Download The regularities of the Rydberg energy levels of many

Vol. 43 No. 2
SCIENCE IN CHINA (Series B)
April 2000
The regularities of the Rydberg energy levels of manyvalence electron atom Al
ZHENG Nengwu (郑能武) & SUN Yujie (孙育杰)
Department of Chemistry, University of Science and Technology of China, Hefei 230026, China
Correspondence should be addressed to Zheng Nengwu
Received September 9, 1999
Abstract Within the scheme of the weakest bound electron potential model theory, the concept of
spectral-level-like series is presented by reasonably classifying the Rydberg energy level of atom Al.
Based on this concept, the regularities of the Rydberg energy levels are systematically studied. The
deviations of the calculated values from the experimental values are generally about several percent of 1 cm, which is of high accuracy.
Keywords: weakest bound electron potential model (WBEPM) theory, spectral-level-like series, multiplet theory of
atomic spectra, quantum defects.
The energy levels of atoms and ions, especially in high Rydberg states, are an interesting research field concerning physics, astrophysics, chemistry and other high techniques (laser, nuclear
fusion, etc.). Since the spectral data of Rydberg states are widely demanded and applied, the corresponding measuring techniques and computing methods are developed quickly. The widely used
experimental means are optical double resonance method, level-crossing method[1], two-photon
Doppler-free spectroscopy method[2], Stark spectroscopy[3] and millimeter spectroscopy method[4],
etc. Due to the eminent contributions of Moore, Corliss, Sugar, Musgrove and Martin, etc., most
of the experimental data of energy levels were systematically compiled[5
9]
. For the lighter ele-
ments that are of interest in the study of celestial bodies, the data are relatively rich. For the elements heavier than Ni (Z>28), only the first three or four energy levels are available, which are
involved in the spectra of solar and other stars, especially stars with special chemical characteristics. During the last few years, a great development was made in the study of the abundant energy
levels of rare earth elements[10]. However, there are still many blanks left for the high Rydberg
energy levels in the periodic table. In respect of the theoretical methods, the relatively refined
ways, such as multiconfiguration self consistent field[11], many-body perturbation theory[12], are
highly praised. Unfortunately, however, the divergence in the calculation not only makes the computation onerous, but makes it difficult to compute high Rydberg states using these methods. The
presentation and development of quantum defect theory (QDT) provided a feasible way to calculate the high Rydberg states theoretically. With QDT, Martin[13] calculated atom Na with high accuracy. However, up to now, its application is still limited to the calculation of one-valence electron systems such as alkali atoms and alkali-like ions. It is still difficult to evaluate the high
Rydberg states of many-valence electron atoms (ions) using QDT. In this paper, based on the
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SCIENCE IN CHINA (Series B)
Vol. 43
weakest bound electron potential model[14,15] (abbreviated to WBEPM) theory, the concept of the
weakest bound electron was introduced into the many-valence electron systems. Through dividing
the electrons in atoms (ions) into the weakest bound electron and non-weakest bound electrons,
appropriate Rydberg state sequences were extracted and then the Martin’s expression was extended ito the many-valence electron atom Al. The results are satisfactory.
1 Evaluation of the quantum defect number
As early as in 1916, Sommerfeld presented the old quantum defect theory. In 1977 Jaffe and
Reinhardt[16] gave a new demonstration to this theory. The energy levels of alkli metal atoms and
alkali-metal-like ions can generally be written as follows (in a.u.):
En = −
1
1
,
2 (n − δ 1 ) 2
(1)
where δl is the quantum defect number, representing the effects of interactions between the valence electron and the atomic core or any short-range spherical perturbation in a many-electron
system. For alkali-metal atoms, δl is nearly a constant for a given angular momentum l.
There are many ways and formulas to evaluate the quantum defect number. An important
formula is Rydberg’s formula:
vn =
R
(n1 − µ1 )
2
−
 1
1 
= R 2 − 2  ,
(n2 − µ 2 )
q 
p
R
(2)
2
where vn is frequency of the given spectrum line, R is Rydberg constant, n1 and n2 are principal
quantum numbers, and µ1 and µ2 are the corresponding quantum defect numbers.
