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[1] So far, levitation of solid particles in the atmosphere by photophoretic forces has been attributed exclusively to gravito-photophoresis. In this paper attention is drawn to magneto-photophoresis, a motion of ferromagnetic particles related to a magnetic field during their irradiation by light. In regions showing an approximately vertically directed magnetic field, magneto-photophoretic levitation is likewise possible. Laboratory experiments with particles produced by an arc discharge in air between iron electrodes have quantitatively shown magneto-photophoretic levitation. Using these results, we show that magneto-photophoresis can be an essential factor for the formation and persistence of particle layers. Under conditions of illumination by the sun, magneto-photophoretic levitation takes place with particles having sizes around and below 0.1 μm whereas gravito-photophoresis occurs for particle sizes around 1 μm.

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[2] If particles suspended in a gas are heated by absorbed radiation, various types of motion may appear, generally subsumed under the term “photophoresi” [Preining, 1966]. One subgroup comprises motions which are oriented to a static field, respectively called gravito-photophoresis, magneto-photophoresis, and electro-photophoresis. These effects occur if a restoring torque related to the direction of the field acts on a solid particle, subject to a particle-fixed photophoretic force.

[3] Recently the possibility was repeatedly discussed that photophoretic forces acting on sunlit particles in the atmosphere might induce levitation and thus contribute to the formation of aerosol layers up to the mesopause. The interest concentrated on gravito-photophoresis, a type of motions related to the direction of gravity. Rohatschek [1984, 1989, 1996] dealt with simulation experiments in the laboratory and the theory of model particles. Applications to actual cases, among them mesospheric aerosol layers at altitudes of about 80 km, were discussed by Pueschel et al. [2000] and Cheremisin et al. [2005].

[4] In this paper we propose a complementary way of levitation, magneto-photophoresis: Particles containing some ferromagnetic component move on helical paths along magnetic lines of force, both north and south. In the polar regions of the Earth, where the direction of the lines of force is nearly vertical, the lifting of magnetic particles by photophoretic forces may be a quite common occurence.

[5] The particles are disturbed in their orientation to the magnetic field by Brownian rotations. The restoring torque is induced by a permanent magnetic moment. Substantial photophoretic effects are possible with particles of sizes of 100 nm and below. We suppose that spontaneous magnetization plays a dominant role and makes these effects possible even in the range below 10 nm.

[6] Owing to its limitation to spaces with inclinations near a right angle, levitation by magneto-photophoresis offers a natural explanation for layers concentrated at high latitudes, in particular, the noctilucent clouds. The material, particles containing ferromagnetic substances, would be provided by disintegration of iron meteorites and satellites. These particles can act as condensation nuclei for ice [Toon and Farlow, 1981; Rapp and Thomas, 2006], but still magneto-photophoretic levitation is possible.

2. Magneto-Photophoresis

[7] Magneto-photophoresis was defined by Ehrenhaft [1930]. The mechanics of this effect [Rohatschek, 1955] is based on two requirements: a particle-fixed photophoretic force and a restoring torque which tends to align the particle with the direction of a magnetic field. A short explanation of the particle-fixed force is given in the appendix. The physical nature of the particle-fixed force is explained as a result of differences in the thermal accommodation coefficients over the surface, the so-called Δα force [Rohatschek, 1985, 1995].

[8] An unambiguous orientation to a magnetic field requires a permanent magnetic moment. This regular effect only occurs with particles containing some ferromagnetic matter [Tauzin and Lespagnol, 1950]. Even a small amount can be sufficient.

[9] For an illuminated particle in a magnetic field the following forces and torques act on the particle: (1) the particle-fixed photophoretic force and torque, (2) the resistance force and torque, (3) the restoring torque related to the magnetic field, and (4) the force of gravity. Molecular perturbations are disregarded for the moment. The particle quickly approximates a stable state of motion where it rotates around the direction of the magnetic lines of force. The photophoretic force rotates jointly with the particle. From these conditions a helical motion in or against the direction of the field results. Because of the action of the photophoretic torque, the body axis m′ aligned to the magnetic field is generally not coincident with the magnetic moment m (except the for very high field strengths). If we designate the angle between m and m′ by η, then we obtain m′ = m cos η.

[10] With particles having sizes below the order of 1 μm the orientation is essentially disturbed by Brownian rotations. In analogy to the theory of paramagnetism, Langevin's function (x) = coth x − 1/x enters here; it basically describes the influence of temperature on the orientation of magnetic dipoles in a magnetic field. If x ≪ 1, then (x) ≈ x/3; if x ≫ 1, then (x) ≈ 1 − 1/x ≈ 1 [see Rohatschek, 1995].

