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 $\text{B}{}_0\text{e}\text{?}{}_{\mathrm{z}}$.     

Analysis of the 2770-Å emission system in I2

A weak emission spectrum of I₂ near 2770 Å is reanalyzed and found to to minate on the A(1u³Π) state. The assigned bands span v″ levels 5–19 and v′ levels 0–8. The new assignment is corroborated by isotope shifts, band profile simulations, and Franck-Condon calculations. The excited state is an ion-pair state, probably the 1g state which tends toward $\text{I}{}^{\mathrm{?}}\text{(}{}^1\text{S) +}\text{I}{}^{\mathrm{+}}\text{(}{}^3\text{P}{}_1\text{)}$. In combination with other results for the A state, the analysis yields the following spectroscopic constants: T″_e = 10 907 cm^?1, $\text{D}$″_e = 1640 cm^?1, ω″_e = 95 cm^?1, $\text{R″}{}_{\mathrm{e}}\text{= 3.06}\text{A}\text{?}$; T′_e = 47 559.1 cm^?1, ω′_e = 106.60 cm^?1, $\text{R′}{}_{\mathrm{e}}\text{= 3.53}\text{A}\text{?}$.     

Functional derivative δTc/δα2 (Ω)F(Ω) for weak coupling superconductors

We present approximate analytic calculation of the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω)}$, where T_c is the superconducting critical temperature and $\text{α}{}^2\text{(Ω)F(Ω)}$ is the electron-phonon spectral function, within the “square-well model” for the phonon mediated electron-electron interaction and weak coupling limit $\text{ω}{}_{\mathrm{D}}\text{(2πT}{}_{\mathrm{c}}\text{)? 1 (ω}{}_{\mathrm{D}}$ is the Debye energy). It is found that $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω) = (1 + λ)}{}^{-1}\text{G(Ω)}$ where λ is the familiar electron-phonon coupling parameter and $\text{G(Ω)}$ is a universal function of the reduced frequency $\text{Ω}\text{=}\text{Ω}\text{T}{}_{\mathrm{c}}$. We compare this formula with accurate numerical results for several weak coupling superconductors. The overall agreement is good     

Rotational analysis of the B3Π(0+) → X1Σ+ band systems of 35Cl35Cl and 35Cl37Cl in the chlorine afterglow emission spectrum

The B³Π(0⁺) → X¹Σ⁺ band system of Cl₂, excited by the recombination of ground state $\text{Cl}{}^2\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ atoms at total pressures near 2 Torr, has been rotationally analyzed in the range 6300–9900 Å. About 30 bands, with 0 ≤ v′ ≤ 6 and 5 ≤ v″ ≤ 14, were investigated, mostly for both ³⁵Cl³⁵Cl and ³⁵Cl³⁷Cl. The band origins and rotational constants for the B state were obtained with the help of the known constants for the ground state. The principal molecular constants (cm^?1) for the B³Π(0⁺) state of ³⁵Cl³⁵Cl are as follows: T_e′ = 17 817.67(3); ω_e′ = 255.38(3); ω_e′x_e′ = 4.59(1); ω_e′y_e′ = ?0.038(8); D_e′ = 3341.17(14); B_e′ = 0.16313(3); α_e′ = 2.42(3) × 10^?3; γ_e′ = ?5.7(7) × 10^?5. The equilibrium internuclear separation is 2.4311(2) Å. The results of Briggs and Norrish on a transient absorption spectrum of Cl₂ assigned as $\text{0}{}_{\mathrm{g}}{}^{\mathrm{+}}\text{← B}{}^3\text{Π}\text{(0}{}^{\mathrm{+}}\text{)}$ are reinterpreted with the present constants.     

