Sommerfeld Theory

11 Key excerpts on "Sommerfeld Theory"

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  The Quantum Revolution

  A Historical Perspective

  • Kent A. Peacock(Author)
  • 2007(Publication Date)
  • Greenwood

    (Publisher)

  Sommerfeld quickly extended the Bohr theory to explain the phenomenon of fine structure. If spectral lines were examined closely using new and more accurate instru- ments that were becoming available, it could be seen that some spectral lines were split, meaning that they were actually combinations of lines very closely spaced in energy. This splitting of spectral lines is called fine structure. Bohr’s theory of 1913 applied only to circular orbits in the hydrogen atom; Som- merfeld extended Bohr’s methods to elliptical orbits. When an object moves on an elliptical orbit it has to move faster when it is closer to the focus. Som- merfeld showed that in many cases the electrons would move fast enough for relativistic effects to be important. The relativistic deviation of spectral lines from Bohr’s predictions were a function of a new number, the fine structure constant, approximately equal to 1/137. This mysterious number would prove central to later developments in quantum electrodynamics. Sommerfeld was also an important teacher (among his doctoral students were the future Nobel winners Werner Heisenberg, Wolfgang Pauli, Hans Bethe, and Peter Debye) and author of textbooks. His book Atomic Structure and Spectral Lines appeared in 1919 and went through several editions over the next few years, expanding rapidly each time, and was up until the mid-1920s the “bible” from which most physicists learned their quantum mechanics. OTHER DEVELOPMENTS AFTER BOHR Moseley and Atomic Numbers Another crucial piece of the atomic puzzle was filled in by the young British physicist Henry G. Moseley (1887–1915). He joined Rutherford, who quickly recognized his talent, at Manchester in 1910. Every element has a character- istic X-ray spectrum, just as it has a spectrum in visible light. Moseley showed that the energy of certain X-ray spectral lines varies with respect to the atomic number of the elements according to a simple formula.

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  Thermocouples

  Theory and Properties

  • DanielD. Pollock, Daniel D. Pollock(Authors)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  3    Solid-State Theories

  T HE THEORETICAL background is now applied to develop the Sommerfeld Theory of metals, the Fermi-Dirac statistics, the band theory, and the Brillouin zone theory in order to understand the physical properties of solids. These are given only to the extent necessary for this book; they are not comprehensive, since they are not intended to be applied in detail to a wide range of the properties of solids. They do, however, provide a reasonable, elementary basis for understanding the properties of thermoelectric materials as well as a foundation for further study.

  3.1 Sommerfeld Theory

  The Sommerfeld Theory of metals marks the beginning of the use of modern physics to explain the physical properties of solids. This theory considers metals simply to be homogeneous, isotropic solids. The free valence electrons are treated as if in a well of constant internal potential similar to that indicated in Fig. 1-2 . The important difference is that the electrons within the potential well obey the quantum mechanics, not the classical Drude-Lorentz model. This model illustrates one of the reasons for the use of the boundary conditions employed to obtain solutions to Schrödinger’s equation (Chapter 2 ).

  It has been shown that large numbers of electrons in a potential well with dimensions of crystal sizes result in a quasi-continuum of energy levels (Fig. 2-4 ). The Pauli exclusion principle determines the filling of these energy states. Including spin, just two electrons can occupy each state. For example, a crystal of a monovalent metal of about one molecular weight contains about 6 x 1023 electrons that must be quantized. Each state in the well can accommodate only two electrons of opposite spin. The states in the well are filled progressively, starting with the lowest energy level, until about 3 ⨯ 1023 states are filled and all of the valence electrons are accommodated. It follows that only a very small fraction of all of the valence electrons can occupy energy states with energies in the neighborhood of Em in Fig. 1-2

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  Einstein, Physics And Reality

  • Jagdish Mehra(Author)
  • 1999(Publication Date)
  • World Scientific

    (Publisher)

