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Question - fine structure

  1. Dec 3, 2007 #1
    Recently I read about element 137 (feynmanium / Untriseptium ), the element which would theoretically have its electrons moving faster than light (if I understood it the right way). I know that if you involve relativity in the analysis, the problem occurs in element 139, but thats kind of irrelevant to my question.

    Here is my first question : The article said the speed of electrons in the 1s electron orbital could be obtained with this equation : v = Z *alpha* c = (Z*c)/(137.036) . Why is that so?

    This leads me to my second question. I did a bit of research myself, and found that the fraction 1/137.036 was the fine structure constant, and thats where I hit a roadblock. What is the fine structure? I know it's the splitting of the spectral lines in atoms, but I'm having trouble understanding this concept. And I need help, please!
     
  2. jcsd
  3. Dec 3, 2007 #2

    Astronuc

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    Staff: Mentor

    The electrons will not be moving FTL!

    Here is a nice background - http://physics.nist.gov/cuu/Constants/alpha.html
     
  4. Dec 8, 2007 #3
    I realize I didn't express the situation correctily. I'm not one implying that electrons would be movint FTL. Instead of saying hte electrons in element 137 would theoretically move FTL (which makes no sense when you do the math), I should've said it would be the last element in which elctrons on th 1s electron orbital would not move FTL. This raises interesting questions, because we know that electrons can'T move FTL. Is element 137 the last possible element? or if it's not, what happens in the following elements?

    Sorry I didn'T express the situation correctly in my first post, by I guess my questions still stand.
     
  5. Dec 8, 2007 #4
    The fine structure constant alpha = e^2/(hbar c 4 pi epsilon_0), e being the electron charge (I hope everything else is self-explanatory).
    I would check out that NIST page, it looks pretty good. The fsc originated as a perturbation parameter, 0th order quantum effects have alpha^0 (= 1) coefficients, fine structure has alpha^2, hyperfine has alpha^4, etc. Later, it was found that this constant determines the interaction between electrons and photons. It kept popping up in places, and since it's dimensionless, people would try to assign all kinds of theoretical importance to it, and make physical interpretations.

    As for an electron moving FTL, don't worry about it. It's not. That expression you gave for velocity (I'm guessing) treats an electron as being in a circular orbit, and goes from there. You can find all sorts of contradictions if you treat QM situations classically. Another example is a single electron, with spin hbar/2. Spin is angular momentum, and treating an electron as a rotating sphere, you find that
    L = hb/2 = m_e r_e^2 w -> v_m = r_e w = hb/(2 * m_e * r_e)
    v_m being the velocity at the equator of the electron. The classical electron radius
    (http://en.wikipedia.org/wiki/Classical_electron_radius) is about 3 picometers, which will give you 2e10 m/s, or ~ 60 c.

    The point I'm trying to make is that one can't always apply classical logic to quantum mechanics. In fact, one often can't. An electron can't be treated as a point particle at a fundamental level. That formula for velocity you gave can be arrived at in the following way:
    For a charged particle of charge e (electron) orbiting at distance r a body of charge Z*e(nucleus), the CLASSICAL velocity would be
    sqrt( (z k e^2) /(m_e r) ), where k is the k in F = ke^2/r^2, Coulomb constant in whatever your system of units is.
    The AVERAGE (not constant) distance for an electron in a 1s orbital is a_b/Z, a_b being the Bohr radius = hb/( m_e c alpha). Work that all through, and you'll get the velocity as Z alpha c.

    I tried finding the expectation value of (p/m)^2= v^2, and square rooting it, and you actually get the same thing. So I guess that's where it came from first. But that was dealing with the non-relativistic Schrodinger equation, it wouldn't be valid for any velocity close to c. Although it would be an expectation value, meaning one would make measurements both higher and lower than it, it comes from an equation which is not relativistically correct.

    Hope that helps.
     
  6. Dec 9, 2007 #5
    The "fine structure" is the small splitting in the spectral emission lines of an element that occur when a magnetic field is turned on.

    The value of the fine structure coefficient is e^2/(4 pi epsilon_0 h-bar c), where epsilon_0 is the permittivity of the vacuum, and e^2 the square of the electron charge.

    It is common, and almost universal practice, to mix in the physics of dielectrics (i.e. permittivity) with charge by redefining the charge e -> e/(4 pi epsilon_0^{1/2}. However this has the effect that discussion about the dielectric behavior of the vacuum near a point-like source now become blurred with discussions about the charge or "coupling constant" exhibiting "frequency dependent" behavior. Thus, one also sees reference to the notion of the "running of the coupling".

