Singlet Energy Levels Research Articles

Synthesis and Photophysical Behaviours of Novel Phenanthro[9,10-d] imidazole Substituted Azo Dyes in Solvent Media

In this study, three novel phenanthroimidazole derivatives containing azo group were synthesized and their photophysical properties were investigated. For this purpose, 2-(4-nitrophenyl)1H-phenanthro[9,10-d]imidazole structure obtained from 9,10-phenanthroquinone and 4-nitrobenzaldehyde was reduced with palladium on activated charcoal 10 % and hydrazine hydrate. Phenanthro[9,10-d]imidazole-2-yl)aniline obtained in this way was converted to phenantroimidazole-azo derivatives containing different aromatic groups. Structural analyses were performed by FT-IR, NMR, LC-MS spectroscopic techniques. Spectral characterizations and photostability studies of phenanthroimidazole derivatives were performed by UV-vis and fluorescence spectroscopy in solvent media. The maximum absorption and emission wavelengths, molar extinction coefficients, singlet energy levels, Stokes’ shift values of phenanthroimidazole-azo derivatives were obtained.

High-order harmonic generation of Li+ with combined infrared and extreme ultraviolet fields**Project supported by the National Natural Science Foundation of China (Grant Nos. 11465016, 11664035, 11764038, and 11765018) and the Foundation of Northwest Normal University, China (Grant No. NWNU-LKQN-17-1).

We investigate high-order harmonic generation (HHG) of Li+ ion driven by an intense infrared (IR) laser field in combination with a weak XUV pulse. To achieve this, we first construct an accurate single-active electron angular-momentum-dependent model potential of Li+ ion, by which the accurate singlet energy levels of Li+ for the ground state and excited states with higher quantum numbers can be obtained. Then, we solve numerically the three dimensional time-dependent Schrödinger equation of Li+ ion by means of the generalized pseudospectral method to obtain HHG. Our results show that the strength of assisted XUV is not amplified during the harmonic generation process, but the yield of HHG power spectrum in the whole plateau has a significant enhancement. Furthermore, the optimal phase delay between the IR and XUV pulses allows the production of ultrabroadband supercontinuum spectra. By superposing some harmonics, a strong new single 27-attosecond ultrashort pulse can be obtained.

Synthesis, photoluminescence, and energy transfer mechanism of a reactive Eu(III)-complex used in white light-emitting diodes

A reactive Eu(III)-complex Eu(TTA)3 ( UA ) was synthesized with 2-thenoyltrifluoroacetone (TTA) as primary ligand and undecylenic acid (UA) as active ligand. The structure of the complex was characterized by elemental analysis, FT-IR spectra, and UV–vis absorption spectra. The results indicate that the Eu(III) ion is successfully coordinated with primary ligand TTA and active ligand UA. Fluorescence spectra suggest that the complex exhibits excellent red emission of Eu(III) ion with a peak at 611 nm being monitored at 365 nm of near ultraviolet. Meanwhile, the complex has also shown a long fluorescence lifetime of 1.4259 ms and a high quantum yield of 86.7%. The singlet and triplet energy levels for the ligands of rare earth complexes were calculated using Gaussian 03 program package. Intramolecular energy transfer diagram was established and an energy transfer process following Dexter electron exchange mechanism was demonstrated, which provides the experimental data and theoretical foundation for the development of high-performance luminescent materials. The result of TGA shows that the complex keeps stable at 156°C or below. All the results demonstrate that the reactive Eu(III)-complex can be used as the red-light material in white light-emitting diodes phosphor.

Luminescence properties and DFT calculations of lanthanide(III) complexes (Ln = La, Nd, Sm, Eu, Gd, Tb, Dy) with 2,6-bis(5-methyl-benzimidazol-2-yl)pyridine

Seven new complexes obtained of hydrated lanthanide(III) nitrates with 2,6-bis(5-methyl-benzimidazol-2-yl)pyridine (bmbp) having the formula [LnIII(bmbp)(NO3)2]NO3n-H2O (where LnIII = LaIII, NdIII, SmIII, EuIII, GdIII, TbIII, DyIII) have been prepared and characterized by ultraviolet–visible and infrared absorption spectroscopies, molar conductivity and elemental composition. The emission spectra of the all complexes have also been analyzed; it has been detected that all complexes revealed the luminescence of bmbp. Samarium and europium complexes also displayed the corresponding lanthanides luminescence. The metal ion is coordinated to seven atoms resultant to three nitrogens of bmbp and to two oxygens for each of the bidentate nitrate groups. Density functional theory calculations were utilized to designate the singlet and triplet excited state energy levels of bmbp.

