Electron Spin Resonance Linewidth Research Articles

Electron Spin Resonance in Quasi-One-Dimensional Quantum Antiferromagnets: Relevance of Weak Interchain Interactions

We discuss universal features on the electron spin resonance (ESR) of a temperature-induced Tomonaga-Luttinger liquid phase in a wide class of weakly coupled $S=1/2$ antiferromagnetic spin chains such as spin ladders, spin tubes and three-dimensionally coupled spin chains. We show that the ESR linewidth of various coupled chains increases with lowering temperature while the linewidth of a single spin chain is typically proportional to temperature. This broadening with lowering temperature is attributed to anisotropic interchain interactions and has been indeed observed in several kinds of three-dimensional (3D) magnets of weakly coupled spin chains above the 3D ordering temperature. We demonstrate that our theory can account for anomalous behaviors of the linewidths in an $S=1/2$ four-leg spin tube compound Cu$_2$Cl$_4 \cdot $H$_8$C$_4$SO$_2$ (abbreviated to Sul-Cu$_2$Cl$_4$) and a three-dimensionally coupled $S=1/2$ spin chain compound CuCl$_2\cdot 2$NC$_5$H$_5$.

Oxygen vacancy related defect dipoles in CaCu3Ti4O12: Detected by electron paramagnetic resonance spectroscopy

Oxygen vacancy associate defects were studied by electron paramagnetic resonance (EPR) for the perovskite oxide CaCu3Ti4O12 (CCTO), which have a colossal dielectric constant. It is found that the EPR line width of the Cu2+–host signal of CCTO correlates with its permittivity. A new signal was found in the second differential of the Cu2+–host signal, which we think is associated with a specific copper–oxygen vacancy defect. This signal shows a negative g factor shift with temperature, suggesting a contribution to the conduction. Assuming solid state reactions between the various defects and using the mass action law, we offer a more specific relation between permittivity, the content of oxygen vacancy and the new signal, which might be useful for an indirect but fast oxygen vacancy content determination.

Gadolinium-Loaded Viral Capsids as Magnetic Resonance Imaging Contrast Agents.

Polymeric nanohybrid P22 virus capsids were used as templates for high density Gd3+ loading to explore magnetic field-dependent (0.5-7.0 T) proton relaxivity. The field-dependence of relaxivity by the spatially constrained Gd3+ in the capsids was similar when either the loading of the capsids or the concentration of capsids was varied. The ionic longitudinal relaxivity, r1, decreased from 25-32 mM-1 s-1 at 0.5 T to 6-10 mM-1 s-1 at 7 T. The ionic transverse relaxivity, r2, increased from 28-37 mM-1 s-1 at 0.5 T to 39-50 mM-1 s-1 at 7 T. The r2/r1 ratio increased linearly with increasing magnetic field from about 1 at 0.5 T, which is typical of T1 contrast agents, to 5-8 at 7 T, which is approaching the ratios for T2 contrast agents. Increases in electron paramagnetic resonance line widths at 80 and 150 K and higher microwave powers required for signal saturation indicate enhanced Gd3+ electron spin relaxation rates for the Gd3+-loaded capsids than for low concentration Gd3+. The largest r2/r1 at 7 T was for the highest cage loading, which suggests that Gd3+-Gd3+ interactions within the capsid enhance r2 more than r1.

Electron spin resonance study on charge ordering and spin ordering in nanocrystalline Bi0.2Ca0.8MnO3

