Rydberg Matter as the diffuse interstellar band (DIB) carrier in interstellar space: the model and accurate calculations of band centres

A model for the interpretation of a large number of the 280 diffuse interstellar bands (DIBs) is presented.

The bands are proposed to be due to electronic excitation processes in Rydberg Matter leading to doubly excited coplanar Rydberg molecules.

Rydberg Matter is composed of atoms and small molecules like H2 in condensed coherent circular Rydberg states.

It has been studied in the laboratory especially with laser time-of-flight experiments and in laser Raman spectroscopy studies.

Rydberg Matter is used as the lasing medium in an ultra wide-band tunable IR laser (Chem.



376 (2003) 812).

Rydberg Matter has recently been proposed to be part of the dark matter in the Universe, to be the source of the so called UIR emission bands from interstellar space and to give rise to the Faraday rotation in intergalactic space.

The potential energy of the doubly excited coplanar Rydberg states, which are the final states in the DIB transitions, is now calculated accurately in the classical limit.

More than 60 of the DIB bands are calculated with good accuracy.

The intensity distributions of the transitions are relatively smooth without unexplained missing bands.

The model used does not predict too many bands.

The calculations are very accurate, in this optimal wavelength region, relative to other calculations in systems of a complexity comparable to Rydberg Matter.


The so called diffuse interstellar bands (DIBs) still do not have accurate assignments more than 75 years after the first observation of the most intense of these bands.

A recent paper compares the possible explanations studied.1

Excellent reviews and extensive sets of measurements exist in this field2–5 and a database of the DIBs is maintained on the World Wide Web.6

The problem of the DIBs is that approximately 280 bands of strongly different widths (FWHM varying between 0.5 and 143 cm−1) and very different intensities (varying within a factor of 50) have been observed by absorption in diffuse interstellar clouds of the light from reddened stars.

The bands are clearly not from free atoms due to the large bandwidths.

The most common assumption has been that they are of molecular origin, and a recent high resolution study7 of four bands shows that they have a doublet structure with a splitting of 0.6 cm−1.

This is assumed to support the molecular character of the features, possibly showing two rotational branches.

Radiation resistant molecules (ions) of polycyclic aromatic hydrocarbons (PAH) have received most interest in laboratory studies.8–11

The PAH proposal has been under active study by several groups during the last 15 years.

However, it has been difficult to find good agreement between the DIB band positions and the PAH bands,1 and a match of one single band for each type of PAH molecule can not be considered to be conclusive.12

Other recent proposals are carbon chain anions13 which was subsequently rejected due to bad matching of the bands at higher resolution,14 and fullerene molecules and ions, where the interpretation of two lines as belonging to C60+ was described.15

Some studies have been directed towards Rydberg states of H2 as a possible source of some of the DIBs,16–19 but the spectroscopic processes involved would probably not be possible at the low photon densities in interstellar space.

Here another description based on Rydberg species is given, which represents the bands well by Rydberg-like formulas based on absorption in ion pairs within Rydberg Matter.

The first accurate calculations of the potential energy in circular coplanar doubly excited Rydberg states, that are the final states in the transitions, are reported.

This means that a large number of the DIB bands can now be calculated directly from theory with no empirical factors, and we give results for around 60 bands.

Of course, matching of even a considerable number of bands can be quite accidental, and other relations like pairs of broad and narrow bands, assignment of almost all DIBs within a range of 500 Å, and trends in the intensity distributions are used as evidence for this theoretical description.

The first experimental observations of Rydberg Matter (RM) mainly gave data on work functions and electrical conductivities.20,21

These results have recently been independently confirmed.22

More refined molecular experiments have been done more recently.

Laser induced fragmentation studies of RM clusters of K give information on both excitation states and sizes of the clusters.23–25

Such studies gave the first direct microscopic proof of the properties of RM.

Similar studies have also been done with formation of RM clusters from small molecules like N2 and H2.26–28

The formation of H2 in RM form is, of course, of great interest for the interpretation of the DIBs.

Note that the main method in these studies is NOT mass spectrometry but neutral time-of-flight since no electric field accelerates any ions for the time-of-flight measurements.

The process giving the neutral particles (clusters) their high quantized kinetic energy is Coulomb explosions induced by the laser pulse passing through the RM cloud.

Several other studies of RM clusters exist, especially studies of small clusters at high temperature carbon or metal oxide surfaces.29–31

In a new type of study, namely laser Raman spectroscopic studies, unique information about the electronic processes in RM32 can be obtained.

Other laser Raman studies have probed RM layers on solid surfaces33 and the stimulated Raman shifting in RM.34

Recently, the first emission spectroscopic study has been performed with RM in the infrared range.35

Stimulated emission in RM was subsequently used in an ultra wide-band tunable IR laser, which may be the first thermally excited laser.36,37

The so called unidentified infrared bands (UIR) were recently shown to be formed in de-excitation processes in RM.38

This identification has been confirmed by Raman33 and IR emission studies35–37 of RM in the laboratory.

