My best blog posts of 2025?

 Best wishes for the New Year!

Here is a list of the posts that I wrote last year that I hope get the most interest.

My review article on emergence. I wrote posts about emergence in a range of systems: thermodynamics, quantum gravity, economics,... They were drafts of sections for my review article. It may be best to just read the article.

Why is the state of universities such an emotional issue for me?

Undergraduates need to learn about the Ising model

I wrote a series of posts on so-called "spin-crossover" compounds. Here are two: Spin crossover is a misnomer, and Elastic interactions and complex patterns in binary systems

2025 Nobel Prize in Physics: Macroscopic quantum effects

As always, I welcome comments, feedback, and suggestions for new posts.

Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.

Elastic frustration in molecular crystals

Crystals of large molecules exhibit diverse structures. In other words, the geometric arrangements of the molecules relative to one another are complex. Given a specific molecule, theoretically predicting its crystal structure is a challenge and is the basis of a competition.

One of the reasons the structures are rich and the theoretical problem is so challenging is that there are typically many different interactions between different molecules, including electrostatic, hydrogen bonding, pi-pi,...

Another challenge is to understand the elastic and plastic properties of the crystals.

Some of my UQ colleagues recently published a paper that highlights some of the complexity.

Origins of elasticity in molecular materials

Amy J. Thompson, Bowie S. K. Chong, Elise P. Kenny, Jack D. Evans, Joshua A. Powell, Mark A. Spackman, John C. McMurtrie, Benjamin J. Powell, and Jack K. Clegg

They used calculations based on Density Functional Theory (DFT) to separate the contributions to the elasticity from the different interactions between the molecules. The figure below shows the three dominant interactions in the family of crystals that they consider.

The figure below shows the energy of interaction between a pair of molecules for the different interactions.

Note the purple vertical bar, which is the value of the coordinate in the equilibrium geometry of the whole crystal. The width of the bar represents variations in both lengths that occur in typical elastic experiments.

What is striking to me is the large difference between the positions of the potential minima for the individual interactions and the minima for the combined interactions.

This is an example of frustration: it is not possible to simultaneously minimise the energy of all the individual pairwise interactions. They are competing with one another.

A toy model illustrates the essential physics. I came up with this model partly motivated by similar physics that occurs in "spin-crossover" materials.

The upper (lower) spring has equilibrium length a (b) and spring constant k (k'). In the harmonic approximation, the total elastic energy is

The equilibrium separation of the two molecules is given by

which is intermediate between a + 2R and b. This illustrates the elastic frustration. Neither of the springs (bonds) is at its optimum length.

The system is stable provided that k + k' is positive. Thus, it is not necessary that both k and k' be positive. The possibility that one of the k's is negative is relevant to reality. Thompson et al. showed that the individual molecular interaction energies are described by Morse potentials. If one is far enough from the minimum of the potential, the local curvature can be negative. 

Multi-step spin-state transitions in organometallics and frustrated antiferromagnetic Ising models

In previous posts, I discussed how "spin-crossover" material is a misnomer because many of these materials do not undergo crossovers but phase transitions due to collective effects. Furthermore, they exhibit rich behaviours, including hysteresis, incomplete transitions, and multiple-step transitions. Ising models can capture some of these effects.

Here, I discuss how an antiferromagnetic Ising model with frustrated interactions can give multi-step transitions. This has been studied previously by Paez-Espejo, Sy and Boukheddaden, and my UQ colleagues Jace Cruddas and Ben Powell. In their case, they start with a lattice "balls and spring" model and derive Ising models with an infinite-range ferromagnetic interaction and short-range antiferromagnetic interactions. They show that when the range of these interactions (and thus the frustration) is increased, more and more steps are observed.

Here, I do something simpler to illustrate some key physics and some subtleties and cautions.

fcc lattice

Consider the antiferromagnetic Ising model on the face-centred-cubic lattice in a magnetic field. 

[Historical trivia: the model was studied by William Shockley back in 1938, in the context of understanding alloys of gold and copper.]

The picture below shows a tetrahedron of four nearest neighbours in the fcc lattice.

Even with just nearest-neighbour interactions, the lattice is frustrated. On a tetrahedron, you cannot satisfy all six AFM interactions. Four bonds are satisfied, and two are unsatisfied.

The phase diagram of the model was studied using Monte Carlo by Kammerer et al. in 1996. It is shown above as a function of temperature and field. All the transition lines are (weakly) first-order.

The AB phase has AFM order within the [100] planes. It has an equal number of up and down spins.

The A3B phase has alternating FM and AFM order between neighbouring planes. Thus, 3/4 of the spins have the same direction as the magnetic field.

The stability of these ordered states is subtle. At zero temperature, both the AB and A3B states are massively degenerate. For a system of 4 x L^3 spins, there are 3 x 2^2L AB states, and 6 x 2^L   A3B states. At finite temperature, the system exhibits “order by disorder”.

On the phase diagram, I have shown three straight lines (blue, red, and dashed-black) representing a temperature sweep for three different spin-crossover systems. The "field" is given by h=1/2(Delta H - T Delta S). In the lower panel, I have shown the temperature dependence of the High Spin (HS) population for the three different systems. For clarity, I have not shown the effects of the hysteresis associated with the first-order transitions.

