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Excitation of highly-lying atomic states: pro ject and rst results
V.F. Bratsev, V.I. Ochkur, A.F. Kholtygin
Abstract
1

The pro ject is aimed at calculation of the cross-sections and rates for electron impact excitation for levels up to n=10 for atoms and their ions with Z 26, and preparing the diagnostic tools for analysis of the spectra of the astrophysical sources using the obtained atomic parameters. As a main instrument of the calculations we plan to use some versions of the Distorted Waveapproximation, that we consider optimal, including e ects of exchange and con guration interaction when necessary. The rst results obtained in the framework of the pro ject - the cross sections and e ective collision strengths for electron impact excitation of considered ions are presented. The impact of the excitations into the high-lying states on the intensity of the C, N and O ion lines in the spectra of low-density plasma and the cooling rates for these ions are demonstrated.

1 Introduction
The electron-impact excitation of highly-lying atomic levels can give a great contribution to the intensities of the UV, optical and IR lines observed in the spectra of various types of the astrophysical sources. For modelling the spectra of these ob jects and obtaining their parameters from comparison of observed and calculated lines and continuum intensities we need a lot of atomic data. The main bulk of those parameters are the transition probabilities (oscillator strengths) and electron impact excitation cross sections (rates of excitation by electron impact). The ma jority of the atomic data which are presented in the literature, in the atomic data catalogue and databases (see, for example, 6, 11]) refer to the ground and low lying states. At the same time, for calculating the spectra of the astrophysical sources the rates of radiative and collision transitions between highly-lying atomic and ionic states are necessary. Here we present the pro ject HILYS (Highly Lying States) aimed at obtaining the atomic parameters of such states for atoms and their ions. The main purpose of the pro ject is calculation of the cross-sections and rates for electron impact excitation for levels up to n=10 for atoms and their ions with Z 26. We also plan to construct the diagnostic tools for obtaining the parameters of the emitting plasma from the observed line intensities in the spectra of astrophysical sources. In this report we present the scope of the pro ject and the rst results. Some previous results can be also found in our recent papers 3, 13].

2 Methods of calculation
Today the methods are developed that permit to obtain probably all characteristics of the scattering process with high accuracy. We have in mind the R-matrix 1] and Convergent Close Coupling (CCC) 5] methods, that make use of big basis sets of pseudofunctions. But these methods require so big computing resources that up to now calculations have been done only for e ectively oneand two-electron targets. As for our pro ject we need a great number of cross sections for di erent atoms and ions, we have to use simpler theories, but try to understand clearly the area of their validity.
1

Send o print request by e-mail: afk@astro.spbu.ru Alexander.Kholtygin@paloma.spbu.ru

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We have accepted as the main method the method of distorted waves (DW approximation), which permits to asses reasonably the cross sections for many processes of excitation for atoms and ions. More then that. For the integral excitation cross sections of neutral atoms we nd it useful even simple Born approximation (with modi cation 2] for exchange), as the DW approximation for such problems is better then simple Born mostly for description of di erential cross sections. Belowwe remind general formulae and explain the notations in the gures.

3 Wave functions of the bound and free electrons
In our program the target wave functions are built as antisymmetrized combinations of one electron functions, which in their turn are de ned as eigenfunctions of one electron Schroedinger equation with appropriate potential, or from the solution of Hartree-Fock equations. In the rst case it is easy to achieve the mutual orthogonality of the initial and nal bound wave functions, and, after calculating in the same potential the functions describing the incoming and scattered electrons, to obtain also their orthogonality to the bound ones. This last propertyis important for correct account of exchange e ects at low energies. In the second case the wave functions of initial and nal states of the target are orthogonalized and then considered as basis on which the energy operator of the target is diagonalized. Such solutions are also mutually orthogonal and in this scheme con guration interaction approximation can be realized. But for free states arti cial orthogonalization was applied.

4 General Formulae
H = H0 + Hat + V H0 = ;1=2r Hat =
N X i
=1 2 0

(1) (2)
N X
=1

;1=2r2 ; Z=ri + i
)= e Z=r0 ;
2

V =V( H H
+

r0 X

i =1

1=rij
!

