Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  10.3.1 Enter Bohr

  Niels Bohr (1885–1962) was one of the giants of twentieth-century physics. He began his theory of the hydrogen atom by taking the classical mechanics of Newton for the solar system but with the gravitational force replaced by the Coulomb force for the positively charged proton attracting the negatively charged electron. This theory leads to a continuum of possible energies for the electron in orbit around the proton. There is also a major problem: according to electromagnetic theory, an electron moving in such an orbit will radiate energy and so gradually spiral down into the proton. (Bohr's 1913 paper is easy to follow and reproduced in Ter Haar's book.)

  Bohr took his cue from Planck's photon theory of radiation and built this into his theory of the atom. Let the electron have energy W, measured as the amount of energy required to remove the electron from its orbit and away to infinity. Building on the discrete photon ideas of Planck, Bohr came to expressions for W and for the diameter 2a of the electron orbit:

  In equation (10.6) m is the mass of the electron and –e is its charge; E is the charge of the nucleus so E = e for hydrogen; h is Planck's quantum constant introduced in section 9.5; and n is an integer. Thus in Bohr's theory, the electron only occupies those orbits labeled by the integer n, and it is assumed that those electron orbits are stable.

  The orbit with n = 1 requires the largest amount of energy to remove the electron from the hydrogen atom leading to what we today call the ground state of the atom. Bohr used the known values of e, m¸ and h to find

  numbers which compared quite well with experimental data.

  Bohr could now make the vital step. He assumed that electrons could move between orbits by emitting or absorbing a photon of frequency ν such that its energy hν was equal to the difference of the electron energies of those orbits. This led to

  This is just the spectral-lines formula in equation (10.4) with an expression for the Rydberg constant given in terms of the fundamental quantities m, e, c, and h. Bohr calculated that cR = 3.1 × 1015 as against the experimental value 3.29 × 1015. This was taken as excellent agreement given the uncertainties in the input data for his calculation.

  Bohr had thus shown that his model for the hydrogen atom, incorporating quantum conditions into classical mechanics, explained the spectral properties of hydrogen so precisely set out by Balmer and his followers. But Bohr could claim another triumph. For ionized helium, there will be one electron orbiting a nucleus with charge 2e (E = 2e in equation (10.6)), and then his theory leads directly to the spectral lines observed by Pickering and Fowler (see equation (10.5) and section 10.2.2). Thus their observations showed the existence of helium in its ionized form and fitted perfectly with Bohr's theory.

  There were many triumphs for Bohr's theory, but there were also problems, especially moving on to the full helium atom and other systems. A new approach was needed.

  10.3.2 Removing the Difficulties with a Quantum Wave

  There is no doubt that Bohr's work is brilliant and deserved the award of the Nobel Prize in 1922, but there is something unsatisfying about the way it cobbles together classical and quantum ideas in an ad hoc sort of way. The resolution to the difficulty takes us into one of the strangest parts of physics, one initiated by a French prince.

  Born of a noble family, Louis de Broglie became Prince de Broglie on the death of an elder brother. After various career changes, including time spent in the French army during World War I, de Broglie became a physicist, and the ideas he presented in his doctoral thesis changed physics forever. They led to the award of the Nobel Prize in 1929, and he sets out his intentions clearly in his acceptance speech (see bibliography). After some discussion of the quantum theory of light, he writes:

  This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity.

  On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations.13

  By this de Broglie means the harmonics found on a vibrating string as in a violin (he was very keen on chamber music), which I discuss further in chapter 12. In this case, only very particular wave configurations occur. De Broglie goes on:

  This suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too.

  I thus arrived at the following overall concept, which guided my studies: for both matter and radiation, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases.

  Here, in de Broglie's final sentence, was the revolutionary step: we must associate a wave with an electron. An electron with momentum p has a wave associated with it with wavelength equal to h/p. Soon, experiments with electrons fired into crystals revealed this wave nature by showing wave-like diffraction effects.

  De Broglie's wave “eigenvibrations” could also be found in the fitting of different waves around Bohr's atomic orbits. Here was an explanation of the fact that only very particular orbits occur in the atom; those orbits must be just right for fitting in a series of waves.

  The origin of other waves in physics was clear and led to equations which embodied their properties and showed how they propagated. De Broglie's ideas about particle waves were taken on by the Austrian physicist Erwin Schrödinger (1887–1961) and incorporated into the equation which bears his name. Finally, the classical equations of Newtonian mechanics were replaced by the quantum mechanics of Schrödinger's equation.

  The wave associated with an electron is conventionally denoted by ψ, and for the electron in a hydrogen atom, the Schrödinger equation takes the form

  The potential V is simply –e2/r for the hydrogen atom, and E is the electron energy. Now for the great discovery: solving the Schrödinger equation for the hydrogen atom and imposing the condition that ψ is finite and tends to zero infinitely far from the proton leads to three wonderful results.