If Delta H is smaller than the values shown in the figure, then at low temperatures, the spin-crossover system will never reach the complete low-spin state.

Main points.

Multiple steps are possible even in a simple model. This is because frustration stabilises new phases in a magnetic field. Similar phenomena occur in other frustrated models, such as the triangular lattice, the J1-J2 model on a chain or a square lattice.

The number of steps may change depending on Delta S. This is because a temperature sweep traverses the field-temperature phase diagram asymmetrically.

Caution.

Fluctuations matter.

The mean-field theory phase diagram was studied by Beath and Ryan. Their phase diagram is below. Clearly, there are significant qualitative differences, particularly in the stability of the A3B phase.

The transition temperature at zero field is 3.5 J, compared to the value of 1.4J from Monte Carlo.

Monte Carlo simulations may be fraught.

Because of the many competing ordered states associated with frustration, Kammerer et al. note that “in a Monte Carlo simulation one needs unusually large systems in order observe the correct asymptotic behaviour, and that the effect gets worse with decreasing temperature because of the proximity of the phase transition to the less ordered phase at T=0”. 

Open questions.

The example above hints at what the essential physics may be how frustrated Ising models may capture it. However, to definitively establish the connection with real materials, several issues need to be resolved.

1. Show definitively how elastic interactions can produce the necessary Ising interactions. In particular, derive a formula for the interactions in terms of elastic properties of the high-spin and low-spin states. How do their structural differences, and the associated bond stretches or compressions, affect the elastic energy? What is the magnitude, range, and direction of the interactions?

[n.b. Different authors have different expressions for the Ising interactions for a range of toy models, using a range of approximations. It also needs to be done for a general atomic "force field".]

2. For specific materials, calculate the Ising interactions from a DFT-based method. Then show that the relevant Ising model does produce the steps and hysteresis observed experimentally.

"Ferromagnetic" Ising models for spin-state transitions in organometallics

In recent posts, I discussed how "spin crossover" is a misnomer for the plethora of organometallic compounds that undergo spin-state phase transitions (abrupt, first-order, hysteretic, multi-step,...)

In theory development, it is best to start with the simplest possible model and then gradually add new features to the model until (hopefully) arriving at a minimal model that can describe (almost) everything. Hence, I described how the two-state model can describe spin crossover. An Ising "spin" has values of +1 or -1, corresponding to high spin (HS) and low spin (LS) states. The "magnetic" field is half of the difference in Gibbs free energy between the two states. 

The model predicts equal numbers of HS and LS at a temperature

The two-state model is modified by adding Ising-type interactions between the “spins” (molecules). The Hamiltonian is then of the form

 The temperature dependence in the field arises because this is an effective Hamiltonian.

The Ising-type interactions are due to elastic effects. The spin-state transition in the iron atom leads to changes in the Fe-N bond lengths (an increase of about 10 per cent in going from LS to HS), changing the size of the metal-ligand (ML6 ) complex. This affects the interactions (ionic, pi-pi, H-bond, van der Waals) between the complexes. The volume of the ML6 complex changes by about 30 per cent, but typically the volume of the crystal unit cell changes by only a few per cent. The associated relaxation energies are related to the J’s. Calculating them is non-trivial and will be discussed elsewhere. There are many competing and contradictory models for the elastic origin of the J’s.

In this post, I only consider nearest-neighbour ferromagnetic interactions. Later, I will consider antiferromagnetic interactions and further-neighbour interactions that lead to frustration. 

Slichter-Drickamer model

This model was introduced in 1972 is beloved by experimentalists, especially chemists, because it provides a simple analytic formula that can be fit to experimental data.

The system is assumed to be a thermodynamic mixture of HS and LS. x=n_HS(T) is the fraction of HS. The Gibbs free energy is given by

This is minimised as a function of x to give the temperature dependence of the HS population.

The model is a natural extension of the two-state model, by adding a single parameter, Gamma, which is sometimes referred to as the cooperativity parameter.

The model is equivalent to the mean-field treatment of a ferromagnetic Ising model, with Gamma=2zJ, where z is the number of nearest neighbours. Some chemists do not seem to be aware of this connection to Ising. The model is also identical to the theory of binary mixtures, such as discussed in Thermal Physics by Schroeder, Section 5.4.

Successes of the model.

good quantitative agreement with experiments on many materials.

a first-order transition with hysteresis for T_1/2 < Tc =z J.

a steep and continuous (abrupt) transition for T_1/2 slightly larger than Tc.

Values of Gamma are in the range 1-10 kJ/mol. Corresponding vaules of J are in the range 10-200 K, depending on what value of z is assumed.

Weaknesses of the model.

It cannot explain multi-step transitions.

Mean-field theory is quantitatively, and sometimes qualitatively, wrong, especially in one and two dimensions.

The description of hysteresis is an artefact of the mean-field theory, as discussed below.

Figure. Phase diagram of a ferromagnetic Ising model in a magnetic field. (Fig. 8.7.1, Chaikin and Lubensky). Vertical axis is the magnetic field, and the horizontal axis is temperature. Tc denotes the critical temperature, and the double-line denotes a first-order phase transition between paramagnetic phases where the magnetisation is parallel to the direction of the applied field.