  Otherwise the classical laws of mechanics and electrostatics (for electrical 4 The ”on-Einsteinian Quantum Theory’ attraction) apply, but the rotation (or, in fact, the angular momentum) becomes ‘quantized.’ The laws of electrodynamics concerning, for instance the radiation, do not apply to these stable states. The radiation occurs only by transition between the states with a well- determined frequency given by the energy difference between the states and Planck’s law.b Bohr’s atomic model of the hydrogen atom could be generalized to hydrogen-like atoms (like the ionized helium) and at least qualitative consequences could be drawn also for multielectron molecules. Arnold Sommerfeld developed Bohr’s model further by including elliptical (Kepler) orbit^.^ In particular, he tried to generalize the quantization condition, his phase integral to several degrees of freedom. This fact did not play a role in the calculation of the hydrogen spectrum, for although we obtain two degrees of freedom in a Kepler ellipse (the motion of the electron occurs in a plane with variable distance from the atomic nucleus and the angle 9)’ the quantum numbers n and n’ (due to the ‘quantization’ of the r and 9 coordinate) appear only as a sum and the spectral lines do not depend on n and n’ separately. On the other hand, Sommerfeld calculated the relativistic mass corrections to the motion of electrons on elliptic orbits and found a fine structure in the lines corresponding to a sum of quantum numbers (n + n’). Further applications of the Bohr-Sommerfeld model were made to the Stark effect of spectral lines.8 In this case, Paul Sophus Epstein showed that one could choose such quantization conditions as explain the empirically found ~plitting.~ It was, however, necessary to restrict the possibility of transitions by ‘selection (Auswahl) pr i nciples. ’ ’ O The calculations of the Zeeman splitting of lines in a magnetic field turned out to be less successful.

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  Self-Field Theory

  A New Mathematical Description of Physics

  • Tony Fleming(Author)
  • 2011(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  Ltd. www.panstanford.com 34 Self-Field Theory than between charge points and applied as a coupled complete electromagnetic field. The atomic self-field motions are obtained using the Maxwell–Lorentz (ML) equations. Quantum theory can be reinterpreted to include the coupled bi-spinorial field to yield the same deterministic closed form eigensolutions as self-field theory. Space–time orthogonality shows the complete self-field theory outer shell electronic structure to be analytic. Self-field theory allows reinterpretation of the weak and strong nuclear forces via a modified system of ML equations. 2.1 Introduction Present-day understanding of the hydrogen atom is linked to the quantum theory that evolved during several decades of effort from the late 19th century until 1927. Bohr put forward a quantum theory of spectroscopy in which angular momentum is a whole number multiplied by Planck’s quantum number = h / 2 π . 2 Using Bohr’s theory, spectral lines can be expressed as a quantum series, for example, the Balmer series, ν mn = R 1 m 2 − 1 n 2 where m 2 = 4 , n = 3 , 4 , 5 , . . . and Rydberg’s number R = q 4 e m e 8 ε 2 0 h 3 c . With this theory, the ground state energy of the hydrogen atom, Bohr’s energy, E B = Rhc , and the electron’s expected position, the Bohr radius, r B = ε 0 h 2 π m e q 2 can be estimated. Spectroscopic experiments led to an understanding of how the hydrogen atom was excited by E-and H-fields. Despite intensive investigation this same period saw a complete failure to find any way in which atomic physics could be based on classical electromagnetic (EM) theory. Maxwell had previously reached the conclusion that gravitation could not be based on EM theory despite the similarity between the inverse-square forms of Coulomb’s electrostatic force and Newton’s gravitational force. Concepts of planetary motion led only to unstable spiraling of the electron into the nucleus at the centres of atoms.

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  Thirty Years that Shook Physics

  The Story of Quantum Theory

  • George Gamow(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  −1 , which is exactly its empirical value obtained by spectroscopic observation. Thus, the problem of quantization of a mechanical system was successfully solved.

