M2 Electron envelope of nuclear atom, their properties and structure (Bohr 1913)

Fundamental postulates for the applicability of the quantum concept to the nuclear atom

The nuclear atom contradicts the classical electro-dynamical theory, because the motion of the electrons should cause the atom to radiate energy continuously, and the radiation should occur with continuously changing (that is, exclude sharp spectral lines) frequency until all electrons have rushed along spirals into the nucleus. Moreover, the structure of the nucleus (similar to the solar system, held together by gravitation) contradicts the constancy of the ordinary physical and chemical properties of the non-radioactive elements, because a strange celestial body, which would travel through the solar system and were to get near to Earth, would, for example, influence its rotation about its axis and the duration of its orbit around the Sun (lengths of day and year). In contrast, the constancy of the properties of the elements demonstrates - for example, a gas atom survives countless impacts without a change - that the state of a single atom, as well as of transversing electrons and a-particles is changed by external action only extremely rarely. This is displayed best by the constancy of radiation, which substances emit under special conditions: The wave lengths of the spectral lines remain unchanged - independently of the preceding treatment of the substances. It is just on this aspect on which rests spectrum analysis and, for example, also the proof, yielded by it, that the elements on the celestial bodies have the same properties as on Earth. This constancy of the atomic properties and the feasibility of spectral lines seem to be, to start with, incompatible with the nuclear atom. However, Bohr has been able to explain them as well by application of the Quantum Theory to the nuclear atom, starting from two basic postulates, which, while appearing at first to be arbitrary, were experimentally justified (at first, through the results of the electron impact experiments of Franck and Hertz 1914). These postulates - the first concerns the constancy of atomic properties, the second the possibility of sharp spectral lines - are based on two assumptions:

1. Certain electron orbits in the nuclear atom are distinguished by the fact that the atom can persist in the states of motion, characterized by them, without radiating energy*; these states are called stationary. The atom can lose or intake energy only by transition from one stationary state to another stationary state. - A certain stationary state of an atom is characterized by the form of the electron orbits, their positions in space, the rotary impulse of the electrons on their orbits (revolution) and the spin of the electrons. Each of these determining items can be quantified and therefore be specified by a quantum number.

*Also the constancy of the para- and ferro-magnetism, generated by circulating electrons, supports the existence of orbits without radiation. - The absence of radiation is a condition for the stability of the properties of atoms; it does not at all explain of what its nature consists in the structure of the atom; that is only explained by the X-ray spectra.

2. The nuclear atom can radiate electro-magnetic waves during a transition between two stationary states. The periodicity n is given** by hn = E_a - E_e (= DE) - Bohr's frequency condition - where E_a and E_e are the energies of the atom in the initial and final stationary states. Corresponding to this frequency condition, the atom can also absorb electro-magnetic waves with the frequency n by which it is irradiated***.

** n = 1/t is the frequency, = 1/l the wave number of a wave length; n has the dimension sec^-1, while has the dimension cm^-1. Since l/t = c (velocity of light), you have n/ = c, that is n = ·c.
*** Absorption of irradiated light is here not absorption in the ordinary sense, but only a storage of the intaken energy for a very short period (sojourn time of order 10^-9 to 1-0^-8 sec); the light is not converted into heat. The atom can radiate it again unchanged, as is demonstrated by fluorescence phenomena.

The kinds of light, which an atom absorbs or radiates, are mono-chromatic, depending on whether it goes from a state with less energy to a state with more energy or conversely. Their frequency follows from the equation hn = DE. (The frequency of the transitions between the different states - transition probability - is very different. Certain regular series of transitions, for example, such which yield views of mono-chromatic emission- or absorption-lines, happen on their own and very frequently, as is proved by the brightness of the corresponding lines; other transitions occur, as a rule, rarely or not at all; however, an electric or a magnetic field or impact of electrons can enforce them.)

These postulates are complemented by a third one - the correspondence principle; it places the transitions accompanied by radiation in parallel with the oscillations, accompanied by radiation, of electric particles of classical electro-dynamics. It says: A transition between two stationary states, which is accompanied by radiation, is uniquely adjoined to one of the components of the oscillation, into which the motion of the electrons can be decomposed. This correspondence demands that the probability of a transition (in other words: Its frequency - it determines the intensity of the radiation) depends on the amplitude of the corresponding harmonic component (which in the classical theory determines the radiation). We will return below to the correspondence principle and deal for the time being only with the first two postulates.

Bohr's theory applies exclusively to the environment of the nucleus (electron swarm). The details of its structure can only be concluded from their properties. These properties can be subdivided into two groups: Into those which adhere to those electrons which are close to the periphery (peripheric properties) and into those which adhere to those close to the nucleus (central properties). For example, the first are the chemical and optical properties, the second the X-ray properties of an atom. For example, the first appear in the optical spectra, the second in the X-ray spectra. The postulates do not at all distinguish between electrons; they relate to the peripheric ones as well as to those close to the nucleus.

Experimental justification of Bohr's postulates

You justify the first two postulates especially by planned, induced phenomena, which occur when electrons collide with atoms and molecules, indeed unelastically with exchange of energy. If an electrom impacts at a sufficiently large velocity* against an atom - more exactly: If the field of an electron with sufficient energy impacts the field of an electron of the shell of electrons, there can occur two actions:

* Let an initially resting electron pass in an electric field through a potential difference of V Volt. Its kinetic eenrgy attains in the process the magnitude 1/2mv² = 1/300eV. With m = 0.899·10^-27g and e = 4.77·10^-10 electro-static units, the velocity becomes v = 5.95·10⁷(V)^1/2 or 595(number of Volt)^1/2.km/sec. As a rule, the velocity of an eelctron is stated in terms of equivalent numbers of Volt. An eelctron has a 4 Volt velocity means: Its velocity is 5.95(4)^1/2 , that is, 1190 km/sec.

1. The impact can transport an electron of the atom from the first quantum orbit to one further outwards, you say: Excite the atom. For this to happen, the impacting electron must have kinetic energy of a certain magnitude, corresponding to a certain number of Volts - excitation tensionthe energy is smaller, the electron rebounds without loss of energy from the atom without . (If exciting it.)
2. The impact can drive off an electron from the electron shell, in other words: Can transport it to an infinitely distant quantum orbit, you say: Ionize the atom. This demands more energy and correspondingly a higher Volt velocity.

Only the (quantum theoretically admissible) atomic state, which is most deficient in energy, is stable, every one with more energy (higher energy) has, in general, shorter medium duration of life (residence time). After its expiration (10^-9 to 10^-8 sec), the atom transcends with emission of mono-chromatic radiation to a lower (energy poorer) quantum state. You call the stable state of an atom normal - also unexcited -, every higher (energy richer) quantum state excited. (All atoms are still unexcited at several 100º C, for example, in Na even at 3000º C only about 0.3 pro mill are excited; the fact that it nevertheless glows visibly in a Bunsen flame (about 1800 C) is due to the myriads of transitions of the altogether present molecules - in the mol altogether 6.06·10²³, that is about 10²⁰ excited ones).

Hence excitation of an atom manifests itself by its radiation as the electron, which has been displaced to an orbit further outwards, returns to its initial orbit. Franck and Hertz 1914 have demonstrated spectrographically, firstly, clearly different stationary states of an atom and, secondly, the link between the energy difference of different states and the frequency of the transition radiation; they have thereby justified Bohr's postulates. Their experiment (Fig. 831) examines the impacting action of electrons on Hg-atoms. The mercury vapour and an electron source (3 - 4 mm long, glowing, electrically heated Pt-wire) are located in a cylindrical glass vessel, which is heated for the generation of the required vapour pressure and held at low pressure by a vacuum pump. Between the glowing electrode and an earthed electrode, which surrounds it radially at about 4 cm distance (receiver, platinum foil) lies an electric field with variable tension. It accelerates the electrons to the receiving electrode and drives them in the process against the mercury atoms. Just ahead of the receiver (1 - 2 mm distance) and coaxially with it lies a fine mesh platinum wire net, through which the electrons must pass on their way to the receiving electrode. However, between it and the net lies an electric field, oppositely directed to the first one (counter field) with a constant tension of ½ Volt. It keeps electrons away from the receiving electrode, but also from the galvanometer, the velocity of which corresponds to less than ½ Volt (about 420 km/sec). In fact, only electrons are to enter the galvanometer, which impact the atom elastically, that is, which have retained their velocity. If the counter field did not exist, also those electrons would reach there, which have impacted the atom inelastically and have thereby lost their velocity. The galvanometer could then not indicate sharply the tension at which the impact becomes inelastic.

