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*cm 1982-04-28*

*pn 0,0**pl 297,16,250,10,10*
*lm0**lw 170* *ld12* *ps15*
*ds !*
*sj*
H. C. Ørsted Institute
Københavns Universitet
Universitetsparken 5
DK-2100 København Ø
Denmark
*qr*
*cd1*
*rj*
*lm10*
The executive editor*nl*
Nils Robert Nilsson
Physica Scripta*nl3**np 20*
*rj*
*nl3**lm0*
Your letter dated 1982-04-20.
*np 10*
The manuscript reg. no. 56 has been revised as suggested by the
referees and is enclosed. I am very grateful for the referees
comments.
*np 10*
*nl6*
*lm100*
Sincerely
*nl4*
Anders Lindgård
*lm 0**ps0*
*sj*
Computer Department
H. C. Ørsted Institute
University of Copenhagen
Universitetsparken 1
2100 København Ø
*nl4*
*ct*
A discussion of mixing coefficients obtained using
the many channel quantum defect theory in
fitting to atomic spectra and their use
for calculation of lifetimes for B I and Al I
*ld12*
*nl 3*
Anders Lindgård
*sj**nl20*
*lm100*
Report 82/1
version *cd1*
*lm 0**ps15*
*rj*
*sj*
version *cd1*
*nl2*
*nl*
*ct*
A discussion of mixing coefficients obtained using
the many channel quantum defect theory in
fitting to atomic spectra and their use
for calculation of lifetimes for B I and Al I
*ld12*
*nl 1*
Anders Lindgård
H. C. Ørsted Institutet
Københavns Universitet
Universitetsparken 5
DK-2100 København Ø
Denmark
*rj*
*ld12**ns 1,6,Running title: Quantum Defect Fitting*
*rh 1,Quantum Defect Fitting*
*ns 1,3,Received by Physica Scripta 82-03-16*
ms 56
*ps0*
*ld12**ns 1,3,Abstract*
--------*ld24**np*
The many channel quantum defect theory (MQDT) has been proposed
as a semiempirical tool for the calculation of mixing coefficients
between interacting Rydberg series in atoms and ions. We have
examined the fitting procedure and estimated the deviations on the
parameters of MQDT. It is found that the parameters in general are rather
badly determined and that the mixing coefficients are rather ambigious.
Results for some two and three valence electron systems are presented.
It is shown that MQDT cannot be used together with coulomb approximation
methods to calculate lifetimes in the B I and Al I homologeous systems
as the mixing coefficients of MQDT are greatly in error.
*pn 5,0*
*ps 0*
*ld12**ns 1,3,Introduction*
------------*ld24**np*
The many channel quantum defect method (MQDT) is based on the
works of Seaton (1) and Fano (2). The most detailed description
of the method is found in the
paper by Lee and Lu (3),
which also gives a set of working formulas for the estimation of
parameters and of mixing coefficients.
*nl**np*
In most of the papers where MQDT has been used, mixing coefficients
have
been used only for the qualitative characterisation of states
(4,5,6,7,8). Little work has been performed, where mixing coefficients
have been used for actual calculation of atomic properties e.g.
oscillator strengths (9,10).
*nl**np*
In this paper we will examine the mixing coefficients obtained from
some calculations on two- and three electron atoms and discuss the
ambiguities in the results obtained. We will estimate the standard
deviations on the parameters and show that these are much larger
than described previously (4,5), indicating
a discrepancy between the rather simple theory and experiment.*np*
We present here calculated lifetimes for the perturbed s-series in B I and
the perturbed d-series in Al I using MQDT together with the numerical
Coulomb approximation (14) and we show that this is meaningless despite
the findings of Weiss (16) that some of these orbitals are
Coulomb-like.
*ps0*