[11] As a result of the superposition of a helical motion with Brownian disturbances, we obtain for the mean force component in the direction of the magnetic field, _{B}, the expression

where F is the photophoretic force, ɛ is the angle between F and m′, B is the magnetic induction (flux density), = B/μ_{o} is the magnetic field strength, k is the Boltzmann constant, and T_{g} is the temperature of the gas.

[12] Some experimental results relevant to this work were obtained by Steipe [1952] and Preining [1953]. Additional important matter from the doctoral dissertations of both authors was quoted by Rohatschek [1955]. The particles were produced by an arc discharge in air between iron electrodes. They most likely consisted of ferromagnetic γ-Fe_{2}O_{3}, were mostly spherical in shape or clusters of spheres, and had sizes on the order of 0.1 μm. At atmospheric pressure the authors measured the mean velocity of the particles and determined the net magneto-photophoretic velocity v and the settling velocity v_{s}. Since v = M_{B}, where M denotes the mobility, inference of _{B} is possible by equation (1).

[13]Steipe [1952] and Preining [1953] measured the velocity v depending on the induction B. Because of many parameters in equation (1), a great variety of curves can be generated, which is not of interest here. We limit our consideration to those cases that can be described with Langevin's function. Considering only cases for which cos ɛ and m′ are constant, only two parameters are needed. We select v_{∞}, the asymptotic value of v for large inductions, and B_{1}, that induction for which the argument of equals unity (Table 1).

Table 1. Data of Magneto-photophoresis From Preining (Pr) and Steipe (St)^{a}

[14] The settling velocity v_{s} permits the estimation of the particle's Stokes radius a. The values are a ≈ 0.15 μm for experiment Pr 8, 0.2 μm for experiments Pr 7, 23, and 24, and 0.3 μm for experiment Pr 22.

[15] We concentrate on an induction of 0.7 G (gauss) = 0.7 × 10^{−4} T (tesla), which are the magnitude and the vertical component of the magnetic field near the magnetic poles of the Earth. The values for the mean magneto-photophoretic velocity v_{0.7}, taken from the graphs, and the ratio v_{0.7}/v_{∞} are given in Table 1. The latter agree well with (0.7/B_{1}).

[16] The settling velocity v_{s} permits conclusions about the forces. If the mobility is a scalar, then

where G denotes the gravitational (weight) force. As an example, for the particle Pr 24, that ratio amounts to 22.

[17] Because of the relation

it is possible to infer from B_{1} the quantity m′ which is connected to the magnetic moment m.

[18] Let us assume data for a particle similar to Pr 23: B_{1} = 0.1 × 10^{−4} T, a = 2 × 10^{−7} m, and T_{g} = 293 K, with μ_{o} = 4π × 10^{−7} N/A^{2}, and k = 1.38 × 10^{−23} J/K. Then we obtain m′ = 5.08 × 10^{−22} T m^{3} ≤ m. Hence, the magnetic polarization J = m/V (V volume) can be calculated. It results that J ≥ 0.0156 T.

[19]Steipe [1950] investigated magneto-photophoresis as a function of the irradiance. Appreciable velocities were obtained between 25 and 300 W/m^{2} (which was an average value). We believe that the relevant irradiance at the site of the particles was actually 1 order of magnitude greater than that average. At any rate, the domain where magneto-photophoresis takes place would include the solar constant.

[20] To sum up, the laboratory experiments have shown that both the induction equivalent to that in the polar regions and an irradiation equivalent to the solar constant can provide the conditions for levitation by magneto-photophoresis in the atmosphere. This applies at least to atmospheric pressure. Experiments at other pressures do not exist for magneto-photophoresis.

[21] The particles used in the laboratory experiments and showing the effect are in a diameter range down to 300 nm. This limit may be due to experimental conditions (observation in the microscope). An arc discharge produces mainly particles in the tens of nanometer size range [see, e.g., Deng, 1998]. We therefore suppose that particles distinctly smaller than 100 nm can be affected. In crystals the size of the Weiss domains is about 10–100 nm, and by nature the domains are magnetized to full saturation (about 1 T).

