On the mobility of very pure semiconductors at very low temperatures

The mobility μ of a very pure semiconductor at very low temperatures is investigated in terms of a model where electrons are scattered by charged impurities distributed uniformly in space, and the electron-electron interaction is taken into account by the Debye-Hueckel screening in the interaction potential. The equation for the current relaxation rate Γ, derived previously by the proper connected diagram expansion, incorporates the quasi-particle effect in a self-consistent manner. The solution of this equation at high carrier concentrations n yields the so-called Brooks-Herring formula. At lower concentrations, the solution deviates significantly from the latter. The solution is in general smaller than the standard expression for the rate based on the Boltzmann equation; and this is consistent with the existing conductivity data available. At the very low concentrations e.g. n = n₃ = 10¹³cm^?3 or lower for Ge, the mobility calculated is inversely proportional to the square-root of the impurity concentration n_s, and has a $\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$-dependence (T: temperature).

$\text{μ = 0.3597\&z.xl;h}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{k(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^1{}^{\mathrm{<mtext>?3</mtext><mtext>4</mtext>}}$

, where k is the dielectric constant. The conductivity data directly comparable with this formula are not available at present. However, the quasi-particle effect which led to this peculiar concentration-dependence should also show itself in the cyclotron resonance width; there, experiment and theory both show the ${}_{\mathrm{n}}{}_{\mathrm{s}}$-dependence for very pure semiconductors.     

Quasiclassical mobility for extrinsic semiconductors

A quasiclassical formulation for mobility in extrinsic semiconductors is presented based on scattering from ionized impurity atoms. Quantum theory enters the otherwise classical Chapman-Enskog expansion of the Boltzmann equation through incorporation of the Thomas-Fermi interaction potential together with the Bom approximation for evaluation of scattering integrals. The following expression results for mobility μ_i, (cgs):

$\text{μ}{}_{\mathrm{i}}\text{3}\text{2}\text{?}{}^2\text{n}{}_{\mathrm{s}}\text{e}{}^3\text{m}{}^1\text{2}\text{2}\text{k}{}_{\mathrm{B}}\text{T}\text{2φ}\text{3}\text{2}\text{1}\text{f(γ)}$

,

$\text{f(γ)=[(1+γ)e}{}^{\mathrm{\gamma }}\text{E}{}_1\text{(γ)?1]}$

, where n_s is impurity concentration, m¹ is effective mass, E₁(γ) is the exponential integral, ? is dielectric constant and γ is dimensionless Thomas-Fermi energy. The structure of the dimensional factor in the preceding expression for μ_i agrees with previous expressions for this parameter.     

Theory of the mobility of electrons limited by charged impurities in two dimensions

The mobility μ of electrons limited by charged impurities in two dimensions is calculated, based on the proper connected diagram expansion of Kubo's formula. For a Coulomb-type interaction, the Boltzmann relaxation rate is $\text{Γ}{}^{\mathrm{(0)}}\text{= 2}\text{π}{}^2\text{h}\text{?}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{I}}\text{m}{}^1\text{z}{}^2\text{e}{}^4\text{κ}{}^{\mathrm{?2}}\text{p}{}^{\mathrm{?2}}$, where n_I, repr constant. This leads to a mobility proportional to temperature T for non-degenerate electrons. The quasi-particle effect, which is self-consistently contained in the proper connected diagram expansion, yields an equation for the relaxation rate Γ. The solution of this equation yields values in general reduced relative to Γ⁽⁰⁾ at higher densities. The values of μ obtained in our calculations, are higher than those predicted by the Boltzmann theory, and deviations are pronounced at low temperatures.     

Exact Fermi energy,susceptibilities, and their singularities for an electron gas in a uniform magnetic field at T = 0 °K

Exact analysis is presented to derive the magnetic response functions and their singularities of free-electron gas in a uniform magnetic field of arbitrary strength at T = 0 °K. The newly defined functions, Λ_μ(s) = ∑₀^[s])(s ? n)^μ of $\text{μ =}\text{?1}\text{2}\text{,}\text{1}\text{2}\text{,}\text{3}\text{2}$, are employed to obtain the Fermi energy, magnetization, and susceptibility as functions of B. It is revealed that the spin susceptibility is composed of two parts, χ_s1 and χ_s2, where χ_s2 is purely oscillatory diamagnetic. A graphical method of finding the Fermi energy ?_F(B) as a function of B has been obtained. The system is shown to become totally one-dimensional electron gas in the field B greater than $\text{B}{}_{\mathrm{\downarrow }}\text{= (2ηn)}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}$ and the total energy satisfies $\text{E}{}_{\mathrm{t}}\text{=}\text{1}\text{3}\text{?}{}_{\mathrm{F}}\text{(B)N}$. The obvious extension of the present theory to the Bloch electrons on the ellipsoidal constant energy surface is also discussed.     