*ld12**ns 1,2,Estimation of standard deviations*
---------------------------------*nl**ld24**np*
To simplify the development we will use the two channel case
in the discrete spectrum.
The work will be based on the Lee and Lu paper (3) and we shall adopt
the notation of Lu (10).
The basic requirement in MQDT is that *nl*
*qr*(Eq. 2.5) in Lee and Lu)
*rj*
*nl*
for all states. Using the method of least squares fitting we obtain
*nl**qr*
(1)
*rj*
*nl*
where n is the number of states, j labels the state,
W is a weight function. In the
ideal case Q should be exactly zero. This is of course never the
case when experimental data
are involved in a fitting
procedure. However, a Q larger than what should be expected from
the uncertainty on the data themselves indicates
that the theory does not describe the data well.
*nl**np*
For the 2-channel case we have:*nl**qr*
(2)
*rj*
*nl*
This is not the most convenient form for actual work. A better form is:
*nl**qr*
(3)
*rj*
*nl* which shows that det F is linear in the parameter cos(2.).
Taking advantage of this fact we reduce the dimensionality in the
minimum search for Q and thus reduce the computing time considerably.
Note also that the fitting procedure cannot determine the sign for ..
*nl**np*
In order to estimate the standard deviations for the 3 parameters
*nl*
*nl*
we need to calculate the covariance matrix (12).
*nl**qr*
(4)
*nl**sj*
where
*nl2*
at the minimum for Q, m is the number of parameters,
and n the number of data.
*rj**np*
Following Cramer (12) eq.(4) may be approximated by:
*nl**qr*
(5)
*nl**sj*
where the M matrix is defined by:
*nl**qr*
(6)
*ps0*
*nl**rj*
The square of the standard deviation on a parameter is then given by
*nl**qr*
(7)
*nl**rj**np*
It may be shown (13) that
*nl**qr*
(6a)
*nl**rj*
is just as good a measure as M in the calculation of the covariance
matrix C.
*nl**np*
In the development sketched above we have assumed that det F is a
proper mean value and that its variance is the same for all states.
Further it is assumed that the data and the parameters are
uncorrelated. The last assumption can be examined by calculation of the
covariance matrix, where large off-diagonal matrixelements
indicates a discrepancy.
*nl*
One could have used (4,5)
*nl**qr*
(8)
*nl**rj*
where .. is the wavelength instead of eq.(1). There is however no physical reason
for regarding eq.(8) as better than eq.(1) or better than any
other norm we may invent provided that the norm is a maximum likelihood
estimator as eq.(1) and eq.(8).
*nl**np*
The crucial point in using MQDT is that we are fitting an approximate
theory to the experimental data. So the covariance matrix C may describe
the deviation between theory and experiment, rather than the uncertanties
in the experimental data.
*nl**np*
When interpreting eq.(7) as standard deviations for the MQDT parameters
one should be very cautious. Due to the approximate nature of the
MQDT theory they do not necessecarily describe
confidence limits, but there is certainly no reason to believe that
we have more significant figures for the parameters than the estimators
in eq.(7) predict.
*nl**np*
To examine the situation in more detail one should compute the
covariance matrix C using both eq.(6) and eq.(6a). They should
give equivalent results. If they do not, a C based on eq.(6a)
will describe, at least in a mathematical sense,
the behaviour
of Q in the neighbourhood of the minimum.
*ps0*
*nl2*
*ld12**ns 1,2,Mixing Coefficients*
-------------------*ld24**np*
The calculation of mixing coefficients z!li is based on
equation 2.11 in Lee and Lu (3) as corrected by Armstrong et al. (4).
*nl**qr*
(9)
*nl*
*rj*
where N!ln is given by 2.7 and 2.8 in Lee and Lu. A!la!u(n)!d
is given by equation 2.6 in Lee and Lu.
*nl**qr*
(10)
*nl**rj*
where "the index i can be choosen arbitrarily for convenience, and
C!dia!u is the cofactor of the element of the ith row and ath column of
the determinant F!dia!u " (cited from Lee and Lu).
*nl**np*
Some properties of eq.(10) should be noted:
*np*
1) If the angle . is zero the coefficients A!l1!u(n)!d and A!l2!u(n)!d are
1 and 0 respectively, but which one is 1 or zero is determined by
the free index i in eq.(10), and they are thus completely indeterminate.
The same of course happens to the mixing coefficients in eq.(9).
*nl**np*
2) If the mixing between the channels is small i.e. . is small, it is
impossible to get unambigious mixing coefficients,
because the two possible
choices of the index i give the expansions
*nl*
*nl*
and
*nl*
*nl*
where k is a constant independant of . . So both A!la!u(n)!d and
z!li are poorly determined.
*nl**np*
The two choices in eq.(10) give information on how parallel the
column vectors in F are. When using the MQDT for calculating
atomic properties where mixing coefficients are needed, a better
fitting choice than eq.(1) or eq.(8) might be to try to get
the column vectors in F as parallel as possible. This has however
not been tried in this work.
*nl2*
*ld12**ns 1,2,Analysis of some two electron systems.*
--------------------------------------*ld24**np*
Lu (15) was the first to give an analysis of the homologous atoms
Be I, Mg I, Ca I, Sr I and Ba I. His work was rather qualitative
and indicates that in most cases the series are only weakly perturbed.
Since then there has been a great deal of more quantitative work mostly
for determining the characters of states in spectroscopic work.
Some examples are Ca I (4), Sr I (5) and Ba I (6,7).
*nl**np*
In this work we have reanalyzed some atomic Rydberg series
