Hydrogenic Atoms and the Bohr Model

As we move up the periodic table atoms get heavier. They do so because the nuclei of heavier atoms contain more protons.  In addition to the protons these heavier nuclei also contain another type of particle; the neutron, roughly equal in mass to the proton, but itself electrically neutral.

So while the hydrogen atom comprises a single electron in orbit around a single proton, helium has two protons and two neutrons in its nucleus.  Hydrogen is electrically neutral, the unit positive charge on the proton being matched by the unit negative charge on the electron. The helium nucleus is surrounded by either one or two electrons.  If it has two electrons the atom is electrically neutral. However it is possible for one of these electrons to be removed, in which case the atom, having two positive protons and one negative electron, has overall positive charge.  In this case the helium atom is said to be ionized.

Next in the periodic table we find Lithium which has three protons, three neutrons and one, two or three electrons. If it has just one electron the Lithium atom is said to be doubly ionized – that is two of the full complement of electrons are missing. If it has two then it is singly ionized and with three electrons it is non-ionized.

Atoms which are ionized in such a way as to have only one orbiting electron are recognised as so called Hydrogenic atoms.  They are an important special case because they contain a nucleus with just one orbiting electron and so are susceptible to simple mathematical analysis.  These are called two body systems.

The position of atoms in the periodic table is determined by the number of protons in the nucleus, usually referred to as the atomic number of the atom and denoted by Z. Hence Helium has Z=2, Lithium; Z=3 and so on up the periodic table to the heaviest atom Ununoctium with Z=118.

In the early part of the 20th century Niels Bohr developed a model for the hydrogen atom.  In it he made an assumption that angular momentum was quantized and could only exist in discrete steps or quanta.  The size of the smallest step was equal to Planck’s constant.  This assumption has formed an integral part of all subsequent theories to the extent that it is now regarded as an article of received wisdom.  And yet this assumption lacks any real justification.

As well as assuming that angular momentum is quantized, Bohr also chose to ignore two other factors which he should have taken into account.  A charged particle which follows a circular path should emit a type of radiation called synchrotron radiation.  Bohr chose to simply ignore this, if he hadn’t the orbital path of the electron would have decayed and the electron eventually collapsed into the atomic nucleus.  Equally, if not more important, he ignored the effects of relativity on the orbiting electron.   The orbital velocity for lighter atoms such as hydrogen and helium is sufficiently low as to not be affected by relativity.  But for heavier atoms it becomes very significant indeed, sufficient to cast doubt not only on the Bohr model,  but also on all of the models which derive from it, including the currently held Standard Model.

The success of Bohr’s model lay in the fact that it produced results for the energy levels of the hydrogen atom and of all hydrogenic atoms which matched those of the empirically derived Rydberg formula.

The basis of Bohr’s model was to balance the centrifugal force, acting to throw the orbiting electron off into space, against the attractive electrical force between the atomic nucleus and the orbiting electron.  In order to make his model work Bohr needed a second equation.  He found it in the work of a colleague, John W Nicholson, that angular momentum could only occur in discrete levels such that it is always an integer multiple of Planck’s constant.

 Hydrogenic EQ 1

Equation 1

This is the statement that angular momentum is quantized and can only take on values which are integer multiples of Planck’s constant.

Note that m0 is the rest mass of the electron – ie it does not take account of relativity.

 Hydrogenic EQ 2

Equation 2

This balances the electrostatic force against the centrifugal force,

 Hydrogenic EQ 3

Equation 3

This is the solution for the orbital radius in the nth energy state.


 Hydrogenic EQ 4

Equation 4

This is the solution for the orbital velocity in the nth energy state.

By substituting for n = 1 we get the Bohr radius and Bohr velocity as

Hydrogenic Function 1

Hydrogenic Function 2

In this case the orbital velocity is some 137 times less than the speed of light, where Gamma has a value of 1.000026629.  The effect of gamma on the orbiting system is to increase the mass of the electron, in this case by a negligible .00266 %.  So Bohr could reasonably justify his assumption that the effects of relativity can be ignored in the case of the hydrogen atom.  However the situation is different as we consider elements which are further up the periodic table.

Rewriting the Bohr equations for the more general case of the hydrogenic atom gives

 Hydrogenic EQ 5

Equation 5

Angular momentum is quantized as before.

 Hydrogenic EQ 6

Equation 6

The charge on the nucleus is now Z q because there are Z protons in the nucleus.

 Hydrogenic EQ 7

Equation 7

The radius is less than that of hydrogen by a factor 1/Z


 Hydrogenic EQ 8

Equation 8

The orbital velocity is more than that of hydrogen by a factor Z.

If we apply Bohr’s model to elements of higher atomic number, we find that the orbital velocity in all energy states is increased, such that for an atom with atomic number Z this orbital velocity is Z times that of the equivalent energy state for the hydrogen atom.  So in the case of hydrogenic helium the velocity of the electron in the base state is 4375607.921.  Here gamma has a value of just 1.000106531, still small enough to be ignored, but as we move further up the periodic table the situation changes.

