Absorption of radiation. The absorption of radiation by hydrogen atoms was shown by experiment to occur normally at only those frequencies that correspond to the lines of the Lyman series (m = 1, n = 2, 3, 4, ... ). But the light of the frequencies corresponding to the Balmer series, the hydrogen atoms must initially be heated or otherwise put in an excited state. These experimental observations are now explained by assuming, consistent with the whole Bohr model, that an atom can absorb a photon only if the energy hf so absorbed brings the orbital electron exactly into a higher permitted orbit. Unexcited hydrogen atoms are in the most stable or ground state corresponding to n = 1. The energy difference between between that level and all the others correspond to the energies giving rise to the Lyman lines. Any photon of less energy (this would includes all those photons having frequencies corresponding to the Balmer seres) cannot be absorbed by an unexcited atom because it cannot make the electron go from the innermost orbit even to the next or second orbit. Only if the gas is initially in an excited state, will sufficient number of atoms be present where their electrons are in the second orbit and will absorb photons having Balmer series frequencies.
Photoelectric effect. The Bohr model also explains the photoelectric effect. In the photoelectric effect it is known that only light above a threshold frequency f0 will cause photoelectric emission. In the case where illuminated gas is hydrogen, the threshold frequency f0 = ΔE/h, where ΔE is the difference of energy levels between the ground state n = 1 of illuminated gas and the state for which n = ∞. Experimentally it is known that above a certain frequency of the incident light there is no longer line absorption, but continuous absorption of all frequencies. According the Bohr model, a photon that corresponds to the highest frequency line in any series is one that, if absorbed, produces completer ionization of the hydrogen atom by carrying the electron to the level n = ∞. That electron is no longer bound to the atom but is free; and since the energy of the free electrons are not quantized, they can accept and carry (as translational kinetic energy) any amount of energy whatever. That is, when a photon with energy larger than sufficient to free the electron is absorbed, part of the energy is used to ionized the atom, the rest is given to the electron as kinetic energy. And this is what the Einstein's photoeclectric equation (7) expresses, where f0 is the threshold frequency and hf0 is the work function. Bohr said, "Obviously, we get in this way the same expression for the kinetic energy of an electron ejected from an atom by photoelectric effect as that deduced by Einstein. ..."
Spectrum of ionized helium. In 1896, an American astronomer E. C. Pickering, who observed in the absorption spectrum of a star a series of previously unknown lines that nearly coincided in wavelength with the Balmer series for hydrogen, except that an additional line existed between every couple of Balmer type lines. At the time this was thought to be evidence for some celestial modification of ordinary hydrogen. In 1912, A. Fowler found these same lines in the spectrum from a discharge tube containing a mixture of hydrogen and helium. Obviously this was not a celestial phenomena. The wavelengths in this spectrum were given by an empirical equation. Bohr in his first paper acknowledged that his model did not provide for these lines. But, he continued, they can be account for by ascribing them to singly ionized helium atoms He+. A neutral atom of helium, according to the Rutherford-Bohr model, would consist of a nucleus of charge (+)2e and surrounded by two electrons. But if one of the two electrons is stripped off through the violence either of an electric discharge or of mutual atomic collision in a hot star, the resulting helium ion He+ is like a hydrogen except for its doubly positively charged nucleus. The frequencies of the radiation emitted by transitions of the single remaining electron can be calculated from the Bohr model. Thus the source of these lines was not hydrogen, but the singly ionized helium atoms.
In his second postulate Bohr quantized the electronic angular momentum, that is, he assumed that angular momentum of an electron in an allowed orbit must be some whole-number n multiple of h/2π. Atomic angular momentum manifests itself directly in spectra by what is called the Zeeman effect. The Dutch physicist Pieter Zeeman (1865-1943) discovered in 1896 that spectral lines split, that is, a single line is replaced by two or more lines, if the radiating atoms are placed between the poles of a magnet. This can be explained qualitatively on the basis of electronic orbits: every orbiting electron is equivalent to a little loop of current, and therefore equivalent to a tiny magnet. Different orientations of such a loop involve different amounts of energy. Therefore, transitions from orbits of different orientation in a magnetic field give rise to radiations of slightly different frequencies, corresponding to these differences in energy. Angular momentum is a measure of the rotation of the electron in an orbit, and is thus proportional to the strength of the microscopic magnet. Thus one should get a quantitative measure of the angular momentum from the study of the Zeeman effect.