Surface cyclotron resonance in Si under uniaxial stress

The cyclotron resonance of inversion-layer electrons on (100)p-type Si is found to depend sensitively on an externally applied compressive stress. At low temperatures (T ? 10 K) we observe a considerable increase of the cyclotron mass m¹_c with stress S along the [001] direction. The effect is most strongly observed at low electron densities n_s. For $\text{S～1.5 × 10}{}^9\text{dyne}\text{cm}{}^2$ and n_s～2 × 10¹¹cm^-2 we obtain $\text{m}{}^1{}_{\mathrm{c}}\text{～0.4 m}{}_0$ instead of the expected 0.2m₀. Along with this change of $\text{m}{}^1{}_{\mathrm{c}}$ a strong narrowing of the resonance is noted. Raising the temperature gives an additional n_s- dependent increase of $\text{m}{}^1{}_{\mathrm{c}}$.     

Glueball spectroscopy from strong coupling expansions in hamiltonian lattice QCD

Masses of all the glueballs which are created by 6- or 7-link operators are calculated to order g^?8 in pure SU(3) hamiltonian lattice gauge theory. Several low-lying states are found with masses $\text{m(0}{}^{\mathrm{++1}}\text{)～ 1.4 m}{}_{\mathrm{<mtext>s</mtext>}}\text{, m(0}{}^{\mathrm{++7}}\text{) ～ 1.7 m}{}_{\mathrm{<mtext>s</mtext>}}$ (¹ and ⁷ stand for radial excitations and m_s is the mass of the lowest 0⁺⁺ state), . These values are obtained at the point g^?2 ? 0.8, which lies near the scaling region.     

Thermal particle production in a contracting and expanding universe without singularity

The creation of massive spin-0 particles described by the conformally coupled Klein-Gordon equation in a 3-flat Robertson-Walker universe ds² = R²(η)(dη² ? dx² ? dy² ? dz² with contraction-expansion law R² = b²η² + R²₀ for ?∞ < η < +∞ is studied. It represents a time-symmetric universe for which the singularity is avoided. Particles are created in the out-region with a nonrelativistic thermal spectrum with temperature T = b(2πR²k_B)^?1 and chemical potential $\text{μ =}\text{?1}\text{2}\text{mR}{}^2{}_0\text{R}{}^{\mathrm{?2}}$. The latter implies an expotential damping for large m.     

Lepton mixing and neutrino oscillations

The present article is a review of phenomena connected with neutrino oscillations. Mixing of two neutrinos (Majorana as well as Dirac) with masses m₁ and m₂ is considered in detail. It is shown that the hypothesis of lepton mixing is not in contradiction with the existing data if |m₁²?m₂²| ? 1 (eV)². Possible experiments designed to reveal neutrino oscillations at reactor, meson factory and high energy accelerator facilities are considered. In such experiments oscillation might be found if $\text{|m}{}_1{}^2\text{?m}{}_2{}^2\text{| ? 0.01}\text{(eV)}{}^2$. The possibilities of searching for oscillations by experiments on cosmic ray neutrinos and especially on solar neutrinos are discussed in detail. The last experiments have an incredible high sensitivity from the point of view of testing the lepton mixing hypothesis (oscillation effects might be observable if $\text{|m}{}_1{}^2\text{?m}{}_2{}^2\text{| ? 10}{}^{\mathrm{?12}}\text{(eV)}{}^2$). The “solar neutrino puzzle” is also discussed from the point of view of lepton mixing. Neutrino oscillations are considered then in the case where in nature there exist N ? 2 neutrino types.In conclusion the case of heavy lepton mixing is considered. It is shown that in a concrete scheme with right-handed currents, the probabilities of such processes as μ → eγ, μ → 3e etc. can be close to existing experimental upper limits, provided the heavy lepton masses are of an order of a few GeV, whereas the probabilities of the above processes are entirely negligible if only neutrinos are mixed.     

Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

Φecчet ДИLcy;н boЛн Л sИЛ osцИЛЛЯtoΦo b ДЛЯ ИЗoЭЛeКtponnoЙ ПosЛeДoBateЛЬn ostИ Li

ФeйnмanoBsкaя диaгpaмnaя teчnи кa пpимenena для pasЧeta длen Bo лn и sил osцилляtopoB osnBoг и nикoto pыч neжnич Boэбyждennыч sstoяn ий Li-пoдoбnыч иonoB. passЧиtanы Bклaды ot диaгp aмм пepBыч пopядкoB длк nepeляtи Bиstsкoй эnepгии, peляtиBиstsкич пoпpaBoк и дипoл ьnыч matpиЧnыч matpиЧnыч элeme ntoB. Для pasЧeta peляtиBиstsкич пoпpaBoк был иsпoльэoBan oпepat op Бpeйta. pяд пo $\text{1}\text{z}$ для эnepгии пpe дstaBлen B sлeдyющeem Bидe

$\text{E = E}{}_0\text{z}{}^2\text{+ΔE}{}_1\text{z+}\text{1}\text{z}\text{ΔE}{}_3\text{+}\text{α}{}^2\text{4}\text{(E}{}_0{}^{\mathrm{p}}\text{z}{}^4\text{+ ΔE}{}_1{}^{\mathrm{p}}\text{z}{}^3\text{),}$

для дипoльnoгo matpиЧoгo элe

$\text{P =}\text{a}\text{z}\text{1+}\text{τ}{}_1\text{z}\text{+}\text{τ}{}_2\text{z}{}^2\text{.}$

ПoлyЧennыe Чиsлennяe эnaЧennы e эnaЧenия кoэффициentoB пpи z^k дaли Boэmoжnostь passЧиtatь длиnы Boлn и sилы osцилляtopoB пe peчoдoB 1s²2s ? 1s²2p, 1s²2s ? 1s²3p, 1s²2p ? 1s²3s, 1s²2p ? 1s²3d, 1s²3s ? 1s²3p, 1s²3p ? 1s²3d для Li-п oдoбnыч иonoB. peэyльtatы pasЧe ta spaBnиBaюtsя s экsпepиmentaльныmи для иэoэлeкtpnnoй пoлeдoBat eльnostи Li. Чopoшee soглasиe s экsпepиment aльnыmи (0,01–0,1%) дaet Boэmoжnostь naдetьsя, Чto pяд пo $\text{1}\text{z}$ sчoдиtsя д ostatoЧno быstpo.Feynman diagram techniques have been applied to the calculation of wavelengths and oscillator strengths of the ground state and of a number of low-lying excited states for Li-like ions (1s²2l, 1s²3l). Contributions have been calculated to the first order for the nonrelativistic energy, relativistic corrections and dipole matrix elements. Relativistic corrections have been obtained by computing the active 〈H_B〉 matrix. Numerical results for the $\text{1}\text{z}$ expansion are presented in the following form: for the energy,

$\text{E = E}{}_0\text{z}{}^2\text{+ΔE}{}_1\text{z+}\text{1}\text{z}\text{ΔE}{}_3\text{+}\text{α}{}^2\text{4}\text{(E}{}_0{}^{\mathrm{p}}\text{z}{}^4\text{+ ΔE}{}_1{}^{\mathrm{p}}\text{z}{}^3\text{),}$

for the dipole matrix elements,

$\text{P =}\text{a}\text{z}\text{1+}\text{τ}{}_1\text{z}\text{+}\text{τ}{}_2\text{z}{}^2\text{.}$

The results were used for calculations of the wavelengths and oscillator ofthe transitions 1s²2s ? 1s²2p, 1s²2s ? 1s²3p, 1s²2p ? 1s²3s, 1s²2p ? 1s²3d, 1s²3s ? 1s²3p, 1s²3p ? 1s²3d for Li-like ions. Results are compared with experimental data for the isoelectronic sequence of Li (Li I-SX IV). Good agreement with experimental data (0·01–0·1%) suggests that the $\text{1}\text{z}$-expansion converges rapidly.     

An inequality for the trace of matrix products

In this paper we consider a product of n complex m×m matrices A_k (k=1,…,n) with singular values ∝^(k)_i ordered in decreasing magnitude. Using the spectral resolution for the operators $\text{A}{}^{\mathrm{dagger;}}{}_{\mathrm{k}}\text{A}{}_{\mathrm{k}}$, it is shown that $\text{|}\text{Tr}\text{A}{}_1\text{…A}{}_{\mathrm{n}}\text{|≤}\text{∑}\text{i=1}\text{m}\text{Φ}\text{i=1}\text{n}\text{α}{}^{\mathrm{(k)}}{}_{\mathrm{i}}\text{.}$This inequality is an extension of an inequality of von Neumann in the simple case that n=2. The necessary and sufficient condition for the equality sign to hold is established. Application of Hölder's inequality leads to further inequalities which can be useful in statistical mechanics.     