3. Magneto-Photophoresis of a Model Particle

[22] We will consider an ideal spherical particle (Figure 1) possessing rotational symmetry with respect to the distribution of thermal accommodation coefficients α and the homogeneous magnetic polarization J as follows: Let the surface be divided into two hemispheres with different accommodation coefficients α_{1} and α_{2}, where α_{1} > α_{2}. If the temperature of that particle is elevated (e.g., due to irradiation by light), then a particle-fixed photophoretic Δα force F, pointing from the side of higher to the side of lower α and acting along the axis, arises.

[23] The mass density is assumed to be homogeneous. Therefore, the center of mass is situated at the geometric center, where also the resistance force R attacks. Consequently, no torque related to gravity and no gravito-photophoresis exist. Furthermore, it is assumed that the material conducts heat so well that any photophoretic force caused by inhomogeneous heating of the surface (ΔT_{s} force), and thus longitudinal photophoresis, are excluded.

[24] The magnetic polarization J produces a magnetic moment m

which is parallel to the axis. For symmetry reasons, m′ is identical with m. In a magnetic field B the particle experiences a torque m × B/μ_{o} which tends to align m parallel to B. In the equilibrium state that torque vanishes. We assume that F is directed opposite to m. This is the basis for levitation at northern latitudes.

[25] In Figure 1 a particle is represented for which disturbances can be neglected. The weight force G, the photophoretic force F = F_{B}, and the resistance force R = −v/M result in a zero vector sum, provided that inertial reactions can be neglected.

[26] In particular, Figure 1 shows the state of a particle in the terrestrial magnetic field in the state of suspension, where the vertical velocity component vanishes. Then

where i denotes the inclination angle. F and G sum up to a force producing a horizontal drift in the plane of the magnetic declination.

[27] Molecular perturbation of the orientation makes the particle-fixed force deviate from the stationary position in a random fashion. As a consequence, merely the average projection _{B} upon the field direction becomes effective. Equation (1) takes the form

[28] The model particle moves on a straight line superimposed with random deviations. The mean velocity v is coincident neither with B nor with G.

[29] It has been shown [Rohatschek, 1995] that the magnitude F of a Δα force is given by

Here, the factor 1/12 holds for diatomic gas and hemispheric distribution of α, is the mean velocity of molecules, p is the pressure, p* is a characteristic pressure inversely proportional to the radius (at p* the corresponding Knudsen number has a value of about 0.5), and H denotes the net energy flux transferred by the molecules.

[30] We will consider two numerical examples for the model particle according to equation (7). The parameters determining F, equation (8), are chosen as follows: Assuming p ≪ p*, _{B} approximates the asymptotic value _{B}^{fm} for the free molecule (fm) limit. We further assume 2(α_{1} − α_{2})/(α_{1} + α_{2}) = 0.2, H = πa^{2}I (where H is equated with the radiation flux hitting an ideal absorber, and the difference of the absorbed and emitted heat radiations is neglected).

[31] For particles used in the laboratory experiments described by Steipe [1952] and Preining [1953], we use the following values as found by those authors: magnetic polarization J = 0.005, 0.01, and 0.02 T; mass density ρ = 2000 kg/m^{3}. Furthermore, B = 0.7 × 10^{−4} T (Earth magnetic field); T = 293 K; hence, = 320 m/s; g = 10 m/s^{2}; and I = 1386 W/m^{2}. In Figure 2_{B}/G is plotted as a function of the particle radius a. The asymptotic behavior of the graphs can readily be understood from the equation _{B}/G = (F/G)(mB/μ_{o}kT_{g}), derived from equation (7). F ∝ a^{2}, G ∝ a^{3}; therefore F/G ∝ a^{−1}. Langevin's function is approximated for small arguments by ∝ a^{3} and for large arguments by = 1. We obtain for small particles _{B}^{fm}/G ∝ a^{2}, and for large particles _{B}^{fm}/G ∝ a^{−1}.

[32] In laboratory arrangements with a vertically directed magnetic field, levitation would be produced if _{B}/G ≥ 1. The region of lifting shown in the diagram agrees well with the experimental results described above. The examples of particles Pr 23 and 24 (Table 1, equation (2)) demonstrate that a force-to-weight ratio even greater than with the supposed parameters is possible.