Lanthanide coordination compounds with benzimidazole-based ligands. luminescence and EPR

LnIII coordination compounds were synthesized with two benzimidazolic ligands, 2 pb = 2-(2-pyridyl)benzimidazole and 4tb = 2-(4-thiazolyl)benzimidazole, with Eu, Gd, Tb and Dy, namely: [Ln (NO3)3 (Ln)2]⋅S, where Ln = 2 pb, 4tb and S = solvent. They were characterized by IR, 1D and 2D 1H and 13C NMR, solid state UV–Vis–NIR absorption and emission and EPR spectroscopies, X-ray powder diffraction, elemental analyses, electrical conductivity and magnetic susceptibility. The compounds of EuIII with 2 pb and 4tb, TbIII and DyIII with 4tb showed luminescence at RT in the solid state; their emission life time ranged from 0.026 to 1.82 ms. 2 pb and 4tb sensitized lanthanides luminescence. Results of DFT theoretical calculations for the ligand (H2L) are used to assign the ligand singlet and triplet excited state energy levels. Additionally, a full analysis of the X-band EPR spectra in the solid state gave the following results, at 297 K for [Gd(NO3)3 (2 pb)2]·CH3COCH3, g = 1.990, |D| = 0.0687 cm−1, E = 0.0165 cm−1 and λ = |E/D| = 0.240; for [Gd(NO3)3 (4tb)2]⋅CH3COCH3, g = 2.000; |D| = 0.0670 cm−1, E = 0.000 cm−1 and λ = |E/D| = 0.000. At lower temperature, 128 K, compounds showed an increase of the rhombicity as the temperature decreased.

Organic planar pn heterojunction light-emitting diodes: Materials and devices

Organic light-emitting diodes (OLEDs) have been widely applicable in the flat-panel display products, such as mobile phones and television. They also possess more outstanding application advantages in the area of wearable display products in the future and have a great potential to serve as a key competitor in the next-generation solid-state lighting technology. For the commercial OLED products, phosphorescent materials that contain the rare metals are essential. This is because that phosphorescent OLEDs can give more efficient radiation, as the result of harvesting both singlet and triplet excitons for radiative utilization, in contrast to traditional fluorescent emitters. However, because of the limited resources of the rare metals, e.g., indium, on the earth, the phosphorescent materials are rather expensive, and their sustainable supply are also questionable. Besides, the long-term stability of blue phosphorescent emitters are also challenging. To further lower the fabrication cost of materials and devices, numerous research efforts have been devoted to developing rare-metal-free, purely organic light-emitting materials, which could be more low-cost and still exhibit excellent electroluminescence performances. In this review, we firstly introduce the evolution of OLED materials and devices. The OLED materials could be roughly classified into three generations. For the 1st generation traditional fluorescent emitters, three quarters of electro-generated excitons are wasted as heat release, due to the spin forbidden rule of triplets. This suggests that the OLEDs employing traditional fluorescent emitters could only achieve a maximum internal quantum efficiency of 25%. For the 2nd generation phosphorescent emitters, the spin orbital coupling due to heavy atom effect results in the mixing of singlet and triplet energy levels, and make triplet radiative transition feasible. Thereby, phosphorescent OLEDs can obtain a maximum internal quantum efficiency close to 100%. Recently, the 3rd generation hyperfluorescence emitters with the thermally activated delayed fluorescence (TADF) character could also make the devices thereof give an internal quantum efficiency close to 100%. This paves a road to realize low-cost and high-performance OLEDs based on purely organic semiconductors. We also outline the evolution of the OLED device configuration. For single-, double- and multi-layer device configuration of conventional OLEDs, their performances are mainly dependent on the emission layer (EML). The current methodology to develop high-performance OLEDs is to maximize the light emission ability of EML. Based on the understanding of the excitonic property of organic semiconductors, we show that efficient EML-free planar pn heterojunction light-emitting diodes (pn-OLEDs) should be realizable, given that the planar heterojunction property is well modulated. This indicates a way to directly construct efficient fluorescent OLEDs without an EML from the perspective of planar heterojunction composed of p type and n type charge transport materials. As following, the progress of OLEDs based on the layer-to-layer charge transfer mechanism using purely organic semiconductors is introduced. Main discussion focuses on pn-OLEDs, including that how this concept was proposed and the progress of relevant materials and devices. We also discuss the challenges of developing high performance pn-OLEDs and their potential applications in other optoelectronic areas, such as organic light-emitting transistors and electrically-pumped organic laser.

