On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀( n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^–2/3 , followed byr _s ^/–4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being –0.481915 Rydbergs.     

Effective spin susceptibility, electron mass and Landé factor in two-dimensional (2D) Si inversion layers

We have studied the silicon (Si) band-structure, electron–electron and electron-ionized donor interaction effects on our accurate and approximate results (AcR and ApR) for renormalized effective spin susceptibitity (RESS), electron mass (EEM), Landé factor and spin polarization in the impure 2D Si (electron system), showing that:(i) our ApR, being strongly deviated from our AcR, reproduces approximately all the data obtained recently by Pudalov et al. (Phys. Rev. Lett. 88 (2002) 196404) [in particular, RESS =4.7 at the critical value of Wigner–Seitz radius r_s: r_s=r_c≈8.5 at which occur the “apparent” metal–insulator transition (MIT)] and can also be compared with other ApRs found in the recent literature,(ii) both the RESS and EEM produce physical singularities at the same critical value: r_s=r_c11.05661 (weakly disordered samples) at which occurs the “true” MIT; the existence of such two “apparent and true” critical values in this impure system agrees with a recent discussion by Abrahams et al. (Rev. Mod. Phys. 73 (2001) 251), and(iii) at r_s=r_c=8.5, at which occurs the “apparent” MIT, our AcR for effective spin polarization and the corresponding result, obtained using a disordered Hubbard model and a determinant quantum Monte Carlo method by Denteneer and Scalettar (Phys. Rev. Lett. 90 (2003) 246401), both give the same result: ξ_eff.=ξ_c0.31 at B0.4 T, which is found to be lower than the critical parallel magnetic field for full spin polarization, B_c=1.29 T, supporting thus the existence of such an “apparent” MIT.     

Diffusion of titanium atoms in Ni-Si-B metallic glass

The diffusion ot Ti atoms in the Ni₇₇Si₃B₂₀ metallic glass was studied by the Auger electron spectroscopy. The temperature dependence of the diffusion coefficient obeys the Arrhenius relation with the activation energyE=1·7 eV and the pre-exponential factorD ₀=0·86 × ×10^–4 m² s^–1. Possible mechanisms of the atomic transport in metallic glasses are discussedThe authors wish to thank Dr. P. Duhaj for providing the metallic glass samples.     

High-resolution infrared and millimeterwave spectra of the v3=1 vibrational state of NF3 at 907 cm

The ν₃^±1 perpendicular band of ¹⁴NF₃ ( cm⁻¹) has been studied with a resolution of 2.5 × 10⁻³ cm⁻¹, and 3682 infrared (IR) transitions (J_max=55, K_max=45) have been assigned. These transitions were complemented by 183 millimeterwave (MMW) rotational lines (J_max=25, K_max=19) in the 150–550 GHz region (precision 50–100 kHz). The kl=+1 level reveals a strong A₁/A₂ splitting due to the l(2,2) rotational interaction (q=−4.05 × 10⁻³ cm⁻¹) while the kl=−2 and +4 levels exhibit small A₁/A₂ splittings due to l(2,−4) and l(0,6) rotational interactions. All these splittings were observed by both experimental methods. Assuming the v₃=1 vibrational state as isolated, a Hamiltonian model of interactions in the D reduction, with l(2,−1) rotational interaction (r=−1.96 × 10⁻⁴ cm⁻¹) added, accounted for the observations. A set of 26 molecular constants reproduced the IR observations with σ_IR=0.175 × 10⁻³ cm⁻¹ and the MMW data with σ_MMW=134 kHz. The Q reduction was also performed and found of comparable quality while the QD reduction behaved poorly. This may be explained by a predicted Coriolis interaction between v₃=1 and v₁=1 (A₁, 1032.001 cm⁻¹) which induces a slow convergence of the Hamiltonian in the QD reduction but has no major influence on the other reductions. The experimental equilibrium structure could be calculated as: r_e(N–F)=1.3676 Å and (FNF)=101.84°.     

