The configuration-fixed deformation constrained relativistic mean field approach with time-odd component has been applied to investigate the ground state properties of 33Mg with effective interaction PK1.The ground st...The configuration-fixed deformation constrained relativistic mean field approach with time-odd component has been applied to investigate the ground state properties of 33Mg with effective interaction PK1.The ground state of 33Mg has been found to be prolate deformed,β2=0.23,with the odd neutron in 1/2[330] orbital and the energy -251.85 MeV which is close to the data -252.06 MeV.The magnetic moment -0.9134 μN is obtained with the effective electromagnetic current which well reproduces the data -0.7456 μN self-consistently without introducing any parameter.The energy splittings of time reversal conjugate states,the neutron current,the energy contribution from the nuclear magnetic potential,and the effect of core polarization are discussed in detail.展开更多
Einstein’s energy-momentum relationship is a formula that typifies the special theory of relativity (STR). According to the STR, when the velocity of a moving body increases, so does the mass of the body. The STR ass...Einstein’s energy-momentum relationship is a formula that typifies the special theory of relativity (STR). According to the STR, when the velocity of a moving body increases, so does the mass of the body. The STR asserts that the mass of a body depends of the velocity at which the body moves. However, when energy is imparted to a body, this relation holds because kinetic energy increases. When the motion of an electron in an atom is discussed at the level of classical quantum theory, the kinetic energy of the electron is increased due to the emission of energy. At this time, the relativistic energy of the electron decreases, and the mass of the electron also decreases. The STR is not applicable to an electron in an atom. This paper derives an energy-momentum relationship applicable to an electron in an atom. The formula which determines the mass of an electron in an atom is also derived by using that relationship.展开更多
If an electron emits all of its rest mass energy mec2, the relativistic energy of the electron will become zero. According to the special theory of relativity, an electron whose relativistic energy is zero does not ha...If an electron emits all of its rest mass energy mec2, the relativistic energy of the electron will become zero. According to the special theory of relativity, an electron whose relativistic energy is zero does not have photon energy. In this paper, however, an electron is regarded as having photon energy mec2 and negative energy −mec2, even when its relativistic energy is zero. The state where relativistic energy is zero is achieved due to the positive energy and negative energy canceling each other out. Relativistic energy becomes zero for an electron in a hydrogen atom when the principle quantum number n is zero. The author has already pointed out the existence of an energy level with n=0. If this model is used, it is possible for an electron in the state with n=0 to emit additional photons, and transition to negative energy levels. The existence of negative energy specific to the electron has previously been nothing more than a conjecture. However, this paper aims to theoretically show the existence of negative energy based on a discussion using an ellipse. The results show that the electron has latent negative energy.展开更多
In the present study,the newly established preformation formula is applied for the first time to study the kinematics of the cluster emission from various radioactive nuclei,especially those that decay to the double s...In the present study,the newly established preformation formula is applied for the first time to study the kinematics of the cluster emission from various radioactive nuclei,especially those that decay to the double shell closure^(208)Pb nucleus and its neighbors as daughters.The recently proposed universal cluster preformation formula has been established based on the concepts that underscore the influence of mass and charge asymmetry(η_(A)andη_(Z)),cluster mass A_(c),and the Q-value,paving the way to quantify the energy contribution during preformation as well as during the tunneling process separately.The cluster-daughter interaction potential is obtained by folding the relativistic mean-field(RMF)densities with the recently developed microscopic R3Y using the NL 3^(*)and the phenomenological M3Y NN potentials to compare their adaptability.The penetration probabilities are calculated from the WKB approximation.With the inclusion of the new preformation probability P_(0),the predicted half-lives of the R3Y and M3Y interactions are in good agreement with the