Tuning the photophysical properties of heteroleptic Ir(III) complexes through ancillary ligand substitution: Experimental and theoretical investigation

A series of new phosphorescent heteroleptic Iridium(III) complexes with 2-phenylpyridine, acetylacetone and 2-picolinic acid as ancillary ligands have been designed and synthesized. The photophysical and electrochemical properties have been investigated experimentally and theoretically. The UV–vis absorption and photoluminescence (PL) emission spectra of the complexes were carried out in solution as well as thin film. The PL study indicates that the Iridium complexes show broadband emission ranging from green to yellowish-green in solution and greenish-yellow to yellow emission in PMMA films with appropriate CIE color gamut. The PLQY of the complexes were found to be in the range of 0.12–0.29. Singlet and triplet energy levels were calculated by using time dependent DFT (TD-DFT) calculations. The excited triplet energy was estimated and correlated to structural and energetic characteristics of the reported host materials. Further, investigations have been performed to explore the optical, electronic, charge transport, and stability properties of Ir(III) complexes as charge transport and emissive materials for organic light emitting devices (OLEDs). Furthermore, all the complexes are promising luminescent and its hole transporting performance is more favorable than electron transport performances. According to theoretical calculations, it is clearly indicating that, Ben-Ir-acac complex is a more potential emitter for OLED applications.

Monitoring of the energy levels by heteroatom substitution to hexacene and controlling over singlet fission and photo-oxidative resistance

The singlet fission is a spin allowed and extremely fast internal conversion process involved in solar cell by which a photo-excited singlet exciton is splitted into two triplet ones. For effective singlet fission and to increase the efficiency of solar cell, designing of new molecules is an interesting area of research and our current interest. The silicon substituted oligocenes, commonly known as silaoligocenes, are found to be the efficient singlet fission material due to their special characteristics. We have shown the SF energy criteria satisfied by the singlet and triplet states of various silahexacene derivatives, and theoretically predicted whether such molecules exhibit fission properties or not. The fluorine atoms have been substituted to various positions of different silahexacenes to manipulate their singlet and triplet energy levels. As fluorine being the most electro-negative substituent, it is capable of lowering frontier molecular orbital energies effectively. Thus, the material can easily match SF energy criteria to compute the SF driving force or triplet–triplet annihilation possibility. The geometries, electronic structures, frontier molecular orbital energies, optimization of excited state and calculation of energies associated with fission process of the substituted hexacene are investigated with well known quantum mechanical methods.

The spectral properties of DNA and RNA macromolecules at low temperatures: fundamental and applied aspects

This paper summarizes the results of studies of the spectral properties—optical absorption, fluorescence and phosphorescence—of DNA and RNA macromolecules and synthetic poly-, oligo- and mono-nucleotides, which have been carried out in our laboratory. The system of first excited singlet and triplet energy levels for DNA and RNA is evaluated using low-temperature (4.2 K–77 K) luminescent measurements. The traps of the singlet and triplet electronic excitations in these compounds are identified. An important self-protection mechanism against photo-damage of DNA and RNA by UV photons or penetrative radiation based on the capture of triplet electronic-energy excitations by the most photostable centers—in DNA, the complex formed by neighboring adenosine (A) and thymidine (T) links; in RNA, the adenosine links—is described. It is confirmed that despite similarities in the chemical and partly energy structures DNA is more stable than RNA. The spectral manifestation of the telomeres (the important functional system) in DNA macromolecules is examined. The results obtained on telomere fragments provide the possibility of finding the configuration peculiarities of the triplet excitations traps in DNA macromolecules. The resulting spreading length of the migrating singlet (ls) and triplet (lt) excitations for DNA and RNA macromolecules are evaluated.