  SOMMERFELDE’S ELLIPTICAL ORBITS

  The original Bohr paper on the hydrogen atom was followed soon by that of a German physicist, Arnold Sommerfeld, who extended Bohr’s ideas to the case of elliptical orbits. The motion of a particle in the field of central force is characterized in general by two (polar) coordinates, its distance r from the center of attraction and its positional angle (azimuth) ɸ in respect to the major axis of the ellipse as indicated in the figure (Fig. 13 ); r has the maximum value when ɸ = 0, decreases to its minimum value at ɸ = π, and increases again to its maximum value at ɸ = 2π. Thus, in contrast to Bohr’s circular orbits where r remains constant and only ɸ changes, the motion along Sommerfeld’s elliptical orbits is characterized by two independent coordinates, r and ɸ. It follows that each quantized elliptical orbit must be characterized by two quantum numbers: azimuthal quantum number nɸ and the radial quantum number nr . Applying Bohr’s quantum conditions that the total mechanical actions for azimuthal and radial components of motion must be integer numbers nɸ and nr of h, Sommerfeld obtained for the energy of the quantized elliptical motion the formula:

  This is exactly the same as Bohr’s original formula except that, instead of the square of an integer, the denominator is the square of a sum of two arbitrary integers which is, of course, an arbitrary integer itself. Putting nr = 0 we get, as a special case, Bohr’s circular orbits. If nr ≠ 0 we get elliptical orbits with different degrees of ellipticity. But the energies of all orbits corresponding to the same sum nɸ + nr is exactly the same in spite of their different shapes. The sum nɸ + nr , usually denoted simply by n, is known as the principal quantum number.

  Fig. 13. Circular and elliptical quantum orbits in the hydrogen atom. The first circular orbit

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  Atomic Physics: 8th Edition

  • Max Born(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  pdq), as we should expect from the correspondence principle (see p. 136).

  As an example of the above rules we shall now give a discussion of the hydrogen atom, the complete quantisation of which was carried out by Sommerfeld. By Kepler’s laws, the orbit of the electron round the nucleus is an ellipse; it is therefore simply periodic. Since the electron has three degrees of freedom, this is a case of double de-generacy. In Appendix XIV (p. 387) we give the quantisation of the Kepler ellipse, which leads to the correct energy levels (Balmer terms).

  Fig. 7.–Orbit (rooette) of the electron about the nucleus, taking into account the relativistic variability of mass; the motion is doubly periodic, the perihelion being displaced by the angle Δϕ per revolution.

  The degeneracy is partly removed by taking into account the relativistic variability of mass, or dependence of the mass of the electron on its velocity. In this case the orbit, according to Som-merfeld, is given by a pre-cessing ellipse (rosette); its major axis revolves in the plane of the ellipse round the nucleus with constant angular velocity (fig. 7 ). The orbit is now doubly periodic; besides the original period of revolution, which remains unchanged if the precession is slight, we have now the period of the precessional motion. In accordance with this, we have here two quantum conditions:

  (compare Appendix XIV , p. 387); n determines the semi-major axis a of the approximate ellipse, k its semi-latus rectum q: a = n2 a0 , q = k2 a0 (fig. 8 ).

  Fig. 8.-Elliptic orbit with the nucleus K

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  The Poetry of Physics and the Physics of Poetry

  • Robert K Logan(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  Another experiment, which demonstrated the existence of Bohr’s energy levels, was devised by Maurice de Broglie (brother of Louis de Broglie, the theoretician, whose work will be discussed in the following chapter). M. de Broglie bombarded the atom with x-rays of a known energy. He then observed the kinetic energy of those electrons ejected from atom as a result of absorbing the x-rays. From the difference of the photon’s energy and the electron’s kinetic energy M. de Broglie was able to determine the energy levels of the atom, which were also in agreement with those obtained from Bohr’s theory.

  In 1913, Moseley, in England, investigated the production of x-rays. His work revealed that the charge of the nucleus increased from one element to another by one unit of charge +e. The relation of the energy of the x-rays emitted by an atom and the charge of its nucleus was found to be exactly that predicted by Bohr’s theory of the atom.

  The experiments of Moseley, M. de Broglie, Franck and Hertz established, beyond a shadow of a doubt, the validity of the basic concepts of Bohr’s model of the atom such as the existence of energy levels and the Bohr frequency condition. As more and more experimental information was gathered, however, it became evident that Bohr’s theory was not sophisticated enough to explain all the data. When the spectroscopists looked closely at the spectral lines of Balmer, they discovered that each line was actually split into a number of finer lines. This fine structure of the spectral lines was explained by Sommerfeld making use of Einstein’s relativity theory. The existence of the fine structure of the spectral lines revealed that, for each of the atomic orbits of a given radius postulated by Bohr, there are actually several orbits each with the same radius and almost the same energy but different ellipsities or different values of angular momentum. The slight energy differences of these orbits arise from relativistic effects and accounts for the fine structure of each line.

