## Monday, February 16, 2015

### Scattering theory

Top: the real part of a plane wave travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of index of refraction higher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact.

In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Scattering also includes the interaction of billiard balls on a table, the Rutherford scattering (or angle change) of alpha particles by gold nuclei, the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the inelastic scattering of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of partial differential equations, propagating freely "in the distant past", come together and interact with one another or with a boundary condition, and then propagate away "to the distant future". The direct scattering problem is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. The inverse scattering problem is the problem of determining the characteristics of an object (e.g., its shape, internal constitution) from measurement data of radiation or particles scattered from the object.

Since its early statement for radiolocation, the problem has found vast number of applications, such as echolocation, geophysical survey, nondestructive testing, medical imaging and quantum field theory, to name just a few.

## Conceptual underpinnings

The concepts used in scattering theory go by different names in different fields. The object of this section is to point the reader to common threads.

### Composite targets and range equations

Equivalent quantities used in the theory of scattering from composite specimens, but with a variety of units.

When the target is a set of many scattering centers whose relative position varies unpredictably, it is customary to think of a range equation whose arguments take different forms in different application areas. In the simplest case consider an interaction that removes particles from the "unscattered beam"at a uniform rate that is proportional to the incident flux I$I$ of particles per unit area per unit time, i.e. that
dIdx=QI
where Q is an interaction coefficient and x is the distance traveled in the target.

The above ordinary first-order differential equation has solutions of the form:
I=IoeQΔx=IoeΔxλ=Ioeσ(ηΔx)=IoeρΔxτ,
where Io is the initial flux, path length Δx ≡ x − xo, the second equality defines an interaction mean free path λ, the third uses the number of targets per unit volume η to define an area cross-section σ, and the last uses the target mass density ρ to define a density mean free path τ. Hence one converts between these quantities via Q = 1/λησρ/τ, as shown in the figure at left.

In electromagnetic absorption spectroscopy, for example, interaction coefficient (e.g. Q in cm−1) is variously called opacity, absorption coefficient, and attenuation coefficient. In nuclear physics, area cross-sections (e.g. σ in barns or units of 10−24 cm2), density mean free path (e.g. τ in grams/cm2), and its reciprocal the mass attenuation coefficient (e.g. in cm2/gram) or area per nucleon are all popular, while in electron microscopy the inelastic mean free path[1] (e.g. λ in nanometers) is often discussed[2] instead.

## In theoretical physics

In mathematical physics, scattering theory is a framework for studying and understanding the interaction or scattering of solutions to partial differential equations. In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or propagate through non-uniform media (such as sound waves, in sea water, coming from a submarine). In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of light or radio waves is studied. In particle physics, the equations are those of Quantum electrodynamics, Quantum chromodynamics and the Standard Model, the solutions of which correspond to fundamental particles.

In regular quantum mechanics, which includes quantum chemistry, the relevant equation is the Schrödinger equation, although equivalent formulations, such as the Lippmann-Schwinger equation and the Faddeev equations, are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and protons. The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and decays occurring. There are two predominant techniques of finding solutions to scattering problems: partial wave analysis, and the Born approximation.

## Elastic and inelastic scattering

The term "elastic scattering" implies that the internal states of the scattering particles do not change, and hence they emerge unchanged from the scattering process. In inelastic scattering, by contrast, the particles' internal state is changed, which may amount to exciting some of the electrons of a scattering atom, or the complete annihilation of a scattering particle and the creation of entirely new particles.

The example of scattering in quantum chemistry is particularly instructive, as the theory is reasonably complex while still having a good foundation on which to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being the bound state solutions of some differential equation. Thus, for example, the hydrogen atom corresponds to a solution to the Schrödinger equation with a negative inverse-power (i.e., attractive Coulombic) central potential. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even ionized, representing an inelastic scattering process.

The term "deep inelastic scattering" refers to a special kind of scattering experiment in particle physics.

## The mathematical framework

In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts.
For example, if a differential equation is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of time. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future".