The asymptotics of the gap in the Mathieu equation

We provide a simple proof that the kth gap, Δ_k, for the Mathieu operator $\text{?d}{}^2\text{dx}{}^2\text{+ 2κ}\text{cos}\text{(2x)}$ is $\text{Δ}{}_{\mathrm{k}}\text{= 8(}\text{κ}\text{4}\text{)}{}^{\mathrm{k}}\text{[(k ? 1)!]}{}^{\mathrm{?2}}\text{(1 + o(k}{}^{\mathrm{?2}}\text{))}$, a result obtained (up to the value of an integral) by Harrell. The key observation is that what is involved is tunneling in momentum space.     

Contributions of operators of twist two and higher in large n moments of structure functions

Though high twist terms are becoming important as x→1, or equivalently, in large n moments, their detection in this regime in deep inelastic lepton scattering needs special caution. The high order terms in the twist two component are strongly dependent on n; one finds that at $\text{Q}{}^2\text{?Q}{}^2{}_7\text{?Λ}{}^2\text{a}{}_{\mathrm{k}}\text{exp}\text{[α}{}_{\mathrm{k}}\text{(}\text{log}\text{n)}{}^{\mathrm{<mtext>2?1</mtext><mtext>k</mtext>}}\text{(}\text{1+b}{}_{\mathrm{k}}\text{log}\text{n}\text{)]}$ the perturbative expansion is invalid whereas higher twist terms are important at $\text{Q}{}^2\text{?Q}{}_1{}^2\text{= Λ}{}^2\text{nC}$. Since $\text{Q}{}_7{}^2$ grows very fast with n the necessary requirement for any deep inelastic phenomenological analysis, namely $\text{Q}{}_1{}^2\text{?Q}{}_7{}^2$, cannot hold for too large moments. The scheme dependence of a_k, α_k and b_k is also discussed.     

Electron-exciton inelastic collision cross sections for different semiconductors

The electron-exciton inelastic collision cross sections for the different semiconductors CdS, ZnO, CdSe, Si, Cu₂O, CuCl, CuBr and CuI have been calculated in the Glauber approximation. The transitions 1s–2s, 1s–3s, 1s–2p and 1s–3p have been considered. The calculations are carried out as function of the different available values of $\text{σ =}\text{m}{}^1{}_{\mathrm{e}}\text{m}{}^1{}_{\mathrm{h}}$ where m¹_e and m¹_h are, respectively, the electron a corresponding semiconductors.     

Influence of self-fields on the Vlasov equilibrium and stability properties of relativistic electron beam-plasma systems

The influence of self-fields on the equilibrium and stability properties of relativistic beam-plasma systems is studied within the framework of the Vlasov-Maxwell equations. The analysis is carried out in linear geometry, where the relativistic electron beam propagates through a background plasma (assumed nonrelativistic) along a uniform guide field $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$, It is assumed that $\text{ν}\text{γ}{}_0\text{? 1}$ for the beam electrons (ν is Budker's parameter, and γ₀mc² is the electron energy), but no a priori assumption is made that the beam density is small (or large) in comparison with the plasma density, or that conditions of charge neutrality or current neutrality prevail in equilibrium. It is shown that the equilibrium self-electric and self-magnetic fields, $\text{E}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{r}}$ and $\text{B}{}_{\mathrm{\theta }}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{\theta }}$, can have a large effect on equilibrium and stability behavior. Equilibrium properties are calculated for beam (j = b) and plasma (j = e, i) distribution functions of the form f_b⁰(H, P_θ, P_z) = F(H ? ω_rbP_θ) × δ(P_z ? P₀)(j = b), and $\text{f}{}_{\mathrm{j}}{}^0\text{(H, P}{}_{\mathrm{\theta }}\text{, P}{}_{\mathrm{z}}\text{) = f}{}_{\mathrm{j}}{}^0\text{(H ? ω}{}_{\mathrm{rj}}\text{P}{}_{\mathrm{\theta }}\text{? V}{}_{\mathrm{j}}\text{P}{}_{\mathrm{z}}\text{?}\text{m}{}_{\mathrm{i}}\text{V}{}_{\mathrm{j}}{}^2\text{2}\text{)}$ (j = e, i), where H is the energy, P_θ is the canonical angular momentum, P_z is the axial canonical momentum, and ω_rj (the angular velocity of mean rotation for j = b, e, i), V_j (the mean axial velocity for j = e, i), and P₀ are constants. The linearized Vlasov-Maxwell equations are then used to investigate stability properties in circumstances where the equilibrium densities of the various components (j = b, e, i) are approximately constant. The corresponding electrostatic dispersion relation and ordinary-mode electromagnetic dispersion relation are derived (including self-field effects) for body-wave perturbations localized to the beam interior (r <R_b). These dispersion relations are analyzed in the limit of a cold beam and cold plasma background, to illustrate the basic effect that lack of charge neutrality and/or current neutrality can have on the two-stream and filamentation instabilities. It is shown that relative rotation (induced by self-fields) between the various components (j = b, e, i) can (a) result in modified two-stream instability for propagation nearly perpendicular to $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$, and (b) significantly extend the band of unstable k_z-values for axial two-stream instability. Moreover, in circumstances where the beam-plasma system is charge-neutralized but not current-neutralized, it is shown that the azimuthal self-magnetic field $\text{B}{}_{\mathrm{\theta }}{}^{\mathrm{s}}\text{(r)}\text{e}\text{?}{}_{\mathrm{\theta }}$ has a stabilizing influence on the filamentation instability for ordinary-mode propagation perpendicular to $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$.     