The properties of RM are such that it is an excellent candidate for the missing dark matter in the universe.39

For example, a hydrogen atom in RM takes up a volume 5 × 1012 times larger than a ground state hydrogen atom.

RM has an intrinsic magnetic field caused by the stacked magnetic dipoles in RM.

The large density of almost free electrons in RM means that a considerable Faraday rotation can be calculated for RM in intergalactic space, in good agreement with observations.40

These facts suggest that RM exists in large amounts in interstellar and intergalactic space, and thus that the interpretation of the DIB bands in terms of processes in RM is based on the actual existence of RM in space.


The experimental methods used to produce RM in the laboratory have recently been described,41 with a short theoretical background.

Some of the background for RM in space is given in .ref. 38

Thus, only the most relevant information for space is repeated below.

Properties of RM

A simple relation exists between the excitation level of RM and the bond distances between the core ions in the RM, as shown by both the classical and QM calculations for RM.

In the classical model, it is found that the interionic distances are 2.9 times larger than the corresponding Rydberg radius.

At the average excitation level of n = 80 in the ISM as given by the interpretation of the UIR bands,38 the interionic distance is 1 μm.

These quite large distances have not yet been measured in the laboratory in the laser fragmentation experiments since the repulsion energies are comparable to thermal energies.

So far, laser induced Coulomb explosions in RM clusters give a bonding distance of 4–6 nm.24–28

This corresponds to n = 3–6 for the outermost electron in a singly charged positive ion.

This is the same range of states as studied in the present contribution.

The RM structure will be the same independent of which atom or molecule it is built up from, since the Rydberg state properties are largely independent of the core ion properties.

A molecule may, of course, be too large to be accommodated into the RM structure for the special common principal quantum number n of the RM cloud.

The lifetime of RM was calculated by Manykin et al. They found it to be long, of the order of 100 years at an excitation level of n = 16.

Auger processes42 were found to be the main de-excitation processes for RM.

The lifetime of RM at n = 80 would be longer than the generally accepted lifetime of the universe from a simple extrapolation of this calculation.

External interactions like energetic particles and quanta will, of course, disturb the RM.

In experimental studies of RM, lifetimes of several minutes to hours are observed20–22,32,34,36,37 even at rather high pressures where the collision rate is high.

Formation of RM in space

Very long-lived circular Rydberg states are formed in space by ionisation and subsequent recombination of ions and electrons.

Their long radiative lifetime means that such states are dark, not interacting readily with radiation in the visible or IR ranges.

However, Rydberg states with n of the order of 100 and more are observed.43

A high Rydberg state with n = 500 will make several collisions during its radiative lifetime of 5 × 104 s since the cross sections for collisions are large.44

Such collisions may not be efficient in forming RM clusters, but they will take part in the build-up of larger RM clouds.

Further, Rydberg–Rydberg collisions are a starting point for condensation in the boundary layer at grain surfaces in space.29

The most rapid route to RM formation is desorption of Rydberg species and small Rydberg clusters from surfaces.

The ground state of an alkali atom is not stable on surfaces of carbon or metal oxide,45 which means that such atoms easily desorb thermally as Rydberg species.23,46,47

Circular Rydberg states are formed through desorption, often in large densities close to the surfaces.

This gives rapid condensation to RM clusters29,30 in the surface sheath, as observed in the experiments.

In subsequent collisions between the desorbing Rydberg species or clusters and gas molecules outside the surface, transfer of excitation energy is possible, as observed in the formation of RM clusters of H2 and N2 at metal oxide surfaces.26–28

Also, RM of H atoms has recently been observed (to be submitted).

In the ISM, particles of graphite or other non-metal surfaces are the most likely ones to form RM since these surfaces work very efficiently also in the laboratory.

This means that there is no limit in the gas phase density below which the RM forming processes can not operate.

In the laboratory, RM clusters are formed from hydrogen molecules even at room temperature.26–28

Photon desorption may also give desorption of Rydberg species.

Photons with rather low energy can desorb alkali metal atoms into Rydberg states from carbon and metal oxide surfaces.

The same process should also be possible to form Rydberg atoms of H.

Electronic structure of RM

The excitation state of RM and thus the width of the conduction band is the main internal parameter of interest.

Laser Raman experiments give rather high excitation levels for dense RM, corresponding to n > 55 (ref. 32) and other spectroscopic studies give similar values of n (refs. 33, 34) for low density RM.

In cold regions of the ISM, higher values of n are likely.

The Rydberg atoms and molecules which condense to RM are in circular states, which means that the excited electron going into the RM conduction band has a large angular momentum quantum number.

In the QM calculations by Manykin et al.,42,48,49 s electrons were used for convenience in the calculations.

The inner effective potential introduced in the calculations removed the electrons from the surroundings of the core ion, thus giving a density distribution similar to the high-l electrons.

This means that the two descriptions are similar.

Clusters of RM in space

Coulomb explosions caused by laser excitation of RM23–28 give clear microscopic proof of the shape of clusters formed by RM.