The emission spectra of H35Cl+, H37Cl+, D35Cl+, and D37Cl+ in the region 2700–4100 Å

The $\text{A}{}^2\text{Σ}{}^{\mathrm{+}}\text{-X}{}^2\text{Π}$ emission spectrum of HCl⁺ has been measured and analyzed for four isotopic combinations. These analyses extend previous work and provide rotational constants for the v = 0–2 levels of the ground state and for the v = 0–9 levels of the excited state. RKR potentials have been determined for both states, although the upper state could not be fitted precisely to such a model. Calculated relative intensities based on these potentials demonstrated that the electronic transition moment must change rapidly with lower state vibrational quantum number. Although considerable caution should be exercised in applying the concept of equilibrium constants to the $\text{A}{}^2\text{Σ}{}^{\mathrm{+}}$ state, the following are the best estimates of these constants (in cm^?1) for the $\text{X}{}^2\text{Π}$ state of H³⁵Cl⁺: B_e = 9.9406, ω_e = 2673.7, A_e = ? 643.7, and $\text{r}{}_{\mathrm{e}}\text{= 1.315}\text{A}\text{?}$. For the $\text{A}{}^2\text{Σ}{}^{\mathrm{+}}$ state of H³⁵Cl: T_e = 28 628.08, B_e ～ 7.505, ω_e ～ 1606.5, and $\text{r}{}_{\mathrm{e}}\text{= 1.514}\text{A}\text{?}$.     

The 2780 Å absorption system of thiophosgene

The vapor phase absorption spectrum of thiophosgene (Cl₂CS) in the 2500–2900 Å region consists of a broad intense band ($\text{log}\text{?}{}_{\mathrm{<mtext>max</mtext>}}\text{= 3.5}\text{at}\text{2540}\text{A}\text{?}$. On the red side of this a vibrationally discrete structure is found which becomes increasingly diffuse and merges into the broad band as the wavelength is decreased. It is shown that this vibrational structure can be explained as due to a $\text{π → π}{}^1$, ${}^1\text{A}{}_1\text{-}\text{X}\text{?}{}^1\text{A}{}_1$ electronic transition between a planar ground state and a pyramidal excited state of the molecule. In the latter state, the CS stretching mode ν₁′(a₁) = 681 cm^?1 and the CCl bending mode ν₃′(a₁) = 147 cm^?1. From the inversion doublet splitting of the out-of-plane mode ν₄′(b₁), the barrier to inversion is calculated to be ～126 cm^?1, with an equilibrium out-of-plane angle of ～20°.     

A new electronic band system of PbO

The chemiluminescence spectrum of atomic Pb reacting with O₃ under single-collision conditions includes a series of 55 bands in the regions 450–850 nm. A vibrational analysis is obtained which shows emission is to the ground state of PbO from excited electronic states not previously analyzed. Forty-nine of the bands are assigned to the a(1)-X(0⁺) transition and the remaining six are tentatively identified as the forbidden b(0^?)-X(0⁺) transition. Both the a and b states are believed to be Hund's case (c) components of the ³Σ⁺ states arising from the configuration $\text{σ}{}^2\text{π}{}^3\text{π}{}^1$. The vibrational parameters of the a state are ν₄ = 16 029 ± 8, ω_e′ = 478.7 ± 1.9, and ω_ex_e′ = 2.292 ± 0.128 cm^?1, where the uncertainties represent two standard deviations of the least-squares fit. Emission is also observed from the PbO B state produced in the reaction of metastable Pb atoms with O₃. Using pulsed laser excitation, an attempt is made to determine radiative lifetimes. We find for the PbO A(0⁺) state τ = 3.74 ± 0.3 μsec, and for the PbO B(1) state τ = 2.58 ± 0.3 μsec, while for the a(1) state τ is estimated to be greater than 10 μsec. From the vibrational analysis, energy conservation arguments place a lower limits to the ground state dissociation energy of $\text{D}{}_0{}^0\text{(}\text{PbO}\text{) ≥ 3.74 ± 0.03}\text{eV}$ (86.2 ± 0.7 kcal/mole). For the Pb + O₃ reaction we find less than 1% of the products are PbO¹ molecules that emit in the visible. Correlations are made with the low-lying states of other Group IV chalconides based on the assignment of the PbO $\text{a}{}^3\text{Σ}{}^{\mathrm{+}}\text{(1)}$ state and the correspondence between the low-lying triplet states of PbO and CO.     