When you increase the tension of the accelerating field beyond ½ Volt, the electrons passing through the net reach the receiving electrode and the galvanometer reacts. If you raise the tension further, it raises the galvanometer current - but only until its has reached 4.9 Volt, a minute increase of the tension beyond this value makes it drop steeply (Fig. 832) to a lowest (non-zero) value. If you now increase the tension further, the galvanometer current rises again until it reaches the tension of 2x4.0 Volt, but drops again for a small further rise in the tension to the lowest value, and then starts again to rise until the tension of 3x4.9 Volt is reached, etc. This process has the explanation: When the velocity of the electron has reached the critical value, which according to the equation ½mv² = eV corresponds to a critical tension V_R, its impact against the atom becomes inelastic and it imparts its entire energy to the atom. This energy lifts one electron of the atom to a higher quantum orbit. The location in the gas (in the section of the gas space, represented in the figure), which has with respect to the glowing electrode the tension of 4.9 Volt, lies at 4.9 Volt total tension close to the platinum net. The impacting electrons lose here their velocity, cannot therefore overcome the counter field between the net and the receiving electrode and cannot reach the electrode. Hence, the current of the electrons, reaching the electrode, must drop. If you raise the total tension, the location, which has with respect to the glowing electrode only 4.9 Volt (or: The location where the electrons have reached the critical Volt velocity V_R), away from the net into the gas space. The electrons, which here impact the Hg-atoms inelastically and lose their velocity, are from here on again accelerated and reach the net with a velocity V', which measured in Volt is V' = V - V_E. As soon as V' > ½ Volt, also the electrons, which have lost 4.9 Volt through the impact, can again overcome the opposing field. Hence the current must again increase. If we further raise the accelerating tension V until the V'-velocity of the electrons has become equal to V_R, they impact ahead of the net for the second time inelastically. That is the case, when the accelerating tension V = 2V_R; close ahead of the net forms then a second location, where the electrons impact inelastically the Hg-atoms. At this instant, the current must drop for the second time. The location of impact shifts with once more increasing tension V again towards the cathode into the gas space, and the first moves yet closer to the the glowing electrode. When V = 3V_R, there arises in the same manner a third such location, etc. Hence we have to expect such locations of impact at distances which are multiples of 4.9 Volt*. You can make them directly visible in the Hg-vapour, if its pressure is high enough. The energy, taken in by the electron impact, is then re-radiated in a band spectrum, which has its brightness maximum in green. You see the green glowing layers at distances similar to those shown in Fig. 833 (Grotian 1921).

*The distance between neighbouring maxima measures in Volt the energy, which was transferred during an inelastic impact to the atom. The efficiency of the impacts is then in the most favourable case (at a maximum of the curve) only 1%, that is, of all electrons, which impact, only every hundredth passes its energy on to the atom (Sponer 1921).

The presence of the differently stationary states of the atom with definite energy values has thereby become very probable**.

**The same is found (Gerlach-Otto Stern 1888-1969 1922) during the splitting of Ag-vapour in a magnetic field into two rays, each corresponding to a definite adjustment of the magnetic moment of the atom - parallel and anti-parallel to the magnetic field. There do not exist "intermediate states" between these two.

The results of these experiments also justify the postulate in the form hn = E_a - E_e. In order to radiate a spectral line, the atom must transit from an energy richer state into an energy poorer state. Hence the excitation tension for a definite spectral line (the tension through which the the electrons have to pass in order to excite the atom to send out this spectral line) is equal to the excitation tension for the higher one of the two states, between which the transition is to occur in order to induce the emission of this spectral line - that is, it is given by eV = E_a - E₀, where e is the charge of the electron and E_a and E₀ are the energies of the atom in the normal and excited state, respectively. A comparison of this equation with Bohr's frequency condition hn = E_a - E_e indicates that, in general, there does not hold a relationship between the frequency n of a spectral line and of its excitation tension V. However, there holds a relationship for those lines, radiated by the atom, when it transits from the higher state E_a to the normal state E₀. These lines demand for their excitation the energy eV = E_a - E₀ and radiate the energy hn = E_a - E₀, whence eV = hn, that is, V is the excitation tension of the spectral line with the frequency n. If the atom returns to the normal state E₀ after it had been brought into that state E₁, which lies closest to the normal state, it can only radiate a single line, namely the one given byn = (E_a - E₀)/h, - there does not exist another transition. And in this line, the atom radiates the entire energy, which it had previously taken in, in order to reach the state E₁. This radiation is called resonance radiation, which we know already from fluorescence; it has been given this name, because the classical theory of optics explains it by the resonance of quasi-elastically linked electrons in the atom. The corresponding line is called resonance line. [It is the first line of the absorption series of the unexcited atom. This has the following interpretation: Not every line, which we know as emission line, exists also as absorption line. A gas or a vapour in its ordinary state, that is, all of the atoms which are in the normal state (not excited), absorbs only such lines which arise in emission during a transition of the atom to the normal state. Only at higher temperatures (also during electric discharges) absorbs it also other lines, because it then contains a large number of atoms above the normal state (excited)]

In a resonance line, the total energy taken in from a radiation with frequency n is again emitted as radiation with the frequency n. If the atom has taken in the energy from impacts of electrons, which have just passed through the excitation tension of the resonance line, it must exclusively emit radiation of the resonance line. Hence Bohr's theory lets you expect: The excitation tension of the resonance line follows according to the hn-relation from its frequency and the atom, impacted by electrons, which have passed freely through this excitation tension, emits the resonance line and only this line. The line
l 2536.7 was known as resonance line of mercury (Wood). Franck and Hertz have demonstrated the excitation of this line (Fig. 834) by electrons, the energy of which equals the frequency of the line multiplied by h. In order that the electron during impact on Hg-atoms releases this radiation, it must be accelerated in accordance with the equation eV = hn by the potential

V =(6.56·10^-27x1.183·10⁺¹⁵)/4.77·10^-10 = 0.0162 abs. electrostatic units = 0.0162·300 = 4.86 Volt.

Hence the computed potential required for the excitation of l 2536 is 4.86 Volt, the measured one 4.9 Volt. - The relationship between the wave length l of a resonance line and the tension its excitation demands is given by the simple formula: V (in Volt) x l (in mm) = 1234. The derivation of this formula starts from h·c/l = eV; you substitute for e/c the value 1.59·10^-20. For example, the resonance tension of mercury is V = 1234/253.6 = 4.9 Volt.

[For the justification of the theory are just those lines important, for which the relations are less simple than for the resonance line. Hertz has also discovered that the appearance of the higher series lines (in Neon, Helium, mercury) agree at characteristic excitation tensions with the computed values.]

Bohr's postulates lead to an explanation of the spectra of the elements from the structure of their atoms (Hg-atom and Hg-spectrum)

The postulates give no information regarding the structure of the atom and the orbits of the electrons. The first postulate assumes for the electrons only the existence of an infinity of discrete orbits about the nucleus, the second the possibility of spectral lines. The quantum theoretical version of the second postulate now allows to relate the spectra of elements to the atomic structure. In fact, the form of the frequency formula, according to the transition postulate in the form n = E_a/h - E_e/h, agrees strikingly with the wave number formula according to the empirically found combination principle (Ritz) = R/n²₂ - R/n²₁. It caused Bohr to suspect that the spectrum forms during transitions of the atom between stationary states, in which the numerical values of the energy of an atom equal the h-fold value of the spectral term (E = h·R/n²). In other words, Bohr linked (with the help of the second postulate) the spectrum of an element to the structure of its atom in that he interpreted the terms R/n² as the values of the energy of the atom in its different quantum states, divided by h. This interpretation of the combination principle by the postulates could, since its origin is only formal, lack* a physical meaning, but the examination of the H-spectrum shows that it is physically justified.

*The interpretation could be without physical meaning because it rests only on the formal agreement of the two expressions for and n and contains none of the established concepts of electro-dynamics, because it lacks any assumption of a connection between inner-atomic motion and atomic radiation. Moreover, according to this interpretation, also the final state, into which the atom transits (not only the state, from which it starts) determines the nature of the radiation it emits (wave length, brightness).

The H-atom - a proton and an electron which circulates around the proton - has the simplest structure, its spectrum is the simplest we know about. The wave number of its lines embraces the formula = R/n²₂ - R/n²₁ of Johann Jacob Balmer 1825-1898. According to Bohr's interpretation, every line forms at a transition of the H-atom between two stationary states, in which its energy has the numerical value h·R/n². However, the justification of this interpretation must first be proved. The proof is as follows: Starting from this interpretation, you reach with the aid of the transition postulate the insight that h·R/n² also represents the work, which must be performed on the atom in order to transport the electron from the n-th quantum orbit to one infinitely far away, in other words: In order to remove the electron from the range of attraction of the nucleus, that is, to separate it from the atom, the atom must be converted into an ion (separation work, initiation work). If we denote this work by A_n, then also A_n = h·R/n². You come from here, first of all, by computations, to the diameter of the innermost quantum orbit, that is, to the diameter of the not excited H-atom, moreover to the magnitude of the separation work (ionization work) and even to the Rydberg-constant of the H-atom. The values of all the three quantities, computed in this manner, agree with values, found by quite different means: The atom's diameter with that known from the Kinetic Gas Theory, the ionization work with the one measured, the Rydberg-constant with the value determined spectroscopically. - this justifies Bohr's interpretation: The terms are energy values of the atom in the stationary states, and hn is the loss of energy of the atom during radiation (energy during absorption) of the corresponding line of the frequency n . The separation work zero is allotted to the infinitely distant orbit - the ionized atom - that is, the term 0. Since now the terms are to signify the energy values of the atom in the stationary states, we must allot to the energy of the ionized atom the value zero. We employ thereby the energy value of the ion as origin and therefore must give the other energy values negative values - negative, because the atom is to take in energy, in order to reach the state of the ion; its energy value rises to that value, which we call for formal reasons zero. This should strictly speaking be: According to Bohr, the terms of the spectral series signify the set of negative and by h divided values of the energy of the atom in its various quantum states, the energy of the atom as positive ion being set equal to zero.