These experiments show that planar RM clusters contain 7, 19, 37, 61 or 91 atoms or molecules.

These magic numbers are characteristic for planar close-packed (hexagonal) monolayer clusters.

Clusters with 10 and 14 atoms also exist, and they probably have a rhombus in their centre.

Classical calculations including electron correlation effects show that RM clusters have a planar structure50 since no bonding exists otherwise.

The very high quantum numbers involved means that a classical description of the electron motion should be possible.51

The electron motion takes place in a plane, and if bonding shall exist, all the highly excited electrons have to move coherently in one plane.50

A classical electron in RM has both a large principal quantum number n and a large angular momentum quantum number l.

Thus, RM has the form of planar monolayer clusters of atoms and molecules.

It has recently been shown39 that there exists an energy barrier for collisions of two planar RM clusters, which means that RM clusters will not de-excite each other in collisions at temperatures typical for the interstellar medium.

At low temperatures as in interstellar space, the magnetic dipoles of the RM clusters40 will interact to form stacks of the clusters with the magnetic momenta in the same direction, giving an enhanced magnetic field in the stacks.

Such stacks will add further clusters, finally forming “nano-wires” with a diameter of the order of μm.

This is the probable form of RM in the low temperature parts of space.

The circular electronic motion in the planes of the RM clusters means that they have their magnetic momenta perpendicular to the cluster plane.

An alignment of the magnetic momentum of each cluster with an external magnetic field in space will take place to some extent.40

As discussed below, the transition momenta in the absorption processes will generally be confined to the cluster planes, but otherwise they will be in arbitrary directions in the plane clusters.

Thus, there will not exist any polarization of the DIB absorptions.


The possible absorptions in Rydberg Matter have been studied by Manykin et al48,49. using a quantum mechanical description.

The absorptions of the electrons in the conduction band of RM are almost continuous, but they will be confined to very long wavelengths.

According to the QM calculations, RM will be transparent up to a limit, which depends on the excitation level in RM.

This absorption wavelength limit is found to be 2 m, from an extrapolation of the data in Manykin et al.48

Thus, no such absorptions will be observed in the visible or NIR.

The average excitation level in interstellar space is n = 80–100, as found for the RM in the interstellar matter from the UIR interpretations.33,35–38

The UIR bands are observed in emission, which means that they are observed from rather warm regions in interstellar space, and higher excitation levels may be found in colder regions.

The core ions in the RM may be able to absorb in the visible range.

In the following, it should be understood that also the electrons in the core ions in the RM are coupled to the surrounding RM cluster with its delocalized electrons.

The transitions described will not be possible if this coupling does not exist.

The rapid electron motion and energy transfer in the cluster are necessary features for the creation of the transition moments interacting with the electromagnetic field.

Since the outermost electron in the atoms or molecules forming the RM is excited to the RM level in the conduction band, it is the second outermost electron in the atom or molecule M (if such an electron exists) which may take part in the absorptions.

This could then be observed as an absorption in the singly ionised species M+.

Such absorptions are quite common also in the ISM, and are not of concern here.

However, some of these transitions are between circular Rydberg states and they are then related to processes in the RM.

The next step to consider is an interaction between the M+ species in the RM and the RM electrons.

Such absorptions within the RM phase could have the form M+(n1l1) + eRM(n2l2) + hν → M(n3l3; n4l4)resembling the emission process shown to give rise to the UIR bands in .ref. 38

All principal quantum numbers n used here, apart from n4, will show the quantum number of an electron in an ion, thus around a doubly charged core.

The quantum number n4 indicates the outermost electron in a neutral state, thus with the electron moving in the field from a singly charged ion.

Since the electron eRM(n2l2) moves in the RM plane, the final state M(n3l3; n4l4) is likely to be coplanar, with both electrons moving in the RM plane.

A process like in eqn. (1) will be indicated as, for example, 64 ← R100R3.

This process is shown in Fig. 1.

The subscripts indicate the quantum number of the electron in an ion, thus the electron moves around a doubly charged core.

The R’s with subscripts indicate that the electron is in a Rydberg atom (or molecule) with the outermost electron in the RM conduction band.

The full number indicates that this electron is in the outermost level in the atom, thus 64 indicates a doubly excited neutral state with the outermost electron in n4 = 6 around an ion with charge +1, and the next highest electron with n3 = 4 around an ion with charge +2.

As described in ref. 38 the electron taking part in a Rydberg-like process as in eqn. (1) can not conserve its angular momentum as shown.

It belongs initially to the conduction band and has no momentum for moving inwards to the ion.

However, the other delocalized electrons coupled to it in the conduction band may compensate for the missing angular momentum; consider, for example, two classical electrons at the rim between two ions which will together have zero angular momentum relative to either one of the ions.

There are relatively few transitions of the type in eqn. (1) that give absorptions in the visible range.

Transitions of this type will not give sharp bands, since the excitation state in the conduction band in a large interstellar cloud will vary and thus give a certain spectral width, of the order of 10–100 cm−1.

