Abstract

The interpretation of the almost 300 diffuse interstellar bands (DIBs) is one of the most long-standing problems in interstellar spectra. The only model showing agreement with a large number of DIBs (>60 bands) is based on transitions in doubly excited atoms embedded in the condensed phase named Rydberg Matter (RM). A similar model is now used to precisely calculate all intense bands from the high-resolution survey by Galazutdinov et al., 63 bands in total. Thus, the RM model interprets at present 120 DIBs accurately in a consistent manner, almost half of the number of DIBs. The origin of almost all intense DIBs is shown to be absorption during electron transitions between co-planar doubly excited He atoms inside RM, and from singly excited He atoms to doubly excited He atoms inside RM. The average error in the assignments is 4 cm−1 (1.4 Å), comparable to the band width. The intense but broad DIB 578.0 nm is part of a series of 10 bands interpreted as RM transitions from n″= 5 to the conduction band limit. The shape of DIB 579.7 nm is consistent with a rotational band from symmetric top RM clusters at approximately 100 K. Good agreement with observational results like band shapes and correlations among various bands is found.

1 INTRODUCTION

The first observation of what is now called diffuse interstellar bands (DIBs) was made by Heger (1921) as ‘stationary’ features from binary stars at 578.0 and 579.7 nm. Since then, many suggestions have been made to explain the numerous DIBs: almost 300 DIBs are detected so far, often using stars obscured by interstellar dust as light sources. Excellent reviews exist (Herbig 1995; Krelowski 1999) and several surveys identifying new bands have been published recently (Jenniskens & Désert 1994; Galazutdinov et al. 2000; Tuairisg et al. 2000; Weselak, Schmidt & Krelowski 2000). There exists at present no consensus regarding the carriers of the DIBs; several proposals have been made previously but only one, the model with doubly excited atomic states embedded in Rydberg Matter (RM) (Holmlid 2004a), has matched a large number of observed bands without any parameters. There exist three main ideas to investigate the DIBs: (1) they can be divided into families of lines which correlate in intensity and thus may belong to the same carrier (e.g. the same type of molecule) (Krelowski 1999), (2) some groups of lines can be singled out since their intensities correlate with different atomic or molecular lines (Weselak et al. 2004) and (3) high-resolution spectroscopy gives the band shapes and thus information on the formation processes (Galazutdinov et al. 2003). None of these principles has so far given any final results concerning the DIB carriers, certainly at least partly due to the large number of lines observed. The recent conclusion (Galazutdinov et al. 2003) that families of lines cannot be found reliably is in line with the results presented here. The results from these lines of investigation will be compared below with the model used in the present study.

The recently published exact calculation of DIBs based on a RM model (Holmlid 2004a) was limited to slightly more than 60 mainly weak and sharp bands in a region of the spectrum between 590 and 650 nm (a few broad-band heads were also identified and calculated). This selection of bands was given by the requirement of providing a very clear proof of the RM model, interpreting bands with a width of only a few cm−1 and with the small average distance of 20 cm−1 between the band centres. The agreement between observations and calculations within 2 cm−1 has a probability of accidental agreement of the order of 0.1 per matching band, and matching a sequence of 20 bands with this precision has a chance of accidental agreement that is vanishingly small. (Some further factors of course enter a strict probability estimation, like the total density of bands predicted by the model.) As described in Holmlid (2004a), there are in fact not too many bands predicted by the RM model, so the total band density in the model is not very high. Thus, the validity of the model was proved. The present study concerns a slight variation of this model, using lower excited states, and was started since the previous model cannot match the most intense DIBs. A high precision in the calculations of the positions of the DIBs is still needed. Since no quantum mechanical description can provide the precision required, a semiclassical treatment with a restricted parametrization in the form of small quantum defects is used, as commonly employed also for other excited states of the Rydberg type (Gallagher 1994; Rau & Inokuti 1997).

In some studies, the DIB carriers are proposed to be large organic molecules like polyaromatic hydrocarbons (PAHs). For such complex molecules, the line positions cannot be calculated with a precision comparable to that for the RM model (imprecision for PAHs probably >100 cm−1). This means that calculations of other parameters like band intensities, using molecular parameters that are uncertain to this degree, are meaningless. Even if the band intensities could be calculated, the temperature is not known, and for such complex molecules the absorption spectra, both the band shapes and band intensities, will be very sensitive to the temperature. This has been shown in numerous studies of cold molecules in molecular beams, free jets and low-temperature gas matrices, as described in standard textbooks on spectroscopy (Hollas 1998, 2004). In fact, the correct reasoning concerning the DIBs is really that since the DIBs are almost unchanged in their width and form in many different astronomical objects even if their intensities change, and thus probably also the temperature of the DIB carriers, they cannot be due to any complex molecule at all. Instead, the DIBs are due to atomic transitions that are not strongly influenced by the temperature or other environmental parameters like the ultraviolet (UV) flux density. This means that the RM model, which provides new atomic-type transitions, is the only feasible model for the DIBs at present.

It should be observed from the start that the RM concept gives the conceptual framework within which the doubly excited atomic states used to interpret the DIBs can exist, but the calculations described below are not dependent on or use any concepts that are unique for the RM phase. The doubly excited states calculated provide the small number of atomic levels (around 10) necessary to explain most of the 63 DIBs interpreted here. These DIBs are not any closely spaced DIBs of the type described in the previous paper on this subject, but they are all the 25 most intense bands described in the latest observational surveys (and 38 weaker as a bonus). Approximately half the number of these bands involves transitions to or from known He levels, so there are many restrictions reducing the number of free parameters involved. Since it is not even possible to match 25 strong lines with 10 parameters if they could be chosen at will (which is not the case here), this study has found a structure of approximately 10 energy levels that describes all the main DIBs (and several more). This is the first important achievement in this paper, and it does not rely on RM theory at all. The second achievement is that several of these energy levels are calculated with high precision with no further assumptions, not for any low-probability atom but for the most abundant one in space that can give doubly excited atomic states, He (no other atom can in fact give these states). The parametrization of these doubly excited He energy levels by <10 reasonable quantum defects is the third and final important achievement in this paper. (A description in terms of quantum defects is the normal method to be used, see further below.) None of these three achievements relies on RM theory. So, even if the RM interpretation is disliked (as it happens), the levels calculated represent the DIBs with a considerably smaller number of parameters than even the number of intense DIBs interpreted, and within a theoretically grounded framework.

