Interaction of Helium Rydberg State Atoms with Superfluid Helium

The pair potentials between ground state helium and Rydberg He $^*(2s,2p,3s)$ atoms are calculated by the full configuration interaction electronic structure method for both the electronic singlet and the triplet manifolds. The obtained pair potentials are validated against existing experimental molecular and atomic data. Most states show remarkable energy barriers at long distances ( $R > 5$ Å), which can effectively stabilize He $^*$ against the formation of He $_2^*$ at low nuclear kinetic energies. Bosonic density functional theory calculations, based on the calculated pair potential data, indicate that the triplet ground state He $^*$ reside in spherical bubbles in superfluid helium with a barycenter radius of 6.1 Å at the liquid saturated vapor pressure. The pressure dependency of the relative He $^*$ $2s$ $^3S$ $\rightarrow $ $2p$ $^3P$ absorption line blue shift in the liquid was obtained through both the statistical line broadening theory as well as the dynamic adiabatic following method. The pronounced difference between the results from the static and dynamic models is attributed to the dynamic Jahn–Teller effect that takes places in the electronically excited state within the dephasing time of 150 fs. Transient non-thermalized liquid surroundings near He $^*$ may contribute to an artificial reduction in the absorption line blue shift by up to 30 cm $^{-1}$ .     

Experimental Study of the Density and Viscosity of n-Heptane at Temperatures from 298 K to 470 K and Pressure upto 245 MPa

The density and viscosity of $n$ -heptane have been simultaneously measured over the temperature range from 298 K to 470 K and at pressures up to 245 MPa using the hydrostatic weighing and falling-body techniques, respectively. The expanded uncertainty of the density, pressure, temperature, and viscosity measurements at the 95 % confidence level with a coverage factor of $k= 2$ is estimated to be 0.15 % to 0.30 %, 0.05 %, 0.02 K, and 1.5 % to 2.0 % (depending on temperature and pressure ranges), respectively. The measured densities were used to develop a Tait-type equation of state for liquid $n$ -heptane. Theoretically based Arrhenius–Andrade and Vogel–Tamman–Fulcher type equations with pressure-dependent coefficients were used to describe the temperature and pressure dependences of the measured viscosities for liquid $n$ -heptane. The measured values of the density and viscosity were compared in detail with reported data and with the values calculated from a reference EOS and correlation models for the viscosity.     

A New Functional Form for Equations of State for Some Weakly Associating Fluids

A new functional form for equations of state for polar and weakly associating fluids was developed. It was established with a simultaneous optimization algorithm developed previously. As a result, equations of state in terms of the Helmholtz energy as a function of temperature and density were developed for hydrogen chloride (HCl) valid within \(T =\) (155–330) K and pressures up to \(p = 20\) MPa, diethyl ether (DEE) valid within \(T =\) (270–500) K and pressures up to \(p = 40\) MPa, and methyl chloride (R40) valid within \(T =\) (230–630) K and pressures up to \(p =\) 100 MPa. Those equations can be used for the calculation of all thermodynamic properties, including density, internal energy, enthalpy, heat capacity, speed of sound, saturation properties, etc.     

({p, \rho,T, x}) Properties of \text{ CO}_{2}/Propane Binary Mixtures at 280 K to 440 K and 3 MPa to 200 MPa

The (p, \(\rho \) , T, x) properties of binary mixtures of CO \(_{2}\) (volume fraction purity 0.99999) and propane (mole fraction purity 0.9999) ( \(x_{1}\) CO \(_{2}+x_{2}\) propane; \(x_{1} = 0.1744\) , 0.3863, 0.5837, and 0.7732) were measured in the compressed liquid phase using a metal-bellows variable volumometer. Measurements were conducted from 280 K to 440 K and 3 MPa to 200 MPa. The expanded uncertainties ( \(k = 2\) ) were estimated to be temperature, \(<\) 3 mK; pressure, 1.5 kPa ( \(p\le 7\)  MPa), 0.06 % (7 MPa \(< p\le 50\)  MPa), 0.1 % (50 MPa \(< p\le 150\)  MPa), 0.2 % ( \(p> 150\)  MPa); density, 0.10 %; and composition, \(4.4\times 10^{-4}\) . At \(p >100\)  MPa and 280 K or 440 K, the uncertainties in density measurements increase to 0.14 % and 0.22 %, respectively. The data were compared with available equations of state. The excess molar volumes, \(v_\mathrm{m}^\mathrm{E}\) , of the mixtures were calculated and plotted as a function of temperature and pressure.     

