There is, however, a difference between the gas and the radiation enclosed in a container. Since the molecules are not mathematical points but have certain finite diameters, they undergo numerous mutual collisions in which their energy can be exchanged. Thus, if we inject into a container some hot gas and some cool gas, mutual collisions between the molecules will rapidly slow down the fast ones and speed up the slow ones, resulting in even distribution of energy in accordance to the Equipartition Principle. In the case of an ideal gas formed by point-molecules, which of course does not exist in nature, mutual collisions would be absent and the hot fraction of the gas would remain hot while the cool fraction would remain cool. The exchange of energy between the molecules of an ideal gas can be stimulated, however, by introducing into the container one or several particles with finite though small diameters (Brownian particles). Colliding with them, fast point-sized molecules will communicate to them their energy, which will be communicated in turn to the other slower point-sized molecules.
In the case of light waves the situation is different, since two light beams crossing each other’s path do not affect each other’s propagation in any way.ov Thus, to procure the exchange of energy between the standing waves of different lengths, we must introduce into the container small bodies that can absorb and re-emit all possible wavelengths, thus permitting energy exchange among all possible vibrations. Ordinary black bodies, such as charcoal, have this property, at least in the visible part of the spectrum, and we may imagine “ideal black bodies” which behave in the same way for all possible wavelengths. Placing into Jeans’ cube a few particles of ideal coal dust, we will solve our energy-exchange problem.
Now let us perform a thought experiment, injecting into an originally empty Jeans’ cube a certain amount of radiation of a given wavelength-let us say some red light. Immediately after injection, the interior of the cube will contain only red standing waves extending from wall to wall, while all other modes of vibrations will be absent. It is as if one strikes on a grand piano one single key. If, as it is in practice, there is only very weak energy exchange among different strings of the instrument, the tone will continue to sound until all the energy communicated to the string will be dissipated by damping. If, however, there is a leak of energy among the strings through the armature to which they are attached, other strings will begin to vibrate too until, according to the Equipartition Theorem, all 88 strings will have energy equal to 1/88 of the total energy communicated.
FIG. 4 A piano with an unlimited number of keys extending into the ultrasonic region all the way to infinite frequencies. The equipartition law would require all the energy supplied by a musician to one of the low-frequency keys to travel all the way into the ultrasonic region out of the audible range!
But if a piano is to represent a fairly good analogy of the Jeans’ cube, it must have many more keys extending beyond any limit to the right into the ultrasonic region (Fig. 4). Thus the energy communicated to one string in an audible region would travel to the right into the region of higher pitches and be lost in the infinitely far regions of the ultrasonic vibrations, and a piece of music played on such a piano would turn into a sharp shrill. Similarly the energy of red light injected into Jeans’ cube would turn into blue, violet, ultraviolet, X-rays, γ -rays, and so on without any limit. It would be foolhardy to sit in front of a fireplace since the red light coming from the friendly glowing cinders would quickly turn into dangerous high-frequency radiation of fission products!
The runaway of energy into the high-pitch region does not represent any real danger to concert pianists, not only because the keyboard is limited on the right, but mostly because, as was mentioned before, the vibration of each string is damped too fast to permit a transfer of even a small part of energy to a neighboring string. In the case of radiant energy, however, the situation is much more serious, and, if the Equipartition Law should hold in that case, the open door of a boiler would be an excellent source of X- and γ -rays. Clearly something must be wrong with the arguments of nineteenth-century physics, and some drastic changes must be made to avoid the Ultraviolet Catastrophe, which is expected theoretically but never occurs in reality.
MAX PLANCK AND THE QUANTUM OF ENERGY
The problem of radiation-thermodynamics was solved by Max Planck, who was a 100 per cent classical physicist (for which he cannot be blamed). It was he who originated what is now known as modern physics. At the turn of the century, at the December 14, 1900 meeting of the German Physical Society, Planck presented his ideas on the subject, which were so unusual and so grotesque that he himself could hardly believe them, even though they caused intense excitement in the audience and in the entire world of physics.
Max Planck was born in Kiel, in 1858, and later moved with his family to Munich. He attended Maximilian Gymnasium (high school) in Munich and, after graduation, entered the University of Munich, where he studied physics for three years. The following year he spent at the University of Berlin, where he came in contact with the great physicists of that time, Herman von Helmholtz, Gustav Kirchhoff, and Rudolph Clausius, and learned much about the theory of heat, technically known as thermodynamics. Returning to Munich, he presented a doctoral thesis on the Second Law of Thermodynamics, receiving his Ph.D. degree in 1879, and then became an instructor at that university. Six years later he accepted the position of associate professor in Kiel. In 1889 he moved to the University of Berlin as an associate professor, becoming a full professor in 1892. The latter position was, at that time, the highest academic position in Germany, and Planck kept it until his retirement at the age of seventy. After retirement he continued his activities and delivered public speeches until his death at the age of almost ninety. Two of his last papers (A Scientific Autobiography and The Notion of Causality in Physics) were published in 1947, the year he died.