Correspondence Principle

The assumption that the terms are the energy values of the atom in the stationary states is supported among others by the agreement of the computed Rydberg-constant with the spectroscopically determined one. The computation of th this constant is of special interest. It is based on a reflection in which the quantum theory of the nuclear atom comes close to classical electro- dynamics. This idea is linked to those orbits without radiation, the numbers of which (in the sequence of the orbits starting from the nucleus) are very large, and the dimensions and the rotation numbers of which change by comparatively little from orbit to orbit. The n of the radiation, which forms during the transition between such neighbouring orbits, coincides very nearly with the n of one of the wave systems (corresponding to fundamental tone and over tones) of the radiation, which the electron would emit classically as a result of its motion. The systematic demand of this agreement yields that the Rydberg-constant can be expressed by R = 2p²e⁴m/h³. Indeed, this relationship now really holds - R becomes with the known values of e and m and h: 109,737.11 0.06 cm^-1, while the spectroscopically determined value is 109,677.69 0.06 cm^-1. This happens as follows: The third Kepler law yields for the circulation frequency on the nth stationary orbit - irrespectively of whether it is a circle or an ellipse - w = r/n². Balmer's formula yields for the oscillation frequency n during the transition between two neighbouring orbits n and n - 1 for very large n: n = 2R/n³. Thus, for orbits with very large n, n is also inversely proportional to n³. Here r is the circulation time on the first orbit, R Rydberg's constant. However, is the assumption that r = 2R justified? This assumption yields R = e²/16ma₁³p² and hence with a₁ = e²/2hR: R = 2p²e⁴m/h³. The values of e, h and m yield the Rydberg-constant in complete agreement with its spectroscopically determined value*. This result justifies the assumption that r = 2R, that is, it proves the agreement of Bohr's radiation with the classical one for large orbit numbers. It supports the correctness of the argument, which guided Bohr (1917) to his Correspondence Principle, which also holds for smaller quantum numbers.

*For circular orbits and with R = 2p²e⁴m/h³, the impulsive moment of the electron in its nth orbit is proportional to n: nh/2p - in words: among the infinitely many circular orbits, which the electron in the H-atom could follow about the nucleus according to Newton's mechanics, its orbits without radiation are characterized by the fact that on them the impulsive moment is an integral multiple of h/2p, that is, that m·v·a_n=n·h/2p. However, this applies only to circular orbits. The formula becomes more complicated for ellipses.

The Correspondence Principle states: A transition between two stationary states, accompanied by radiation, is uniquely allotted to one of the harmonic components of oscillation, into which the motion of electrons can be decomposed. This correspondence demands that the probability of a transition (its frequency - it determines the intensity of the radiation) depends on the amplitude of the corresponding harmonic component (which determines in the classical theory the intensity), and indeed in such a manner that for large quantum numbers the intensity of the (per unit time) emitted radiation (in the mean) is the same as it would be according to classical electro-dynamics. The polarization of the radiation should display a similar analogy with classical electro-dynamics: If the corresponding harmonic oscillation in all states of the atom is rectilinear and parallel (circular and perpendicular, respectively) to a fixed line, then the radiation will be just like that which a likewise oscillating electron emits (rectilinearly and circularly polarized, respectively).

Like the first two postulates, also the correspondence principle is justified by experience - impressively, for example, by its performance in the explanation of the Stark-effect (Johann Stark 1874-1957). It explains the polarization of the individual components, into which the lines are split, as well as the distribution of the intensity between the individual components. The theory returns all essentials of the experimental results. In the light of the correspondence principle, the Stark-effect displays in the smallest details the action, which an external electric field exercises on the electron orbits in the hydrogen atom.

Structure and spectrum of the hydrogen atom. Fine structure of its lines

Starting from A = h·R/n², the separation work from the 1. orbit is hR, from the 2.,3.,4., ··· orbit is 1/4, 1/9. 1/16 ··· of it. From the charge of the nucleus and the electron - both are equal to e - and the separation work from a given orbit simple mechanical considerations * yield the radius of this orbit a_n = e²n²/2·h·R. With the numerical values for e, h and R, you find for n = 1 the radius of the innermost quantum orbit a₁ = 0.532·10^-8 cm. For n = 1, 2, 3, ···, you find orbits the diameters of which are related to each other like 1 : 4 : 9, that is, like the squares of the orbit numbers.

*Bohr starts by assuming that A_n = h·R/n². A_n, the energy required to bring the electron from the nth orbit to infinity, can be represented by the radius a_n of the circular orbit (under the assumption that the ordinary mechanics and electro-statics apply to the attraction between the electron and the nucleus). This energy is the difference of two quantities of energy (we omit the computation): The work e²/a_n, which must be performed against the attraction of the nucleus along the track between the distances a_n and , reduced by the energy of the motion e²/2a_n, which the electron had on the nth orbit; this energy becomes available, because the velocity vanishes on the orbit at , whence A_n=e²/a_n-e²/2a_n=e²/2a_n.

The fact that the structure of the H-atom and the sequence of the lines of its spectrum are interlinked is demonstrated as follows: If you leave in c = R/n₂² - R₁/n² n₂= 2 unchanged and then set sequentially n₁ = 3, 4, 5 · · ·, you find for successively R(1/2² - 1/3²), R(1/2² - 1/4²), R(1/2² - 1/5²) · · ·. These are exactly the wave numbers, which the spectroscope yields for the Balmer series (H_aH_bH_g). According to Bohr, the quantities (1/2² - 1/3²), · · · are the wave numbers, which the atom emits, as the electron moves from the 3., 4., 5., ··· orbit to the 2. orbit and it itself transits from the 3.,4.,5.,· · · orbit to the 2. orbit. Hence we arrive at the conclusion: To the electron transitions from the 3.,4.,5.,··· orbit to the 2. orbit correspond in the H-spectrum the (H_aH_bH_g) lines. H_a and H_b correspond to the transits of the atom 3 - 2 and 4 - 2 in Fig. 835.

However, only under exceptional circumstances, the orbits of the electrons are circles; in general, they are ellipses. This insight is arrived at as follows: The H-spectrum looks differently than in Fig. 807 in a very strongly resolving spectroscope. Every line ( previously seen individually) becomes a group of closely spaced lines, and these groups are separated by comparatively wide gaps, displayed in Fig. 807. The splitting of a line into a group of lines is referred to as fine structure, and Sommerfeld has explained this fine structure on the basis of the assumption that the electron tracks are ellipses. [This discovery expanded Bohr's theory not only in its significance for spectroscopy. It also opened up a path for the explanation of a period in the periodic system of the chemical elements.] Also the lines of the fine structure are interpreted as transits of the atom between stationary states. However, the minuteness of the difference (0.364 cm^-1) in the wave numbers of the fine structure lines, for example, of the lines which you see at the location of the H_a-line, can only be due to transitions between states of energy, which lie close together and must be quite different from the transit 32, which would generate the H_a line. According to Sommerfeld's interpretation of the fine structure, the electron in the H-atom described an ellipse. Due to its enormous velocity, which changes from one point on the orbit to another, its mass changes (relativistically *), whence the ellipse rotates in its plane (Fig. 838).

*While in the circular orbits the velocity and also the electron mass remain constant, it changes greatly on the elliptic tracks. especially along such tracks with large eccentricity. The velocity at the aphe;ion is small and rises as the perihelion is approached. As the electron on such an orbit passes close to the nucleus and is exposed to the strong force of the field near it, there increases with the velocity also its mass. Hence you will understand that the energy of the electron differs from that on the circular orbit the more, the more eccentric is the elliptic orbit, and that for the line components, corresponding to the different orbits, there must arise different positions in the spectrum (Sommerfeld: Atomic structure and spectral lines, 4.Ed.)