Rheology of weakly wetted granular materials: a comparison of experimental and numerical data

Shear cell simulations and experiments of weakly wetted particles (a few volume percent liquid binders) are compared, with the goal to understand their flow rheology. Application examples are cores for metal casting by core shooting made of sand and liquid binding materials. The experiments are carried out with a Couette-like rotating viscometer. The weakly wetted granular materials are made of quartz sand and small amounts of Newtonian liquids. For comparison, experiments on dry sand are also performed with a modified configuration of the viscometer. The numerical model involves spherical, monodisperse particles with contact forces and a simple liquid bridge model for individual capillary bridges between two particles. Different liquid content and properties lead to different flow rheology when measuring the shear stress-strain relations. In the experiments of the weakly wetted granular material, the apparent shear viscosity $\eta _g$ η g scales inversely proportional to the inertial number $I$ I , for all shear rates. On the contrary, in the dry case, an intermediate scaling regime inversely quadratic in $I$ I is observed for moderate shear rates. In the simulations, both scaling regimes are found for dry and wet granular material as well.     

Ion exchange synthesis and thermal characteristics of some \left[ \rm{\mathbf{N}}_{{2222}}^{{+}} \right] based ionic liquids

Eight salts were obtained by reacting tetraethylammonium cation $\big[ {{\rm {\bf N}}_{{\rm {\bf 2222}}}^{{+}} } \big]$ with inorganic anions like BF $_{{4}}^{{-}}$ , NO $_{{3}}^{{-}}$ , NO $_{{2}}^{{{-}}}$ , SCN^???, BrO $_{{3}}^{{-}}$ , IO $_{{3}}^{{-}}$ , PF $_{{6}}^{{-}}$ and HCO $_{{3}}^{{-}}$ using ion exchange method. These ionic liquids (ILs) were characterized using thermal methods, infrared spectroscopy and densitometry. Thermophysical properties such as density, coefficient of volume expansion, heat of fusion, heat capacity and thermal energy storage capacity were determined. Thermal conductivity of the samples was determined both in solid and liquid phases. Owing to high values of thermal energy storage capacity coupled with handsome liquid phase thermal conductivity, ILs under investigation were recommended as materials for thermal energy storage (TES) as well as heat transfer applications.     

Densities, Viscosities, Speeds of Sound, and Refractive Indices of Binary Mixtures of 2-Ethyl-1-hexanol with Benzene and Halobenzenes

Densities, $\rho $ , viscosities, $\eta $ , speeds of sound, $u$ , and refractive indices, $n_\mathrm{D} $ , of binary liquid mixtures of 2-ethyl-1-hexanol with benzene, chlorobenzene, and bromobenzene have been measured over the entire range of composition at 298.15 K, 303.15 K, and 308.15 K and at atmospheric pressure. From the experimental data of the density, speed of sound, viscosity, and refractive index, the values of the excess molar volume, $V^\mathrm{E}$ , isentropic compressibility, ${\kappa _{S}}$ , and deviations in molar refraction, $\Delta R$ , have been calculated. The viscosity data have been correlated using McAllister’s three-body interaction model at different temperatures. The calculated excess and deviation functions have been analyzed in terms of molecular interactions and structural effects.     

Multi-vortex State Induced by Proximity Effects in a Small Superconducting Square

The influence of the different negative values of the deGennes parameter \(b\) in the thermodynamic properties of a superconducting infinitely long prism of square cross section area \(S=9\xi ^{2}(0)\) in the presence of a magnetic field is investigated theoretically by solving numerically the nonlinear Ginzburg-Landau equations; \(\xi (0)\) is the coherent length at zero temperature. We obtain the vorticity, magnetic induction, Cooper pair density, magnetization and phase of the order parameter as functions of the external applied magnetic field and the \(b\) parameter. Our results show that a multi-vortex state appear in the sample choosing a convenient value of \(b<0\) parameter, even for such small system. Also, we study a superconducting parallelepiped of volume \(V=Sd\) by means of true \(3D\) numerical simulations; \(d\) is the height of the parallelepiped. We focused our analysis on the way which the magnetization curves approximate from \(d\) finite to the characteristic curve of \(d\rightarrow \infty \) . This is the case for which the magnetic field and the order parameter are invariant along \(z\) -direction. For a superconductor of size \(S=9\xi ^2(0)\) we find that the limit below which the system should be considered a real three-dimensional sample when is \(d=8\xi \) .     