Curves show the free energy as a function of the order parameter (magnetisation) in mean-field theory. The dashed lines are the lines of metastability deduced from these free-energy curves. Inside these lines, the free energy has two minima: the equilibrium one and a metastable one. The lines are sometimes referred to as spinodal curves.

The consequences of the metastability for a field sweep at constant temperature are shown in the Figure below, taken from Banerjee and Bar.

How does this relate to thermally induced spin-state transitions?

Consider the phase diagram shown above of a ferromagnetic Ising model in a magnetic field. The red and blue lines correspond to temperature scans for two SCO materials that have different values of the parameters Delta H and DeltaS.

The occurrence of qualitatively different behaviour is determined by where the lines intercept the temperature and field axes, i.e. the values of T_1/2 /J and Delta H/J. If the former is larger than Tc/J, as it is for the blue line, then no phase transition is observed. 

The parameter Delta H/J determines whether at low temperatures, the complete HS state is formed.

The figure below is a sketch of the temperature dependence of the population of HS for the red and blue cases.

Note that because of the non-zero slope of the red line, the temperature  T_1/2 is not the average of the temperatures at which the transition occurs on the up and down temperature sweeps.

Deconstructing hysteresis.

The physical picture above of metastability is an artefact (oversimplification) of mean-field theory. It predicts that an infinite system would take an infinite time to reach the equilibrium state from the metastable state.

(Aside: In the context of the corresponding discrete-choice models in economics, this has important and amusing consequences, as discussed by Bouchaud.)

In reality, the transition to the equilibrium state can occur via nucleation of finite domains or in some regimes via a perturbation with a non-zero wavevector. This is discussed in detail by Chaikin and Lubensky, chapter 4.

The consequence of this “metastability” for a first-order transition in an SCO system is that the width of the hysteresis region (in temperature) may depend on the rate at which the temperature is swept and whether the system is allowed to relax before the magnetisation (fraction of HS) is measured at any temperature. Emprically, this is observed and has been highlighted by Brooker, albeit without reference to the theoretical subtleties I am highlighting here. She points out that up to 2014, chemists seemed to have been oblivious to these issues and reported results without testing whether their observations depended on the sweep rate or whether they waited for relaxation.

(Aside. The dynamics are different for conserved and non-conserved order parameters. In a binary liquid mixture, the order parameter is conserved, i.e., the number of A and B atoms is fixed. In an SCO material, the number of HS and LS is not conserved.)

In the next post, I will discuss how an antiferromagnetic Ising model can give a two-step transition and models with frustrated interactions can give multi-step transitions.

The two-state model for spin crossover in organometallics

Previously, I discussed how spin-crossover is a misnomer for organometallic compounds and proposed that an effective Hamiltonian to describe the rich states and phase transitions is an Ising model in "magnetic field".

I introduce the two-state model that defines the model without the Ising interactions. To save me time on formatting in HTML, here is a pdf file that describes the model and what comparisons with experimental data (such as that below) tells us.

Future posts will consider how elastic interactions produce the Ising interaction and how frustrated interactions can produce multi-step transitions.

Spin crossover is a misnomer

There are hundreds of organometallic compounds that are classified as spin-crossover compounds. As the temperature is varied the average spin per molecule can undergo a transition between low-spin and high-spin states.

The figure below shows several classes of transitions that have been observed. The vertical axis represents the fraction of molecules in the high-spin state, and the horizontal axis represents temperature.

Taken from Gutlich, Garcia, and Spiering and reproduced by Nicolazzi and Bousseksou

a) A smooth crossover. At the temperature T_{1/2} there are equal numbers of high and low spins.

b) There is sharp transition with the curve having a very large slope at T_{1/2}.

c) There is a discontinuous change in the spin fraction at the transition temperature, the value of which depends on whether the temperature is increasing or decreasing, i.e., there is hysteresis. The discontinuity and hysteresis are characteristic of a first-order phase transition.

d) There is a step in the curve when the high-spin fraction is close to 0.5. This is known as a two-step transition.

e) Although a crossover occurs, the system never contains only low- or high-spins.

But, there is more. Over the past decade, multiple-step transitions have been observed. An example of a four-step transition is below.

The figure is taken from a paper by Clements, Price, Neville, and Kepert.

Hysteresis is present and is larger at lower temperatures.

In a few cases of multiple-step transitions on the down-temperature sweep, the first step is missing compared to the up-temperature step.

Given the diverse behaviour described above, including sharp transitions and first-order phase transitions, spin "crossover" is a misnomer.

More importantly, given the chemical and structural complexity materials involved, is there a simple model effective Hamiltonian that can capture all this diverse behaviour?

Yes. An Ising model in a field. A preliminary discussion is here. I hope to discuss this in future posts. But first I need to introduce the simple two-state model and show what it can and cannot explain.

Emergence talk at University of the Philippines

Today I am giving a seminar on emergence at the University of Philippines Los Banos, in the Institute of Mathematical Sciences and Physics.

Here are my slides.

A model for light-induced spin-state trapping in spin-crossover materials

 An important challenge required to understand the physical properties of materials that are chemically and structurally complex is to ascertain which microscopic details are important. A related question is at what scale (length, number of atoms, energy) models should be developed.

