Understanding the static structure factor

The structure factor \(S(q)\) is an important quantity for characterizing disordered systems of particles, like colloids. Its significance comes from the fact that it can be directly measured in a light scattering experiment and is related to other quantities that characterize a system's microscopic arrangement and inter-particle interactions. However, it's difficult to learn about it the context of disorder because it's primarily used in crystallography. Crystals are far from being disordered.

In this post, I'll explore the nature of the static structure factor, which is something of an average structure factor over many microscopic configurations of a material. A dynamic structure factor describes the statistics of a material in time as well as space.

The structure factor of a disordered material can be measured by illuminating the material with a beam of some type of radiation (usually X-rays, neutrons, or light). The choice of radiation depends on the material. It is also important that the material should not scatter the incident beam too strongly because the structure function is typically found in singly-scattered radiation (see the first Born approximation for a discussion about a related concept). If the material is multiply scattering, the information about the material's structure, which is largely carried by the singly scattered light, is washed out.

In a measurement, the sample is usually placed at the center of rotation of a long rotating arm. A detector for the radiation is placed at the opposite end of the arm. The arm is rotated about this axis and the intensity of the scattered radiation as determined by the detector is recorded as a function of the angle. This data set essentially contains the structure factor, but must be transformed and corrected for, accordingly.

First, the structure factor is usefully represented as a function of the scattering wave number, \(q\), and not as a function of the scattering angle. In optics, \(q\) is usually given by

\[ q = \frac{4 \pi n}{\lambda} \sin( \theta / 2) \]

where \(n\) is the refractive index of the background material (usually a solvent like water) and \(\lambda\) is the wavelength of the light. \(\theta\) is the scattering angle.

Additionally, the structure factor must be corrected for a large number of confounding factors, such as scattering from the sample cell and radiation frequency-dependent detectors. A classic paper that details all these corrections to find \(S(q)\) in a neutron scattering experiment is given here.

Once the structure factor is found in an experiment, it may be Fourier transformed numerically to give the radial distribution function, \(g(r)\) (see Ziman for the proper conditions for which this applies) of particles. This function gives the probability of finding a particle at a radial distance from another particle in the system. Many important thermodynamic properties are related to \(g(r)\). Importantly, the pair-wise interaction potential between any two particles is related to \(g(r)\), and the pair-wise interaction determines many macroscopic system properties.

The structure factor as \(q\) (or equivalently the scattering angle) goes to zero is also an important quantity in itself. \(S(0)\) is equal to the macroscopic density fluctuations of particles in the medium (see Ziman, Section 4.4, p. 130). But density fluctuations can be calculated from thermodynamics and leads to the isothermal compressibility of a material.

In the language of optics, which I'll stick to for the rest of this post, the density fluctuations would correspond to large regions of refractive index variations across the sample.

This leads to an interesting problem, the resolution of which reminds me of the fallibility in taking some models too literally: for a homogeneous and non-scattering optical material, like a very nice piece of glass, the value of the density fluctuations in the refractive index are essentially zero (this is true because the disorder in a glass is at a length scale that is much smaller than the wavelength of light). This means that \(S(0) = 0\). At the same time, the scattered intensity in the type of experiment measured above is directly proportional to the structure factor:

\[I(q) \sim S(q).\]

So, if I illuminate a nice piece of glass with a laser beam, and I know that \(S(0)\) is equal to zero, the above expression means that there should be no scattered intensity in the forward direction. But this is a silly conclusion, because when I do this experiment in the lab I see the laser beam shining straight through the glass! In other words, \(I(0)\) is not zero.

The problem is that this expression is for the scattered intensity. In random media, we often talk about the scattered light and the ballistic light. The latter of these two is not scattered but directly transmitted through the material as if the material were not there. So, even though no light is scattered into the forward direction, there is still the ballistic, unscattered beam, that is passing straight through the sample.

Most small angle light scattering experiments measure as close as they can to \(q=0\) and extrapolate to the structure function's limiting value. \(S(0)\) can't actually be measured. But, it's determination is important for materials with significant long-range order, such as those near a phase transition, because the small angles correspond to large distances, due to their inverse Fourier relationship.

One can also engineer a material to not transmit any light into the forward direction. To do this, \(S(q)\) must be zero AND there must be no ballistic light passing through the material. This can be achieved with a crystal that diffracts all the light into directions other than the forward direction, such as a blazed grating.

On a final note, the structure factor can sometimes be related to important material properties beyond the radial distribution function. Ziman says in section 4.1, pg. 126 that the direct correlation function (which measures interactions between pairs of particles) can be derived directly from the structure factor. This correlation function is related to the Percus-Yevick model for liquids.

