⟦65c469783⟧

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*cm 1982-02-06*

 *pn 0,0**pl 297,28,240,4,10*
*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 editor*nl*
Physica Scripta*nl3**np 20*
*rj*
*nl3**lm0*
*np 10* 
Enclosed is a manuscript I would like to have considered for
publication  
in Physica Scripta.
*np 10*
*nl6*
*lm100*
Sincerely
*nl4*
Anders Lindgård
*lm 0**ps15*
*rj*
*sj*
submitted to Physica Scripta *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:*
*ps0*
*ld12**ns 1,3,Abstract*
--------*ld24**np*
The many channel quantum defect theory (MQDT) has been proposed
as a semiempirical tool for 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 fail.
*pn 5,0*
*ps 0*
*ld12**ns 1,3,Introduction*
------------*ld24**np*
The many channel quantum defect method (MQDT) is based on the
work 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 only been used for the qualitative characterisation of states
(4,5,6,7,8). Only little work has been performed, where mixing coefficients
have been used for actual calculation of atomic properties e.g.
oscillatorstrengths (9,10).
*nl**np*
In this paper we will examine the mixing coefficients obtained for
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 theese are much larger
than described previously (4,5), indicating
a discrepancy between the rather simple theory and experiment.*np*
We will try to calculate liftimes in 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 show that this is meaningless despite
the findings of Weiss (16) that some of theese orbitals are 
coulomblike.
*nl2*

*ld12**ns 1,2,Estimation of standard deviations*
             ---------------------------------*nl**ld24**np*
To simplify the development we will use the two channel case only
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*(2.5) in Lee and Lu
*rj*
for all states. Using the method of least squares fitting we obtain
*nl**qr*
(1)
*rj*
where n is the number of states, W is a weight function. In the
ideal case Q should be exactly zero. This is of course never the
case when experimental datapoints are involved in a fitting 
procedure. However a Q larger than what should be expected from
the uncertanty on the datapoints themselves does indicate
that the theory does not describe the datapoints well.
*nl**np*
For the 2-channel case we have:*nl**qr*
(2)
*rj*
*np*
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 we reduce the dimentionality 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, and m is the number of parameters,
and n the number of datapoints.
*nl4*
*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)
*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 meanvalue and that its variance is the same for all states.
Further it is assumed that the data and the parameters are
uncorrelated. One could   have used
*nl**qr*
(8)
*nl**rj* 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 norm is a mximum 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 our experimental data. So our covariancematrix C may describe
the deviation between theory and experiment rather than the uncertanties
in the experimental data.
*nl**np*
When interpreting eq.(7) as a standard deviation for the MQDT parameters
we should be very cautious. Due to the approximate nature of the
MQDT theory they do not necessecarily describe a
confidence limit, but there is certainly no reason to believe that
we have more significant figures for the parameters than the estimator
in eq.(7) predicts.
*nl**np*
To examine the situation in more detail one should compute the
covariancematrix C using both eq.(6) and eq.(6a). They should
give equivalent results. If they do not a C based on eq.(6a)
will at least in a mathematical sense describe the behaviour
of Q in the neighbourhood of the minimum.
*nl2*
*ld12**ns 1,2,Mixing Coefficients*
             -------------------*nl**np*
The calculation of mixing coefficients  z!li is based on
formula 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 formula 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*
However some properties of eq.(10) should be noted:
*np*
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*
If the mixing between the channels is small i.e. . is small, it is
impossible to get unambigious mixing coefficients as two possible
choices of the index i gives the expansion
*nl*
*nl*
and
*nl*
*nl*
where k is a constant which does not depend on . . So both A!la!u(n)!d and
z!li are poorly determined.
*nl**np*
The two choices in eq.(10) gives 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.*
             --------------------------------------*nl**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.
For Ca I (4), for Sr I (5) and Ba I (6,7).
*nl**np*
In this work we have reanalyzed the rydberg series in theese atoms 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 exitation 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  energydependance for u!l1.
In general we find that the energy dependant part du!l1/dE is
highly uncertain. Usually it has zero significant figures. The
other energy independant parameters usually are found to have
2-3 significant figures.
*nl**np*
Theese findings are quite contrary to the findings of Armstrong et al.
(4,5) who quotes many more signigficant figures, but does not tell
how they have calculated the number of figures they publish.
*nl**np*
Further analysis reveals that the minimum value of Q is
not really small. Typically from 0.0001 to 0.00005.
The parameters are also somewhat correlated, Which we
*ds!*
assumed they were not. Worst is u!l1 and . , with a
correlation coefficient of 0.4 for the !h1S-series of
Ca I. The mixing coefficient
actually calculated does depend on our choice of
index i in eq. (10). If we again take a look at
!h1S-series of Ca I the situation is not too bad.
For the higher states z!l2 may vary 20%, but z!l2 is rather
small compared with z!l1. If we however take a look at the !h1P-series
in Sr I the results are quite discourageing as shown in table II.
*nl2*
*ld12**ns 1,2,Lifetimes in B I and Al I*
             -------------------------*ld24**np*
Theese two systems and their isolectronic sequences has been
very difficult to handle both by ab-initio methods and by
semi-empirical methods. Until resently 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 theese 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 has the same problem.
The other series in the neutrals are unperturbed, as far known.
For the isolectronic ions the perturbations increase rapidly
in all series as the 2p and 2s becomes more degenerate.
MQDT qualitatively agrees with this picture.
*np*
When doing an MQDT analysis of theese three-electron systems one
problem arises: which ionization potential should be used for
the secon channel?. The singlet-triplet separation is quite large,
placing 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.
*ns 1,3,B I**np*
The MQDT results are found in table III. MQDT predicts that
the labels on 2p!h2 and 7s are arbitrary; they should both
preferrably be called 2p!h2. It further predict 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 is 10s, and that 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.
*ps0*
*ns 1,3,Al I*
*np*
The MQDT results for the D-series are found in table IV.
The fit is quite poor, despite that the number of states
known is rather large. For the highest members the
quantum defects are erratic as noted by Erikson and Isberg (25).
Excluding the states 25d to 35d gives a much better fit without
changing the parameters much.*np* 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 better than the 