Realizing that p and q in Rydberg’s formula (2) should be the functions n, Ritz
cally suggested that p and q are in the form of infinite series
 1
1 
v n = R 2 − 2  ,
q 
p
p = n1 + a1 +
q = n2 + a2 +
b1
n12
b2
n 22
+
c1
n14
+
c2
n 24
+
+
[17]
theoreti-
(3.1)
d1
n16
d2
n 26
+ L,
(3.2)
+ L.
(3.3)
This is the well-known form of Ritz formula, which is now still applied in many fields. Only using
the first two terms of (3.2) and (3.3), will Ritz formula reduce to Rydberg’s formula. With a slight
transformation, one can get the quantum defect number µ1
µ1 = n1 − p = − a1 −
b1
n12
−
c1
n14
−
d1
n16
− L.
(4)
Ritz formula can be used to precisely evaluate the spectral data of lower energy levels of alkali
atoms; for high Rydberg states so doing is very inconvenient. In 1980, a qualification for Ritz
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REGULARITIES OF RYDBERG ENERGY LEVELS
115
formula was made by Martin[13], who set δ=n-n*, m=n-δ0, where δ0 is the quantum defect number
of the lowest energy level in a Rydberg state series. Thus
δ = a + bm −2 + cm −4 + dm −6 ,
(5)
δ is the quantum defect corresponding to an energy level with principal quantum number n. With
(5), Martin evaluated the energy levels of several Rydberg series of atom Na, with deviations from
experimental results within only several percent of 1 cm-1.
In addition, there are other methods to calculate the quantum defect number, including the
formula presented by Hicks[17], the dynamical improvement of quantum defects by Ganesh
Vaidyanathan, et al.[18], the core polarization qualification for quantum defects in high-angularmomentum states of alkali atoms by Freeman et al.[19], perturbation WKB calculation of quantum
defects of alkali atoms by Lu[20].
2 The weakest bound electron and spectral-level-like series
The weakest bound electron is such an electron that most weakly connects with the system
among all the electrons in the system. So it is also the most active electron that is excited or ionized most easily in the system. In the systems of alkali atoms or alkali-like ions with one valence
electron, the weakest bound electron assuredly is the only valence electron, while in the systems
with many-valence electrons such as atoms C and Al, one has to determine which of the electrons
is the weakest bound electron by separating the valence electrons. Let us take atom C as an example. The ground electronic configuration of atom C is [He]2s22p2. Its valence electrons are two
electrons in 2s orbits and two in 2p orbits, in which the two 2p electrons are easier to be excited
with the same probability. In the viewpoint of stepwise excitation or ionization, however, the two
electrons in 2p orbits can only be excited or ionized one by one. Then the first excited or ionized
electron can be viewed as the electron that most weakly connects with the system. So one can
appoint the excited 2p electron as the weakest bound electron in the present system. Exciting the
weakest bound electron to various energy levels will produce various electronic configuration
series, such as series 2p3s, 2p4s,
2pns, and series 2p3d, 2p4d,
2pnd, etc. Note in an elec-
tronic configuration series, the only difference lies in principal quantum number n of the weakest
bound electron. As is well known, in an electronic configuration series each electronic configuration may usually possess some spectral terms and further several spectral levels. Therefore, the
concept of “spectral-level-like series” is suggested here: a spectral-level-like series is a series
composed of energy levels with the same spectral level symbol in a given electronic configuration
series of a system, such [He]2s23pns (n
3) ns 3P0 as one spectral-level-like series of atom C.
Note that a spectral-level-like series always depends on a given electronic configuration series,
because the same spectral term can be generated from different electronic configurations. Starting
with this definition, it is easy to define a spectral-level-like series for the electronic configuration
series 3s3p, 3s4p
3snp of atom C in double excited state.