The A′(3Π2) state of ICl

Although $\text{A′(}{}^3\text{Π}{}_2\text{) ← X(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)}$ is forbidden in near case c molecules the A′ ← X transition can be efficiently accomplished by the three-step sequence $\text{A′(}{}^3\text{Π}{}_2\text{) ← D′(2) ← A(}{}^3\text{Π}{}_1\text{) ← X(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)}$. Transitions to a range of levels of A′, v_A′ = 2–38, have been recorded by this means, using J-selective polarization-labeling spectroscopy. Principal constants of the A′ state of I³⁵Cl are T_e = 12682.05, ω_e = 224.57, ω_eχ_e = 1.882, ω_ey_e = ?0.0107, B_e = 0.08653, and α_e = 0.000675 cm^?1. The A′ state is therefore similar in its physical characteristics to two other (relatively) deep states, $\text{A(}{}^3\text{Π}{}_1\text{)}$ and $\text{B(}{}^3\text{Π}{}_0\text{+)}$, of the 2431 configuration.     

Determination of molecular constants of nitric oxide from (1-0), (2-0), (3-0) bands of the 15N16O and 15N18O isotopic species

The (1-0), (2-0), and (3-0) transitions of ¹⁵N¹⁶O and ¹⁵N¹⁸O are investigated. The wavenumbers of the rotation-vibration lines are reported for the overtone bands and the ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{-}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ (1-0) subband. It is shown that in the data reduction it is advantageous to calculate first merged spectroscopic constants ignoring the Λ-type doubling. The vibrational constants ω_e, ω_ex_e, ω_ey_e and the vibrational dependence of the rotational constants are determined. The study of ¹⁵N¹⁸O allows the determination of the equilibrium values of the centrifugal distortion correction A_De to the spin-orbit constant and of the spin-rotation constant γ_e from the isotopic invariance of the ratios $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}$. It is found that $\text{A}{}_{\mathrm{<mtext>De</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?3.9 ± 1.3) × 10}{}^{\mathrm{?6}}$ and $\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{B}{}_{\mathrm{<mtext>e</mtext>}}\text{= (?4.00 ± 0.05) × 10}{}^{\mathrm{?3}}$.     

High resolution Fourier spectroscopy of P2 infrared emission spectrum: Analysis of two new systems b3Πg → w3Δu and A1Πg → W1Δu

The investigation of the emission infrared spectrum of P₂ was performed with a high resolution Fourier spectrometer. Two new electronic systems were attributed to b³Π_g → w³Δ_u and A¹Π_g → W¹Δ_u transitions. The molecular parameters are obtained by a complete fitting procedure. The main equilibrium constants of the new states are (in cm^?1):

$\text{ω}{}^3\text{Δ}{}_{\mathrm{u}}\text{T}{}_{\mathrm{e}}\text{= 243228.07 ω}{}_{\mathrm{e}}\text{= 591.3 ω}{}_{\mathrm{e}}\text{X}{}_{\mathrm{e}}\text{= 2.5}$

$\text{B}{}_{\mathrm{e}}\text{= 0.256040 δ}{}_{\mathrm{e}}\text{= 0.001409 D}{}_{\mathrm{e}}\text{= 19.0 X 10}{}^{\mathrm{?8}}$

$\text{W}{}^1\text{Δ}{}_{\mathrm{u}}\text{T}{}_{\mathrm{e}}\text{= 31096.64 W}{}_{\mathrm{e}}\text{= 627.206 W}{}_{\mathrm{e}}\text{X}{}_{\mathrm{e}}\text{= 2.331}$

$\text{B}{}_{\mathrm{e}}\text{= 0.2628 δ}{}_{\mathrm{e}}\text{= 0.0014 D}{}_{\mathrm{e}}\text{= 23 X 10}{}^{\mathrm{?8}}$

     

Multiplicities in lepton-hadron processes

The average multiplicity in deep inelastic electro- and neutrinoproduction at large ω(ω～s/Q² + 1) is related in Feynman's version of the parton model to the average multiplicities in high-energy electron-positron annihilation and hadron-hadron scattering. The relation is: , where C_e⁺e^? and C_h are, respectively, the coefficients of ln(s/M_1⊥²) in the multiplicities from e⁺-e^? and P-P in to hadrons, and M_1⊥ is an average transverse mass.     

On the electronic spectrum of the Mo2 molecule observed after flash photolysis of Mo(CO)6

Absorption and emission spectra of Mo₂ were investigated using flash photolysis of the Mo(CO)₆ molecule. Tentative vibrational and rotational analyses of the ⁹⁸Mo₂ spectra were performed. For the ground state, ${}^1\text{Σ}{}_{\mathrm{g}}{}^{\mathrm{+}}$ type was proposed with ω_e = 477.1 cm^?1, $\text{r}{}_{\mathrm{e}}\text{= 1.929}\text{A}\text{?}$, and D₀(Mo₂) = 95 ± 15 kcal mole^?1. The results were compared with theoretical calculations for Mo₂ and experimental results for Cr₂ obtained previously. It seems reasonable that the transition metal diatomic molecules of this type have a high bond order.     

