Rotational analysis of some bands of C1 Σ u + -X1 Σ g + system of P2 molecule involving lowν′,ν″ values

Ten ultraviolet bands of the C¹ Σ _u ⁺ -X¹ Σ _g ⁺ system of P₂ involving lowv′ andv″ values have been photographed at dispersion of 0·38 and 0·56 Å/mm. and analysed for their rotational structure. While four of these bands were analysed earlier, six of them,viz., 0–10, 1–12, 2–7, 2–14, 4–8 and 6–9 have been analysed for the first time during the present studies. The rotational constants, B _v S with lowv″ quantum numbers are obtained from which value of B _θ ^″ has been derived. The value of B _θ ^″ is found to be in agreement with the value obtained by Douglas and Rao from their study of A¹ Π _g-X¹ Σ _g ⁺ bands of P₂. Earlier findings on the perturbation ofν′=2 level of the C¹ Σ _u ⁺ state have been confirmed from the analysis of the 2–7, 2–14 and 2–15 bands. Theν ₀₀ values of the bands show large deviations from the expected values.     

Structure and analysis of some bands of theβ-system of PO molecule

Rotational analysis of (4–4), (7–6), (7–8) bands of the² Σ?² II _1/2 sub-system and (3–3) band of² Σ–² II _3/2 sub-system of theβ-system of PO have been made. Vibrational assignments of fifteen new bands of this system have also been reported.     

F2Δr−A2Πi band system of CN

The spectrum of CN has been studied in condensed electrical discharge through flowing CH₃CN vapour. Ten new bands of F²?,?A²Π_i system in the region 2100–2700 Å are obtained. The vibrational constantsω ₀′ = 1229.7 cm.^?1 and ω₀′χ₀′= 14.0 cm.^?1 are obtained for the first time for the²Δ state of CN.     

A new2 Π-X2 Π band system of NS

Vibrational and rotational analysis of some bands forming a new band system of NS is given. It is also shown that the system involves the ground X² Π _reg. state of the molecule, and is due to the transition² Π _reg.→X² Π _reg. The bands form a singlev″=0 progression withv′=7, 8, 9 and 10. The assignment of these quantum numbersv′, v″ is supported by (1) Δ₂F″ (J) values which are identical with those for thev″=0 bands of theβ andγ systems and (2) the isotopic shift data from¹⁵NS bands, respectively. The derived vibrational and rotational constants for the new² Π _reg. state are as follows (cm.^?1 units):

	T e	ω e	ω e x e	B e	D e
2 Π 3/2..	30364·8	803·3	3·82	0·6030	2·0×10?6
2 Π ½..	30292·3	797·0	3·63	0·5898	2·0×10?6

     

Isotope shift studies of the B3 Σ −−X3 Σ − bands of SO

The B³ Σ ^??X³ Σ ^? bands of SO were excited in an electrodeless microwave (2450 Mc/s) discharge through trace amounts of sulphur enriched with 37% of³⁴S and oxygen enriched with about 40% of¹⁸O. The emission spectra of³²S¹⁶O,³⁴S¹⁶O and³²S¹⁸O were photographed on a Jarrell-Ash 3·4 m. grating spectrograph at a dispersion of 5 Å/mm. Isotope shift data obtained from the present studies support the revision of the earlier vibrational scheme (Henri and Wolff, 1929; Martin, 1932) by increasingv″-quantum numbering by two units as suggested by Norrish and Oldershaw (1959).     

Emission spectrum of bromine excited in the presence of argon

The wavelengths and wavenumbers of the band heads of the system 2660-2590 Å as obtained from the plates taken on the first order 21-feet grating spectrograph are given along with its vibrational analysis. This system is shown as the transition from an upper state at T _e =56776 cm.^?1 withω _e = 108·0 cm.^?1 to the³ Π _u (O _u ⁺) state at T _e =15918 cm.^?1 The lower state is the same as that of the two systems in the regions 2950-2670 Å and 3150-2970 Å reported earlier.     