2 PROPERTIES OF RM

The discovery of RM (Manykin, Ozhovan & Poluéktov 1992a,b; Holmlid 1998) means that new types of spectroscopic atomic transitions exist in the interstellar medium (ISM), where the conditions are almost ideal for the formation and existence of RM. RM is a condensed type of matter formed from long-lived circular Rydberg species, mainly atoms with the outermost electron in an almost circular orbit. These states may be the result of a recombination of ions and electrons. The most facile path to formation of RM is, however, through desorption of RM clusters from non-metal particle surfaces like carbon and metal oxide surfaces (Kotarba et al. 1995; Wang, Andersson & Holmlid 1998; Wang, Engvall & Holmlid 1999). RM is a state of matter comparable to liquid or solid matter and can be formed from all atoms and small molecules. A wealth of experimental results on RM exists, and the reader is referred to the more than 50 publications on this. A few examples are provided in Badiei & Holmlid (2002a, 2003, 2006) and Holmlid (2004c). The theory of RM was first developed by Manykin et al. (1992a,b). The RM phase is almost metallic and has a very long radiative lifetime. It consists mainly of six-fold symmetric planar clusters with so-called magic numbers equal to N= 7, 19, 37, 61 and 91. Such clusters have recently been observed directly by radio-frequency rotational spectroscopy (Holmlid 2007a), providing bond lengths with a precision as good as 1 × 10−4. Recently, RM composed of H atoms in n= 1 was studied from many different aspects (Badiei & Holmlid 2006). Heavy atoms in RM cannot be in n states lower than 5–6, since the inner electrons occupy the space corresponding to low values of n. A theoretical study by LaViolette, Godin & Switendick (1995) of a condensed excited He phase similar to RM only treated the condensation of the He (1s 2s 3S) state. This type of excited matter does not give any transitions of the DIB type, as far as we can find.

Several other spectral features from interstellar and intergalactic space agree well with predictions and experimental results on RM. The first type of bands are the IR emission bands described as the UIR bands (Holmlid 2000). The IR emission spectra from comets, and the polarization measurements of light scattering from comets, are also well described by RM theory and experiments (Holmlid 2006a). The alkali atom atmospheres on the Moon and Mercury can be interpreted as due to photo-dissociation of RM clusters, with the clusters at high altitudes acting as the reservoir of alkali in the atmospheres (Holmlid 2006b). Several features of the atmosphere of Mercury are explained in this study, like its large electrical conductance.

Several different methods for forming RM exist (Holmlid 2002). It can be expected that laboratory verification of the DIBs will eventually be possible with sensitive laser-based absorption methods like cavity ring-down. In the meantime, we have computed a large fraction of the band positions (and will continue to do so). Such a procedure will in the end be as necessary for the understanding of the DIBs as experimental studies will be, since almost every DIB line has its own RM carrier as shown by the calculations in Holmlid (2004a) and below. The experimental work cannot be successful without information on how to form the carrier cluster or pair of states responsible for each DIB. Thus, accurate assignment of each band is needed before any experimental progress can be made. In this respect, the RM model is different from other (inaccurate) DIB models where one species is assumed to give rise to several different bands (which is highly unlikely from the observational evidence). This means that the notion of families of DIBs is not expected to be realistic, as noted by observational groups (Galazutdinov et al. 2003).

3 CALCULATIONAL METHOD

The atomic states that give agreement with the observed DIBs are co-planar doubly excited He states. That such states give unique agreement with DIBs was shown in Holmlid (2004a), so from a pragmatic point of view there is nothing new in the use of such states. That the states are considered to be co-planar, i.e. to have both electrons in the same plane (within the framework of the Bohr model of the atom), may seem as an unnecessary complexity. From a pragmatic point of view, only such states will be in agreement with the DIBs and this argument is of course sufficient. Further, the doubly excited states behave in a way that can only be expected in contact with RM (or another similar condensed phase). Since the RM clusters are planar, there is a planar environment around the He atoms. For example, the orbital angular momenta of the Rydberg electrons in a RM cluster are aligned perpendicular to the clusters plane and strongly coupled, as can be seen from the rotational spectroscopy studies of RM clusters (Holmlid 2007a). Thus, the absorption of one photon giving a DIB will only be able to slightly change the direction of the total angular momentum for the cluster or for the excited electron. This means that also the final doubly excited state will have its electrons very close to the plane, as described here in the semiclassical limit by the co-planar form of the He atoms.

The calculational method used to find the energies of the doubly excited co-planar states, both the initial and the final states, is almost the same as used previously (Holmlid 2004a). The inner electron is in its Bohr orbit with principal quantum number n3 and with l=n3, thus in a classical circular orbit. The correct time-independent quantum mechanical description is assumed to show that the outer electron is in a stable circular state outside the inner electron. Since a time-independent solution is sought, the description should be independent of the special position of the electrons during the motion. The neutral states are thus modelled as a core ion with charge +2, an inner electron in a circular orbit in the field from the ion and an outer electron in a circular orbit in the same plane as the inner electron. Since no accurate quantum description of this type of state is yet possible, we have to rely on a semiclassical approach with a quantum defect description. The quantum defect covers the effects due to the many-body effects in the interaction between the electron and core ion with its electron, which is often referred to as core penetration of the outermost electron (Gallagher 1994; Rau & Inokuti 1997). Such a description is necessary for the case of interacting electrons as studied here, but it will only be useful (small effect) in well-behaved cases.

The shielding of the core ion due to the inner electron is calculated here by averaging the interaction potential energy during one period of the electronic motion, or alternatively by considering the inner electron charge to be dispersed along the orbit. The integral for the potential energy of the outer electron is given by  

1
formula
where r4 is the orbit radius for the outer electron and r3 the same for the inner electron. In effect, the outer electron is relaxed outwards relative to the case with a spherically distributed inner electron charge, as shown by Holmlid (2004a), table 2.

To evaluate this integral recursively, the initial guess of r4 is n24a0 where a0 is the Bohr radius. When a positive quantum defect is included, n4 is decreased by this amount. The momentum in the circular orbit for the outer electron is calculated as forumla. This gives the kinetic energy and the total energy for the outer electron. The derivative of the potential energy in equation (1) is used to determine the stable circular orbit with the correct quantized orbit angular momentum. This new value of r4 is used as a better value of r4 for a renewed calculation of the potential energy until a stable value of r4 is found. This means that the calculation of the potential energy V4 is for a fixed quantum number n4 for the outer electron, and that the orbit is circular for the electron with n4=l4. It should be observed that the potential for the outer electron in the present case is not easily calculated in an exact classical way in the form of a closed orbit since the force is non-central.

The precision and accuracy of the calculations can be estimated from the results in Holmlid (2004a). For the 36 high quantum number transitions assigned in table 7 in this reference, the average error is 2.9 cm−1. Thus, the precision and accuracy of the energy calculations for high quantum numbers is remarkably good. A comparison with the expected precision of quantum mechanical calculations is made in Holmlid (2004a), indicating much worse (orders of magnitude) results for such a type of treatment. For the low quantum numbers applicable in the present case, the precision is certainly not equally good. This is the reason for introducing quantum defects as parameters for calculating the energies of the eight main states involved. Spin states are also included to cover all the states that should exist in the energy range of interest.