Thus, they are initially unsuitable to prove the interpretation of the DIBs as due to such transitions: thus, we limit this report to sharper transitions, even if transitions of the type in eqn. (1) certainly are found among the broad DIB bands, especially at short wavelengths.

Another possible process corresponds to a direct interaction between two electrons encircling two different ions in the RM, with the total process giving one neutral state and one doubly charged ion, M+(n1l1) + N+(n2l2) + hν → M(n3l3; n4l4) + N2+

The most probable form of this absorption process is found when the low state in the final doubly excited state does not change its quantum state.

Such a transition is given by M+(n3l3) + N+(n2l2) + hν → M(n3l3; n4l4) + N2+which should give more intense absorptions than the process in eqn. (2) which has n1 different from n3.

This process implies transfer of an electron from one of the ions to the other ion.

It is depicted in Fig. 2.

Also in this case, the final atomic state is coplanar with both electrons in the RM plane.

In the following, such a process will be indicated as, for example, 64 ← R4R5.

We will only study this type of transition below.

It should be noted that both N and M must have two electrons, thus excluding H atoms from this type of process.

Also, in the case of eqns. (2) and (3) angular momentum restrictions will apply, similar to the case in eqn. (1).

While the angular momentum mismatch is smaller here than in the case of eqn. (1), it is not directly obvious that the same remedy is possible here, i.e. that the other conduction electrons take care of the excess angular momentum.

However, the electron transfer from N to M involves also the conduction band, and this could have been indicated by a transfer via the RM states in Fig. 2.

The electrons in the ions are coupled to the conduction band, and the absorption does not take place by electron transfer between two isolated ions: instead, they are inside a condensed phase with many highly excited electrons, in a case where the Born–Oppenheimer approximation is not valid.

This means that the core ions interact with the electrons in the formation of the transition moments.

It is not necessary that N and M are neighbours in the RM cluster, since the electron (energy) transfer implies a change in state of excitation of the entire RM cluster.

A variation of this type of transition which may involve H atoms as N(n2l2) could have the form M+(n3l3) + N(n2l2) + hν → M(n3l3; n4l4) + N+but such a transition would either (i) not take place within RM, the N(n2l2) hydrogen atom being outside the planar RM clusters and thus not giving the coplanar form of M(n3l3; n4l4), or (ii) be restricted to high states n2l2 and thus be similar to the process in eqn. (1).

We will restrict this study to the form of process given in eqn. (3).

In the final state in eqns. (1)–(4) an interaction will exist between the excited Rydberg electrons in the atom or molecule M. If n3 ≪ n4 is valid, this interaction may accurately be taken into account as a change in effective charge state of the core ion, thus with n3 corresponding to a doubly charged ion and n4 to a singly charged ion.

This means, in the case of low-l3 orbitals, that the inner electron shields the outer electron efficiently from the doubly charged core ion.

RM has a planar structure due to the bonding between the high-l Rydberg species.

Thus, the electron orbits in the final doubly excited state are in the same plane in the classical limit, and the shielding by the inner electron is then much stronger than for a three-dimensional case.

It means that the potential energy of the outer electron is increased, thus giving a smaller ionisation energy in this case with two coplanar Rydberg orbits.

Since atomic physical or condensed physics methods are not yet able to cope with this type of problem efficiently, a quasi-classical method has been adopted.

It is modelled after the successful quasi-classical methods used in reaction dynamics and kinetics, and in surface dynamics.

Methods like quasi-classical trajectory calculations have been highly successful for many classes of problems, and have long become standard tools.

In a high circular Rydberg state, the electrons behave almost classically, and the Born–Oppenheimer approximation is not valid due to the low velocity of the orbiting electrons.

This situation is further accentuated in the doubly excited coplanar Rydberg atoms of interest here, since correlation effects between the electrons will be important.

The classical (or, more correctly, quasi-classical) theory for RM50 is successful, and has, for example, given the interionic distances in RM correctly, in agreement with the kinetic energy release observed in RM fragmentation experiments.25–28

Thus, it should be possible to handle the doubly excited coplanar Rydberg states in a similar way.

Of course, we have in mind that a description in terms of the Bohm pilot wave description of quantum mechanics is better for Rydberg states than the conventional forms of Q.M.,52 since in this formulation the classical trajectory and the wave packet both exist and are interlinked.

Thus, the energy state of the outer electron in a doubly excited coplanar Rydberg atom should be found.

The correct time-independent Q.M. description is assumed to show that this electron is in a stable state outside the core and the inner electron.

Thus, this state is likely to be circular.

More complex penetrating orbits (passing inside the inner electron) may of course exist, but they will probably not be periodic but chaotic due to coupling to this inner electron and the core ion.

Since a time-independent solution is sought, the description should be independent of the special position of the electrons during the motion.

The final state is thus modeled 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.

The whole planar state is still surrounded by an RM plane.

The shielding of the charge on the core ion due to the inner electron has been calculated by averaging the interaction potential energy during one period of the electronic motion or, alternatively, considering the electron charge to be dispersed along the orbits.

