Niels Bohr was 19 years old in 1905 when Einstein had his miracle year. The Danish scientist would be second only to Einstein in his scientific contribu­tions in the twentieth century, and he became an unmatched scientist-statesman. He founded the Institute of Theoretical Physics in Copenhagen, which was the breeding ground for most of the ideas that became the new physics of quantum mechanics. Everyone wanted to run their ideas past Bohr to see what he thought. But that is getting ahead of the story. Bohr went to Cambridge in 1911 to study under J. J. Thomson, but he soon became disenchanted with the work going on there, and Thomson was not particularly interested in him. Rutherford was in Manchester by then and he came to Cambridge to speak. Bohr was immediately impressed by Rutherford and wanted to study with him. In the spring of 1912, Bohr moved to Manchester and began studying radioactivity with Rutherford’s group. He also began thinking deeply about the problem of the nuclear atom that Rutherford had just published (7).

Bohr was familiar with the idea of the quantum that had been developed by Planck and Einstein in which nature became discontinuous and only certain dis­crete or quantized values could exist. He began thinking of stable orbits of an elec­tron around the nucleus as being quantized, that is, having only certain discrete values, but he had no theory to explain why they should. Then he heard about work on atomic spectra that had actually been done decades before. Since the mid-1800s it had been known that if you heated an element such as hydrogen or carbon or oxygen, it emitted light that was not continuous but rather consisted of very specific, discrete frequencies. It was possible to identify a specific element by measuring its atomic spectrum—the specific frequencies of light it emitted—but the cause of the spectrum was not understood. In 1885 the Swiss physicist and mathematician Johann Balmer developed a formula that precisely matched the atomic spectrum of hydrogen gas. The frequency of the emitted light in cycles per second was proportional to the difference of the ratios of two integer numbers squared. Mathematically,

f = 3.29 X1015 ( -1 —l— I V n m )

By setting n = 2 and letting m be 3, 4, or higher, this formula precisely agreed with the actual observed frequencies of light from the hydrogen atom (7). And Balmer predicted that n could also be any integer and m could be any integer larger than n, which turned out to be true also. But no one had a clue why the formula worked.

Bohr had gotten married and returned to Copenhagen in 1912 while he was thinking about the problem of the stable atom. A spectroscopy expert there asked Bohr to explain the Balmer formula. Bohr had never heard of it, but as soon as he saw the formula “the whole thing was immediately clear to me” (7). Bohr reasoned that there must be only certain orbits allowed for an electron as it revolved around the nucleus. These orbits are represented by the integers. He postulated that these orbits are stable and are not able to emit radiation, thus solving the problem of Rutherford’s atom, where the electron should collapse into the nucleus. In his quantum view of the hydrogen atom, the electron is most stable in the lowest orbit with n = 1. It can only exist in orbits with n = 1, 2, 3, and so on (Figure 6.2).

When the electron is in its lowest orbit, it is most tightly bound to the nucleus. The electron can be “excited” with a collision or absorption of a photon to move up to a higher orbit (a higher value of n). His critical insight was that when an electron is excited to a higher orbit m, it is unstable and will drop down to a lower orbit n. In the process it will emit a quantum of light (a photon) with an energy hf equal to the difference in energy between the two orbits. This is the same energy formula that Einstein discovered in his theory of the photoelectric effect.

A different way of thinking about the electron orbiting a nucleus is that the electron is trapped in an energy well and has negative energy (Figure 6.3). An electron with negative energy is bound to a nucleus. The energy well has differ­ent energy levels that are also quantized—that is the electron can only have spe­cific energies that vary by the integer n—so the inner orbit has the most negative energy. An electron is normally in the lowest (most negative) energy level with n = 1, which is also the innermost orbit and is called the “ground state” When the electron is given energy by a collision or by absorbing a photon, it moves up to another discrete energy level with n equal to 2 or 3 or higher. It can never be between energy levels, only in one or another of them.

After an electron has been excited to a higher level, it will then jump back down to the lower level and emit a quantum of energy in the form of a photon of light (the wave packets shown in Figure 6.3) that has an energy exactly equal to the difference between the two energy levels and is equal to hf. The larger the jump (e. g., from n = 3 to n = 1), the higher the frequency of the photon and the more

energy it has. This is, in fact, where some X-rays come from. If the electron is given enough energy, it can completely leave the energy well, and it is then a free electron with positive energy. In this case, the atom is ionized.

These two different pictures of the Bohr atom completely account for the light spectrum formula given by Balmer, where the values for n and m are the initial and final integer values for the energy levels of the electron. Bohr’s model also predicted spectra that had not yet been seen but were later measured exactly as predicted. It also fits in very well with Einstein’s photoelectric theory that photons can kick electrons out of the atom. This is the reverse situation to that shown in Figure 6.3; in the photoelectric effect, a photon of light with energy hf can give its energy to an electron and kick it up into a higher orbit or kick it completely out of the energy well.

Bohr understood that he had to make some assumptions that were clearly in violation of classical physics: namely, that the electron could have stable, quan­tized orbits in which it would not radiate energy, and that it emitted photons of light with energy hf when it jumped from one energy level to a lower one. These assumptions explained the observations and allowed predictions, but he did not have an underlying theory of why this should be. That would have to wait.