Bi-based manganites Bi0.2Ca0.8MnO3 samples with different paiticle sizes were prepared by the sol-gel technique. The effect of paiticle size on the charge ordering (CO) and spin correlations of Bi0.2Ca0.8MnO3 was investigated by electron spin resonance (ESR). The variation in ESR intensity with temperature shows that the long-range CO transition is suppressed by the size reduction, and completely disappears as the paiticle size is reduced to 40 nm. In the paramagnetic (PM) region, the ESR intensity is fitted by the Arrhnius formula. The result shows that the activation energy is significantly enhanced with decreasing of paiticle size, especially in the 40 nm sample, indicating the enhancement of ferromagnetic (FM) correlations. However, the temperature dependence of ESR line width displays a typical CO characteristics for all samples. Thus, it is suggested that there is the short-range CO state in the 40 nm sample, while the long-range CO transition is completely suppressed. It is found that the onset temperatures of CO states are almost the same in all samples, indicating that the strength of CO correlations is not influenced by size reduction in this compound. A positive Curie-Weiss (CW) temperature is obtained from the line width in high-temperature PM regime, which confirms the existence of FM correlations in this system. Moreover, the value of CW temperature shows a significant decrease with particle size reduction, which indicates that the FM interactions can be weakened by size reduction in this system. Based on the research of ESR intensity and line width, it is concluded that the suppressed CO cannot be attributed to the enhancement of double-exchange FM interactions in Bi0.2Ca0.8MnO3 nanoparticles. To explain these behaviors, a core-shell model based on surface effect is proposed. In nanosized CO manganites, the disordered surface spins destroy the collinear antiferromagneitc (AFM) configuration, and favor FM surface layer coupled with the inner AFM core. With the reduction of paiticle size, the weakening of long-range AFM CO is more significant than that of short-range FM ordering due to the increase of surface spin disorders. With the reduction of paiticle size, the FM ordering will gradually dominate in the competition between FM ordering and AFM ordering, which results in the disappearance of CO transition peak in the ESR intensity curve.

High Frequency Electron Spin Resonance in Mn1-xFexSi

High frequency (60 GHz) electron spin resonance (ESR) has been studied for the quantum critical system Mn1−xFexSi (0 < x < 0.24) at temperatures up to 50 K by using the original experimental technique. The analysis of ESR line shape allowed detecting the full set of spectroscopic parameters (oscillating magnetization, line width and g factor). The violation of classical Korringa-type relaxation and scaling behavior of the ESR line width was found to a result from iron doping. Non-Fermi-liquid effects in the temperature dependence of the ESR line width observed at quantum critical points x* = 0.11 and xc = 0.24 are shown to be described quantitatively in the theory by Wölfle and Abrahams.

Electron Paramagnetic Resonance from a Single Biomolecule

Electron paramagnetic resonance (EPR) is an established and powerful tool for studying atomic-scale biomolecular structure and dynamics. Yet it requires a homogenous sample size of ∼10∧15 spin labeled biomolecules. Single molecule measurements provide improved insights into heterogeneous behaviors that can be masked by ensemble measurements and are often essential for illuminating the molecular mechanisms behind the function of a biomolecule. We report EPR measurements of single labeled biomolecule demonstrating the merging of these two powerful techniques. We selectively labelled individual double-stranded DNA molecules with nanodiamonds containing nitrogen-vacancy (NV) centers, and optically detected the paramagnetic resonance of the single NV nanodiamond probe. Analysis of the spectrum reveals that the characteristic time scale for reorientation of the labeled molecule relative to the applied magnetic field is slow compared to the inverse of the EPR linewidth. This demonstration of EPR spectroscopic determination of the dynamics of an individual labeled biomolecular construct provides the foundation for single molecule magnetic resonance studies of complex biomolecular systems.

Paramagnetic relaxation in anisotropic materials in zero and weak constant fields

Paramagnetic relaxation in strongly anisotropic materials is analytically investigated in zero and weak constant magnetic fields. The objectives of the microscopic analytical investigation are (i) the weak-field electron paramagnetic resonance (EPR) linewidth and (ii) the electron spin relaxation rates given by a calorimetric Gorter type experiment in the zero constant field at the arbitrary low-frequency field directions, respectively, to the sample crystallographic axes. The EPR linewidth is calculated under the suggestion of its spin-phonon nature at the one-phonon mechanism of the spin-lattice relaxation in the case of the strong isotropic exchange interaction for the arbitrary direction Z of the constant magnetic field. The EPR linewidth is presented as the half sum of the zero-field relaxation rates, measured by the Gorter experiment with the low-frequency field oriented along the X, Y axes. With the help of the macroscopic consideration, it is shown that the zero-field relaxation rates describe the relaxation of the X and Y magnetization components in a zero or weak constant magnetic field. The relaxation rates of the magnetizations created along a,b,c crystallographic axes by a low-frequency field in a Gorter type experiment follow the obtained expressions in the particular cases and are in the experimentally confirmed relations with the EPR linewidth.

