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I'm currently reading about nuclear and particle physics in B.R.Martins 'An introduction to nuclear and particle physics'.
In chapter 7.3 he introduces the shell model of the nucleus and draws an analogy to the periodic table and how we think of it being constructed of progressively placing more and more electrons in 'orbitals' given by the quantum numbers n,l and ml.
From the study of the hydrogen atom we know that the energy levels are degenerate with 2n^2. Martin further notes that if a shell or a subshell is filled then
\sum m_s = \sum m_l = 0
which implies that
\vec L = \vec S = 0 = \vec J = \vec L + \vec S.
He then states that 'For any atom with a closed shell or a closed sub-shell, the electrons are paired off and thus no valence electrons are avaiable. Such atoms are therefore chemically inert. It is straight forward to work out the atomic numbers at which this occurs. These are
Z = 2,10,18,36,54.'
But I do not understand how these numbers are obtained. If one follows the 'hydrogen model' I would think that the numbers at which we have a closed shell or subshell would be
Z = 2,4,10,12,18,28, \ldots
corresponding to the electron configurations
1s^2, 1s^2 2s^2, 1s^2 2s^2 2p^6,1s^2 2s^2 2p^63s^2, \ldots
What have I missunderstood here?
In chapter 7.3 he introduces the shell model of the nucleus and draws an analogy to the periodic table and how we think of it being constructed of progressively placing more and more electrons in 'orbitals' given by the quantum numbers n,l and ml.
From the study of the hydrogen atom we know that the energy levels are degenerate with 2n^2. Martin further notes that if a shell or a subshell is filled then
\sum m_s = \sum m_l = 0
which implies that
\vec L = \vec S = 0 = \vec J = \vec L + \vec S.
He then states that 'For any atom with a closed shell or a closed sub-shell, the electrons are paired off and thus no valence electrons are avaiable. Such atoms are therefore chemically inert. It is straight forward to work out the atomic numbers at which this occurs. These are
Z = 2,10,18,36,54.'
But I do not understand how these numbers are obtained. If one follows the 'hydrogen model' I would think that the numbers at which we have a closed shell or subshell would be
Z = 2,4,10,12,18,28, \ldots
corresponding to the electron configurations
1s^2, 1s^2 2s^2, 1s^2 2s^2 2p^6,1s^2 2s^2 2p^63s^2, \ldots
What have I missunderstood here?