Solutions to differential equations are often posed on manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold. As a result, the solutions often have a spectrum that can be identified with a Hilbert space, and scattering is described by a certain map, the S matrix, on Hilbert spaces. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

An important, notable development is the inverse scattering transform, central to the solution of many exactly solvable models.

### Spectroscopy

Analysis of white light by dispersing it with a prism is an example of spectroscopy.

Spectroscopy is the study of the interaction between matter and radiated energy.[1][2] Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to comprise any interaction with radiative energy as a function of its wavelength or frequency. Spectroscopic data is often represented by a spectrum, a plot of the response of interest as a function of wavelength or frequency.

## Introduction

Spectroscopy and spectrography are terms used to refer to the measurement of radiation intensity as a function of wavelength and are often used to describe experimental spectroscopic methods. Spectral measurement devices are referred to as spectrometers, spectrophotometers, spectrographs or spectral analyzers.

Daily observations of color can be related to spectroscopy. Neon lighting is a direct application of atomic spectroscopy. Neon and other noble gases have characteristic emission frequencies (colors). Neon lamps use collision of electrons with the gas to excite these emissions. Inks, dyes and paints include chemical compounds selected for their spectral characteristics in order to generate specific colors and hues. A commonly encountered molecular spectrum is that of nitrogen dioxide. Gaseous nitrogen dioxide has a characteristic red absorption feature, and this gives air polluted with nitrogen dioxide a reddish brown color. Rayleigh scattering is a spectroscopic scattering phenomenon that accounts for the color of the sky.

Spectroscopic studies were central to the development of quantum mechanics and included Max Planck's explanation of blackbody radiation, Albert Einstein's explanation of the photoelectric effect and Niels Bohr's explanation of atomic structure and spectra. Spectroscopy is used in physical and analytical chemistry because atoms and molecules have unique spectra. As a result, these spectra can be used to detect, identify and quantify information about the atoms and molecules. Spectroscopy is also used in astronomy and remote sensing on earth. Most research telescopes have spectrographs. The measured spectra are used to determine the chemical composition and physical properties of astronomical objects (such as their temperature and velocity).

## Theory

One of the central concepts in spectroscopy is a resonance and its corresponding resonant frequency. Resonances were first characterized in mechanical systems such as pendulums. Mechanical systems that vibrate or oscillate will experience large amplitude oscillations when they are driven at their resonant frequency. A plot of amplitude vs. excitation frequency will have a peak centered at the resonance frequency. This plot is one type of spectrum, with the peak often referred to as a spectral line, and most spectral lines have a similar appearance.

In quantum mechanical systems, the analogous resonance is a coupling of two quantum mechanical stationary states of one system, such as an atom, via an oscillatory source of energy such as a photon. The coupling of the two states is strongest when the energy of the source matches the energy difference between the two states. The energy (E)$(E)$ of a photon is related to its frequency (ν)$(\nu )$ by E=hν$E=h\nu$ where h$h$ is Planck's constant, and so a spectrum of the system response vs. photon frequency will peak at the resonant frequency or energy. Particles such as electrons and neutrons have a comparable relationship, the de Broglie relations, between their kinetic energy and their wavelength and frequency and therefore can also excite resonant interactions.

Spectra of atoms and molecules often consist of a series of spectral lines, each one representing a resonance between two different quantum states. The explanation of these series, and the spectral patterns associated with them, were one of the experimental enigmas that drove the development and acceptance of quantum mechanics. The hydrogen spectral series in particular was first successfully explained by the Rutherford-Bohr quantum model of the hydrogen atom. In some cases spectral lines are well separated and distinguishable, but spectral lines can also overlap and appear to be a single transition if the density of energy states is high enough.

## Classification of methods

Spectroscopy is a sufficiently broad field that many sub-disciplines exist, each with numerous implementations of specific spectroscopic techniques. The various implementations and techniques can be classified in several ways.