which
are believed to be 2-channel cases. We have calculated standard deviations
for the MQDT parameters and the alternative sets of mixing coefficients.
For a weight function we have used the excitation energy. (This is roughly
the same weight function as used by Armstrong et al. (4).)
*nl**np*
In most cases we have performed the analysis both with and without
a linear energy dependence for u!l1.
In general we find that the energy dependent part du!l1/dE is
highly uncertain. Usually it has zero significant figures. The
other energy independent parameters are usually found to have
2-3 significant figures.
*nl**np*
The present findings are quite different from the findings of Armstrong et al.
(4,5) who quote many more significant figures (2-3 more),
but they do not explain
how they calculated the number of figures they published.
It is interesting to note that in a later paper (28) they compare
MQDT parameters given only with two significant figures,
which is in agreement
with our findings.
*nl**np*
Further analysis reveals that the minimum value of Q
(or equivalent s!h2) is
not really small. s!h2 varies between
10!u-5!d and 10!u-8!d where we from the analysis of spectra
performed by the spectroscopists should expect s!h2<10!u-10!d.
The parameters are also somewhat correlated,
contrary to what was initially assumed.
*ds!*
Worst are u!l1 and . , with a
correlation coefficient of 0.4 for the !h1S-series of
Ca I. The mixing coefficient
actually calculated depends on the choice of
index i in eq. (10). For the
!h1S-series of Ca I, the situation is not too bad, see table I.
For the higher states z!l2 may vary 20%, but z!l2 is rather
small compared with z!l1. However, for the !h1P-series
in Sr I the results are quite discourageing as shown in table II.
*ps0*
*ld12**ns 1,2,Lifetimes in B I and Al I*
-------------------------*ld24**np*
These two systems and their isoelectronic sequences have been
very difficult to handle both by ab-initio methods and by
semi-empirical methods. Until recently the situation was
quite unsatisfactory, but with the latest ab-initio calculations
the agreement between experiment and theory is good enough
for most purposes. The semiempirical methods are still unable
to handle these systems and their isoelectronic seqences,
which is not surprising as one need to take the detailed
correlations between the three valence electrons into account.
*np*
In the neutrals, the !h2S-series in B I is highly perturbed
by 2p!h2 and in Al I the !h2D-series is disturbed by the
3p!h2 configuration.
The other series in the neutrals are unperturbed.
For the isolectronic ions the perturbations increase rapidly
in all series as the s and p valence shell terms become more degenerate.
MQDT qualitatively agrees with this picture.
*np*
In the MQDT analysis we have used 2-channel theory only.
For B I the only configuration which can be expected to perturb is the
2s2p!h2 configuration, as the coupling is pure LS. This is supported
by the analysis of the spectrum (29,30). For Al I the situation is
more dubious as configurations with 3p!h23d might perturb together
with the 3s3p!h2. Weiss (17) has pointed out that it should be possible
to describe the wavefunctions and transition probabilities by a
rydberg type wavefunction and a single perturber originating from
the 3s3p!h2 configuration implying that a 2-channel treatment
is sufficient.
*np*
When doing an MQDT analysis of these three-electron systems one
problem arises: which ionization potential should be used for
the second channel?. The singlet-triplet separation
for the ground state of the ion (e.g. 3s3p in Al II) is quite large,
with the singlet ionization limit 50% above the triplet.
We have selected to use the triplet for the folloving reasons
a) the statistical weight favors the triplet b) the singlet
is so far away that the interaction is very much reduced for
energy reasons.
*ps0*
*ns 1,1,B I**np*
The MQDT results for B I are found in table III. MQDT predicts that
the labels on 2p!h2 and 7s are arbitrary; they should both
preferably be called 2p!h2. It further predicts that 3s is
a pure configuration, contrary to Weiss(16) and
Dankwort and Trefftz(17). The mixing coefficients in the
vicinity of the perturber are quite unstable for the
selection of index i in eq. (10). The highest level
included in the analysis was 10s, which should be sufficient.
*np*
Using the numerical Coulomb approxiamation we obtain a pure
configuration lifetime for the 3s state of 1.8 ns, which is
less than 50% of the experimental value (26) obtained
using beam-foil. The ab-inito result is 4.1 ns using the
SOC (superposition of configurations) method(16), where
3s by no means can be called a pure configuration.
As MQDT predicts that 3s is pure no combination of that
method with a semiempirical method for matrixelements
will give sensible results for the ns-series in B I, see also table III.
*ps0*
*ns 1,0,Al I**ld12**np**ld24*
The MQDT results for the D-series are found in table IV.
The fit is quite poor,
despite the fact that the number of known states
is rather large, but even so eq. 6 and eq. 6a still give
equivalent covariance matrices C. For the highest members the
quantum defects are erratic as noted by Erikson and Isberg (25).
Excluding the states 23d to 35d gives a much better fit without
changing the parameters much. Still the fit is so poor
that it can only be explained as a serious discrepancy between
experiment and theory. The errors on the experimental data
are several orders of magnitude smaller than the