experimental data.Furthermore,a careful inspection reflects slight differences in the decay half-lives,which arise from their respective barrier properties.The P_(0)for systems with double magic shell closure^(208)Pb daughter are found to be an order of≈10^(2)higher than those with neighboring Pb daughter nuclei.By exploring the contributions of the decay energy,the recoil effect of the daughter nucleus is evaluated,in contrast to several other conjectures.Thus,the centrality of the Q-value in the decay process is demonstrated and redefined within the preformed cluster-decay model.Additionally,we have introduced a simple and intuitive set of criteria that governs the estimation of recoil energy in the cluster radioactivity.展开更多
Evolution of an electrostatic plasma wave driven by a low-density ultra-relativistic electron beam in dense inhomogeneous plasma is considered. In particular, the wavelength variation as observed at fixed locations in...Evolution of an electrostatic plasma wave driven by a low-density ultra-relativistic electron beam in dense inhomogeneous plasma is considered. In particular, the wavelength variation as observed at fixed locations in the plasma is analyzed in terms of the wave characteristics. It is shown that for a negative density gradient, the observed local wavelength decreases monotonically with time, but for a positive density gradient, it first increases and then decreases with time, accompanied by reversal of the wave phase. However, in both cases the local wavelength eventually decreases with time since Landau damping becomes significant as the wavelength becomes of the order of the plasma Debye length. Results from particle-in-cell simulations agree well with theoretical analyses of the wavelength variation.展开更多
In this paper we substantiate a necessity of introduction of a concept the counterpart of rapidity into the framework of relativistic physics. It is shown, formulae for energy and momentum defined via counterpart of r...In this paper we substantiate a necessity of introduction of a concept the counterpart of rapidity into the framework of relativistic physics. It is shown, formulae for energy and momentum defined via counterpart of rapidity are regular near the zero-mass and speed of light states. The representation for the energy-momentum is realized as a mapping from the massless-state onto the massive one which looks like as a 'q'-deformation. Quantization of the energy, momentum and the velocity near the light-speed is presaged. An analogue between the relativistic dynamics and the statistical thermodynamics of a micro-canonical ensemble is brought to light.展开更多
In geostationary orbits and other quasi-vacuum environments,relativistic electron beams are affected by the initial emittance and space charge effects during the propagation process,resulting in beam quality degradati...In geostationary orbits and other quasi-vacuum environments,relativistic electron beams are affected by the initial emittance and space charge effects during the propagation process,resulting in beam quality degradation.Furthermore,axial energy distribution change in the beam and the axial transient electromagnetic effect caused by current changes in the head and tail regions of the beam also cause the beam to expand and affect its quality.In this study,the particle-in-cell method was used to construct a long-range propagation model of a relativistic electron beam in a vacuum environment.By calculating and simulating the axial energy distribution of the beam and the changes in the transient electromagnetic field,the axial effect during the propagation process was analyzed,and the parameter change law of the effective propagation of the beam was explored.This provided a theoretical reference for a more accurate assessment of the beam quality during propagation.展开更多
Density-dependent nuclear symmetry energy is directly related to iso spin asymmetry for finite and infinite nuclear systems.It is critical to determine the coefficients of symmetry energy and their related observables...Density-dependent nuclear symmetry energy is directly related to iso spin asymmetry for finite and infinite nuclear systems.It is critical to determine the coefficients of symmetry energy and their related observables because they hold great importance in different areas of nuclear physics,such as the analysis of the structure of ground state exotic nuclei and neutron star studies.The ground state bulk properties of Scandium(Z=21) and Titanium(Z=22) nuclei are calculated,such as their nuclear binding energy(B.E.),quadrupole deformation(β_2),two-neutron separation energy(S_(2 n)),differential variation in the two-neutron separation energy(dS_(2 n)),and root-mean-square charge radius(r_(ch)).The iso spin properties,namely