(3) (4) (5) (6) (7)

1=r0

i

at n

="
+

nn

X

n a

j n ih n j =1
) a(X)+ scatt:wav es
+

a

=E

+

a

a

exp(i

k r0

= ; 21

Z

fba (kb ka)= ; 21 kb b jV j
exp(;i
k r0

a

= ( )dr0 d
X

b

) b (X)V (

r0 X

)
2

+r X a0

(8) (9) (10)

d ba = kb jf (k k )j d ka ba b a Z d ba d ba = d
2


total
q

=

X

b b

ba

= 4 Imfaa(# =0) ka

(11) (12) (13)

=

k

a

;

k

2 2 q 2 = ka + kb ; 2kakb cos #

qdq = kakb sin #d#

4.1 First Born Distorted Wave Approximation (DWB1)
If and then in DW approximation
a

V = V1(r0 )+ V2(r0 X )
(K0 + V1) ka (r)= 0 (r0 X ) is put equal to
pm a

(14) (15)

(r0 X )= ka (r0 ) a(X ) (16) It can be substituted into (8), but much more consistent approximation 4, 17] can be obtained rewriting (8) by the Gell-Mann and Goldberger "two-potential formula" what results in
dir 2 fba (

;

k

for direct part of the scattering amplitude, and 2f for the exchange one
exch ( ba Dkb ka)= ; ;(r1 r0 b

Db ka )= ;h kb (r0) b(r1 ; + b (r0 X )jV2(r0 X )j a

:::RN )EV1(r0 )j +(r0 X )i j a (r0 X )

(17)

;h kb (r1) b(r0 :::RN )jV1(r0)E +(r0 X )i ja +(r X ) r2:::rN )jV2(r0 X )j a 0

(18)

For the inelastic scattering matrix element with V1 in direct part of the scattering amplitude is zero because of orthogonality of atomic stats a and b . For the exchange one this is so only when free states k are either calculated in the same potential as atomic wave functions, or arti cially made orthogonal to them. It seems very natural not to havecontribution to the exchange part of the scattering amplitude from the interaction with the core. This idea was advocated byDayetal. 4], but was not accepted by the atomic collision community,may be because of the habit not to have functions with coulomb asymptotic behavior for neutral targets. But simple antisymmetrization led often to unrealistically big increase of the exchange ampliexch tude. So, forced orthogonalization became popular. It was mostly used only for fab and its dir in uence on the fab is not stressed. But the scattering amplitude is a single whole and such a procedure cannot be regarded as a satisfactory solution. Below on some simple examples of direct and exchange transitions in H and He we tried to demonstrate the e ects of "forced" and "natural" orthogonalization. We hope the results are not widely recognized. But they should be investigated whenever one wants to use the DW approximation. At the present stage of the pro ject wehave programs to calculate target and free wave functions, Born and di erent versions of DWintegral and di erential cross sections and collision rates. We 3


are in the stage of accumulating experience, comparing the results of our calculations with those available in the literature. For this seminar we present several results demonstrating the role of forced and natural orthogonalization whichwebelieve are not clearly appreciated. We also show the case of 1s1 S ; 2s3 S transition in He for which also CCC (I.Bray) result is given and an example of rst order DW result for di erential cross section for 1s ; 2s +2p transition in atomic hydrogen for which new experimentisavailable and compared with CCC and Madison second order DW calculations 8]. To nish with let us explain (on example of H atom) notations in the gures connected with di erentchoices of distorting potential: Born approximation- B EP - for DW approximation with distorting potential same, as seen byactive atomic electron DW approximation - distortion potential AP1: Z V1(r0)= a r ; r1 a 0 01 AP1 NN - distortion potential AP1, No orthogonalization, No exchange. AP1 YN - distortion potential AP1, with orthogonalization, but no exchange and so on. (19)
The curves are lab eled as:

AP2 - for DW approximation with distortion potential (recommended by D.Madison 12]): Z V1(r0)= b r ; r1 b (20) 0 01 AP2 NN, AP2 YN, AP2 YY have the same meaning as for AP1: rst letter N or Y means account for orthogonalization, second - for exchange.

5 Plasma diagnostics
The problem of the plasma diagnostics is a determination of the plasma parameters: atom and ion density and temperature distribution over the all emitting volume. Here we consider the case of the optically thin in the line frequencies (excluding in some case the line of the resonance transitions) medium.

5.1 Diagnostic of the homogeneous plasma

For the sake of the simplicitywe consider only the lines controlled by electron collisions and photo and dielectronic recombinations. The total energy emitted by the studied ob ject in a recombination or collisionally excited line k ! i of the ion X is

Eki = h

Z
ki

V

e nF rki dV :

(21)

Here ki is the frequency of the line V is the total volume of the region emitting in the line nF = n(XF) is the number density of the ion XF, which is responsible for the formation of the line. e For collision lines XF X, but for recombination ones XF X+ . The coe cient rki is known as 4


e the e ective line formation coe cient (see 16]). In the case of recombination lines, rki = ne e , ki e is the e ective recombination coe cient of the where ne is the electron number density and ki e line. For collision lines, rki = ne qe . Here qe is the e ective coe cient of the collision excitation ki ki for the line k ! i, determined in such a way that (4 );1 ne n(X) qe is the emission coe cientof ki the line. In the homogeneous case plasma diagnostics reduces to determination of mean electron temperature and density and the partial ion abundances:

Xij =(Nij )=N

j

Nj = Aj N
Z

(22)

where Nij isanumber densityofion i of element j , Aj is a relative element j abundance.