The final state of one of the atoms participating in the RM two-ion absorption process (Holmlid 2004a) is a doubly excited atom, with the two electron orbits co-planar, possibly within a RM cluster. The process giving this state is a one-electron transfer from one ion to another in the RM cluster, and the interaction between the two electrons in the final doubly excited state is strong. Two-electron systems are inherently difficult to treat exactly in quantum mechanical terms. The best-known system of this type is the He atom, which is abundant in space and which takes part in the DIB absorption processes. The energy levels of a singly excited He atom can be calculated quite accurately by the semiclassical method used here as shown in Table 1. This is of course good evidence that the calculational process is accurate. (The ground state with strong spin interaction cannot be calculated by the same method.) This method is here used primarily to calculate the energy levels of the final co-planar doubly excited states which are not calculable by other methods; this method gave good agreement with observations for the >60 DIBs calculated previously (Holmlid 2004a), where the outer electron is in a state n= 7–17. Thus, it has been tested and found to give accurate results for large outer electron quantum numbers n. States with low outer quantum number n will only be described approximately by the formula in equation (1) due to core penetration, and a quantum defect scheme must here be introduced to give a consistent description of the low n states, as expected.

Table 1

Energies in cm−1 of the second, outermost electron with principal quantum number n4 in a co-planar Rydberg state in RM, compared to an ordinary singly excited Rydberg atom and a singly excited He atom with maximum orbital angular momentum. The Rydberg constant for He mass was used. The quantum defect δ is zero.

State n4/n3 Calculated with n3= 1 Rydberg state He (1s n4(n4− 1)) 
1/1 −106985.0 −109707.2 −198310.8 (1S) 
2/1 −27221.9 −27426.8 −29223.9 (3P2
−27175.8 (1P) 
3/1 −12172.6 −12191.4 −12209.2 (3D3
4/1 −6854.3 −6857.6 −6858.8 
5/1 −4388.0 −4388.9 −4389.0 
State n4/n3 Calculated with n3= 1 Rydberg state He (1s n4(n4− 1)) 
1/1 −106985.0 −109707.2 −198310.8 (1S) 
2/1 −27221.9 −27426.8 −29223.9 (3P2
−27175.8 (1P) 
3/1 −12172.6 −12191.4 −12209.2 (3D3
4/1 −6854.3 −6857.6 −6858.8 
5/1 −4388.0 −4388.9 −4389.0 

4 RESULTS AND DISCUSSION

4.1 Transitions involving co-planar He states

Initially, the energies of the co-planar doubly excited states of He atoms were calculated with zero quantum defects. Such results with the inner electron in state n3= 1 are compared to singly excited Rydberg state energies and to the relevant experimental values for singly excited He atoms in Table 1. The symbols used to describe these states are of the form n4/n3, where n4 is the quantum number for the outer electron as clearly indicated in the tables. (The spin state is appended to the upper left corner of this symbol, as normally done.) Note the good agreement between the various values. It is obvious that the classical calculations used here agree very well with the singlet states of the He atom. Thus, the energies with zero quantum defect are used as a start also for singlet levels of the doubly excited states, but they have to be modified slightly in the comparison with the observed bands, as seen in Table 2. It was also realized that triplet states must be included, and the levels giving good agreement with observed bands are also given in Table 2, together with the corresponding quantum defects. It should be observed that the quantum defects have a strongly limited effect on the energy levels. For example, a comparison of Tables 1 and 2 shows that for the state 33/1, a quantum defect of 0.0095 changes the energy by only 80 cm−1 from a defect value of zero. Thus, the defects are certainly not free parameters but a refinement needed due to the interaction of the two electrons (core penetration). It is further assumed that transitions from singly excited He atoms to the doubly excited states should be observed as DIBs, and the very accurately known values for free He atoms were used to calculate the transitions. A close agreement was found with singlet states, and the triplet energies were modified into agreement with the transitions observed as DIBs, using reasonable values of the quantum defect as shown in Table 2. These transitions are of the form  

2
formula
where ‘−’ indicates an electron missing in the He atom. Transitions between two co-planar doubly excited states (possibly within the RM phase) are of the form  
3
formula
where n1 and n3 are the quantum numbers of the inner electrons. Such a transition is shown schematically in Fig. 1. If n1=n3 is valid in the transition, it may even be intra-atomic of the form  
4
formula

Table 2

Energies in cm−1 of the second, outermost electron with principal quantum number n4 in a co-planar Rydberg state in RM with the value of n3 given. The quantum defect δ is given in parentheses. The Rydberg constant for He is used.

n4n3 /1 /2 /3 /4 
3−28075 −25188 −18194 −12497 
(0.0313) (0.0367) (0.01) (0.003) 
1−27235 −24720 −17880 −12363 
(0.0005) (0.013) (−0.03) (−0.038) 
3−12250 −11978 −10891 −9049 
(0.0095) (0.011) (0.03)  
1−12124 −11893 (0) −10881 (0) −9039 
(−0.006)    
n4n3 /1 /2 /3 /4 
3−28075 −25188 −18194 −12497 
(0.0313) (0.0367) (0.01) (0.003) 
1−27235 −24720 −17880 −12363 
(0.0005) (0.013) (−0.03) (−0.038) 
3−12250 −11978 −10891 −9049 
(0.0095) (0.011) (0.03)  
1−12124 −11893 (0) −10881 (0) −9039 
(−0.006)    
Figure 1

The DIB absorption process with electron transfer He(n2, n1) + He+(−, n3) +hν→ He+(−, n1) + He (n4, n3).

Figure 1

The DIB absorption process with electron transfer He(n2, n1) + He+(−, n3) +hν→ He+(−, n1) + He (n4, n3).

The calculated transitions are shown in Fig. 2, and a detailed comparison with the observed bands is given in Table 3. In general, the agreement is quite satisfactory with an average error of 4.2 cm−1 for the 49 bands interpreted, similar to the half-width of the sharp bands. (It is obvious that the He atoms taking part in such transitions in contact with the RM phase may be influenced by the field from the RM, and thus that the free He atom energies may not be exactly applicable. However, the level of agreement found is certainly sufficient for accurate assignment of most of the bands.)

Figure 2

Calculated transitions in the RM model for He atoms. The calculated values for the doubly excited states from Table 2 are given at the top, and each He level to the left-hand side is subtracted to give the transitions in the panels to the right-hand side. Transitions outside the range in Galazutdinov et al. (2000) are excluded (blank spaces). Full borders indicate relatively strong and sharp lines, and broken borders indicate matches to weaker bands.

Figure 2

Calculated transitions in the RM model for He atoms. The calculated values for the doubly excited states from Table 2 are given at the top, and each He level to the left-hand side is subtracted to give the transitions in the panels to the right-hand side. Transitions outside the range in Galazutdinov et al. (2000) are excluded (blank spaces). Full borders indicate relatively strong and sharp lines, and broken borders indicate matches to weaker bands.