errors on the fitted parameters.
Weiss(17) found that the d-orbitals derived from the
SOC calculation are very close to coulomb wavefunctions.
So if good mixing coefficients could be found, there should
be the hope that one is able to determine lifetimes in good agreement with
experiment and ab-initio methods, by using the NCA method.
In table IV the mixing coefficients from MQDT and from
SOC(17) are compared. We note that the sign change for z!l2
in the wavefunction is reproduced by MQDT, but otherwise
the overall agreement is very poor.
*np*
When comparing lifetimes for the 3d-state, the situation is
even worse as seen from table IV. Both experiment and
ab-initio methods predict a lifetime
above 12 ns. A pure configuration calculation using the numerical
Coulomb approximation gives around 8 ns, but if one attempts
to improve it using MQDT mixing coefficients, we get
a lifetime as small as 5 ns.
This result is not surprising as the sign of
z!l1 and z!l2 leads to constructive interference,
and also because the matrix elements for both channels are approximately
equal as are the effective quantum numbers in both channels.
*ps0*
*ld12**ns 1,1,Conclusion*
----------*ld24**np*
In the first papers on MQDT there was an optimistic hope that the
method would describe perturbed series
quantitatively and
there was some success for the inert gases(10). Thus
it would be natural to combine MQDT with simple
semiempirical methods to obtain lifetimes etc. in
two and three-electron systems. Both MQDT and
semi-empirical methods have the same conceptual
framework based on effective quantum numbers.
Merging these to calculate lifetimes is a resonable
thing to attempt.
*np*
However for three electron systems MQDT cannot predict mixing
coefficients quantitatively in agreement with ab-initio methods.
This is also the case for two-electron systems. Froese Fischer
and Hansen (19) have analyzed the situation for Ca I and Sr I
using MCHF wavefunctions. MQDT, as it has been presented,
cannot describe mixing coefficients correctly for these systems
and hence it has no meaning to calculate lifetimes using MQDT
and some semi-empirical method for these series.
*np*
Still MQDT may be adequate for the qualitative analysis of
optical spectra to find where series perturbations occur and to
disentangle Rydberg series.
*np*
It should be noted that the the NCA method gives resonable results
for lifetimes in the unperturbed series of B I and Al I,
see table V.
*np*
Recently Fano (20) has suggested a new method to describe
interacting series. It will be interesting to see whether
this will be a more quantitative tool than MQDT for two
and three-electron systems.
*ps0*
*ld12**ns 1,2,Acknowledgement*
---------------*np*
Many fruitful discussions with Richard Crossley and Susan Richards
on these matters made this work possible. Erling Veje critically read the manuscript and improved
it substantially. The work has in part
been supported by a NATO travel grant RG1648. The computations
were carried out on the RC4000 and RC8000 computers of the
H. C. Ørsted Institute.
*ps0**ct*Table I*sj*
MQDT parameters for !h1S!l0 in Ca I