Types of spectroscopy are distinguished by the type of radiative energy involved in the interaction. In many applications, the spectrum is determined by measuring changes in the intensity or frequency of this energy. The types of radiative energy studied include:

### Nature of the interaction

Types of spectroscopy can also be distinguished by the nature of the interaction between the energy and the material. These interactions include:[1]
• Absorption occurs when energy from the radiative source is absorbed by the material. Absorption is often determined by measuring the fraction of energy transmitted through the material; absorption will decrease the transmitted portion.
• Emission indicates that radiative energy is released by the material. A material's blackbody spectrum is a spontaneous emission spectrum determined by its temperature. Emission can also be induced by other sources of energy such as flames or sparks or electromagnetic radiation in the case of fluorescence.
• Elastic scattering and reflection spectroscopy determine how incident radiation is reflected or scattered by a material. Crystallography employs the scattering of high energy radiation, such as x-rays and electrons, to examine the arrangement of atoms in proteins and solid crystals.
• Impedance spectroscopy studies the ability of a medium to impede or slow the transmittance of energy. For optical applications, this is characterized by the index of refraction.
• Inelastic scattering phenomena involve an exchange of energy between the radiation and the matter that shifts the wavelength of the scattered radiation. These include Raman and Compton scattering.
• Coherent or resonance spectroscopy are techniques where the radiative energy couples two quantum states of the material in a coherent interaction that is sustained by the radiating field. The coherence can be disrupted by other interactions, such as particle collisions and energy transfer, and so often require high intensity radiation to be sustained. Nuclear magnetic resonance (NMR) spectroscopy is a widely used resonance method and ultrafast laser methods are also now possible in the infrared and visible spectral regions.

### Type of material

Spectroscopic studies are designed so that the radiant energy interacts with specific types of matter.

#### Atoms

Atomic spectroscopy was the first application of spectroscopy developed. Atomic absorption spectroscopy (AAS) and atomic emission spectroscopy (AES) involve visible and ultraviolet light.
These absorptions and emissions, often referred to as atomic spectral lines, are due to electronic transitions of outer shell electrons as they rise and fall from one electron orbit to another. Atoms also have distinct x-ray spectra that are attributable to the excitation of inner shell electrons to excited states.

Atoms of different elements have distinct spectra and therefore atomic spectroscopy allows for the identification and quantitation of a sample's elemental composition. Robert Bunsen and Gustav Kirchhoff discovered new elements by observing their emission spectra. Atomic absorption lines are observed in the solar spectrum and referred to as Fraunhofer lines after their discoverer. A comprehensive explanation of the hydrogen spectrum was an early success of quantum mechanics and explained the Lamb shift observed in the hydrogen spectrum led to the development of quantum electrodynamics.

Modern implementations of atomic spectroscopy for studying visible and ultraviolet transitions include flame emission spectroscopy, inductively coupled plasma atomic emission spectroscopy, glow discharge spectroscopy, microwave induced plasma spectroscopy, and spark or arc emission spectroscopy. Techniques for studying x-ray spectra include X-ray spectroscopy and X-ray fluorescence (XRF).

#### Molecules

The combination of atoms into molecules leads to the creation of unique types of energetic states and therefore unique spectra of the transitions between these states. Molecular spectra can be obtained due to electron spin states (electron paramagnetic resonance), molecular rotations, molecular vibration and electronic states. Rotations are collective motions of the atomic nuclei and typically lead to spectra in the microwave and millimeter-wave spectral regions; rotational spectroscopy and microwave spectroscopy are synonymous. Vibrations are relative motions of the atomic nuclei and are studied by both infrared and Raman spectroscopy. Electronic excitations are studied using visible and ultraviolet spectroscopy as well as fluorescence spectroscopy.

Studies in molecular spectroscopy led to the development of the first maser and contributed to the subsequent development of the laser.

#### Crystals and extended materials

The combination of atoms or molecules into crystals or other extended forms leads to the creation of additional energetic states. These states are numerous and therefore have a high density of states. This high density often makes the spectra weaker and less distinct, i.e., broader. For instance, blackbody radiation is due to the thermal motions of atoms and molecules within a material. Acoustic and mechanical responses are due to collective motions as well.