Of course, this description is an approximation of the complex three-body system and may be refined further to enable the interpretation of more DIBs.

The integral for the potential energy of the outer electron takes the form where r4 is the orbit radius for the outer electron and r3 the same for the inner electron in the excited atom or molecule M. In effect, the outer electron is relaxed outwards relative to the case with a single charge representing the doubly charged core ion together with the inner electron.

To evaluate this integral, r4 is initially put equal to n24a0, where a0 = 52.9 pm is the Bohr radius for the state n = 1.

This is a reasonable starting point and the calculation is recursive, giving a new value of r4.

The momentum in the circular orbit for the outer electron is calculated as p = n4ħ/r4, and the kinetic energy is found from this value of the momentum.

This gives the total energy for the outer electron.

The derivative of the potential energy in eqn. (5) is used to determine the stable circular orbit with the correct quantized orbit angular momentum.

This new value of r4 is used to guess a better initial value of r4 for a renewed calculation of the potential energy until a stable value of the radius r4 is found.

At that point, the transition energy is finally calculated.

This means that the calculation of the potential energy is done for the fixed known quantum number n4 for the outer electron, and that the orbit is circular for the electron.

The calculation is thus accurate in the classical limit, with the added benefit of the correct quantisation of the angular momentum n4 = l4.

It should be observed that the potential for the outer electron in such a case is not due to a central force so it is not easily calculated in an exact classical way in the form of a closed orbit.

This type of calculation does not include the spatial structure of the inner electron orbitals from a quantum mechanical description.

If the inner electron orbits are slightly inclined to the plane of the outer electron, the absolute value of the potential energy calculated may not be correct.

This effect should be observable at low n3 values for the inner electron.

Also, interactions due to the particle structure of the electrons may exist, such that the averaging in eqn. (5) is not absolutely exact due to a periodicity in the near-circular motion.

This effect should be important when n3/2 and n4 are close or have the same value.

These effects should be included to give the model a larger range of validity.

With no corrections included, it is expected that the model should work for a relatively large difference in orbits for the inner and outer electrons, thus for relatively large values of n4 > n3/2.

The classification schemes sometimes used for doubly excited states53,54 employ approximate quantum numbers, which are not very useful for circular doubly excited Rydberg states.

For example, the approximate quantum numbers K and T used in these contributions describe the angular correlation of the two electrons.

For a doubly excited coplanar Rydberg state with strongly different principal quantum numbers, the number K will not exist.

Thus, such a parametrization is not useful in the present case.

A comparison of the calculation procedure used here for the doubly excited coplanar states with experiments is not possible due to lack of such data besides the DIBs.

A comparison with other theoretical descriptions will, of course, be possible when such treatments exist.

Results and discussion

The results are concerned with the data in the DIB database6 which is an updated version of the information found in .ref. 4

Rydberg doublets

One important observation concerning the DIB bands is easily made.

This observation is that most broad bands with FWHM > 7 cm−1 are accompanied by a sharp band that is displaced from the centre of each broad band towards longer wavelengths.

This displacement is 6.3 ± 5.3 cm−1, which is an average for 18 pairs of lines at wave numbers below 18 400 cm−1 (5487–8621 Å).

The large uncertainty in the displacements is mainly due to the difficulty in determining the band centres for the broad bands accurately, since the average FWHM of the broad bands is 33 cm−1.

This kind of doublet structure is expected if the transitions take place in Rydberg species, since the different values of the Rydberg constant will give rise to such doublets.

For the hydrogen molecule mass, the Rydberg constant is 30 cm−1 smaller than for infinite mass, thus similar to the width of the broad bands involved.

That some DIBs are broader than this can be motivated by: (i) they are of the type in eqn. (1), most of them at short wavelengths (this may give a width due to the width of the conduction band, of the order of 100 cm−1); (ii) they are of the band head type, as described below.

In the calculations, the energy of N+(n2l2) (i.e., the ion within RM acting as an electron donor) is the most important factor giving a variation with the value of the Rydberg constant.

The other terms in the energy expression for the transition are smaller, and, in fact, the two terms corresponding to n3 cancel in the present approximation where the energy state of the inner electron in the final doubly excited state is not influenced by the outer electron.

The term corresponding to this outer electron is small in comparison.

Thus, since the process in eqn. (3) is used for the calculations, the Rydberg constant for the hydrogen atom should not be employed.

This is concluded because the core ions N+(n2l2) must be H2+ or heavier.

The average difference between the band centres for infinite mass and for the hydrogen molecule mass is found to be 4.7 cm−1 for n2 = 5 with RH2 = 109 707.25 cm−1.

This value is clearly within the error limits from the observed band pairs.

In the results presented below, the Rydberg constant value for each calculated band is given as R or RH2.

The Rydberg constant for the hydrogen atom gives good agreement in many cases for the sharp bands, which may signify a systematic error in the calculations of 2–4 cm−1, not unexpected for a quasiclassical calculation.

