Quantum defect theory

The quantum defect theory of physical chemistry is used for the model calculation of the characteristic line spectra of atoms with several electrons , especially those of the 1st main group .

Johannes Rydberg had already established in 1889 that the Rydberg formula , which was later named after him, should be corrected for the alkali metals in such a way that instead of the main quantum numbers, the effective main quantum numbers reduced by an amount must be used; the later were called quantum defects . ${\ displaystyle n}$ ${\ displaystyle \ delta _ {n}}$ ${\ displaystyle n '= n- \ delta _ {n}}$ ${\ displaystyle \ delta _ {n}}$

In the context of the orbital models, the deviations are to be understood in such a way that the inner electrons do not completely shield the atomic core from the outer electron , since its orbital overlaps with those of the inner electrons.

Comparison of the external electrons of lithium and hydrogen

If one compares the 2s electron of lithium ( external or luminous electron, in the ground state ; 5.37 eV ) and a 2s electron of hydrogen ( excited state ; 3.4 eV), then the 2s lithium electron is ionized more energy is required than for that of the 2s hydrogen electron. This means that the 2s lithium electron is more strongly attracted to its own atomic nucleus than the 2s hydrogen electron to its own .

This stronger attraction can be explained using the probabilities of the location of the respective orbitals. Orbitals are spaces with a non-negligibly small probability of their presence: a 2s electron or the 2s orbital has a characteristic wave function that gives an indication of the probability of finding the electron at precisely this distance at any distance from the nucleus. This distribution is spherically symmetric for s orbitals , but also depends on the direction for p orbitals, d orbitals etc.

The further structure of the atom plays no role for the wave function, so that the 2s electrons of lithium and hydrogen have a similar wave function. This has a maximum close to the core , i.e. In other words, the 2s electron can "dive" into the area between the inner 1s electrons and the nucleus with a small but important probability. There it experiences a significantly higher attraction due to the positive charge of the nucleus ; the shielding of the core from the outside, which had previously caused the two 1s electrons, is now incomplete.

The 2s electron of the lithium atom experiences a triple positively charged nucleus in the area close to the nucleus, but a single positively charged nucleus in the distant nucleus ( effective nuclear charge ); averaged over time, it is attracted to a core of the charge . In stark contrast to this, the 2s electron of hydrogen is only attracted by one nucleus of the charge , since the hydrogen atom has only one proton in its nucleus. ${\ displaystyle 1e <Q <3e}$ ${\ displaystyle Q = 1e}$

The quantum defect

A stronger attraction leads to a higher ionization energy , which is why the electrons of the alkali atoms are shown lower in the diagram of the energy levels than comparable orbitals of the hydrogen atom; the difference to the ionization continuum increases. This relationship is expressed by the effective principal quantum number by subtracting the quantum defect δ from the principal quantum number of the respective orbital , which reflects the strength of the additional attraction: ${\ displaystyle n '}$ ${\ displaystyle n}$

{\ displaystyle n '= n- \ delta _ {n, l}.}

The quantum defect and thus the energy level depends not only on the main quantum number (the size of the orbital), but also significantly on the orbital angular momentum quantum number and thus the wave function or the shape of the orbital (s, p, d or f). The spherical s orbital has a pronounced maximum of the wave function near the core, the dumbbell-shaped p orbital less, etc. The shielding becomes more “complete” with increasing secondary quantum numbers, the outer electron “dips” less often into the core near area and is less strongly attracted . In other words: the larger the secondary quantum number l, the smaller the quantum defect δ or the more approximated the energy level of the orbital at that of a comparable hydrogen atom. ${\ displaystyle n}$ ${\ displaystyle l}$

In the case of the hydrogen atom, the electron only sees the nucleus, which has a radially symmetrical potential . From this it follows that a 4s electron of hydrogen is energetically equal to a 4d electron of hydrogen: there is an l-degeneracy , i.e. H. the different shapes of the orbitals are not reflected in different energies. In the case of alkali metals, the l-degeneracy is eliminated, since the inner electrons do not all have radially symmetrical location probabilities.

If one determines the individual transitions and their energies by means of spectroscopy , one can determine the quantum defect for each orbital of an atom. ${\ displaystyle \ delta _ {n, l} \! \,}$

Line spectra

If the Rydberg formula derived from Bohr's atomic model for hydrogen-like one-electron systems is adapted to the quantum defect theory, the formula for optical transitions ( spontaneous emission or absorption of light ) results :

{\ displaystyle {\ frac {1} {\ lambda _ {\ mathrm {vac}}}} = Z ^ {2} R \ left ({\ frac {1} {n_ {1} '^ {2}}} - {\ frac {1} {n_ {2} '^ {2}}} \ right).}

Are there

${\ displaystyle \ lambda _ {\ mathrm {vac}}}$ the wavelength of light in a vacuum
${\ displaystyle Z}$ the atomic number
${\ displaystyle R}$ $R.$ the Rydberg constant for the respective element: with ${\ displaystyle R = {\ frac {R _ {\ infty}} {1 + {\ frac {m_ {e}} {M}}}}}$ $R = {\ frac {R _ {{\ infty}}} {1 + {\ frac {m_ {e}} {M}}}}$
- ${\ displaystyle m_ {e}}$ the mass of the electron
- ${\ displaystyle M}$ the nuclear mass (depending on the isotope present )
- ${\ displaystyle R _ {\ infty}}$ the Rydberg constant for hydrogen.
${\ displaystyle n '= n- \ delta _ {n, l}}$ the effective principal quantum numbers

Quantum Defects in Laser Physics

The term used here is to be distinguished from the quantum defects , which in laser physics describe the energy difference between the stimulating pump energy and the signal energy.

Individual evidence

^ MJ Seaton: Quantum defect theory . In: Rep. Prog. Phys. tape 46 , 1983, pp. 167-257 , doi : 10.1088 / 0034-4885 / 46/2/002 ( iop.org [accessed November 28, 2014]).

↑ Ch. Junge: Molecular Applications of Quantum Defect Theory . Taylor & Francis Group, 1996, ISBN 978-0-7503-0162-6 ( limited preview in Google Book Search).

^ TY Fan: Heat generation in Nd: YAG and Yb: YAG . In: IEEE Journal of Quantum Electronics . tape 29 , no. 6 , 2002, pp. 1457-1459 , doi : 10.1109 / 3.234394 ( ieee.org [accessed November 28, 2014]).

[1] MJ Seaton: Quantum defect theory . In: Rep. Prog. Phys. tape 46 , 1983, pp. 167-257 , doi : 10.1088 / 0034-4885 / 46/2/002 ( iop.org [accessed November 28, 2014]).

[2] Ch. Junge: Molecular Applications of Quantum Defect Theory . Taylor & Francis Group, 1996, ISBN 978-0-7503-0162-6 ( limited preview in Google Book Search).

[3] TY Fan: Heat generation in Nd: YAG and Yb: YAG . In: IEEE Journal of Quantum Electronics . tape 29 , no. 6 , 2002, pp. 1457-1459 , doi : 10.1109 / 3.234394 ( ieee.org [accessed November 28, 2014]).