Table 3

Transitions for DIBs, observed from Galazutdinov et al. (2000) and calculated. The Rydberg constant is for He. Note that 33D is an upper state. Transmission (T) is from the website http://www.sao.ru/ha/coude/DIBwavelength.htm. Bold type indicates strong and sharp lines.

See Table 2  See Table 1  Observed λ (Å)  T  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
33/1  21       19783    5053 Å excluded range 
13/1          19909    5022 Å small peak 
33/2    4984.81  0.973  20055  20055   
13/2    4963.90  0.956  20140  20140   
32/3    7224.00  0.858  13839  13839   
12/3    7062.65  0.972  14155  14153   
33/3    (4728.3)  0.977  21145  21142  Flank of 4726 Å 
13/3    4726.27  0.946  21152  21152   
32/4          19536    5117 Å excluded range 
12/4          19670    5082 Å excluded range 
33/1  23P2        16974    5890 Å excluded range 
13/1    5849.80  0.917  17090  17100  −10   
33/2    5796.96  0.821  17245  17246  −1   
13/2    5769.04  0.983  17329  17331  −2   
33/3          18333    5453 Å excluded range 
13/3          18343    5450 Å excluded range 
32/4    5975.74  0.981  16730  16727   
12/4    5927.68  0.979  16865  16861   
33/4          20175    4955 Å no peak 
13/4          20185    4953 Å excluded range 
33/1  21 6699.24  0.960  14923  14926  −3   
13/1    6646.03  0.990  15042  15052  −10   
33/2    6572.84  0.990  15210  15198  12   
13/2    6543.20  0.987  15279  15283  −4   
33/3    6139.94  0.978  16282  16285  −3   
13/3          16295    6135 Å no peak 
32/4    6810.49  0.943  14679  14679   
12/4    6747.80    14816  14813  Weselak et al. (2000) (uncertain) 
33/4          18127    5515 Å peak exists 
13/4    5512.64  0.969  18135  18137  −2   
32/1  33D3        15866    6301 Å no peak 
12/1    6660.64  0.937  15009  15026  −17   
33/1    7699    12985  12979  Within K i 
13/2          12511    7992 Å small peak 
33/1  32/1  6317.06  0.980  15826  15825   
13/1    6269.75  0.910  15945  15951  −6   
33/2    6211.67  0.983  16094  16097  −3   
13/2    6181.25    16174  16182  −8  Weselak et al. (2000) 
33/3    5818.75  0.973  17181  17184  −3   
13/3    5815.71  0.985  17190  17194  −4   
32/4    6413.93  0.983  15587  15578   
12/4    6362.30  0.982  15713  15712   
33/4          19026    5255 Å excluded region 
13/4          19036    5252 Å excluded region 
33/1  12/1  6672.15  0.965  14984  14985  −1   
13/1    6613.56  0.784  15116  15111   
33/2    6553.82  0.974  15254  15257  −3   
13/2    6520.56  0.967  15332  15342  −10   
33/3    6116.80  0.979  16344  16344   
13/3    6113.20  0.961  16354  16354   
32/4    6778.99  0.985  14747  14738   
12/4          14872    6722 Å peak but noise? 
33/4    (5496)  0.978  18190  18186  5497 Å flank of 5494 Å 
13/4    5494.10  0.948  18196  18196   
33/1  32/2  7721.85  0.966  12947  12938   
13/1          13064    7653 Å excluded range 
33/2    7562.25  0.954  13220  13210  10   
13/2          13295    7520 Å small peak 
33/3    6993.18  0.885  14296  14297  −1   
13/3          14307    6988 Å peak exists 
32/4          12691    7877 Å abs. interferes 
12/4          12825    7795 Å peak exists 
33/4    6195.96  0.896  16135  16139  −4   
13/4    6194.73  0.976  16138  16149  −11   
33/1  12/2  8026.27  0.947  12456  12470  −14   
13/1          12596    7937 Å noisy range 
33/2    7845  0.986  12744  12742   
13/2          12827    7794 Å peak exists 
33/3    7228.49  0.979  13830  13829   
13/3    7224.00  0.858  13839  13839   
32/4          12223    8179 Å peak exists 
12/4          12357    8090 Å peak exists 
33/4    6379.29  0.854  15671  15671   
13/4    6375.95  0.944  15680  15681  −1   
See Table 2  See Table 1  Observed λ (Å)  T  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
33/1  21       19783    5053 Å excluded range 
13/1          19909    5022 Å small peak 
33/2    4984.81  0.973  20055  20055   
13/2    4963.90  0.956  20140  20140   
32/3    7224.00  0.858  13839  13839   
12/3    7062.65  0.972  14155  14153   
33/3    (4728.3)  0.977  21145  21142  Flank of 4726 Å 
13/3    4726.27  0.946  21152  21152   
32/4          19536    5117 Å excluded range 
12/4          19670    5082 Å excluded range 
33/1  23P2        16974    5890 Å excluded range 
13/1    5849.80  0.917  17090  17100  −10   
33/2    5796.96  0.821  17245  17246  −1   
13/2    5769.04  0.983  17329  17331  −2   
33/3          18333    5453 Å excluded range 
13/3          18343    5450 Å excluded range 
32/4    5975.74  0.981  16730  16727   
12/4    5927.68  0.979  16865  16861   
33/4          20175    4955 Å no peak 
13/4          20185    4953 Å excluded range 
33/1  21 6699.24  0.960  14923  14926  −3   
13/1    6646.03  0.990  15042  15052  −10   
33/2    6572.84  0.990  15210  15198  12   
13/2    6543.20  0.987  15279  15283  −4   
33/3    6139.94  0.978  16282  16285  −3   
13/3          16295    6135 Å no peak 
32/4    6810.49  0.943  14679  14679   
12/4    6747.80    14816  14813  Weselak et al. (2000) (uncertain) 
33/4          18127    5515 Å peak exists 
13/4    5512.64  0.969  18135  18137  −2   
32/1  33D3        15866    6301 Å no peak 
12/1    6660.64  0.937  15009  15026  −17   
33/1    7699    12985  12979  Within K i 
13/2          12511    7992 Å small peak 
33/1  32/1  6317.06  0.980  15826  15825   
13/1    6269.75  0.910  15945  15951  −6   
33/2    6211.67  0.983  16094  16097  −3   
13/2    6181.25    16174  16182  −8  Weselak et al. (2000) 
33/3    5818.75  0.973  17181  17184  −3   
13/3    5815.71  0.985  17190  17194  −4   
32/4    6413.93  0.983  15587  15578   
12/4    6362.30  0.982  15713  15712   
33/4          19026    5255 Å excluded region 
13/4          19036    5252 Å excluded region 
33/1  12/1  6672.15  0.965  14984  14985  −1   
13/1    6613.56  0.784  15116  15111   
33/2    6553.82  0.974  15254  15257  −3   
13/2    6520.56  0.967  15332  15342  −10   
33/3    6116.80  0.979  16344  16344   
13/3    6113.20  0.961  16354  16354   
32/4    6778.99  0.985  14747  14738   
12/4          14872    6722 Å peak but noise? 
33/4    (5496)  0.978  18190  18186  5497 Å flank of 5494 Å 
13/4    5494.10  0.948  18196  18196   
33/1  32/2  7721.85  0.966  12947  12938   
13/1          13064    7653 Å excluded range 
33/2    7562.25  0.954  13220  13210  10   
13/2          13295    7520 Å small peak 
33/3    6993.18  0.885  14296  14297  −1   
13/3          14307    6988 Å peak exists 
32/4          12691    7877 Å abs. interferes 
12/4          12825    7795 Å peak exists 
33/4    6195.96  0.896  16135  16139  −4   
13/4    6194.73  0.976  16138  16149  −11   
33/1  12/2  8026.27  0.947  12456  12470  −14   
13/1          12596    7937 Å noisy range 
33/2    7845  0.986  12744  12742   
13/2          12827    7794 Å peak exists 
33/3    7228.49  0.979  13830  13829   
13/3    7224.00  0.858  13839  13839   
32/4          12223    8179 Å peak exists 
12/4          12357    8090 Å peak exists 
33/4    6379.29  0.854  15671  15671   
13/4    6375.95  0.944  15680  15681  −1   