A specific example is understanding the magnetic properties and state transitions of spin-crossover materials. This is difficult for equilibrium properties, let alone for non-equilibrium properties such as Light-Induced Excited Spin-State Trapping (LIESST). At low temperatures irradiation with light can induce a transition from the equilibrium low-spin state to a long-lived high-spin state, which is only an equilibrium state at higher temperatures. (LIESST gets a lot of attention because of the potential to make optical memories for information storage).

Some of my UQ colleagues recently published a nice paper that elucidates some of the key physics with the proposal and analysis of a (relatively simple) model that captures many details of the experimental data.

Toward High-Temperature Light-Induced Spin-State Trapping in Spin-Crossover Materials: The Interplay of Collective and Molecular Effects

M. Nadeem, Jace Cruddas, Gian Ruzzi, and Benjamin J. Powell

Hiding a Devil's staircase in spin-crossover materials

 Collaborators and I just posted a preprint.

Hidden Devil's staircase in a two-dimensional elastic model of spin crossover materials  Gian Ruzzi, Jace Cruddas, Ross H. McKenzie, Ben J. Powell

Condensed matter physicists study diverse states of matter (crystals, antiferromagnets, liquid crystals, superfluids, …), the types of ordering and symmetry associated with each state, and the transitions that can occur between different states when external parameters such as temperature and magnetic field are varied. A challenge for theorists is to create simple models that can describe the orderings in broad classes of materials, taking into account properties at the microscopic (atomic) level. Spin-crossover materials are a broad class of materials that are chemically and structurally complex: composed of transition metal ions surrounded by large molecules. The quantum spin (magnetic) state of the metal ion can change as the temperature is varied, leading to changes in magnetism, colour, and structure. A diversity of spatially periodic orderings of the spin states has been observed and transitions between these ordered states can be smooth, or discontinuous with hysteresis. Chemists love these materials because of their chemical and structural complexity and their potential technological applications, including as switchable magnetic memories. 

We present a general procedure to derive a simple lattice Hamiltonian that can describe the diversity of spin-state orderings and transitions, based on physically realistic microscopic interactions. We show that two classes of models that have been considered previously, Ising models and ball-and-spring elastic models are equivalent. Furthermore, our work shows that the Ising interactions arise from elastic interactions, whereas previously the Ising interactions were just treated as empirical parameters with no clear physical origin. These interactions compete with one another (they are frustrated). At zero temperature there appears to be an infinite number of possible spin-state orderings and transitions between them are described by a Devil’s staircase. At finite temperature, some of this structure is washed out, but there are first-order transitions between some of these ordered states. This complexity is similar to that found in spin-crossover materials, and here is produced by a theoretical model with just a single elastic parameter, the ratio of the bulk and shear moduli. 

The physical insight and methodology advances in our work provide a foundation for more detailed microscopic descriptions (such as using electronic structure methods) and the establishment of structure-property relations that may guide the development of new materials tailored for specific technological applications.

 

This supersedes an earlier preprint.  (Aside: given the new paper is so different we thought this should be a new submission to the arxiv, but they did not agree).

We got so much helpful feedback on that work that we did more calculations and discovered the hidden Devil's staircase.

We welcome any comments.

Isotope effects in spin crossover materials

A range of isotope substitution experiments have been performed on spin-crossover materials. 

Just like for other systems such as superconductors their interpretation is subtle.

The first studies are reviewed in Section 2.3.5 of this review article.

Isotopic exchange was investigated for a tris(picolylamine)iron(II) system which exhibits a two-step spin transition. Results are shown in the figure below. Significant changes in the spin-state transition curve were observed only when the isotopic substitution (H/D and 14N/15N) was made for atoms directly involved in the hydrogen-bonding network that connects the spin-crossover molecules. For example, with C2H5OD/ND2 the crossover temperature was shifted to higher temperatures by about 15 K and the middle step was no longer present. 

I would not have expected such a large effect given the chemical complexity of these systems and that the H atoms are not immediately bonded to the iron atoms which undergo the spin-state transition.

I now mention two other studies. They are particularly helpful because they also measured how the enthalpy and entropy change associated with the spin-state transtion changed with isotopic substitution.

Weber et al. studied the iron(II) spin-crossover complex [FeL1(HIm)2] and the deuterium-substituted [FeL1(DIm)2] where Him is (not a man but) imidazole. Both exhibit a single-step transition with hysteresis. H/D exchange decreased both the transition temperature and the hysteresis width by a few K. Deuteration decreased the value of the enthalpy and entropy differences between the low spin and high spin states (determined from differential scanning calorimetry) by about twenty and ten percent, respectively. (See Table 2 in the paper). They estimated an interaction parameter J = 560 K, indicating strong intermolecular interactions, which they attributed to a hydrogen bond. They reference some earlier studies showing how the magnitude of the ligand field in a transition metal complex can be modified by hydrogen bonds involving the complex. 

Very recently, Jornet-Mollá et al. studied the iron(ii) salt [Fe(bpp)2](isonicNO)2·HisonicNO·5H2O, which with decreasing temperature undergoes a transition at 162 K. There is a width of about 5 K, associated with hysteresis. With deuteration, the transition temperature decreases to 155 K, the width increases to 7 K, and the enthalpy and entropy differences both increase by about fifteen percent. “Annealing the compound at lower temperatures results in a 100% LS phase that differs from the initial HS phase in the formation of a hydrogen bond (HB) between two water molecules (O4W and O5W) of crystallisation. Neutron crystallography experiments have also evidenced a proton displacement inside a short strong hydrogen bond (SSHB) between two isonicNO anions.” 

