Quantum defect

The term quantum defect refers to two concepts: energy loss in lasers and energy levels in alkali elements. Both deal with quantum systems where matter interacts with light.

In laser science

In laser science, the term "quantum defect" refers to the fact that the energy of a pump photon is generally higher than that of a signal photon (photon of the output radiation). The energy difference is lost to heat, which may carry away the excess entropy delivered by the multimode incoherent pump.

The quantum defect of a

lasing.^[1]

At given frequency

\omega _{\rm {p}}

of pump and given frequency

\omega _{\rm {s}}

of

lasing

, the quantum defect

q=\hbar \omega _{\rm {p}}-\hbar \omega _{\rm {s}}

. Such a quantum defect has dimensions of energy; for the efficient operation, the

gain medium

(measured in units of energy) should be small compared to the quantum defect.

The quantum defect may also be defined as follows: at a given frequency $\omega _{\rm {p}}$ of pump and given frequency $\omega _{\rm {s}}$ of

lasing

, the quantum defect

q=1-\omega _{\rm {s}}/\omega _{\rm {p}}

; according to this definition, quantum defect is dimensionless.^{[citation needed]} At a fixed pump frequency, the higher the quantum defect, the lower is the upper bound for the power efficiency.

In hydrogenic atoms

In an idealized Bohr model alkali atom (such as sodium, pictured here), the single outer-shell electron stays outside the ionic core and it would be expected to behave just as if in the same orbital of a hydrogen atom.

The quantum defect of an

hydrogen wavefunction. A simple model of the potential experienced by the single valence electron of an alkali atom is that the ionic core acts as a point charge with effective charge e and the wavefunctions are hydrogenic. However, the structure of the ionic core alters the potential at small radii.^[2]

The

1/r potential in the hydrogen atom leads to an electron binding energy

given by

E_{\text{B}}=-{\dfrac {Rhc}{n^{2}}},

where

R

is the Rydberg constant,

h

is Planck's constant,

c

is the speed of light and

n

is the principal quantum number.

For

Coulomb potential with an effective charge of e no longer describes the potential. The spectrum is still described well by the Rydberg formula

with an angular momentum dependent quantum defect,

\delta _{l}

:

E_{\text{B}}=-{\dfrac {Rhc}{(n-\delta _{l})^{2}}}.

The largest shifts occur when the orbital angular momentum is equal to 0 (normally labeled 's') and these are shown in the table for the

alkali metals:^[3]

Element	Configuration	$n-\delta _{s}$	$\delta _{s}$
Li	2s	1.59	0.41
Na	3s	1.63	1.37
K	4s	1.77	2.23
Rb	5s	1.81	3.19
Cs	6s	1.87	4.13

References

doi:10.1109/3.234394
.

^ http://www.phy.davidson.edu/StuHome/joesten/IntLab/final/rydberg.htm, Rydberg Atoms and the Quantum Defect at the site of Davidson College, Physics department

ISBN 978-0-19-850695-9

Retrieved from "https://en.wikipedia.org/w/index.php?title=Quantum_defect&oldid=1147632494"

[LaserQD-1] :10.1109/3.234394
.

[2] ttp://www.phy.davidson.edu/StuHome/joesten/IntLab/final/rydberg.htm, Rydberg Atoms and the Quantum Defect at the site of Davidson College, Physics department

[3] ISBN 978-0-19-850695-9

[1]

[2]

[3]

In laser science

In hydrogenic atoms

See also

References