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Simple Spectral Lines Data Model

 

Version 1.0

Working Draft 13 July 2009

 

 

This version:

            WD-SSLDM-1.0-20090713

Latest version:

            http://www.ivoa.net/Documents/SSLDM

Previous versions:

 

Editors: Pedro Osuna, Jesus Salgado

                     

Authors:

            Pedro Osuna

            Matteo Guainazzi

            Jesus Salgado

Marie-Lise Dubernet

            Evelyne Roueff

           

Status of This Document

This is an IVOA Working Draft for review by IVOA members and other interested parties. It is a draft document and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use IVOA Working Drafts as reference materials or to cite them as other than “work in progress”.

 

 

 

Abstract

 

This document represents a proposal for a Data Model to describe Spectral Line Transitions in the context of the Simple Line Access Protocol defined by the IVOA (c.f. Ref[] IVOA Simple Line Access protocol)

 

The main objective of the model is to integrate with and support the Simple Line Access Protocol, with which it forms a compact unit. This integration allows seamless access to Spectral Line Transitions available worldwide in the VO context.

 

This model does not deal with the complete description of Atomic and Molecular Physics, which scope is outside of this document.

 

In the astrophysical sense, a line is considered as the result of a transition between two levels. Under the basis of this assumption, a whole set of objects and attributes have been derived to define properly the necessary information to deal with lines appearing in astrophysical contexts.

 

The document has been written taking into account available information from many different Line data providers (see acknowledgments section).

 

Acknowledgments

 

The authors wish to acknowledge all the people and institutes, atomic and molecular database experts and physicists who have collaborated through different discussions to the building up of the concepts described in this document.
 

 

Contents

 

 