In a few cases, even an intermediate band is observed within a doublet giving a triplet, which is here interpreted as due to Rydberg electrons in a slightly heavier ion mass like He giving an intermediate Rydberg constant for the ion N+(n2l2).

The He mass may give rise to a peak with RHe = 109 722.18 cm−1.

Heavier masses for the core ions will give a distribution of Rydberg constants and thus an unresolved broad band with its extreme edge at R = 109 737.11 cm−1.

Many of the broad bands are of the bandhead type described below, which means that they contain several unresolved bands.

This gives them a larger FWHM than what corresponds to the range of Rydberg constant values.

That DIB bands have strongly different shapes has been observed previously.

However, the recent high resolution study of four DIBs7 shows a common form with a doublet structure, with a peak spacing of 0.6 cm−1.

The bands studied are not broad but sharp with FWHMs of 2–5 cm−1.

This doublet structure could possibly be due to different Rydberg constants, but it is more likely due to other effects, possibly spin–orbit interactions.

Energy levels

The procedure using eqn. (5), as described above, gives the energy levels of the electron bound in the coplanar doubly excited atom.

The resulting energy levels and distances for the outermost electron to the core ion are given in Tables 1 and 2.

The potential energy is also depicted in Fig. 3 as curves at constant values of n4.

At the lowest energy level of the inner electron n3 = 1, the energy levels should agree approximately with the levels for an ordinary Rydberg electron, and with the corresponding levels of a He atom.

This comparison is made in Table 3.

At the lowest levels calculated, a small difference exists in the values.

Thus, the procedure used is correct in this limit, including the correct quantisation of the angular momentum n4 = l4.

Note that this calculation still assumes coplanar states, thus that the electron at n3 = 1 is in a planar orbit encircling the ion core, thus not in an s state.

The field from the outer electron and from the surrounding RM will prevent the inner electron from being in an s state.

This means that the energy shift from the He atom at n4 = 3 is considerable.

The calculation procedure employed gives the correct energies in the classical limit which agrees with the Rydberg description and describes at least approximately the non-s state of the inner electron.

Unfortunately, there is no way at present to check the accuracy of the other results in Table 1.

It is apparent that the errors in the calculations will be most important at small n4, and also when n3/2 ≈ n4 is true.

In these cases, either the bonding energy is large, or the interaction between the two electrons is strong.

Other methods to calculate the energies of these coplanar states will be studied in the future.

Possible methods include perturbation theory methods.

However, the change in energy for the outer electron at the higher energy levels of the inner electron is not small, and thus such methods may not be the best to use.

It is also believed that electron correlation is an important factor, as it is for the energy states of RM, which means that a quantum mechanical method must be used.

Heads of band series

In the DIB data, many broad and intense bands exist.

From the theory presented here, it is apparent that many band heads should exist with their extrema at n3 = 1.

Consecutive n3 values = 2, 3, 4… will give results which change very little, and thus a series head (“foot”) should be observed.

Some of the series heads observed are shown in Table 4.

The main series of band head series is the one of the form X1 ← R5(4)R1, which is found in the DIB spectral range with at least five entries, as seen in Table 4.

This agrees with the proposal that the bands observed are due to Rydberg species.

Of course, also non-coplanar transitions will contribute intensity to these bands.

Since the bands are broad, it is apparent that the Rydberg species can not exist isolated but must be incorporated in some material, like RM discussed here.

That the Rydberg constant for H2 was used in some of the entries in the table is due to the fact that only the sharp component of the Rydberg doublet is regularly observed for large values of n4.

Series of bands

The comparison of calculated values with observations is strikingly accurate in a certain wavelength range and becomes considerably worse outside this range.

The results given in Tables 5–7 in the wavelength range 5800–6540 Å are calculated accurately with no approximations or empirical corrections for the bands.

The series of most interest are all series between 12n ← RnR5 and 7n ← RnR5.

Higher series are also observed, and lower series 6n ← RnR5 and 5n ← RnR5 exist but some of the transitions are not accurately calculated since n3/2 is close to n4, and thus reliable assignments of individual bands in these series have to await improved calculations.

See further Tables 5 and 6 which include >40 bands.

All transitions in these tables are of the type where the quantum number of the inner electron in the final doubly excited Rydberg state is constant in the process, as in eqn. (3).

Good agreement between calculations and observed bands is found.

In Table 5, the standard deviation of Δ is 4.6 cm−1, thus a relative standard deviation of 2.8 × 10−4 relative to .

In Table 6, the standard deviation is larger since lower quantum numbers are involved, with a value of 13.8 cm−1.

This means a relative standard deviation of 8.6 × 10−4.

These values are found without any adjustable parameters.

The results here can be compared with Q.M. calculations done on systems of comparable complexity.

Ionization energies of molecules may be a field suitable for comparison.

In a recent study,55ab initio methods in the Gaussian 3 form were used to determine several different energy values for molecules, including ionization potentials.

The accuracy (average deviation) found was of the order of 1 kcal mol−1, or 350 cm−1.

Another object for comparison may be doubly excited atoms.