Stick-slip behaviour of model granular materials in drained triaxial compression

Drained triaxial axisymmetric compression tests are performed on water-saturated short cylindrical samples of nearly monodisperse glass beads, initially assembled in a loose state by a moist tamping technique. Both deviator stress $q$ and volumetric strain $\epsilon _v$ , measured as functions of axial strain $\epsilon _a$ , for different strain rates, are affected by stick-slip events of very large amplitude, while the classical behavior of loose, contractant granular assemblies, approaching the critical state for large $\epsilon _a$ , corresponds to the upper envelop of the stress-strain behaviour. Those events consist in $(i)$ a very fast (slip) part in which a drop of $q$ coincides with a jump of $\epsilon _v$ (contraction), while loss of control of $\epsilon _a$ and generation of pore pressure signal a dynamic collapse of the material structure triggered by an instability; and then $(ii)$ a quasi-static (stick) part in which the sample regains its strength and, over a short strain interval, behaves similarly to a denser system that dilates before reaching its critical state. A unique stress-dilatancy relation applies to all stick-slip events. Apparent internal friction angles and effects of strain rate and confining pressure are discussed, and it is argued that stick-slip instabilities originate in physico-chemical aging phenomena coupled to contact mechanics.     

Viscosity Measurements on Ionic Liquids: A Cautionary Tale

The vibrating-wire viscometer has proven to be an exceedingly effective means of determining the viscosity of liquids over a wide range of temperature and pressure. The instrument has a long history but a variety of technological and theoretical developments over a number of years have improved its precision and most recently have enabled absolute measurements of high accuracy. However, the nature of the electrical measurements required for the technique has inhibited its widespread use for electrically conducting liquids so that there have been only a limited number of measurements. In the particular context of ionic liquids, which have themselves attracted considerable attention, this is unfortunate because it has meant that one primary measurement technique has seldom been employed for studies of their viscosity. In the last 2 years systematic efforts have been made to explore the applicability of the vibrating-wire technique by examining a number of liquids of increasing electrical conductivity. These extensions have been successful. However, in the process we have had cause to review previous studies of the viscosity and density of the same liquids at moderate temperatures and pressures and significant evidence has been accumulated to cause concern about the application of a range of viscometric techniques to these particular fluids. Because the situation is reminiscent of that encountered for a new set of environmentally friendly refrigerants at the end of the last decade, in this paper the experimental methods employed with these liquids have been reviewed which leads to recommendations for the handling of these materials that may have consequences beyond viscometric measurements. In the process new viscosity and density data for 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide [ \({\mathrm{C}}_{6}\) mim][ \({\mathrm{NTf}}_{2}\) ], 1-ethyl-3-methylimidazolium ethyl sulfate [ \({\mathrm{C}}_{2}\) mim][ \({\mathrm{EtSO}}_{4}\) ], and 1-ethyl-3-methylpyridinium ethyl sulfate [ \({\mathrm{C}}_{2}\) mpy][ \({\mathrm{EtSO}}_{4}\) ] have been obtained.     

Electrolytic Conductivity of Four Imidazolium-Based Ionic Liquids

In this article, electrolytic (ionic) conductivity measurements of four ionic liquids (ILs), namely, 1-ethyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl] imide ([C $_{2}$ 2 mim][NTf $_{2}$ 2 ]), 1-ethyl-3-methylimidazolium trifluoromethanesulfonate ([C $_{2}$ 2 mim][OTf]), 1-hexyl-3-methylimidazolium bis[(trifluoromethyl)sulfonyl]imide ([C $_{6}$ 6 mim][NTf $_{2}$ 2 ]), and 1-ethyl-3-methylimidazolium ethyl sulfate ([C $_{2}$ 2 mim][EtSO $_{4}$ 4 ]) (ECOENG212 $^\circledR $ ® ), were performed in a temperature range of (288.15 to 333.15) K. [C $_{6}$ 6 mim][NTf $_{2}$ 2 ] was chosen to be a reference ionic liquid for several properties, including the electrolytic conductivity by the IUPAC Project 2002-005-1-100. For that reason, the measurements performed with that ionic liquid primarily serve the purpose to validate the instrumentation and the experimental procedure used in this work. The measurements were carried out using a complex impedance method, applying a novel electronic device designed and constructed for this purpose. The complete setup includes a Schott Instruments LF 913 T, used as a four-electrode conductivity cell, and a lock-in amplifier. The cell was calibrated using standard reference KCl aqueous solutions. The measurements of the impedance of the conductivity cell were carried out along a range of frequencies from (0.2 to 30) kHz, and the results were extrapolated to infinite frequency, in order to determine the electrolytic conductivity of the liquid samples. The results obtained for the ionic liquid [C $_{6}$ 6 mim][NTf $_{2}$ 2 ] were compared to reference data, and it was estimated that the overall uncertainty of the present results is better than 2 %. All the data obtained were compared with available literature data, and were analyzed and discussed in respect to the effect of temperature, cation alkyl chain length, and anion.     