the coefficient of nuclear symmetry energy and its components,such as the surface and volume symmetry energy of a finite isotopic chain,from the corresponding quantities of infinite nuclear matter,are also estimated.Finally,we correlate the neutron-skin thickness with the coefficient of symmetry energy and the related observables corresponding to the isotopic chains of these nuclei.The coherent density fluctuation model(CDFM) is used to estimate the iso spin-dependent properties of finite nuclei,such as symmetry energy,surface symmetry energy,and volume symmetry energy,from their corresponding component in infinite nuclear matter.The relativistic mean-field(RMF) formalism with non-linear NL3 and relativistic-Hartree-Bogoliubov theory with density-dependent DD-ME2 interaction parameters are employed in the analysis.The weight function |F(x)|^(2) is estimated using the total density of each nucleus,which in turn is used with the nuclear matter quantities to obtain the effective symmetry energy and its components in finite nuclei.We calculate the ground state bulk properties,such as nuclear binding energy,quadrupole deformation,two-neutron separation energy,differential variation in the two-neutron separation energy,and root-mean-square charge radius,for the Sc-and Ti-isotopic chains using the non-l展开更多
We investigate 158A GeV 207Pb projectile fragmentation on Al target using CR-39 nuclear track detectors. A stack containing target-detectors assembly was irradiated at SPS-CERN. After chemical etching, detectors were ...We investigate 158A GeV 207Pb projectile fragmentation on Al target using CR-39 nuclear track detectors. A stack containing target-detectors assembly was irradiated at SPS-CERN. After chemical etching, detectors were scanned using an optical microscope to collect the data in the form of etched cone heights. From the cone height measurements, total and partial charge changing cross sections of 207Pb projectiles on aluminum target are determined. The results are compared with both relevant published measurements available in the literature and model predictions. Odd-even effect in the formation of the fragments of 158A GeV 207Pb pro-jectiles is observed in a consistent and a clear manner over the observed range of fragments (Z=63-81). This is achieved by using optimized etching and track measurement conditions.展开更多
In this paper,we include the density dependence behavior of the symmetry energy in the improved quark mass density dependent （IQMDD） model.Under the mean field approximation,this model is applied to investigate neut...In this paper,we include the density dependence behavior of the symmetry energy in the improved quark mass density dependent （IQMDD） model.Under the mean field approximation,this model is applied to investigate neutron star matter and neutron stars successfully.Effects of the density dependence of the symmetry energy on neutron stars are described.展开更多
This study presents the systematics of energy staggering for magnetic rotational bands with M1 and E2 transition properties,which are strictly consistent with the features of good candidates of magnetic rotational ban...This study presents the systematics of energy staggering for magnetic rotational bands with M1 and E2 transition properties,which are strictly consistent with the features of good candidates of magnetic rotational bands in the A^80,110,130,and 190 mass regions.The regularities exhibited by these bands with respect to the staggering parameter,which increases with increasing spin,are in agreement with the semiclassical description of shears mechanism.Moreover,the abnormal behaviour in the backbend regions or close to band termination has also been discussed.Taking the magnetic dipole bands with same configuration in three N=58 isotones,i.e.,103Rh,105Ag,and 107In,as examples,the transition from chiral to magnetic rotation with the proton number approaching Z=50 is presented.Moreover,the self-consistent tilted axis and principle axis cranking relativistic mean-field theories are applied to investigate the rotational mechanism in the dipole band of 105Ag.展开更多
We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within ...We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz,?where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.展开更多