Deformation Defects Supporting Quantum Readout of 29 Si Nuclear Spins in Si: P Deformed Crystals

High temperature (950 °C) plastic deformation of the 29Si enriched silicon crystals generates ˜10 15 cm−3 paramagnetic defects of the new type. The multiline spectrum of electron paramagnetic resonance (EPR) of the defects corresponds to S = 1 electronic spin of the clusters containing oxygen. Strong anisotropy and large linewidth of the EPR spectra (˜1 kOe) were observed. The Pake doublet split by spin-spin nuclear dipole interaction was analyzed in the nuclear magnetic resonance (NMR) spectra of the crystals. Contribution of the electron-nuclear interaction to the EPR linewidth is negligibly small, while the NMR linewidth is controlled by the dipole-dipole nuclear relaxation.

Coexistence of localized and collective magnetism in the coupled-spin-tetrahedra systemCu4Te5O12Cl4

We report high-field magnetization, electron spin resonance (ESR), and Raman scattering measurements of the coupled spin-tetrahedra system ${\mathrm{Cu}}_{4}{\mathrm{Te}}_{5}{\mathrm{O}}_{12}{\mathrm{Cl}}_{4}$ with magnetic ordering at ${T}_{N}=13.6$ K. We find thermodynamic and spectroscopic signatures for the concomitant occurrence of localized and collective magnetism. Magnetization measurements up to 60 T exhibit a spin-flop transition at ${\ensuremath{\mu}}_{0}{H}_{\mathrm{SF}}=16$ T only for $H\ensuremath{\parallel}c$ as well as periodic magnetization steps at ${\ensuremath{\mu}}_{0}H=16.5,\phantom{\rule{0.28em}{0ex}}24.8,\phantom{\rule{0.28em}{0ex}}33.8,\phantom{\rule{0.28em}{0ex}}42.3,\phantom{\rule{0.16em}{0ex}}\text{and}\phantom{\rule{0.16em}{0ex}}49.7$ T, which are independent of the crystallographic orientations. For $T&gt;{T}_{N}$, the temperature dependence of ESR linewidth is described by a critical power law, $\ensuremath{\Delta}{B}_{pp}(T)\ensuremath{\propto}{(T\ensuremath{-}{T}_{N})}^{\ensuremath{-}0.56\ifmmode\pm\else\textpm\fi{}0.02}$. For $T&lt;{T}_{N}$, an antiferromagnetic resonance mode is observed for $H\ensuremath{\parallel}c$, and its linewidth is given by $\ensuremath{\Delta}{B}_{pp}(T)\ensuremath{\propto}{T}^{3.13\ifmmode\pm\else\textpm\fi{}0.04}$, being close to ${T}^{4}$ expected for a classical magnet. Raman spectra show three one-magnon-like excitations superimposed on a broad two-magnon continuum. While the two higher frequency modes show an intensity variation in accordance to a three-dimensional Heisenberg antiferromagnet, the lower frequency mode clearly deviates. These results suggest that ${\mathrm{Cu}}_{4}{\mathrm{Te}}_{5}{\mathrm{O}}_{12}{\mathrm{Cl}}_{4}$ is a unique material which shows a dual character of zero-dimensional, localized and three-dimensional, collective magnetic behaviors.

Investigation of radical locations in various sesame seeds by CW EPR and 9-GHz EPR imaging

We investigated the location of radical in various sesame seeds using continuous-wave (CW) electron paramagnetic resonance (EPR) and 9-GHz EPR imaging. CW EPR detected persistent radicals (single line) for various sesame seeds. The EPR linewidth of black sesame seeds was narrower than that of the irradiated white sesame seeds. A very small signal was detected for the white sesame seeds. Two-dimensional (2D) imaging using a 9-GHz EPR imager showed that radical locations vary for various sesame seeds. The paramagnetic species in black sesame seeds were located on the seed coat (skin) and in the hilum region. The signal with the highest intensity was obtained from the hilum part. A very low-intensity image was observed for the white sesame seeds. In addition, the 2D imaging of the irradiated white sesame seeds showed that free radicals were located throughout the entire seed. For the first time, CW EPR and 9-GHz EPR imaging showed the exact location of radical species in various sesame seeds.

An addressable quantum dot qubit with fault-tolerant control-fidelity.