Simulation of fatigue crack growth with a cyclic cohesive zone model

Fatigue crack growth is simulated for an elastic solid with a cyclic cohesive zone model (CZM). Material degradation and thus separation follows from the current damage state, which represents the amount of maximum transferable traction across the cohesive zone. The traction–separation relation proposed in the cyclic CZM includes non-linear paths for both un- and reloading. This allows a smooth transition from reversible to damaged state. The exponential traction–separation envelope is controlled by two shape parameters. Moreover, a lower bound for damage evolution is introduced by a local damage dependent endurance limit, which enters the damage evolution equation. The cyclic CZM is applied to mode I fatigue crack growth under \(K_{\mathrm{I}}\) -controlled external loading conditions. The influences of the model parameters with respect to static failure load \(K_{\mathrm{0}}\) , threshold load \(\varDelta K_{\mathrm{th}}\) and Paris parameters \(m, C\) are investigated. The study reveals that the proposed endurance limit formulation is well suited to control the ratio \(\varDelta K_{\mathrm{th}}/K_{\mathrm{0}}\) independent of \(m\) and \(C\) . An identification procedure is suggested to identify the cohesive parameters with the help of Wöhler diagrams and fatigue crack growth rate curves.     

Twenty Years of Magnon Bose Condensation and Spin Current Superfluidity in 3He-B

20 years ago a new quantum state of matter was discovered and identified (Borovik-Romanov et al. in JETP Lett. 40:1033, 1984; 45:124, 1987; 47:478, 1988; Fomin in JETP Lett. 40:1037, 1984; Borovik-Romanov et al. in Sov. Phys. JETP 61:1199, 1985; Fomin in Sov. Phys. JETP 61:1207, 1985; Bunkov et al. in JETP Lett. 43:168, 1986). The observed dynamic quantum state of spin precession in superfluid ³He-B bears the properties of spin current superfluidity, Bose condensation of spin waves—magnons, off-diagonal long-range order and related phenomena of quantum coherence.     

High-Frequency Shear Viscosity of Low-Viscosity Liquids

A thickness shear quartz resonator technique is described to measure the shear viscosity of low-viscosity liquids in the frequency range from 6 MHz to 130 MHz. Examples of shear-viscosity spectra in that frequency range are presented to show that various molecular processes are accompanied by shear-viscosity relaxation. Among these processes are conformational variations of alkyl chains, with relaxation times \(\tau _{\eta }\) of about 0.3 ns for \(n\) -pentadecane and \(n\) -hexadecane at 25  \(^{\circ }\) C. These variations can be well represented in terms of a torsional oscillator model. Also featured briefly are shear-viscosity relaxations associated with fluctuations of hydrogen-bonded clusters in alcohols, for which \(\tau _{\eta }\) values between 0.3 ns ( \(n\) -hexanol) and 1.5 ns ( \(n\) -dodecanol) have been found at 25  \(^{\circ }\) C. In addition, the special suitability of high-frequency shear-viscosity spectroscopy to the study of critically demixing mixtures is demonstrated by some illustrative examples. Due to slowing, critical fluctuations do not contribute to the shear viscosity at sufficiently high frequencies of measurements so that the non-critical background viscosity \(\eta _\mathrm{bg}\) of critical systems can be directly determined from high-frequency shear-viscosity spectroscopy. Relaxations in \(\eta _\mathrm{bg}\) appear also in the shear-viscosity spectra with, for example, \(\tau _{\eta }\,\approx \) 2 ns for the critical triethylamine–water binary mixture at temperatures between 10  \(^{\circ }\) C and 18  \(^{\circ }\) C. Such relaxations noticeably influence the relaxation rate of order parameter fluctuations. They may be also the reason for the need of a special mesoscopic viscosity when mutual diffusion coefficients of critical polymer solutions are discussed in terms of mode-coupling theory.     