Linkage modes on phthaloyl/triphenylamine hybrid compounds: Multi-functional AIE luminogens, non-doped emitters and organic hosts for highly efficient solution-processed delayed fluorescence OLEDs

Two soluble AIE-TADF emitters (m-DTPACO and p-DTPACO), comprising hole-transporting triphenylamine and electron-transporting phthaloyl moiety, have been designed and synthesized by a simple Friedel-Crafts reaction. Through altering the linkage modes (from para- to meta-position) on the central phenyl ring of terephthaloyl or isophthaloyl, optoelectronic properties can be finely tuned for two compounds. Both m-DTPACO and p-DTPACO exhibit aggregation-induced emission and delayed fluorescecene emission characteristics, with prompt/delayed fluorescent emission life times of 2.7 ns/8.3 μs and 7.4 ns/9.6 μs, respectively. p-DTPACO shows singlet/triplet energies of 2.37/2.32 eV and solution/solid-state photoluminescence quantum yield of 2/39% compared to those of 2.59/2.38 eV and 46/75% for m-DTPACO. External quantum efficiency of 2.4 and 3.7% has been obtained in solution-processed non-doped OLEDs based on AIE-TADF sky blue m-DTPACO and green p-DTPACO luminogens, respectively. Furthermore, solution processed green TADF OLEDs with double host/dopant TADF system by using the AIE-TADF materials as the host exhibit high efficiencies of 43.5 cd A−1, 33.3 lm W−1 and 13.0% for meta-positioned m-DTPACO and 29.3 cd A−1, 25.4 lm W−1 and 9.0% for para-linked p-DTPACO, respectively. The significantly higher performance for m-DTPACO than p-DTPACO host could be ascribed to its higher singlet and triplet energy levels for more efficient energy transfer as well as more efficiently preventing reverse energy transfer from the green TADF dopant to the TADF host.

Phenyleneanthracene derivatives as triplet energy acceptor/emitter in red light excitable triplet-triplet-annihilation upconversion

A series of anthracene derivatives with 9,10-substituents were prepared as triplet acceptors/emitters for triplet-triplet-annihilation (TTA) upconversion. Different linkages of CC single bonds and CC triple bonds were used to tune the singlet and triplet state energy levels, which may enhance the TTA upconversion. The study of the photophysical properties of the compounds indicates that the CC linker does not alter the T1 state energy level substantially, whereas the CC linker significantly reduced the T1 state energy levels. With nanosecond transient absorption spectroscopy, the intermolecular triplet-triplet-energy-transfer (TTET) process was studied. The lack of the upconversion for some anthracene derivatives was attributed to the inappropriate T1 energy levels thus the lack of TTET. On the other hand, different upconversion quantum yields were observed for some acceptors, although the TTET processes are similar. This result is due to the different TTA. These studies will be useful for future development of the TTA upconversion and for study of the triplet state properties of organic chromophores.

A Dual‐Characteristic Bidentate Ligand for a Ternary Mononuclear Europium(III) Molecular Complex – Synthesis, Photophysical, Electrochemical, and Theoretical Study

The new bipolar ligand 4‐{1‐(9,9‐diethyl‐9H‐fluoren‐2‐yl)‐1H‐imidazo[4,5‐f][1,10]phenanthrolin‐2‐yl}‐N,N‐diphenylbenzenamine (Phen‐Fl‐TPA) and its β‐diketonate EuIII complex have been designed, synthesized, characterized, and their photophysical and electrochemical properties have been investigated. The UV/Vis absorption and photoluminescence (PL) emission spectra of the Eu complex and Phen‐Fl‐TPA in solution as well as in the solid state and as thin films were recorded. The PL study indicated that the Eu complex emits narrow‐band red emission with an appropriate International Commission on Illumination (CIE) color gamut. Moreover, it also confirmed the efficient energy transfer from the ligand to the Eu3+ ion. Time‐dependent DFT (TD‐DFT) calculations were performed for the ligand to determine the exact positions of the excited singlet and triplet energy levels. The EuIII complex shows a dominant pathway involving energy transfer between the ligand triplet level and the excited (5D0) level of the EuIII ion. In addition, the theoretically calculated UV/Vis spectrum of the ligand is similar to the experimental one. The Judd–Ofelt theory was applied to the emissive properties of the EuIII complex, and the lifetime was found to be 0.66 ms. The ground‐state optimized ligand structure was a good match with the single‐crystal structure. The photoluminescence quantum yield (PLQY) of Eu(TTA)3Phen‐Fl‐TPA (TTA = thenoyltrifluoroacetone) in solution was 34.1 %, and it possess a high thermal decomposition temperature (317 °C), as determined by differential scanning calorimetry/thermogravimetric analysis (DSC‐TGA). Electrochemical analysis revealed highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) energy levels for the EuIII complex of 5.6 and 2.8 eV, respectively.