a) energy independant (ref. 4)
u!l1 = 0.347(9)
u!l2 = 0.172(6)
. =-0.195(29)
Q!dmin!u = 0.0001
s = 0.003

b) energy independant (this work)
u!l1 = 0.344(7)
u!l2 = 0.172(4)
. = 0.175(26)
Q!dmin!u = 0.00008
s = 0.002

c) energy dependant (ref. 4)
u!l1 = 0.349(3) - 0.33(21)de
u!l2 = 0.172(3)
. =-0.235(8)
Q!dmin!u = 0.00006
s = 0.002

d) energy dependant (this work)
u!l1 = 0.349(3) - 0.32(12)de
u!l2 = 0.172(3)
. = 0.235(7)
Q!dmin!u = 0.00005
s = 0.002
*rj**nl*
The uncertainty is given in paranteses as the variation on the
least significant digit given. The uncertainties in a) and c)
are those calculated by us. The authors give many more significant
digits. de is the energy variation as measured up to the first
ionization limit (conf. ref. 4).
*ps0**ct*Table II*sj*

Mixing coefficients for !h1P-series in Sr I.

state index ref. (5) this work

z!l1 z!l2 z!l1 z!l2

16p 1 0.49 0.92 0.46 0.93
2 0.56 0.87 0.52 0.90

17p 1 0.78 0.69 0.76 0.68
2 0.72 0.73 0.69 0.76
*rj**ld24*
*nl*
u!l1 with linear energy dependance,
and other parameters energy independant
*ld12**ps0**ct*Table III*sj*
MQDT parameters for !h2S in B I *ld24*

u!l1 = 0.94(1) - 0.27(11)de
u!l2 = 0.356(3)
. = 0.142(6)
Q!dmin!u = 0.00008
s = 0.004
*ld12*
Selected mixing coefficients:

state index z!l1 z!l2

3s 1 1.02 0.02
2 1.02 0.01

4s 1 1.00 0.09
2 1.00 0.07

5s 1 0.99 -0.17
2 0.99 -0.17

6s 1 0.92 0.40
2 0.89 0.46

2p!u2 1 0.59 0.80
2 0.63 0.78

7s 1 0.60 0.80
2 0.50 0.87

8s 1 0.98 -0.22
2 0.96 -0.27

Lifetimes for 3s
NCA (pure) 1.8 ns
NCA+MQDT 1.8 ns
experiment (beam foil) 3.6 (2) ns (ref. 26)
ab-initio (SOC) 4.2 ns (ref. 16)
*rj**nl*
The uncertainty is given in paranteses as the variation on the
least significant digit given.
de is the energy variation as measured up to the first
ionization limit (conf. ref. 4).
Energy level data are taken from ref. (23) where states up to 10s
are reported.
*ps0**ct*Table IV*sj*
*ld24*MQDT parameters for !h2D in Al I
u!l1 = 0.50(9) - 0.003(1800)de
u!l2 = 0.10(3)
. = 1.00(5)
Q!dmin!u = 0.0003
s = 0.004
*ld12*
Selected mixing coefficients:
This work Weiss (ref. 17)
state index z!l1 z!l2 z!l1 z!l2

3d 1 0.67 0.75 0.86 0.46
2 0.67 0.75

4d 1 0.91 -0.41 0.85 -0.46
2 0.91 -0.41

5d 1 0.82 0.57 0.90 0.36
2 0.82 0.57

6d 1 0.87 0.50
2 0.86 0.52

7d 1 0.99 -0.13
2 0.99 -0.13

Lifetimes for 3d

NCA (pure) 7.8 ns
NCA+MQDT 4.8 ns
Experiment(laser) 12.3 (5) ns (ref. 22)
ab-initio (SOC) 13.8 ns (ref. 17)
ab-initio (MCHF) 12.5 ns
*rj**nl*
The uncertainty is given in paranteses as the variation on the
least significant digit given.
de is the energyvariation as measured up to the first
ionization limit (conf. ref. 4).
Energy level data are taken from ref. (24) where states up to 35d
are reported, but only up 22d included in the calculation. See text.
*ps0**ct*Table V*sj*
Lifetimes for unperturbed states in B I and Al I.