The prevailing cosmological constant and cold dark matter (ΛCDM) cosmic concordance model accounts for the radial expansion of the universe after the Big Bang. The model appears to be authoritative because it is base...The prevailing cosmological constant and cold dark matter (ΛCDM) cosmic concordance model accounts for the radial expansion of the universe after the Big Bang. The model appears to be authoritative because it is based on the Einstein gravitational field equation. However, a thorough scrutiny of the underlying theory calls into question the suitability of the field equation, which states that the Einstein tensor <strong><em>G</em></strong><span style="white-space:nowrap;"><sub><em><span style="white-space:nowrap;">μv</span></em></sub></span> is a constant multiple of the stress-energy tensor <em> <strong>T</strong></em><span style="white-space:nowrap;"><sub><em><span style="white-space:nowrap;">μv</span></em></sub> </span>when they both are evaluated at the same 4D space-time point: <strong style="white-space:normal;"><em>G</em></strong><sub><em><span style="white-space:nowrap;">μv</span> </em></sub>= 8<span style="white-space:nowrap;">π</span>k<strong style="white-space:normal;"><em>T</em></strong><sub><em><span style="white-space:nowrap;">μv</span></em></sub>, where k is the gravitational constant. Notwithstanding its venerable provenance, this equation is incorrect unless the cosmic pressure is <em>p</em> = 0;but then all that remains of the Einstein equation is the Poisson equation which models the Newtonian gravity field. This shortcoming is not resolved by adding the cosmological constant term to the field equation, <strong style="white-space:normal;"><em>G</em></strong><sub><em><span style="white-space:nowrap;">μv</span> </em></sub>+<span style="white-space:nowrap;">Λ</span> <strong style="white-space:normal;"><em>g</em></strong><sub><em><span style="white-space:nowrap;">μv</span> =<span style="white-space:normal;">8<span style="white-space:nowrap;">π</span></span><span style="white-space:normal;">k</span><strong style="white-space:normal;"><em>T</em></strong><sub style="white-space:normal;"><em><span style="white-space:nowrap;">μv</span></em></sub><span style="white-space:norma展开更多
基金Supported by the Major State Basic Research Development Program (Grant No.2007CB815000)the National Natural Science Foundation of China (Grant Nos.10775004,10221003,10720003,and 10705004)
文摘The configuration-fixed deformation constrained relativistic mean field approach with time-odd component has been applied to investigate the ground state properties of 33Mg with effective interaction PK1.The ground state of 33Mg has been found to be prolate deformed,β2=0.23,with the odd neutron in 1/2[330] orbital and the energy -251.85 MeV which is close to the data -252.06 MeV.The magnetic moment -0.9134 μN is obtained with the effective electromagnetic current which well reproduces the data -0.7456 μN self-consistently without introducing any parameter.The energy splittings of time reversal conjugate states,the neutron current,the energy contribution from the nuclear magnetic potential,and the effect of core polarization are discussed in detail.
文摘Einstein’s energy-momentum relationship is a formula that typifies the special theory of relativity (STR). According to the STR, when the velocity of a moving body increases, so does the mass of the body. The STR asserts that the mass of a body depends of the velocity at which the body moves. However, when energy is imparted to a body, this relation holds because kinetic energy increases. When the motion of an electron in an atom is discussed at the level of classical quantum theory, the kinetic energy of the electron is increased due to the emission of energy. At this time, the relativistic energy of the electron decreases, and the mass of the electron also decreases. The STR is not applicable to an electron in an atom. This paper derives an energy-momentum relationship applicable to an electron in an atom. The formula which determines the mass of an electron in an atom is also derived by using that relationship.
文摘If an electron emits all of its rest mass energy mec2, the relativistic energy of the electron will become zero. According to the special theory of relativity, an electron whose relativistic energy is zero does not have photon energy. In this paper, however, an electron is regarded as having photon energy mec2 and negative energy −mec2, even when its relativistic energy is zero. The state where relativistic energy is zero is achieved due to the positive energy and negative energy canceling each other out. Relativistic energy becomes zero for an electron in a hydrogen atom when the principle quantum number n is zero. The author has already pointed out the existence of an energy level with n=0. If this model is used, it is possible for an electron in the state with n=0 to emit additional photons, and transition to negative energy levels. The existence of negative energy specific to the electron has previously been nothing more than a conjecture. However, this paper aims to theoretically show the existence of negative energy based on a discussion using an ellipse. The results show that the electron has latent negative energy.