The conclusion that that the electron travels along an ellipse (in motion) is decisive for the explanation of the fine structure. In fact, if the electron were to travel along an ellipse at rest, its energy constant would only depend on the large axis A and would be the same as when the electron travels along a circle with the diameter of the large axis. It is quite different in the case when the elliptic orbit rotates! Then the energy content depends also on the form of the ellipse (its larger or smaller slenderness), that is, on its eccentricity e or - what results in the same - also on the small axis b [b/a = (1 - e ²)^½]. In summary:
1. The elliptic form of the orbit causes its rotation;
2. the rotation of the orbit causes the dependence of the energy content on the eccentricity e of the orbit. And now follows something very important:
3. The eccentricity of the orbit is not arbitrary, but there are only allowed certain quantum-like, stepped eccentricities - in other words: Only certain small axes b.
Having denoted the quantum orbits by n and numbered them 1, 2, 3, · · ·, we denote the admissible small axes by k and number them in the same way. We call n the primary quantum number, k the secondary quantum number, and will speak of n_k orbits. The relationship between the small (b) and large (a) axis, on the one hand, and the quantum numbers k and n, on the other hand, is b/a = (1 - e ²)^½ = k/n. Besides the circular orbit 3, we have still two elliptic orbits - 3₁ and 3₂ - and the corresponding stationary states. (The quantum orbit 3₃ is the circular orbit hitherto denoted by 5, since for k = n you have b = a). You can now conceive the transitions

but only

are allowed In fact, the transitions are subject to certain conditions (selection rules): They follow from the correspondence principle. According to the selection rule, which applies here, only such lines appear with detectable brightens, which correspond to transitions between n_k orbits, the k of which differ by 1. For n, every change Dn is admitted, for k only Dk = 1.

Stationary orbits with the same primary quantum number like 3_³3_²3_¹ give the atom almost the same energy content - denote it by E_³³, E_²³, E_¹³, respectively. During transitions such as 3_³2_², 3_²2_¹, 3_¹ 2_², its loss of energy is therefore almost the same, that is, almost E_³³ - E_²² = E_²³ - E_¹² = E_¹³ - E_²². Hence, the atom emits during all these transitions almost the same n. In other words: Closely spaced spectral lines correspond to all these transitions. To the transition 3_³2_², there corresponds so small an energy hn, that is so small a n, that the corresponding line lies undetectably in the ultrared. Indeed, the spectroscope displays each of the lines H_aH_bH_g · · ·as double lines - and not only this happens: Closeby each of the two lines lie still other lines, arranged symmetrically. The wave numbers of the double lines differ at all locations of the spectrum, according to Sommerfeld's theory, by the constant distance D_H = 0.3636 cm^-1. Measurements yield approximately 0.32 cm^-1 (Michelson 1887).

Symbols for the electron terms and their combinations

The perception that the spectral terms of the series formula represent states of energy determines the task of spectroscopy in the service of Atomic Physics: It is to discover the electron terms. The spectroscopyist proceeds for it as follows: In general, the lines, which he sees in the spectrum of the atom, differ by their appearance and behaviour. (Appearance: Simple lines or double- or triplet-system, intensity ratio, characteristic burning, line-length. - Behaviour: At a temperature change, a pressure change, in a magnetic field, according to capability of being excited.) However, some lines have some details in common. The spetroscopyist attempts to sort the lines according to these details, to order every individual sort* in a series. The criterion for a line sequence to really form a series is its ability to be summarized in a formula - indeed a formula for the wave numbers ( = T₂ - T₁ according to the presentation of every wave number as the difference of two electron terms T; it returns the wave numbers. The electron terms are what the atomic physicist must have.

*Most series have been discovered accidentally. Most difficult is the discovery of simple lines, especially vacuum light sources, as there is no lack of sharpness (Paschen).

In the H-spectrum, the wave numbers of all lines can be collected in the single formula = R(1/n² - 1/m²), in which R is the Rydberg-constant and n,m are integers. If n or m passes through the series of integers, there forms an infinite series of T-values - a term sequence. For hydrogen, there exists only a single term sequence (because the m-term-sequence T(m) and the n-term-sequence T(n) are identical). The formulae for the individual series (Balmer, Lyman, Paschen, etc.) arise by choosing for T₂ a value T(n) with a definite n (= n₁) and combination, according to the combination principle, with a term sequence T₁(m) in which you can set m to all values larger than n. The frequency of the border of the series (m = ) - the limiting term - equals T₂. The term T₁ passes through a (with increasing m to 0 converging) infinite sequence of values and is therefore called the running term. The variable numbers are called running numbers.

The H-spectrum can be dominated by a single series formula. It is quite different for the other spectra, for example, the alkali-metals, although these have the greatest similarity with the H-spectrum. In general, the lines of the spectrum of an alkali-metal, for example, of Li, Na, K can be ordered in 4 series. They are called the principal series, the 1. (diffuse) subordinate series, the 2. (sharp) subordinate series, the Bergmann-series (called p-, d-, s-, f-series, respectively**). According to the combination principle, everyone is to obey the formulae = T₁ - T(m), where T₁ represents a fixed term, belonging to the corresponding series, and T(m) the running term. T(m) is a function of m, similar to the function R/m² in the formula of the series of the H-spectrum. Many series (found by Rydberg by trial and error) can be well represented by the function T(m) = R/(m + a)² as running term, where R is Rydberg's constant, m the running number and a a constant real fraction. (It is an additive correction to m, which decides at which number m starts its run. In order to be quite strict, the formula demands further corrections, however, it will be sufficient for the present purpose.) For everyone of the 4 series, the fraction a has another value; it must be determined empirically. You denote it in the principal series by p, in the primary subordinate series by d, in the secondary such series by s, in the Bergmann-series by f. The running term of the principal series is therefore R/(m + p)², that is, the value which it assumes for m = 1. The formula for the principal series is therefore
= R/(1 + s)² - R/(m + p)²; for example, p for Lithium is still very small, which means that the corresponding term sequence is very close to that for hydrogen. In contrast, s has about the value +0.6, which means a very strong deviation from the similarity with hydrogen - a peculiarity which is characteristic for most spectra for the running term of the second subordinate series.

**This is in agreement with the English terminology: Principal, diffuse, sharp, fundamental.

The running number starts at m = 2. It is similar for the other series. For the sake of abbreviation, the terms are denoted by symbols. For example, Paschen abbreviates R/(m + a)² by m a. In this abbreviation, the formulae of the series become

term		running number			series
= 1s - mp		m = 2, 3, 4			principal
= 2p - ms		m = 2, 3, 4			II. subordinate series
= 2p - md		m = 3, 4, 5			I. subordinate series
= 3d - mf		m 4, 5, 6			Bergmann-series

In words, this means, for example for the principal series: The wave number of the m-th line of the p-series is equal to the value of the function s(m) for m = 1, reduced by the value of the function p(m) for the value m, which belongs to the m-th line of the p-series - m can be any integer between 2 and . You say briefly: s-term, f-term, also s-term-sequence, etc.

These formulae yield a system of series of simple lines, a singulet series. In order to be able to reproduce the doublet-, triplet-, etc, to multiplet-series, the symbols for the terms had to be changed. In general, the changes, introduced into the notation above (Paschen) by Russell and Saunders is employed. Paschen's symbols nS, nP, nD, nF for the singulet term (capital letters instead of the lower case letters used previously) are now n¹S₀, n¹P₁, n¹D₂, n¹F₃ (the superscript 1 points to the singulet-series, the subscripts 0, 1, 2, 3 re the quantum numbers, which we need not discuss here). The doublet terms

(1 and 2 are the levels of the doublet components).

Consequences of Bohr's interpretation of the terms as energy values

The justified interpretation of the terms of the H-spectrum suggest to do the same for the series spectra of the other elements. However, only a single electron circulates around the nucleus of the H-atom (atomic number 1), while two electrons circulate around that of the He-atom (atomic number 2). If the He-atom loses one of them, so that it becomes a helium-ion (He⁺), it has the same simple structure as the neutral H-atom (Fig. 828); the frequencies of its spectral lines can also be computed and agree with experiment. However, if we must deal with several electrons on orbits, as already in the case of the neutral He-atom, the result of the calculations contradicts the facts, and even more so when more electrons have to be allowed for. However, nevertheless, the path followed so far brings an advance: Also the spectra of the elements with larger atomic numbers are in a certain sense similar to the H-spectrum: Also their lines can be adjoined, in spite of their apparent lack of clarity, - indeed not to a single series of integers - to several series and everyone of them is very similar to that of the H-spectrum. The similarity manifests itself objectively in that the empirical expression for the atomic term in every individual series can be represented by the same function T(m)=R/(m + a)², and above all in that R is everywhere almost equal to the Rydberg-constant of hydrogen.

For this reason, the theory essentially ascribes a spectrum to an electron - the emitting electron - and to the terms of the spectrum its stationary orbits. In order to be able to ascribe to the emitting electron a periodic motion (without change of its energy while it traverses the shell of electrons, the theory replaces the atom's body by a centrally symmetric field of force with the nucleus as centre. For the emitting electron, which circulates a nucleus with N times as large charge as has the H-nucleus, the computation yields the conclusion: In a given stationary state, the work required to remove the electron from the range of attraction of the nucleus is proportional to N². The spectrum, emitted during transitions between stationary states, can therefore be represented by

= N²R(1/n²₂ - 1/n₁²).