On the charged-impurity-limited mobility for a very pure semiconductor

The low temperature mobility μ limited by charged impurities in very pure semiconductors (Ge) is interpreted in terms of the Coulomb collision in medium. The theory yields a peculiar dependence in temperature T and charged impurity concentration $\text{n}{}_{\mathrm{s}}\text{: μ ∞n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>-</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$.     

Microwave optical double resonance and continuous wave dye laser excitation spectroscopy of NO2: Rotational assignment of the K = 0−4 subbands of the 593 nm band

A rotational assignment of approximately 80 lines with K_a′ = 0, 1, 2, 3, and 4 has been made of the 593 nm ${}^2\text{A}{}_1\text{→}{}^2\text{B}{}_2$ band of NO₂ using cw dye laser excitation and microwave optical double-resonance spectroscopy. Rotational constants for the ²B₂ state were obtained as A = 8.52 cm^?1, B = 0.458 cm^?1, and C = 0.388 cm^?1. Spin splittings for the K_a′ = 0 excited state levels fit a simple symmetric top formula and give $\text{(?}{}_{\mathrm{bb}}\text{+ ?}{}_{\mathrm{cc}}\text{)}\text{2}\text{= ?0.0483}\text{cm}{}^{\mathrm{?1}}$. Spin splittings for K_a′ = 1 (N′ even) are irregular and are shown to change sign between N′ = 6 and 8. Assuming that the large inertial defect of 4.66 amu Ų arises solely from A, a structure for the ²B₂ state is obtained which gives $\text{r}\text{(NO) = 1.35}\text{A}\text{?}$ and an ONO angle of 105°. Alternatively, weighting the three rotational constants equally gives $\text{r = 1.29}\text{A}\text{?}$ and θ = 118°.     

The nuclear quadrupole interaction in Pr3+:LaF3 — An optical-RF double resonance measurement of the ground electronic state

The nuclear quadrupole hyperfine splittings of Pr³⁺ in LaF₃ have been measured for the ground electronic state using a RF optical resonance technique. A hamiltonian $\text{H}\text{= P[(I}{}^2{}_{\mathrm{z}}\text{?}\text{1}\text{3}\text{I(I+1) + (η/6)(I}{}^2{}_{\mathrm{+}}\text{?I}$²_-)] was fitted to the data with zP=4.185 ± 0.003 MHzandη = 0.105 ± 0.010. Linewidths of 180 kHz were observed.     

3Σu−-3Σu+ coupling in the O2B3Σu− predissociation

The role of $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ spin-orbit mixing in the O₂ Schumann-Runge predissociation is investigated. The $\text{2}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ state is found to cross the $\text{B}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}$ state near 2.0 Å with an interaction matrix element of approximately 55 cm^?1. This state contributes to the widths of the Bv ≥ 6 levels, but introduces only small level shift perturbations. When the partial widths due to the ${}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{?}}\text{-}{}^3\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ interaction are added to the previously calculated widths due to the ⁵Π_u, ³Π_u, and ¹Π_u states, reasonable agreement is obtained with experimental measurements on O¹⁶O¹⁶ and O¹⁸O¹⁸. The possibility of non-Lorentzian line profiles and the dependence of the width on rotational quantum number is investigated. The approximation of the spin-orbit matrix element by its value at the crossing point is shown to be a good approximation for calculating the second difference perturbations.