Analogous study in low magnetic field between Cu–Ge and Cu–Si ferrites

Two series of mixed copper ferrites, Cu_1+x Ge_x Fe_2−2x O₄ and Cu_1+x Si_x Fe_2−2x O₄, have been analogously investigated for x=0.0, 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3. The two systems were prepared using the standard ceramic techniques. X-ray diffraction analysis indicates that both systems formed in a single phase cubic spinel structure. The lattice parameter has a constant value (0.838 nm±0.001) for the two series. The grain diameter was estimated from the scanning electron microscope micrographs for the two series. Some magnetic properties were measured at room temperature. The magnetization M was measured in the range of magnetizing field up to 5500 Am⁻¹. The relative permeability (μ_r) was calculated from the B–H relation. The B–H loops were measured at constant magnetizing current (I=2.5 A which is equivalent to 900 Am⁻¹). Also, the hystersis area and the magnetic parameters B_r, B_s, m_R (B_r/B_s) and apparent energy loss (E) were estimated from the B–H loops; μ_r, B_r, B_s and E are composition dependent.     

Volume and grain boundary diffusion of implanted 113Sn in aluminium

Volume and grain boundary diffusion of ¹¹³Sn in aluminium was investigated with the radiotracer method. The implantation technique was used for tracer deposition to avoid problems of tracer hold-up caused by the oxide layer always present on aluminium. The diffusion penetration was chosen large enough to permit serial sectioning of samples with the aid of a microtome.The temperature dependence of the volume diffusivity was determined as D(T)=4.54×10^–5×exp[–(114.5±1.2)kJmol^–1/RT] m ² s ^–1. This confirms previous measurements from our group which already showed that Sn is the fastest foreign metal diffusor so far investigated in aluminium.Grain boundary diffusion of ¹¹³Sn in Al polycrystals was measured in the type-B kinetic regime. The grain boundary diffusion product P=sD _gb (s=segregation factor, =grain boundary width, D _gb=grain boundary diffusivity) was found to be strongly affected by the impurity content of aluminium. For Al polycrystals of 99.9992% nominal purity we obtained P _5N(T)=1.08×10^–8exp [–(96.9±7.5) kJ mol^–1/RT] m³ s^–1 and for less pure Al polycrystals of 99.99% nominal purity P _4N(T)=3.0×10^–10 exp [–(90.1±4.2) kJ mol^–1/RT] m³ s^–1 was determined. The grain boundary diffusion product in the purer material is more than one order of magnitude higher than in the less pure material. Very likely this is an effect of co-segregation of non-diffusant impurities into the grain boundaries.     

Energieverlust und Lichtausbeute bei axialem Channeling von α-Teilchen in Anthrazen

In an atomic beam experiment we measured the hyperfine structure (hfs) of the metastable states 3d ² 4s ⁴ F _3/2,9/2 of Scandium, which were populated by electron impact. A single mode cw dye laser was used to create a population difference between neighbouring hfs states by optical pumping. Rf transitions between these states, equalizing the population numbers, could then be detected by repeating the optical pumping experiment and looking for the laser induced fluorescent light. The following constants of the magnetic dipole and electric quadrupole interactions have been obtained by this method:A _3/2=–158.513(3)MHzB _3/2=–5.19(2)MHzA _9/2=285.967(9)MHzB _9/2=–15.46(70)MHzFrom these constants and the corresponding values forJ=5/2, 7/2, as measured by [1], the following parameters can be calculated: r ^–3 _d ⁰¹ =0.911(4)a ₀ ^–3 r ^–3 _d ¹² =2.395 (62)a ₀ ^–3 and:=r ^–3 _d ⁰¹ /r ^–3 _d ¹² =0.38(1).     

Shallow acceptor population and free hole concentration in Si: In and Si:Ga with IR-photoexcitation

Hall measurements at low temperaturesT<50 K have been performed on Si:In (N _In10¹⁷ cm^–3) and Si:Ga (N _Ga10¹⁸ cm^–3) with infrared photoexcitation of holes into the valence band. It is shown in quantitative agreement with a theoretical model that the population of shallow acceptors, e.g. B and Al, which are present as impurities in concentrations ofN _B,Al10¹²-10¹⁴ cm^–3 strongly affects the photoexcited hole concentration. Photo-Hall measurements can, therefore, serve as a tool for the determination of low impurity acceptor concentrations in the case of high In- or Ga-doping. Hole capture coefficientsB _In=6×10^–4 (T/K)^–1,8 cm³ s^–1 andB _Ga=2×10^–4 (T/K)^–1 cm³ s^–1 have been determined.     