Fourier spectrometry: Rovibrational analysis of an infrared 1Π → 1Σ system of yttrium monoiodide

The infrared spectrum of yttrium monoiodide has been excited in an electrodeless microwave discharge and explored between 2500 and 12 000cm^?1 with a high-resolution Fourier transform spectrometer. A unique system is observed (ν₀₀ = 9905.520 cm^?1), which we attribute to a ${}^1\text{Π}\text{→}{}^1\text{Σ}$ transition and an extensive analysis is made. Rovibrational constants are obtained for both states mainly from a simultaneous multiband fitting. This procedure is applied to the whole set of 2231 observed line wavenumbers in the 1-0, 0-0, and 0–1 bands, yielding a final weighted standard deviation of 0.0038 cm^?1. Furthermore, a partial analysis of the 2-0 and 3-1 bands is performed. The following equilibrium constants are derived (cm^?1):

$\text{ω′}{}_{\mathrm{e}}\text{=192.210 ω′}{}_{\mathrm{e}}\text{x′}{}_{\mathrm{e}}\text{=0.463}$

$\text{B′}{}_{\mathrm{e}}\text{=0.0399133 α′}{}_{\mathrm{e}}\text{=0.0001150}$

$\text{ω″}{}_{\mathrm{e}}\text{=215.815 ω″}{}_{\mathrm{e}}\text{x″}{}_{\mathrm{e}}\text{=0.514}$

$\text{B″}{}_{\mathrm{e}}\text{=0.0422163 α″}{}_{\mathrm{e}}\text{=0.0001125}$

High-order constants D_v and H_v are also calculated for the various vibrational levels (v′ = 0, 1, 2, 3; v″ = 0, 1).     

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H₀ and H = H₀ + V(r) acting in L²(R³, dx) with V(r) a radial potential satisfying certain technical conditions, and for ? a real function on R with ?′ > 0 except on a discrete set, we prove that the Moller wave operators

exist and are independent of ?. The scattering operator

$\text{S = (Ω}{}^{\mathrm{+}}\text{)1Ω}{}^{\mathrm{?}}$

is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. $\text{V(r) =}\text{sin}\text{r}\text{r}$ and many others.     

Overtone bands of 14N16O and determination of molecular constants

The wavenumbers of the rotation-vibration lines of ¹⁴N¹⁶O are reported for the (2-0) and (3-0) bands. The full set of spectroscopic constants for the three bands (1-0), (2-0), and (3-0) has been determined with the method developed by Albritton, Schmeltekopf, and Zare for merging the results of separate least-squares fits. The vibrational constants ω_e, ω_ex_e, ω_ey_e, and the vibrational dependence of the rotational constants have been deduced. The apparent spin-orbit constant $\text{A}\text{?}{}_{\mathrm{<mtext>v</mtext>}}$ and its centrifugal correction $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext>}}$ (including the spin-rotation constant) have a vibrational dependence of the following form: $\text{A}\text{?}{}_{\mathrm{v}}\text{=}\text{A}\text{?}{}_{\mathrm{e}}\text{? α}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + γ}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{)}{}^2$ and $\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>v</mtext>}}\text{=}\text{A}\text{?}{}_{\mathrm{<mtext>D</mtext><mtext>e</mtext>}}\text{? β}{}_{\mathrm{A}}\text{(v +}\text{1}\text{2}\text{) + δ}{}_{\mathrm{A}}\text{(v +}\text{built+1}\text{2}\text{)}{}^2$; the values of the constants in these two equations have been determined.     

