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Anharmonicity

Anharmonicity refers to the deviation of a physical system from ideal , in which the restoring force is not strictly proportional to the from , leading to nonlinear behaviors in vibrational motion. Unlike the simple harmonic oscillator, where the is purely and energy levels are equally spaced, anharmonic systems feature higher-order terms in the expansion, such as cubic or quartic contributions, resulting in amplitude-dependent frequencies and non-uniform energy spacing. This nonlinearity arises in real-world systems due to finite interatomic distances and interactions, making anharmonicity a fundamental correction to the approximation in classical and quantum descriptions of . In , anharmonicity significantly influences molecular vibrations, particularly in diatomic and polyatomic molecules, where it causes energy levels to converge at higher quantum numbers and enables transitions that are forbidden in the model. These effects are crucial for interpreting vibrational , as anharmonic perturbations shift fundamental frequencies and introduce combination bands, allowing for more accurate modeling of and Raman spectra in chemical analysis. is commonly employed to incorporate anharmonicity into frameworks, providing corrections to wavefunctions and energies that align theoretical predictions with experimental observations in systems like polypeptides and high-resolution molecular spectra. In , anharmonicity governs - interactions, which are responsible for key thermal properties such as and finite thermal conductivity. These interactions arise from the anharmonic terms in the lattice potential, enabling processes like that limit heat transport in materials, with implications for thermoelectrics and . For instance, in crystals like , cubic anharmonicity contributes to shifts in phonon modes, while effects further modulate these interactions, influencing macroscopic behaviors in metals and insulators. Overall, anharmonicity bridges microscopic vibrational dynamics to observable thermodynamic phenomena across diverse physical systems.

Fundamentals

Definition and Principles

Anharmonicity refers to the deviation of an oscillator from , in which the restoring force is no longer exactly proportional to the from . In ideal harmonic oscillators, this proportionality arises from , resulting in periodic motion with a independent of . Anharmonicity introduces nonlinearity, leading to phenomena such as amplitude-dependent oscillation , where larger displacements alter the effective of . The general principle underlying anharmonicity stems from the form of the function. For a , the potential energy is quadratic, given by
V(x) = \frac{1}{2} k x^2,
where k > 0 is the force constant and x is the . Anharmonicity arises when higher-order terms are included in the potential, such as cubic (\lambda x^3) or quartic (\mu x^4) contributions, yielding forms like V(x) = \frac{1}{2} k x^2 + \lambda x^3 + \mu x^4. These terms cause asymmetry in the , resulting in either stiffening (increased with for positive quartic terms) or softening (decreased for negative effective quartic terms). Early recognition of anharmonicity dates back to 17th-century studies of mechanical systems like s, where Galileo observed approximate isochronism and Huygens noted amplitude-dependent periods in 1656, with systematic investigations in the for more complex systems like springs. Key developments occurred in acoustics through Lord Rayleigh's The Theory of Sound (1877–1878), which analyzed non-linear vibrations in strings, pipes, and other sound-producing mechanisms, highlighting how anharmonic effects influence wave propagation and . In quantum mechanical contexts, anharmonicity leads to qualitative effects such as unequal spacing between vibrational levels, which decreases for higher quantum numbers, and the appearance of —excited states beyond the that are not exact integer multiples due to the perturbed potential. These deviations contrast with the equally spaced levels of the harmonic approximation and enable observations of vibrational structure.

Harmonic Approximation Limitations

The harmonic approximation models vibrational motion by assuming a quadratic potential energy surface, which confines the description to small-amplitude oscillations where higher-order terms (cubic, quartic, and beyond) in the Taylor expansion of the potential are negligible. This simplification yields exact solutions for frequencies and energy levels but holds only near the equilibrium position and for low-energy excitations, as larger displacements activate these neglected anharmonic contributions, leading to inaccuracies in predicted dynamics. Anharmonicity originates from two primary sources: intrinsic, stemming from the inherently non-quadratic form of the interatomic or molecular potential even for isolated oscillators, and extrinsic, arising from interactions in multi-particle systems that couple modes and introduce additional nonlinearities beyond single-mode potentials. Intrinsic anharmonicity is fundamental to the shape of the , while extrinsic effects become prominent in condensed phases or crowded environments where mode-mode couplings dominate. To quantify these limitations, anharmonic effects are often incorporated via , treating the higher-order potential terms as small corrections to the . perturbation yields direct shifts in the unperturbed energies from diagonal matrix elements of the anharmonic , while second-order contributions capture mixing between harmonic states through off-diagonal elements, providing a systematic expansion for error estimation when the perturbation parameter is small. Experimentally, breakdowns in the harmonic approximation manifest as nonlinear dependence of oscillation on in classical measurements, deviating from the constant frequency predicted by the quadratic model, and as linewidth broadening in vibrational spectra, reflecting finite from anharmonic processes. These indicators highlight the regime where anharmonicity alters observable properties, such as spectral shifts and rates.