Pure crystals, though, can have distinct spectral transitions and the crystal arrangement also has an effect on the observed molecular spectra. The regular lattice structure of crystals also scatters x-rays, electrons or neutrons allowing for crystallographic studies.

#### Nuclei

Nuclei also have distinct energy states that are widely separated and lead to gamma ray spectra.
Distinct nuclear spin states can have their energy separated by a magnetic field, and this allows for NMR spectroscopy.

## Other types

Other types of spectroscopy are distinguished by specific applications or implementations:

## Applications

UVES is a high-resolution spectrograph on the Very Large Telescope.[9]

## History

The history of spectroscopy began with Isaac Newton's optics experiments (1666–1672). Newton applied the word "spectrum" to describe the rainbow of colors that combine to form white light and that are revealed when the white light is passed through a prism. During the early 1800s, Joseph von Fraunhofer made experimental advances with dispersive spectrometers that enabled spectroscopy to become a more precise and quantitative scientific technique. Since then, spectroscopy has played and continues to play a significant role in chemistry, physics and astronomy.

### The Argument For Nuclear Energy In Australia: Scientist, Energy Consultant

In New Matilda's ongoing debate around nuclear energy, Barry Brook and Ben Heard make their case for nuclear power's role in tackling climate change.

By now, most of you would have heard that the Premier of South Australia, Labor’s Jay Weatherill, has announced a Royal Commission into an expanded future role for the state in nuclear energy. For people like us, who are both strongly focused on tackling climate change by eliminating Australia’s dependence on fossil fuels, and who consider nuclear to be an essential tool, this is real progress.

In a recent article on The Conversation, we explained the types of issues we think the Royal Commission might consider. These obviously only represent our opinions and perspectives, albeit well-informed and researched.

We cover most of the well-trodden ground on radioactive waste management and energy generation. We also explain a number of reasons, ranging from political to economic to geological, why we think South Australia is a particularly good place to kick-start any deeper foray by our nation into the nuclear fuel cycle.

One thing that particularly frustrated us was the immediate condemnation of the news by the SA Greens Party, and disappointingly, also by the Australian Youth Climate Coalition.

The whole point of Royal Commissions is the rigorous uncovering of facts, based on solid research and deep consultation with experts, government and public representatives. So why the objection?

Well, the arguments are well rehearsed and endlessly debated. Nuclear is too costly, unsafe, produces dangerous and intractable waste, is connected with weapons proliferation, is unsustainable, and besides, is unneeded.

Such a ‘washing list’ of objections is superficially convincing, and the last one in particular appeals to most people’s sensibilities. Australia is large, sunny and sparsely populated country with long, windswept coastlines. Surely then, we can (and should) do it all with wind and solar, and forget about dirty and technically complex alternatives like nuclear fission?

The thing is, with an issue as serious and immediate as climate change, we can’t afford to be carried away by wishful thinking, nor get trapped into thinking that ‘hope’ is a plan. We owe it to the future to be ruthlessly pragmatic about solutions, and accept that trade-offs are inevitable.

So, in as brief a summary as we can put it, here is the state of play was we see it.

Nuclear is expensive, at least compared to coal. But when coal pays its environmental costs (especially for air pollution and greenhouse gas emissions) nuclear is not expensive at all.

Electricity from some renewables is now comparatively cheap. But when renewables pay their full system costs to overcome variability, a renewable system is very expensive indeed.

In this context, a ‘nuclear intensive’ strategy is still likely to underpin the most viable, scalable and cost-effective pathway to replace coal.

Nuclear is the safest form of large-scale energy production, when evaluated on the basis of deaths per unit of generation.

Nuclear accidents like Chernobyl and Fukushima, although awful, are actually far less environmentally hazardous than many claim, and become ever less probable with newer, inherently safer designs.

Chernobyl Reactor 4, which melted down in April 1986. The other three reactors at Chernobyl continued to generate power, with the last reactor decommissioned in 2000.

Nuclear produces radioactive waste, but this is captured almost completely and isolated, and it can be recycled many times.