Metamaterials for heat

There's a really cool experiment described in a recent PRL and summarized here about creating a metamaterial for cloaking objects from heat flow.

What connections to metamaterials for light can be drawn? The transport of heat is governed by a diffusion equation, which is very different from the wave equation and Maxwell's equations for governing light transport.

However, the diffusion equation can apply to light transport in disordered materials (see Ishimaru or van Rossum and Nieuwenhuizen). Is there some way, then to cloak objects in randomly scattering media from light?

The trouble with this thought is that one would have to add structure to a random material, and the only way I can think of doing this would be to create large-scale structures from a material with small-scale disorder.

A beautiful experiment on a nonequilibrium thermodynamic system

I just finished reading an impressive article from 2003 entitled "Observing Brownian motion in vibration-fluidized granular matter". In the article's beginning, the authors established a simple question: can linear response theory describe a nonequilibrium thermodynamic system? This question is important because systems that are not in thermodynamic equilibrium are both difficult to analyze and serve as appropriate models for most natural phenomena. However, very powerful mathematical tools exist for systems that are in equilibrium, so it would be very convenient if their mathematical formalism could be extended to nonequilibrium cases.

In particular, the authors explore whether a torsion oscillator driven by a "heat bath" of randomly vibrating glass beads can be described by the fluctuation-dissipation theorem (FDT). The FDT is arguably the hallmark of linear response theory and describes the return to equilibrium of a many-body system subjected to a small perturbation. (A small perturbation means that the response is linearly proportional to the perturbation.)

A canonical example where the FDT finds use is in describing the motion of ions in a fluid between two plates of a capacitor after a voltage difference has been applied to the plates. Prior to the voltage being applied, the motion of the ions is erratic and Brownian. A long time after the constant voltage is applied, they move with an average velocity that is proportional to the electric field between the plates, the proportionality constant being called the mobility. The FDT describes the very short times immediately after the field is applied. It also links the noise (the random movement of the ions in equilibrium) to the mobility of the ions.

Returning to the article, the authors find that the motion of the oscillator is described by the FDT so long as an "effective" temperature is adopted. Effective temperatures are very appealing as analytical tools for describing nonequilibrium systems because they are very, very simple modifications to the FDT. Simply replace T with T_eff and you're done.

As Cugliandolo points out, to be a good thermodynamic descriptor, an effective temperature should be measurable by a thermometer. I'm not sure what the thermometer is in this system, but I suspect that it's the torsion oscillator itself. Furthermore, she stresses that not all nonequilibrium phenomena are describable by effective temperatures. It seems that one requires coupling between fast processes and slower observables, among other requirements. The beauty of the Nature article is that the authors not only confirmed this point (which seems to currently be an area of contention), but did so convincingly by measuring the relevant quantities directly and under a number of different conditions.

I'm not sure whether the effective temperature is a universal property of nonequilibrium systems; I'm inclined to say it is not. Hopefully more experiments like this one will be done that may further elucidate the current maze of theoretical papers on the topic.

Notes on "Biological Physics," Part I

There is an article from 1999 in Reviews of Modern Physics entitled "Biological Physics." This review summarizes research during the twentieth century where "physics has influenced biology and where investigations on biological systems have led to new physical insights." The exchange of ideas between the two fields has not been of equal magnitude, the authors note. Many tools from physics have found their way into the biological sciences, though some biological systems have led to new physics, usually in the form of providing experimental testbeds for new physical theories. The article is primarily concerned with molecular biological physics.

The seven primary sections of the review are

1. The structures of biological systems
2. Complexity in biology
3. Dynamics, mostly within proteins
4. Reaction theory, where biology has provided testbeds for new physical theories
5. Bioenergetics
6. Forces
7. Single-molecule experiments.

Most of the interesting ideas I've found so far in the article are associated with the complexity and dynamics of biomolecules. Particularly, there is an idea known as the principle of minimal frustration. From the Wikipedia article,

"This principle says that nature has chosen amino acid sequences so that the folded state of the protein is very stable. In addition, the undesired interactions between amino acids along the folding pathway are reduced making the acquisition of the folded state a very fast process. Even though nature has reduced the level of frustration in proteins, some degree of it remains up to now as can be observed in the presence of local minima in the energy landscape of proteins."

This idea came from a theory of energy landscapes for proteins that was developed by Bryngelson and Wolynes. In language that I'm more familiar with, the potential energy of the molecules has some fractal-like structure, because from Section III in the article the authors state that

"The kinetic observations suggest that the energy landscape might have a hierarchical structure, arranged in a number of tiers, with different tiers having widely separated average barrier heights."