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SCIENCE IN CHINA (Series B)
Vol. 43
According to multiplet theory for the atomic spectra[21], the energy level T, illustrated by L-S
coupling scheme, can be represented as a function of parameters of spectral level,
T = T (nl , config. - nwbe, L, S , J , Z ),
(6)
where n, l are principal quantum number and angular quantum number of the weakest bound
electron, respectively; the config.-nwbe means the electronic configuration of the non-weakest
bound electrons; the whole of nl and config.-nwbe jointly represent the electronic configuration of
an atom (ion) system; L, S and J are the total orbital angular quantum number, the total spin angular quantum number and total angular quantum number respectively; Z is the nuclear change
number.
For a spectral-level-like series as we defined, config-nwbe L, S, J and Z of each member of
the series are all the same. Then the corresponding energy level T becomes a unique functional of
config. And for the unique function of principal quantum number n of the weakest bound electron,
we express the relationship as follows:
T = Tl ,config - nwbe, L , S , J , Z [n]
(7.1)
T = T [n],
(7.2)
or abbreviate it to
where T represents energy level; n is the principal quantum number of the weakest bound electron.
In WBEPM theory, the relationship between spectral level and the effective parameters of the
weakest bound electron has been provided[15]
RZ ′ 2
.
n′ 2
T=
where T denotes spectral level, Z
(8)
and n are the effective nuclear charge and effective principal
quantum number of the weakest bound electron, respectively, and R is Rydberg constant.
As a matter of fact, for a spectral-level-like series of a many-electron atom (ion), the effective
nuclear charge Z in eq. (8) is apparently not 1, even so is the case of the alkali atoms. The situation is more complex in many-valence electron systems. However, in terms of the weakest bound
electron, we continue following the concept of quantum defects theory and attribute various effects to the change in the effective principal quantum number n, i.e. the change in the quantum
defect number. The following equation expresses the transformation between WBEPM theory and
QDT:
Z ′ / n ′ = Z net / n * ,
(9)
where Znet=1, n*= n ′ / z ′ . Then (8) becomes
T=
2
RZ net
,
(10)
n* = n − δ ,
(11)
n *2
where δ is the quantum defect number. Thus in the framework of WBEPM theory, through sepa-
No. 2
REGULARITIES OF RYDBERG ENERGY LEVELS
117
ration of the weakest bound electron and non-weakest bound electrons, using the concept of spectral-level-like series, one can extend the Martin’s expression from alkali atomic systems to manyvalence electron system Al and predict the energy levels of high Rydberg states of many-valence
electron Al.
3 Examples and analysis
According to the above discussion, we calculated three spectral-level-like series 3s2ns(n
ns2S1/2, 3s2np(n 4) np2P1/2 and 3s2nd(n
4)
4) nd2D3/2 of atom Al. The experimental values were
selected from ref. [8], the energy unit is wave number (in cm-1). Extending the Martin’s expression
to this many-valence electron atomic system, we have
δ = a + bm −2 + cm −4 + dm −6 ,
(12)
where δ=n-n*, m=n-δ0, δ0 is the quantum defect number of the lowest energy level in a given series, and a, b, c, d are fitted parameters (table 1).
Table 1 The fitted parameters in Martin’s expression for three spectral-level-like series of atom Al
Al: 3s2ns(ns2S1/2)
Al:3s2np(np2P1/2)
Al:3d2ns(nd2D3/2)
a
1.757 5
1.275 3
0.977 8
b
0.205 9
0.252 1
-2.685 7
c
0.337 7
0.243 2
-72.002 5
d
-0.322 0
0.693 2
425.128 1
δ0
1.812 4
1.492 3
0.368 1
Using the extended Martin’s expression to calculate the quantum defect numbers of each energy level in a given spectral-level-like series, one can predict the energy of each level with (10).
In tables 2 4, the calculated level energy values are given. For comparison, the corresponding
experimental values are also listed.
