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 nonionized.
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 20^{th} 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.
Equation 2 This balances the electrostatic force against the centrifugal force, 
Equation 3 This is the solution for the orbital radius in the n^{th} energy state. 



Equation 4 This is the solution for the orbital velocity in the n^{th }energy state. 
By substituting for n = 1 we get the Bohr radius and Bohr velocity as
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
Equation 5 Angular momentum is quantized as before. 
Equation 6 The charge on the nucleus is now Z q because there are Z protons in the nucleus. 
Equation 7 The radius is less than that of hydrogen by a factor 1/Z 



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 nonrelativistic 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.