In Table 2, the lowest energy states are usually taken as the triplet states even for states/3 and/4, where the adjustment process, in fact, gives slightly negative values of the quantum defects for the singlet states. (In the case of triplet states, the electrons will mainly avoid each other in space.) The intermediate doubly excited states used in the transitions in Table 3 and Fig. 2 would also be expected to give transitions to the Rydberg limit in absorption. It is found that bands slightly broader than most bands interpreted here exist for transitions to upper levels approximately 50 cm−1 below the ionization limit for almost all relevant energy levels given in Table 2. Such results are shown in Fig. 2 and in Table 4 with detailed data for the transitions: these band values just fall out from the calculations. The existence of these (not adjusted) bands is strong support for the accuracy of the results found here. It is especially remarkable that accurate values are predicted for the four singular DIBs in the range 803–850 nm. The results in Table 2 that give rise to these DIBs are calculated using quite small values of the quantum defects, which means that the theory is accurate.

Table 4

Relatively broad DIBs corresponding to transitions from doubly excited states to the Rydberg continuum in RM, observed by Galazutdinov et al. (2000) and calculated with an average Rydberg level at −53 cm−1. The Rydberg constant used is for He.

See Table 2    Observed λ (Å)  Tλ (Å)  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
33/1  Ry        12197    8196 Å noisy range 
13/1    8283.29  0.961  12069  12071  −2   
33/2          11925    8383 Å no clear peak 
13/2    8439.38  0.933  11846  11840   
32/3    5508.35  0.967  18149  18141   
12/3    5609.73  0.980  17821  17827  −6   
32/4    8037.90  0.975  12438  12444  −6   
12/4    8125.75  0.988  12303  12310  −7   
See Table 2    Observed λ (Å)  Tλ (Å)  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
33/1  Ry        12197    8196 Å noisy range 
13/1    8283.29  0.961  12069  12071  −2   
33/2          11925    8383 Å no clear peak 
13/2    8439.38  0.933  11846  11840   
32/3    5508.35  0.967  18149  18141   
12/3    5609.73  0.980  17821  17827  −6   
32/4    8037.90  0.975  12438  12444  −6   
12/4    8125.75  0.988  12303  12310  −7   

One important result from the matching of the calculated transitions with observations is that a few of the strong DIBs cannot be matched. An overview of the DIBs is given in Fig. 3, where two of the unassigned bands are indicated with stars. These not interpreted bands are 578.0, 620.3, 628.3* and 862.1* nm. They are all relatively broad. DIBs 620.3 and 628.3 are assigned in Holmlid (2004a), and they will not be discussed further here. DIB 862.1 nm is outside the range studied in the previous publication, but is probably of the same type. DIB 578.0 has a different shape, being broad and skewed, surrounded by several smaller equidistant bands. It is proposed to be unique in the DIB range in that it corresponds to a transition n″≈ 40 ←n″= 5, where n″ is the principal quantum number around an ion (Z= 2). The upper level of such transitions is probably close to the conduction band of a more continuous RM phase, but still within a core ion in the material. In Table 5, the bands in this range are interpreted as a series of transitions from n″= 5 to an upper level at 38 < n″ < 47. The conduction band of RM at typical excitation levels for RM in interstellar space is close to n″= 40 as derived from the UIR bands (Holmlid 2000). This description of the bands around DIB 578.0 adds an interpretation for a further 10 bands, increasing the total number of bands interpreted here to 63. The average error for these bands is 3.8 cm−1.

Figure 3

Some of the most intense DIBs interpreted here with the RM model, with the bands observed by Heger (1921) in boxes. Two intense but broad bands within the range studied by Galazutdinov et al. (2000) that are not interpreted here are marked with stars. The synthesized DIB spectrum is from Sorokin, Glownia & Ubachs (1998). The bands are assigned in Table 3.

Figure 3

Some of the most intense DIBs interpreted here with the RM model, with the bands observed by Heger (1921) in boxes. Two intense but broad bands within the range studied by Galazutdinov et al. (2000) that are not interpreted here are marked with stars. The synthesized DIB spectrum is from Sorokin, Glownia & Ubachs (1998). The bands are assigned in Table 3.

Table 5

Interpretation of the bands around 578.0 nm as transitions close to the conduction band limit in RM. n″ is the principal quantum number at an ion (Z= 2).

n″  Observed λ (Å)  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
47  5760.40  17355  17359  −4   
46  5762.70  17348  17351  −3   
45  5766.16  17338  17341  −3   
44  5769.04  17329  17331  −2   
43  5772.6  17318  17320  −2   
42  5775.78  17309  17309   
41  5780.37  17295  17297  −2  Strong 
40  5785.05  17281  17284  −3   
39  5788.9  17270  17269  Weselak et al. (2000) 
38  5793.22  17257  17254   
n″  Observed λ (Å)  Observed forumla (cm−1 Calculated forumla (cm−1 Difference (cm−1 Comment 
47  5760.40  17355  17359  −4   
46  5762.70  17348  17351  −3   
45  5766.16  17338  17341  −3   
44  5769.04  17329  17331  −2   
43  5772.6  17318  17320  −2   
42  5775.78  17309  17309   
41  5780.37  17295  17297  −2  Strong 
40  5785.05  17281  17284  −3   
39  5788.9  17270  17269  Weselak et al. (2000) 
38  5793.22  17257  17254   

The two first DIBs observed 86 years ago may also finally be interpreted now: the 578.0 nm band is found to be due to a transition from a RM level n″= 5 to the limit of the conduction band as described above. The 579.7 nm band is due to the transition 33/2 ← 2 3P, thus from a singly excited He atom in contact with a RM cluster to a co-planar doubly excited He atom within the RM cluster. It is thus obvious that these two bands will vary very differently with the conditions in the interstellar matter. This will be discussed below. To complement Tables 3–5 which are organized after the assignments of the bands, Table 6 gives the 63 assigned bands ordered after wavelength. In all tables, the wavelengths are given in angstrom (Å) rather than in nanometre (nm) since the data even in recent observational studies are given in this obsolete unit.

