Measurements of the scattering of pulsar radio emission

Measurements of the broadening of pulsar pulses by scattering in the interstellar medium are presented for a complete sample of 100 pulsars with Galactic longitudes from 6° to 311° and distances to three kiloparsec. The dependences of the scattering on the dispersion measure (τ _sc(DM) ∝ DM^α), frequency (τ _sc(v) ∝ v ^?γ ), Galactic longitude, and distance to the pulsar are analyzed. The dependence of the scattering on the dispersion measure in the near-solar neighbourhood can be represented by the power law τ _sc(DM) ∝ DM^2.2±0.1). Measurements at the low frequencies 111, 60, and 40 MHz and literature data are used to derive the frequency dependence of the scattering (τ _sc(v) ∝ V ^?γ ) over a wide frequency interval (covering a range of less than 10: 1) with no fewer than five frequencies. The index for the frequency dependence, γ = 4.1 ± 0.3, corresponds to a normal distribution for inhomogeneities in the turbulence in the scattering medium. Based on an analysis of the dependence of the scattering on the distance to the pulsar and on Galactic longitude, on average, the turbulence level C _n ² is the same in all directions and at all distances out to about three kpc, testifying to the statistical homogeneity of the turbulence of the scattering medium in the near-solar region of the Galaxy.     

Synchrotron spectra of short-period pulsars

A model with synchrotron radiation near the light cylinder is proposed to explain the observed spectra of short-period pulsars (P≤0.1 s). These spectra can be described if a power-law energy distribution of the emitting electrons with exponent γ=2–8 is assumed. For most pulsars, the peak frequency ν_m is below 10 MHz. The ν_m(γ) dependence is derived, and shows that the peak frequencies for pulsars with spectral indices α<1.5 may fall in the observable range. In particular, ν_m may be ν_m ～ 100 MHz for PSR J0751 + 1807 and PSR J1640 + 2224. The observed radio spectrum of Geminga (PSR J0633 + 1746) can be described by a synchrotron model with a monoenergetic or Maxwellian distribution of relativistic electrons and a small angle β between the spin axis and magnetic moment (β ～ 10°).     

Deviation of the arrival-time delay of pulses for the Crab pulsar from a quadratic frequency region

Our measurements of the arrival-time delays of radio pulses from the Crab pulsar, PSR B0531+21, at low frequencies 111, 63, and 44 MHz revealed additional delays compared to the usual quadratic frequency relation, Δt(v) ∝ v ^?2. These additional delays are 65 ms between 63 MHz and 111 MHz—i.e., a factor of two longer than the pulsar’s period, i.e., a factor of five longer than the pulsar period—and cannot be explained by the “twisting” of the magnetic-field lines by the rotation of the pulsar. We suggest the model in which a previously unknown high-density plasma layer with a high electron concentration is present along the line of sight in the Crab nebula, causing an additional frequency-dependent delay of the observed radio pulses at low frequencies due to the contribution of the n _e ² v ^?4 term in the dispersion-delay formula. The parameters of this inferred layer have been derived: emission measure EM ? 4 × 10⁶ pc/cm⁶, electron density n _e ? 10⁶ cm^?3, depth along the line of sight d ? 4 × 10^?6 pc, and electron temperature T _e ≥ 2 × 10⁶ K.     

The radio pulsar J0205+6449 in the supernova remnant 3C 58

The detection of pulsed radio emission from the recently discovered X-ray pulsar J0205+6449 in the young supernova remnant 3C 58 is reported together with the results of first studies of this emission. The observations were carried out at 111 and 88 MHz on radio telescopes of the Pushchino Observatory. The pulsar period, 65.68 ms, and period derivative, \(\dot P = 1.9 \times 10^{ - 13} \), have been confirmed. The integrated pulse profile at 111 MHz has been obtained and the flux density and spectral index α=2.8 measured. The pulsar dispersion measure DM=141 pc cm^?3 has been confirmed. This dispersion measure yields a distance to the pulsar of d=6.4 kpc, a factor of two or more greater than the previously favored distance to the supernova remnant 3C 58 (2.6 kpc). The problem of the age and distance of the pulsar-SNR system is discussed. If the age of the pulsar J0205+6449 is equal to that of the SNR (820 years), this pulsar is the youngest known radio pulsar. The synchrotron mechanism for the radio and X-ray emission is proposed to explain the lower radio and X-ray luminosity of this new pulsar compared to the Crab pulsar, which is similar to it in many ways. Optical emission with luminosity L_opt=10³¹ erg/s and gamma-ray emission with L_γ=7×10³⁵ erg/s are predicted, and the steep radio spectrum (α≈3) can be explained.     