基金Supported by the Fundamental Research Grant Scheme(FRGS/1/2019/STG02/UNIMAP/02/2)from the Ministry of Education Malaysia stipulated with the Institute of Engineering Mathematics(IMK)of the Faculty of Applied and Human Sciences UniMAP as the beholder,Science and Engineering Research Board(SERB,CRG/2021/001229)FOSTECT Project(FOSTECT.2019B.04)FAPESP Project(2017/05660-0)
文摘In the present study,the newly established preformation formula is applied for the first time to study the kinematics of the cluster emission from various radioactive nuclei,especially those that decay to the double shell closure^(208)Pb nucleus and its neighbors as daughters.The recently proposed universal cluster preformation formula has been established based on the concepts that underscore the influence of mass and charge asymmetry(η_(A)andη_(Z)),cluster mass A_(c),and the Q-value,paving the way to quantify the energy contribution during preformation as well as during the tunneling process separately.The cluster-daughter interaction potential is obtained by folding the relativistic mean-field(RMF)densities with the recently developed microscopic R3Y using the NL 3^(*)and the phenomenological M3Y NN potentials to compare their adaptability.The penetration probabilities are calculated from the WKB approximation.With the inclusion of the new preformation probability P_(0),the predicted half-lives of the R3Y and M3Y interactions are in good agreement with the experimental data.Furthermore,a careful inspection reflects slight differences in the decay half-lives,which arise from their respective barrier properties.The P_(0)for systems with double magic shell closure^(208)Pb daughter are found to be an order of≈10^(2)higher than those with neighboring Pb daughter nuclei.By exploring the contributions of the decay energy,the recoil effect of the daughter nucleus is evaluated,in contrast to several other conjectures.Thus,the centrality of the Q-value in the decay process is demonstrated and redefined within the preformed cluster-decay model.Additionally,we have introduced a simple and intuitive set of criteria that governs the estimation of recoil energy in the cluster radioactivity.
基金supported by the National Key R&D Program of China (No. 2016YFA0401100)National Natural Science Foundation of China (Nos. 12175154, 11875092, and 12005149)the Natural Science Foundation of Top Talent of Shenzhen Technology University (Nos. 2019010801001 and 2019020801001)。
文摘Evolution of an electrostatic plasma wave driven by a low-density ultra-relativistic electron beam in dense inhomogeneous plasma is considered. In particular, the wavelength variation as observed at fixed locations in the plasma is analyzed in terms of the wave characteristics. It is shown that for a negative density gradient, the observed local wavelength decreases monotonically with time, but for a positive density gradient, it first increases and then decreases with time, accompanied by reversal of the wave phase. However, in both cases the local wavelength eventually decreases with time since Landau damping becomes significant as the wavelength becomes of the order of the plasma Debye length. Results from particle-in-cell simulations agree well with theoretical analyses of the wavelength variation.
文摘In this paper we substantiate a necessity of introduction of a concept the counterpart of rapidity into the framework of relativistic physics. It is shown, formulae for energy and momentum defined via counterpart of rapidity are regular near the zero-mass and speed of light states. The representation for the energy-momentum is realized as a mapping from the massless-state onto the massive one which looks like as a 'q'-deformation. Quantization of the energy, momentum and the velocity near the light-speed is presaged. An analogue between the relativistic dynamics and the statistical thermodynamics of a micro-canonical ensemble is brought to light.
基金National Natural Science Foundation of China(Nos.61372050,U1730247)the HighPower Microwave Key Laboratory Foundation Program(No.6142605200301)。
文摘In geostationary orbits and other quasi-vacuum environments,relativistic electron beams are affected by the initial emittance and space charge effects during the propagation process,resulting in beam quality degradation.Furthermore,axial energy distribution change in the beam and the axial transient electromagnetic effect caused by current changes in the head and tail regions of the beam also cause the beam to expand and affect its quality.In this study,the particle-in-cell method was used to construct a long-range propagation model of a relativistic electron beam in a vacuum environment.By calculating and simulating the axial energy distribution of the beam and the changes in the transient electromagnetic field,the axial effect during the propagation process was analyzed,and the parameter change law of the effective propagation of the beam was explored.This provided a theoretical reference for a more accurate assessment of the beam quality during propagation.