I am particularly interested in this because of previous work I have done on strong hydrogen bonds.

Again I am surprised at the magnitude of these effects because the zero-point energy associated with the relevant H atoms is only a small fraction of the total zero-point energy and the entropy contribution from vibrations.

I now start a preliminary discussion of how these experiments might be interpreted in terms of an Ising model picture, such as in a recent preprint. The Hamiltonian is

 where the pseudospin sigma=+1/-1 corresponds to high spin and low spin states.

The crossover temperature is independent of the Ising interactions J's and given by 

Thus, it is a property of the individual molecules and not directly dependent on the intermolecular interactions.

The presence of hysteresis and the difference between the transition temperature for increasing and decreasing temperature sweeps are determined by the magnitude of the Ising interactions. 

Our results in Appendix A of the preprint imply that there should be no dynamical isotope effects on the J’s, i.e., provided other parameters such as structural details and bond lengths do not change with isotope substitution.

This does not rule out changes in the crossover temperature. Both the enthalpy and entropy differences can change with isotope substitution (as is observed). The former due to changes in zero-point energies, and the latter due to changes in the vibrational contribution to the entropy change. 

Interpretation of isotope effects can be subtle

 Isotopic substitution has provided significant insights into molecular and solid-state physics. This involves the substitution of particular atoms in a compound by the same chemical element with a different nuclear mass (i.e. a nuclear isotope). An example is hydrogen/deuterium substitution which has shown the significant role that quantum nuclear motion can play in hydrogen bonding, particularly in strong hydrogen bonds. Of particular relevance to the discussion below is that isotopic substitution does not only change vibrational frequencies but can also change bond lengths. 

 A key piece of evidence on the road to the BCS theory of superconductivity in 1957 was the observation of an isotope effect. In 1950 a shift in the transition temperature of mercury was observed, suggesting that superconductivity resulted from electron-phonon interactions, as argued by Frohlich that same year. In particular, the magnitude of the shift was consistent with theoretical work by Herbert Frohlich. (Whether he predicted or postdicted the observed effect is a matter of debate, as discussed by Jorge Hirsch.) BCS theory gives that $\Delta T_c/T_c = - {1/over 2} \Delta M /M$, which arises from the fact that phonon frequencies scale with $1/\sqrt{M}$, consistent with the mercury experiments. 

However, in the 1960s there were many observations of “anomalous” isotope effects, particularly in transition metals, that were inconsistent with the prefactor in this equation. These anomalies were resolved by going beyond the BCS theory and allowing for strong-coupling effects. Following the discovery of cuprate superconductors in 1986, isotope effects were observed in some cuprates. However, the consensus now is that these observations do not support an electron-phonon mechanism for superconductivity but rather are due to structural changes due to the isotope substitution. For example, isotopic substitution changes the zero-point energy, and that can alter the unit cell volume and the hopping parameter t in a Hubbard model. 

This illustrates that there are subtleties in interpreting isotope experiments. This is because there are both structural and dynamical isotope effects. Changes in isotope can lead to changes in structure, such as bond lengths or lattice constants, and even in changes in crystal symmetry. These structural changes arise because the equilibrium structure of the system is that which minimises the total energy of the system. The contribution to this energy from the zero-point energy of the atomic vibrations changes with isotope substitution and with bond lengths. Dynamical effects are those that involve exchange of phonons such as in superconductivity. 

I am not sure how to sharpen this structural/dynamical or static/dynamical distinction. Or is it secondary and primary effects?

In the next post, I will discuss observations of isotope effects in spin-crossover materials and how that relates to recenttheoretical work with my collaborators.

The Devil is not in the details

Condensed matter physics aims to understand different and describe states of matter. Each state (phase) is associated with a particular type of order and phase diagrams encode the external parameters (temperature, pressure, chemical composition, magnetic field,...) that are necessary for each of the possible orderings to be stable.

Phase diagrams of even the simplest systems, such as binary alloys, can be quite rich, with many competing phases. Nevertheless, in many cases, simple microscopic models, with just a few degrees of freedom and a few parameters can describe these rich diagrams. Often a key is for the model to involve competing interactions, which can arise from different forces or from geometric frustration. Earlier, I wrote about how an Ising model on a hexagonal close-packed lattice could describe the plethora of distinct orderings that are observed in binary alloys. There the orderings are defined by the ordering wave vector and the composition of the unit cell. Another example occurs in spin-crossover materials and is described in recent work by my UQ colleagues.

Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Jace Cruddas and Ben J. Powell

A 2016 chemistry paper was First Step Towards a Devil's Staircase in Spin‐Crossover Materials

Another rich example is where there are two different spatial scales associated with interactions between the components of the system. In a lattice system, this can lead to ordering wavevectors that are incommensurate with the lattice. About forty years Per Bak wrote a nice series of papers that explored this situation.

Ising model with solitons, phasons, and "the devil's staircase", Per Bak and J. von Boehm 

This is an elegant and clear study of a simple Ising model in three dimensions with a frustrating interaction J_2 in the vertical direction. This is an example of an ANNNI model.