When fully recycled, its half-life is 30 years, and its already tiny volume is reduced by 50 times.

Nuclear power and nuclear weapons both work by using technology to split atoms, but beyond that the relationship is complex. The international safeguards set up to constrain proliferation are extensive and one has to draw a very long bow to link weapons acquisition to commercial power generation.

Nuclear fuel is not in short supply today, and long before it does become scarce, we will be recycling the waste to produce over 100 times more zero-carbon energy that will last millennia.

To set the goal of a “100 per cent renewables grid” is, at best, logistically and economically ‘courageous’, and at worst, a foolhardy strategy that is doomed to fail.

Either way, it is detached from what we consider the actual goal. If we really want to guarantee that we can rid ourselves of fossil fuels, then renewables come together, in a combined package with nuclear fission.

That last paragraph contains a whole lot of assertions. Yet we stand by all of them, because we have looked deeply at each of those statements. We have probed them critically for flaws, tested them in consultation with experts, exposed them repeatedly to the peer-reviewed energy literature, and debated them with opponents endlessly.

It’s almost all on the public record, in our scientific publications, lectures, blogs (Brave New Climate and Decarbonise SA), books, articles and videos.

Why have we done this? If you could roll back the calendar enough years, you would find one of us (Brook) was perfectly ambivalent regarding nuclear power, and the other (Heard) was an outright opponent.

Change did not come easily, and it did not come without challenge. The biggest challenge always came from within, to make sure we were moving beyond just having opinions, and moving towards having informed and reasoned positions.

The thing is, we still are. We make mistakes and get things wrong. Our positions continue to evolve and, we hope, improve, with greater nuance, understanding and balance.

We keep learning from each other, our “opponents”, our colleagues, our students, our research, from other experts in a variety of fields and, of course, when the facts change. Our position is being tested constantly. Learning does not end.

Moreover, we reckon we’ve heard all of the counter-arguments (slanted from a variety of viewpoints!), and thought carefully about them.

Some we’ve taken onboard, some disputed, some rejected. We understand the failings of nuclear energy, and we acknowledge that it is hardly a ‘perfect solution’. But we still hold that, when balanced against the alternatives, nuclear fission is a real winner.

The biggest win will be found by using everything to get an important job done.

This short essay is definitely not the place for us to try and convince the doubters. We’ve put briefly what we consider to be the ‘key facts’ and we’ve drawn what we think are robust conclusions from them. But you should all be skeptical of our claims — and those of anyone else — until you’ve looked hard at the evidence yourselves, and ideally, tried hard to disprove your cherished beliefs, rather than comfortably prop up the world-view that you already think you ‘know’ to be right.

It’s a fun intellectual exercise to try and show yourself why you’re wrong (on any number of things), and it’s the kind of strategy that scientists use every day to learn about how things work. If you do this and still disagree with us, then that’s fine — we place great value on rigorous challenges and evidence-based rebuttals. For dealing with climate change, the bottom line is, we need a plan that will work!

To conclude, below are some sources of information that we think are particularly valuable if you want to really understand nuclear energy and the plausibility of alternative options.

There is obviously plenty more out there, but please apply critical judgment when you consider the robustness of your source material.

The new Royal Commission is going to be following a similar process of judicious knowledge acquisition, albeit a most exhaustive one. Relish the journey.

Barry Brook is an Australian scientist. He is a professor and Chair of Environmental Sustainability at the University of Tasmania in the Faculty of Science, Engineering & Technology. He was formerly an ARC Future Fellow in the School of Earth and Environmental Sciences at the University of Adelaide, Australia, where he held the Sir Hubert Wilkins Chair of Climate Change from 2007 to 2014. He was also Director of Climate Science at the Environment Institute and co-ran the Global Ecology Lab.

Ben Heard is an independent environmental consultant. He holds a Masters of Corporate Environmental Sustainability Management from Monash University. He is currently undertaking doctoral studies at the University of Adelaide, examing pathways for optimal decarbonisation of Australian electricity using both nuclear and renewable sources.