Exciting progress towards spin-based quantum computing has recently been made with qubits realized using nitrogen-vacancy centres in diamond and phosphorus atoms in silicon. For example, long coherence times were made possible by the presence of spin-free isotopes of carbon and silicon. However, despite promising single-atom nanotechnologies, there remain substantial challenges in coupling such qubits and addressing them individually. Conversely, lithographically defined quantum dots have an exchange coupling that can be precisely engineered, but strong coupling to noise has severely limited their dephasing times and control fidelities. Here, we combine the best aspects of both spin qubit schemes and demonstrate a gate-addressable quantum dot qubit in isotopically engineered silicon with a control fidelity of 99.6%, obtained via Clifford-based randomized benchmarking and consistent with that required for fault-tolerant quantum computing. This qubit has dephasing time T2* = 120 μs and coherence time T2 = 28 ms, both orders of magnitude larger than in other types of semiconductor qubit. By gate-voltage-tuning the electron g*-factor we can Stark shift the electron spin resonance frequency by more than 3,000 times the 2.4 kHz electron spin resonance linewidth, providing a direct route to large-scale arrays of addressable high-fidelity qubits that are compatible with existing manufacturing technologies.

Dynamical spin-orbital correlations versus random singlets inBa3CuSb2O9investigated by magnetization and electron spin resonance

High-field magnetization and high-frequency electron spin resonance (ESR) are employed to differentiate magnetism between an orthorhombic and a hexagonal majority phase of ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$. For the orthorhombic sample, an ESR signal changes its temperature dependence at ${T}_{S}\ensuremath{\sim}200$ K, suggesting a static Jahn-Teller (JT) ordering. A magnetization curve follows a power-law behavior $M\ensuremath{\sim}{H}^{{\ensuremath{\alpha}}_{m}}$ with the exponent ${\ensuremath{\alpha}}_{m}=0.72\ifmmode\pm\else\textpm\fi{}0.06$ for $8lHl26$ T and ${\ensuremath{\alpha}}_{m}=1.06\ifmmode\pm\else\textpm\fi{}0.04$ for $Hg26$ T. The ESR linewidth exhibits a critical-like divergence, $\ensuremath{\Delta}{H}_{pp}(T)\ensuremath{\propto}{T}^{\ensuremath{-}\ensuremath{\alpha}}$ with the exponents of $\ensuremath{\alpha}=0.22\ifmmode\pm\else\textpm\fi{}0.07$ and $0.32\ifmmode\pm\else\textpm\fi{}0.04$. The sublinear magnetization and the critical ESR line broadening are taken as evidence of a random singlet state. For the hexagonal sample, both $\ensuremath{\Delta}{H}_{pp}(T)$ and $g$ factor are described by the same thermally activated process with the energy barrier of 300 K. This evidences intrinsic coupling of spins to orbital degrees of freedom and thereby gives support for a dynamic spin-orbital entangled state. Our results demonstrate that magnetism in the spin-orbital coupled compound ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$ is dictated by a spatiotemporal structure of the JT distortions.

Spectroscopic Studies on Valine-Functionalized Single-Walled Carbon Nanotubes

The oxidized single-walled carbon nanotubes (SWCNTs) and valine-functionalized SWCNTs were prepared and characterized by using X-ray diffraction (XRD), ultraviolet-visible (UV-Vis), Fourier transform infrared (FT-IR), electron paramagnetic resonance (EPR), scanning electron microscopy (SEM), and energy dispersive X-ray (EDX) techniques. The XRD pattern of valine-functionalized SWCNTs was similar to the oxidized SWCNTs. UV-Vis study confirms that the formation of charge transfer complex, valine-functionalized SWCNTs. FT-IR study reveals that valine molecules have been bonded to SWCNTs. The line shape analysis shows that the EPR line shape is a Gaussian line shape. The EPR line width and g-factor values indicate the presence of impurities, defects or magnetic ion site in the samples. The surface morphology of the samples was observed by SEM images. EDX elemental micro analysis confirms that the nitrogen element exists in valine-functionalized SWCNTs.