A mean-field theory for a state with a sinusoidally varying magnetisation (with wavevector 2 pi q) gives the phase diagram on the right above. The point P is a Lifshitz point, a tricritical point where the incommensurate (modulated) phase becomes stable. 

[Aside: in fermion models, a Lifshitz point is quite different: where the volume of the Fermi surface vanishes].

However, there is much more to the story. The authors then construct a mean-field theory where the magnetisation is commensurate, allowing for large unit cells. This leads to the phase diagram below.

The fractions p/q correspond to states with wavevector 2 pi p/q.

For example, the 1/4 state is below.

What does this have to do with a "devil's staircase"?

If for fixed J_2/J_1 the wavevector is plotted as a function of temperature it has steps of varying size and width.

A paper by Bruinsma and Bak considers an AFM Ising chain with 1/n^2 interaction, in a magnetic field, at zero temperature.

Note: In the figure below q is NOT the wavevector but rather the ratio of up to down spins. 

The magnetisation vs field curve has a fractal structure.

Bak also wrote a Physics Today article (that compares the phenomena to frequency mode locking) and a general review that includes examples of experimental realisations, ranging from magnets to atoms on surfaces.

This illustrates an important point that is often made in complexity science. Simple rules (theoretical models) can produce complex behaviour. 

This also illustrates characteristics of emergent phenomena. A wide range of physical systems can exhibit the same phenomena. Many of the details do not matter.

The devil is not in the details.

Minimal effective Hamiltonian for spin-crossover materials

My colleagues and I just put a preprint on the arXiv. I am particularly proud of it.  As always, comments would be appreciated.

Equivalence of elastic and Ising models for spin-crossover materials

Gian Ruzzi, Jace Cruddas, Ross H. McKenzie, Ben J. Powell 

Spin crossover (SCO) materials are reversible molecular switches; and occur in a wide range of near octahedral transition metal complexes and frameworks with d4−d7 electron configurations. SCO systems present collective spin-state phase transitions that show hysteresis, multistep transitions, gradual transitions, and anti-ferroelastic phases. Ising models have often been employed to model these behaviors, as they are far easier to solve than more realistic elastic models. However, previously Ising models have required phenomenological parameters that do not have a clear physical origin. 

We present an exact mapping from an elastic model of balls and springs to the Ising model. The resulting Ising coupling constants arise only from the elastic interactions, and are independent of the lattice dynamics, i.e., there are no isotope effects. The elastic interactions, and hence the Ising coupling constants can be determined from the measurements of the bulk and shear moduli. The Ising coupling constants can be frustrated, their signs can be negative or positive, and their magnitude agrees well with previous estimates from fits of experimental spin-transition curves. The Ising coupling constants follow a power law for large separations between metal centers, in particular an inverse square law for the square lattice. For the square lattice with nearest neighbor elastic interactions this model predicts a diverse range of spin-state orders including multistep transitions.

 

The quantum physics of life in red and green

Life is truly amazing!
Life is beautiful!
...and it involves quantum many-body physics...

There is a beautiful (short) review
Heme: From quantum spin crossover to oxygen manager of life 
Kasper Kepp

The article involves a plethora of topics that I have discussed before on this blog. I have included relevant links.

Kepp starts with the unique (chemically fine-tuned) properties of both iron and porphyrin that enable them to play a central role in two of the most important processes in life: respiration and photosynthesis. He has a beautiful paragraph (perhaps in the style of Roald Hoffmann):

Such ligand-field transitions of iron in porphyrin were familiar to our ancestors as the characteristic red color of blood that largely defines the human psychological and cultural connotations of the color representing courage, war, danger, and suffering. 

Incidentally, pi-pi* transitions within the porphyrin-derived chlorophylls are also responsible for the green color of plants, associated with nature, life and hope, so the reader may perhaps agree that porphyrin has had vast (but alas! rarely appreciated) cultural consequences.

The oxygen molecule is a spin triplet.
Iron(II) porphyrin is in a triplet spin state (S=1). The Fe(II) is a d6 configuration in a D_4h crystal field.
When they bind together the ground state is a spin-singlet.

There are two fundamental quantum chemistry questions that are discussed.

1. What is the electronic structure (many-body wave function) of the ground state for oxygen bound to heme?

2. What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

The first question has a long history. Like almost anything important and profound in quantum chemistry it goes back to Linus Pauling! In 1936 Pauling and Coryell argued that the ground state is

essentially a neutral O=O binding with two of its electrons to iron to produce a formally iron(II) if both the bonding electrons were confined to O2, corre- sponding to the non-bonding limit of neutral parts, but a formally iron(I) if the Fe–O bond were to be considered covalent. 

In 1960, McClure suggested a valence-bond formulation based on triplet–triplet coupling, which is appealing by the low promotion energies required to access these states, rather than the singlet states. In 1964, Weiss suggested, based on analogy to chemical reactions in aqueous solution, that the true ferrous hemeO2 adduct was mainly of the superoxo-iron(III) type caused by ‘‘electron transfer” from iron to O2. 

Goddard and Olafson suggested an ozone model of the adduct in 1975 which emphasized the four-electron three-center bond with maintained triplet state of dioxygen as in the McClure model with less electronic reorganization to explain the reversible binding. 