  First, there is only a discrete set of values En for the energy E that accompanies a suitable finite form ψn for ψ (x, y, z) that can represent an electron bound to the proton to make the hydrogen atom. This is what is behind Bohr's discovery that only certain electron orbits should be chosen. The form of ψ can be interpreted as representing a standing wave in the space around the proton; these are de Broglie's waves.

  Second, those energies En are called the eigenvalues, or eigenenergies, and they are labeled by an integer n so that

  These are exactly the energies given by Bohr in his theory of the hydrogen atom. (The minus sign indicates that these are binding energies, and energy must be added to remove an electron from an orbit.)

  Third, the electron can only be in one of those wave-states (or eigenstates), and moving between them gives the emission or absorption of energy as a photon exactly as Balmer and others require. The hydrogen spectrum is now explained with the various series of lines linked to the quantum numbers n as shown in figure 10.3.

  Figure 10.3. Transitions between quantum states showing how the various spectral series are generated. From Wikimedia Commons, user OrangeDog.

  This is one of the very greatest results in physics, and it goes on my list as calculation 39, the new mechanics explain atoms. It paves the way for the whole of atomic physics and beyond that, to modern particle physics.

  In general, it is not easy to solve the Schrödinger equation for more complex, multi-electron systems, and approximate methods had to be developed. However, the basic theory was now available for understanding the properties of atoms and molecules, and its validity was proven by the results of the first simple calculations. (The recent book by Fayer uses pictures and largely nonmathematical arguments to show how quantum theory is used for atoms and to understand how molecules
may be constructed.)

  Another case that is simple but still impressive involves the deuterium spectrum. Deuterium is “heavy hydrogen,” an isotope of hydrogen with the nucleus consisting of the proton and also a neutron tightly bound to it. The nucleus now has mass (very close to) 2M, where M is the proton mass. Now the m in equations (10.8) and (10.9) is really the reduced mass,

  where me is the mass of the electron and Mn is the mass of the nucleus. Because the electron mass is so small (the proton mass is about 1,836 times larger), the difference between m and me is also small. However, calculating the spectrum for hydrogen uses m = 0.99945me, while for deuterium it must be m = 0.99973me. The outcome is a spectrum for deuterium that is shifted from that of hydrogen by a tiny amount. Yet this tiny difference was still detected in experiments first conducted in 1931 and so deuterium was discovered (see the article by Clark and Reader for the fascinating story).

  10.3.3 Quantum Physics

  After more than two hundred years, a challenge was made to the fundamentals of Newtonian mechanics, and now we accept that quantum mechanics must be used to properly describe the microscopic, atomic, and nuclear worlds. The price to be paid is the introduction of the wave function ψ and the ensuing struggle to interpret its role in the theory. The probability interpretation (where ψ gives the probability of finding the electron at a particular location) championed by Max Born is widely accepted. (There is an enormous literature on this subject; one of my favorites is the book by Wallace.)

  10.4 CHADWICK DISCOVERS THE NEUTRON

  The early part of the twentieth century was an exciting time for physicists as they struggled to find the fundamental building blocks for matter. The research of Rutherford and others led to the model of the atom considered in the previous section with a heavy, positively charged nucleus surrounded by very much lighter, negatively charged electrons. It gradually became clear that while the hydrogen nucleus was a single proton, there were particles other than protons present in the heavier nuclei. These other particles carry no charge; they are neutral. Around 1920, Rutherford suggested there might be some other electron-proton combination, but much more closely bound than is the case for hydrogen.

  In this same time period, there was much activity in nuclear physics involving the scattering of alpha particles (helium nuclei) from the nuclei of a range of elements. The experimental data gathered revealed that sometimes new elements were produced accompanied by other particles and electromagnetic radiation in the form of gamma rays, for example. By 1930, it had been shown that bombarding beryllium with alpha particles produced carbon and some other particle or radiation. Irène Curie and her husband Frédéric Joliot discovered that this output radiation could knock loose protons from a material such as paraffin.

  The decisive steps were taken by James Chadwick (1891–1974), and they confirmed the existence of the particle now known as the neutron. Chadwick was awarded the Nobel Prize in 1935, and both his address on receiving the award and his 1932 Nature letter give a good, readable introduction to his work in his own words (see bibliography).

  Chadwick considered the various possibilities for explaining the alpha particle-beryllium results, and in his Nature letter, he concluded:

  These results, and others I have obtained in the course of the work, are very difficult to explain on the assumption that the radiation from beryllium is a quantum radiation, if energy and momentum are to be conserved in the collisions. [And it would have been a brave person who challenged those conservation laws.] The difficulties disappear, however, if it be assumed that the radiation consists of particles of mass 1 and charge 0, or neutrons.14

  10.4.1 Settling the Argument

  The charge-neutral nature of the particle was relatively easy to demonstrate, but to show that it was the nucleus constituent required by nuclear physics meant that its mass had to be determined.

