Abstract

Observations of the magnetic field in galaxy clusters by Faraday rotation give new important information. Since the magnetic field is coupled to matter, the Faraday rotation gives direct information on the intracluster matter. It was recently proposed that at least part of the dark matter in the Universe is so-called Rydberg matter. The magnetic field which arises as a result of Rydberg matter in space at low temperature is described. It is found to be of considerable strength, and strongly inhomogeneous, owing to the filamentary structure of Rydberg matter. The very low work function of Rydberg matter means that the electrons in the matter are almost free. The inhomogeneous distributions of magnetic field strength and electron density are calculated to give a Faraday rotation which easily covers the range of observed values.

Introduction

The observations of magnetic fields of considerable strength, deduced to be of the order of >10−10 T (>1 μG) in the medium inside galaxy clusters, are both intriguing and exciting (Clarke, Kronberg & Böhringer 2001). The source of this kind of magnetic field is under investigation at present (Colgate, Li & Pariev 2001). It seems that most interest is directed towards possible mechanisms by which galactic magnetic fields could be drawn out into intergalactic space and amplified to the levels observed, but that no explanation of this phenomenon exists at present.

In this contribution, we would like to draw the attention to a simple and direct solution to this problem, namely the likely existence of a material in the intracluster medium which by itself gives rise to a magnetic field. This material is called Rydberg matter (RM) (Manykin, Ozhovan & Poluéktov 1980, 1981; Holmlid 1998). It has recently been suggested that RM is at least part of the missing dark matter in the Universe (Badiei & Holmlid 2002). The existence of RM in space is inferred from the interpretation of the so-called unidentified infrared bands (UIR) (Tokunaga 1997; Geballe 1997) as caused by de-excitation processes in this material (Holmlid 2000, 2001). Also the so-called diffuse interstellar bands (DIBs) can be calculated from an RM model (Holmlid 2002a). Of course, the magnetic fields which are maintained in the ICM by other astrophysical processes will interact strongly with the normally dark RM. Since RM will also be a source of thermal electrons, it seems likely that the electron density in intergalactic space is much larger than deduced (Clarke et al. 2001) from the hot electron fraction observed by its X-ray or UV emission.

Theory of Rydberg matter

Rydberg matter (RM) is a nearly metallic phase of low density, at the densities of interest in interstellar and intergalactic space built up by planar clusters with a thickness of one atomic layer. Its existence was predicted by Manykin et al. (1980, 1981). RM is a condensed phase containing atoms and small molecules in Rydberg-like states with interparticle distances of the order of μm. The Rydberg states of interest here are the most long-lived ones, the circular Rydberg states with large principal quantum number n and also large angular momentum quantum number l slightly less than n (Gallagher 1994; Stebbings & Dunning 1983). The RM clusters are planar since this is the lowest energy state of the excited matter (Holmlid 1998). This can also be understood since RM is formed by circular thus planar Rydberg states. Calculations show (Badiei & Holmlid 2002) that a potential energy barrier exists for addition of atoms (molecules) to an RM cluster from outside the plane. Thus, the planar form is retained. Quite complete quantum mechanical calculations were performed to predict a range of properties (see especially Manykin, Ozhovan & Poluéktov (1992a,b). The bonding interaction is almost metallic, with the valence electrons delocalized in the conduction band, physically surrounding the core ions, as shown in Fig. 1. The bonding is due to Coulomb forces supplemented by exchange-correlation forces as for most chemical systems. The first macroscopic experimental proof was found 10 yr after the prediction of its existence (Svensson, Holmlid & Lundgren 1991; Svensson & Holmlid 1992), and detailed microscopic proofs concerning the formation and properties of planar clusters of RM were presented in the last few years (Wang & Holmlid 1998, 2000, 2002). RM can be produced in various pressure regimes, and macroscopic amounts of so-called overdense Rydberg matter have been produced at pressures of 1 mbar and at high temperatures in a large number of experiments during the last 10 yr (Svensson & Holmlid 1999). Methods to produce RM in cluster form at low pressures and low temperatures have been developed somewhat later (Wang & Holmlid 1998, 2000). The recent studies of H2 Rydberg state condensation to RM (Wang & Holmlid 2000, 2002) are of direct relevance for the RM processes in interstellar space. The experimental and theoretical background necessary to understand the formation, structure and radiative lifetime of RM in space was recently given by Holmlid (2000) and will not be repeated here. Recently, the direct IR emission from decaying RM has been measured spectroscopically in the laboratory (Holmlid 2002b) in good agreement with the UIR bands (Holmlid 2000).