Table 1 lists the orbital velocity of the base state and second energy state of the first 30 hydrogenic atoms and that of Ununoctium in the periodic table together with the corresponding values for gamma.

Z Velocity n=1 Gamma n=1 Velocity n=2 Gamma n=2 ΔGamma
1 2187803.961 1.000026629 1093901.98 1.000006657 0.0000
2 4375607.921 1.000106531 2187803.961 1.000026629 0.0001
3 6563411.882 1.000239742 3281705.941 1.000059919 0.0002
4 8751215.842 1.000426327 4375607.921 1.000106531 0.0003
5 10939019.8 1.000666376 5469509.901 1.000166469 0.0005
6 13126823.76 1.000960004 6563411.882 1.000239742 0.0007
7 15314627.72 1.001307352 7657313.862 1.000326358 0.0010
8 17502431.68 1.001708588 8751215.842 1.000426327 0.0013
9 19690235.64 1.002163906 9845117.822 1.000539662 0.0016
10 21878039.61 1.002673526 10939019.8 1.000666376 0.0020
11 24065843.57 1.003237695 12032921.78 1.000806484 0.0024
12 26253647.53 1.003856689 13126823.76 1.000960004 0.0029
13 28441451.49 1.00453081 14220725.74 1.001126953 0.0034
14 30629255.45 1.005260389 15314627.72 1.001307352 0.0040
15 32817059.41 1.006045783 16408529.7 1.001501222 0.0045
16 35004863.37 1.006887382 17502431.68 1.001708588 0.0052
17 37192667.33 1.007785602 18596333.66 1.001929473 0.0059
18 39380471.29 1.008740892 19690235.64 1.002163906 0.0066
19 41568275.25 1.009753729 20784137.62 1.002411913 0.0073
20 43756079.21 1.010824624 21878039.61 1.002673526 0.0082
21 45943883.17 1.011954119 22971941.59 1.002948775 0.0090
22 48131687.13 1.013142788 24065843.57 1.003237695 0.0099
23 50319491.09 1.01439124 25159745.55 1.003540321 0.0109
24 52507295.05 1.01570012 26253647.53 1.003856689 0.0118
25 54695099.01 1.017070106 27347549.51 1.004186839 0.0129
26 56882902.97 1.018501913 28441451.49 1.00453081 0.0140
27 59070706.93 1.019996296 29535353.47 1.004888646 0.0151
28 61258510.89 1.021554046 30629255.45 1.005260389 0.0163
29 63446314.86 1.023175997 31723157.43 1.005646085 0.0175
30 65634118.82 1.024863022 32817059.41 1.006045783 0.0188
118 258160867.3 1.967026747 129080433.7 1.107960681 0.8591

 Table 1

The Right Hand column shows the difference in Gamma between base and second energy states and so represents the error in the mass term from the Bohr model and the Rydberg formula for the first spectral line of these atoms.

The spectral lines of these atoms are calculated by taking the difference between the energy levels.  Here we have calculated the effect of Gamma in the base energy state and the second energy state for each of the atoms.  In the Bohr model the orbital velocity falls off with increasing energy state, but rises with increasing atomic number.  For atoms with low atomic number the value of gamma in the base state is very small and even smaller in the second energy state, hence the first spectral line in such atoms is does not deviate significantly from the Rydberg and non-relativistic Bohr models for these types of atom.

The situation changes however as we look at atoms with higher atomic numbers. By the time we get to Zinc with atomic number 30, the mass of the electron in the base state would have increased by 2.4% due to relativity while in the second energy state it would have increased by only 0.6%.  Hence there is a discrepancy between the frequency of the first spectral line that is predicted by the Rydberg formula or that observed directly and that predicted by the Bohr model if we take relativity correctly into account of around 1.8%.

For heavier atoms the situation gets worse, to the point where the first spectral line of Ununoctium would be 85% in error if we used Bohr’s model and correctly took the effects of relativity into account.

The Bohr model is fundamentally flawed, almost everybody is agreed on that.  The question is to determine where the flaw lies and at least one flaw lies in the fact that it ignores the effects of special relativity on the mass of the orbiting electron.

The response of the physics community to this – has been either to ignore it, or to say that the Bohr model is no longer current and has been superseded by other more up to date models and so this is irrelevant.  Or to say that angular momentum is different on the quantum scale and effectively to invent a new type of angular momentum called quantum angular momentum which somehow ignores the effects of relativity when it needs to and takes account of it when it needs to.

We cannot fault Bohr’s dynamics or his calculations, and the problem has to lie in the assumption on which the model rests; that angular momentum is quantized. There is no doubt that something in the atom is quantized, but I don’t believe that it is angular momentum.  For Bohr this was expedient, but he was never able to explain just why it should be the case.  His model, and by implication all succeeding models which are based on it, lacks a mechanism to explain just how and why angular momentum should be quantized.

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