Anomalous spin relaxation and quantum criticality in Mn1 − x Fe x Si solid solutions

We report results of the high frequency (60 GHz) electron spin resonance (ESR) study of the quantum critical metallic system Mn1 − xFexSi. The ESR is observed for the first time in the concentration range 0 < x < 0.24 at temperatures up to 50 K. The application of the original experimental technique allowed carrying out line shape analysis and finding full set of spectroscopic parameters, including oscillating magnetization, line width and g factor. The strongest effect of iron doping consists in influence on the ESR line width and spin relaxation is marked by both violation of the classical Korringa-type relaxation and scaling behavior. Additionally, the non-Fermi-liquid effects in the temperature dependence of the ESR line width, which may be quantitatively described in the theory of Wolfle and Abrahams, are observed at quantum critical points x* ∼ 0.11 and xc ∼ 0.24.

Development and performance of a 129-GHz dynamic nuclear polarizer in an ultra-wide bore superconducting magnet.

We sought to build a dynamic nuclear polarization system for operation at 4.6 T (129 GHz) and evaluate its efficiency in terms of (13)C polarization levels using free radicals that span a range of ESR linewidths. A liquid helium cryostat was placed in a 4.6 T superconducting magnet with a 150-mm warm bore diameter. A 129-GHz microwave source was used to irradiate (13)C enriched samples. Temperatures close to 1 K were achieved using a vacuum pump with a 453-m(3)/h roots blower. A hyperpolarized (13)C nuclear magnetic resonance (NMR) signal was detected using a saddle coil and a Varian VNMRS console operating at 49.208 MHz. Samples doped with free radicals BDPA (1,3-bisdiphenylene-2-phenylallyl), trityl OX063 (tris{8-carboxyl-2,2,6,6-benzo(1,2-d:4,5-d)-bis(1,3)dithiole-4-yl}methyl sodium salt), galvinoxyl ((2,6-di-tert-butyl-α-(3,5-di-tert-butyl-4-oxo-2,5-cyclohexadien-1-ylidene)-p-tolyloxy), 2,2-diphenylpicrylhydrazyl (DPPH) and 4-oxo-TEMPO (4-Oxo-2,2,6,6-tetramethyl-1-piperidinyloxy) were assayed. Microwave dynamic nuclear polarization (DNP) spectra and solid-state (13)C polarization levels for these samples were determined. (13)C polarization levels close to 50 % were achieved for [1-(13)C]pyruvic acid at 1.15 K using the narrow electron spin resonance (ESR) linewidth free radicals trityl OX063 and BDPA, while 10-20 % (13)C polarizations were achieved using galvinoxyl, DPPH and 4-oxo-TEMPO. At this field strength free radicals with smaller ESR linewidths are still superior for DNP of (13)C as opposed to those with linewidths that exceed that of the (1)H Larmor frequency.

Collective excitations in the metallic triangular antiferromagnetPdCrO2

We report a combined Raman-scattering and electron spin resonance (ESR) study of the metallic triangular compound $\mathrm{Pd}\mathrm{Cr}{\mathrm{O}}_{2}$ along with lattice dynamic calculations. Our Raman data reveal (i) an exponential increase of the 706 cm${}^{\ensuremath{-}1}$ ${A}_{g}$ mode involving the out-of-plane motions of the oxygen atoms and (ii) electronic Raman scatterings whose scattering rate is strongly suppressed as the temperature approaches ${T}_{N}=37.5$ K. The temperature dependence of the ESR linewidth is described by a critical broadening $\ensuremath{\Delta}{H}_{pp}(T)\ensuremath{\propto}{(T\ensuremath{-}{T}_{N})}^{\ensuremath{-}p}$ with the exponent $p=0.63\ifmmode\pm\else\textpm\fi{}0.05$. These are interpreted in terms of coherence of interplane hoppings along the $c$ axis and a reconstruction of electronic bands on approaching ${T}_{N}$. Our results suggest a significant role of an interference between metallic and magnetic layers in explaining collective excitations in $\mathrm{Pd}\mathrm{Cr}{\mathrm{O}}_{2}$.