[33] At least sometimes ferromagnetic particle occur in the mesosphere. Gerding et al. [2003] describes the detection of an “unusual” aerosol layer in the mid-stratosphere. The particles can originate from micrometeors, which pervade the atmosphere unmelted, or from parts of a larger meteor splitting into smaller particles due to mechanical stress or within the trail of large meteors [Megner et al., 2006]. The material could be soot, Al_{2}O_{3}, SiO_{2}, or iron. Only the last substance has ferromagnetic properties and will show magneto-photophoresis. Other magnetic materials which may occur in meteorite debris or smoke could be iron oxides. The magnetization of the material has a big influence on the phoretic force. Single domain iron particles have lengths between <8 and 23 nm; for magnetite the values are 25–50 nm [Dunlop, 1981; Ranna, 2007]. The magnetizations are 2.2 and 0.5 T, respectively [Ranna, 2007]. The upper size limit for the single domains is rather well established; the lower limit is not so well definable and is usually given as less than 10 nm, which also depends on shape. We have assumed a minimum domain diameter for spherical particles of 6 nm. Once the particle is larger than the single domain size, a transition from pseudo single domain to multiple domains occurs, and the magnetization decreases steadily to less than 10% of the single domain value. For magnetizations of 2, 0.5 (single domain), 0.2, 0.05 0.02, 0.01, and 0.005 T and a temperature of 140 K graphs have been obtained, which are shown in Figure 3. The particles are exposed to the solar radiation as well as to the solar radiation diffusely reflected from the Earth (scattering by cloud and aerosol particles, reflections from the ground, sea, snow fields, etc.) This amounts to 30% of the incoming solar light; i.e., the albedo is 0.3. Therefore, the calculations were made for an illumination of 1.3 times the solar constant. Due to the lower temperature, the disorientation is less; thus the relative force becomes bigger compared to Figure 2. For the same reason the maximum reduced force shifts to smaller particles. The magnetizations of 2 and 0.5 T would apply to single domain iron and iron oxide particles, respectively. Olivines (Fe_{x}Mg_{1−x}SiO_{4}), abundant in meteorides and cosmic dust, have similar magnetic properties [Belley et al., 2009]. Obviously their size is limited to the size of the domain, and this range is shown as thick lines in Figure 3; for completeness the continuation is shown in a thin line. Figure 3 clearly shows that magnetic particles will have phoretic lifting force which is larger than gravity in the size range between radii of 3 nm to 3 μm.

[34] In the atmosphere, where the condition for ascension or suspension is (_{B}/G) sin i ≥ 1, a laboratory specimen would likewise show levitation, since F does not decrease with the pressure, and the lower air temperature raises F by and (equations (7) and (8)). Implicitly we assumed only a minor change of H. This holds approximately over the middle atmosphere up to about the mesopause. At the very low pressures near that limit, enhancement of the particle temperature causes appreciable losses in H due to thermal radiation. Finally, an altitude is attained where the lifting photophoretic and the weight forces come into balance and the particles accumulate at this altitude.

[35]Gerding et al. [2003] have reported an unusual occurrence of a mid-stratospheric layer in the Arctic. The particles had sizes around 30 nm and the composition could be soot, oxides, or iron. The authors argue that the particles could be of meteoritic origin, and in that case magnetic properties are to be expected. As is evident from Figure 3, the magneto-photophoretic force greatly exceeds gravity.

4. Ice Particles in Noctilucent Clouds

[36] Les us consider an ice particle with a magnetic core. The supersaturation of water vapor in the mesosphere is not sufficient for homogeneous ice particle formation: a value of >100 would be necessary [Lübken, 1999]. Since this is not the case, the condensation of water vapor needs to take place on condensation nuclei. For the usual supersaturation the size of the nuclei needs to be a few nanometers. Several substances acting as condensation nuclei are under discussion: large proton hydrate clusters, soot particles, sulfuric acid droplets, sodium bicarbonate, sodium hydroxide, and meteoritic smoke (for an overview, see Rapp and Thomas [2006]). The most widely accepted substance is meteoritic smoke. It is formed by glow up of meteors or satellite debris [Strelnikova et al., 2008]. Rapp and Thomas [2006] discuss size distributions of meteor smoke particles obtained by various authors: all show abundant particle numbers below 1 nm but also a small number of particles above 1 or 2 nm. The critical radius for ice particle formation is given as 1.3 nm. An effective radius of meteoritic smoke particles of ∼1.3 nm has been found, e.g., by Fentzke et al. [2009] at 85 km, which is in agreement with previous findings.