But most of the Zeeman patterns did not fit such a simple explanation. For this reason, and to account for the multiplicity of lines in the spectra of many-electron atoms, Arnold Sommerfeld (1868-1951) introduced in 1915 elliptical orbits into the Bohr theory. An ellipse of the same average radius as that of a circular orbit yields very nearly the same total energy, but will have a smaller angular momentum. This is clear from the correlation between angular momentum and area-per-unit-time swept over by the radius. The introduction of elliptical orbits solved some of the problems of the Bohr theory, but by not all. It made clear that more than one "quantum number" is needed to describe an orbit. The integer n, called the principal quantum number, which gave a good value for the energy in equation (12), cannot specify the angular momentum of an electron in an elliptical orbit, even though Sommerfeld introduced it originally for just circular orbits. It was found that four quantum numbers were needed. Sommerfeld's argument for ellipical orbits led to the introduction of an orbital quantum number to characterize the angular momentum.
The orbital quantum number l specifies the number of units of angular momentum (h/2π) associated with an electron in a given orbit. This quantity can be represented vectorially by an arrow which is parallel to the axis of rotation and whose positive sense is given by the direction of advance of a right hand screw rotated with the motion of rotation. It was found that the integer l could take on only positive values from zero to (n - 1). Thus the electron in the smallest orbit (n = 1) will have no angular momentum, since for it l must equal zero. For a larger orbit, say with n = 3, there are three possible orbital shapes or eccentricities corresponding to l = 0, l = 1, l = 2. Now the total or principal quantum number n gives the energies of the hydrogen atomic orbits, as in Bohr's elementary theory. For a particular n the energy of a Bohr hydrogen-like orbit is the same whether the orbit is circular or ellipical. For multi-electron atoms there may be considerable difference between the energies of circular and elliptical orbits with the same value of n. For circular orbits, inner electrons shield the nucleus and reduce the effective attractive force it may exert on outer electrons. Ellipical orbits penetrate near the nucleus and the electron is strongly attracted over some part of its path. Each value of n corresponds to a particular value of the average orbital distance from the nucleus and determines a "shell" of orbits about the nucleus. The concept of shells originated in the interpretation of x-ray spectra, and the x-ray vocabulary is still used to denote them: the shells are designated K, L, M,... correspond to n = 1, 2, 3, ... Thus the K shell is that nearest the nucleus, L the next, and so on.
A third quantum number, designated m, is called the magnetic quantum number. In an external magnetic field, an orbit with l units of angular momentum may be oriented only in certain "allowed" ways: the vector at right angles to the orbit, representing l, may be directed parallel to the magnetic field, against it (antiparallel), or in any way such that the number of units of angular momentum in the direction of the field is a whole number. For one unit of angular momentum l it may be oriented so that m = 1, 0, or -1. When l = 2, m may be 2, 1, 0, -1, or -2, and so on. When l = 0, m may have only the value 0. The restriction of m to whole numbers signifies that the amount of orbital angular momentum measured along any particular direction in space is restriced to integral multiples of (h/2π).
In 1925 the Dutch physicists George Uhlenbeck and Samuel Goudsmit realized that the Zeeman and related effects could not be systematically accounted for without assuming that an electron has intrinsic angular momentum. This angular momentum was independed of the orbit in which the electron may be and had been graphically called spin. Electron spin is analogous to the angular momentum of the earth due to rotation about its own axis, which the earth has in addition to that due to revolution about the sun. With this discovery of spin, the number of quantum numbers required to specify the behavor of an electron in an orbit came to four, when the various possibilities of orienting angular momentum in a magnetic field are included. This fourth quantum number is called the electron spin quantum number, s. The rotating electron also has a magnetic moment, and spectroscopic observation show the spin vector is capable of orientation only in either of two ways, with (parallel to) or against (antiparallel to) an external magnetic field. Therefore, s can have only two values, +1/2 or -1/2. In absence of a magnetic field, the positive direction of the l vector is taken as the reference direction for m and s. The spin momentum has the single numerical value 1/2(h/2π).
Since the Pauli exclusion rule does not apply to hydrogen, the
one electron in hydrogen may have any set of the quantum number
of their possible values.
n = 1, 2, 3, 4, ...
l = 0, 1, 2, 3, ..., (n - 1)
m = l, (l + 1), ..., (l - 1), l
s = +1/2, -1/2
When this electron is in its lowest energy state, these numbers
must be: n = 1, l = 0, m = 0, s = +1/2.
Helium has two electrons. The lowest-energy electron has
those quantum numbers already assigned to hydrogen, and the second
electron has the next lowest energy set permitted by the Pauli
principle, namely, n = 1, l = 0, m = 0,
s = -1/2. Lithium has three electrons, two of which can have
the quantum number already assigned; these two sets constitute a helium
core. The third electron must have considerably more energy than
either of the first two. The permitted variations of s
have already been exploited. The next lowest energy quantum number
m cannot be increased without increasing l, but
since l must remain less than n, which thus far
has been 1, then the only option is to increase the most important
quantum number n. Thus the lowest set quantum numbers
that can be assigned to the third electron of lithium is
n = 2, l = 0, m = 0, s = +1/2. This procedure
can be continued until the element potassium, where the irregularities
begin. Although the structure becomes more complex with potassium,
it is possible to extend the application of the principle through all
the irregularies and assign quantum numbers to all the electrons
of the unexcited elements.