Strmömgren zones of type II supernovae

The emission measures EM in the directions of supernova remnants and pulsars are considered as functions of their ages t. The resulting plot has a well-defined lower boundary, which can be approximated by the expression EM_min∝1/t. The quantity EM_min increases with decreasing age t and does not level off or reach a maximum until t?500 yr. It is concluded that the bulk of the radiative energy that goes into ionizing and heating the interstellar gas is released at early stages of the supernova remnant’s evolution. We suggest that most of the kinetic energy of the supernova shell is converted into thermal energy and radiated at remnant ages t<100 yr, when the supernova shell, which is expanding at an enormous speed (about 10⁴ km/s), overtakes the shell produced by the presupernova in the supergiant stage. We have estimated the ionization energy E?10⁵¹ erg, diameter L?60 pc, and electron density N_e?7 cm^?3 of the HII regions around the supernovae (the supernova Strömgren zones). A list of objects that can be reliably identified as Strömgren zones of type II supernovae is presented. The plot of pulsar pulse broadening τ as a function of the pulsar age t also has a well-defined lower boundary, for which τ∝t^?2 when t≥1000 yr. This suggests that turbulence develops during the first thousand years after the supernova outburst. It is also concluded that turbulence plays an important role in the formation and evolution of the Strömgren zones of type II supernovae.     

Empirical law for fault-creep events

Fault-creep events measured on the San Andreas and related faults near Hollister, California, can be described by a rheological model consisting of a spring, power-law dashpotand sliding block connected in series. An empirical creep-event law, derived from many creep-event records analyzed within the constraints of the model, provides a remarkably simple and accurate representation of creep-event behavior. The empirical creep law is expressed by the equation: D(t)= D_f [1?1/{ct(n?1)D_f^n?1+1}/^(n?1)] where D is the value of displacement at time t following the onset of an event, D_f is the final equilibrium value of the event displacementand C is a proportionality constant. This discovery should help determine whether the time—displacement character of creep events is controlled by the material properties of fault gouge, or by other parameters.     

Integrated radio luminosities of pulsars

The integrated radio luminosities of 311 long-period (P > 0.1 s) and 27 short-period (P < 0.1 s) pulsars have been calculated using a new compilation of radio spectra. The luminosities are in the range 10²⁷ ? 10³⁰ erg/s for 88% of the long-period pulsars and 10²⁸ ? 10³¹ erg/s for 88% of the short-period pulsars. We find a high correlation between the luminosity L and the estimate L ₁ = S ₄₀₀ d ² from the catalog of Taylor et al. The factor η for the transformation of the rotational energy of the neutron star into radio emission increases-decreases with increasing period for long-period and short-period pulsars. The mean value of η is ?3.73 for the long-period and ?4.85 for short-period pulsars. No dependence was found between L and the pulsar’s kinematic age t _k = |z|/〈v _z〉, where |z| and 〈v _z〉 = 300 km/s are the pulsars’ height above the plane of the Galaxy and mean velocity. A dependence of L on the rate of rotational energy losses ? was found for both groups of pulsars. It is shown that L ∝ ? ^1/3 for the entire sample. The pulsar luminosity function is constructed, and the total number and birth rate of pulsars in the Galaxy are calculated.     

The decay of triple systems

Numerical simulations have been carried out in the general three-body problem with equal masses with zero initial velocities, to investigate the distribution of the decay times T based on a representative sample of initial conditions. The distribution has a power-law character on long time scales, f(T) ∝ T ^?α , with α = 1.74. Over small times T < 30T _cr (T _cr is the mean crossing time for a component of the triple system), a series of local maxima separated by about 1.0T _cr is observed in the decay-time distribution. These local peaks correspond to zones of decay after one or a few triple encounters. Figures showing the arrangement of these zones in the domain of the initial conditions are presented.     