  In his Nobel Prize–winning address, Chadwick explained how he had done that using simple ideas from classical mechanics. Here was another case of making a calculation to interpret experimental data and so reach an extremely important conclusion.

  Assume that particle 1 with mass m1 is moving with speed V to collide with particle 2 of mass m2 as shown in figure 10.4. After the collision, the particles have speeds v1 and v2. Applying the conservation of energy and momentum laws (see Pask's chapter 13 or any classical mechanics textbook) shows that the second particle leaves the collision with speed

  Figure 10.4. Particles 1 and 2 before and after a collision. Figure created by Annabelle Boag.

  Chadwick used this result for his neutrons (with mass m1 = M) colliding with protons (m2 = 1 and v2 = Up) and with nitrogen nuclei (m2 = 14 and v2 = Un) to obtain

  These two equations may be combined to give

  If the speeds are measured, this equation may be solved to find the neutron mass M. In his address, Chadwick quotes early results for the approximate speeds as Up = 3.7 × 109 cm/sec and Un = 4.7 × 108 cm/sec, and using those numbers leads to M = 0.9. This first effort showed that the neutron mass is of the same order of magnitude as the proton mass; later experiments have given the neutron mass as 1.0014 times the proton mass. (This excess mass over the proton will be important when we return to the neutron story in section 11.3.) After reviewing the experimental and related theoretical work, Chadwick, in his 1932 Nature letter, concluded that “Up to the present, all the evidence is in favor of the neutron.”15 Chadwick's announcement of the discovery of the neutron came just seven weeks after the work on the deuterium spectrum described in section 10.3.2.

  Thus calculation 40, discovering the mass of the neutron proved to be of Nobel Prize–winning importance. The discoverer of the neutron, James Chadwick, went on to be a major figure in nuclear physics and a member of the team developing atomic weapons at Los Alamos. He received many honors and was knighted in 1945.

  10.5 THE TRIUMPH OF THE ATOMIC HYPOTHESIS

  With the discovery of the neutron in 1932, the list of particles forming atoms (electrons, protons, and neutrons) was complete. The atomic hypothesis was now generally accepted, and the structure of the atoms was revealed, even though it did require a leap into a new and extremely strange aspect of physics. Of course, there were complications and refinements that had to be addressed, but basically the world of atoms and molecules was now established by that powerful combination of theory and experiment. It seemed that Lucretius had got it right:

  The bodies themselves are of two kinds: the particles

  And complex bodies constructed of many of these

  Which particles are of an invincible hardness

  So that no force can alter or extinguish them.16

  But before too long, and as we will see in the next chapter, doubts emerged about the properties so eloquently described in the last two lines.

  It has not been easy to leave out other calculations involving atoms, particularly for crystals and other areas of solid state physics and chemistry. There are times when the limit of fifty calculations seems quite unfair.

  in which we see how properties of particles were explored using the theory-experiment combination and how the physics of the nucleus solved old mysteries and gave mankind some terrifying options.

  Atoms were discovered to consist of negatively charged electrons orbiting a positively charged nucleus built from protons and neutrons. The force holding atoms together is the electric, or Coulomb, force between the electrons and the positively charged protons. The Schrödinger equation allows properties of atoms to be calculated, and theory and experiment are in agreement. The theory can then be applied in the realm of molecules. However, going in the opposite direction—down into the nucleus of the atom—opens up a whole new world of physics. Experiments began to reveal strange properties of nuclei and to cast doubt on the comfortable idea that electrons, protons, and neutrons are the only contenders in the structure of the world at its most fundamental level. Also it was found that not just light, electromagnetic radiation, or photons, reached the earth from
space; there are also mysterious “cosmic rays” which had to be fitted into the scheme of things.

  Although the theory of atomic physics is impressive, there too lurked some tricky problems. In particular, Einstein's theory of relativity stood apart from atomic theory, and Schrödinger's equation seemed to be based on the concepts of classical mechanics rather than the relativistic approach. Relativity treats space and time on a more equal footing than is found in classical mechanics; how should that show up in quantum theory? Einstein had shown that there is a certain equivalence between mass and energy summarized in his famous E = mc2 equation; what are the implications of that in the atomic and subatomic worlds?

  The development of those experimental and theoretical challenges led to an undreamed of expansion in the nature of physics and the number of objects to be observed and accounted for. I have chosen six calculations as examples of the steps into modern physics.

  11.1 A STRANGE MAN OPENS UP A WEIRD WORLD

  We are now about to enter the weird and wonderful world of quantum field theory. Some people find the concepts mind-blowing, and the mathematics is difficult to follow. Nevertheless, I want to include this section because it leads to one of the most stunning examples of agreement between theory and experiment at an incredible level of accuracy. And I think the main parts of the story can be followed without worrying too much about the mathematics.

  The first calculation to be discussed in this chapter tells us about a property of one of the fundamental building blocks, the electron. It also introduces you to a pictorial approach to the organization of calculations.