Helium (Z = 2) is a chemically inert element, belonging to the family of inert gases; it does not form compounds and its gases are monatomic. All this indicates that the helium atom is highly stable, having both of its electron closly bound to the nucleus. It seems to make sense to regard both electrons as moving in the same innermost shell around the nucleus when the atom is excited. But in addition, because of the great stability and chemical inertness of the helium atom, it is reasonable to assume that this shell, which has come to be called the K-shell, cannot accommodate more than two electrons. The single electron of the hydrogen is also in the K-shell when the atom is unexcited. For the element lithium, two electrons are in the K-shell, filling it to capacity, and the third electron starts a new one, called the L-shell. To this single outlying and loosely bound electron must be ascribed the strong chemical affinity of lithium for oxygen, chlorine, and many other elements.
Sodium (Z = 11) is the next element in the periodic table that has chemical properties similar to those of hydrogen and lithium, and this suggests that the sodium atom also is hydrogen-like in having a central about which one electron revolves. And in addition, just as lithium follows an inert gas helium in the periodic table, so does sodium follows another inert gas, neon (Z = 10). For the neon atom, we may assume that 2 of its 10 electrons in the first (K) shell, and that the remaining 8 electrons are in the second (L) shell; and these 8 electrons may be expected to fill the L-shell to capacity because of neon's great stability and inertness. For sodium, then, the 11th electron must be in a third (M) shell. Passing on to postassium (Z = 19), the next alkali metal in the periodic table, we again have the picture of an inner core consisting of a nucleus of charge (+)19e, and 2, 8, 8 electrons occupying the K-, L- and M-shells, respectively, and revolving around this core is the 19th electron, in a fourth (N) shell. The inert argon atom just ahead of it in the periodic table (Z = 18) again represents a distribution of electrons in a tight and stable pattern, with 2 in K-shell, 8 in L-shell, and 8 in the M-shell.
These qualitative considerations have led us to the conception that the electrons are arranged in groups, or shells, concentric about the nucleus. The arrangement of electrons for the inert gases can be taken to be particularly stable and preferred, and each time we encounter a new alkali metal in Group I of the periodic table, a new shell is started with a single electron around a core which resembles the pattern for the preceding inert gas. We may expect that this outlying electron will easily come loose under the attraction of neighboring atoms, and this corresponds with the facts. The alkali metals in compounds or in solution (as in electroysis) spontaneously form ions such as Li+, Na+, and K+, each with one positive net charge (+)e. In the solid state, the outer electrons are relatively free to move about, which accounts for the fact that alkali metas are good electrical conductors.
Several months before Moseley's paper was published, Bohr followed up a suggestion made previously by Thomson, namely that the x-ray lines should be ascribed to the "settling down" of the atoms after the electrons in the inner rings have been completely removed from the atoms, say by the impact of a beam of high-speed electrons. Moseley in his first paper extended this argument beautifully. He reasoned that if sufficient energy was supplied to the electron in the innermost orbit or K-shell to remove it completely from the atom, a vacant place would be left. If an electron from some outer shell could be drop into the vacancy, the result would be the emission of an x-ray photon. Moseley assumed that a photon corresonding to the K line is emitted when, subsequent to the removal of an electron from the K-shell, one from the next (L) shell shall fall into its place; similarly, the second or Kβ line of the K-series is abscribed to the fall of an electron from the M-shell into the K-shell, and so on. With the x-ray lines thus attributed to the inner electrons lying close in the strong field of the nuclear charge Ze, the dependence of the x-ray wavelengths on Z immediately becomes understandable. The electron in hydrogen is a K-shell electron held by the attraction of one proton in the nucleus. The two K electrons in helium are held by two protons and thus their binding energies are greater than (more negative) than that of the electron in hydrogen. The two K electrons of a heavy element are bound by the attraction of many protons. Suppose that one of the two K-shell electrons of an heavy element is moved to infinity by a high speed electron. The remaining outer electrons will now be attracted to the nucleus not by the nuclear charge Ze as Bohr assumed , but by a charge (Ze - e) or e(Z - 1). The remaining K electron may be thought as "screening" the outer elctrons from the full nuclear attraction. Thus the coefficient b in Moseley's law is a nuclear screening constant and it is equals one for K-series lines.
Bohr's model was applied to the spectra of other elements. The spectral lines of a singly ionized helium atom were calculated by Bohr and found to agree with experimental measurement of them. The helium atom, which is normally has two electrons, is singly ionized, if one of the two electrons is stripped from it. But in the case of the heavier elements the exact calculations of frequencies become extremely difficult. There are so many electrons involved that interact with each other, causing such large perturbations in each other's orbit, that ordinary mathematical analysis breaks down. Other methods had to be devised.