Sequential two-photon spectroscopy of some ion-pair states of IBr

Molecular constants of the first E 0⁺ ion-pair state of IBr vapor have been determined using polarization-labeling spectroscopy applied to the sequential transitions E 0⁺ ← B′ 0⁺ ← X 0⁺, while the second f 0⁺ ion-pair state is reported and characterized for the first time. A least-squares, simultaneous analysis of data for the I⁷⁹Br and I⁸¹Br isotopes gives the following constants (in cm^?1) for I⁷⁹Br:

$\text{E}\text{state}\text{: T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 39487.32(12), ω}{}_{\mathrm{<mtext>e</mtext>}}\text{= 119.518(21), ω}{}_{\mathrm{<mtext>e</mtext>}}\text{ξ}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.2109(12)}$

,

$\text{ω}{}_{\mathrm{<mtext>e</mtext>}}\text{y}{}_{\mathrm{<mtext>e</mtext>}}\text{= ? 2.34(22) × 10}{}^{\mathrm{?4}}\text{, B}{}_{\mathrm{<mtext>e</mtext>}}\text{= 2.9701(14) × 10}{}^{\mathrm{?2}}$

,

$\text{α}{}_{\mathrm{<mtext>e</mtext>}}\text{= 5.43(59) × 10}{}^{\mathrm{?5}}\text{,}\text{and}\text{γ}{}_{\mathrm{<mtext>e</mtext>}}\text{= ? 6.8(16) × 10}{}^{\mathrm{?7}}$

.

$\text{F}\text{state}\text{: T}{}_{\mathrm{<mtext>e</mtext>}}\text{= 45382.58(17), ω}{}_{\mathrm{<mtext>e</mtext>}}\text{= 128.805(66), ω}{}_{\mathrm{<mtext>e</mtext>}}\text{ξ}{}_{\mathrm{<mtext>e</mtext>}}\text{= 0.3630(69)}$

,

$\text{ω}{}_{\mathrm{<mtext>e</mtext>}}\text{y}{}_{\mathrm{<mtext>e</mtext>}}\text{= ? 9.7(22) × 10}{}^{\mathrm{?4}}\text{, B}{}_{\mathrm{<mtext>e</mtext>}}\text{= 3.0073(30) × 10}{}^{\mathrm{?2}}\text{,}\text{and}\text{α}{}_{\mathrm{<mtext>e</mtext>}}\text{= 8.52(48) × 10}{}^{\mathrm{?5}}$

. Preliminary data for the first $\text{Ω}\text{= 1}$ ion-pair state, accessed in the sequence $\text{1(}{}^3\text{P}{}_2\text{) ← A(}\text{Ω}\text{= 1) ← X 0}{}^{\mathrm{+}}$, indicate that T_e is ?30 cm^?1 higher in energy than that of the E state.     

The electronic isotope shift in the A2Π-X2Σ+ bands of BeH,BeD and BeT

The 0-0, 1-1, 2-2, and 3-3 bands of the A²Π-X²Σ⁺ transition of the tritiated beryllium monohydride molecule have been observed at 5000 Å in emission using a beryllium hollow-cathode discharge in a He + T₂ mixture. The rotational analysis of these bands yields the following principal molecular constants.

$\text{A}{}^2\text{Π:}\text{B}{}_{\mathrm{e}}\text{= 4.192}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.333}\text{A}\text{?}$

$\text{X}{}^2\text{Σ:}\text{B}{}_{\mathrm{e}}\text{= 4.142}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.341}\text{A}\text{?}$

$\text{ω}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″ = 16.36}\text{cm}{}^{\mathrm{?1}}\text{; ω}{}_{\mathrm{e}}\text{′X}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″X}{}_{\mathrm{e}}\text{″ = 0.84}\text{cm}{}^{\mathrm{?1}}$

From the pure electronic energy difference (E_Π - E_Σ)_BeT = 20 037.91 ± 1.5 cm^?1 and the corresponding previously known values for BeH and BeD, the following electronic isotope shifts are derived

$\text{ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?4.7 ≠ 1.5}\text{cm}{}^1\text{, ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?1.8 ≠ 1.5}\text{cm}{}^1$

and related to the theoretical approach given by Bunker to the problem of the breakdown of the Born-Oppenheimer approximation.     

