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Home The comparison of the models The quantum-mechanical shell model

The comparison of the models

The quantum mechanical states The periodic table from Mendelejew and Meyer The Periodic Table by Bettermann Discussion of both models The quantum-mechanical shell model The gradient term Limits of Quantum Mechanics

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The quantum-mechanical shell model

Another order for the structure of the elements is the quantum-mechanical shell model. The atomic shells are characterized by the four quantum numbers: main quantum number (N), the angular momentum quantum number (l), the magnetic quantum number (m) and the spin (± 1/2). According to the Pauli principle the four quantum numbers must vary in minimum one quantum value in each quantum mechanical system, i.e. that a state can always only be occupied by one electron. Combinatorial, possible states show that for each main quantum number N, states are possible. But the elements of the periodic table show that not every possible state is occupied. In the following table all theoretical and the found values of the traditional periodic table are listed:

The reason for the differences in the shell occupation is not known.
The following figure shows the current grading of shell in association with the periodicity of the periodic table. This representation illustrates each element according to the stated filling sequential arrangement. On the left side of the figure, the periods of the periodic table are plotted.
The beginning of each new shell is characterized by the principal quantum number at the left edge of the graph. All the following sub-shells are connected by the black dotted lines. It is evident that the quantum-mechanical sub-shells from shell M on continue over several periods. In the fourth period, for example, the elements of the 3d sub shell can be found. Also the sequential arrangement does not correspond with the periods or the classification of the shells. Furthermore it is remarkable I that each shell has a fixed number of occupied states. The sub shells is able to collect two electrons, the sub shell p six, the sub shell d ten and the sub shell f 14.

Here, the states (s) from the N shell on are marked g yellow and the last two last elements of the states (d) green. This identification shows which changes need to be made to get to the “novel” periodic table.

For better understanding of the changes in the adapted shell model, please look at figure belowe.
Here also the periods are defined by the blue dashed lines. The black lines represent the demarcation of the sub shells. It is easy to see that there is only one shell left for each period.

The first change takes place in the third period. That is why the 3 d sub shell, which was previously assigned to the fourth period, now belongs to the third period. This sub shell contains the two elements potassium and calcium. The last two elements, copper and zinc, are removed from the sub shell. So by this time the third sub shell corresponds to the third period. The sub shells 3 s and 3 p remain unaffected.

The fourth period begins with the two elements copper and zinc; both had filled the sub shell s, they come from third period, from the 3d sub shell. So the 4 d shell is now merely attached to the fourth period. From this fourth period the two elements silver and cadmium are removed; instead of them, rubidium and strontium from the previous 4 s sub shell are attached. The 4 p sub shell remains unaffected. The fourth period is filled as the third period.

In the fifth and sixth period there are greater rearrangements in the sub shells d and f than before. In the traditional presentation, the fifth period holds 18 elements. These are enlarged to the elements of Lantanoide. So now the fifth period holds 32 elements.

The fifth period is made up of the following now: Both the states are filled by silver (Ag) and cadmium (Cd), both come from the 4d sub shell. The 5 p sub shell remains unchanged and contains six elements. The new 5 d sub shell does not have ten elements anymore but 14; these are the elements of the lanthanides. The last three elements of the sub shell are attached to the 5 f sub shell. The new 5 f sub shell comes from the old 5d sub shell without the metals gold (Ag) and mercury (Hg), which fill the 6 s subshell.

In the sixth period, there have been similar changes like in the fifth. Here, the two 6 s states are occupied by gold and mercury. The 6 p sub shell remains, as in other periods, unaffected. The new 6 d sub shell contains, like the 5 d sub shell, now 14 elements. The sub-group of actinides is assigned the 5 d sub shell. This sub-group of actinides is again enlarged by the two elements of the 7 s state: francium (Fr) and radium (Rd). At the end of this sub shell both the two elements Mendelevium (Md) and nobelium (No) are missed. The 6 f sub shell completes and closes the 6th period and ends with the 110th element.

The seventh period starts with the 111th element, which is Roentgenium (Rg), the second element in the s 7 sub shell is Copernicium (Co).

At this point we are already deep going into the subject of artificially produced elements and for this reason I interrupt the description of changes in the shell model here. The filling of the further shells will continue as described.