The Rydberg Constant and Rydberg Series

Here I take a look at the Rydberg formula in more detail and derive the Rydberg series, which is useful in calculating the change in energy level between any two states of the hydrogen atom and can be used to generate all of the other series:  Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys etc.  I go on to look at how Rydberg’s constant and series can be used to develop a more credible model for the hydrogen atom than that of the Standard Model.

Rydberg’s formula is the more general case of a formula developed by Balmer to describe the emission spectra of hydrogen.  Where Balmer only dealt with one set of spectral lines, Rydberg dealt with all the emission lines of the hydrogen atom.

Balmer’s formula dealt with the wavelength of the emitted light, whereas Rydberg found it more convenient to deal with the reciprocal of the wavelength or wavenumber and developed the formula:

 Equation 20-1

Equation 1

Ris known as the Rydberg constant.  Rydberg originally used a value based on Balmer’s formula, which was derived empirically, but Niels Bohr was able to calculate a value analytically for this constant based on his model for the hydrogen atom.

 Equation 20-2

Equation 2

Bohr had assumed that the proton around which the electron orbits was much more massive than the electron itself.  In such a case the electron orbits around the centre of gravity of the proton.  In fact the electron orbits around the combined centre of gravity of the proton and the electron, which is somewhat closer to the electron than the centre of gravity of the proton alone.  In 1917 Arnold Somerfield provided a correction called the reduced mass:

 Equation 20-3

Equation 3

Where mp is the rest mass of the proton and me is the rest mass of the electron and me/mp=1/1847.

The use of wavenumber in Rydberg’s formula provides very little insight into what is actually going on and so it is useful to express the formula in terms of frequency or energy multiplying both sides by either by c for frequency or by 2πħc for energy.  Expressing the Rydberg formula in terms of energy is particularly useful:

 Equation 20-4

Equation 4

If we ignore the slight error due to the effect of the reduced mass then:

 Equation 20-5

Equation 5

Recognizing that K, the Coulomb constant or electrostatic force constant, is given by:

 Equation 20-6

Equation 6

That

 Equation 20-7

Equation 7

And that Alpha, the fine structure constant is given by:

 Equation 20-8

Equation 8

Equation 5 can be simplified to:

 Equation 20-9

Equation 9

This in itself is interesting because ½mc2 is the kinetic energy that an electron would have if it were travelling at the speed of light and cα is the Bohr velocity, the velocity at which the electron supposedly travels in the Bohr model.

By setting n2 = ∞  and allowing n1 to take on successive integer values we obtain the difference between the energy level in any particular energy state and the maximum possible energy that the atom can contain.  This forms a series of values called the Rydberg series.

 Equation 20-10

Equation 10

The Rydberg series is useful because it allows us to calculate the energy, frequency or wavelength associated with any energy transition within the atom.

By setting n =1 in the Rydberg series we obtain the maximum possible energy that the hydrogen atom can absorb or emit, this is called energy potential [1]of the atom, which for the base energy state is

 Equation 20-11

Equation 11

All of this is consistent with the Bohr model, however the Bohr model is fraught with difficulties which are also present in other later models.  Most notable in the Bohr model is the quantum leap; the need for the electron to move instantly from one place to another without occupying anywhere in between.  Other, later models seek to avoid this by asserting that the state of the electron is inherently uncertain, until it is subject to some sort of observing process, at which time its wave function (whatever that is) collapses (however that works) to reveal either its position or velocity.  Of course the collapsing wave function is merely a euphemism for the quantum leap and is equally unrealizable and unrealistic.

Since nothing can travel faster than light, the maximum kinetic energy that the orbiting electron in the hydrogen atom could ever possibly achieve is given by:

 Equation 20-12

Equation 12

If we are to avoid the problem of the quantum leap, then the electron must orbit at the same orbital radius in all energy states, since anything else implies that the quantum leap exists.  If the orbital radius remains constant for all energy states then there can be no change in potential energy between energy states and all of the change in energy must therefore be kinetic in nature.  In which case the energy in the base state must be given by

 Equation 20-13

Equation 13

Where v is the orbital velocity in the base energy state.