Isotope shifts in the second negative bands of O 2 +

The second negative bands of O ₂ ⁺ , lying in the region 5000–1850 Å, have been photographed using oxygen enriched in¹⁸O. A study of the isotope shifts in the band heads revealed that thev″ numbering is to be increased by one unit whereas thev′ numbering remains unaltered. This change in vibrational numbering leads to a revision of the molecular constants of O ₂ ⁺ , especially of those in its ground state. These are discussed and the extension of the band system in the vacuum ultra-violet region reported. The possibility of developing a simple spectrographic method for the isotopic assay of oxygen using the isotope shifts in this band system is pointed out. The revised rotational and vibrational constants in theX ² II _g ground state of O ₂ ⁺ molecule are as follows :

$$\begin{gathered} \omega _e = 1903 \cdot 85 cm.^{ - 1} , \hfill \\ \omega _e x_e = 16 \cdot 18 cm.^{ - 1} , \hfill \\ B_e = 1 \cdot 6920 cm.^{ - 1} \hfill \\ and \hfill \\ r_e = 1 \cdot 1161 {\AA}. \hfill \\ \end{gathered} $$     

Emission spectrum of chlorine excited in the presence of argon

The spectrum of chlorine excited in the presence of argon has been photographed with a 21-ft. grating spectrograph in the first order. Two band systems in the region 2600–2390 Å and 2365–2239 Å are observed which appear to be respectively analogous to the 2950–2670 Å and 2660–2590 Å systems of bromine reported earlier by Venkateswarlu and Verma. The wavelengths and the wavenumbers of all the bands in the system 2600–2390 Å are given. The vibrational scheme along with the corresponding Franck-Condon parabola is also given. The analysis suggests that the lower state of the system is the 3π(O _u ⁺_ state established by Elliott at 17658 cm.^?1 and that the upper state is at 67773 cm.^?1 The vibrational constants obtained arew ₀′ = 246·6 cm.^?1,w ₀′x ₀′ = 0·615 cm.^?1,w ₀″ = 255·2 cm.^?1,w ₀″x ₀″ = 5·5 cm.^?1,w ₀″y ₀″ = ?0·0155 cm.^?1 andw ₀″z ₀″ = 0·00115 cm.^?1     

Emission spectrum of bromine excited in the presence of argon

The wavelengths and wavenumbers of the band heads of the system 3150–2970 Å as obtained from the plates taken on the first order 21′ grating spectrograph are given along with the vibrational analysis. This system is shown to be due to a transition from an upper electronic state at T_e = 48516 cm.^-1 with ω′ _e = 162·0 cm.^?1 and ω′ _e χ′ _e = 0·29 cm.^?1 to the well-known³ Π _u (O _u ⁺) state at T_e = 15918 cm.^-1 This lower state is common with that of the system 2950–2670 Å.     

Emission spectrum of chlorine excited in the presence of argon

The weak band system in the region 2365-2239 Å is discussed in this paper. The wavelengths and the wavenumbers of the bands photographed with the first order of a 21-ft. grating spectrograph are recorded. The vibrational analysis of the bands and their corresponding intensity distribution are also given. The analysis shows that the lower state of the system is the same as that of the 2600-2390 Å system discussed in the earlier paper and is the 3π (O_u ⁺) state established by Elliott at 17658 cm.^?1 The constants of the upper state arew ₀′ = 261·5 cm.^?1,w ₀′x ₀′ = 0·812 cm.^?1, T₀=61290 cm.^?1.     

Rotational analysis of some of the bands of the orange system of FeO

The orange bands of FeO are excited in a low pressure arc in oxygen and photographed at dispersions of 0·8 and 0·5 Å/mm. respectively. Rotational analysis of five of the bands shows that they involve a¹ Σ-¹ Σ transition. The vibrational and rotational constants (in cm.^?1) of the upper and lowerΣ states are found to be

	Lower state	Upper state
B e	0·3491	0·3063
α e	0·0029	0·0032
w e	870·7	(ΔG½=770·0)
w e x e	4·39	..

     

Compactification of the space of branched coverings of the two-dimensional sphere

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S ² of the two-dimensional sphere. We complete X_Σ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S ², the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ?P¹ ≌ S ². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ? X _Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.     