In a recent DFT calculation on doubly excited Rydberg states of He atoms, Roy and Chu56 reached an accuracy of the energy of within 3.6%, i.e. several thousand cm−1 compared to other theoretical values for doubly excited Rydberg states.

The only comparison with experimental data that they could do showed a difference (deviation) of 1650 cm−1.

The DFT method was concluded to be accurate.

These examples of modern calculations show that the accuracy of the results for systems in some ways resembling RM is low relative to the results presented here.

One interesting observation is that in the range 5800–6180 Å almost every known DIB line is interpreted with an accuracy within the FWHM of the bands (see Table 7).

This is excellent evidence for the correctness of the present description of the DIBs.

Number of bands predicted

In the model used here it is assumed that the process in eqn. (3) dominates, that is that the quantum number for the inner electron in the final Rydberg state is not changed during the transition.

This means that the number of bands from this model in the DIB range, between 7500 and 25 000 cm−1, is only 440.

Since many of them form band heads which are unresolved and the number of coincidences using the observed band widths even otherwise is large, this number of bands is not enough to explain all the observed bands.

The situation may, however, improve with more accurate calculations.

Approximately 160 bands can still not be assigned, mainly in the range 15 000–12 000 cm−1.

If the condition that the quantum number for the inner electron should be constant is dropped, the number of bands may be sufficient to assign the observed bands.

The model shown in eqn. (1) and Fig. 1 must also be included to explain the very broad bands observed at short wavelengths, which can not be band heads of the same type as described above.

With the condition for the inner electron in effect, the model in eqn. (1) predicts no bands in the DIB range.

With this condition removed, a few broad bands can be assigned.

Most unassigned bands are, however, sharp and weak.

To make these assignments of the closely spaced bands between 15 000 and 12 000 cm−1 correctly, improved calculations are needed, since in this spectral range the two quantum numbers in the final Rydberg state approach n3/2 ≈ n4.

Thus, the model used does not predict too many bands but it may be able to predict most DIB bands.

Intensity and intensity distributions

The observed intensity distributions of the bands assigned to all transitions of the type (n4)n3 ← Rn3R5 are shown in Table 8 and in Fig. 4.

The agreement between calculated bands and the observed DIBs in Table 8 is not as good as in Tables 4–7 due to the larger spectral range used, and the added assignments are thus not as reliable as in the earlier tables.

Entries marked X in the table are outside the spectral range studied for the DIBs.

The blank part of the table is partly excluded since no state can exist with n3 ≫ 2n4, with these parameters defined in the heading of the table.

Several general points are apparent:

(i) Transitions (n4)1 ← R1R5 are observed even if low transitions like (n4)2 ← R2R5 are not observed.

This is expected since these lowest transitions in the series also correspond to non-coplanar final Rydberg states, and to transitions that may pass through states in the RM continuum.

(ii) Transitions (n)n ← RnR5 are less intense than surrounding transitions ending at (n ± 1)n.

This is likely to be due to an interaction between the two electrons in the final coplanar Rydberg state, which will give a lower stability when the electrons have an integer relation between their periods of motion.

It may be noted that this is contrary to initial expectations.

(iii) No unexplained gaps exist in the quantum number series.

(iv) The intensity distributions are reasonable with relatively smooth behaviour, excluding some intense overlaps with other bands.

Some intense bands in the DIB catalogue may be of a different origin.

(v) Two different parts of parameter space are observed, one at 3 ≤ n4 ≤ 8 with large intensities and often broad bands, and one at n4 ≥ 9 where the intensities are much lower and the peaks are sharp.

One possible explanation is that the high n4 range corresponds to atoms with high excitation levels, while the low n4 range corresponds to small molecules that may be more rapidly de-excited in the RM phase.

Another possibility is that many broad bands at low signal levels, even with a considerable integrated intensity, are not observed above the noise level.

In Fig. 4, the band intensities marked with X have contributions from other types of transitions, possibly of the type in eqn. (1).

Note the black line showing the position of the lower intensity contributions from transitions to states (n)n.

Detailed discussions of the structure of the intensity distribution that are of mainly astrophysical interest will be given elsewhere.

Fundamental to the proposed nature of the DIBs is the assumption that the number of low excited core ion pairs in the RM, i.e. the lower state in the transitions in the tables, is high enough to give the observed absorptions.

The UIR bands are related to the DIBs, since the final states in the UIR transitions with n = 12–24 (ref. 38) may be one of the initial states in the DIB transitions, according to the RM description of these processes.

The other state for the DIBs is a lower state of a core ion in the RM, mainly with n = 5, as described here.

The lifetimes of these doubly excited atomic states within the RM are unknown, as are also their densities in space.

A comparison with free atoms may be both misleading and meaningless, since the coupling of the outer delocalized electrons in the conduction band of RM is lost.

If we anyway attempt to make such a comparison, the states of interest would be doubly excited coplanar states n4 = 80, n3 = 5.

No studies have been made of such states but only of other non-coplanar states.

Some states studied like n4 = 21, 5d have lifetimes of 400 μs,57 a factor 105 higher than for singly excited states.