Influence of magnetic field on cesium thermionic energy converter

This article deals with the calculation of the influence of the magnetic field upon the electric current of a thermionic converter presupposing the approach to conditions in a low-pressure cesium converter. The distribution of the starting velocities of the emitted electrons is considered firstly as independent of the angle from the perpendicular to the emitter plane, and secondly according to the cosine law.The magnetic field effect from the converter current is calculated and compared with the calculations in the papers by Schock [1] and Block [2]; the effect of the external magnetic field is verified by measurements on a solar thermionic converter prototype.Symbols F=I/I ₀ factor of current reduction from magnetic field effect - ¯F value of factorF (when the magnetic field is not constant) - I [A/m²] density of collector current (real current influenced by magnetic field) - I ₀ [A/m²] theoretical density of collector current (in ideal case equals electron emission current) - T _e [°K] electron gas temperature; assumed equal to emitter temperatureT _E [°K] - B[Wb/m²] magnetic induction (field) - D[m] distance from emitter to collector - R[m] radius of electrodes, emitter and collector - r[m] variable radius in the limits 0 toR - V [m/s] random velocity of electron - v _xz [m/s] component of the vectorV inx-z plane - v _m =2kT _E /m most probable velocity in the velocity distribution according to Maxwell and Boltzmann - w-v _xz /v _m relatively expressed electron velocityv _xz - the angle of any vectorV - [m] radius of circular electron path - n [m^–3] number (density) of electrons with certain value of random velocity - n ₀ [m^–3] total electron number (density) - n ₁ [m^–3] number of electrons returned to emitter by means of magnetic field - N ₀ [m^–2s^–1] total flow of thermionic electrons emitted from a unit surface - N ₁ [m^–2s^–1] partial flow of electrons returned to emitter - P=N ₁/N₀ relatively expressed flow of electrons returned to emitter (whenB = const.) - ¯P mean value ofP (whenB const.) - F _cos, ,P _cos, values asF,¯F,P,¯P in case of velocity distribution according to cosine law - m=9·107×10^–31 [gk] electron mass - e=1·60×10^–19 [C] electron charge - k×1·38×10^–23 [J/grad] Boltzmann's constant - ₀ 1·257×10^–6 [H/Vs] permeability of vacuum     

The sub-micrometer thickness n-InSb/i-GaAs epilayers for magnetoresistor applications at room temperatures of operation

Magnetotransport at fields up to 500 mT and LF-noise characteristics are reported for miniature magnetoresistors with ferrite concentrators based on Sn-doped n-InSb/i-GaAs heterostructures grown by MBE. The thickness of the InSb epilayers lie in the range 0.55–1.5 μm giving room temperature mobilities of 2.5–5.5 m² V⁻¹ s⁻¹ with carrier densities of (0.5–1.5)×10¹⁷ cm⁻³. The room temperature magnetoresistance (MR) for our two terminal devices could be as high as 115% at 50 mT which is comparable to the extraordinary MR (ExMR) recently reported in microscopic composite van der Pauw disks four terminal devices [Science 289 (2000) 1530]. In addition, a high signal-to-noise ratio and a good temperature stability of R(B)/R₀=0.5–0.83% K⁻¹ was observed for B<60 mT (below the saturation field B_sat for ferrite). Device resistance stability R₀(T) was equal to 0.27–0.66% K⁻¹ in zero field with a nominal device resistance R₀=197–224 Ω for DC currents in the range I=0.01–1.0 mA. The minimum detectable magnetic field is estimated from the reduced differential MR (∂R/∂B)/R=2000% T⁻¹ at B=31 mT and normalised 1/f current noise power spectral density measured at the same field. The resolution limit B_min=2.6 nT at 10² Hz and B_min=0.82 nT at 10³ Hz. These resolution limits are seven times better than those recently reported for the same material n-InSb/i-GaAs and ferrite fabricated Hall sensors [Magnetotransport and Raman characterization of n-InSb/i-GaAs epilayers, for Hall sensors applications over extremely wide ranges of temperature and magnetic field, Proceedings NGS 10, IPAP Conference Series 2, IPAP, Tokyo, 2001, pp. 151–154].     