1

A cluster of Rydberg matter with 19 atoms (or molecules). This type with six-fold symmetry axis is one of the main types found in experiments on Rydberg matter clusters in ultra-high vacuum. The circular Rydberg states are planar and form the planar clusters by delocalization of the valence electrons in the conduction band.

1

A cluster of Rydberg matter with 19 atoms (or molecules). This type with six-fold symmetry axis is one of the main types found in experiments on Rydberg matter clusters in ultra-high vacuum. The circular Rydberg states are planar and form the planar clusters by delocalization of the valence electrons in the conduction band.

However, the behaviour of RM and especially of RM clusters in space has to be described briefly. H atoms are probably the most important constituent of the RM clusters there, but H2 molecules may also be important. It is likely that planar clusters characterized by so-called magic numbers N= 7, 10, 14, 19, 37, 61, 91 … will exist in space, since these are the stable (or more correctly metastable) forms found in ultrahigh vacuum environments (Wang & Holmlid 1998, 2000, 2002, see Fig. 1). The bonding distances between the atoms and molecules in the clusters at an excitation level of n= 80 are around 0.5 μm, and the diameters of the clusters are a few μm. Thus, they are of the same size as that deduced for many grains in space. All the atoms (molecules) in such a cluster normally have the same excitation state n, since this is the lowest energy state with coherent motion of the RM electrons (Holmlid 1998). This final state is reached rapidly since the energy flows easily between the coupled delocalized electrons. The clusters form large clouds in the vacuum chamber in the experiments, and it is shown that they exist in the boundary layer of the sample which emits the primary Rydberg states (Wang, Engvall & Holmlid 1999). A more complete description of the formation processes of RM is given in Holmlid (2000).

The question of the stability in space of the RM and RM clusters is often posed. It should be observed that Rydberg states may be formed by recombination of electrons and ions, and that RM is further formed by the condensation of circular Rydberg states. Thus, one of the origins of RM is energetic ionizing radiation. From the quantum mechanical calculations on the stability of RM (Manykin et al. 1992b) it is possible to extrapolate to the RM excitation level in space. This gives a radiative lifetime for RM of the order of the presently estimated lifetime of the Universe. In the laboratory, the RM clusters formed that are thermally excited still survive intense focused laser light in the form of ns pulses. Only multiphoton processes appear to be important for the fragmentation processes observed using visible light (Wang & Holmlid 1998, 2000). One reason is that the RM electrons are delocalized, which will give collective transient excitations and finally two-electron excitation, not just ionization and bond destabilization through the removal of a single electron. Another argument is that short-wave radiation will not be able to couple to the RM electrons and a core ion simultaneously, since the distance between them is too large. Other arguments based on the small size of the cavities surrounding the core ions in RM have also been used to explain the very small coupling between RM and electromagnetic radiation (Badiei & Holmlid 2002).

Of course, at high enough levels of radiation where multiphoton processes become important the RM is destroyed faster than it is formed, and this phenomenon is observed in the laboratory experiments. Most parts of intergalactic space will probably be much calmer than the laboratory vacuum chamber with a laser intensity of 109 W cm−2 in the pulses, so the stability of RM is probably sufficient for it to survive for a very long time. The contact with ground-state atoms and molecules seems in the laboratory to give incorporation of these ground-state species into the RM clusters after excitation by energy transfer from the Rydberg matter. This is observed as the formation of clusters with ion masses corresponding to mixed clusters. Thus, it is expected that RM is formed and survives for a long time in interstellar space.