Table 6

All 63 DIBs assigned, in order of increasing wavelength. Values in parentheses not marked as DIBs in Galazutdinov et al. (2000) or uncertain in Weselak et al. (2000).

High level Low level Wavelength (Å) High level Low level Wavelength (Å) High level Low level Wavelength (Å) 
13/3 214726.27 13/3 32/1 5815.71 13/1 12/1 6613.56 
33/3 21(4728.3) 33/3 32/1 5818.75 13/1 216646.03 
13/2 214963.90 13/1 23P2 5849.80 33D3 12/1 6660.64 
33/2 214984.81 12/4 23P2 5927.68 33/1 12/1 6672.15 
13/4 12/1 5494.10 32/4 23P2 5975.74 33/1 216699.24 
33/4 12/1 (5496) 13/3 12/1 6113.20 12/4 21(6747.80) 
Ry 32/3 5508.35 33/3 12/1 6116.80 32/4 12/1 6778.99 
13/4 215512.64 33/3 216139.94 32/4 216810.49 
Ry 12/3 5609.73 13/2 32/1 6181.25 33/3 32/2 6993.18 
RM 47 5760.40 13/4 32/2 6194.73 12/3 217062.65 
RM 46 5762.70 33/4 32/2 6195.96 32/3 217224.00 
RM 45 5766.16 33/2 32/1 6211.67 13/3 12/2 – 
13/2 23P2 5769.04 13/1 32/1 6269.75 33/3 12/2 7228.49 
RM 44 – 33/1 32/1 6317.06 33/2 32/2 7562.25 
RM 43 5772.6 12/4 32/1 6362.30 33D3 33/1 7699 
RM 42 5775.78 13/4 12/2 6375.95 33/1 32/2 7721.85 
RM 41 5780.37 33/4 12/2 6379.29 33/2 12/2 7845 
RM 40 5785.05 32/4 32/1 6413.93 33/1 12/2 8026.27 
RM 39 5788.9 13/2 12/1 6520.56 Ry 32/4 8037.90 
RM 38 5793.22 13/2 216543.20 Ry 12/4 8125.75 
33/2 23P2 5796.96 33/2 12/1 6553.82 Ry 13/1 8283.29 
33/2 216572.84 Ry 13/2 8439.38 
High level Low level Wavelength (Å) High level Low level Wavelength (Å) High level Low level Wavelength (Å) 
13/3 214726.27 13/3 32/1 5815.71 13/1 12/1 6613.56 
33/3 21(4728.3) 33/3 32/1 5818.75 13/1 216646.03 
13/2 214963.90 13/1 23P2 5849.80 33D3 12/1 6660.64 
33/2 214984.81 12/4 23P2 5927.68 33/1 12/1 6672.15 
13/4 12/1 5494.10 32/4 23P2 5975.74 33/1 216699.24 
33/4 12/1 (5496) 13/3 12/1 6113.20 12/4 21(6747.80) 
Ry 32/3 5508.35 33/3 12/1 6116.80 32/4 12/1 6778.99 
13/4 215512.64 33/3 216139.94 32/4 216810.49 
Ry 12/3 5609.73 13/2 32/1 6181.25 33/3 32/2 6993.18 
RM 47 5760.40 13/4 32/2 6194.73 12/3 217062.65 
RM 46 5762.70 33/4 32/2 6195.96 32/3 217224.00 
RM 45 5766.16 33/2 32/1 6211.67 13/3 12/2 – 
13/2 23P2 5769.04 13/1 32/1 6269.75 33/3 12/2 7228.49 
RM 44 – 33/1 32/1 6317.06 33/2 32/2 7562.25 
RM 43 5772.6 12/4 32/1 6362.30 33D3 33/1 7699 
RM 42 5775.78 13/4 12/2 6375.95 33/1 32/2 7721.85 
RM 41 5780.37 33/4 12/2 6379.29 33/2 12/2 7845 
RM 40 5785.05 32/4 32/1 6413.93 33/1 12/2 8026.27 
RM 39 5788.9 13/2 12/1 6520.56 Ry 32/4 8037.90 
RM 38 5793.22 13/2 216543.20 Ry 12/4 8125.75 
33/2 23P2 5796.96 33/2 12/1 6553.82 Ry 13/1 8283.29 
33/2 216572.84 Ry 13/2 8439.38 

Further narrow DIBs, both strong and weak, agree well with transitions starting from singly excited H and Li atoms, instead of He atoms as studied here. The number of such reasonable fits is >15. These results are not included in the present treatment since the main principles are probably most easily observed in the He case.

4.2 Selection rules

Since the transitions observed are proposed to take place in RM clusters, selection rules valid for isolated atoms are not expected to apply. The strongest selection rule in light atoms is the spin selection rule, giving transitions within classes of triplet and singlet states, but not allowing spin transitions triplet ↔ singlet. In the present case, such a selection rule cannot be very strong, since the atoms are more or less embedded in RM clusters, but it may be hoped that some features will remain. In the case of transitions from almost free He atoms to atoms inside the RM clusters, a spin selection rule is not expected to be strong since the system is not even isolated. This can also be seen in Fig. 2, where transitions to both spin states of almost all levels are observed from atomic states 2 1S, 2 3P, 2 1P and 3 3D. However, for the transitions in the lower part of Fig. 2, i.e. transitions within the RM clusters, spin selection rules may have more influence. This is observed especially in transitions from 32/2 that are much stronger (full borders in Fig. 2) to triplet states 33/X than to singlet states 13/X. A similar but weaker pattern is also found for the other doubly excited states 32/1, 12/1 and 12/2. If some of the transitions would be mainly intra-atomic, as 13/1 ←12/1 might be, the spin selection rule could be very strong. There are, however, not so many cases where this assumption can be tested. The transition 13/1 ←12/1 is much stronger than 13/1 ←32/1, as should be the case with a strong selection rule, but 33/1 ←3 2/1 is not very strong. In the same way, 33/2 ←32/2 is much stronger than 33/2 ←12/2, but 13/2 ←12/2 is hardly observed. Of course, this may just indicate that these two weak transitions are not intra-atomic. Thus, the main indication of a spin selection rule is found for the transitions from the states 32/2, 12/1, 32/1 and 12/2.