Thermo-elastic behaviour of Be2BO3OH (hambergite) up to 7 GPa and 1,100 K

The thermo-elastic behaviour of Be₂BO₃(OH)_0.96F_0.04 (i.e. natural hambergite, Z = 8, a = 9.7564(1), b = 12.1980(2), c = 4.4300(1) Å, V = 527.21(1) ų, space group Pbca) has been investigated up to 7 GPa (at 298 K) and up to 1,100 K (at 0.0001 GPa) by means of in situ single-crystal X-ray diffraction and synchrotron powder diffraction, respectively. No phase transition or anomalous elastic behaviour has been observed within the pressure range investigated. P?V data fitted to a third-order Birch–Murnaghan equation of state give: V ₀ = 528.89(4) ų, K _T0 = 67.0(4) GPa and K′ = 5.4(1). The evolution of the lattice parameters with pressure is significantly anisotropic, being: K _T0(a):K _T0(b):K _T0(c) = 1:1.13:3.67. The high-temperature experiment shows evidence of structure breakdown at T > 973 K, with a significant increase in the full-width-at-half-maximum of all the Bragg peaks and an anomalous increase in the background of the diffraction pattern. The diffraction pattern was indexable up to 1,098 K. No new crystalline phase was observed up to 1,270 K. The diffraction data collected at room-T after the high-temperature experiment showed that the crystallinity was irreversibly compromised. The evolution of axial and volume thermal expansion coefficient, α, with T was described by the polynomial function: α(T) = α ₀ + α ₁ T ^?1/2. The refined parameters for Be₂BO₃(OH)_0.96F_0.04 are: α ₀ = 7.1(1) × 10^?5 K^?1 and α ₁ = ?8.9(2) × 10^?4 K ^?1/2 for the unit-cell volume, α ₀(a) = 1.52(9) × 10^?5 K^?1 and α ₁(a) = ?1.4(2) × 10^?4 K ^?1/2 for the a-axis, α ₀(b) = 4.4(1) × 10^?5 K^?1 and α ₁(b) = ?5.9(3) × 10^?4 K ^?1/2 for the b-axis, α ₀(c) = 1.07(8) × 10^?5 K^?1 and α ₁(c) = ?1.5(2) × 10^?4 K ^?1/2 for the c-axis. The thermo-elastic anisotropy can be described, at a first approximation, by α ₀(a):α ₀(b):α ₀(c) = 1.42:4.11:1. The main deformation mechanisms in response to the applied temperature, based on Rietveld structure refinement, are discussed.     

The radio sources CTA 21 and OF+247: The hot spots of radio galaxies

The physical conditions in the radio sources CTA 21 and OF+247 are studied assuming that the low-frequency spectral turnovers are due to synchrotron self-absorption. The physical parameters of the radio sources are estimated using a technique based on a nonuniform synchrotron source model. It is shown that the magnetic-field distributions in the dominant compact components of these radio sources are strongly inhomogeneous. The magnetic fields at the center of the sources are B ～ 10^?1 G, and the fields are two to three orders of magnitude weaker at the periphery. The magnetic field averaged over the compact component is B ～ 10^?3 G, and the density of relativistic electrons is n _e ～ 10^?3 cm^?3. Assuming that there is equipartition of the energies of the magnetic field and relativistic particles, averaged over the source, 〈E _H 〉 = 〈E _e 〉 ～ 10^?7–10^?6 erg cm^?3. The energy density of the magnetic field exceeds that of the relativistic electrons at the centers of the radio sources. The derived parameters of CTA 21 and OF+247 are close to those of the hot spots in the radio galaxy Cygnus A. On this basis, it is suggested that CTA 21 and OF+247 are radio galaxies at an early stage of their evolution, when the hot spots (dominant compact radio components) have appeared, and the radio lobes (weak extended components) are still being formed.     

Gamma-ray bursts as a result of the interaction of a shock from a supernova and a neutron-star companion

A supernova explosion in a close binary system in which one of the components is a compact magnetized object (neutron star or white dwarf) can form a narrow “tail” with length l _t ～10⁹ cm, width h _t ～10⁸ cm, and magnetic field B _t ～10⁶, due to the resulting shock wave flowing around the magnetosphere of the compact object. The energy released by the reconnection of magnetic field lines in this tail can accelerate electrons to relativistic speeds (γ≈10⁴), creating the conditions required for powerful synchrotron radiation at energies from hundreds of keV to several MeV, i.e., for a gamma-ray burst (GRB). The duration of this radiation will depend on the power of the shock that forms during the supernova. If the shock is not sufficiently powerful to tear off the magnetosphere tail from the compact object, the duration of the GRB will not exceed l _t /V _A ≤1 s, and the conditions necessary for an “afterglow” at softer energies will not arise. If the shock is more powerful, the tail can be torn from the magnetosphere, forming a narrow ejection, which is perceived in its relativistic motion toward the observer(Γ～10⁴) as an afterglow whose duration grows from tens of seconds at gamma-ray energies to tens of days in the optical. This may explain why afterglows are observed only in association with long GRBs (T ₉₀>10 s). Very short GRBs (T ₉₀<0.1 s) may be local, i.e., low-power, phenomena occurring in close pairs containing compact, magnetized objects, in which there is again an interaction between the magnetosphere of the compact object and a shock wave, but the shock is initiated by a flare on the companion, which is a red-dwarf cataclysmic variable, rather than by a supernova.     