  For a given value of n, which determines the radius of the orbit or half the distance of the major axis of the ellipse, the possible values of the angular momentum in units of h/2π are ℓ = 1, 2, 3 … n – 1. Studies of the splitting of spectral lines of atoms in magnetic fields revealed that the component of the angular momentum or the plane of the electron’s orbit can take on 2ℓ + 1 different orientations with respect to some external magnetic field. These studies also revealed that the electron has in addition to its angular momentum an intrinsic spin of 1/2 of h/2π, which can be oriented either up or down with respect to the external magnetic field. An electron in an atom can therefore be defined by four quantum numbers, namely, n that determines the radius of its orbit, ℓ that determines its angular momentum or the eccentricity of its elliptical orbit, m that determines the orientation of its orbital plane and ms that determines the orientation of its spin. The number n is an integer, which does not exceed 7; ℓ is an integer ranging from 0 to n – 1; m is an integer with 2ℓ + 1 values that range from –ℓ to +ℓ and ms

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  A Student's Guide to Atomic Physics

  • Mark Fox(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Hydrogen The quantum theory of hydrogen is the starting point for the whole subject of atomic physics. Bohr’s derivation of the quantized energies was one of the tri-umphs of early quantum theory, and makes a useful introduction to the notion of quantized energies and angular momenta. We, therefore, give a brief review of the Bohr model before moving to the main subject of the chapter, namely: the solution of the Schr¨ odinger equation for the electron-nucleus system. 2.1 The Bohr Model of Hydrogen The Bohr model is part of the “old” quantum theory of the atom (i.e., pre-quantum mechanics). It includes the quantization of energy and angular momentum, but uses classical mechanics to describe the motion of the electron. With the advent of quantum mechanics, we realize that this is an inconsistent approach, and therefore should not be pushed too far. Nevertheless, the Bohr model does give the correct quantized energy levels of hydrogen, and also gives a useful parameter (the Bohr radius) for quantifying the size of atoms. Hence, it remains a useful starting point to understand the basic structure of atoms. It is well known from classical physics that planetary orbits are characterized by their energy and angular momentum. We shall see that these are also key quantities in the quantum theory of the hydrogen atom. In 1911, Rutherford discovered the nucleus, which led to the idea of atoms consisting of electrons in classical orbits where the central forces are provided by the Coulomb attraction to the positive nucleus, as shown in Figure 2.1 . The problem with this idea is that the electron in the orbit is constantly accelerating. Accelerating charges emit radiation called bremsstrahlung , and so the electrons should be radiating all the time, losing energy. This would cause the electron to spiral into the 20 2.1 The Bohr Model of Hydrogen 21 -e + Ze v F r Figure 2.1 The Bohr model of the atom considers the electrons to be in orbit around the nucleus.

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  Poetry Of Physics And The Physics Of Poetry, The

  • Robert K Logan(Author)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  In classical electromagnetic theory, on the other hand, the frequency of Bohr’s Atom 179 the periodic motion and the frequency of the subsequent electromagnetic radiation are identical. Finally, in classical mechanics, an electron must orbit the nucleus in an infinite number of paths, differing by only an infinitesimal amount of energy. In Bohr’s scheme, however, the number of orbits is severely limited by restricting the allowed orbits to those for which the angular momentum is equal to an integer times Planck’s constant, h, divided by 2 π . If we represent the angular momentum by L, then L = l h/2 π where l is an integer. The angular momentum of the electron is equal approximately to the product of its momentum times the radius of its orbit. This definition is exact if the orbit is a perfect circle. By placing this restriction on the angular momentum, Bohr was able to obtain Balmer’s formulae for the radiated frequencies of the hydrogen atom. Bohr was also able to calculate Rydberg’s constant, R y and showed that it is simply related to the mass of the electron, m e , the charge of the electron, e and Planck’s constant, h, by the formula R y = 2 π 2 m e e 4 /h 3 . This result, in which one of the fundamental constants of nature was related to the others, was a great success and insured the acceptance of Bohr’s model. This model not only explained Balmer’s formula for the hydrogen atom but it also explained Ritz’s combination principle. Let us label the quantum states or energy levels of the atom by E 1 , E 2 , ... , E n where E 1 is the energy of the ground state, E 2 is the energy of the first excited state, ... , and E n is the energy of the (n-1) th excited state. (See Fig. 19.1). Here, we refer to the higher energy orbits of the electrons of the atom as excited states. These electrons have absorbed energy, but do not retain the additional energy very long.