E

ki

0 Eki = h

ki

e rki (T0 n0) nF dV = e 0 e

h

e ki ki

r (T0 n ) N
Z

V

F

(23) (24)

where NF is the total number of the ions XF . The value of T0 has been determined by 14] and 15]:

T0 = Te ne nF dV = ne nF dV :
V V

Z

More elaborately the methods of homogeneous plasma diagnostics are consider in 16]. It should be mention that the values of n0 and Te0 obtained from line intensities of di erent ions e can be strongly di erent. This di erences are connected mainly with the temperature and density uctuations inside the emitting volume. The impact of such uctuations on the line intensities and plasma diagnostics is considered in the next subsection.

5.2 Diagnostic of the inhomogeneous plasma
t = Te =104K
where parameter Let us suppose small relatively to is responsible for t

Here we consider both the temperature and density uctuations in the linear approximations. Instead of the values of Te and ne we can use the dimensionless parameters

s = lg (ne)+ :

=0 { 2. that the values of the temperature and electron number density uctuations are the values of Te and ne and determine the mean (for the considered ion, which he line formation) values of the parameters:

htiF = t = tnFdV = nF dV = tnFdV =N
V V V

Z

Z

Z

F

(25) (26)

and

hsiF = s = snFdV = nFdV = snF dV =NF :
V V V
R

Z

Z

Z

Let us determine the following parameters describing the rms uctuations of Te and ne :
2

=

V

(t ; t)2 nF dV

t2 N
5

F

(27)


(t ; t)(s ; s) nF dV = (28) t sNF R (s ; s)2 nF dV 2=V : (29) s2 NF In the linear approximation for small amplitude uctuations of Te and ne , the total line intensity can be presented in the form:
V

R

Eki = E

ki

0

;

1+

tt

2

+

ts

+

ss

2

(30) and
ss

0 Where Eki is described by expression (23). The parameters following expressions: " # e 1 @ 2 rki (re );1 t2 tt = 2 @t2 ki t=t

tt ts s=s

are determined bythe (31) (32)

ts ss

2e e = @ r@ki (rki );1 ts @t s

"

#

re : (33) = 1 @@ski (rki );1 s2 22 t=t s=s By using the parameter s instead of the value of ne we can describe the signi cant (up to 2 times) deviations on the mean electron number density in the linear approximation. The dependencies of the parameters tt ts and ss on Te and ne for OIII lines are plotted in the Figs. D1 and D2. As this gures shows the coe cients ts and ss are well below the tt . Only for the high density nebulae (ne > 104cm;3 ) the contribution of the electron number density variations in the total line intensities can be essential.
2e

"

t=t s=s #

We use an empirical model of a emitting ob ject (in this paper { planetary nebulae) to nd the plasma parameters from the observed line intensities. In this model the ob ject is described by its mean electron temperature T0 , mean electron number density n0 and rms temperature and e electron number density uctuations 2 and 2 as well as the correlation parameter . The relative element abundances are assumed to be constantinthe wholevolume of the nebula. In general each ion Xn+ has to be described byits own values of plasma parameters T0(Xn+ ), s(Xn+ ), 2(Xn+ ), (Xn+ ) and 2(Xn+ ). However, as numerous calculations have shown those parameters for ions with the similar ionization potentials are very close, so we do not consider these di erences. Finally we list the parameters of the model: T0 s 2 , , 2 and fN(X)/Hg { the relative abundances of elements. The atomic data which are necessary for line intensities were taken from the catalogue 6], databases, cited in 11] or were calculated in the framework of the considered pro ject. The contribution of both the photo and dielectronic recombination into the total e ective recombination coe cients has been taken into account. For tting the calculated and observed line intensities we use the procedure proposed in 9,10]. This procedure is illustrated in Fig. D3.