Theβ- andγ-band systems of14NS and15NS

The spectra of¹⁴NS and¹⁵NS molecules are obtained in a 2450 mc./s. microwave oscillator discharge. Vibrational assignments of theβ- andγ-systems are studied in more detail. Several hitherto unrecorded bands are satisfactorily explained as belonging to higherv′,v″ levels in the Deslandres schemes of bothβ- andγ-systems. The observed isotope shifts (¹⁴NS-¹⁵NS) provide confirmatory evidence for the proposed vibrational analysis. In the case of theγ-system, thev′=2 level of the upper state appears to be strongly perturbed showing a shift of about 38 cm.^?1 from the expected position. In theβ-system, the isotopic shifts in the band-heads involving thev′=0 and 1 levels of the²Δ_5/2 of the upper state show small deviations from expected values. The reality of these small deviations is established beyond doubt by the occurrence of the effect on the 0–0 sub-band which exhibits the isotopic head in a wavelength direction opposite to the expected one.     

The emission spectrum of iodine bromide excited in the presence of Argon

IBr vapour was excited in the presence of argon by an uncondensed transformer discharge. Four band systems were obtained in the regions 5425–5360 Å, 4520–4415 Å, 4120–4010 Å and 3915–3540 Å of which the first three are discussed in this paper. The wavelengths and wavenumbers of the band heads in three systems as measured from the plates obtained with a 3-prism Steinheil glass spectrograph are given along with their visually estimated relative intensities. The three band systems, which are new, are analysed and the following vibrational constants expressed in cm.^?1 are obtained:

Band system	v e	w e ″	w e ″x e ″	w e ″y e ″	w e ′	w e ′x e ′
5425-5360 Å	18613	65·5	0·24	?0·01	43·0	0·026
4520-4415 Å	22312	65·5	0·24	?0·01	77·0	0·5
4120-4010 Å	24540	160·6	1·125	..	128·4	0·1

     

The near ultraviolet absorption spectra ofortho- andmeta-thiocresols

The absorption spectra ofortho- andmeta-thiocresols have been studied in the present investigation. The ortho-thiocresol spectrum consists of about forty-five bands of rather a diffuse nature and in general low intensity in the region from 2873 Å to 2600 Å. The maximum number of bands is obtained by using a path length of 330 cm. for absorption, the temperature of the bulb being maintained at 14°C. Several of these bands are assigned as due tov-v-transitions. The (0, 0) band is chosen at 35386 cm.^?1 which is the strongest band on the longer wavelength side. Vibrational frequencies in the excited state have values 729, 957 and 1159 and combinations and overtones of these are present. Themeta-thiocresol spectrum consists of about forty bands of rather a diffuse nature and very weak intensities in the region from 2900 Å to 2590 Å. The maximum number of bands is obtained by using a path length 200 cm. for absorption and by keeping the temperature of the bulb at 20° C. The (0, 0) band is chosen to be that at 34793 cm.^?1 which is the strongest band on the longer wavelength side. Vibrational frequencies in the excited state have values 492, 611, 720, 845, 965, 1016 and 1155 cm.^?1 and combinations and overtones of these are present.     

The influence of isobar production on charge ratio and transverse momentum of secondary particles in high energy collisions

An isobar model in which collision between two particles leads to the creation of only two bodies which by subsequent decay give rise to the observed secondaries has been considered. On the basis of such a model, the charge ratios of pions, kaons andΣ-hyperons inp?p andπ?p collisions have been computed and compared with the available experimental data. Some features of transverse momentum of pions and protons in 24 GeV/cp?p collisions have also been studied. The main conclusions can be summarised as follows:

1. (1)
   The observed positive excess among pions produced in high energyp?p collisions leading toπ ⁺/π ^? andπ ⁺/π ⁰ ratios of ～3 and 1·6 respectively for high momentum pions can be explained on the basis of the isobar model. Further, the fast increase of K⁺/K^? ratio as the kaon momentum increases, the high ratio (～4) ofΣ ⁺/Σ ^? in 24 GeV/cp?p collisions and the existence of a strong positive (negative) excess amongΣ-hyperons produced inπ ⁺?p(π ^??p) collisions at various primary energies result, in a natural way, from such a model. The agreement results mainly from the restriction of only two bodies in the final states and does not critically depend on the isospins of produced isobars.     
   