基金Supported by Science and Engineering Research Board (SERB)(CRG/2021/001229)FOSTECT (FOSTECT. 2019B.04)FAPESP (2017/05660-0)。
文摘Density-dependent nuclear symmetry energy is directly related to iso spin asymmetry for finite and infinite nuclear systems.It is critical to determine the coefficients of symmetry energy and their related observables because they hold great importance in different areas of nuclear physics,such as the analysis of the structure of ground state exotic nuclei and neutron star studies.The ground state bulk properties of Scandium(Z=21) and Titanium(Z=22) nuclei are calculated,such as their nuclear binding energy(B.E.),quadrupole deformation(β_2),two-neutron separation energy(S_(2 n)),differential variation in the two-neutron separation energy(dS_(2 n)),and root-mean-square charge radius(r_(ch)).The iso spin properties,namely the coefficient of nuclear symmetry energy and its components,such as the surface and volume symmetry energy of a finite isotopic chain,from the corresponding quantities of infinite nuclear matter,are also estimated.Finally,we correlate the neutron-skin thickness with the coefficient of symmetry energy and the related observables corresponding to the isotopic chains of these nuclei.The coherent density fluctuation model(CDFM) is used to estimate the iso spin-dependent properties of finite nuclei,such as symmetry energy,surface symmetry energy,and volume symmetry energy,from their corresponding component in infinite nuclear matter.The relativistic mean-field(RMF) formalism with non-linear NL3 and relativistic-Hartree-Bogoliubov theory with density-dependent DD-ME2 interaction parameters are employed in the analysis.The weight function |F(x)|^(2) is estimated using the total density of each nucleus,which in turn is used with the nuclear matter quantities to obtain the effective symmetry energy and its components in finite nuclei.We calculate the ground state bulk properties,such as nuclear binding energy,quadrupole deformation,two-neutron separation energy,differential variation in the two-neutron separation energy,and root-mean-square charge radius,for the Sc-and Ti-isotopic chains using the non-l
文摘We investigate 158A GeV 207Pb projectile fragmentation on Al target using CR-39 nuclear track detectors. A stack containing target-detectors assembly was irradiated at SPS-CERN. After chemical etching, detectors were scanned using an optical microscope to collect the data in the form of etched cone heights. From the cone height measurements, total and partial charge changing cross sections of 207Pb projectiles on aluminum target are determined. The results are compared with both relevant published measurements available in the literature and model predictions. Odd-even effect in the formation of the fragments of 158A GeV 207Pb pro-jectiles is observed in a consistent and a clear manner over the observed range of fragments (Z=63-81). This is achieved by using optimized etching and track measurement conditions.
基金Supported by National Natural Science Foundation of China (10535010, 10675090, 10775068, 10735010, 10975072)National Major State Basic Research and Development of China (2007CB815004)+1 种基金CAS Knowledge Innovation Project (KJCX2-SW-N02)Research Fund of Doctoral Point (RFDP) (200700284016)
文摘In this paper,we include the density dependence behavior of the symmetry energy in the improved quark mass density dependent （IQMDD） model.Under the mean field approximation,this model is applied to investigate neutron star matter and neutron stars successfully.Effects of the density dependence of the symmetry energy on neutron stars are described.
基金Supported by National Natural Science Foundation of China(11675063,11205068,11847310,11775098)
文摘This study presents the systematics of energy staggering for magnetic rotational bands with M1 and E2 transition properties,which are strictly consistent with the features of good candidates of magnetic rotational bands in the A^80,110,130,and 190 mass regions.The regularities exhibited by these bands with respect to the staggering parameter,which increases with increasing spin,are in agreement with the semiclassical description of shears mechanism.Moreover,the abnormal behaviour in the backbend regions or close to band termination has also been discussed.Taking the magnetic dipole bands with same configuration in three N=58 isotones,i.e.,103Rh,105Ag,and 107In,as examples,the transition from chiral to magnetic rotation with the proton number approaching Z=50 is presented.Moreover,the self-consistent tilted axis and principle axis cranking relativistic mean-field theories are applied to investigate the rotational mechanism in the dipole band of 105Ag.