Thepin structure employed as current amplifier

Injection of excess carriers into thei region of a forward biasedpin diode diminishes proportionally its resistivity (primary circuit). Resistivity variations in thei region are used to control higher currents and powers in the secondary circuit. This basic idea is developed quantitatively for a simplified symmetrical model of thepin structure in a stationary regime and then generalized for the asymmetrical case. The frequency characteristics of the electronic device are studied. For demonstration of the theoretical results thepin structure in silicon with known parameters is used.Notation 2d [m] length ofi region - D [m² s^–1] ambipolar diffusion constant - e [C] electron charge - E ₂ [Vm^–1] electric field strength iny direction - i ₁ [Am^–2] current density inx direction - i ₂ [Am^–2] current density iny direction - i _m [Am^–2] current density due to recombination of carriers ini region - i _1ef ,i _2ef [Am^–2] effective values of currentsi ₁,i ₂ - i _ns ,i _ps [Am^–2] saturated current densities from the heavily dopedn, p regions - i intrinsic region - I [A] total current throughp-n junction - I ₁ [A] total current in pin diode - I ₂ [A] current in secondary circuit - k[J grad^–1] Boltzmann's constant - L=(D) [m] ambipolar diffusion length of carriers in middle region - n(x) [m^–3] excess electron concentration in middle region - ¯n [m^–3] average value of electron concentration in middle region - n _i [m^–3] intrinsic electron concentration - n _A [m^–3] acceptor concentration inp region - n _D [m^–3] donor concentration inn region - p(x) [m^–3] excess hole concentration in middle region - q [m²] area of electrodes 3 and 4 - Q [C] charge stored ini region - R [m^–3 s^–1] recombination rate - s [m] width of diode - t [m] thickness of diode - T [K] absolute temperature - U [V] voltage acrossp-n junction - U ₁ [V] voltage acrosspin diode - U ₂ [V] voltage across terminals of secondary circuit - U _m [V] voltage drop acrossi region - V _D [V] voltage drop acrossn — i andp — i junctions at zero load - W ₁ [W] power inpin diode circuit - W ₂ [W] power in secondary circuit - x [m] distance from center of diode - coefficient in current amplification factor - [rad] phase shift of diode current with respect to applied voltage - [s] life time of excess carriers ini region - [m² V^–1 s^–1] carrier mobility ini region in the symmetrical model - _n [m² V^–1 s^–1] electron mobility ini region - _p [m² V^–1 s^–1] hole mobility ini region - [^–1 m^–1] conductivity     

Electrical transport, magnetic and thermal properties of icosahedral Al–Pd–Mn quasicrystals

We report measurements of electrical resistivity (ρ), Hall coefficient (R_H), magnetization (M) and specific heat (C_p(T)) of high-quality icosahedral Al_70.4Pd_20.8Mn_8.8 phases with different thermal treatment. An improvement in the quasi-crystallinity upon the annealing treatment caused a drastic increase in ρ up to 7000 μΩ cm accompanied by a very small electronic specific heat coefficient γ. The low temperature ρ(T) data has been analyzed in terms of weak localization and electron–electron interaction effects. The Hall resistivity (ρ_H) is found to be strongly temperature-dependent and varies linearly with the magnetization (M) for the same field and temperature. Magnetization measurement reveals that more conductive samples are more magnetic and vice versa. Magnetic susceptibility (χ) data of all the annealed samples agrees with the Curie–Weiss-like behavior implying the existence of localized moments. The negative Curie–Weiss temperature (θ) indicates strong antiferromagnetic coupling between individual Mn atoms. The magnetic Mn concentration is found to be small, ranging from 1.73×10^-4 for the less magnetic sample studied up to 3×10^-3 for the more magnetic one. The small electronic specific heat coefficient obtained for all the samples suggests a significant reduction in the electronic density of states (DOS) at the Fermi level (E_F) upon thermal annealing treatment.     

