Ultraviolet catastrophe and

Ernest Rutherfords proposed atomic structure added to the problems posed to nineteenth century physics by the ultraviolet catastrophe and the photoelectric effect. Rutherfords atom had a negatively charged electron circling a positively charged nucleus. The physics of the day predicted that the atom would emit radiation, causing the electron to lose energy and spiral down into the nucleus. Theory predicted that Rutherfords atom could not exist. Clearly, science needed new ideas to explain these three anomalies. 

Q.7.10 What was the ultraviolet catastrophe and how did Planck s proposal resolve it ... 

Describe the ultraviolet catastrophe and its empirical resolution by the Planck radiation law. 

For nineteenth-century scientists, the obvious way to account for the laws of black-body radiation was to use classical physics to derive its characteristics. However, much to their dismay, they found that the characteristics they deduced did not match their observations. Worst of all was the ultraviolet catastrophe classical physics predicted that any hot body should emit intense ultraviolet radiation and even x-rays and y-rays According to classical physics, a hot object would devastate the countryside with high-frequency radiation. Even a human body at 37°C would glow in the dark. There would, in fact, be no darkness. 

Rayleigh found the v2 dependence, Jeans later supplied the rest). Their distribution function g(v) increases as v2, with no provision for a fall-off to zero as the frequency and the energy go to infinity ("ultraviolet catastrophe"). 

These calculated results, shown for 5000 and 7000 K by the dashed curves in Figure 4.6, agree well with experiment at lower frequency. But the theory does not predict a maximum in the intensity distribution, and even worse, it disagrees badly with the experimental results at high frequencies. This feature of the result was called the ultraviolet catastrophe because it predicts an infinite intensity at very... 

Quantum mechanics represents one of the cornerstones of modem physics. Though there were a variety of different clues (such as the ultraviolet catastrophe associated with blackbody radiation, the low-temperature specific heats of solids, the photoelectric effect and the existence of discrete spectral lines) which each pointed towards quantum mechanics in its own way, we will focus on one of these threads, the so-called wave-particle duality, since this duality can at least point us in the direction of the Schrodinger equation. 

C.S. Cockell, A.R. Blaustein (2000). Ultraviolet spring and the ecological consequences of catastrophic impacts. Ecol. Lett., 3, 77-81. 

This is Wien s displacement law. Classical principles had failed to explain the shape of the curve in Fig. 19.4 and f ailed to predict the displacement law. The application of the classical law of equipartition of energy between the various degrees of freedom by Rayleigh and Jeans was satisfactory at long wavelengths but failed at short wavelengths, in the ultraviolet ( ultraviolet catastrophe ). 

It can be seen that the Rayleigh-Jeans law reproduces the experimental data at low frequencies fairly well. However, at high frequencies, the Rayleigh-Jeans law diverges as v. Since the frequency increases in the ultraviolet region of the spectrum, this divergence was called the ultraviolet catastrophe, a phenomenon that classical physics was unable to explain theoretically. This was the first such phenomenon to be observed in physics and did in fact mark a major milestone in the annals of physics. 

This hypothesis about energy quanta led to the agreement of theory with experiment and the elimination of the ultraviolet catastrophe. 

Max Planck wanted to understand black botty radiation. The black body may be modelled by a box, with a small hole. Fig. 1.1. We heat the box up, wait for the system to reach a stationary state (at a fixed temperature) and see what kind of electromagnetic radiation (intensity as a function of frequency) comes out of the hole. In 1900 Rayleigh and Jeans tried to apply classical mechanics to this problem, and calculated correctly that the black bod would emit electromagnetic radiation having a distribution of frequencies. However, the larger the frequency the larger its intensity, leading to what is known as ultraviolet catastrophe, an absurd conclusion. Experiment contradicted theory (Fig. 1.1). 

The famous Planck constant h followed soon after. (The actual equation for the Planck constant ish = 6.62607 10 " J s, but in this book, we will use a more convenient constant h =. ) This hypothesis about energy quanta led to the agreement of theory with experiment and the elimination of the ultraviolet catastrophe. 

In Eq. (12) V is an effective nearest neighbor electron transfer matrix element ( V > as B > ), m is the effective mass of the reference model, and d is the dimension of the model. No ultraviolet catastrophe occurs for d < 2, but it does occur for d > 2 leading to a shift of the unperturbed band edge to When energies are measured relative to... 

Ultraviolet catastrophes do not occur for d < 2, and the desired universality is easily shown to exist.17 In d > 2, the divergences exist and lead to a shift of the original, underlying band edge. is... 

Planck s theory removed the ultraviolet catastrophe because short wavelengths correspond to large frequencies and large energy spacings. The excited states therefore have very small populations. The result of Planck s theory is that... 

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