Results

Magnetic field strength

The delocalized electrons in the conduction band of an RM cluster move coherently and in one plane. Their orbital motion gives a magnetic field arising from the separate but collinear dipoles formed by the circular motion. The magnetic field outside an isolated RM cluster can be calculated easily. The magnetic dipole of a cluster is  

1
formula
where e is the electron charge, me the electron mass, N the number of Rydberg atoms (molecules) in the cluster, n the principal quantum number, graphic and μB the Bohr magneton. The magnetic field strength B from the magnetic dipole of the cluster varies with the distance r to the cluster as r−3. If an ensemble of clusters is lined up with their magnetic momenta in the same direction, and if they are dispersed homogeneously in space, it is straightforward to determine the average value of B, integrating up to the edge of the cluster only to find the average. One finds  
2
formula
where μ0 is the vacuum permeability, d the average distance between the clusters, and rCL the radius of the cluster. The average density of atoms in space D is related to d through d= (N/D)1/3. Using Equation (2) gives a rather low average magnetic field strength of 3 × 10−9μG, using n= 80–100 (Holmlid 2000) and a relatively low value of D= 107 m−3, which is 10 times larger than the average luminous density in space (Badiei & Holmlid 2002). In this calculation, N = 19 was used for the cluster size. The average distance between the clusters becomes 1.2 cm.

However, this picture with dispersed magnetic momenta is not the most probable for the situation in space, since the magnetic momenta of the clusters will interact and the planar RM clusters will attract each other even at extremely large distances. Owing to this interaction, the clusters will condense to form long columns (physical shape of filaments) where all the clusters have the same direction, much like a (flexible) stack of coins, or similar to a wire coil, where each cluster represents one turn of the coil (see Fig. 2). We have shown (Badiei & Holmlid 2002) that there exists a stabilizing potential energy barrier, which prevents the clusters from approaching each other too closely. Owing to this barrier, the clusters do not disturb each other, avoiding de-excitation processes. Thus, the RM clusters probably exist in the form of long filaments in undisturbed form in space at low temperature. See further in the Discussion.

2

The form of packing of six-fold symmetric RM clusters which is believed to exist in space. The magnetic momenta of the clusters are directed along the filaments, all in the same direction.

2

The form of packing of six-fold symmetric RM clusters which is believed to exist in space. The magnetic momenta of the clusters are directed along the filaments, all in the same direction.

Using this kind of description, the magnetic field inside the filaments is easy to calculate as arising from a circulating current along the circumference of the filament. The field strength is  

3
formula
where re is the electron orbit in each Rydberg state, considering that each Rydberg state in each cluster is part of a separate inner filament in the larger cluster filament, and df is the average intercluster distance in the filament. A reasonable value of re is 0.3–0.5 μm. To relate this distance to the average density D, one can introduce a filling factor η < 1 such that  
4
formula

The magnetic field strength within the filaments is then of the order of 1 μG with D= 107 m−3, a filling factor of 10−10 and n= 80–100. This corresponds to a value of df= 10–20 μm. The orbit diameter of the Rydberg electrons is approximately 0.8 μm.

The electrical resistivity has been measured in several cases with dense and overdense RM (Svensson et al. 1991; Svensson & Holmlid 1992; Holmlid & Manykin 1997) and it is found to be low, of the order of 10−2–10−3Ω m. However, such low values are not expected for the more ordered filamentary structure described here since the electrons have to move from one cluster to the other to flow along the filament. Anyway, with external electric or magnetic fields which for example vary in time, an interaction with current flow along the RM filaments will probably be an important factor. The motion of the filaments owing to such external fields may give both high voltages and currents in the filaments.

In this description, the electron spins are not included at all. Thus, the magnetic field is only assumed to come from the orbital motion in the RM. If and how the spins are coupled in the system is not known from experiments. The large distance between the electrons, of the order of 1 μm, means that a strong interaction between the spins is not likely. Thus, RM is not ferromagnetic. It is also likely that the considerable resistivity related to electron transport between clusters means that the complete cloud of RM at the extremely low densities of interest here is not so metallic in character, even if each cluster still is metal-like. Thus, RM will not behave as a highly conducting material seen in large quantities and at low electric and magnetic field strengths.