errors on the fitted parameters.
Weiss(17) found that the d-orbitals derived from the
SOC calculation are very close to coulomb wawefunctions.
So if good mixing coefficients could be found, there should
be hope to determine lifetimes in good agreement with
experiment and ab-initio methods, when using then 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 wawefunction is reproduced byf 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 predicts a lifetime
above 12 ns. A pure configuration calculation using the numerical
coulomb approximation gives only around 8 ns, but if one attempts
to improve it using MQDT mixing coefficients, we get only 5 ns.
This result is not surprising as z!l1 and z!l2 have the same sign,
and as the matrixelements for both channels are approximately
equal as are the effective quantumnumbers.
*ld12**ns 1,6,Conclusion*
             ----------*ld24**np*
In the first papers on MQDT there was a great hope to the
method for describing perturbed series quantatively and
there was some success in 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 has the same conceptual
framework based on effective quantumnumbers.
Merging theese 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) has analyzed the situation for Ca I and Sr I
using MCHF wawefunctions. MQDT, as it has been presented,
cannot describe mixing coefficients correcly for theese systems
and hence it has no meaning to calculate lifetimes using MQDT
and some semi-empirical method for theese series.
*np*
Still MQDT may be adequate for the qualitative analysis of
optical spectra to find where series are perturbed and to
disentangle rydberg series.
*np*
It should be noted the the NCA method gives good 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.
*ld12**ns 1,6,Acknowledgement*
Many fruitful discussions with Ricard Crossley and Susan Richards
on theese matters made this work possible. 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 gives many more significant
digits. de is the energyvariation as measured up to the first
ionization limit (conf. ref. 4).
*ps0**ct*Table II*sj*

Mixing coefficients for !u1P-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*
u!l1 with linear energydependance, and other parameters energyindependant
*ps0**ct*Table III*sj*
MQDT parameters for !h2S in B I

   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

Selected mixing coefficients:

state   index    z!l1    z!l2

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
*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 gives many more significant
digits. de is the energyvariation 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*
MQDT parameters for !h2D in Al I
   u!l1     = 0.49(9) - 0.03(18)de
   u!l2     = 0.10(4)
   .      = 1.00(6)
   Q!dmin!u   = 0.002
   s      = 0.01

Selected mixing coefficients:
                 This work                         Weiss
state   index     z!l1     z!l2                     z!l1      z!l2
3d          1    0.67   0.75                   0.86   0.46
            2    0.66   0.75 
4d          1    0.91  -0.41                   0.85  -0.46
            2    0.91  -0.41
5d          1    0.83   0.57                   0.90   0.36
            2    0.87   0.50

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.
*ps0**ct*Table V*sj*
Lifetimes for unperturbed states in B I and Al I.
*ps0**nl2*
*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. Aynar, 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

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   At. Data & Nucl. Data Tables _«bs»1_«bs»9 534

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   J. Opt. Soc. Am. _«bs»6_«bs»4 706

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

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
*ef*
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