Mathematical Descriptions

Classical Models

In classical mechanics, the motion of an anharmonic oscillator is described by the second-order differential equation m \ddot{x} + f(x) = 0, where m is the mass and f(x) is the nonlinear restoring force. For small displacements, f(x) approximates the harmonic form -kx, but anharmonicity arises from higher-order terms in the Taylor expansion: f(x) \approx -kx + \alpha x^2 + \beta x^3 + \gamma x^4 + \cdots, with coefficients \alpha, \beta, \gamma quantifying deviations from linearity. These terms lead to amplitude-dependent frequencies and potentially asymmetric or chaotic dynamics. A prominent classical model is the Duffing oscillator, originally introduced by Georg Duffing to describe systems with nonlinear stiffness, such as buckled beams or electrical circuits. Its is given by V(x) = \frac{1}{2} k x^2 + \frac{1}{4} \beta x^4, leading to the equation \ddot{x} + \frac{k}{m} x + \frac{\beta}{m} x^3 = 0. The sign of \beta determines the behavior: positive \beta yields a hardening oscillator, where the effective increases with due to the steeper potential at large displacements; negative \beta results in softening, with decreasing . Asymmetric potentials, incorporating odd-powered terms like \beta x^3, model systems such as the simple pendulum, where the equation expands from \ddot{\theta} + \frac{g}{\ell} \sin \theta = 0 to include a cubic anharmonicity -\frac{g}{6\ell} \theta^3 for moderate angles. Exact analytical solutions for these models are generally unavailable, necessitating approximate methods. techniques, such as the Lindstedt-Poincaré method, address frequency shifts by introducing a strained time \tau = \omega t, where \omega = \omega_0 (1 + \epsilon \omega_1 + \cdots) expands in powers of a small nonlinearity parameter \epsilon, eliminating secular terms that would otherwise cause unbounded growth in the solution. This yields periodic approximations valid for weak anharmonicity, such as amplitude-dependent angular frequencies. For stronger nonlinearities or driven systems like the forced Duffing oscillator, methods (e.g., Runge-Kutta) are essential to capture chaotic regimes, where sensitive dependence on initial conditions emerges. The -amplitude relation exemplifies anharmonicity's effects, as seen in the simple pendulum where large swings deviate from predictions. The approximate is T \approx \frac{2\pi}{\omega_0} \left(1 + \frac{1}{16} \frac{A^2}{\ell^2}\right), with \omega_0 = \sqrt{g/\ell}, A, and \ell, demonstrating an increase in T with A due to the softening cubic term. This relation, derived via , highlights how anharmonicity stretches the cycle for greater excursions.