Bohr's interpretation of the terms as values of energy leads to further insight. The transition of an electron from its ground orbit to one with higher quantum changes the atom's interior; it probably acquires thereby another chemical property. In general, it cannot be observed, because its life time (resident time of the excited state) is too short (according to the theory and measurements on an average of the order of magnitude of 10^-8 sec). But we can say about it that in different stationary states of the atom the emitting electron is bound to it differently strongly. This conclusion is reached as follows: Every term yields the difference between the energy of its corresponding stationary state of the atom and the energy of the ionized atom - in other words: Every term yields the energy required to convert the atom from that stationary state into that of the ion. The term of the normal state must therefore be equal to the ionization work of the atom, divided by h. This term corresponds in the optical spectrum to the frequency of the border of the first series, that is, to the frequency to which the frequencies of the lines of the principal series converge as the atomic number increases. Hence, the frequency of the border of the principal series, multiplied by h, must yield the atom's ionization work. Measurements of the ionization tension have confirmed this for many elements. Thus, the series of the stationary atomic states converges indeed towards the state of the positive ion, that is, higher stationary states of an atom correspond always to less strong binding of the emitting electron to the atom's trunk.

Let the energy eV_inf (ionization energy) be required to raise the electron (Fig. 840) from the normal orbit to the infinitely far away orbit - to convert the atom into an ion and a free electron - , where e is the electron's charge and V_inf the ionization tension (in Volt: V_inf/300). Then eV_inf = hn_inf, whence: The ionization energy is equal to the quantum, which corresponds to the last line of the absorption series (border of the series) with the frequency n_inf. This relationship has been confirmed throughout. For example, for Na, measurements yield V_inf = 5.13 Volt, whence eV_inf = = 8.17·10^-12; for mercury, the border of the principal series is l 2413, whence V_inf = 10.4 Volt, while measurements yield 10.2 - 10.3 Volt.

Structure of a chemical element according to Bohr

For N = 2, the formula = N²R(1/n² - 1/m²) yields series of lines similar to the hydrogen series; definite ones of them are known from the light of certain stars, which were at first interpreted as H-spectra. However, this was incorrect. According to Rutherford, the nuclear charge N = 2 belongs to helium, according to Bohr the lines arise from an ionized helium-atom; while in the neutral He-atom two electrons circulate around the doubly charged nucleus, in the H⁺-atom we have only a single circulating electron; hence, as Rutherford nuclear atom, the He-ion differs from the H-atom only by its nuclear charge. Fowler has succeeded in producing the H⁺-spectrum experimentally. It is a spectrum the elements of which only emit when they are exposed to especially strong discharges - a spark-spectrum (the opposite is the arc-spectrum, for the formation of which weaker electric discharges suffice). - Bohr conceives - according to his second postulate - a spectrum as a sign of capture and binding of an electron by the nucleus - binding to a definite orbit, that is, retained in a definite stationary state. Hence a stationary state signifies a definite grade - a definite strength of binding (measurable by the ionization tension). The spectrum accompanies the process of capture. Hence, according to Bohr, the build-up of the electron-cloud of a neutral atom is the action of a positively charged nucleus, which in steps catches and binds as many electrons as its number of nuclear charges indicates; during this process the electrons emit electro-magnetic radiation. A neutral atom is built completely, a simply ionized atom lacks the last stage. Hence the H-spectrum accompanies the only and last phase in the construction of a neutral H-atom, the He⁺-spectrum the last but one phase in the construction of a neutral He-atom, for the He-atom forms through binding of two electrons to a two-fold charged nucleus; hence it must catch a second electron, in order to conclude the formation of a neutral He-atom.

A generalization of this is: The optical series-arc-spectrum witnesses the last phase of the formation of the neutral atom, the nucleus with the charge number N has already captured (N - 1) electrons and binds now the last one. In contrast, the spark spectrum witnesses the last but one phase of the neutral atom, the nucleus with the charge number N has hitherto captured (N - 2) electrons and now bonds the (N - 1)-th. You can imagine of two neighbouring elements in the periodic system that the one with the larger atomic number has arisen from the preceding one - in fact as follows: First of all, the nuclear charge of the preceding one increases by one unit and the thus created ion neutralizes itself by accumulation of a new added electron - Bohr says: By capture and binding of an electron.

The effect of the nucleus on this finally captured electron is weakened by the already present cloud of electrons - it is screened off. In order to separate it from the atom, that is, to ionize the atom of the newly formed element, less work is sufficient (ionization tension) than to separate the last but one, because the one captured earlier is influenced by the nucleus more strongly - hence the increase in the ionization tension from simple to double ionization, for example, for nitrogen from N⁺ 14.49 Volt to N⁺⁺ 25.56 Volt, for oxygen from O⁺ 13.35 Volt to O⁺⁺ 34.99 Volt. The insight that the spark spectra originate from ionized atoms and their emission corresponds to the last but one phase in the formation of the neutral atom (by capture and binding of electrons), the arc spectra, in contrast, to the completion of the neutral atom guided Bohr to a general Formation Principle. He applies it now to the atoms of the various elements. The capture of the first electron by the nucleus of each element is always accompanied by a hydrogen-similar spectrum (H, He⁺, Li⁺⁺, etc.), the capture ends in that the electron circulates around the nucleus along a unit quantum orbit. Also the capture of each electron, following the first one, manifests itself in the emission of a series spectrum; each time, the capture is finished, when the series electron reaches the normal state. (All frequencies of the corresponding spectrum are to be expected only when many atoms are formed - by no means only one! However differently the binding in the various atoms may take place, the final result will be the same everywhere - the normal state.) The so-called higher spark spectra verify the binding of the third-last, fourth-last, etc. electron, the X-ray spectra the maintenance of the atomic structure in the vicinity of the nucleus. But we may assume that the external electron moves during all bindings of this kind along an n_k-orbit, that is, it belongs to a stationary state, which is characterized by a primary quantum number n and a secondary quantum number k. This also applies to the normal state. Hence we conclude: In the finished atom, each of the electrons circulates the nucleus along an orbit, which in first approximation is characterized by two quantum numbers n and k. Theoretical considerations on the basis of the postulates and the Correspondence Principle and the spectroscopic facts guided Bohr to the allotment of the quantum numbers n and k to the individual elements. The following table does not contain all elements, but correctly reflects the allotment.

The number (for example 20) to the left of the symbol (Ca) of the element is its atomic number, simultaneously the number of electrons in the cloud of the neutral atom. The numbers (2 26 26 2) on the right hand side of the symbol (Ca) of the element indicate the distribution of its (20) electrons in (6) groups of orbits; the primary and secondary quantum numbers (1₁2₁2₂3₁3₂4₁) of the orbits of each group are located above and at the foot of each column of numbers.

An important confirmation of Bohr's theory (especially of the Synthesis Principle) on the part of X-ray-spectroscopy is the discovery of the chemical element with the atomic number 72 (Hevesy 1923). The theory links the appearance of certain irregularities in he periodic system to the formation of internal groups of electrons during the construction of the atom. For example, the appearance of the rare earths [(57) to (71)] corresponds to the development of four-quantum electrons from a (preliminarily concluded) group of 18 electrons in three subgroups into a (finally concluded) group of 32 in 4 subgroups. According to Bohr, in the element (71), the four-quantum electron group is completely formed. Hence, for the element (73), the number of the five- or six-quantum electrons must be larger by one than for the rare earths. This mean chemically: The element 72 does not belong to the rare earths, but is tetra-valent and homologous with zircon. Since the chemically homologous elements appear in Nature very frequently in unison, it suggested itself, in view of this insight, to look for the element 72 among the zircon minerals. According to X-ray spectroscopic results, all investigated zircon minerals contained indeed 5 - 10% of the element 72. Attempts to separate it from zircon led, on the one hand, to zircon, completely free of element 72, on the other hand, to samples which contained about 50% of the element. The discovery of the element 72 - it was given the name hafnium - was therefore due to Bohr's theory, especially due to the construction principle.

X-ray spectra in the light of Bohr's theory

While X-ray spectra are formed like optical ones (electron impact, irradiation of electromagnetic waves, here X-ray waves), they differ fundamentally from them. Already externally: An optical series, that is, for example, the Balmer-series of the H-atom has very many, computationally even an infinity of lines and a location, at which the lines accumulate; however, the X-ray series have only a very few lines and no location of accumulation. The optical spectra start at a definite excitation tension with a single line (resonance line), and as the tension rises (velocity of the impacting electrons), the number of lines grows until at the ionization tension the entire spectrum has been formed. In contrast, the X-ray spectrum knows neither a resonance line nor an augmentation of the number of lines; when its excitation tension is exceeded, all lines are simultaneously present. Moreover: The optical absorption line spectrum is a copy of the emission line spectrum, while the X-ray absorption spectrum is a continuous spectrum which runs from an absorption edge towards shorter wave lengths without any absorption line. However, the continuous absorption X-ray spectrum and its edge recall the optical continuous absorption spectrum, which runs from its accumulation location (series boundary) towards shorter wave lengths. The optical series boundary is, in a certain sense, similar to the X-ray absorption edge, the more so, since for both eV = hn, except that in the one case n is the frequency of the optical series boundary, in the other case it is the frequency of the absorption edge, while in both cases V is an ionization tension. The ionization tension for the light spectrum corresponds to the separation of an external electron, that is, the X-ray spectrum to the detachment of an internal electron from the part of the electron cloud, close to the nucleus, because for an excitation of the X-ray spectrum thousands to hundred thousands of Volt are required, for the excitation of the ionization spectrum of light only several or tens of Volts. The attraction of the nucleus on an electron in its vicinity is enormous, but already much smaller on a perioherical electron due to its large distance from the nucleus, and this attraction is yet substantially reduced (screening) by the intermediate electrons. In atoms with large nuclear charge, the innermost orbits lie very close to the nucleus, as a result of its strong attraction. The electrons, which circulate there cannot be reached by external actions, for example, by chemical, thermal and such like forces; the electrons between them and the atomic periphery act as like a protective wall. Only a and b-rays and very fast cathode rays penetrate deeply into the atom and remove electrons close to the nucleus from their orbits.