Other selection rules could also be operative, like the angular momentum selection rule ΔL=±1. In the transitions involving singly excited He atoms, the system is not isolated and the angular momentum selection rule is probably not applicable. As noted above, almost all possible transitions of this type are observed, so no selection rule exists. In the transitions between doubly excited atoms interpreted here, l for the outermost electron normally changes by unity but in cases where l for the outermost electron is unchanged as in 32/4 ←32/1. Several such bands exist even if some are quite weak, and there is no clear pattern resembling an angular momentum selection rule. It is concluded that an angular momentum selection rule is not applicable due to the coupling between several atoms during the DIB transition. That angular momentum conservation is not an important restriction for the RM transitions is supported by the spectroscopic results in Holmlid (2007b).

4.3 Band shapes

The observed shapes of the DIBs may give important information about the nature of the bands. One of the bands interpreted here has been studied with very high resolution by Kerr et al. (1998), namely the 579.7 nm DIB. It shows a simple profile resembling a rotational spectrum for a symmetric rotor or possibly a type B spectrum for an asymmetric rotor with a distance between the P and R branch maxima of Δ= 0.70 cm−1. This small distance indicates a molecule or cluster with a large moment of inertia, or a too low temperature of the ISM. The band shape was confirmed by Galazutdinov et al. (2003). The band is only slightly asymmetric, and if it is a rotational structure, the B rotational constant is given by B2hc/(8kT). Its uncomplicated shape and the lack of a clear Q branch indicate that the absorbing entity does not change its structure dramatically in the electronic transition (the rotational B constant is almost unchanged) and thus the transition takes place in a localized orbital. It also seems that the absorbing entity has a simple structure, most likely being a symmetric rotor like a RM cluster. The corresponding temperature in the local ISM is not known accurately, but is estimated by Rouan, Léger & Le Coupanec (1997) to be 100 K. (Their interpretation of the rotational structure as due to PAH molecules requires 18 < T < 35 K, thus infrathermal.) The transition for this band is here proposed to be 33/2 ← 23P, thus from a He atom in contact with a RM cluster to a co-planar doubly excited He atom. Assuming that all atoms in the cluster are He in excitation state n= 2 and that the cluster is six-fold symmetric with the magic number N= 61, the temperature from the branch separation becomes 140 K. If instead the masses in the cluster are H2, the temperature becomes 71 K. Thus, a temperature of 100 K is compatible with RM clusters, and the clusters discussed are symmetric rotors giving simple P and R branches. Since the transition is local, i.e. not changing the electronic state of the entire cluster appreciably but only of one of its atoms, the value of rotational B constant is almost unchanged, which agrees with the observation that the band is only slightly asymmetric.

The RM cluster structures discussed have properties quite different from van der Waals (vdW) bonded clusters of He and H2. So-called vdW clusters only exist at very low temperature. In the RM state n= 2 implied above, the binding energy is probably close to 1.4 eV per atom, by extending the results by Manykin et al. (1992a) to low n values. This means that such RM clusters will be stable up to temperatures of 1600 K, using the rule-of-thumb that the energy barrier should be 10 times larger than the thermal energy to avoid decomposition. Rotational spectra of RM clusters formed from K atoms at n= 4–6 have recently been studied for the first time (Holmlid 2007a) at temperatures of a few hundred K with no stability problems.

4.4 Correlations

The studies of DIBs in nearby and only slightly obscured stars by Galazutdinov et al. (1998) show variable ratios between the intensities of the bands at 579.7 and 578.0 nm. A distinction is made between ‘sigma-’ and ‘zeta-’type ISM clouds in which the absorptions take place. In the sigma case, the ratio I(578.0)/I(579.7) is high, while in zeta clouds the 579.7 nm band is strong. Thus, these two strong bands are due to different carriers. In the assignment used here, DIB 579.7 is due to a transition from a He atom in contact with a RM cluster, 33/2 ← 2 3P, while DIB 578.0 is due to a transition to the RM limit at n= 41. Thus, the transitions are very dissimilar. Another interesting observation in the same study is that the DIBs probably do not originate in dust grains, since there is no clear relation between the observed dust absorption and scattering (reddening) and the strength of the DIBs. In the RM model, the RM clusters are formed on particle surfaces but the lifetime of RM is so long that a direct relation between particle densities and the RM density is not necessarily found.

Another study by Galazutdinov et al. (2004) investigates the correlation between DIBs and atomic line intensities. The four lines they study in detail are due to very different transitions, according to the RM model. Anyway, they are all shown to be related to the K i line, with correlation coefficients of 0.76–0.92. The correlation to K atom densities is obvious in the RM model, since RM cluster formation is promoted by K at the particle surfaces from which RM clusters desorb (Wang et al. 1999; Wang & Holmlid 2002) (as a few examples). The K atoms are included in the RM as ions with n″ at least down to 5 as found experimentally (Holmlid 2007b). The smallest correlation is found for DIB 578.0, which is due to a transition in RM from n″= 5. During the formation of such a RM phase, K atoms are included as ions in the RM, and they can no longer be observed as K i. Thus, K i and DIB 578.0 nm will not be observed simultaneously in RM. For the other strong and sharp DIBs described here, the inner quantum number n″, here corresponding to n3, is ≤4. In such cases, K atoms cannot be incorporated in the RM as ions but will be observed in the atomic state K i. Thus, this quite intriguing low correlation of DIB 578.0 is remarkably well described by the detailed physics discovered experimentally for RM.

The authors also investigate the correlation with Ca ii, which is much lower, with correlation coefficients of 0.46–0.64. Ca in any form does not take part directly in the formation of RM clusters, as far as we know from experiments. A weak correlation is anyhow expected since ionization always leads to the formation of Rydberg species by recombination; the Rydberg species will attach to the RM clusters and share excitation energy with them, thus increasing the lifetime and density of RM which gives more light absorption in the form of DIBs.

A study by Weselak et al. (2004) compares the column densities of small molecules like H2, CH and CO with the intensity ratio I(579.7)/I(578.0) and with the colour excess E(BV). Their conclusions are that narrow DIBs and the column densities of the molecules studied are related. The explanation offered by the RM model is that RM clusters act as energy and H atom transfer centres. This is especially obvious in the case of H RM, which is a source of quite reactive H atoms, bound by approximately 1.4 eV (at n= 2) instead of 4.5 eV in the case of an H2 molecule. Thus, many DIBs will correlate with the amount of RM and especially H RM that is the only other type of RM besides He RM that may exist in excitation state n= 2 (the same level as n″= 4). The same will also be true for small molecules H2 and CH that are at least partially formed by H transfer from H RM. That DIB 578.0 nm is less apparent in such an environment with high densities of RM clusters in low excitation levels is expected since this DIB is due to a transition at higher excitation level n″= 41 ← 5.