B I 3d !h2D

Beam-foil 4.15(25) ref. 26
ab-initio (SOC) 4.27 ns ref. 16
NCA (This work) 4.9 ns

Al I 4s !h2S

Laser experiment 6.78(6)ns ref. 27
ab-initio (SOC) 7.1 ns Weiss, reported in ref. 27
NCA (This work) 6.2 ns

NCA lifetimes for other states in B I and Al I

B I Al I

4s 5.2 ns 5s 16.0 ns
5s 11.6 ns 6s 34 ns
6s 21.7 ns 7s 62 ns
3p 57 ns 4p 64 ns
4p 203 ns 5p 207 ns
4d 9.0 ns 4d 31 ns
5d 20.0 ns 5d 182 ns
6d 97 ns 6d 89 ns
4f 74 ns 4f 35 ns
5f 157 ns 5f 65 ns

*ps0**nl2**rj*
*ld12**ns 1,2,References*
----------*ld12**nl**sj*

1. Seaton, M. J. 1966
Proc. Phys. Soc. (London) _«bs»8_«bs»8 801

2. Fano., U. 1975
J. Opt. Soc. Am. _«bs»6_«bs»5 979

3. Lee, C. M. and Lu, K. T. 1973
Phys. Rev. _«bs»A_«bs»8 1241

4. Armstrong, J. A., Esherick, P. and Wynne, J. J. 1977
Phys. Rev. _«bs»A_«bs»1_«bs»5 1920

5. Esherick, P. 1977
Phys. Rev. _«bs»A_«bs»1_«bs»5 180

6. Aymar, M., Camus. P, Dieulin, M. and Morrillon, C. 1978
Phys. Rev. _«bs»A_«bs»1_«bs»8 2173

7. Aymar, M. and Robeaux, O. 1979
J. Phys. _«bs»B _«bs»1_«bs»2 531

8. Brown, C. M. and Ginter M. L. 1978
J. Opt. Soc. Am. _«bs»6_«bs»8 817

9. Crossley, R. J. S. and Richards, S. M. 1976
Beam-Foil Spectroscopy (Ed. Selllin and Pegg)
p.83, Plenum Press, New York

10.Starace, A. F. 1973
J. Phys. _«bs»B _«bs»3 76

11.Lu, K. T. 1971
Phys. Rev. _«bs»A_«bs»4 579

12.Cramer, H. 1966
Mathemathical Methods of Statistics,
Princeton University Press
Princeton

13.Heilmann, O. J. 1968
Mat. Fys. Medd. Dan. Vid. Selsk. _«bs»3_«bs»7 214

14.Lindgård, A. and Nielsen S. E. 1977
At. Data & Nucl. Data Tables _«bs»1_«bs»9 534

15.Lu, K. T. 1974
J. Opt. Soc. Am. _«bs»6_«bs»4 706

16.Weiss, A. W. 1969
Phys. Rev. _«bs»1_«bs»8_«bs»8 119
*ps*

17.Weiss, A. W. 1974
Phys. Rev. _«bs»A_«bs»9 1524

18.Dankwort, W. and Trefftz, E. 1977
J. Phys. _«bs»B _«bs»1_«bs»0 2541

19.Froese Fischer, C. and Hansen, J. 1981
Phys. Rev. _«bs»A_«bs»2_«bs»4 631

20.Fano, U. 1981
Phys. Scr. _«bs»2_«bs»4 656

21.Froese Fischer, C. 1981
Phys. Scr. _«bs»2_«bs»3 38

22.Selter, K. D. and Kunze, H.-L. 1978
Astrophys. J. _«bs»2_«bs»2_«bs»1 713

23.Odintzova, G. A. and Striganov, A. R. 1979
J. Phys. Chem. Ref. Data _«bs»8 63

24.Martin, W. C. and Zalubas, R. 1979
J. Phys. Chem. Ref. Data _«bs»8 817

25.Erikson, K. B. and Isberg, H. B. S. 1967
Ark. Fys. _«bs»3_«bs»3 593

26.Kernahan, J. A.,Pinnington, E. H.,Livingston, A. E. and
Irwin, D. J. G. 1975
Phys. Scr. _«bs»1_«bs»2 319

27.Klose, J. Z. 1979
Phys. Rev. _«bs»A_«bs»1_«bs»9 678

28.Wynne, J. J. and Armstrong, J. A. 1979
IBM J. Res. Develop. _«bs»2_«bs»3 490

29.Brown, C. M., Tilford, S. C. and Ginter, M. L. 1974
J. Opt. Soc. Am. _«bs»6_«bs»4 877

30.Edlen, B. 1981
Phys. Scr. _«bs»2_«bs»3 1079
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