The X-ray spectra of the elements are much more clearly arranged and more similar to each other than their optical spectra (Fig. 822). Also their lines can be ordered in series (K-, L-, M-, N-, O- series) and the frequency of each line can be represented by a difference of two terms. Moreover, you can join the spectra with the nuclear atom by the assumption, that each term, multiplied by h - the X-ray energy-level - is equal to the energy which must be supplied to the atom, in order to separate one of its electrons - however, in this case, an electron of one of those close to the nucleus. This energy, which is to benefit the close vicinity of the atom's nucleus, can be taken in by absorption of radiation energy; it detaches an electron from a definite energy level and drives it out of the atom. The irradiated energy, required for this action, must be a radiation with a minimum frequency (n = eV/h). As soon as this is exceeded, the absorption edge displays itself in the spectrum; starting from it towards smaller waves, there appears a continuous absorption spectrum.

The electrons, bound in the atoms, sort themselves into groups according to the strength of their linkage to the nucleus. For each member of a definite group, the binding by the nucleus is equally large; it differs from group to group. The electrons, which lie between it and the nucleus, reduce the action of the nucleus to a certain extent (screening). The orbits of the electrons of each group lie within a shell-like space, which encloses the nucleus concentrically. Every shell demands its own excitation, in order to tear away an electron. The shell which demands the largest excitation is called the K-shell, the following ones are the L-, M- shells, etc. Hence the K-electrons are bound more tightly than the L-electrons, the L-electrons more tightly than the M-electrons, etc. The K-shell lies closest to the nucleus, it is followed by the L-shell, etc. The further away lies the nucleus, the larger is the screening of the nuclear action, that is, the weaker is the binding. The conclusion regarding this shell-formed structure of the nuclear environment arises from the theory of the generation of X-ray spectra, formulated and experimentally confirmed by Kossel (1914).

Also the of the X-ray lines can be represented by differences of terms. The wave number of the strongest K-line, the K_a-line. of an element with the atomic number Z can be represented approximately by _Ka = R(Z - 1)²(1/1² - 1/2²). According to Bohr, this has the interpretation: The energy is radiated as an electron transits from the second innermost shell to the innermost one. While the nuclear charge does not act on the electron with its full strength Z, it is only reduced by one unit. It is different for the softer L-radiation. The wave number of an L-line is given by _La = R(Z - 7.4)²(1/2² - 1/3²). The energy is radiated as an electron transits from the third innermost shell to the second innermost one. The nuclear charge acts then on the atom with a force reduced by more than 7 units. This has the explanation: The L-radiation forms further away than the K-radiation, and in-between the radiating electron and nucleus interfere with other electrons, which (by their negative change) reduce the action of the nuclear charge on the radiating electron. A radiation, which is formed yet further outwards, forms under yet stronger screening of the nuclear action. In the innermost environment of the nucleus, we must deal with the complete nuclear action, at the periphery only with the charge 1, because then the nuclear charge is screened to a single unit - the rest of the atom as a simple charged ion, when the most external one is removed from the total electron cloud.

Fig. 841 schematizes the K-, L-, · · · shells where form the K-, L-, ··· radiations. For example, the K-radiation forms as follows: An electron is torn away from the K-shell and driven out of the atom. An electron from the L-shell enters the shell, perturbed by the removal of an electron. As replacement, an electron from the M-shell enters the L-shell, etc. These electrons's transit from the L-shell into the K-shell, from the M- into the L-shell, etc. is accompanied by X-ray radiation. Thus, the X-rays manifest the reinstatement of the atomic structure after a preceding perturbation of its inner nuclear environment. Hence X-ray excitation is an immediate consequence of the expulsion of central ( to the excitation of light spectra peripheral!) cloud electrons from the atom. However note: Electrons, close to the nucleus are removed from the atom, that is, the atom is ionized from inside; during the radiation of the X-ray waves, the ionization moves outwards. Hence the X-ray radiation originates from atomic ions. In order to overcome the strong attraction by the nucleus, it demands high tension (during light-electric irradiation of the large frequency of the absorption edge).

Once a K- electron has been driven out and an L-electron has replaced it, the atom radiates the K_a-line. However, it is also possible that an M- or an N-electron replaces it; the atom then radiates the K_a- or K_b- line. The first happens, because in the case when an M-electron move to the K-shell more energy is released than in the case when an L-electron moves there. The second happens, because the probability of a transition into the K-shell from the further away M-shell is much smaller than that from the closerby L-shell. The conditions are similar when an L-electron is driven out of the atom and an M- or an N-electron take its place. In that case, there form the corresponding lines of the L-spectrum. Fig. 842 presents the K-series for elements with neighbouring atomic numbers. The larger is the atomic number, that is, the larger is the nuclear charge (in other words: the heavier is the element), the more move the lines into the short wave spectral area from element to element. This corresponds to Moseley's Law, because the larger is the nuclear charge (atomic number), the closer lie the electron orbits to the nucleus, the larger are the differences in energy and therefore the frequencies of the radiation, belonging to the transits, and the shorter are the wave lengths.

You call the absorption edge (absorption boundary) the straight cam border, visible in a photographic spectral record, to which the blackening gradually rises, just to end there abruptly, whenever the blackening is due to X-ray radiation after its passage through an absorbing layer, for example, aluminium foil. The edge is, whenever the radiation is continuous, the cam edge between gradual blackening rise and steep descent. According to Bohr, its formation is as follows: For example, let there be required the work W for the removal of one electron from the K-shell (of the Al-atom). Those rays of the continuously distributed X-ray frequencies can perform this (photo-electric) work, for which n > W/h; they are weakened by their performance of work (absorbed). The rays of frequencies < W/h cannot perform this work, whence they pass through the foil much less weakened. The border frequency is the frequency n = W/h which lies on the border between the two. They are linked to the ionization work W by the equation nh = W. Hence it yields for the absorption edge (or border) the characteristic frequency. [A measure for the frequency and energy is also given by the minimum potential difference V, which a cathode ray must have traversed, in order to be able to ionize the shell: V = nh/e = W/e. (e = charge ot the electron). According to experiments, there belong to every shell, with the exception of the K-shell, several : To the L-shell 3, the M-shell 5, the N-shell 7, the O-shell 5, the P-shell 3. The L-shell contains accordingly 3 partial shells, the M-shell 5, etc. Every individual shell demands its own excitation tension. Among all of them , the K-border is the one with the shortest waves (hardest). The position of the absorption edges is as characteristic for an element as are the X-ray emission lines; it changes just as regularly with the atomic number of the element: The absorption edges represent simultaneously the series borders of the emission lines.

Hence W = nh is the energy of the X-ray quanta, which is sufficient to ionize the K shell. You will realize now the link between the absorption edge and the emission of X-rays. You compute from the wave length of the edge the minimum tension required for the excitation of the spectrum V = 12.34/l (V in kilo-Volt, l in Å.-E.). The L-spectrum has 3 edges (Fig. 843) to which correspond 3 excitation tensions, whence the L-shell decomposes into 3 partial shells and 3 energy levels, the M-shell into 5, the N-shell into 7, the O-shell into 5 and the P-shell into 3 partial shells.

We can here draw a general conclusion regarding the relationship between the frequencies of the absorption edges and the frequencies of the emission lines. The neutral, not excited atom exists only in a single, completely defined configuration, in contrast to an atom, from the nucleus near area of which an electron has been removed, can exist in several states with different energy values. We will relate these energy values to the neutral atom as zero (in contrast to what is done in the optical case). Then every absorption edge corresponds to such a state of the ionized atom, and the frequency of this edge, multiplied by h, yields directly the energy value of the corresponding state, referred to as the zero point. These states, which are characterized by the actual empty place in the atom, we will call, like the corresponding edges in the sequence of decreasing energy values, by K-, L-, etc. states. Every X-ray line can then be viewed (Kossel) to be the transition between two such states of the ionized atom, provided you pay attention, instead of to the jumping of an electron, to the level of the empty place in the atom. On the basis of Bohr's frequency condition, there follows now directly that the wave number of every X-ray line must be equal to the difference of the wave numbers of two absorption edges; hence these have in the field of X-rays the role of the spectroscopic term.