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  Physical Properties of Materials for Engineers

  • Daniel D. Pollock(Author)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Section 11.2 ).

  Where Es and E are of normal energies, so that Es − μ, ≅ E − EF , and both are sufficiently larger than kB T, it is approximated that the Bose-Einstein and Fermi-Dirac functions give virtually the same results. This is shown as

  1 exp [ ( E s   −   μ ) / k B T ] ≅ 1 exp [ ( E   −   E F ) / k B T ]

  (5-50b)

  FIGURE 5-5. Fermi-Sommerfeld distribution of electrons. (a) Sommerfeld distributions; (b) Fermi-Dirac functions; (c) products of (a) and (b) at 0 and T K.

  This relationship has important implications that are useful in understanding phase equilibria. As a result of Equation 5-50b,

  μ  ≅ E F

  (5-50c)

  Equation 5-50c is accurate for metal-alloy systems at those temperatures and energies to which these are normally exposed. It constitutes a bridging relationship between thermodynamics and quantum mechanics. The understandings of the factors affecting EF (particularly N/V in Equation 5-24) give greater insight into the reasons for changes in the chemical potential and the corresponding changes in physical, metallurgical, and chemical properties.

  5.7 FERMI-Sommerfeld Theory OF METALS

  The Sommerfeld Theory provides a means for computing the density of states (Equation 5-21). The Fermi-Dirac function (Equation 5-50) shows how the electrons are distributed in those states as well as those that can enter into physical processes. These relationships, along with Equations 5-28a and b, are used to show their combined effects graphically in Figure 5-5

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  Quantum Concepts in Physics

  An Alternative Approach to the Understanding of Quantum Mechanics

  • Malcolm Longair(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  Bohr also realised from the outset that the structure of atoms could not be understood on the basis of classical physics. The obvious way forward 81 4.5 The Bohr model of the hydrogen atom was to incorporate the quantum concepts of Planck and Einstein into the models of atoms. Einstein’s statement, quoted in Sect. 3.4, ‘. . . for ions which can vibrate with a definite frequency, . . . the manifold of possible states must be narrower than it is for bodies in our direct experience.’ (Einstein, 1906c) was precisely the type of constraint which Bohr was seeking. Such a mechanical con- straint was essential to understand how atoms could survive the inevitable instabilities according to classical physics. How could these ideas be incorporated into models of atoms? In the summer of 1912, Bohr wrote an unpublished memorandum for Rutherford, in which he made his first attempt at quantising the energy levels of the electrons in atoms (Bohr, 1912). He proposed relating the kinetic energy T of the electron to the frequency ν  = v/2π a of its orbit about the nucleus through the relation T = 1 2 m e v 2 = K ν  , (4.19) where K is a constant which he expected would be of the same order of magnitude as Planck’s constant h . Bohr believed there must be some such non-classical constraint in order to guarantee the stability of atoms. Indeed, his criterion (4.19) absolutely fixed the kinetic energy of the electron about the nucleus. For a bound circular orbit, mv 2 a = Ze 2 4π 0 a 2 , (4.20) where Z is the positive charge of the nucleus in units of the charge of the electron e. As is well known, the binding energy of the electron is E = T + U = 1 2 m e v 2 − Ze 2 4π 0 a = − Ze 2 8π 0 a = −T = U 2 , (4.21) where U is the electrostatic potential energy. The quantisation condition (4.19) enables both v and a to be eliminated from the expression for the kinetic energy of the electron.

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