Chapter 5 Alkali atoms
Properties of elements in periodic table
The Alkali metals
Alkali atoms
Alkali atoms: atoms only has a valence electron which is weakly
bound outside, and all other electrons are in closed shells.
Shell structures:
3Li
11Na
(1s)2(2s)1
(1s)2(2s)2(2p)6(3s)1
19K (1s)2(2s)2(2p)6(3s)2(3p)6(4s)1
37Rb (1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10(4p)6(5s)1
55Cs (1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10(4p)6(5s)2(4d)10(5p)6(6s)1
87Fr (1s)2(2s)2(2p)6(3s)2(3p)6(4s)2(3d)10(4p)6(5s)2(4d)10(5p)6(6s)2
(4f)14(5d)10(6p)6(7s)1
Work of ionisation for the elements with Z=1 to 20.
Values are given for the neutral atom, and for singly, doubly and triply charged
ions. The ionisation energy is always especially large for a noble gas
configuration (closed shell). It is especially low if there is only one electron
more than a noble gas configuration.
Lifting of the l degeneracy
The alkali atoms have the next simplest spectra after those of
hydrogen atom.
The comparison shows that in the alkali atoms, the l
degeneracy (orbital degeneracy) is lifted. States with the
same principal quantum number n and different orbital
angular momentum quantum numbers l have different
energies.
We would like to understand this effect qualitatively.
The l degeneracy is lifted
Simplified term diagram for the alkali metal atoms, showing the empirical positions of the most important
energy terms. The principal quantum number n is indicated by numerals, the secondary quantum numbers l
by the letters S, P, D, and F. For comparison, the levels of the H atom are given on the right.
1) Relative to the terms of H atom, those of alkalis lie lower;
2) For larger n, the terms are only slightly different from those of Hydrogen;
3) For small l are more strongly bound and their terms lie lower in the term diagram;
4) This effect becomes stronger with increasing Z.
Screening effect
R
r
+Ze
-(Z-1)e
-e
Model of an alkali atom
The valence electron is screened from the
nuclear charge +Ze by the (Z-1) inner
electron.
The closed shell is spherically symmetric, and is strongly bound to the
nucleus. The valence electron is located at a relatively large distance r from
the nucleus. It moves in the electrostatic field of the nuclear charge +Ze,
which is for the most part screened by the (Z-1) inner electrons.
We describe the screening effect of the inner electrons together with the
nucleus potential by means of an effective potential Veff(r) for the valence
electron. In this way we reduce the original many-body problem to a singleparticle system, and we can treat the energy levels of an alkali atom as
terms of a single-electron atom.
Effective potential Veff for an Alkali atom
Effective potential Veff(r)
for an alkali atom
At small electron-nuclear distances, Veff has the shape of the
unscreened nuclear Coulomb potential; at large distances, the nuclear
charge is screened to one unit of charge.
The screening effect of the inner
electrons can be quantitatively calculated,
if one knows their charge distribution
with sufficient accuracy. Here we only
understand it qualitatively.
The energy term of alkali atoms,
compare to H atom, are determined by the
quantum numbers n and l.


1
1
En,l   Rhc 2   Rhc 
2
neff
 n  (n, l ) 
neff: an effective principle quantum number, not an integer;
(n,l) = n – neff: the quantum defect, associated with the
quantum numbers n and l.
The quantum defect (n,l) are empirical expressions of the
different degrees of screening of the s, p, d, etc. electrons by
the electrons of the inner shells.
The quantum defects (n,l) for the spectra
of the Na atom:
The empirically determined numerical values for the quantum
defect are largest for s electrons, decrease with increasing orbital
angular momentum quantum l, and are largely independent of the
principal quantum number n.
They increase down the column of alkali atoms from lithium to
cesium, or with increasing nuclear charge number Z.
Term diagram of the
Lithium atom with the most
important transitions --Grotrian diagram
The term diagram permits a
classification of the spectral
lines to series.
With selection rule for optical
transitions: Δl = ± 1,
The quantum number l must
change by 1.
Term scheme (Grotrian
diagram) of the Sodium atom.
The numbers in the diagram
indicate the wavelength of the
transition in Angstrom units.
The term diagram indicated on
the upper edge of the figure
also represent the quantum
numbers for the multiplicity
and the total angular
momentum.
With selection rule for optical
transitions: Δl = ± 1,
The quantum number l must
change by 1.
For the sodium atom, the three shortest-wave spectral
series decomposed from the total spectrum
The emission spectrum is a composition of these series. In absorption spectra,
normally only the principal series is observed, because in the ground state of
the Na atom the highest occupied term is the 3s term.
Four most important series
The principal series, with transitions from p to s electron terms:


1
1

, n  n0
2
2
 n0  (n0 ,0) n  (n,1) 
 p  R
n0: the integral principal quantum number of the lowest state,
2 for Li, 3 for Na, 4 for K, 5 for Rb, 6 for Cs
The sharp or second secondary series with the transitions
from s to p electron terms:


1
1

, n  n0  1
2
2
 n0  (n0 ,1) n  (n,0) 
 s  R
Four most important series
The diffuse or first secondary series, with transitions from d to p
electron terms:


1
1
 d  R

, n  n0
2
2
 n0  (n0 ,1) n  (n,2) 
The Bergmann or fundamental series with the transitions
from f to d electron terms:


1
1
 f  R

, n  n0  1
2
2
 n0  (n0 ,2) n  (n,3) 
Term scheme (Grotrian
diagram) of the Sodium atom.
For the electron with l = 0, 1,
2, 3 correspond to s, p, d, f
are historic,
p: for principal, ps;
s: for sharp, sp;
d: for diffuse, dp;
f: for fundamental, fd.
Double D lines
(yellow)
capital letters for terms with
several electrons in an atom;
lower case letters for the
terms for individual electrons.
The absorption spectrum
Under normal conditions, only the principal series is observed by absorption
spectroscopy, because only the ground state of the atoms is sufficiently
populated for transitions into higher states to be observed.
The lines of the principal series are resonance lines.
The best known is the D line of sodium, which is the transition 3s — 3p.
The sum of the s terms can also be designated S, and of the p terms, P, so
that the sodium series can be written:
Principal series: 3S — nP
Secondary series: 3P — nS
Difuse series: 3P — nD, with n  3.
Capital letters are used for terms which apply to several electrons in an atom,
and lower case letters for the terms for individual electrons. In the alkali
atoms, which have only one valence electron, the two notations are
equivalent.
Electrons in inner shells
Left: Grotrian diagram for the neutral potassium in the visible and infrared regions.
Right: term scheme for the K atom in the infrared, visible, ultraviolet and x-ray
regions. K, L, M… are inner shells.
homework
11.1, 11.3