5.3 Determination of the plasma parameters

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6 Cooling rates for inhomogeneous plasma
One of the most important parameters of the plasma is its cooling rate. This value { L is determined as an energy, emitted by the unit volume in the unit time. We consider an optical thin plasma, controlled by the collisions of atoms and ions with electrons. In this case the local cooling rate is determined by local electron temperature and density of the medium. We can express the cooling rate in term of the so named "cooling function" = L=n2 , where n is the total number density. Using the partial ion abundances Xij determined by the equation 22 we nd: = where
ij
X

ij

Xij

ij

(34) (35)

is the partial cooling function:
ij

=

X

kl

nk h kl A

kl

For nding the total cooling function we need the distributions of atom on their ionization states. Those are determined by solving the equations of the ionization equilibrium: xi;1j = ij (T ) (36) xij Ci;1j (T ) where ij is the total recombination rate for ion i of atom with number j and Cij is the collision ionization rate for the same ion together with condition
Z X j
=0

6.1 Ionization equilibrium

xij =1

(37)

where Z = Z (j ) is the atomic number of element j . The typical dependencies of the relative ionic abundances are presented in Fig. D4 for O ions. The sources of the atomic data are described in the previous section.

6.2 Results

As an example, wehave calculated the cooling functions both for homogeneous and inhomogeneous plasma. The solar abundances of elements were taken from 7]. In Fig. D5 is presented the cooling functions for H+He+O mixture both for solar abundances and for values of 0.1 and 10.0 of them. One can see that in the presence of temperature uctuations the cooling function can strongly exceed its value for homogeneous plasma. It means, that the cooling times for inhomogeneous plasma are less then homogeneous one. Acknowledgements The authors aregrateful for the support provided by the RFBR grant 99-02-17207
and by Russian Federal Program Astronomy (project 2.2.1.4)

References

1] Bartschat K. Comp. Phys. Commun. 114,1998 2] Burkova L., Ochkur V.I. Vestnik Leningr. Univers. No. 10, 5, 1979 3] Bratsev V.F., Ochkur V.I.inProc. of "III Seminar on Atomic Data for astrophysical investigations", St.Petersburg, 8-10, 2000 (in russian).

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4] Dai T.B. et al. Phys. Rev. 123, 1051, 1961 5] Fursa V., Bray I. J.Phys. B30,757,1997 6] Golovatyj V.V., Sapar A., Feklistova T., Kholtygin A.F., "Catalogue of atomic data for low-density Astrophysical plasma", Astro.Astroph.Transact., 12, 85-262 (1997) 7] Grevesse N., Noels A., Sanval A.J., ASP Conference Series, 99, 117, 1996. 8] Khakoo M.A. et al. Phys. Rev. Lett., 82, 3980, 1999 9] Kholtygin A.F., 1998a, A&A, 329,691 10] Kholtygin A.F., 1998b, A&SSci 255,513 11] Kholtygin A.F., in Proc. of "III Seminar on Atomic Data for astrophysical investigations", St.Petersburg, 31-35, (2000) (in russian). 12] Madison D. et al. J.Phys.B24, 3861, 1991 13] Ochkur V.I. in Proc. of "III Seminar on Atomic Data for astrophysical investigations", St.Petersburg, 26-29, 2000 (in russian). 14] Peimbert M., 1967, ApJ 150, 825 15] Peimbert M., Torres-Peimbert S., Luridiana V., 1995, Rev. Mex. Astron. Astrofis. 31, 131 16] Rudzikas Z.B., Nikitin A.A., Kholtygin A.F., 1990, Theoretical Atomic spectroscopy, Izd. LGU, Leningrad 17] Tailor J. Scattering Theory,chpt. 22, x5, 1966

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Fig. 1. Cross sections for electron impact excitation of transition 1s-2p HI.
9


Fig. 2. The same as in Fig. 1, but for ion He II.
10


Fig. 3. The same as in Fig. 1, but for ion Be IV.
11


Fig. 4. The same as in Fig. 1, but for ion Ne X.
12


Fig. 5. The same as in Fig. 1, but for transition 1s-3p HI.
13


Fig. 6. The same as in Fig. 1, but for transition 1s-5p HI.
14


Fig. 7. The same as in Fig. 1, but for transition 1s 1S- 1s 3S HeI.
15


Fig. 8. The parameters ne (lower panel).

tt

,

ts

,and

ss

for the OIII] 5007 line as a function of Te (upper panel) and

16


Fig. 9. The same as in Fig. 1 but for the OIII] 88 m and 52 m lines.

17


Fig. 10. Line tting for planetary nebula NGC 7027. Calculated line intensities (solid lines) are normalized to the observed ones. The dashed line represents the probability distribution function normalized to its maximal value. Upper panel: Te t, lower panel: ne t

18


Fig. 11. The dependencies of the relative abundances of oxygen ions on electron temperature Te in low-density plasma.

19


Fig. 12. Upper Panel: cooling function (eV s;1 cm;6) for H+He+O and 2 = 0.01, 0.02, 0.04, 0.08, 0.16 (from bottom to top). Lower Panel: the same as in upper panel, but for di erent oxygen abundances (Z/Zsun ) relatively the solar value (7.4 10;4).
20