Quantum Mechanical Treatment

In , anharmonicity is treated by considering the \hat{H} = \frac{\hat{p}^2}{2m} + V(x), where the potential V(x) = \frac{1}{2} k x^2 + \lambda x^3 + \gamma x^4 includes non-quadratic terms that introduce deviations from behavior. These cubic (\lambda x^3) and quartic (\gamma x^4) contributions capture asymmetric and stiffening/softening effects, respectively, and are essential for modeling real systems beyond the harmonic approximation. Perturbation theory provides a systematic approach to compute corrections to the harmonic oscillator eigenstates and energies, treating the anharmonic part V_{\text{anh}}(x) = \lambda x^3 + \gamma x^4 as a small perturbation relative to the unperturbed harmonic Hamiltonian \hat{H}_0 = \frac{\hat{p}^2}{2m} + \frac{1}{2} k x^2. The first-order energy shift for the nth level is E_n^{(1)} = \langle n | V_{\text{anh}} | n \rangle, where |n\rangle denotes the harmonic eigenstates; this vanishes for the odd cubic term due to parity but yields a nonzero positive contribution for the even quartic term, proportional to n^2 for large n. Second-order corrections arise from off-diagonal matrix elements, given by E_n^{(2)} = \sum_{m \neq n} \frac{|\langle m | V_{\text{anh}} | n \rangle|^2}{E_n^{(0)} - E_m^{(0)}}, which mix nearby states and typically reduce energy levels for repulsive anharmonicity. This perturbative expansion, however, diverges or breaks down at high n, where anharmonic effects dominate the level spacing \hbar \omega. The perturbed energy levels E_n are thus non-equispaced, with a leading approximation E_n \approx \hbar \omega (n + \frac{1}{2}) + \chi n(n-1), where \chi is the anharmonicity constant determined by the perturbation strength (e.g., \chi \propto \gamma for quartic terms). This quadratic correction in n reflects the compression or expansion of level spacings, essential for understanding quantum dynamics in anharmonic systems. The corresponding wavefunctions exhibit mixing of the unperturbed harmonic states, \psi_n(x) \approx \phi_n(x) + \sum_{m \neq n} c_{nm} \phi_m(x), where coefficients c_{nm} \propto \langle m | V_{\text{anh}} | n \rangle / (E_n^{(0)} - E_m^{(0)}) introduce distortions and asymmetry, particularly from odd perturbations that break parity symmetry. For diatomic molecular vibrations, the Morse potential V(x) = D_e (1 - e^{-\alpha (x - x_e)})^2 offers an exact solvable model, with bound-state energies E_v = \hbar \omega (v + \frac{1}{2}) - \frac{(\hbar \omega)^2}{4 D_e} (v + \frac{1}{2})^2 and wavefunctions expressible via associated Laguerre polynomials, naturally limiting the number of levels to v_{\max} < D_e / (\hbar \omega) - 1/2.

Physical Manifestations

In Molecular Systems

In molecular systems, anharmonicity manifests prominently in the vibrational dynamics of diatomic and polyatomic molecules, where the harmonic approximation fails to capture essential features like bond dissociation and mode interactions. For diatomic molecules, the provides a foundational anharmonic model for the interatomic interaction, given by V(r) = D_e \left(1 - e^{-a(r - r_e)}\right)^2, where D_e is the well depth, r_e the equilibrium bond length, and a a parameter controlling the width of the potential. This form accurately reproduces the finite dissociation energy and the decreasing spacing between vibrational energy levels at higher quanta, unlike the parabolic harmonic potential that predicts infinite bond strength. The was introduced to solve the for nuclear motions in diatomic systems, enabling exact analytical solutions for vibrational levels. In polyatomic molecules, vibrations are described through normal modes, but anharmonic couplings between these modes introduce interactions that mix overtone and combination states. A classic example is Fermi resonance, where near-degeneracy between a fundamental mode and an overtone or combination leads to energy level repulsion and intensity borrowing. In carbon dioxide (CO₂), the symmetric stretch fundamental ν₁ resonates with the bending overtone 2ν₂, resulting in split and shifted bands observed in spectra; this phenomenon was first explained for CO₂'s Raman spectrum to account for unexpected multiple peaks. Such couplings are widespread in linear and nonlinear polyatomics, altering the simple harmonic picture of independent oscillators. Bond anharmonicity further explains the observed weakening and red-shifting of higher vibrational overtones relative to harmonic predictions, as the potential asymmetry reduces transition moments and compresses energy spacings at elevated levels. This effect is crucial for accurate thermochemistry, where anharmonic corrections to zero-point energies (ZPEs) can shift enthalpies of formation by several kcal/mol, tipping the balance in reaction energetics for systems like hydrocarbons. For instance, including anharmonicity in ZPE calculations via vibrational perturbation theory refines predicted stabilities, highlighting its role beyond the harmonic baseline. Experimentally, anharmonicity is evident in infrared (IR) and Raman spectra of molecules, where hot bands—transitions from thermally populated excited vibrational states—appear as weaker features at lower frequencies due to the contracted level spacing. Similarly, combination tones arise from coupled excitations of multiple modes, gaining intensity through anharmonic mixing and providing insights into intramolecular interactions without requiring high excitation energies. These features are routinely analyzed in diatomic and polyatomic spectra to quantify anharmonic constants, as seen in studies of simple gases like and .