Oxide ion conductivity and relaxation in CaREZrNbO7 (RE = La, Nd, Sm, Gd, and Y) system

CaREZrNbO₇ (RE = La, Nd, Sm, Gd and Y) system changed from fluorite (F)-type to pyrochlore (P)-type structure when the ionic radius ratios, r(Ca²⁺–RE³⁺)_av/r(Zr⁴⁺–Nb⁵⁺)_av were larger than 1.34. Thus, the La, Nd, and Sm compounds have a cubic P-type structure and the Gd and Y ones have a defect F-type structure. The electrical conductivity was measured using complex-plane impedance analysis over a wide temperature (300–750 °C) and frequency (1 Hz–1 MHz) ranges. The conductivity relaxation phenomenon was observed in these compounds and the relaxation frequencies were found to show Arrhenius-type behavior and activation energies were in good agreement with those obtained from high temperature conductivity plots. These results support the idea that the relaxation process and the conductivity have the same origin. The ionic conductivity of CaREZrNbO₇ (RE = La, Nd, Sm, Gd and Y) system showed the maximum at the phase boundary between the F-type and P-type phases. On the other hand, the activation energy for the conduction decreased in the F-type phase and increased in the P-type phase with increasing ionic radius ratio. Among the prepared compounds, CaGdZrNbO₇ showed the highest ionic conductivity of 9.47 × 10^− 3 S/cm at 750 °C which was about twice as high as that observed in Gd₂Zr₂O₇ (4.2 × 10^− 3 S/cm at 800 °C). The grain morphology observation by scanning electron microscope (SEM) showed well-sintered grains. AC impedance measurements in various atmospheres further indicated that they are predominantly oxide ion conductors at elevated temperatures (> 700 °C).     

Electrical properties of doped 3-tetradecylpolypyrrole/metal devices

The electrical properties of devices made of doped 3-tetradecylpolypyrrole (PPy-C₁₄) thin films sandwiched between indium-tin-oxyde (ITO) and gold metal electrodes are reported. The current density–voltage (J–V) curves are asymmetric and nonlinear implying a non Ohmic rectifying contact. Using standard thermionic emission theory (Schottky) J–V characteristics were satisfactorily fitted with a saturation current of J₀=1.5×10^-5 A cm^-2, a barrier height of ϕ_b=0.7 eV, and an ideality factor of n=5.3. Characteristics from the plot of J–V versus 1/T show that the activation energy of the thermionic emission process is higher below the glass transition temperature of PPy-C₁₄ (T_g=45 °C) than above, which seems to indicate that the hopping conduction process is enhanced at T>T_g. The carrier concentration has been calculated from capacitance–voltage (C-V) measurements (N=1.9×10¹⁷ cm^-3) allowing estimation of the carrier mobility μ=2.6×10^-2 cm² V^-1 s^-1. PACS 73.61.Ph; 73.40.Sx; 73.30.+y     

Flavor symmetry, K– mixing and new physics effects on CP violation in D and Ds decays

Flavor symmetry and symmetry breaking, K⁰– mixing and possible effects of new physics on CP violation in weak decay modes D^±→K_S,L+X^±, (K_S,L+π⁰)_K^*+X^± (for X=π,ρ,a₁) and D^±_s→K_S,L+X^±_s, (K_S,L+π⁰)_K^*+X^±_s (for X_s=K,K^*) are analyzed. Relations between D^± and D^±_s decay branching ratios are obtained from the ds subgroup of SU(3) and dominant symmetry-breaking mechanisms are investigated. A CP asymmetry of magnitude 3.3×10⁻³ is shown to result in the standard model from K⁰– mixing in the final-state. New physics affecting the doubly Cabibbo-suppressed channels might either cancel this asymmetry or enhance it up to the percent level. A comparison between the CP asymmetries in D^±_(s)→K_SX^±_(s) and D^±_(s)→K_LX^±_(s) can pin down effects of new physics.     

Resonances in at rest

Data on at rest show two resonant processes: (a) f₀(1370)η,f₀(1370)→σσ and ρρ, (b) η(1440)σ, η(1440)→ηπ⁺π⁻. The branching ratio BR[f₀(1370)→ρρ]/BR[f₀(1370)→σσ]=0.98±0.25 in the mass range available here. Using data on , the ratio Γ_4π/Γ_2π5 for f₀(1370). The effects of the strongly s-dependent width of f₀(1370) are discussed in some detail.The η(1440) is observed decaying to ησ and a₀(980)π, with strong destructive interference between them. In its decay to a₀(980)π, a narrow peak appears in the ηπ mass spectrum, but 30–50 MeV above that usually attributed to a₀(980) and significantly above the KK threshold. This effect is explained naturally by a two-step process: η(1440)→K^*(890)K followed by rescattering of the two kaons through a₀(980) to ηπ above the KK threshold.     