The stationary states of the nuclear atom classified in terms of quantum numbers

The quantum numbers n and k only describe a stationary state in first approximation. More of them are required, in order to characterize it completely, for example, on behalf of the properties, which appear during the splitting of the terms in doublettes, triplets, etc. - in general, called multi-plets - as well as in the splitting of the spectral lines in an external magnetic field. You must take into consideration: There move electrons around the nucleus; they react to a magnetic field and give the orbit a definite position with respect the the direction of the field; every individual electron acts like a magnet (with the moment of 1 magneton) due to its own rotation. Hence you require 4 quantum numbers n, l, j, m (the place of k is taken by l = k - 1 as quantum number for the impulsive moment of the electron, circulating about the nucleus, m determines the orientation in the external field) and selection rules. They are interrelated as follows:

n = 1, 2, 3, 4 ,5, ···				I
l = 0, 1, 2, 3, ···		0 <= l <= (n - 1)		II
j = l 1/2		(for l = 0 only j = 1/2)		III
m = -j, -j + 1, -j + 2,··· j - 2, j - 1		(-j <= m <= +j) for a given j 2j +1 values of m are possible		IV

These four quantum numbers define the stationary state of an electron. The enormous variety of stationary states and the corresponding quantum numbers are limited by the Exclusion Principle of Pauli (Pauli-Prohibition 1925): "Never have two electrons, bound to the same atom, stationary states with the same four quantum numbers n, l, j, m". For example:

If n = 1, then (according to II) there is only possible l = 0. and therefore (according to III) j = 1/2. The possible values of m (according to IV) are 1/2 and -1/2. For n = 1, by the Pauli Principle, there are only two 4-combinations and hence two electrons possible:

If n = 2, then l = 0 and l = 1 must be distinguished. For l = 0, j = 1/2, and m = 1/2, whence with n = 2 and l = 0. there are the 4-combinations:

For n = 2 and l = 1 (according to III) j has the values 1/2 and 3/2. For j = 1/2, m = 1/2. For j = 3/2, m = 3/2, +1/2, -1/2, - 3/2.Altogether with n = 2 and l = 1, there are the six 4-combinations

In other words, the Pauli Principle admits:

Aa an obvious generalization of this scheme, there follows from the Pauli Principle: The number of electrons, which with equal values of n and l can be bound to a nucleus (equivalent electrons), is always limited. The highest number of equivalent electrons with the quantum numbers n and l is

N_l = 2(2l + 1), l = 0, 1, 2, 3, · · · (n - 1).

Also the number of electrons with equal n is limited. Their maximum numbers are

Period subdivision of chemical elements in the light of Bohr's theory

T|he Pauli Principle thus leads to a 2-group of electron orbits with n = 1, an 8-group with n = 2, an 18-group with n = 3, etc. (cf. table). The group of electron orbits corresponds in its number of members 2, 8, 18 to the period (horizontal row) in the periodic system of the chemical elements (Julius Lothar Meyer 1830-1895, Mendeleev 1869). The periodicity of the properties of the elements comes thereby related to the structuring laws of the electron clouds surrounding their atomic nucleus.

What is the meaning of periodicity of the nature of the chemicals elements? If you order the elements, starting from hydrogen, according to their rising atomic number (H, He, Li, Be, B, C, etc. up to Th, Pa, U, this monotone series demands on its own structuring: Certain chemical and physical properties of neighbouring elements grade themselves noticeably; counted from any given element (for example, Na), there follows after a certain number (8) of neighbours a closely related element (K), which is closely related to that element (Na). The series of elements demands therefore a classification such as the following:

In a single horizontal line (period), there occurs a classification in the change of the properties of the elements; elements which are similar through their properties (homologous) appear vertically below each other (group). The periodicity of a single property appears distinctly, for example, the the periodicity of the atomic volume of the elements (atomic weight/density), that is, in the up and down of the atomic volumes (Fig. 844 along the monotonous series of the numbers of atomic weights; the curve shows that the start of a horizontal line follows without a gap the end of the preceding line. At the locations, where the curve has a minimum, lie the elements with para-magnetic salts - in between those with dia-magnetic salts. At the minima, there lie also elements the positive ions of which are coloured (in solutions, glasses, jewels). Also the compressibility of the solid elements has almost the same periodicity like the atomic volume. Moreover: Along the horizontal rows, the valence of the elements changes (against halogenes, hydrogen, oxygen); in general, it is the same in a vertical column.

Periodicity is not a strict law, just a certain regularity which suggests a law. Moseley's Law arranges the elements strictly according to their atomic number and very closely approaches the series of atomic weight numbers. However, the periodicity of the properties of the elements becomes compatible with the monotony of the atomic numbers by joining (Bohr) the subdivision of the periods of the elements to a subdivision into groups of lectron orbits according to rising quantum numbers. According to Bohr, these properties rest above all on the outside electron groups of the atomic cloud, and the periodicity of the properties rests on a kind of repetition of these groups. For example, the alkali metals (apart from Li) agree each and all with one another in the last electron groups (cf. table at Na, K, Rh, Cs and the, in 1935 unknown, element 87). The primary quantum numbers of these groups increase by 1 for each successive alkali metal. The electrons, which travel along extended ellipses, transverse also parts close to the nucleus, and this affects the inner part of their orbits.The difference between the outer electron groups of the homologous elements is above all due to the difference in the inner sections of the orbits, the outer sections being very similar to each other. A purely chemical periodicity is probably displayed most distinctly by the chemical peculiarities of the noble gases (helium, neon, argon, krypton, xenon, emanation; the lower strong line in Fig. 845 corresponds to the Group 18 in the table above); they are spaced from each other by 8, 8, 18, 18, 32 elements (corresponding to the atomic numbers 2, 10. 18, 36,54, 86). They do not form chemical bonds. This means for the nuclear atom: In a noble gas, the coherence of the electrons with each other and with the nucleus cannot be changed chemically; the electron cloud allows neither removal nor addition of electrons. Hence the atomic numbers 2, 10. 18, 36,54, 86 form natural incisions in the monotonous series of the atomic numbers and weights. Stable electron configurations arise in that a new, closed and therefore especially stable electron group adds itself to the present one. According to Bohr, an atom constructs itself by the charged nucleus progressively catching an electron and binding it to itself. After conclusion of the catching process, every electron is in the stationary state, which corresponds to the strongest bond among all possible ones, that is, to the smallest energy. Since the bond is the firmer, the smaller is the primary quantum number, the first two electrons are bound with n = 1. Hence, according to the Pauli principle, helium closes the group of single quantum electrons completely. Further electrons cannot be added with n = 1, but only still with n = 2. Hence, from Li onwards, follows the accumulation of the 8 two-quantum electrons, which is closed at the noble gas neon with formation of an especially stable configuration. In the case of the now following binding of 8-quantum electrons, there is reached again at the noble gas Argon a quite similar configuration and the system continues analogously at the remaining noble gases, as can be seen from the table.

Hence we understand: In general, the sequence of the additions of single electrons is the growing primary quantum number. However, there exist exceptions to this rule. The first arises at potassium. Up to the preceding element Ar, 8 3-quantum electrons are bound, whence there are still lacking 10 of the total number 18. However, before these are added, it can be shown by spectra that there are added first of all at the elements K and Ca 4-quantum electrons. However, immediately afterwards, at the elements Sc to Ni, this defect is made up, so that starting from the Cu⁺-ion all 3-quantum electrons are bound. The group of elements, at which such processes occur, are framed in Fig. 845. These occasional violations of the normal sequence in the addition of electrons explain the apparent irregularities of the periodic system. This insight is also of greatest importance for the explanation of the chemical valence conditions (Kossel).

Frequency condition DE = hn, a form of Einstein's equation ½mv² = hn - P

The frequency condition E_e - E_a = hn complements the equation ½mv² = hn - P, which regulates the light-electric effect (Einstein 1905), that is, the exit of electrons from metals, irradiated by short wave light. According to Einstein's concept, this process is: Energy quanta hn enter the surface layer of the body, their energy converts itself there (at least partially) into kinetic energy of electrons - the simplest concept is that one quantum of light passes its entire energy to a single electron as kinetic energy. As such an electron reaches from inside the body the surface, it has lost a part of its kinetic energy. In addition, during the passage through the surface, every electron performs work which is characteristic for the body (exit work). Those electrons will have the largest velocity, which have been excited immediately below the surface and exit perpendicularly to the surface. Their kinetic energy is ½mv² = hn - P, where v is the exit velocity, m the electron mass and P the work performed during the exit. This is the energy which can be measured outside the surface. Measurements have shown that this law is strictly realized for the entire spectral range. For most metals, the exit work lies between 1 and 3 Volt. Bohr's theory of the nuclear atom allots to the exit work P a frequency n_p by setting P = hn_p; the term hn_p corresponds to the tension V_p, which is just sufficient to remove the electron from the atom - to ionize it, whence n_p is also the frequency of the border of the absorption series.