Scaling properties of corner frequencies of Kamchatka earthquakes

Scaling properties of earthquake populations bear the major information on the physics of the source process of an earthquake. To determine scaling properties, source spectra of more than 400 earthquakes of Kamchatka were determined in a frequency range 0.1–30 Hz using materials of digital registration of PET station, and characteristic frequencies of earthquakes were estimated. The range of magnitudes is 4–6.5, the range of distances is 80–220 km. To enable reduction of a spectrum to the source, attenuation properties of the medium around PET were determined beforehand. It is revealed that source spectra show several corner (characteristic) frequencies: f _c1, f _c2 and f _c3; where the spectral trend changes: from f ⁰ to f ^?1, from f ^?1 to f ^?2, and from f ^?2 to f ^?3, respectively. Although in some cases f _c1 ≈ f _c2 in agreement with the usual ω^?2 spectral model, the main part of spectra has more complicated character. For a large part of the studied earthquakes a source-controlled upper cutoff of acceleration spectrum, or corner frequency f _c3, is observed. This is an important fact, as the existence of f _c3 (source-controlled f _max) is not recognized in the bulk of the seismological literature. For f _c1, the observed scaling agrees with the usual hypothesis of similarity of the earthquake sources of different size (magnitude); it is close to f _c1 ～ M ₀ ^?1/3 , where M ₀ is seismic moment. For f _c2, scaling is close to f _c2 ～ M ₀ ^?0.17 ～ f _c1 ^0.5 , that indicates an expressed violation of similarity. For f _c3, scaling is close to f _c2 ～ M ₀ ^?0.08 ～ f _c1 ^0.25 , so that similarity is broken even sharper in this case. Hypotheses about possible causes of the observed scaling are discussed.     

The pulsar J1852+0040 in Kes 79 and the nature of central compact objects in supernova remnants

A possible model for the pulsar PSR J1852+0040 associated with the supernova remnant Kes 79 and detected in place of a central compact object in this remnant is discussed. The main observational properties of the pulsar can be understood as consequences of its weak surface magnetic field (B _s < 3 × 10¹¹ G) and short rotational period (P ～ 0.1 s). Its X-ray emission is thermal, and is generated in a small region near the surface of the neutron star due to cooling of the surface as the surface accretes matter from a relict disk surrounding the pulsar. The radio emission is generated in the outer layers of the pulsar magnetosphere by the synchrotron (cyclotron) mechanism. The optical luminosity of J1852+0040 is estimated to be L _opt < 10²⁸ erg/s. If the spectral features in another central compact object, 1E 1207.4+5209, are interpreted as electron cyclotron lines, this provides evidence for a weak surface magnetic field for this neutron star as well (B < 6 × 10¹⁰ G). The hypothesis that all central compact objects have weak surface fields makes it possible to explain the number of detected central compact objects, the absence of pulsar-wind nebulae associated with these objects, and the fact that no pulsar has yet been detected at the position of SN 1987a. We suggest that, after the supernova remnant has dissipated, the central compact object becomes a weak X-ray source (XDINS), whose weak emission is also due to the weakness of its magnetic field.     

Experimental investigation of shear strength of sands with inherent fabric anisotropy

Loading direction-dependent strength of sand has been traditionally characterized in the principal stress space as a direct extension of the Mohr–Coulomb criterion. A recent study found that it is more appropriate to define anisotropic strength of sand on failure/shear planes, but this proposition has only been demonstrated with discrete element method (DEM) simulations. The present study experimentally investigates anisotropic shear strength of sands in this new framework. Three granular materials with distinct grain characteristics ranging from smooth and rounded particles to flaky and angular particles are tested with the bedding plane inclination angle ψ _b varying over the full range of 0°–180°. The main objective is to study how the peak friction angle ? _p of sand is affected by the ψ _b angle and how the ψ _b–? _p relationship evolves with the change of characteristics of constituent sand particles. We find that the general trend of ψ _b–? _p curves for real sands resembles what was predicted by DEM in a previous study, whereas rich anisotropic strength behavior is revealed by the laboratory data. The effects of normal stress and initial density, as well as shear dilation behavior at different shear directions, are also studied.     