Inversion of electron spins in CaO with the reversal of the magnetic field

We have obtained population inversion of a system of polarized (P=80%) electron spins in a solid during a very fast (dB _z dt=4·10⁵ T·s ^–1) reversal of the external magnetic field. The electrons were trapped at oxygen vacancies in CaO single crystals. This method, whch does not exploit any high frequency electromagnetic field, has been for the first time successfully used to achieve an inverted state of electron spins in a solid. The negative temperature of an electron spin ensembleT=–23 mK has been obtained.     

Examination of two new low-lying electronic states of SiS observed in chemiluminescent flame spectra

Diatomic silicon sulfide molecules have been produced in the chemiluminescent reaction of silicon atoms with OCS. Spectra of the resulting flame consist mainly of two new band systems in the region 350–400 and 385–600 nm. These systems have been assigned as b ³Π_r-X ¹Σ⁺ and a ³Σ⁺-X ¹Σ⁺ on the basis of band structure, spin-orbit splitting, molecular constants, and comparison with chemiluminescent spectra of isovalent molecules. Vibrational assignments were made with the help of the isotope effect and vibrational constants were obtained. Rotational structure was observed in some a-X bands and a partial analysis yielded an approximate rotational constant, B 0.247 ± 0.007 cm⁻¹, for the a ³Σ⁺ state. Franck-Condon factors, calculated for the a-X system, are shown to fit the general trend of the intensity distribution. Irregularities in spin-orbit splitting and in relative intensities of the spin-orbit components of the b ³Π-X ¹Σ⁺ system were observed and an attempt is made to explain them in terms of interactions with neighboring states. Addition of active nitrogen to the flame was shown to greatly increase the intensity of the b-X system relative to the a-X system. Constants (in cm⁻¹) obtained for the new state are: a³σ⁺: T_e=24 582.1 ± 1.3, ω_e=503.8±1.0, ω_ex_e=1.86±0.21b³σ⁺: T_e=27 314.5 ± 2.2, ω_e=619.4±2.0, ω_ex_e=5.75±0.52b³σ⁺: T_e=27 407.9 ± 1.1, ω_e=524.3±1.2, ω_ex_e=3.97±0.28     

Conductometric and thermodynamic studies on the ionic association of HCOONH4, phCOONH4, HCOONa and phCOONa in aqueous–organic solvents

The specific conductance of ammonium formate, ammonium benzoate, sodium formate and sodium benzoate in (10%, 20% and 30% (W/W)) methanol–water, ethanol–water and glycerol–water mixtures at different temperatures (293, 298, 303 and 308 K) was measured.The molar conductance (Λ), limiting molar conductance (Λ₀), limiting ionic conductance (λ₀), association constants (K_A), the activation energy of the transport process (E_a), Walden product (Λ₀η₀), hydrodynamic radii (1/r_s⁺ + 1/r_s⁻)^− 1, transfer numbers of the studied ions (t), standard thermodynamic parameters of association (ΔG_A, ΔH_A and ΔS_A) were calculated and discussed.The results show that, the molar conductance and the limiting molar conductance values were decreased as the relative permittivity of the solvent decreased while, the association constant increased. Also the results show that the molar conductance, the limiting molar conductance and the association constant values were increased as the temperature increased indicating that the association process is an endothermic one.     

Ferromagnetism of the electron gas

The ground-state energy of the ferromagnetic electron gas is calculated for the relative polarizationζ=0−1 and the interelectron separationr _s =5−12. The method consists in describing the electron gas approximately by a quadratic boson Hamiltonian, and contains the random-phase approximation as a special case. Numerical studies show that in both the random-phase and the present approximations the paramagnetic state has the lowest energy: the energy increases withζ for all values ofr _s considered. In the present approximation instabilities are found to occur forr _s above a critical value, due to exchange processes of finite momentum transfers. Forζ=0 this critical value ofr _s is 9.4; it decreases with increasingζ. However, the fully-polarized state (ζ=1), which lies above the rest, is always stable. The conclusions are as follows: (1) Forr _s <9.4 the electron gas is paramagnetic. (2) Atr _s =9.4 it goes over to the fully-polarized ferromagnetic state. (3) This phase transition requires an energy absorption of 0.03 rydberg per electron. (4) The fully-polarized state is not obtainable as the limitζ→1.