[37] As soon as the size of these meteoritic smoke particles exceeds the size for heterogeneous water vapor nucleation, condensation takes place on the seed particles and an ice particle is formed. The larger particles are preferred for condensation since less supersaturation is needed. Obviously also further meteoritic smoke can be incorporated on the particle. Finally, other substances can condense on the particle's surface. Thus a likely particle in the mesosphere will have a core of a few nanometers, which can be a ferromagnetic material, surrounded by ice, containing impurities and other substances condensed on the surface. The particle most likely is not spherical. Owing to the magnetic core the particle can be oriented in the Earth's magnetic field and photophoresis can occur. In order to calculate the relative force as previously, only a few parameters have to be changed in equation (4): (1) the radius of the core is used instead of the particle's radius in order to determine the torque in the magnetic field (we assume the core to be a Weiss domain with saturated magnetization (2 T)); (2) the density of ice is ρ = 500 kg/m^{3} (accounting for hollow spaces and nonsphericities); (3) a particle having a (smaller) magnetic core and a (larger) ice mantle obviously will have other light-absorbing properties as a particle formed by one single magnetic substance, as shown in Figure 3. In order to account for this, the single scattering albedo (ω) of the particles has been calculated. This is the ratio of the light scattered by the particle divided by the light attenuated by the particle. The value (1 − ω) can be interpreted as the absorbed fraction of the light interceped by the particle. Therefore, the values obtained by the modified equation (4) were multiplied by (1 − ω). As an approximation, we have considered spherical particles and used the volume mixing rule to obtain the effective medium refractive index, using optical constants and Mie programs listed by Bohren and Huffman [1983].

[38] Results are shown in Figure 4 for various core diameters. Photophoretic levitation is possible if the core radius is 3 nm or larger, which is a likely size for ice nuclei to start with. For elongated core particles as found, e.g., by Saunders and Plane [2006], even a slightly smaller size of the core particle would give levitation.

[39] It is obvious that only a selection of particles in the mesosphere will qualify for magneto-photophoresis. For example, particles having a silica core will show no lifting. The lifetime of particles is limited by their settling velocity and amounts to a few days. If the photophoretic levitation force is larger than gravity, the particles have a theoretically indefinite lifetime. Thus particles which show levitiation will be more concentrated, even if they are only few in number to begin with.

5. Conclusion

[40] Using data available in the literature, it has been shown that levitation by magneto-photophoresis of certain 100 nm sized iron oxide particles and iron particles tens of nanometers in size is possible in the stratosphere and the mesosphere of polar regions.

[41] We suggest applying photophoretic levitation to noctilucent clouds. It is generally accepted that the particles are composed of a magnetic substance and water ice and are very small (below 100 nm). In our notion ferromagnetism plays a dominant role, in particular, by spontaneous magnetization. A lattice of iron forms small complexes (order of 3–10 nm) forming a single magnetic domain. Such complexes, and not single atoms, combine with bulk ice matter (order of 10–100 nm).

[42] The magnetized part provides the magnetic moment. If that part penetrates the surface of the ice bulk, differences in the accommodation coefficients will arise. In this way the necessary preconditions for magneto-photophoresis can be fulfilled. Favorable circumstances as to the polarity, etc., and adequate irradiation by the sun can make particles of that kind be lifted or held in suspension.

Appendix A:: Particle-Fixed Photophoretic Force

[43] The particle is assumed to have a surface structure such that a variation of the accommodation coefficient over the surface occurs (e.g., by different roughness). A molecule with kinetic energy T_{0} hitting the surface having a temperature T_{w} > T_{0} leaves the surface with a temperature T, with T_{0} < T < T_{w}. The accomodation coefficient is defined by α = (T − T_{0})/(T_{w} − T_{0}) and 0 < α < 1 [Knudsen, 1911]. If the particle has another temperature as the surroundings, which always is the case, the difference in reflection of the gas molecules causes a net force acting on the particle as well as a torque. Its direction depends on the location of the surfaces on the particle with different accommodation coefficients. If the particle changes its orientation, the force does as well. Thus this force is called the particle-fixed force. A typical everyday example for a particle-fixed force is the thrust produced by the jets of an airplane.

α

accommodation coefficient.

B

magnetic flux density.

ɛ

angle between m′ and F.

F

photophoretic force.

F_{B}

force component in direction of magnetic field.

fm

free molecular limit.

G

weight force.

ℋ

magnetic field strength.

i

inclination angle.

J

magnetic polarization.

k

Boltzmann constant.

m′

body axis of particle.

m

magnetic moment.

η

angle between m′ and m.

ℒ

Langevin function.

p

pressure.

p*

characteristic pressure (corresponding to a Knudsen number of 1/2).