Microwave spectrum of dimethylketene: Centrifugal distortion and Coriolis interaction

For the prolate asymmetric (κ = ?0.58) top molecule dimethylketene the centrifugal constants have been determined in terms of the Δ constants by means of ΔK_? ≠ 0 rotational transitions. Two earlier assigned torsional excited states (01, 10)₁ and (01, 10)₂ of symmetry B₂ and A₂ of C_2v could be confirmed. A third fundamental vibration v_r(b₁), probably the inplane skeletal rock, has been found and analyzed in the v_r = 1 and v_r = 2 states. The variation of the centrifugal constants and of the inertial defects with the state of excitation could be explained by Coriolis coupling between the v_r(b₁) = 1 vibrational and v_t(b₂) = 1 torsional excited states. The Coriolis constant $\text{ζ}{}_{\mathrm{rt}}{}^{\mathrm{a}}$ and the torsional frequency ω_t(b₂) have been estimated to be $\text{ζ}{}_{\mathrm{rt}}{}^{\mathrm{a}}\text{≈ 0.24}$ and ω_r(b₁) - ω_t(b₂) ≈ 5.8 cm^?1, respectively.     

The breakdown of the Born-Oppenheimer approximation for a diatomic molecule: The dipole moment and nuclear quadrupole coupling constants

In the Born-Oppenheimer approximation the dipole moment of the vibrational levels of a ¹Σ electronic state of a heteronuclear diatomic molecule can be expressed as a power series in [$\text{(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v +}\text{1}\text{2}\text{)}$], where v is the vibrational quantum number, and to order $\text{(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)}{}^2$ this expression is

$\text{μ}{}_{\mathrm{v}}\text{=[μ}{}_{\mathrm{e}}\text{+(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)}{}^2\text{μ}{}_{\mathrm{c}}\text{]+μ}{}_1\text{[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]+μ}{}_2\text{(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]}{}^2$

Similarly the nuclear quadrupole coupling constant eQq of each nucleus in the molecule can be expressed as

$\text{eQq=[eQq}{}_{\mathrm{e}}\text{+(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)}{}^2\text{eQq}{}_{\mathrm{c}}\text{]+eQq}{}_1\text{[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]+eQq}{}_2\text{(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]}{}^2$

In this paper the effect of the breakdown of the Born-Oppenheimer approximation on the expressions for μ_v and eQq for an isolated ¹Σ ground electronic state of a heteronuclear diatomic molecule is determined. The effect is to change only μ_c and eQq_c and, therefore, to alter the relationship between the μ_v or eQq values of two isotopes of a molecule. The intensities of the lines in the rotation and rotation-vibration spectrum are also slightly modified by this effect.For the HCl molecule we find that

$\text{μ}{}_{\mathrm{v}}\text{=[1.0908+(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(164)]+8.6[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]?9.5[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]}{}^2\text{D}$

where the second term (+164 D) would have the value ?4 D in the Born-Oppenheimer approximation. Similarly for the ³⁵Cl nucleus of the HCl molecule we have

$\text{eQq=[?66.806+(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(2460)]?472.23[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]+750[(}\text{B}{}_{\mathrm{e}}\text{ω}{}_{\mathrm{e}}\text{)(v+}\text{1}\text{2}\text{)]}{}^2\text{MHz}$

where the second term (+2460 MHz) would be ?110 MHz in the Born-Oppenheimer approximation.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. I

We consider the class of non-integrable, non-linear equations,

$\text{L}{}_{\mathrm{q}}\text{K=K}{}^2\text{, L}{}_{\mathrm{q}}\text{=? +}\text{∑}\text{1?i+j?q}\text{aij?}{}^{\mathrm{i}}{}_{\mathrm{x}}{}^{\mathrm{i}}\text{?}{}^{\mathrm{j}}{}_{\mathrm{t}}{}^{\mathrm{j}}\text{, ?≠0,}$

in 1+1 dimensions. We seek rational solutions K(ω₁,ω₂), which we call bi-solitons, with exponential type variables ω_i = exp(γ_ix + ρ_it). In this paper, we restrict to q = 2 and 3, and investigate the general q case in the following paper. We find that these bi-solitons exist when the operator L_q (with ± ?) can be factorized as the product of smaller order differential operators. Besides the trivial factorized bi-solitons, we show that there exist non-trivial ones whenever K may be written as Σ^lmaxx ω^l₂F_l(Z = ω₁ + ω₂). In order to understand the origin of the factorization property, to any polynomial K = Σω^l₂F_l(Z) we associate a linear transformation such that L_qK has only the power ω^l₂ of K². For q = 2 and 3, we find that there exist particular polynomials of this type restraining L_q to be a product of smallr order operators. For the full non-linear equations we verify that all the bi-solitons can be obtained from these particular polynomials.