If the absorbed radiation energy is insufficient to liberate an electron from an atom, nevertheless the atom's interior changes itself by raising the electron to a higher quantum orbit. It returns again the potential energy, which the atom has gained in this manner, according to the equation DE = hn as it reverts. Then no exit work is performed, whence ½mv² = hn. The energy ½mv² can also be given to the electron by the work eV, whence eV = ½mv² = hn. This equation regulates - reading from the right hand side to the left hand side - the electron release by light of the frequency n, and - reading from the left hand side to the right hand side - the generation of radiation of light (ultraviolet, X-rays) by electrons at the Volt velocity V.

For example, the electron impact experiment explains the meaning of reading the equation from the left hand side to the right hand side. In Fig. 846, A is a metal plate and at a cm from it is B, a wire netting with a small mesh. A is earthed, P charged to the potential V⁺. Between A and B lies a uniform electric field of strength V/a Volt/cm. We generate at the plate A electrons - say, light-electrically. They move with acceleration towards the wire netting. The force, acting on a single electron, equals the charge e times the field strength V/a. Hence it performs along the path a the work ae·V/a, so that eV measures the kinetic energy, with which the electron arrives at the wire netting, so that the left hand side of the equation represents the energy; then also the right hand side must represent energy - it is the energy of the electro-magnetic radiation at the frequency n. Thus, Einstein's equation links the energy of a moving electron and that of a radiation. Its interpretation is: If the entire kinetic energy of an electron is converted into electro-magnetic radiation, it has a definite frequency, computed from eV = hn. The impact of sufficiently fast moved electrons is used to excite mercury atoms, which then radiate light waves. The ultraviolet line l 2536, characteristic for the spectrum of the ultraviolet, has the frequency n = 1.183·10¹⁵. The potential, required for the excitation of l 2536, is computed at 4.86 Volt - measurement (Franck and Hertz) yields 4.9 Volt. However,why does the electron only at reaching 4.86 Volt give energy to the atom, which is converted into radiation? For example, the equation would still hold when only half the energy (V = 2.43) were transferred; the radiation would then have half the frequency, that is, twice the wave length. Answer: The Hg-atom can only take in amounts of energy, which are sufficient for the transport of an electron to a higher quantum state. The smallest such quantity of energy is 4.9 Volt. The line l 2536 is formed by the transition of an electron, which belongs to the electron cloud of the Hg-atom, which returns to its base orbit after an external impact has removed it form there.

Also a process in X-ray radiation demonstrates the meaning of Einstein's equation. The continuous X-ray spectrum (impulse white radiation) ends suddenly and sharply on the side of short waves. This phenomenon is also explained by the equation eV = hn. The electrons, coming from the cathode of the X-ray tube (cathode rays) are accelerated in the field between the cathode and anti-cathode. The tension between them determines the maximum energy, which the electrons can reach. The role of the Hg-vapour in the last example is here assumed by the metal of the anti-cathode. Its atoms radiate the wave length demanded by the equation eV = hn. If you operate the tube with V = 20,000 Volt, the computation yields that the energy of the electrons, accelerated by 20,000 Volt, can in the most favourable case convert only into the radiation l 0.61·10^-8 cm.Yet shorter wave lengths cannot arise, since the energy of the electrons is limited by the tension (however, larger ones can be conceived, since the electrons on their way from the cathode to the anti-cathode can lose energy, for example, by collision with gas particles). The short wave border is so sharp that one has been able to determine by measurement of the length of the limit wave and the tension in the tube Planck's action quantum. This method yields its most accurate value.

Electron cloud around the atomic nucleus under magnetic influence (Zeeman-effect 1896) and electric influence (Stark-effect 1913)

Irradiation of light and electronic impact affect mechanically the electrons circulating around the atomic nucleus. However, the electrons, as carriers of electric charge, also react to a magnetic field and an electric field of appropriate strength (magnetic field: several 10,000 Gauss, electric field: several 10,000 Volt/cm). The atoms, excited into incandescence, manifest the effect of the field by a change of their spectral lines (splitting of one into several) - the effect of the magnetic field as Zeeman-effect, that of the electric field as Stark-effect.

You observe the Zeeman-effect in the line spectrum of incandescent gases and vapours. You place the source of light (most frequently because it being most convenient as a spark in air, otherwise as a vacuum arc, also in a Geissler tube) between the specially designed clamps of an electro-magnet. You form the spectrum with a strong, resolving* concave lattice or also an interference spectroscope, every individual line as sharply as possible. If you now switch on the magnet, there appear, viewed perpendicularly to the lines of force (lateral-effect), instead of a single line three lines (triplet), the central one at the location of the line in the absence of a field, the outer ones displaced equally towards red and violet, in most cases by about 1 Å.-E.; they are linearly polarized. If you turn off the magnetic field, there appears again only a single line (without a field). If you view along the magnetic lines of force (longitudinal effect), there appear at excitation of the magnetic field two lines, which are mutually, opposite, circularly polarized.However, only a few lines split into normal triplets, most of them split anomalously, that is, either into a triplet with other distances Dn or into many more components than three. The magnetic splitting of the absorption lines is called the Zeeman-effect. According to Lorentz's electron theory, a spectral line of the frequency n splits into two components with the frequencies n Dn and a components with the unchanged frequency n. The splitting is determined by Dn = H·e/4p c·m, where e and m are the charge and mass of the oscillating particle, H is the intensity of the external magnetic field, c the velocity of light. The values of e/m, determined by the splitting Dn, agree well for these spectral lines with those determined for the electron in another way - it is a proof that the emission of the spectral lines rests on the motions of electrons.

*The following question leads to a definition of the resolution capability of a spectral apparatus: By what fraction of their medium wave length must neighbouring wave lengths l and l + Dl at least differ, in order to be able to see clearly their corresponding lines? The resolution capability is given by l/Dl (Lord Rayleigh). This number (without a name) allows a comparison of the performance of various kinds of spectral equipment. As a rule, a small prism set displays the D-line of sodium (l_D1 = 5895.932 Å.-E., l_D2 = 5889.965 Å.-E., dl = 5.967, 6Å.-E. still just as double line; its resolution capability is: l/Dl = 5893/5.967 6000/6 = 1000. The magnetic resolution of the D-lines demands a resolution of about 100,000.

The Zeeman-effect widens our insight into the nature of sun spots. These are probably (Erve Faye 1814-1902 1877) funnels, which are formed by vortices and comparable to the cyclones of our atmosphere; also the spectro-heliograph images of the spots suggest vortices. If you now assume that the gases of the photo-sphere are ionized, you must view the vortices on the sun's surface as circulating, electric currents. Such a current must generate a magnetic field, the lines of force of which are essentially parallel to the axis of the vortex. The spectrum of sun spots displays now as most frequent phenomena iron-, titanium- and chrome-lines (Hale 1908), which are characteristic for the Zeeman-effect, Hence the spots may indeed be areas of vorticity of electrically charged gases.

You observe the electric splitting of spectral lines (Stark-effect) in light emitted by canal-ray particles. It is a phenomenon similar to the magnetic effect, but more involved. Unusual experimental difficulties faced its discovery: As a rule, incandescent gases and vapours conduct too well to be able to maintain in them an electric field of appreciable intensity. It is different at extremely low pressure. That is why Stark generated the field in a wide enough, evacuated discharge tube, in the space behind the cathode. He employed the canal rays, entering it through the holes in the cathode, as source of light. He was able to generate between two, at the most 3 mm apart plates 100,000 V/cm without there occurring a discharge. (Lo Surdo employed the high tension field inside the cathode dark room; it is inhomogeneous and only leads to qualitative observations.) The electric field influences the various spectral lines very differently (in contrast to what the magnetic field achieves). The strongest effects display the hydrogen- and hydrogen-similar lines of ionized helium He⁺. The H_d-line displays in a field of 30,000 V/cm about 30 components, the outermost ones about 18 Å.-E. apart; in the Zeeman-effect, the distance for a line of equal wave length is at 30,000 Gauss about 0.6 Å.-E.

While Bohr's theory yields an almost complete qualitative description of the atomic structure, it does not yield a quantitative one of atomic phenomena. For example, the theory encounters great difficulties during applications to dispersion or brightness of the spectral lines - essentially, because it is a synthesis of quantum and classical physics which it cannot completely combine. Heisenberg's Quantum Mechanics (1925) and Schrödinger's (1926) Wave Mechanics arose from such considerations. The orbits of Bohr's theory and the associated electro-magnetic oscillations recall the behaviour of vibrating, elastic structures (with fundamental and over tones). Wave Mechanics has advanced this analogy. In the place of electron orbits, it introduces oscillations with nodal lines, the number of which corresponds to the number of Bohr's orbits 1, 2, 3, · · · in Fig. 835. This theory yields the frequencies of the hydrogen lines, their brightness and other properties of the atom in agreement with experiments, a treatment of which by Bohr's theory is impossible. Whatever advances the new quantum theory and wave mechanics may yield beyond the earlier achievements (their presentation lies outside the limits of this volume) and "what changes our fundamental concepts in the time to come may experience, it is certain that the quantum theory and Bohr's atomic models in any form whatever will remain a inalienable part of physics" (Sommerfeld).

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