文摘We calculate the energy levels of He+ ion placed in a uniform magnetic field directed perpendicular to the direction of its center of mass (CM) velocity vector, correct to relative order . Our calculations are within the frame work of an approximately relativistic theory, correct to relative order , of a two-particle composite system bound by electromagnetic forces, and written in terms of the position, momentum and spin operators of the constituent particles as proposed by Krajcik and Foldy, and also by Close and Osborn. Since the He+ ion has a net electric charge, the total or the CM momentum is not conserved and a neat separation of the CM and the internal motion is not possible. What is new in our approach is that, for the basis states in a first order degenerate perturbation theory to calculate the effects of the external magnetic field, we use the direct product of the coherent state of the Landau Hamiltonian of the He+ ion in a uniform magnetic field and of the simultaneous eigenstate of the internal Hamiltonian h, j2, l2, s2 and jz,?where j, l and s are the internal total, orbital and spin angular moments of the He+ ion. The coherent state is an excellent approximation to the expected classical circular motion of the center of mass (CM) of the He+ ion. In addition to the Z2 a2 corrections to the usual nonrelativistic results, including the small corrections due to the nuclear motion, we also obtain corrections which depend on the kinetic energy (ECM ) of the CM circular motion of the He+ ion, in a nontrivial way. Even though these corrections are proportional to , where M is the mass of the He+ ion, and are small for nonrelativistic CM motion, the results should be verifiable in careful experiments. Our results may also have application in astrophysical observations of the spectral lines of He+ ions in magnetized astrophysical objects.
文摘The prevailing cosmological constant and cold dark matter (ΛCDM) cosmic concordance model accounts for the radial expansion of the universe after the Big Bang. The model appears to be authoritative because it is based on the Einstein gravitational field equation. However, a thorough scrutiny of the underlying theory calls into question the suitability of the field equation, which states that the Einstein tensor <strong><em>G</em></strong><span style="white-space:nowrap;"><sub><em><span style="white-space:nowrap;">μv</span></em></sub></span> is a constant multiple of the stress-energy tensor <em> <strong>T</strong></em><span style="white-space:nowrap;"><sub><em><span style="white-space:nowrap;">μv</span></em></sub> </span>when they both are evaluated at the same 4D space-time point: <strong style="white-space:normal;"><em>G</em></strong><sub><em><span style="white-space:nowrap;">μv</span> </em></sub>= 8<span style="white-space:nowrap;">π</span>k<strong style="white-space:normal;"><em>T</em></strong><sub><em><span style="white-space:nowrap;">μv</span></em></sub>, where k is the gravitational constant. Notwithstanding its venerable provenance, this equation is incorrect unless the cosmic pressure is <em>p</em> = 0;but then all that remains of the Einstein equation is the Poisson equation which models the Newtonian gravity field. This shortcoming is not resolved by adding the cosmological constant term to the field equation, <strong style="white-space:normal;"><em>G</em></strong><sub><em><span style="white-space:nowrap;">μv</span> </em></sub>+<span style="white-space:nowrap;">Λ</span> <strong style="white-space:normal;"><em>g</em></strong><sub><em><span style="white-space:nowrap;">μv</span> =<span style="white-space:normal;">8<span style="white-space:nowrap;">π</span></span><span style="white-space:normal;">k</span><strong style="white-space:normal;"><em>T</em></strong><sub style="white-space:normal;"><em><span style="white-space:nowrap;">μv</span></em></sub><span style="white-space:norma