Planck was a typical German professor of his time, serious and probably pedantic, but not without a warm human feeling, which is evidenced in his correspondence with Arnold Sommerfeld who, following the work of Niels Bohr, was applying the Quantum Theory to the structure of the atom. Referring to the quantum as Planck’s notion, Sommerfeld in a letter to him wrote:You cultivate the virgin soil,
Where picking flowers was my only toil.
and to this answered Planck:You picked flowers–well, so have I.
Let them be, then, combined;
Let us exchange our flowers fair,
And in the brightest wreath them bind.ow
For his scientific achievements Max Planck received many academic honors. He became a member of the Prussian Academy of Sciences in 1894, and was elected a foreign member of the Royal Society of London in 1926. Although he made no contribution to the science of astronomy, one of the newly discovered asteroids was called Planckiana in his honor.
Throughout all his long life Max Planck was interested almost exclusively in the problems of thermodynamics, and the many papers he published were important enough to earn him the honorable position of full professor in Berlin at the age of thirty-four. But the real outburst in his scientific work, the discovery of the quantum of energy, for which, in 1918, he was awarded the Nobel Prize, came rather late in life, at the age of forty-two. Forty-two years is not so late in the life of a man in the usual run of occupations or professions, but it usually happens that the most important work of a theoretical physicist is done at the age of about twenty-five, when he has had time to learn enough of the existing theories but while his mind is still agile enough to conceive new, bold revolutionary ideas. For example, Isaac Newton conceived the Law of Universal Gravity at the age of twenty-three; Albert Einstein created his Theory of Relativity at the age of twenty-six; and Niels Bohr published his Theory of the Atomic Structure at the age of twenty-seven. In his small way, the author of this book also published his most important work, on natural and artificial transformations of the atomic nucleus, when he was twenty-four. In his lecture Planck stated that according to his rather complicated calculations the
paradoxical conclusions obtained by Rayleigh and Jeans could be remedied and the danger of the Ultraviolet Catastrophe avoided if one postulates that the energy of electromagnetic waves (including light waves) can exist only in the form of certain discrete packages, or quanta, the energy content of each package being directly proportional to the corresponding frequency.
Theoretical considerations in the field of statistical physics are notoriously difficult, but by inspecting the graph in Fig. 5 one can get some notion of how Planck’s postulate “discourages” radiant energy from leaking into the limitless high-frequency region of the spectrum.
In this graph the frequencies possible within a “one-dimensional” Jeans’ cube are plotted on the abscissa axes and marked 1, 2, 3, 4, etc.; on the ordinate axes are plotted the vibration energies that can be allotted to each possible frequency. According to classical physics any value of energy (that is, any point on the vertical lines drawn through 1, 2, 3, etc.) is permitted, the distribution resulting statistically in the Equipartition of Energy among all possible frequencies. On the other hand, Planck’s postulate permits only a discrete set of energy values, equal to one, two, three, etc., energy packages corresponding to the given frequency. Since the energy contained in each package is assumed to be proportional to the frequency, we obtain the permitted energy values shown by large black dots in the diagram. The higher the frequency, the smaller is the number of possible energy values below any given limit, a fact which restricts the capacity of the high-frequency vibrations to take up more additional energy. As a result, the amount of energy that can be taken by high-frequency vibrations becomes finite in spite of their infinite number, and everything is dandy.
FIG. 5 If, according to Planck’s hypothesis, the energy corresponding to each frequency ν must be an integer of the quantity hv, the situation is quite different from that shown in the previous diagram. For example, for ν = 4 there are eight possible vibration states, whereas for ν = 8 there are only four. This restriction reduces the number of possible vibrations at high frequencies and cancels Jeans’ paradox.
It has been said that there are “lies, white lies, and statistics,” but in the case of Planck’s calculations the statistics turned out to be well-nigh true. He had obtained for energy distribution in thermal radiation spectrum a theoretical formula that stood in perfect agreement with the observation shown in Fig. 2.
While the Rayleigh-Jeans formula shoots sky high, demanding an infinite amount of total energy, Planck’s formula comes down at high frequencies and its shape stands in perfect agreement with the observed curves. Planck’s assumption that the energy content of a radiation quantum is proportional to the frequency can be written as:E = h ν
where ν (the Greek letter nu) is the frequency and h is a universal constant known as Planck’s Constant, or the quantum constant. In order to make Planck’s theoretical curves agree with the observed ones, one has to ascribe to h a certain numerical value, which is found to be 6.77 × 10−27 in the centimeter-gram-second unit system.ox
The numerical smallness of that value makes quantum theory of no importance for the large-scale phenomena which we encounter in everyday life, and it emerges only in the study of the processes occurring on the atomic scale.