Physical properties of EuPtIn4 intermetallic antiferromagnet

We report the physical properties of EuPtIn4 single crystalline platelets grown by the In-flux technique. This compound crystallizes in the orthorhombic Cmcm structure with lattice parameters a=4.542(1)A., b=16.955(2)A. and c=7.389(1)A.. Measurements of magnetic susceptibility, heat capacity, electrical resistivity, and electron spin resonance (ESR) reveal that EuPtIn4 is a metallic Curie–Weiss paramagnet at high temperatures with an effective moment of μeff≈7.8(1)μB due to divalent Eu ions. At low temperatures, antiferromagnetic (AFM) ordering is observed at TN=13.3K followed by a successive anomaly at T⁎=12.6K. In addition, within the magnetic state, a spin-flop transition is observed with Hc~2.5T at T=1.8K when the magnetic field is applied along the ac-plane. In the paramagnetic state, a single divalent Eu2+ Dysonian ESR line with a Korringa relaxation rate of b=4.1(2)Oe/K is observed. Interestingly, even at high temperatures, both ESR linewidth and electrical resistivity reveal a similar anisotropy. We discuss a possible common microscopic origin for the observed anisotropy in these physical quantities likely associated with an anisotropic magnetic interaction between Eu2+ 4f electrons mediated by conduction electrons.

Super-linear frequency dependence of ac conductivity in nanocrystalline lithium ferrite

An attempt has been made to synthesize nanocrystalline high dense lithium ferrite, namely, Li0.5MxFe2.5−xO4 (M = Bi, Pb; x = 0, 0.02) by a solution combustion technique. Various characterization methods like XRD and TG-DTA were employed to validate the structure and phase purity. UV–vis diffuse reflectance spectroscopic studies were conducted to obtain the oxidation states of metal ions. Raman analysis was used to identify the phonon modes involved in specific atomic motions present in the synthesized materials. Morphology of sample surface was explored by electron microscope and elemental analysis was done by energy dispersive analysis. The nanocrystalline nature of the materials were confirmed through transmission electron microscopy. Strong temperature dependence of electron spin resonance linewidth was exhibited by the samples and a resonance field was observed for all the materials. Ac electrical properties were investigated using the super-linear power law and activation energies were calculated for all compositions. The conductivity data were consistent with an asymmetric double well potential model.

Electron paramagnetic resonance study of paramagnetic centers in carbon-fumed silica adsorbent

Fumed silica A-300 was carbonized by means of pyrolysis of CH2Cl2. The obtained initial SiO2:C nanopowders of black color, with an average diameter of 14–16 nm and carbon (C) concentration 7 wt. %, subjected to the oxidation and passivation treatment were studied by electron paramagnetic resonance (EPR) in the temperature range 4–400 K. Two EPR signals of Lorentzian lineshape with nearly equal g-factors and different linewidth were observed in the initial, oxidized, and passivated SiO2:C nanopowders. The two-component EPR spectrum was explained by the presence of C in two electronic states. The intensive narrow EPR signal, which has a temperature-dependent intensity, linewidth, and resonance field position, was attributed to the carbon-related defect with non-localized electron hopping between neighboring C-dangling bonds. The striking effect is that the temperature dependence of the EPR linewidth demonstrates the motional narrowing of the EPR signal at very low temperatures from 4 K to 20 K, which is not typically for nonmetallic materials and was explained by the quantum character of C layer conductivity in the SiO2:C. The observed peaks in the temperature dependence of the conduction electron EPR signal integral intensity in the high-temperature range 200–440 K was explained by the presence of the C nanodots at the surface of SiO2 nanoparticles and the ejection of electrons from the confinement energy levels of C quantum dot when the temperature becomes comparable to the confinement energy.

Empirical Monod-Beuneu relation of spin relaxation revisited for elemental metals

Monod and Beuneu [Monod and Beuneu, Phys. Rev. B 19, 911 (1979)] established the validity of the Elliott-Yafet theory for elemental metals through correlating the experimental electron spin resonance line-width with the so-called spin-orbit admixture coefficients and the momentum-relaxation theory. The spin-orbit admixture coefficients data were based on atomic spin-orbit splitting. We highlight two shortcomings of the previous description: i) the momentum-relaxation involves the Debye temperature and the electron-phonon coupling whose variation among the elemental metals was neglected, ii) the Elliott-Yafet theory involves matrix elements of the spin-orbit coupling (SOC), which are however not identical to the SOC induced energy splitting of the atomic levels, even though the two have similar magnitudes. We obtain the empirical spin-orbit admixture parameters for the alkali metals by considering the proper description of the momentum relaxation theory. In addition, we present a model calculation which highlights the difference between the SOC matrix element and energy splitting.