It seems like structural determination of proteins and other biomolecules has become something akin to bookkeeping. The tools exist and are refined to find static structures, like neutron scattering and NMR. Additionally, the energy landscape theory for protein folding seems to be mature at this point as well. So what open-ended questions still exist in biological physics? After reading up to section V, I've compiled the following grand problems in biological physics as I've interpreted them from this paper only:

1. "A synthesis that connects structure, energy landscape, dynamics, and function has not yet been achieved." This seems to suggest that there is some degree of incoherence between these individual fields of study, so ideas that link them together are required.
2. Biochemists can now synthesize their own proteins, but can they do this in a useful manner, for, say, molecular and microscopic engineering purposes?
3. Sensing and characterizing phase transitions, especially in glassy systems, could lead to better experimental investigations into protein folding.
4. "Understanding protein folding can improve the capability of predicting protein structure from sequence." Apparently there's a lot of DNA sequence information, but predicting what proteins come from it is nontrivial.

What is a good description for entropy?

"Insight into Entropy," by Daniel F. Styer, is a nice paper that appeared in the American Journal of Physics in 2000. In the paper, he argues for a qualitative explanation of entropy that involves two ideas: disorder and freedom.

Entropy as disorder is a common analogy given to students who are learning about thermodynamics, but Styer provides several arguments for why this qualitative description fails to adequately explain the idea. One such argument involves a glass of shredded and broken ice. Despite the fact that the ice has been shattered into many pieces, the entropy of the bowl of ice is less than that of an identical bowl filled with water. The water may seem to be more ordered because it is homogeneous, but it does not possess a lower entropy.

Styer's idea of entropy as freedom attempts to explain how systems can possess multiple classes of states (commonly known as macrostates) and how entropy limits the microscopic details of each class. In the game of poker, the probability of getting a royal flush is identical to any other five-card selection without replacement. However, the number of configurations that form a royal flush is extremely small, so the entropy of the class of hands forming a royal flush is low. This very low entropy class of poker hands restricts the possible configurations of the microstate—the description of what five cards are in one's hand—and completes the analogy with freedom. High entropy macrostates have greater freedom in choosing their microstate by having a larger number of microstates to choose from; low entropy macrostates (royal flushes, for example) have less freedom.

Styer does propose retaining the "entropy as disorder" description by suggesting that both the freedom and disorder analogies be presented simultaneously to negate any emotions commonly associated with either word. His example of such an analogy goes as "For macrostates of high entropy, the system has the freedom to choose one of a large number of microstates, and the bulk of such microstates are microscopically disordered."

Finally, on a different train of though: teaching ideas by analogy apparently must be done with sensitivity to the common emotions associated with a word. I've never considered this idea before, but will surely be mindful of it in the future.

A point about negative temperatures

Negative temperatures occur when the derivative of entropy with respect to a system's energy is negative. In other words, the entropy decreases with added energy. As Daniel Schroeder points out in his Introduction to Thermal Physics, this may only occur when the total energy that a system may take is limited, such as a two-state paramagnet. In other, more common systems, such as a gas in a container, the total energy that the system may absorb is practically unlimited. This is why negative temperatures are not observed.

I think a lot of confusion in learning thermodynamics is that the common sense notion of temperature is very different from its thermodynamic definition. Other ideas, such as force or work, do not contradict common sense quite so much and are more readily adopted.

Finally, Schroeder references this article for an experiment in which negative temperature was observed.

Momentum and position—it's all you need

I've had a bit of downtime recently which has led to thumbing through my undergrad QM book, Griffiths' Introduction to Quantum Mechanics, out of curiosity.

In the very first chapter he makes the statement "The fact is, all classical dynamical variables can be expressed in terms of position and momentum." To be honest, I never fully realized this as an undergrad, and if I did, it certainly did not leave such an impression on me to have remained in my memory.

Is this bit of knowledge a common oversight in the education of physics students, or something that simply went unnoticed by me? Furthermore, how important is it to the development of a student's understanding of physics?

Question everything

As you may know, my major field is optics, which concerns the study and application of light. Throughout my studies I've been constantly amazed that Maxwell's electromagnetic theory of light, which has been around since the late 1800's, still contains features that have not been settled or have been simply overlooked by scientists. One such artifact is the dissimilarity between Minkowski's and Abraham's descriptions of the momentum carried by an electromagnetic wave.

