scattering. In addition to the neutron scattering already discussed, Fig. 6 also shows comparisons of Mie scattering and NOM calculations for protons. Again, only the normalization, the index of refraction parameters, and the value of D (and thus λ/D) for the Mie scattering calculations are adjusted to reproduce the main features of the NOM calculations. The NOM calculations employing global parameters^24,27 again closely correspond to measured data (Fig. 5), where the proton cross sections (in contrast to Fig. 6) are shown as ratio to Rutherford-model calculated cross sections. While the effect of the nuclear Coulomb potential is noticeable at some angles, the overall features are nonetheless reproduced. The values of λ/D shown (or extracted from similar data) correspond closely to the values of D expected for the NOM nuclear potentials and, in particular, to the radius of W(r), the absorptive potential where R_W is given by R_W = r_W A_T^1/3 with rw about 1.3 fm.^23,24,27

In Fig. 7, we show alpha-particle (⁴He) elastic-scattering cross sections, which appear to be highly diffractive with pronounced minima and maxima (again shown without dividing by the calculated Rutherford-model scattering). A reduced value for the complex part of the index of refraction is needed in the Mie-model calculations indicated. Recall that ⁴He is tightly bound and the nuclear reactions possible, e.g., nucleon transfers, are limited at the bombarding energies shown by their highly negative reaction Q values. This corresponds to a reduced absorptive-potential strength for W(r), and hence a low value for the equivalent n_I. Again, the impact of the repulsive Coulomb potential at forward angles is evident, but this is minimized in the calculation for low-Z targets (e.g., Si, Z = 28). We also note the low value of λ/D needed (i.e., large D), approaching that for Fraunhofer diffraction (i.e., λ≪D). However, the values of D still correspond closely to the NOM absorptive potential dimensions for the target nuclei shown (e.g., Table 1, Ref. 25, and listed in the supplementary material²⁷).

Returning to Fig. 3, we see that ³He scattering is much less diffractive compared to ⁴He scattering. ³He is loosely bound relative to ⁴He and subject to many more nuclear reactions, i.e., those with positive or small reaction Q values, and hence an increase in W(r). Thus, as anticipated, the ³He data can then be simulated in the Mie-model with an increase in the complex component of the index of refraction relative to that needed for ⁴He. However, otherwise, reasonable Mie-model scattering parameters with suitable λ/D values can be used to fit the data, corresponding to reasonable nuclear-potential dimensions.

Thus, somewhat unexpectedly, we find even for charged ions such as protons, ³He, and ⁴He that simple Mie scattering calculations still simulate quite well the key features of the multi-parameter NOM calculations that reproduce measured nuclear elastic scattering for these projectiles. It can be verified, e.g., by students as exercises employing the NOM and Mie models to fit experimental data;^9,27,33,34 this also applies to deuteron- and triton-nucleus elastic scattering.²³ 

A new frontier in nuclear physics is the production and study of nuclear reactions with exotic nuclei Z ≫ N or N ≫ Z, such as ⁸B, ⁸He, ¹¹Li, and others. These are often short-lived radioactive nuclei that can be produced and studied as secondary beams.³¹ This is typically done using fragmentation or nuclear reactions in a production target with an incident intense, stable primary nuclear beam. One can then study traditional nuclear reactions such as nucleon transfers, elastic and inelastic scattering, and fusion–fission, by utilizing inverse kinematics, i.e., a heavy incident short-lived secondary beam on a low-mass nuclear target.

An example is shown in Fig. 8 where a 576-MeV N ≫ Z secondary ⁸He beam (Z = 2, N = 6; and T_1/2 = 119 ms) has been produced and scattered from a hydrogen target to measure and analyze p+⁸He elastic and inelastic scattering.³² This scattering event is equivalent to elastic scattering of a 72 MeV proton from a fixed ⁸He target (see kinematics equations given in Ref. 27). In the center-of-mass frame, this corresponds to Ec.m. = (A_T E_I)/(A_T  + A_I) = 64 MeV.²⁷ This experiment was designed to test a “neutron halo” model where ⁸He consists of a compact ⁴He (α-particle) core surrounded by an extended 4-neutron halo.³² This model can be tested using the model's nucleon distributions to generate NOM potentials using a version of the reformulated NOM. In order to fit the data, a very diffuse, extended NOM potential and neutron distribution must be used, confirming the neutron halo model for ⁸He.³² 

Applying the Mie scattering model also provides support for the neutron-halo model. Figure 8 shows two Mie optical-scattering calculations.⁹ One (dashed curve) assumes a uniform index of refraction out to a fixed radius, corresponding to that expected for a conventional nucleus. The other (solid curve) assumes a non-uniform, graded index of refraction falling uniformly from a maximum value to zero at a large radius. The latter is consistent with the neutron-halo model proposed for ⁸He and provides a better description of the data. (Details of the calculation are given in the supplementary material.²⁷)

We have shown that Mie optical-scattering calculations with a few simple parameters can accurately model nucleon–nucleus and other nuclear elastic scattering data, producing results that are similar to the related multi-parameter nuclear optical-model calculations used to describe such data. Mie-scattering calculations can provide insight into the nuclear optical model, its parameters, and the related nuclear scattering including scattering from exotic nuclei. A key parameter characterizing these and similar scattering processes over a wide range of dimensions is the ratio of incident wavelength to the size of the scattering object (λ/D). Our results suggest that table-top undergraduate laboratories^10,11 demonstrating Mie optical scattering, with appropriate values of λ/D for the light source and scattering objects, can simulate large-scale nuclear-scattering measurements. These would normally require access to a large nuclear particle accelerator.²⁷ 

An extended version of this paper, additional tutorial and historical material, some suggested student exercises, nuclear kinematics equations, together with descriptions of MiePlot, NOM, and DWBA computer programs,^9,33,34 including typical parameters for nuclear-scattering calculations, can be found in the supplementary material.²⁷ 

Thanks to P. Laven, R. Torres-Isea, J. D. Cossairt, Professor Roberto Merlin, Professor Georg Raithel, Professor Paul Berman, Professor Filomena M. Nunes, and the anonymous referees for their useful comments and suggestions.

The author has no conflicts of interest to disclose.