In 1977, Pauling maintained his original view again, the same year that Huynh, Case, and Karplus did a first attempt to bridge these views by performing early quantum chemical calculations that diplomatically emphasized the importance of both Weiss and Pauling resonance forms. 

However, interpretation depends on model language, orbital localization, and transformation between valence bond and orbital formalisms:  

In terms of molecular orbital theory, the wave function was a multi-configurational state dominated by the Pauling configuration; however, if one uses valence bond theory considerations, it can be interpreted as having large Weiss character. Thus, the multi-configurational state produced from CASPT2 is interpreted differently by different models. This partly explains why the trenches were so deeply dug during the exchange between Pauling, Goddard, McClure, and Weiss; all were right, and all were wrong. 

This is just another example of unnecessary conflicts about valence bond vs. molecular orbital (VB vs. MO). 

In terms of valence structures, the ground state was summarized by Shaik and Chen as having contributions from both Weiss, Pauling, and McClure forms, the first .. dominating. 

Ironically DFT ends up providing a useful language after all! 

The charge assignments to O2 are very dependent on calculation scheme, and both the orbitals, valence structures, and atomic charges that defined the Weiss-Pauling debate are non-observable. In contrast, the electron density is observable as are the geometries and spectroscopic data...

Molecular orbitals are not physical observables but calculational constructs. MO's don't exist.

In different words, one can take a many-body wave-function and make a linear unitary transformation of the molecular orbitals. The Slater determinants do not change. [The value of a determinant is invariant to a change of basis.]

Now. Question 2.
What is the mechanism for the ``spin-forbidden'' transition of the oxygen binding?

Kepp talks about spin-orbit coupling and the fact that it is small for oxygen, motivating a discussion of a "broad crossing mechanism".  However, I am not sure this is relevant. I don't see the binding as necessarily spin forbidden. As the oxygen approaches the heme the two triplet states can mix to form a total spin singlet.
This is analogous to bringing two hydrogen atoms (each of which is spin 1/2) together to form a hydrogen molecule (which is spin zero). A multi-configurational wavefunction has no problem with this. But DFT-based approximations, which use a single determinant cannot describe this smooth crossover.

Other things of particular interest to me that are discussed in the review include the central role of back bonding and the success of the TPSSh functional in DFT calculations for organometallics.

Unfortunately, the review does not mention recent work by Weber et al, applying DMFT to the problem of oxygen binding to haemoglobin.

Ising model basics

The Ising model is a paradigm in both statistical mechanics and condensed matter physics. Today for most theorists it is so familiar that some of its historical and conceptual significance is lost.
Previously, I posted about what students can learn from computer simulations of the Ising model.

If you had to talk about the Ising model to an experimental chemist what would you say?
[Last week I had to do this].

The Ising model is the simplest effective model Hamiltonian that can describe a thermodynamic system that undergoes a first-order phase transition and has a phase diagram containing a critical point.

On each site i of a lattice one defines a spin sigma_i= +1 or -1, representing spin up or spin down.

The Hamiltonian H is

J_ij describes the interaction between spins on sites i and j. In the simplest version the interactions are only between nearest neighbours, and have the same value J.
h is the external magnetic field.

If J is positive, the ground state at h=0 is a ferromagnet.
If J is negative, the ground state at h=0 is an anti-ferromagnet for a bipartite lattice.

[Caution: just like for the Heisenberg model, some authors define the Hamiltonian with the opposite sign of J].

For h=0 there is a critical point at a finite temperature Tc, for lattices of dimension two and higher.

The spins sigma_i= +/- 1 defined at each lattice site i, were originally to represent the atomic magnetic moments in a ferromagnetic material. However, the sigma's can represent any two states of the site i. For example, the ``spin'' or pseudo-spin can represent the presence or absence of an atom or molecule in a ``lattice gas'', atom A or atom B in a binary alloy (mixture), or the low-spin and high-spin states in a spin-crossover material.

The mean-field theory of the Ising model is mathematically equivalent to the thermodynamic theory of binary mixtures with an entropy of an ideal mixture.
There is a nice discussion of such mixtures in Section 5.4 [and the associated problems] of Introduction to Thermal Physics by Schroeder.
[Here are the slides for a lecture I have given based on that text].
Chapter 15 of the text by Dill and Bromberg is also helpful as it has more detail.
Neither text makes an explicit connection to the Ising model. Following this paper on alloys, one has

This is shown in Section 8.1.2 of James Sethna's text, Statistical MechanicsEntropy, Order Parameters and Complexity.

When interactions beyond nearest-neighbours are included in the Ising model or when the lattice is frustrated (e.g. fcc or triangular) a richer phase diagram is possible. Examples include the ANNNI model and some models for spin-state ice considered by Jace Cruddas and Ben Powell.

Tuning the dimensionality of spin-crossover compounds

An important question concerning spin-crossover compounds concerns the origin and the magnitude of the interactions between the individual molecular units.