In 1922, the year in which Bohr received the Nobel Prize for his theory of atomic spectra, he made another great theoretical triumph. He developed a more general theory of atoms that explained the chemical properties of elements in the peri­odical table. By then, Rutherford had shown that the nucleus is made up of pro­tons—the nucleus of the hydrogen atom—with a positive charge. Bohr proposed that the electrons in atoms are arranged in concentric shells, as in Figure 6.2, with only a certain number of electrons allowed in each shell. Once a shell was filled, electrons would have to go into the next shell until the total number of electrons was the same as the number of protons in the nucleus. The electrons in any shell are associated with four different quantum numbers (n, l, m, and s), and Wolfgang Pauli proposed that no two electrons can have the same values for all four quan­tum numbers. This was known as the Pauli Exclusion Principle, and it explained why only a certain number of electrons can exist in any given shell (2). Bohr showed that only the electrons in the outer occupied shell determined the chemi­cal properties of elements; this explained the ordering of chemical properties that Mendeleev first outlined in his periodic table of the elements (12).

Werner Heisenberg was a young German from Bavaria who became captivated by Bohr’s ideas when he heard him lecture in Gottingen, Germany, one of the great scientific centers in Europe where numerous Nobel Prizes were hatched. Heisenberg went to Copenhagen to work with Bohr on the atom. However, Heisenberg was not enamored with the semi-classical model of orbits envisioned by Bohr. After returning to Gottingen in 1925, he focused entirely on a mathemat — ical—rather than physical—approach to the atom and developed a new theory based on arrays of numbers that were multiplied together by certain rules. This form of mathematics is known as matrix algebra; using it, he was able to develop a coherent theory that completely described the Bohr atom but had no physical model associated with it. In 1926 another student of Bohr’s, the Austrian Erwin

Schrodinger, developed another mathematical approach to describing the atom that was based on probability waves rather than jumping electrons. Schrodinger’s wave equations described the probability that an electron could be in a particu­lar orbit. Together, the contrasting mathematical approaches of Heisenberg and Schrodinger were called “quantum mechanics" and Schrodinger proved that they were mathematically equivalent (9, 10). Quantum mechanics, not the Bohr the­ory, provides the true self-consistent description of atoms.

Heisenberg made another great theoretical contribution to quantum mechan­ics, known as the Uncertainty Principle. Fundamentally, Heisenberg did not believe in the reality of objects but only in the reality of measurements. If you cannot measure something, then it does not exist. Classical physics described by Newton assumes that you can know to any degree of certainty where an object is and what momentum it has. But Heisenberg said that this is true only if you can measure it. If you were trying to measure where an electron is, you would try to shine light on it to see it, but you can only see something if it is larger than the wavelength of light used to observe it, so to see an electron you would have to use short wavelength light. Recall that the frequency of light is inversely related to the wavelength, so a short wavelength has a high frequency, which has a high energy (hf). So, in the very process of trying to determine exactly where an electron is, you have to use high frequency light with energy that then moves the electron so you no longer know its momentum (which is its mass times its velocity). The more accurately you try to determine where the electron is, the less accurately you can know how fast it is moving (13). According to Heisenberg, you can only simultaneously determine the momentum and location of an object to an accuracy given by Planck’s constant. Mathematically, the uncertainty (Д) in momentum (Ap) times the uncertainty in position (Ax) is greater than or equal to h, or ДpДx > h (9). Philosophically, the uncertainty principle means that we cannot know the precise details of the world to infinite precision and can therefore not predict future events precisely. It also means that it is impossible to state precisely where an electron’s orbit is around the nucleus. The Heisenberg Uncertainty Principle transformed our way of thinking about what we can know about reality.

Quantum mechanics describes a world that is very hard to think about in visual terms, and it is a world that seems contrary to all our experience. Electrons are not really hard charged balls that circulate around a solid nucleus but are prob­ability waves that can amplify or interfere with each other. In a quantum world, positions and energies of electrons are quantized, so they have a maximum prob­ability of existing in certain places but not in other places. Depending on how you measure them, objects can have properties of waves or properties of particles. Furthermore, there are fundamental limits to how accurately you can measure anything. The world of Newton and Galileo simply does not work in the realm of the atom. Fortunately, we don’t have to worry about it in our macroscopic world.

The semi-classical, semi-quantum picture of an atom developed by Bohr is not really an accurate picture for complex atoms. Quantum mechanics provides the true picture, but it is mathematically complex and difficult to think about.

However, it is easy to conceptualize Bohr’s atom, and it is good enough for our purposes. A specific atom is characterized by a certain number of electrons, which exactly matches the number of protons in the nucleus. All of chemistry is based on the electrons in the outer shell. It is possible to kick an electron out of its orbit and completely out of the atom. In this case, you have a free electron and an atom that has a net positive charge. These are called ions and the process is called ionization. Most of the effects of radiation on cells are determined by the ionization of atoms and molecules. We will look at this in more detail in Chapter 7.