Electron densities

One fact which probably is very important for the observed Faraday rotation in intergalactic space is the very small electron binding energy (low work function) of the RM. The work function of RM has been measured in several different experiments (Svensson & Holmlid 1992; Holmlid & Manykin 1997), and this quantity has also been calculated with quantum mechanical (Manykin et al. 1981, 1992a; Svanberg & Holmlid 1994) and classical methods (Holmlid 1998). The experimental studies give work functions of <0.7 eV in hot (so-called overdense) RM, and theory gives a value of <0.1 eV at an excitation level n > 20. At an assumed RM (atom) density of 107 m−3 as used above and a work function of 0.02 eV (200 K) that probably is valid for n= 100, the density of free electrons in equilibrium with RM at 5 K is 4 × 104 m−3 from the Saha equation. Thus the density of free electrons is considerable even at the very low temperatures characteristic for intergalactic space. These electrons exist outside the RM filaments and move in the magnetic field that is created by the RM filaments.

The electrons in the RM itself are delocalized as in a metal and almost free, with a binding energy of the order of 0.02 eV as used above. Their motion is coupled to the magnetic field, and they are likely to interact with the electromagnetic radiation field giving the Faraday rotation effect. With an RM density of 107 m3 the free electron density which is of concern to us for the Faraday rotation observations may approach the atom density. The magnetic field interacting with these electrons has the field strength of 1 μG as calculated from equation (4).

Faraday rotation

To estimate the Faraday rotation from the RM model, we use a few different approaches. The average magnetic field of 2 × 10−9μG calculated above can be combined with the average thermal electron density of 4 × 104 m3 in open space given by the Saha equation from an RM atom density of 107 m3. Integrating over a distance of 104 pc gives a Faraday rotation value of the order of 5 × 10−4 rad mm2 according to the formula used in Clarke et al. (2001). (The basic formulae may be found in Bittencourt 1986.) This is several orders of magnitude smaller than the typical observed rotations. This calculation does not take into account the filamentary structure of RM or that the electrons in the RM are almost free.

Using the average RM density of 10^{7} m3 and approximating the free electron density with the same value of 10^{7} m3 as an average in the entire volume gives a Faraday rotation of 0.15 rad m−2. This value is considerably lower than the ones observed, but the filamentary structure of RM has still not been included.

The RM density in the filaments is of the order of 1017 m3 (filling factor 10−10 and average RM density of 107 m3). Then the corresponding free electron density from the Saha equation becomes 4 × 109 m3. Taking into account the low filling factor and using the internal magnetic field strength in RM of 1 μG gives a Faraday rotation of 3 rad m−2.

Finally, using the electron density in the RM instead of the value from the Saha equation one finds a rotation of the order of 108 rad m−2. This is considered too high since the electrons in RM are still bound and not completely free. However, the fraction of the electrons above the Fermi level is 1.5 × 10−2 at a work function of 0.02 eV and a temperature of 5 K, so a large number of electrons are quite free to move in the RM. Assuming that a fraction of 10−6 of the electrons in RM are more or less free, the Faraday rotation found is approximately 100 rad m−2 which agrees with observations. Thus, the inhomogeneity of the magnetic field and the almost free electron density in RM lead to values of the rotation in agreement with the observed values.

Discussion

Short distance interaction

The potential interaction between the RM clusters when they are at a relatively large distance from each other can be calculated as the electrostatic Coulomb force in the classical limit, in a way similar to the classical calculation for the bonding within a cluster (Holmlid 1998). It is crucial for the description given here that the clusters are stable if they collide: they should of course not de-excite and decompose when they approach each other. The interactions between two coplanar RM clusters in which the electrons move coherently and with the same angular phase has been studied (Badiei & Holmlid 2002). When two such clusters approach each other, a repulsive potential energy exists which will prevent close collisions. In a small range of relative lateral positions of the two clusters, the potential is attractive at short distances. However, there always exists a barrier in the potential energy surface at an intermediate distance. This barrier will prevent close collisions of the clusters. For an excitation level n= 80, the lowest barrier has a height of 42 K for two 19-atom clusters, with a turning distance in the collisions close to 1 μm. The electrostatic barrier calculated in this way is probably only a lower bound to the true quantum mechanical barrier which may be found by more elaborate calculations. This means that RM clusters will be stable during cluster–cluster collisions at low temperature in space.