But the difference between these two values (Equation 12 and Equation 13) is the energy potential of the atom and so we can write:

 Equation 20-14

Equation 14

Simplifying and rearranging this gives:

 Equation 20-15

Equation 15

Readers will be familiar with the LHS of this equation as the Lorentz factor Gamma (γ) associated with the effects of special relativity. Hence

 Equation 20-16

Equation 16

And from this we can calculate the value of v as 0.999973371c.

What this reveals is an atom in which the electron in the base energy state is orbiting at 99.9973371%c and in the maximum possible energy state is orbiting at velocity c.  There is no substantial change in orbital radius across this dynamic range and hence no need to introduce the idea of the quantum leap or its latter day equivalents – collapsing wave functions.  The angular momentum too remains substantially the same for all energy levels and if we equate that to Planck’s constant we find that the orbital radius is

 Equation 20-17

Equation 17

We stationary observers then see the orbital frequency as

 Equation 20-18

Equation 18

However the orbiting electron is in an environment where time is slowed down due to the effects fo relativity by a factor Gamma and therefore where frequency is multiplied by the same factor Gamma and hence, as far as the orbiting electron is concerned, its orbital frequency is

 Equation 20-19

Equation 19

We can repeat this exercise for the values in the Rydberg series and not just the base energy state, in which case we will obtain the orbital velocities and corresponding orbital frequencies for each respective energy level.

 Equation 20-20

Equation 20

The orbital frequency seen by us stationary observers remains substantially unaltered as in Equation 18, however that for the orbiting electron is different for each energy level and is given by

 Equation 20-21

Equation 21

From Equation 21 it can be seen that the frequencies experienced by the orbiting electron form a harmonic series, starting with a base frequency experienced in the base energy state an rising in integer multiples of Gamma with each succeeding energy state.  We stationary observers, on the other hand, always see the orbital frequency as being substantially constant.  This shows that at the heart of the discrete energy levels of the atom lies a harmonic series, much as de Broglie suggested, only rather than appearing directly in our observing domain, it appears instead in the domain of the moving electron.

Equation 20 shows us that the variable of quantisation within the atom is not angular momentum as postulated by Niels Bohr and incorporated into subsequent models, but instead is Gamma, the Lorentz factor.  In the base energy state Gamma has a value of 137.03, in the second energy state this rises to 2*137.03 and so on.  It is important to note that Gamma is not inherently quantised, that is it can vary continuously, rather that the dynamics of the atom are such that the atom is only stable when it takes on values which are an integer multiple of 1/α.

The model proposed here therefore gives a physical significance to α, the Fine Structure Constant.  In the past the precise nature of what this constant represents has been missing. This has led to some wild speculation as to its true nature and significance.  Here it is simply seen as the ratio of the orbital circumference as traversed at relativistic speeds and so foreshortened, to that traversed at non relativistic speeds.

The stability of such an atom depends on their being a force balance between the orbiting electron and the nuclear proton. It must be stable in each of the various energy states.

We also know that wherever we see a harmonic series in the frequency domain there must be a corresponding sampling process in the time domain.  This comes about because a harmonic sequence in the Fourier space appears as a series of impulses or spikes, equally spaced along the frequency axis, forming what is known as a Dirac comb.  The inverse Fourier transform of such a Dirac comb in the time domain is itself a Dirac comb formed of a series of impulses or spikes, only this time in the time domain and spaced T seconds apart where T=1/F and where F is the fundamental frequency of the harmonic series.  Such a Dirac comb in the time domain represents a sampling signal and so we can expect to find some sort of sampling process taking place in the time domain when we examine the electro dynamics of the atom.

Such an analysis is beyond the scope of this post, however if you want to understand the role of the sampling process and the internal dynamics of the hydrogen atom then this is covered in much more detail in Sampling the Hydrogen Atom.


 

[1] Note that energy potential should not be confused with potential energy.  Energy potential is the difference between the energy contained within the atom and the maximum possible energy that the atom can absorb, while potential energy is energy associated with the difference in orbital radius between a particular energy state and the base energy state in the presence of an electric field and is analogous to gravitational potential energy associated with the height of an object above some datum level.

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