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States of Matter

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This book is a very good intro, discussing topics which are rarely found elsewhere. A must have for Graduate/PhD in Solid State Physics.

Years ago, I opened States of Matter looking for a text to supplement to a course in Statistical Physics I was taking. What I took away was different and far more valuable: a new method -- dimensional analysis -- of simplifying and solving complex problems of physics. Goodstein presents this method as a brief historical anecdote, one entertaining and memorable. In short, Goodstein showed that sometimes one does well to ignore the detail of a problem and instead focus on the data at hand. Let the data and their units (or, dimensions) lead one to guessing the answer.

Here is the anecdote in its entirety. You will find it at the beginning of chapter 6, "Critical Phenomenon and Phase Transitions", pp. 436-7 in the Dover edition.

Dimensional analysis is a technique by means of which it is possible to learn a great deal about very complicated situations if you can put your finger on the essential features of the problem. An example is the well-known story of how G.I. Taylor was able to deduce the yield of the first nuclear explosion from a series of photographs of the expanding fireball in Life magazine. He realized that he was seeing a strong shock expanding into an undisturbed medium. The pictures gave him the radius as a function of time, r(t). All that could be important in determining r(t) was the initial energy release, E, and the density of the undisturbed medium, p (i.e., the Greek letter rho). The radius, with the dimension of length, depended on E, p, and t, and he constructed a distance out of these quantities. E and p had to come in as E/p to cancel the mass. E/p has the dimensions (length)^5 / (time)^2, so the only possible combination was

r(t) ~ ( (E / p) t^2 )^(1/5)

A log-log plot of r versus t (measured from the pictures) gave a slope of 2/5, which checked the theory, and E/p could be obtained from extrapolation to the value of log r when log t = 0. Since p, the density of undisturbed air, was known, E was determined to within a factor of order one. For the practitioner of the art of dimensional analysis, the nation's deepest secret had been published in Life magazine. [End of Goodstein anecdote.]

This anecdote can be worked as an example. For the density of air at 5000 feet (i.e., roughly the elevation of Alamogordo, NM, near where the explosion happened) I used 1.05 kg/m^3, which is extrapolated from 1.2 kg/m^3 value at sea level. To convert units of energy from joules (J) to kilotons of TNT, I used 4600 J per gram TNT. There is one difficulty. Life magazine never published the pictures. (Well, if they did, my extensive searching never turned them up.) However, I did find them elsewhere on the internet (the history section of the Los Alamos National Lab's website). Using an image editing program, I measured the following radii from the panel of four photos: 75, 109, 182, and 245 meters for times of 6, 16, 53, and 100 milliseconds. Least squares fitting of log distance vs log time yields a slope of 0.419, which well approximates the expected 2/5. Solving for E yields an explosive yields of 23 kilotons (kT), which is close the officially reported yield of 20 to 22 kT.

This colorful anecdote is something one would expect from Richard Feynman. So it is hardly surprising that Goodstein wrote the book at Caltech when Feynman was there. Perhaps Goodstein learned from Feynman, or vice versa.