DEM analysis of the influence of the intermediate stress ratio on the critical-state behaviour of granular materials

The critical-state response of granular assemblies composed of elastic spheres under generalised three-dimensional loading conditions was investigated using the discrete element method (DEM). Simulations were performed with a simplified Hertz–Mindlin contact model using a modified version of the LAMMPS code. Initially isotropic samples were subjected to three-dimensional stress paths controlled by the intermediate stress ratio, \(b=[(\sigma '_{2}-\sigma '_{3})/\) \((\sigma '_{1}-\sigma '_{3})]\) . Three types of simulation were performed: drained (with \(b\) -value specified), constant volume and constant mean effective stress. In contrast to previous DEM observations, the position of the critical state line is shown to depend on \(b\) . The data also show that, upon shearing, the dilatancy post-peak increases with increasing \(b\) , so that at a given mean effective stress, the void ratio at the critical state increases systematically with \(b\) . Four commonly-used three-dimensional failure criteria are shown to give a better match to the simulation data at the critical state than at the peak state. While the void ratio at critical state is shown to vary with \(b\) , the coordination number showed no dependency on \(b\) . The variation in critical state void ratios at the same \(p'\) value is apparently related to the directional fabric anisotropy which is clearly sensitive to \(b\) .     

Containerless Measurements of Density and Viscosity for a Cu_{48}Zr_{52} Liquid

The densities of solid and liquid Cu \(_{48}\) Zr \(_{52}\) and the viscosity of the liquid were measured in a containerless electrostatic levitation system using optical techniques. The measured density of the liquid at the liquidus temperature (1223 K) is (7.02 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) and the density of the solid extrapolated to that temperature is (7.15 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) . The thermal expansion coefficients measured at 1223 K are (6.4 \(\pm \) 0.1) \(\,\times \,10^{-5}\) K \(^{-1}\) in the liquid phase and (3.5 \(\pm \) 0.3) \(\,\times \,10^{-5}\) K \(^{-1}\) in the solid phase. The viscosity of the liquid, measured with the oscillating drop technique, is of the form \(A\exp \left[ \left( {{E}_{0}}+{{E}_{1}}\left( 1/T-1/{{T}_{0}} \right) \right) \times \left( 1/T-1/{{T}_{0}} \right) \right] \) , where \({{T}_{0}}=1223\) K, \(A= (0.0254 \pm 0.0004)\) Pa \(\cdot \) s, \({{E}_{0}}\) =  (8.43 \(\pm \) 0.26) \(\,\times \,10^3\) K and \({{E}_{1}}\) =  (1.7 \(\pm \) 0.2) \(\,\times 10^7\) K \(^{2}\) .     

Nonlinear correction to Darcy’s law for channels with wavy walls

For low Reynolds numbers ${\mathcal{R}}$ , the flow of a viscous fluid through a channel is described by the well-known Darcy’s law which corresponds to a linear relation between the pressure gradient ${\overline{\nabla p}}$ and the average velocity ${\overline{u}}$ . When the channel is not straight and when the Reynolds number is not negligible, additional terms appear in this relation. Some previous authors investigated the first three coefficients in the expansion of ${|\overline{\nabla p}|}$ in the powers of ${\overline{u}}$ and they showed that the coefficient of ${\overline{u}^2}$ vanishes for moderate ${\mathcal{R}}$ . Other authors demonstrated that this coefficient can be non-zero. This question is addressed and solved. It is demonstrated that both cases occur; Forchheimer’s law has a cubic correction for small ${\mathcal{R}}$ and a quadratic one for large ${\mathcal{R}}$ . Two analytical–numerical algorithms are constructed to prove this property. These algorithms are applied to the Navier–Stokes equations in three-dimensional channels enclosed by two wavy walls whose amplitude is proportional to ${b{\varepsilon}}$ , where 2b is the mean clearance of the channels and ${\varepsilon}$ is a small dimensionless parameter. The first algorithm is applied for small ${\mathcal{R}}$ by representing the velocity and the pressure in terms of a double Taylor series in ${\mathcal{R}}$ and ${\varepsilon}$ . The accuracy ${O(\mathcal{R}^2)}$ and ${O(\varepsilon^6)}$ following Padé approximations yield analytical approximate formulae for Forchheimer’s law. The first algorithm is applied to symmetric channels on the theoretical level (all terms on ${\mathcal{R}}$ and ${\varepsilon}$ are taken into account) to show that ${|\overline{\nabla p}|}$ is an odd function of ${\overline{u}}$ . This observation yields, in particular, a cubic correction to Darcy’s law. Numerical examples for non-symmetrical channels yield the same cubic correction. The second algorithm is based on the analytical–numerical solution to the Navier–Stokes equations for arbitrary ${\mathcal{R}}$ up to ${O(\varepsilon^{3})}$ . This algorithm yields, in particular, a quadratic correction to Darcy’s law for higher ${\mathcal{R}}$ .     