1        Introduction   4

2        Data model 5

2.1     Objects Description   7

2.1.1       PhysicalQuantity  7

2.1.2       Unit 7

2.1.3       Line  8

2.1.4       Species  12

2.1.5       Level 12

2.1.6       QuantumState  14

2.1.7       QuantumNumber 16

2.1.8       Process  17

2.1.9       Environment 18

2.1.10     Source  19

3        UCDs  19

4        Working examples  22

4.1     The Hyperfine Structure of N2H+  22

4.1.1       The values in the model 23

4.1.2       JSON representation  25

4.1.3       UML instantiation diagram   29

4.2     Radiative Recombination Continua: a diagnostic tool for X-Ray spectra of AGN   30

4.2.1       The values in the model 31

4.2.2       JSON representation  32

4.2.3       UML Instantiation diagram   35

4.3     References  36

5        Appendix A: List of Atomic Elements  38

6        Appendix B: List of quantum numbers  41

6.1     Various Quantum numbers  42

6.1.1       totalNuclearSpinI 42

6.1.2       totalMagneticQuantumNumberI 42

6.1.3       totalMolecularProjectionI 42

6.1.4       nuclearSpin  42

6.1.5       parity  42

6.1.6       serialQuantumNumber 42

6.2     Quantum numbers for hydrogenoids  42

6.2.1       nPrincipal 42

6.2.2       lElectronicOrbitalAngularMomentum   43

6.2.3       sAngularMomentum   43

6.2.4       jTotalAngularMomentum   43

6.2.5       fTotalAngularMomentum   43

6.2.6       lMagneticQuantumNumber 43

6.2.7       sMagneticQuantumNumber 43

6.2.8       jMagneticQuantumNumber 43

6.2.9       fMagneticQuantumNumber 44

6.3     Pure rotational quantum numbers  44

6.3.1       asymmetricTAU   44

6.3.2       asymmetricKA   44

6.3.3       asymmetricKc  44

6.4     Quantum numbers for n electron systems (atoms and molecules) 44

6.4.1       totalSpinMomentumS   44

6.4.2       totalMagneticQuantumNumberS   45

6.4.3       totalMolecularProjectionS   45

6.4.4       totalElectronicOrbitalMomentumL  45

6.4.5       totalMagneticQuantumNumberL  45

6.4.6       totalMolecularProjectionL  45

6.4.7       totalAngularMomentumN   45

6.4.8       totalMagneticQuantumNumberN   45

6.4.9       totalMolecularProjectionN   46

6.4.10     totalAngularMomentumJ  46

6.4.11     totalMagneticQuantumNumberJ  46

6.4.12     totalMolecularProjectionJ  46

6.4.13     intermediateAngularMomemtunF  47

6.4.14     totalAngularMomentumF  47

6.4.15     totalMagneticQuantumNumberF  47

6.5     Vibrational and (ro-)vibronic quantum numbers  47

6.5.1       vibrationNu  47

6.5.2       vibrationLNu  47

6.5.3       totalVibrationL  47

6.5.4       vibronicAngularMomentumK   48

6.5.5       vibronicAngularMomentumP   48

6.5.6       rovibronicAngularMomentumP   48

6.5.7       hinderedK1, hinderedK2  48

7        APPENDIX C:Description of couplings for atomic Physics  48

7.1     LS coupling  48

7.2     jj coupling  49

7.3     jK coupling  49

7.4     LK coupling  50

7.5     Intermediate coupling  50


 


1         Introduction

 

Atomic and molecular line databases are a fundamental component in our process of understanding the physical nature of astrophysical plasmas. Density, temperature, pressure, ionization state and mechanism, can be derived by comparing the properties (energy, profile, intensity) of emission and absorption lines observed in astronomical sources with atomic and molecular physics data.

The latter have been consolidated through experiments in Earth's laboratories, whose results populate a rich wealth of databases around the world. Accessing the information of these databases in the Virtual Observatory (VO) framework is a fundamental part of the VO mission.

 

This document aims at providing a simple framework, both for atomic and molecular line databases, as well as for databases of observed lines in all energy ranges, or for VO-tools, which can extract emission/absorption line information from observed spectra or narrow-band filter photometry.

 

The Model is organized around the concept of "Line", defined as the results of a transition between two levels (this concept applies to bound-bound and free-bound transitions, but not free-free transitions). In turn each "Level" is characterized by  one (or more) "QuantumState". The latter is characterized by a proper set of "QuantumNumber".

 

The object “Species” represents a placeholder for a whole new model to represent the atomic and molecular properties of matter. This will take form in a separate document. We reserve here one single attribute for the time being, the name of the species (including standard naming convention for ionised species), and shall be pointing to the future model whenever available.

 

Any process which modifies the intrinsic properties of a "Line" (monochromatic character, laboratory wavelength etc.) is described through the attributes of "Process", which allows as well to describe the nature of the process responsible for the line generation, whenever pertinent. The element "Environment" allows service providers to list physical properties of the line-emitting/absorbing plasma, derived from the properties of the line emission/absorption complex. Both "Process" and "Environment" contain hooks to VO “Model”s for theoretical physics (placeholders for future models).

 

The present Simple Spectral Line Data Model does not explicitly address non-electromagnetic transitions.

 

2         Data model

 

We have attempted to create a Simple Data Model for Spectral Lines that would be useful to retrieve information from databases both of observed astronomical lines and laboratory atomic or molecular lines. 

 

We give in what follows a standard UML diagram describing a Line.

UML Data Model

 

 


2.1      Objects Description

2.1.1      PhysicalQuantity 

 

Class used to describe a physical measurement. This could be superseded by a general IVOA Quantity DM definition. It contains the basic information to understand a Quantity.

 

Although the definition of the physical quantity object is out of the scope of the present DM, here we attach a UML description of it as per DM working group discussion.

 

 

 

2.1.1.1                                                             PhysicalQuantity.value

Value of the measure. General Number format.

2.1.1.2                                                             PhysicalQuantity.error

General error of the measure. General Number format. Please note that this is the total error. A more formal description should be provided in a general IVOA Physical Quantity Data Model.