Antiphotobleaching: A Type of Structurally Rigid Chromophore Ready for Constructing Highly Luminescent and Highly Photostable Europium Complexes

Antiphotobleaching is a critical challenge in the field of luminescent lanthanide complexes (LLCs) as well as in many disciplines concerning organic luminescent processes. In this work, a type of structurally rigid organic ligand, 4‐hydroxy‐1,5‐naphthyridine (ND), is developed, which can not only efficiently sensitize the europium emission but also demonstrate unique photostability. A series of ND derivatives with different substituent groups are synthesized and their singlet and triplet excited state energy levels are systematically investigated. Photophysical characterizations of the corresponding europium complexes reveal that the sensitization efficiencies (ηsens) are close to 100% and the total photoluminescence quantum yields can reach up to 84%. Most importantly, these structurally rigid luminescent europium complexes exhibit outstanding photostability and thermostability. Unlike the widely used β‐diketone complexes that are easily photodegraded, ND‐based chelates show no obvious degradation during the UV aging test (10 W m−2 340 nm Ultraviolet A irradiation) within 200 h. Such superior UV resistance is even better than that of the famous compound tris(8‐hydroxyquinolate)aluminum (Alq3). Possible reasons are discussed and a general rule for designing photostable LLCs is proposed. Such a chromophore is very promising for introducing luminescent materials with good photostability in potential application in many disciplines.

The effect of heavy atom to two photon absorption properties and intersystem crossing mechanism in aza-boron-dipyrromethene compounds

New aza-boron-dipyrromethene compounds containing bromine atoms at various positions were designed and synthesized to enhance the triplet state population and two photon absorption properties for applications such as two-photon photodynamic therapy, triplet–triplet annihilation up-conversion. Steady state fluorescence and ultrafast pump probe spectroscopy techniques revealed that only 2, 6 positions of aza-boron-dipyrromethene core contribute to triplet state population significantly. Density function theory calculations showed that when bromine atoms introduced to 2, 6 position of aza-boron-dipyrromethene core, singlet and triplet energy levels get closer therefore probability of intersystem crossing increases. Z-scan experiments at 800 nm wavelengths revealed considerably large (610 GM) two photon absorption cross section value with respect to literature for compounds showing intersystem crossing mechanism. The efficient intersystem crossing and enhanced two-photon absorption properties make the investigated aza-boron-dipyrromethene compounds good candidates for two-photon photodynamic therapy application.

Impact of non-symmetric 2,9,10-aryl substitution on charge transport and optical properties of anthracene derivatives

A new methodology for the synthesis of non-symmetric 2,9,10-arylanthracenes is proposed. The anthracenes were intentionally decorated with non-symmetric aryls to improve charge transport properties. The impact of non-symmetric aryl substituents on optical and electrical properties was thoroughly studied. Non-symmetric 9,10-aryl substituents ensured blue emission in the range of 422–436 nm and fluorescence quantum efficiency of ca. 0.45, while the type of substituents had a minor impact on excited state deactivation due to their twisted configuration. Optical properties of the compounds obtained in dilute solution and in a rigid polymer matrix revealed the dominating non-radiative decay mechanism to be due to the intersystem crossing (ISC), which was mainly governed by 2-phenyl substitution. Favourable alignment of the singlet and triplet energy levels for efficient ISC was confirmed by density functional theory calculations. Some of the non-symmetrically substituted anthracene derivatives showed very high hole drift mobilities reaching 10−2 cm2/Vs in solution-processed neat films.