There is a nice paper
Evolution of cooperativity in the spin transition of an iron(II) complex on a graphite surface
Lalminthang Kipgen, Matthias Bernien, Sascha Ossinger, Fabian Nickel, Andrew J. Britton, Lucas M. Arruda, Holger Naggert, Chen Luo, Christian Lotze, Hanjo Ryll, Florin Radu, Enrico Schierle, Eugen Weschke, Felix Tuczek, and Wolfgang Kuch

An impressive achievement is the control of the number of monolayers (ML) of SCO molecules deposited on a highly oriented surface pyrolytic graphite. The coverage varies between 0.35 and 10 ML. The shape of the spin-crossover curve changes significantly as the number of monolayers varies, as shown in the upper panel below.

The natural interpretation is that as the number of monolayers increases the interaction between molecules (co-operativity) increases. This can be quantified in terms of the parameter Gamma in the Slichter-Drickamer model [which is equivalent to a mean-field treatment of an Ising model], with Gamma = 4 z J where z=number of nearest-neighbours and J=Ising interaction.
The blue curve in the lower panel shows the variation of Gamma with ML.

The figure above and Table 1 shows that for ML=0.35, Gamma=-0.44 kJ/mol is almost zero for ML=0.7, and then monotonically increases to 2.1 kJ/mol for the bulk.

Does that make sense?

The magnitude of the Gamma values is comparable to those found in other compounds.

The negative value of Gamma for ML=0.35 might be explained as follows. Suppose a monolayer consists of SCO molecules arranged in a square lattice. Then ML=0.33 will consist of chains of SCO molecules that interact in the diagonal direction. If the J_nnn for this next-nearest neighbour interaction is negative then the Gamma value will be negative.

For a monolayer on a square lattice, Gamma= 16 (J_nn + J_nnn). J_nn will be positive and so if it is comparable in magnitude to J_nnn then Gamma will be small for a monolayer.

For a bilayer, Gamma = 16 (J_nn + J_nnn) + 4 J_perp, where J_perp is the interlayer coupling.
For the bulk, Gamma = 16 (J_nn + J_nnn) + 8 J_perp.

This qualitatively explains the trends, but not quantitatively.

The authors also note that the values of Delta E and Delta S obtained from their data vary little with the coverage, as they should since these parameters are single-molecule properties. This also means that the crossover temperature, T_sco also varies little with coverage.

A more rigorous approach is to not use mean-field theory, but rather consider a slab of layers of Ising models. The ratio of the transition temperature T_c to J_nn increases from 2.27 for a single layer to 4.5 as the dimensionality increases from d=2 to d=3.
[In contrast, for mean-field theory the ratio increases from 4 to 6].

If the crossover temperature T_sco is larger than T_c, [as it must be if there is no hysteresis] and assuming J_nn does not change with coverage, then as the coverage increases the crossover temperature becomes closer to the critical temperature and the transition curve will become steeper, reflected in a smaller transition width Delta T (and a correspondingly larger effective Gamma in the Slichter-Drikamer fit). This claim can be understood by looking at the last Figure in this post.

Estimating the Ising interaction in spin-crossover compounds

I previously discussed how one of the simplest model effective Hamiltonians that can describe many physical properties of spin-crossover compounds is an Ising model in an external "field". The s_i=+/-1 is a pseudo-spin denoting the low-spin (LS) and high-spin (HS) states of a transition metal molecular complex at site i.

The ``external field" is one half of the Gibbs free energy difference between the LS and HS states. The physical origin of the J interaction is ``believed to be'' elastic, not magnetic interactions. A short and helpful review of the literature is by Pavlik and Boca.

Important questions are:

1. What is a realistic model that can explain how J arises due to elastic interactions?
2. How does one calculate J from quantum chemistry calculations?
3. How does one estimate J for a specific material from experimental data?
4. What are typical values of J?

I will focus on the last two questions.
One can do a mean-field treatment of the Ising model, leading to a model free energy for the whole system that has the same form as that of an ideal binary mixture of two fluids where
x = (1 + av(s_i))/2, is the relative fraction of low spins. 
This model free energy was proposed in 1972 by Slichter and Drickmamer.
The free energy of interaction between the two "fluids" is of the form -Gamma x^2.
Gamma is often referred to as the ``co-operativity" parameter.
Minimising the free energy versus x gives a self-consistent equation for x(T).
This can be compared to experimental data for x vs T, e.g. from the magnetic susceptibility, and a Gamma value extracted for a specific material.

Values for Gamma obtained in this way for a wide-range of quasi-one-dimensional materials [with covalent bonding (i.e. strong elastic interactions) between spin centres] are given in Tables 1 and 2 of Roubeau et al. The values of Gamma are in the range 2-10 kJ/mol. In temperature units this corresponds to 240-1200 K.

My calculations [which may be wrong] give that Gamma = 4 J z, where z is the number of nearest neighbours in the Ising model. This means that (for a 1d chain with z=2) that J is in the range of 0.3-1.5 kJ/mol, or 30-150 K.

In many spin-crossover materials, the elastic interactions are via van der Waals, hydrogen bonding, or pi-stacking interactions. In that case, we would expect smaller values of J.
This is consistent with the following.
An analysis of a family of alloys by Jakobi et al. leads to a value of Gamma of 2 kJ/mol.
[See equation 9b. Note B=Gamma=150 cm^-1.  Also in this paper x is actually denoted gamma and x denotes the fraction of Zn in the material.].

I thank members of the UQ SCO group for all they are teaching me and the questions they keep asking.