In Condensed Matter

In condensed matter physics, anharmonicity plays a crucial role in the behavior of lattice vibrations, particularly through phonons, which are quantized collective oscillations in crystals. Anharmonic interactions lead to phonon-phonon scattering processes that limit thermal conductivity by enabling energy transfer between modes. Among these, Umklapp processes—where the total wave vector changes by a reciprocal lattice vector—dominate at high temperatures and are responsible for the finite thermal resistivity in insulators and semiconductors, as opposed to the momentum-conserving normal processes that merely redistribute heat without net resistance. For instance, in materials like silicon, these scattering events result in thermal conductivities that decrease with increasing temperature, following a T^{-1} dependence in the high-temperature limit due to the cubic anharmonicity in the lattice potential. To account for volume-dependent effects while approximating lattice dynamics, the quasi-harmonic approximation is widely employed, treating phonons as harmonic at each fixed volume but allowing frequencies to vary with lattice constant. This approach captures anharmonicity indirectly through thermal expansion and is quantified by the mode-specific Grüneisen parameter, defined as \gamma = -\frac{d \ln \omega}{d \ln V}, where \omega is the phonon frequency and V is the volume; positive values indicate softening of modes under expansion, reflecting asymmetric interatomic potentials. In crystals such as diamond, average \gamma values around 1-2 highlight moderate anharmonicity, enabling accurate predictions of thermodynamic properties like the equation of state from first-principles calculations. A key macroscopic consequence of anharmonicity is thermal expansion, stemming from the coupling between vibrational modes and lattice volume; asymmetric potentials cause the average interatomic distance to increase with temperature as higher-energy anharmonic terms populate. In most materials, this yields positive expansion coefficients, but exceptional cases exhibit negative thermal expansion (NTE), where the lattice contracts upon heating due to transverse vibrational modes that effectively pull atoms closer. Zirconium tungstate (ZrW_2O_8), a prototypical NTE material, displays this behavior from 0.3 K to nearly 1050 K, with a volumetric coefficient of -29 \times 10^{-6} K^{-1} at room temperature, attributed to low-frequency rocking modes of ZrO_6 octahedra involving strong anharmonic coupling. Anharmonicity also influences defect dynamics in solids, where impurities introduce local vibrational modes that deviate from the host lattice spectrum due to altered force constants. These modes, often appearing as resonances or gaps in the phonon density of states, interact with lattice anharmonicity to produce broadened or split features in neutron scattering spectra, reflecting lifetime shortening from scattering. In impure anharmonic solids, such as those with isotopic defects, this "impurity-anharmonicity interference" manifests as additional low-frequency tails or asymmetric line shapes, altering thermal and transport properties. For example, hydrogen impurities in palladium deuteride exhibit broad peaks around 68 meV in inelastic neutron scattering, superimposed on host lattice features, due to coupled local and anharmonic vibrations.