Rydberg matter lifetime

The radiative lifetime of a Rydberg state is large, increasing with excitation level as n5 for circular states. At n= 100, the lifetime of a circular state is of the order of 20 s, and such states are thus metastable (Beigman & Lebedev 1995). The long lifetime is due to the extremely small overlap between the Rydberg orbitals and the lower states. When RM is formed by attractive interaction and condensation, the system is further stabilized and the radiative lifetime is increased strongly. The mutual attraction and condensation processes of Rydberg states are much faster than any other such processes for example for ground-state H atoms, since the interaction forces and the collision cross sections are much larger for the Rydberg states. The most important long range forces, the dispersion forces, vary almost proportional to the polarizability, which in turn varies roughly as the volume of the state. It is then easy to understand that the force between two H* Rydberg states with n= 100 and a diameter of 1 μm is >108 times larger than for two ground-state H atoms with a diameter of 0.1 nm. Thus, the condensation processes to RM are correspondingly faster than for recombination to H2 molecules. The recombination of ground-state H atoms to H2 molecules is not possible without a third body to remove the large bond energy, because of the short collision time relative to the radiative lifetime of the complex formed. The corresponding process for Rydberg states on to Rydberg clusters is strongly favoured owing to the very large number of degrees of freedom in the cluster and the resulting long collision time, which means that collisions with third bodies and emission processes become more probable.

Manykin et al. calculated the lifetime of RM in different excitation levels, and found it to be long, of the order of 100 yr at an excitation level of n= 16. The main de-excitation processes for Rydberg matter were found to be so-called Auger processes (Manykin et al. 1992b), involving two electrons which simultaneously change their energy and orbital angular momentum. At n= 80–100 which probably is correct for the interstellar and intergalactic space, the lifetime from a simple extrapolation would be extremely large according to this calculation, longer than the lifetime of the Universe. Of course, the background radiation at a few K will interact with RM and possibly shorten the lifetime, an effect which is well documented for isolated Rydberg atoms (Gallagher 1994). Other external interactions like energetic particles and photons will of course disturb the RM stronger. At the same time, such energetic processes will also deposit energy in the RM by recombination which gives highly excited Rydberg states. These states then add on to existing RM clusters.

Magnetic interactions

The clusters in the filaments will probably interact to reinforce the magnetic field within the filaments, but no calculations have been done to incorporate this effect, which is of unknown size. If a magnetic field penetrates the clusters, either in the form of the internal filament field, or in the form of an external magnetic field, a few additional effects will exist. The case with a magnetic field oriented in any other direction than along the RM filaments will not be studied here, since the clusters will align themselves rapidly with the external field direction, and it is likely that also the RM filaments will more or less follow any external magnetic field. Since the magnetic field is always at right angle to the electron orbit motion, the resulting force on the RM electrons counteracts the electrostatic attraction towards the core ion in each Rydberg state. Thus, for the same radius of the electron orbit the velocity in the orbit will decrease, resulting in a slightly lower value of the orbital angular quantum number, and thus also in the principal quantum number for an unchanged orbit. The velocities of the Rydberg electrons are small, decreasing as 1/n, and the Lorentz force caused by the magnetic field will also be small. The magnetic field will thus only have a marginal influence on the behaviour of the RM clusters. For example, at a field strength of 1 μG the magnetic force is only 10−5 of the centrifugal force on the electron. Thus, the RM clusters will not be disturbed by the magnetic field in the RM filaments.

Conclusions

We have shown that the observations of magnetic field strengths of considerable size in galaxy clusters by Faraday rotation observations can be caused by RM formed by hydrogen atoms and molecules in intergalactic space. This material will give rise to an inhomogeneous magnetic field with maximum strength of about 1 μG, but it also produces a large thermal electron density in addition to the large density of delocalized almost free electrons within the RM filaments. The interaction between the radiation and the electrons in the magnetic field can give the observed order of magnitude of the Faraday rotation in the intracluster medium.

Acknowledgments

This study has been supported by the Swedish Research Council for Engineering Sciences and by the Swedish Research Council for Natural Sciences.

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