    What this means, in fact, is that the fine structure constant, near a point-like source, actually is NOT a constant. It exhibits an effective radial dependence. In scattering experiments, one normally does the analysis under a Fourier transform, so this radial dependence translates into a dependence of "frequency" or "energy" or "scale".

    In fact, what's happening is that the classical Maxwell-Lorentz constitutive relation (D = epsilon_0 E) that links electric induction (D) with the electric force (E) is no longer valid. In its place, one finds a relation of the form (D = epsilon E), where epsilon may be a function of the radius r from the center of the point-like source. That is, the vacuum near the source behaves like a dielectric medium.

    Quantum theorists explain this phenomenon as the polarization of the vacuum surrouning the point-like source, and -- because e has been redefined with epsilon in it -- make a distinction between the "bare" charge vs. "dressed" charge. In fact, e is well-defined, itself, being 1.6 x 10^{-19} Coulombs. Rather, it's epsilon that begins to exhibit its variability around the source.

    In quantum field theory, the effective constitutive relation actually arises from what is known as the Heisenberg-Euler Lagrangian and it takes on a yet more general form: D = epsilon E + theta B, H = epsilon c^2 B - theta E, where theta = 7/2 (epsilon - epsilon_0) c up to the 2nd order in Planck's constant. The theta term can be eliminated, though, by redefining H -> H + theta E, D -> D - theta B, since the Maxwell equations retain their form (however, things are not so simple, if theta, too, has dependence on position and is not constant).

    The result of all this is that the fine structure coefficient takes on the form
    alpha = e^2/(4 pi epsilon h-bar c)
    and also becomes a function of the radius. Its asymptotic value at large radii is around 1/137. It gets larger the closer to the point source you reach. In fact, the quantum theory of the electromagnetic field even surmises that there is a finite positive radius where alpha -> infinity -- the Landau Pole.

    Classical theory does not have a clear-cut analogue to all of this, though it is (in fact) possible to attempt to model one. For instance, one might adopt a dynamic law of the form (del^2 - (1/c)^2 d^2/dt^2) log(epsilon) = A epsilon (E^2 - B^2 c^2) for a suitable choice of the constant A. In fact, such an effective law is motivated by linking electromagnetic fields to a Kaluza-Klein model. Interetingly, the solution this entails for a point-like source DOES exhibit a Landau Pole! (It also exhibits other phases, including an "anti-screening" phase, where alpha -> infinity as r -> infinity, but alpha -> 0 as r -> 0 ... just like one expects in quark physics).

    In a more general gauge theory, the gauge field has an electric part E^a and magnetic part B^a, where a is now an index to the "complexion" of the field (e.g. a = 1,2,3 for an SU(2) gauge field). Here, the corresponding dual fields are D_a, H_a and the constitutive laws would have the form D_a = epsilon_{ab} E^b, H_a = epsilon_{ab} c^2 B^b for a vacuum. The coefficients k_{ab} = epsilon_{ab} c play the role of a METRIC for the gauge group (e.g. k_{ab} = delta_{ab}/g^2 for an SU(2) gauge field, where g is the coupling coefficient).

    The corresponding "fine structure" coefficient for a gauge charge e_a would be
    alpha = sum (e_a e_b k^{ab})/(4 pi h-bar).

    For an SU(2) force, corresponding to the charge e_a = (I_1,I_2,I_3) of the weak nuclear force ("isospin"), this works out to alpha_I = g^2 I^2/(4 pi h-bar), and so involves the "coupling coeffcient" g. This coefficient g plays the analogous role that the vacuum permittivity epsilon_0 does in electromagnetism.

    So, it is surmised that this "fine structure" constant alpha_I, that the one for quantum chromodynamics and the one for the U(1) part of the standard model (which is only indirectly related to the fine structure constant of electromagnetism) will all assume a common value at a critical radius or critical "scale". To a fair degree of approximation, they do. Nobody knows what the significance of that is.
     
  7. Dec 10, 2007 #6
    The fine structure constant was first introduced to us by Sommerfeld, and its name is a little confusing. It is really just the velocity of an electron in Bohr's n=1 quantum orbit divided by the speed of light.

    [tex] \alpha = \frac{v}{c}[/tex]

    And while there is the other equation related above, the first definition is just as definitive.
    Fine structure relates to the subtle differences between electron states in the spectra of the elements. Wikpedia is a good place to start, or you could try Hyperphysics.

    A word of caution about the fine structure constant, it pops up alot. There are many crackpot attempts to reproduce 1/137... You should try and avoid these. Bethe worked on the fine structure of the elements- again the spectra.
     
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