Effects and Consequences

Oscillation Period Dependence

In classical mechanics, anharmonicity leads to an oscillation period that varies with the amplitude or energy of the motion, unlike the constant period of a harmonic oscillator. This dependence arises because the restoring force is nonlinear, altering the effective frequency as the displacement changes. For potentials softer than quadratic, such as the simple pendulum where the potential is approximately V(\theta) \approx \frac{1}{2} m g \ell \theta^2 - \frac{1}{24} m g \ell \theta^4, the period increases with amplitude due to the negative quartic term reducing the effective stiffness at larger displacements. The approximate period for small but finite amplitudes \theta_0 (in radians) is given by T \approx 2\pi \sqrt{\frac{\ell}{g}} \left[1 + \frac{1}{16} \theta_0^2 + \cdots \right], where \ell is the pendulum length and g is gravitational acceleration; this shows a quadratic increase in T with \theta_0, with the correction term becoming significant for \theta_0 > 10^\circ. For symmetric anharmonic potentials like the quartic V(x) = \frac{1}{2} m \omega_0^2 x^2 + \lambda x^4 with \lambda > 0, the period decreases with increasing because the steeper walls at large |x| accelerate the particle more, shortening the cycle time; the scales as T \propto E^{-1/4}, where E is the total , diverging as E \to 0 and decreasing for higher E. In contrast, for a cubic perturbation V(x) = \frac{1}{2} m \omega_0^2 x^2 + \frac{1}{3} k_2 x^3 with k_2 > 0, the \omega' decreases with h according to \frac{d\omega}{dh} = -\frac{15 k_2^2}{ \pi m \omega_0^5 }, implying a longer at higher amplitudes as the particle spends more time in the softer potential region./22%3A_Resonant_Nonlinear_Oscillations/22.02%3A_Frequency_of_Oscillation_of_a_Particle_is_a_Slightly_Anharmonic_Potential) In the quantum mechanical treatment, anharmonicity causes energy level spacings \Delta E_n = E_{n+1} - E_n to vary with the principal n, unlike the constant \hbar \omega_0 in the case. An effective classical-like period can be defined as T_n = 2\pi \hbar / \Delta E_n, which increases with n because spacings typically decrease for anharmonic potentials (e.g., due to the softening effect in molecular vibrations modeled by potentials). This quantum analogy mirrors the classical dependence, with larger n corresponding to higher effective energies. The variation in period with amplitude serves as a key test for isochronism, where harmonic systems maintain constant periods across amplitudes, while anharmonic ones do not; experimentally, this is assessed by measuring oscillation times at different initial displacements, revealing nonlinear effects through period shifts. In precision timekeeping, such as pendulum clocks, anharmonicity limits accuracy unless amplitudes are kept small (\theta_0 \lesssim 5^\circ) to approximate isochronism, as larger swings introduce timing errors up to several percent. Similar period dependencies appear in seismology, where anharmonic oscillations in Earth's crust during large earthquakes lead to amplitude-dependent wave periods, complicating long-period seismic signal analysis. In driven anharmonic systems, higher-order effects can produce bifurcations, such as period-doubling cascades, where the oscillation period doubles successively as driving amplitude increases, eventually leading to ; this was experimentally observed in a parametrically driven oscillator at twice the natural , with the route to chaos following a universal scaling law independent of the specific nonlinearity.

Energy Level Shifts

In the approximation, the energy levels are equally spaced with a constant separation of \Delta E = \hbar \omega, where \omega is the . Anharmonicity perturbs this equidistance, resulting in level spacings that decrease with increasing n, as higher levels experience stronger nonlinear effects from the potential. This shift arises primarily from cubic and quartic terms in the potential , leading to a of levels toward the limit in bound systems. The modified level spacing can be expressed as \Delta E_n = \hbar \omega + 2 \chi (n + 1/2), where \chi is the anharmonicity constant, which is negative for most molecular and solid-state systems, causing the spacings to narrow progressively. In diatomic molecules, this is quantified using the spectroscopic constants \omega_e (the frequency) and \omega_e x_e (the anharmonicity correction), derived from transitions in vibrational spectra; for example, the vibrational energy is E_v = \hbar \omega_e (v + 1/2) - \hbar \omega_e x_e (v + 1/2)^2, yielding \Delta E_v = \hbar \omega_e - 2 \hbar \omega_e x_e (v + 1/2). These constants are determined experimentally by fitting observed frequencies, which deviate from predictions, with typical values of x_e \approx 0.01 - 0.03 for common diatomics like HCl or . In coupled anharmonic systems, such as those involving light-matter interactions or multi-mode oscillators, level anticrossings occur due to off-diagonal terms that mix states near degeneracy. These avoided crossings manifest as a splitting in the , with the size proportional to the coupling strength; in regimes of strong anharmonicity, this leads to vacuum Rabi splitting, where the minimum separation at the anticrossing point equals the \Omega_R. For instance, in anharmonic vibrational , tuning parameters across degeneracy points reveals true or avoided crossings, enhancing state hybridization and observable splittings on the order of meV. Anharmonicity also introduces non-radiative relaxation pathways by enabling energy transfer to other , such as through intramolecular vibrational redistribution (IVR) or , which broadens the energy levels via lifetime effects. In multilevel systems, this results in Lorentzian linewidths that increase with excitation v, as higher levels couple more strongly to decay channels; for example, in liquid-phase vibrational , dephasing rates scale linearly with v due to anharmonic mode interactions, yielding broadenings from homogeneous relaxation on picosecond timescales. This broadening is distinct from pure and directly ties to the perturbed induced by anharmonicity.