In situ observation of crystal growth in a basalt melt and the development of crystal size distribution in igneous rocks

The speciation of CO₂ in dacite, phonolite, basaltic andesite, and alkali silicate melt was studied by synchrotron infrared spectroscopy in diamond anvil cells to 1,000 °C and more than 200 kbar. Upon compression to 110 kbar at room temperature, a conversion of molecular CO₂ into a metastable carbonate species was observed for dacite and phonolite glass. Upon heating under high pressure, molecular CO₂ re-appeared. Infrared extinction coefficients of both carbonate and molecular CO₂ decrease with temperature. This effect can be quantitatively modeled as the result of a reduced occupancy of the vibrational ground state. In alkali silicate (NBO/t = 0.98) and basaltic andesite (NBO/t = 0.42) melt, only carbonate was detected up to the highest temperatures studied. For dacite (NBO/t = 0.09) and phonolite melts (NBO/t = 0.14), the equilibrium CO₂ + O^2? = CO₃ ^2? in the melt shifts toward CO₂ with increasing temperature, with ln K = ?4.57 (±1.68) + 5.05 (±1.44) 10³ T ^?1 for dacite melt (ΔH = ?42 kJ mol^?1) and ln K = ?6.13 (±2.41) + 7.82 (±2.41) 10³ T ^?1 for phonolite melt (ΔH = ?65 kJ mol^?1), where K is the molar ratio of carbonate over molecular CO₂ and T is temperature in Kelvin. Together with published data from annealing experiments, these results suggest that ΔS and ΔH are linear functions of NBO/t. Based on this relationship, a general model for CO₂ speciation in silicate melts is developed, with ln K = a + b/T, where T is temperature in Kelvin and a = ?2.69 ? 21.38 (NBO/t), b = 1,480 + 38,810 (NBO/t). The model shows that at temperatures around 1,500 °C, even depolymerized melts such as basalt contain appreciable amounts of molecular CO₂, and therefore, the diffusion coefficient of CO₂ is only slightly dependent on composition at such high temperatures. However, at temperatures close to 1,000 °C, the model predicts a much stronger dependence of CO₂ solubility and speciation on melt composition, in accordance with available solubility data.     

Physical properties of molecular clouds as functions of their Galactocentric distance from CS and C<Superscript>34</Superscript>S observations

Twenty-eight CS molecular clouds toward HII regions with Galactocentric distances from ～ 4 to 20 kpc have been studied based on observations obtained in the J=2→1 lines of CS and C³⁴S on the 20-meter radio telescope of the Onsala Space Observatory (Sweden) in March 2001. All 28 clouds have been mapped with an angular resolution of ～40″. The peak intensity in the C³⁴S line has been measured for 20 objects. An LTE analysis has been performed and the parameters of the molecular cloud cores derived. The core sizes are d_A=0.3–4.8 pc, with a median value of ～1.6 pc. The mean hydrogen densities in the cloud cores are n_H2=3.5×10²–3.7 × 10⁴ cm^?3, with a median value of ～7.2×10³ cm^?3. The value of n_H2 ends to decrease with increasing Galactocentric distance of the cloud. The masses of most clouds are 10²?6×10³M_⊙, with the most probable value being M_CS～10³M_⊙. The data follow the dependence M_CS ∝d _A ^(2.4–3.2) . As a rule, the cloud masses are lower than the virial masses for M_CS<10³M_⊙.     

Determination of quantitative cation distribution in orthopyroxenes from electronic absorption spectra

Electronic and Mössbauer absorption spectra and electron microprobe data are correlated for iron-bearing orthopyroxenes. The correlation provides a means of quantitatively determining the distribution of Fe²⁺ between the M(1) and M(2) sites of orthopyroxene crystals from electronic spectra and electron microprobe analysis. The electronic spectra are used to analyze the changes in the Fe²⁺ distribution produced during heating experiments and confirm earlier results from Mössbauer spectra. Two components of the spin-allowed transition of Fe²⁺ in the M(1) site are identified at about 13,000 cm^?1 and 8,500 cm^?1 in γ. Molar absorptivity (?) values for all spin-allowed Fe²⁺ absorption bands in the near-infrared region are determined. The M(2) Fe²⁺ band at ～5,000 cm^?1 in β is the analytically most useful for site occupancy determinations. It remains linear with concentration (?=9.65) over the entire compositional range. The band at ～10,500 cm^?1 in α is the most sensitive to M(2) Fe²⁺ concentration (?=40.8), but deviates from linearity at high iron concentrations. The origins of spin-forbidden transitions in the visible region are examined.     