Table 2 Comparison between the calculated and experimental energy values of spectral-level-like
series Al: 3s2ns (n 4) ns2S1/2/cm-1
n
Eexp
Ecal
δexp
δcal
4
22 930.614
1.812 4
22 930.574
1.812 4
5
10 588.957
1.780 8
10 588.960
1.780 8
6
6 133.968
1.770 3
6 134.970
1.770 3
7
4 005.248
1.765 7
4 006.245
1.765 7
8
2 821.137
1.763 2
2 821.137
1.763 2
9
2 094.474
1.761 6
2 094.481
1.761 7
10
1 616.470
1.760 6
1 616.491
1.760 7
11
1 285.370
1.760 6
1 285.328
1.760 0
12
1 046.470
1.759 7
1 046.448
1.759 6
13
868.470
1.759 1
868.483
1.769 2
14
732.370
1.759 1
732.347
1.759 0
15
625.870
1.758 6
625.887
1.758 7
16
541.070
1.758 6
541.063
1.758 6
17
472.385
1.758 4
18
416.000
1.758 3
19
369.141
1.758 2
20
329.775
1.758 2
25
203.144
1.757 9
30
137.581
1.757 8
35
99.305
1.757 7
40
75.035
1.757 7
50
47.152
1.757 6
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SCIENCE IN CHINA (Series B)
Vol. 43
Table 3 The comparison between the calculated and experimental energy values of spectral-level-like
series Al: 3s2np(n 3)np2P1/2/cm-1
δexp
Ecal
δcal
3
48 278.370
1.492 3
48 278.650
1.492 4
4
15 328.566
1.324 4
15 328.628
1.324 4
5
8 006.405
1.297 8
8 006.408
1.297 8
6
4 943.357
1.288 4
4 943.356
1.288 4
7
3 358.716
1.284 0
3 358.625
1.283 9
8
2 431.097
1.281 4
9
1 841.231
1.279 9
10
1 442.811
1.278 9
15
582.692
1.276 7
20
313.012
1.276 1
35
96.486
1.275 6
n
Eexp
Table 4 The comparison between the calculated and experimental energy values of spectral-level-like
series Al: 3s2nd(n 3)nd2D3/2/cm-1
n
Eexp
Ecal
δcal
3
15 842.935
0.368 2
δexp
15 842.880
0.368 1
4
9 348.965
0.573 9
9 195.529
0.545 5
5
6 044.648
0.739 2
6 044.625
0.739 2
6
4 111.953
0.834 0
4 113.287
0.834 8
7
2 934.206
0.884 5
2 934.202
0.884 5
8
2 184.946
0.913 1
2 184.649
0.912 6
9
1 684.950
0.929 8
1 684.948
0.929 8
10
1 337.400
0.941 7
1 337.196
0.941 0
11
1 086.070
0.948 1
1 086.199
0.948 7
12
899.670
0.955 8
899.413
0.954 2
13
757.270
0.962 1
756.789
0.958 2
14
645.770
0.964 2
645.487
0.961 3
15
557.070
0.964 7
556.994
0.963 7
16
485.370
0.963 7
485.494
0.965 6
17
427.470
0.977 7
426.908
0.967 2
18
378.270
0.967 6
378.308
0.968 4
19
337.670
0.972 7
337.550
0.969 5
20
303.035
0.970 4
21
273.552
0.971 1
22
248.169
0.971 7
23
226.160
0.972 3
24
206.952
0.972 7
25
190.091
0.973 2
30
130.256
0.974 6
50
45.661
0.976 7
4 Discussion
Tables 2 4 show that the calculated values are very close to the corresponding experimental
No. 2
REGULARITIES OF RYDBERG ENERGY LEVELS
119
values, with an absolute deviation generally no more than several percent of 1 cm-1 except for
level 4d2D3/2. The much higher energy levels, without the corresponding experimental values being compared, are predicted and listed in the tables. Considering the high accuracy of the foregoing calculated values, the forecasted values are reliable. The results prove that the extended Martin’s expression is suitable for the spectral-level-like series of the many-valence electron atom Al.
The study on how to extend the Martin’s expression to other many-valence electron systems is in
progress. The related work will be reported in succession.
In SCF methods, the concept of indistinguishability of all electrons in a system prevails. In
pseudopotential model based on the behavior of the valence electron there is the idea of separation
of valence electrons and core electrons; in QDT, the valence shell and the core are also treated
separately in one-valence electron atoms (ions); while in WBEPM theory, through separation of
the weakest bound electron and non-weakest bound electrons, one can solve the one-electron
Schrödinger equation of the weakest bound electron. This is very convenient for the discussion
about the excited states of atomic systems. It is these conceptual breakthroughs and the advancement of the concept of spectral-level-like series within an electronic configuration series that make
it possible to transform (9) and extend Martin’s expression. This paper provides a new way of
studying the regularities in the Rydberg energy levels of many-valence electron atoms Al.