Thus, long lifetimes even for free atoms are observed.

Very highly excited He atoms (65 eV, singly or doubly excited) have been shown to have lifetimes of >10 μs.58,59

Note that lifetimes given are for autoionization.

This type of process will not take place easily in RM since the outer electrons are delocalized.

The probable de-excitation process is, instead, the one giving the UIR bands.38

Thus, it is expected that the lifetimes of the relevant states within the RM are much longer.

Large enough densities of the low states may thus exist, giving the observed DIB absorptions.

Correlation with elemental concentrations

There exist several reports about correlations in intensity between the DIBs and atomic lines, primarily due to H, C, Na, K and Ca.3,60,61

Such correlations are certainly expected from the point of view of formation of RM.

As described above, Rydberg states of alkali atoms are easily formed at carbon or metal oxide surfaces.

Excitation energy transfer from alkali atoms has been shown to exist to other atoms, and also to molecules like H2 and N2.

RM clusters can quite easily be formed in the laboratory from alkali atoms and also from small molecules like H2 and N2.26–28,39

Energy transfer is also possible to other molecules even without RM formation, as in the case of transfer to H2O.62

The correlation with hydrogen atom densities is expected since the RM in space is mainly built up by H atoms and H2 molecules.

Other atoms and small molecules are also incorporated into the RM structure.

The transitions discussed in this study all require two interacting ions with excited electrons, i.e. of any kind of atom but H. RM consisting mainly of H and H2 together with a few non-hydrogen atoms is the likely object giving rise to the DIB bands.

The total density of RM is believed to be much larger than the visible hydrogen density, either in the form of atoms or molecules, as discussed in .ref. 39

The Rydberg doublets discussed above are also of interest from the elemental concentration point of view.

In the pairs due to different Rydberg constants R and RH2, the integrated intensities of the broad and sharp bands are often similar, but the sharp bands peak above the broad bands since they would otherwise not be detectable.

Due to the very high hydrogen molecule density relative to heavier atoms and molecules, a lower intensity of the broad bands might be expected.

However, there is reason to believe that heavier atoms and molecules due to their large number of electrons, slow motion and resulting stronger transient multipoles, might take part more efficiently in the transitions described here.

Further, also non-coplanar transitions will contribute intensity to these bands which are generally of the bandhead type, so the intensity relations may not be easy to find from the elemental composition.

Consequences of the proposed model

The model proposed here is a logical continuation of our earlier studies on the origin of the UIR bands,38 on the form of the dark matter in the universe39 and on the origin of the Faraday rotation observation in intergalactic space.40

These proposals fit well the observed data from these very different fields of study.

Thus, we suggest that interstellar and intergalactic space contains large amounts of RM, mainly consisting of H atoms and H2 molecules.

It is now possible to form and study RM in the laboratory in its lowest possible energy state n = 1 (to be submitted).

In this state RM is composed solely of H atoms since even H2+ is too large to be incorporated in the RM structure.

The interpretation of the DIBs as due to transitions in RM has been studied intensively for some time.

The types of transition studied, primarily eqn. (3), probably do not give a complete interpretation of all DIB bands.

Thus, other explanations for some of the intense and relatively sharp bands may be possible.

It is apparent that one can rely safely on a model that agrees with a large number of bands of practically the same intensity, as in the present case.

It is much more uncertain with a model that agrees with the remaining broad bands only, since the information content in the broad band positions is not large and many models may agree reasonably well.

Thus, it is not considered reliable to use the RM model to assign just a few broad or intense bands, or indeed any other model as sometimes done.

What is required in the future is instead calculational procedures able to cope with the transitions in RM more accurately and to predict the remaining sharp bands, including methods giving reliable information about the relative band intensities.

On the other hand, the information found in the shape of the DIB bands may be useful also with simpler models, like the one used here.

The direct consequences for the studies of the ISM may also be important.

Since the RM model does not require any special composition of the matter in the ISM, this does not give any restrictions on the amount of, for example, carbon in the ISM.

On the other hand, a large amount of carbon is needed to exist in graphite particles and PAH molecules in the ISM if the previous interpretation of the various spectral features from interstellar space are correct.

This gives some kind of carbon crisis where the amount required is unreasonably large.63

The RM model for the DIBs does not require a large amount of carbon in the ISM.


The interpretation of the diffuse interstellar bands, as due to electronic transitions in Rydberg Matter (RM), is quite successful.

A large number of bands in a well studied selected spectral region is described by accurate calculations.

The interaction during light absorption of two excited ions in the RM gives one doubly excited Rydberg species and a doubly ionised atom or molecule.

The energy of the final coplanar doubly excited Rydberg state is calculated quasi-classically, giving good agreement with observations, especially for 60 bands in the range 5800–6540 Å.

The intensity distributions of the transitions are relatively smooth with no unexplained missing bands.

The model used does not predict too many bands.

The accuracy of the calculations is high relative to other recent calculations on systems showing some similarity with the RM transitions.