p31318notes12.pdf - 1 P313 Notes 12 The Alkali Metals Li Na...

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1 P313 Notes 12 The Alkali Metals Li Na K Rb Cs The spectrum of these metals looks a bit like this, drawn for sodium: σ
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2 This looks like complete irregularity, but in fact it is made up of four overlapping Rydberg series . That is to say, series with wavenumbers of the form - 2 2 1 1 b a R - except that a and b are not integers. They are of the form α - n , where α is the quantum defect for a particular level, and α - n is the effective quantum number . A very fine point: Note that the Rydberg constant contains the reduced electron mass, M m mM + = μ . Here m is the electron mass, but what is M ? The alkali metals have a single electron outside a number of closed shells. Thus the ground state configurations of the first three are Li: 1 s 2 2s Na: 1 s 2 2 s 2 2 p 6 3 s K: 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s The atoms (in a Bohr model) thus look like this (schematically, and not to scale): The radius of the outermost orbit is much larger than the radii of the inner ones. (Recall that the radii go as n 2 .) Thus M is the mass of the nucleus plus the masses of the inner electrons. However, the radius of the core is much larger than the radius of the nucleus, so the possibility arises that not only s electrons, but other electrons, with 0 > l may penetrate the core. *
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3 Here is a Bohr-Sommerfeld representation of a potassium atom. There is a single electron with n = 4, outside a core consisting of filled KLM shells. If the optical electron is in an f orbital, it does not penetrate the core at all. The lower the orbital angular momentum, the greater the penetration of the core, and the greater the quantum defect. s p d f
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4 Here is a Schrödinger representation of a sodium atom. The continuous curve shows the charge density in the K and L shells. The dashed curve shows the square of the radial part of the eigenfunction - i.e. the probability density for the electron in a 3 s orbital. For hydrogen R 2 peaks at about 14 a 0 . It peaks at a rather shorter distance in Na. You can see that the electron spends most of its time outside the core, but a non-negligible time inside the core. Just inside outer boundary of the core, the electron experiences a potential r e 0 4 πε . Just inside the inner boundary of the core, it experiences a potential r Ze 0 4 πε . [From Atoms and Molecules Interacting with Light by van der Straten and Metcalf] Even when the optical electron is outside the core there is a small complication. If the core is spherically symmetric, the field and potential outside the core is the same as if all the charge (which is +1 e ) is concentrated at a point at the centre. But the core is not quite spherically symmetric.
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