Applications and Implications

In Spectroscopy

In vibrational , anharmonicity relaxes the strict of the model, which permits only transitions with Δv = ±1, allowing (Δv = ±2, ±3, ...) and bands that provide insights into higher vibrational states. These intensities are weaker than fundamentals and scale approximately as (v+1)v due to mixing of states, enabling the probing of anharmonic potential curvatures. Anharmonicity also induces spectral shifts, where the transition ω_{01} from v=0 to v=1 is lower than the prediction ω_e because spacings decrease with v, as described by the anharmonic term -ω_e x_e (v + 1/2)^2 in the vibrational expression (with x_e > 0). Additionally, hot bands arise from thermally populated upper levels (e.g., v=1 to v=2), appearing as red-shifted features relative to the due to reduced spacings, which complicates but enriches at elevated temperatures. Techniques like () and exploit these effects to quantify molecular properties; for instance, anharmonicity parameters ω_e and ω_e x_e from observed fundamentals and yield the bond dissociation energy via the relation D_e = \frac{\hbar \omega_e^2}{4 \omega_e x_e}, derived from the that models the asymmetric well. spectroscopy, particularly for polyatomic molecules, reveals local-mode anharmonicity in CH-stretching vibrations, as seen in where a single anharmonicity constant of approximately -55 cm^{-1} describes a series of bands across IR and visible regions. In advanced applications, anharmonicity enhances two-photon processes by altering resonances, as in of CO_2 overtones where off-resonance shifts of ~0.1 cm^{-1} due to anharmonicity enable sensitive detection limits down to parts-per-quadrillion. Furthermore, in , anharmonic IR emission spectra of polycyclic aromatic hydrocarbons (PAHs) in interstellar environments—modeled via —aid the identification of neutral molecules through shifted CH-stretching and bending bands matching astronomical observations.

In Nonlinear Phenomena

Anharmonicity plays a central role in the emergence of nonlinear wave phenomena, where deviations from harmonic behavior lead to interactions that alter wave propagation. In , anharmonic effects within the medium cause distortion as higher-amplitude portions of the wave travel faster than lower-amplitude regions, resulting in frequency mixing and the generation of harmonics. This process is evident in sound waves, where initial sinusoidal signals develop higher harmonics, contributing to audible in audio systems when amplitudes exceed linear limits. In optical systems, anharmonicity underlies the Kerr nonlinearity, a third-order effect that modulates the with light intensity. This nonlinearity balances in optical fibers, enabling the formation and stable propagation of solitons—self-reinforcing pulses that maintain their shape over long distances. Seminal studies have demonstrated how Kerr-induced supports fundamental and higher-order solitons, crucial for high-bit-rate . Chaotic dynamics arise prominently in anharmonic oscillators under external driving, exemplified by the Duffing oscillator, which models systems with cubic stiffness nonlinearity. As driving amplitude increases, the system undergoes a period-doubling cascade of bifurcations, transitioning from periodic to aperiodic motion and eventual chaos. This route to chaos is characterized by positive Lyapunov exponents, quantifying exponential sensitivity to initial conditions, as observed in numerical and experimental analyses of driven anharmonic systems. In nonlinear optics, anharmonicity drives second-harmonic generation (SHG) in non-centrosymmetric crystals through the second-order nonlinear susceptibility \chi^{(2)}, arising from the anharmonic potential experienced by bound electrons. When intense light at frequency \omega interacts with the crystal, it produces output at $2\omega, with efficiency enhanced by phase-matching conditions. This process, first theoretically framed using classical anharmonic oscillator models, underpins frequency conversion in lasers and has been experimentally verified in materials like . Acoustic shock waves in fluids exemplify anharmonicity's role in wave steepening, where nonlinear speed variations compress the wave front until balances the distortion, forming a discontinuous . In gases and liquids, this leads to via heating and increase, limiting distance and influencing applications like ultrasonics. From an engineering perspective, anharmonicity imposes limits on microelectromechanical systems () resonators by introducing Duffing-like nonlinearities that cause amplitude-dependent shifts and potential instabilities, reducing signal-to-noise ratios in sensors and oscillators. Feedback control strategies, such as phase-locked loops or delayed feedback, mitigate these effects by linearizing the response or stabilizing bifurcations, enabling operation beyond linear regimes with improved stability.