The correlations between the far-UV curvature of the UV extinction curve, the DIB 579.7 nm strength and the density of small molecules are discussed further in Weselak et al. (2004) and Rachford et al. (2002). It appears that the far-UV curvature is linked to very small particles (Cardelli, Clayton & Mathis 1989). The explanation provided by the RM model is that the RM density in the form of small RM clusters is the common factor behind these observations. The diameter of a H2 RM cluster with N= 61 and n= 5 is 31 nm, while the lengths of the filaments formed are unknown. (The excitation state n= 5 is the most common state for the visible DIBs in general; see Holmlid 2004a.) Further, the 217.5 nm UV bump may correspond to the DIB-type transitions XyR3Ry (notations from Holmlid 2004a) that start from two Rydberg states in RM, in the same way as in the general two-ion RM model in Holmlid (2004a). The typical absorption 61R3R1 will be at 218.8 nm, thus a very good agreement, while the corresponding DIB in the visible from the transition 61R5R1 is at 693.9 nm. Rachford et al. (2002) find a strong correlation between the width of the UV bump and the hydrogen molecular fraction, which agrees with the assumption that the UV bump is due to transitions in RM.

Krelowski & Greenberg (1999) studied the behaviour of several DIBs in the Orion association and observed strong departures from the average properties of DIBs. For example, they observed that sharp DIBs like 579.7 nm were missing, while 578.0 nm was weak. The broad 577.8 nm on the other hand was of normal intensity (described in their fig. 7 and related text), as also 601.0, 570.5 and 620.5 nm. These broad bands are all probably due to cluster transitions of the type described in Holmlid (2004a), and are not closely related to the sharp atomic-type transitions discussed in the present study. Thus, the observations in the Orion association are in good agreement with the RM model used. One interesting comparison is possible between 628.4 and 579.7 nm in the Orion association. DIB 628.4 nm is broad and interpreted in Holmlid (2004a) as a two-ion transition in RM, while 579.7 nm is sharp and here proposed to be due to 33/2 ← 2 3P. The first of these is not attenuated in the Orion while the second one is. This means that low excited He RM clusters and doubly excited He atoms, that are involved in the formation of DIB 579.7 nm, are quite rare in this cloud.

The RM model also gives a simple explanation of the redshifting observed for the DIBs in the Orion study. Clouds of RM will give redshifting of the light passing through from stimulated Raman scattering (Holmlid 2004b,d, 2005). The redshifting observed in Orion is of the order of 3 cm−1 which requires rather small amounts of RM for the light to pass through; in the laboratory, a redshift of this size requires a passage length of 15 m or less through the (admittedly much denser) RM produced there.

4.5 Intensities and densities

The energies of doubly excited atomic states cannot yet be calculated quantum mechanically. This is not a special feature of the RM or of the DIB transitions, but a general observation. As discussed in Holmlid (2004a), the precision of quantum mechanical calculations for systems of a complexity similar to RM clusters is at least two orders of magnitude (thus a factor of 100) worse than the classical calculations used in the present study. To assign the transitions is what is attempted in Holmlid (2004a) and also in the present study: when reasonable assignments have been found, the next step is to determine band intensities. It is of course necessary to assign the transitions first, before any extensive calculations of the rates of transition, thus of the band intensities, should be attempted: otherwise, the calculations cannot be done since the states involved and the transition processes are not known. However, no accurate method is known to calculate the rates of the various transitions giving the DIBs. Since the transitions take place in the condensed RM phase, the problem is quite difficult to solve theoretically. No theoretical relation between the intensities of the DIBs and the densities of RM in the interstellar clouds can thus be found at present. The relative intensities of the various DIBs cannot be determined either, but accurate assignments based on precise calculations are of course absolutely required before any intensity calculations can be done. It is also necessary to assign all possible DIBs before intensity calculations are investigated, since the number of random overlap of different DIBs is likely to be quite large. Thus, any conclusions attempted from intensity calculations will have a low validity before all bands are correctly assigned.

It is likely that RM is quite common in the ISM (Badiei & Holmlid 2002c), but it will mainly exist in undisturbed form with very long radiative lifetimes and thus be ‘dark’. A typical average density of RM is assumed to be 106 atoms m−3, corresponding to most of the dark matter in interstellar space (Badiei & Holmlid 2002b,c). RM is almost dark even in the laboratory, if left undisturbed. The details of the processes preceding the DIB transitions are not known, but it appears possible that the RM clouds and clusters must be disturbed by, for example, energetic photons or charged particles before the optical absorptions can take place. That DIBs are most often observed in absorption from stars embedded in dense clouds is in agreement with this. In some of the transitions in the present electron transfer model, a low doubly excited state is formed initially. This initial state is not the normal form of RM but a fragment of disturbed RM formed by external influences like energetic photons, fast particles or possibly strong electric fields due to shocks or other processes. Thus, the problem of relating the intensities of the absorptions to RM densities has to await a detailed theoretical description of the RM phase. However, the experimental approach might be faster and better. Since it is possible to form RM in enough quantities and under controlled conditions in the laboratory, both from H2 molecules (Wang & Holmlid 2002) and from H atoms (Badiei & Holmlid 2006), it might be possible to measure the DIB absorptions by sensitive laser methods. Unfortunately, He RM has not yet been reported, and studies of such a material will probably be quite demanding. The problem of DIB intensities is a complex matter even observationally, since the intensities of the various DIBs vary strongly for different objects (ISM clouds). Thus, not so much general information can in fact be obtained from the strengths of DIBs, but probably only local information about the state and composition of the RM in each cloud can be obtained.

The possibility of experimental verification of the DIBs of course exists at least in principle since RM can be produced relatively easily in the laboratory. However, absorption measurements in RM are still not possible and may appear quite unlikely to ever be possible, due to the assumed very low rate of the DIB processes. In the laboratory, typically 1 cm passage through RM in the lab is used, while in space the absorption lengths may be of the order of 1 pc, thus a factor of 1018 longer. A RM density of 1015 m−3 can be reached in the lab, and the density in space is probably 106 m−3, thus the nl factor for absorption is still a factor of 109 larger in space. Laboratory measurements of this type may thus not be the best option to investigate the transitions in the doubly excited states.

5 CONCLUSIONS

The RM model is the first model that gives the possibility to calculate a large number of DIBs with a precision better than or comparable to the observed band widths. The method used here was shown previously to give agreement with >60 observed lines with a precision of a few cm−1 for transitions between states typically with values of n″ > 5. A quantum defect scheme is now introduced to describe the energies of co-planar doubly excited He atoms at lower values of n″. Transitions between such states, and also between singly excited He atoms and co-planar doubly excited states are used to accurately interpret all intense DIBs. The precision of the assignment of the 63 bands interpreted here is on average 4 cm−1 (1.4 Å), thus in most cases smaller than the half-widths of the bands. The strong DIBs at 579.7 and 578.0 nm are now interpreted and are found to arise by very different formation processes. This gives the varying behaviour of them in different ISM clouds. Several other observational results like the rotational structure in DIB 579.7 nm are discussed in relation to the RM model.

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