In a 2010 PRA Rapid Communication, Chaumet et al. expand on earlier work by Hinds and Barnett that examines the force on a dipole in a time-varying (i.e. pulsed) plane wave. This force is written completely as

 

where Pj is the dipole moment, E is the electric field, B is the magnetic induction, and ε is the Levi-Civita tensor. The first term in the sum relates to both the radiation pressure and the gradient force. The second term, according to Hinds and Barnett, is usually absent in laser trapping and cooling texts because it is proportional to the time-derivative of the Poynting vector, which is zero in common cooling setups. This term is responsible for repulsion of systems such as a two-level atom from the leading edge of the wave when the first term alone predicts an attraction.

Works like this make me wary of blindly using formulas when performing calculations since it reminds me that a theory may not be complete or its assumptions explicit when presented to a niche audience.

Absolutes vs. relatives

Though my advisor has stressed this for the entirety of my grad school career, I today finally appreciated the significance of relative measurements over absolute ones.

An absolute measurement is one in which a value is extracted from a data set that is physically important in a particular model. A relative measurement, on the other hand, is one that extracts the effect of varying a parameter amongst two or more data sets.

Model specific parameters are obtained from absolute measurements. Curve fitting is usually performed to find the values of parameters. Alternatively, relative measurements establish relationships between two variables. For example, the reading on a scale will increase proportionally with the mass added on top of it. From this observation, one can infer that weight is linear with mass. A constant, namely the acceleration of a particle due to gravity at the earth's surface, is needed to obtain the absolute value of the weight from a single measurement.

The practical problem with absolute measurements is that they require certain standards to have any significance. At the start of graduate school, I would often puzzle over why a parameter from a curve fitting routine would so often differ from theory. I would often vary different parameters in my calculation and struggle in vain to determine which independent quantity I had measured "wrongly." However, I failed to realize that each measurement was against some standard. A time is measured relative to an internal clock in a circuit; a length is measured relative to a ruler; mass is measured relative to a scale which was calibrated relative to some mass standard.

From the above it seems that the nature of measurement itself is a relative process, and as such a measurement can not be "wrong." If standards differ between two measurements, the measured variable will differ as well. And no one can say which measurement produced the "correct" value. Both conclusions are correct so long as they are logically consistent with how they are derived from the measurement.

I am aware of the definitions of the second and other fundamental quantities, but the definitions are simply agreed to based upon the precision of the measurement that produced the standard. They are arbitrary.

If I have to assert anything from this, it is that I value relative measurements above absolute measurements in scientific papers. Relative measurements reveal physical truths where as absolute ones tell us how well data fit into some theory.

I hope to write more on this in the future once my thoughts have more fully materialized.

A physical standard for time

I've now completed Chapter 2 of Cook's "The Observational Foundations of Physics," my current lunchtime reading. In this chapter, Cook describes a thought experiment in which a beam of caesium atoms is polarized by a strong magnetic field, then enters a region where a strong RF field is applied. Following the RF region, the beam passes through another magnetic field such that atoms whose magnetic dipole moments are not flipped by the RF field are deflected into a beam block. Those atoms that do undergo an electronic transition that is accompanied by a flip of the magnetic dipole moment reach a detector that reports the intensity of the beam. A feedback mechanism adjusts the frequency of the RF field so that the beam intensity at the detector is maximized; in this way, a quantum standard of time is established through the frequency of the RF field that maximizes the atomic beam intensity (you may note the similarity to the Stern-Gerlach appartus).

Cook then proceeds to argue for his thesis, namely that the experiments and observations that are available to us dictate the form of our physical theories. He starts first with the theory. The time-evolution of the caesium atoms is described by the Schroedinger equation. This equation contains a first order time derivative which is a consequence of the wavefunction containing all information about the system at any one point in time. If only one initial condition is required to establish the wavefunction, then it must be first order in time (this is in contrast to the wave equation which is second order in time and whose solution requires an initial condition on the wavefunction and its derivative).

Cook next mathematically defines the operations of the experiment described above, postulating that the two magnetic states of the atoms are described by stationary states of a wavefunction. Using only mathematical arguments derived from the nature of the experiment, he obtains the form of the equations governing the time evolution of the system; the wavefunction is affected by a first order time derivative. This suggests that how we perform experiments determines the form of our theories. The time standard need not be quantum in nature as he repeats the argument for a classical, mechanical oscillator. Again he stresses that once the time standard is set, it is meaningless to ask whether or not its period remains invariant with time, since the standard defines time itself. It is recognized that differences between the same apparatus for establishing the standard exist when the apparatus are spatially separated due to the geometry of spacetime.