(Invited) Singlet Exciton Fission in Molecular Solids

Singlet fission (SF) is the process by which a singlet exciton in a molecular material is energetically down-converted into two independent triplet excitons. Thermodynamic modeling predicts that using a SF material can theoretically increase the Shockley-Queisser limit for power conversion efficiency in solar cells from 32% to 44%, assuming that SF results in the formation of two triplet excitons, each of which produce an electron-hole pair quantitatively. We are examining new dye molecules, which undergo SF, using guidance from electronic structure calculations to assure the requisite relationships between molecular singlet and triplet energy levels. We are preparing hierarchical assemblies from these chromophores, starting from covalent dimers and trimers, then developing supramolecular assemblies, and engineered crystalline materials to investigate SF in bulk, ordered materials. We are using femtosecond transient spectroscopy as well as time-resolved electron paramagnetic resonance spectroscopy to characterize the SF mechanism and the factors that determine its efficiency at time scales down to 100 fs. We are also using X-ray diffraction to investigate the single crystal structures of these materials and X-ray scattering on their thin polycrystalline solid films to determine how their structures correlate with their ability to carry out SF. Our results suggest that a p-p slip-stacked geometry is important for maximizing SF efficiency.

Achieving the photon up-conversion thermodynamic yield upper limit by sensitized triplet-triplet annihilation.

Triplet-triplet annihilation (TTA) based up-conversion is a promising strategy for light harvesting the low-energy tail of the solar spectrum with photovoltaic technologies. Here, we present a bi-component system for photon managing via TTA that allows bypassing the classical statistic limit of 2/5 in the singlet generation, achieving a near unitary conversion efficiency. This result is obtained because of the peculiar relative position of the triplet and singlet energy levels of perylene, used as up-converter and emitter. The system shows a record red-to-blue external up-conversion yield of ∼10% under an irradiance of 1 sun.

Rationally investigating the influence of T1 location on electroluminescence performance of aryl amine modified phosphine oxide materials.

The correspondence between triplet location effect and host-localized triplet-triplet annihilation and triplet-polaron quenching effects was performed on the basis of a series of naphthyldiphenylamine (DPNA)-modified phosphine oxide hosts. The number and ratio of DPNA and diphenylphosphine oxide was adjusted to afford symmetrical and unsymmetrical molecular structures and different electronic environments. As designed, the first triplet (T1 ) states were successfully localized on the specific DPNA chromophores. Owing to the meso- and multi-insulating linkages, identical optical properties and comparable electrical performance was observed, including the same first singlet (S1 ) and T1 energy levels to support the similar singlet and triplet energy transfer and the close frontier molecular orbital energy levels. This established the basis of rational investigation on T1 location effect without interference from other optoelectronic factors.

Highly efficient long-life blue fluorescent organic light-emitting diode exhibiting triplet–triplet annihilation effects enhanced by a novel hole-transporting material

A novel hole-transporting material with high singlet and triplet excitation energy levels was developed. Quantum efficiency of a fluorescent organic light-emitting diode (OLED) using this material as a hole-transporting layer can be increased because of facilitated triplet–triplet annihilation (TTA) due to exciton confinement in an emission layer. Furthermore, this material has a deep highest occupied molecular orbital level because of the absence of triarylamine structure. This feature also contributes to the increase in the quantum efficiency, owing to inhibition of a low-energy exciplex formed between the material and a host in the emission layer. Achieved consequently was a blue fluorescent OLED exhibiting a high external quantum efficiency of 11.9% and a long half-decay time of 8,000 h at 1,000 cd/m2. By the device analysis including time-resolved electroluminescence measurements, it was confirmed that TTA contributes to the high efficiency.

Preparation of Nuclear Spin Singlet States Using Spin-Lock Induced Crossing

We introduce a broadly applicable technique to create nuclear spin singlet states in organic molecules and other many-atom systems. We employ a novel pulse sequence to produce a spin-lock induced crossing (SLIC) of the spin singlet and triplet energy levels, which enables triplet-singlet polarization transfer and singlet-state preparation. We demonstrate the utility of the SLIC method by producing a long-lived nuclear spin singlet state on two strongly coupled proton pairs in the tripeptide molecule phenylalanine-glycine-glycine dissolved in D(2)O and by using SLIC to measure the J couplings, chemical shift differences, and singlet lifetimes of the proton pairs. We show that SLIC is more efficient at creating nearly equivalent nuclear spin singlet states than previous pulse sequence techniques, especially when triplet-singlet polarization transfer occurs on the same time scale as spin-lattice relaxation.