The wonders of gallium

 A friend recently showed me that solid gallium can melt in your hand.

I did not know this. I was quite familiar with liquid mercury, but not gallium. 

The existence of elemental gallium was predicted by Mendeleev in 1869 after he constructed the periodic table. It was discovered within six years. He was able to predict that it would have a low-melting temperature, based on extrapolations from the known melting temperatures of elements close to it in the periodic table.

Solid gallium is soft enough to be cut with a knife.

Three different stable crystal structures for solid gallium are shown below.

The phase diagram of pure gallium is shown below.

Note the negative slope of the phase boundary between the liquid and the solid alpha-Ga. This is like water. It follows from the Clausius-Clapeyron equation that the solid state has lower density than the liquid state. Gallium is the only elemental metal with this property. (The semi-metals antinomy and bismuth also do).

Gallium remains liquid over a wider range of temperatures (2373 K) than any other known substance.

The figures above are taken from the following paper from 2020.

Ab initio phase diagram and nucleation of gallium  Haiyang Niu, Luigi Bonati, Pablo M. Piaggi, and Michele Parrinello

Unfortunately, that paper does not provide much insight into the low melting temperature. The key is that the solid state contains dimers of Ga, that are weakly bonded to each other. A helpful discussion is the introduction to the following paper.

On the bonding of Ga2, structures of Gan clusters and the relation to the bulk structure of gallium 

N. Gaston and A.J. Parker

The image above is from the entry on Gallium in the beautiful book The Elements by Theodore Gray.

I thank my young friend Alexey for introducing me to the wonders of gallium.

Very Short Introduction can be pre-ordered

 

I am currently working on the proofs and index for Condensed Matter Physics: A Very Short Introduction. It is wonderful to have got to this stage.

It is slated for release on December 29. It can be pre-ordered from Oxford UP (GDP 9) , Amazon (US $12), Book Depository (US $16), ...

The value of "simple" models for complex systems

Significant understanding of emergent phenomena in quantum materials has come from the study of model Hamiltonians such as those associated with the names Hubbard, Anderson, Kondo, Heisenberg, Kitaev, Haldane, BCS,...

I had not appreciated until recently that an early key to the Modern Synthesis of evolutionary biology (that brought together Darwinian natural selection with Mendelian genetics) was the development of simple mathematical models. The discussion below is taken from

Towards a unified science of cultural evolution 

Alex Mesoudi, Andrew Whiten and Kevin N. Laland 

Significant advances were made in the study of biological [micro]evolution before its molecular basis was understood, in no small part through the use of simplified mathematical models, pioneered by Fisher (1930), Wright (1931), and J.B.S. Haldane (1932)... 

Mathematical models such as [those for cultural evolution and gene-culture coevolution] are often treated with suspicion and even hostility by some social scientists, who consider them to be oversimplifications of reality... The alternatives..., however, are usually either analysis at a single (purely genetic or purely cultural) level or vague verbal accounts of “complex interactions,” neither of which we believe to be productive. Gene-culture analyses have repeatedly revealed circumstances under which the interactions between genetic and cultural processes lead populations to different equilibria than those predicted by single level models or anticipated in verbal accounts... as illustrated by the aforementioned examples of dairy farming and handedness.  

Interestingly, fifty years ago the same reservations about simplifying assumptions were voiced about the use of population genetic models in biology by the prominent evolutionary biologist Ernst Mayr (1963). He argued that using such models was akin to treating genetics as pulling coloured beans from a bag (coining the phrase “beanbag genetics”), ignoring complex physiological and developmental processes that lead to interactions between genes. 

 

In his classic article “A Defense ofBeanbag Genetics,” J. B. S. Haldane (1964) countered that the simplification of reality embodied in these models is the very reason for their usefulness. Such simplification can significantly aid our understanding of processes that are too complex to be considered through verbal arguments alone, because mathematical models force their authors to specify explicitly and exactly all of their assumptions, to focus on major factors, and to generate logically sound conclusions. Indeed, such conclusions are often counterintuitive to human minds relying solely on informal verbal reasoning. 

Haldane (1964) provided several examples in which empirical facts follow the predictions of population genetic models in spite of their simplifying assumptions, and noted that models can often highlight the kind of data that need to be collected to evaluate a particular theory. Ultimately, Haldane won the argument, and population genetic modelling is now an established and invaluable tool in evolutionary biology (Crow 2001). We can only echo Haldane’s defence and argue that the same arguments apply to the use of similar mathematical models in the social sciences.

A more recent version of J.B. S. Haldane's argument is Not Just a Theory—The Utility of Mathematical Models in Evolutionary Biology Maria R. Servedio,Yaniv Brandvain, Sumit Dhole, Courtney L. Fitzpatrick, Emma E. Goldberg, Caitlin A. Stern, Jeremy Van Cleve, D. Justin Yeh 

All models are wrong but some are useful. I first learnt this aphorism from Scott Page, in his wonderful course Model Thinking at Coursera.  This short talk discusses how models help us think more clearly. Simple quantitative models, such as agent-based models, in the social sciences, have the value that their assumptions can be clearly stated, and then the consequences of these assumptions can be investigated in a rigorous manner.

There is also a nice discussion of the importance of model building for science in John Holland's beautiful book, Emergence.