These are some of the thoughts I had while reading this chapter:

1. Many times physical theories are developed first and then experiments follow that verify their predictions. Does this fact weaken Cook's argument that experiments shape our theories? If the purpose of theory is to predict experimental outcomes, then why argue for the reverse? Which came first, the chicken or the egg?
2. Cook was careful to explain that his arguments are based on a physical world that is independent of a subjective observer. Still, I wonder how the human perception of time can be reconciled with these arguments. As stated earlier, it makes no sense to ask whether or not the time standard is invariant within the context of observation and theory. But a human can perceive large changes in the period of a slow mechanical oscillator. What is it that acts as an internal time standard for a subjective observer and can it be related to the physical standard?
3. Cook only obtains the form of the equations of motion for the systems he describes. On the other hand, the theories give meaning to his unspecified parameters, such as energy and the unit of electronic charge. What determines how these mental concepts are developed? Energy is a relatively easy concept to understand. Was this why the fathers of thermodynamics used it as a core concept in physics as opposed to some other mental construct?

Notes from "The Observational Foundations of Physics"

Section 1.3, Measurements and Standards, is a continuation of the setup for the arguments for Cook's thesis on how measurement affects the logical structure of physics. First, Cook states that the equations of physics are simply relationships between physical states or quantities. These relationships are congruent to the relationships between observations. I am a bit unclear as to what congruent means here, but aside from that the setup so far seems fairly obvious.

He continues onto a more lengthy discussion of the role that standards play in measurement. Every measurement consists of comparing some quantity to a standard quantity. When measuring the length of an object, for example, one simply compares the object's length to the length of a ruler (the standard). Our system of standards plays a significant role in shaping the nature of physical theories.

What was very surprising is that, traditionally, standards for all physical measurements can be derived from four independent standards: length, mass, time, and current. These standards have since been replaced by other physical constants and quantities, but the number of independent standards has remained the same. For example, length is measured as a ratio between the speed of light in free space to a unit of time, which is derived from a standard of frequency from a certain atomic process.

The standard of voltage comes from the standard of frequency and the Josephson effect with help from another fundamental constant, the ratio of Planck's constant to the unit of electronic charge. Mass currently (as of the book's publishing) escapes a relation to the standard of frequency, but it's conceivable that it could be related to energy, voltage, and current through the quantum Hall effect.

The shift from mechanical standards to electronic and quantum standards has greatly increased the precision with which we can measure physical quantities. It has also changed the nature of our physical theories, Cook claims.

Lunchtime reading: The Observational Foundations of Physics

I have started reading "The Observational Foundations of Physics" by Sir Allan Cook during my lunch breaks. The book's purpose, as Cook states in the first sentence of Section 1.1, "is to attempt to unravel some ways in which the practice of physics determines the form and content of physics and physical theory." In other words, Cook wishes to understand how the practices found in physics affect physical theories and the practices themselves. It is as if there existed a feedback loop such that performing experiments changed not simply the theory used to describe a phenomenon but the nature of theory itself.

Further in Section 1.1, he poses these questions that are central to his analysis:

1. "Why should physics be so effective, and what does that tell us about the world of physics and our ways of gaining knowledge of it?"
2. "Is there a real world that exists independently of whether I or anyone else is looking at it, or are all the ideas I have about a world external to me just the construction of my mind?"

He defers a thorough answer to the second question until the end of the book, but does offer that he believes that most physicists, while working at the bench or on a computer, act as if an external world existed independent of their attention.

Section 1.2 deals with observations and sets many of the premises of his arguments. Observation and experiment are decided to be equivalent. Observations also consist of two aspects: objective and subjective. Of the subjective aspect, only the communal nature of observation is of consequence to his arguments. Science is a social construct and scientists hold great influence over each other such that the act of observation is never truly independent of people other than the experimenter.

Cook goes to some length to explain that physics is empirical, "with observation primary and theory secondary," but he concedes that rarely can observation be performed without some theory underlying the act of observing. He gives the example of reading a voltage from a digital multimeter. The direct observation is of figures on a LCD readout, a consequence of numerous electronic circuits that respond to potential differences between two probes and relates to the potential energy difference of electrons between two points in a circuit. Of course, electrons are theoretical constructs. The theories underlying an observation can in some ways assure an experimenter that the results are telling us something of the real world and not subject to some extraneous errors or misinterpretations. For simplicity, an observation is defined as the operations that lead to a measurement and result in  "raw data." The data is considered "raw" regardless of the complexity of the measurement.

Finally, theories are models of observations, not a model of the real world itself. "I take a theory to be a mathematical realisation of an abstract system that has properties corresponding to those of a set of observations... It is in that sense that I take a theory to be a model of the world of observations, with the implication that there is a more fundamental correspondence than just giving the right answers..." Theory is an abstraction of the real world, not vice versa.