   

Microwave spectrum of pentafluorobenzonitrile

The absorption spectrum of pentafluorobenzonitrile has been investigated in the frequency range of 18–26·5 GHz using a 100 KHz stark modulated microwave spectrometer. The analysis of the spectrum is based on the rigid asymmetric rotor theory. The rotational constants obtained are A=1026·82±0·3 MHz, B=776·34±0·1 MHz, C=442·06±0·1 MHz and the asymmetry parameterχ=+0·1433. The inertial defect is I _o ?I _a ?I _b =0·081 amu Ų. The bond distances ared _CF=1·328 Å andd _CN=1·157 Å. The results are in good agreement with the assumed planarity of the molecule and the normal values of bond distances.     

The D2Σ+ − X2Σ+ system of A10

Eleven bands of A10 belonging to the system (D² Σ ⁺-X² Σ ⁺) in the ultra-violet region have been analysed for their rotational structure. These are the 0-2, 1-3, 2-4, 0-3, 1-4, 3-0, 4-1, 3-1, 4-0, 5-1 and 6-1 bands lying at 2611.8Å, 2620.7Å, 2629.4Å, 2677.4Å, 2685.7Å, 2347.7Å, 2358.3Å, 2402.2Å, 2305.8Å, 2316.7Å and 2277.3Å respectively. The spin-splitting for the D² Σ ⁺ state has been determined from the doubling of the rotational lines observed for the bands 0-2, 1-3, 2-4 and 1-4. The rotational and vibrational constants (in cm^?1) evaluated for the D² Σ ⁺ state are: $$\begin{gathered} T_a = 40267 \cdot 6 \hfill \\ G(V) = 817 \cdot 47 (v + 1/2) - 4 \cdot 795 (v + 1/2)^2 - 0 \cdot 1107 (v + 1/2)^3 \hfill \\ B_v = 0 \cdot 56522 - 0 \cdot 0046 (v + 1/2) - 0 \cdot 00005 (v + 1/2)^2 \hfill \\ \gamma = 0 \cdot 004 \pm 0 \cdot 002 \hfill \\ \sigma = - 0 \cdot 4 \pm 0 \cdot 1 \times 10^{ - 6} \hfill \\ \end{gathered} $$      

The emission spectrum of IBr excited in the presence of Argon

Wavelengths and wavenumbers of the band heads in the region 3915–3540 Å are recorded as obtained from the measurements of the plates taken on a first order 21-feet grating spectrograph. Earlier workers recently reported 40 bands of this system covering the region 3900–3800 Å. All the bands of this system obtained in the present experiments are analysed as involving the³ Π (1) state for lower state. The constants for the lower state are such that they represent well the ΔG (v+1/2) values obtained in the present experiments fromv=0 tov=26 as well as those obtained by Brown fromv=9 tov=43. The vibrational constants of the two states involved are:

$$\begin{gathered} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } } \\ {137 \cdot 8 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } x_e ^{\prime \prime } } \\ {0 \cdot 571 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } y_e ^{\prime \prime } } \\ { - 0 \cdot 1156 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e z_e ^{\prime \prime } } \\ {3 \cdot 09 \times 10^{ - 3} cm.^{ - 1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\omega _e ^{\prime \prime } t_e ^{\prime \prime } } \\ { - 2 \cdot 5 \times 10^{ - 5} cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime } \\ {90 \cdot 1 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime x_e ^\prime } \\ {0 \cdot 15 cm.^{ - 1} } \\ \end{array} \hfill \\ \end{gathered} $$     

The limit absorption principle and homogenization procedure for periodic elliptic operators

For a periodic matrix elliptic operator \(A_\varepsilon \) with (x ?-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent 〈x〉^?1/2?· (\(A_\varepsilon \) ? (η ± i? ^σ )I)^?1〈x〉^?1/2?· has a limit in the operator norm in L ₂ as ? → 0 provided that η > 0, · > 0, and 0 < σ < 1/2.