LIGHT QUANTA AND THE PHOTOELECTRIC EFFECT
Having let the spirit of quantum out of the bottle, Max Planck was himself scared to death of it and preferred to believe the packages of energy arise not from the properties of the light waves themselves but rather from the internal properties of atoms which can emit and absorb radiation only in certain discrete quantities. Radiation is like butter, which can be bought or returned to the grocery store only in quarter-pound packages, although the butter as such can exist in any desired amount (not less, though, than one molecule!). Only five years after the original Planck proposal, the light quantum was established as a physical entity existing independently of the mechanism of its emission or absorption by atoms. This step was taken by Albert Einstein in an article published in 1905, the year of his first article on the Theory of Relativity. Einstein indicated that the existence of light quanta rushing freely through space represents a necessary condition for explaining empirical laws of the photoelectric effect; that is, the emission of electrons from the metallic surfaces irradiated by violet or ultraviolet rays.
An elementary arrangement for demonstrating photoelectric effect, shown in Fig. 6a, consists of a negatively charged ordinary electroscope with a clean metal plate P attached to it. When a light from an electric arc A, which is rich in violet and ultraviolet rays, falls on the plate, one observes that the leaves L of the electroscope collapse as the electroscope discharges. That negative particles (electrons) are discharged from the metal plate was demonstrated repeatedly, by the American physicist Robert Millikan (1868–1953) among others. If a glass plate, which absorbs ultraviolet radiation, is interposed between the arc and the metal plate, electrons are not given off, conclusive evidence that the action of the rays causes the emission. A more elaborate arrangement used for the detailed study of the laws of photoelectric effect is shown schematically in Fig. 6b. It consists of:1. A quartz or fluoride prism (transparent for ultraviolet) and a slit permitting the selection of a monochromatic radiation of desired wavelength.
2. A set of rotating discs with triangular openings of various sizes, permitting a change in the intensity of radiation.
3. An evacuated container somewhat similar to the electron tubes used in radio sets. A variable electric potential is applied between the plate P, from which the photoelectrons are emitted, and the grid G. If the grid is charged negatively, and potential difference between the grid and the plate is equal to or larger than the kinetic energy of photoelectrons expressed in electron volts, no current will flow through the system. In the opposite case there will be a current, and its strength can be measured by the galvanometer GM. Using this arrangement, one can measure the number and the kinetic energy of electrons ejected by the incident light of any given intensity and wavelength (or frequency).
FIG. 6 Experimental Studies of the Photoelectric Effect. In (a) a primitive method for a demonstration of the photoelectric effect is illustrated. The ultraviolet radiation emitted by an electric arc ejects the electrons from a metal plate attached to an electroscope. The negatively charged leaves L, which have been repelling each other, lose charge and collapse. In (b) the modern method is shown. Ultraviolet radiation from an electric arc passes through a prism allowing only one selected frequency to fall on the plate. Turning the prism, one can select a monochromatic light and direct it to the plate. The energy of photoelectrons is measured by their ability to get through from plate to receiver, moving against the electric force produced by a potentiometer between plate and grid.
The study of photoelectric effect in different metals resulted in two simple laws:I. For light of a given frequency but varying intensity the energy of photoelectrons remains constant while their number increases in direct proportion to the intensity of light (Fig. 7a).
II. For varying frequency of light no photoelectrons are emitted until that frequency exceeds a certain limit ν0, which is different for different metals. Beyond that frequency threshold the energy of photoelectrons increases linearly, being proportional to the difference between the frequency of the incident light and the critical frequency ν0 of the metal (Fig. 7b).
FIG. 7 The Laws of Photoelectric Effect. In (a) the number of electrons is plotted as the function of the intensity of the incident monochromatic light. In (b) the energy of photoelectrons is shown as the function of the frequency of the incident monochromatic light for three different metals: A, B and C.
These well-established facts could not be explained on the basis of the classical theory of light; in some points they even contradicted it. Light is known to be short electromagnetic waves, and the increase of intensity of light must mean an increase of the oscillating electric and magnetic forces propagating through space. Since the electrons apparently are ejected from the
metal by the action of electric force, their energy should increase with the increase of light intensity, instead of remaining constant as it does. Also, in the classical electromagnetic theory of light, there was no reason to expect a linear dependence of the energy of photoelectrons on the frequency of the incident light.
Using Planck’s idea of light quanta and assuming the reality of their existence as independent energy packages flying through space, Einstein was able to give a perfect explanation of both empirical laws of photoelectric effect. He visualized the elementary act of the photoelectric effect as the result of a collision between a single incident light quantum and one of the conductivity electrons carrying electric current in the metal. In this collision the light quantum vanishes, giving its entire energy to the conductivity electron at the metallic surface. But, in order to cross the surface and to get into the free space, the electron must spend a certain amount of energy disengaging itself from the attraction of metallic ions. This energy, known by the somewhat misleading name of “work function,” is different for different metals and is usually denoted by a symbol W. Thus, the kinetic energy K with which a photoelectron gets out of the metal is:K = h (ν − ν0) = h ν − W
The Dreams That Stuff is Made of Page 88