Diverging Thermodynamic Derivatives Associated with Heterogeneous Chemical Equilibrium in a Binary Liquid Mixture with a Consolute Point

The solubilities of tin(II) oxide, copper(II) oxide, and cobalt(II) oxide have been determined in the liquid mixture, isobutyric acid + water, along the critical isopleth. When plotted in van’t Hoff form with \(\ln s\) versus \(1/T\) , the solubility measurements, \(s\) , lie on a straight line for values of the temperature, \(T\) , which are sufficiently in excess of the critical solution temperature, \(T_\mathrm{c}.\) In the case of SnO, the dissolution reaction is exothermic, and the slope of the van’t Hoff plot diverges toward positive infinity as \(T\rightarrow T_\mathrm{c} .\) In the case of both CuO and CoO, the dissolution reaction is endothermic, and the slope of the van’t Hoff plot diverges toward negative infinity as \(T\rightarrow T_\mathrm{c} .\) Analysis of these ternary, heterogeneous equilibria using finite dimensional vector space stoichiometry theory shows that each contains two linearly independent components. According to the Gibbs phase rule, two-phase equilibria of this type can be described by two fixed, intensive variables, which are accounted for by the temperature and the pressure, respectively. The Gibbs–Helmholtz equation and the principle of critical-point universality can be combined to predict under conditions of fixed temperature and pressure that when dissolution is exothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward positive infinity in the critical region, while when dissolution is endothermic, \((\partial \ln s/\partial (1/T))\) should diverge toward negative infinity. Our experiments include examples confirming both these predictions.     

Influence of Magnetic Field and LO Phonon Effects on the Spin Polarization State Energy of Strong-Coupling Bipolaron in a Quantum Dot

On the basis of Lee–Low–Pines unitary transformation, the influence of magnetic field and LO phonon effects on the energy of spin polarization states of strong-coupling bipolarons in a quantum dot (QD) is studied by using the variational method of Pekar type. The variations of the ground state energy $E_0$ and the first excited state the energy $E_1$ of bipolarons in a two-dimensional QD with the confinement strength of QDs $\omega _0$ , dielectric constant ratio $\eta $ , electron–phonon coupling strength $\alpha $ and cyclotron resonance frequency of the magnetic field $\omega _{c}$ are derived when the influence of the spin and external magnetic field is taken into account. The results show that both energies of the ground and first excited states ( $E_0$ and $E_1)$ consist of four parts: the single-particle energy of electrons $E_\mathrm{e}$ , Coulomb interaction energy between two electrons $E_\mathrm{c}$ , interaction energy between the electron spin and magnetic field $E_\mathrm{S}$ and interaction energy between the electron and phonon $E_{\mathrm{e-ph}}$ ; the energy level of the first excited state $E_1$ splits into two lines as $E_1^{(1+1)}$ and $E_1^{(1-1)}$ due to the interaction between the single-particle “orbital” motion and magnetic field, and each energy level of the ground and first excited states splits into three “fine structures” caused by the interaction between the electron spin and magnetic field; the value of $E_{\mathrm{e-ph}}$ is always less than zero and its absolute value increases with increasing $\omega _0$ , $\alpha $ and $\omega _c$ ; the effect of the interaction between the electron and phonon is favorable to forming the binding bipolaron, but the existence of the confinement potential and Coulomb repulsive energy between electrons goes against that; the bipolaron with energy $E_1^{(1-1)}$ is easier and more stable in the binding state than that with $E_1^{(1+1)}$ .