High-pressure polymorphism and structural transitions of norsethite,BaMg(CO3)2

In situ high-pressure investigations on norsethite, BaMg(CO₃)₂, have been performed in sequence of diamond-anvil cell experiments by means of single-crystal X-ray and synchrotron diffraction and Raman spectroscopy. Isothermal hydrostatic compression at room temperature yields a high-pressure phase transition at P _c ≈ 2.32 ± 0.04 GPa, which is weakly first order in character and reveals significant elastic softening of the high-pressure form of norsethite. X-ray structure determination reveals C2/c symmetry (Z = 4; a = 8.6522(14) Å, b = 4.9774(13) Å, c = 11.1542(9) Å, β = 104.928(8)°, V = 464.20(12) Å³ at 3.00 GPa), and the structure refinement (R ₁ = 0.0763) confirms a distorted, but topologically similar crystal structure of the so-called γ-norsethite, with Ba in 12-fold and Mg in octahedral coordination. The CO₃ groups were found to get tilted off the ab-plane direction by ~16.5°. Positional shifts, in particular of the Ba atoms and the three crystallographically independent oxygen sites, give a higher flexibility for atomic displacements, from which both the relatively higher compressibility and the remarkable softening originate. The corresponding bulk moduli are K ₀ = 66.2 ± 2.3 GPa and dK/dP = 2.0 ± 1.8 for α-norsethite and K ₀ = 41.9 ± 0.4 GPa and dK/dP = 6.1 ± 0.3 for γ-norsethite, displaying a pronounced directional anisotropy (α: β _a ^?1  = 444(53) GPa, β _c ^?1  = 76(2) GPa; γ: β _a ^?1  = 5.1(1.3) × 10³ GPa, β _b ^?1  = 193(6) GPa β _c ^?1  = 53.4(0.4) GPa). High-pressure Raman spectra show a significant splitting of several modes, which were used to identify the transformation in high-pressure high-temperature experiments in the range up to 4 GPa and 542 K. Based on the experimental series of data points determined by XRD and Raman measurements, the phase boundary of the α-to-γ-transition was determined with a Clausius–Clapeyron slope of 9.8(7) × 10^?3 GPa K^?1. An in situ measurement of the X-ray intensities was taken at 1.5 GPa and 411 K in order to identify the nature of the structural variation on increased temperatures corresponding to the previously reported transformation from α- to β-norsethite at 343 K and 1 bar. The investigations revealed, in contrast to all X-ray diffraction data recorded at 298 K, the disappearance of the superstructure reflections and the observed reflection conditions confirm the anticipated \(R\bar{3}m\) space-group symmetry. The same superstructure reflections, which disappear as temperature increases, were found to gain in intensity due to the positional shift of the Ba atoms in the γ-phase.     

Electronic absorption spectra of Cr3+ ions in natural clinopyroxenes

The polarized (E‖a′, E‖b and E‖c) electronic absorption spectra of five natural chromium-containing clinopyroxenes with compositions close to chromdiopside, omphacite, ureyite-jadeite (12.8% Cr₂O₃), jadeite, and spodumene (hiddenite) were studied. The polarization dependence of the intensities of the Cr³⁺ bands in the clinopyroxene spectra cannot be explained by the selection rules for the point groups C ₂ or C _2v but can be accounted for satisfactorily with the help of the higher order pseudosymmetry model, i.e. with selection rules for the point symmetry group C _3v. The trigonal axis of the pseudosymmetry crystal field forms an angle of 20.5° with the crystallographic direction c in the (010) plane. D _q increases from diopside (1542 cm^?1) through omphacite (1552 cm^?1), jadeite (1574 cm^?1) to spodumene (1592 cm^?1). The parameter B which is a measure of covalency for Cr³⁺-O bonds at M1 sites in clinopyroxene depends on the Cr³⁺ concentration and the cations at M2 sites.