Acknowledgements This work was supported by the Research Foundation of Ministry of Science and Technology and
the National Natural Science Foundation of China (Grant No. 59872039).
References
1. Svanberg, S., Tsekeris P., Happer, W., Hyperfine-structure studies of highly excited D and F levels in alkali atoms using a
cw[continuous wave]tunable dye laser, Phys. Rev. Lett., 1973, 30: 817.
2. Harvey, K. C., Stoicheff, B. P., Fine structure of the n2D series in rubidium near the ionization limit, Phys. Rev. Lett.,
1977, 38: 537.
3. Ducas, T. W., Zimmerman, M. L., Infrared stark spectroscopy of sodium Rydberg states, Phys. Rev. A, 1977, 15: 1523.
4. Fabre, C., Haroche, S. Goy, P., Millimeter spectroscopy in sodium Rydberg states: Quantum defect, fine-structure, and
polarizability measurements, Phys. Rev., A, 1978, 18: 229.
5. Moore, C., Atomic Energy Levels, Washington D. C.: NSRDS-NBS 35, 1971, 1
6.
7.
8.
9.
10.
309.
Corliss, C., Sugar, J., Energy levels of potassium, J. Phys. Chem. Ref. Data, 1979, 8: 1111.
Sugar, J., Musgrove, A., Energy levels of krypton Kr I through Kr XXXVI, J. Phys. Chem. Ref. Data, 1991, 20: 859.
Martin, W. C., Zalubas, R., Energy levels of aluminum, J. Phys. Chem. Ref. Data, 1979, 8: 820.
Martin, W. C., Zalubas, R., Energy levels of silicon, J. Phys. Chem. Ref. Data, 1983, 12: 325.
Pell. F. E, Gardant, N., Auzel, F., Effect of excited-state population density on nonradiative multiphonon relaxation rates
of rare-earth ions, J. Opt. Soc. Am., B, 1998, 15: 667.
11. Roos, B. O., Advances in Chemical Physics, Vol. 69, New York: Wiley, 1987, 339 348.
12. Lauderdale, W. J., Stanton J. F., Gauss, J. et al., Many-body perturbation theory with a restricted open-shell Hartree-Fock
reference, J. Chem. Phys., 1992, 97: 6606.
13. Martin, W. C., Series formulas for the spectrum of atomic sodium (Na I), J. Opt. Soc. Am., 1980, 70: 784.
14. Zheng, N. W., Xin, H. W., Successive ionization potentials of 4f n electrons within
WBEPM theory, J. Phys. B: At. Mol.
Opt. Phys., 1991, 24: 1187.
15. Zheng Nengwu, A New Outline of Atom (in Chinese), Nanjing: Jiangsu Educational Press, 1988, 1
332.
16. Jaffe, C., Reinhardt, R. P., Semiclassical theory of quantum defects: alkali Rydberg states, J. Chem. Phys., 1977, 66: 1285.
120
SCIENCE IN CHINA (Series B)
17. White, H. E., Introduction to Atomic Spectra, New York and London: McGraw-Hill Book Company, 1934, 16
Vol. 43
89.
18. Vaidyanathan, A. G., Shorer, P., Dynamical contributions to the quantum defects of calcium, Phys. Rev., A, 1982, 25:
3108.
19. Freeman, R. R., Kleppner, D., Core polarization and quantum defects in high-angular-momentum states of alkali atoms,
Phys. Rev. A, 1976, 14: 1614.
20. Lu Shucheng, Jiang Xiong, Jiang Junchao et al., The perturbation WKB calculation of the quantum defects of alkali atoms,
J. Atomic and Molecular Physics (in Chinese), 1996, 13: 478.
21. Slater, J. C., Quantum Theory of Atomic Structure, Vol. II, New York-Toronto-London: McGraw-Hill Book Company,
Inc., 1960, 95 127.