2.1.1.3                                                             PhysicalQuantity.unit

Unit in which the measure is expressed. Type, unit (see definition in next section). Both value and error should be expressed in the same units

 

 

2.1.2      Unit

 

Class used to describe a physical unit. This could be superseded by a general IVOA Unit DM definition. It contains the basic information to understand a Unit:

 

2.1.2.1                                                             Unit.expression

String representation of the unit

2.1.2.2                                                             Unit.scaleSI

Scaling reference of the unit described to the international system of units analogue, i.e., to the unit in the IS with the same dimensional equation

 

2.1.2.3                                                             Unit.dimEquation

Dimensional equation representation of the unit. The format is a string with the dimensional equation, where M is mass, L is length, T is time, K is temperature and where the “^” has been sustracted.

 

 Examples:

1 Angstrom = 1.E-10 m

1.E-10 L

 

Unit.expression= Angstrom

Unit.scaleSI=1.E-10

Unit.dimEquation=L

 

1 erg/cm^2/s/Angstrom = 1.E7 Kg/m/s^3

1.E7 ML-1T-3

 

Unit.expression= erg/cm^2/s/Angstrom

Unit.scaleSI= 1.E7

Unit.dimEquation= ML-1T-3

 

 

 

See, e.g., IVOA SSAP for more examples

 

2.1.3      Line

 

This class includes observables, e.g. measured physical parameters, describing the line, as well as the main physical properties of the transition originating it. Recombination and dissociations are expressed through atomic coefficients rather than through global properties.

2.1.3.1                                                             Line.title

A small description title identifying the line. This is useful when identification is not secure or not yet established.

 

2.1.3.2                                                             Line.initialLevel

A full description of the initial level of the transition, originating the line.

 

2.1.3.3                                                             Line.finalLevel

A full description of the final level of the transition, originating the line

 

2.1.3.4                                                             Line.initialElement

A full description of the initial state of the atom (including its ionization state) or molecule, where the line transition occurs.

 

2.1.3.5                                                             Line.finalElement

A full description of the final state of the atom (including its ionization state) or molecule, where the line transition occurs. For bound-bound atomic transitions, it follows: "initialElement"="finalElement".

 

2.1.3.6                                                             Line.wavelength

Wavelength in the vacuum of the transition originating the line.

 

2.1.3.7                                                             Line.frequency

Frequency in the vacuum of the transition originating the line.

 

2.1.3.8                                                             Line.wavenumber

Wavenumber in the vacuum of the transition originating the line.

 

2.1.3.9                                                             Line.airWavelength

Wavelength in the air of the transition originating the line.

 

2.1.3.10                                                         Line.einsteinA

Einstein A coefficient, defined as the probability per unit time  for spontaneous emission in a bound-bound transition from "initialLevel" to "finalLevel".

 

2.1.3.11                                                         Line.oscillatorStrength

If positive ("absorption oscillator strength"): the quantity defined by the relation:

 

 

where  is the Einstein A-coefficient for spontaneous emission between "initialLevel" and "finalLevel" - characterized by the energy difference -,   , , , and h are the usual symbols for the fine-structure constant, electron mass, speed of light and Planck constant, respectively;  is the statistical weight of the i-th level. The subscripts "i" and "f" refer to the "initialLevel" and "finalLevel", respectively. As usual throughout this document, units are S.I. with  e0 expressed in .

 

If negative ("emission oscillator strength") the quantity (fif)  is defined by:

 

 

whereis the weighted oscillator strength.

 

2.1.3.12                                                          Line.weightedOscillatorStrength

The product between "oscillatorStrength" and the statistical weight  of the "initialLevel"

 

2.1.3.13                                                          Line.intensity

This is a source dependent relative intensity, useful as a guideline for low density sources. These are values that are intended to represent the strengths of the lines of a spectrum as they would appear in emission. They may have been normalized. They can be expressed in absolute physical units or in relative units with respect to a reference line.

 

The difficulty of obtaining reliable relative intensities can be understood from the fact that in optically thin plasmas, the intensity of a spectral line is proportional to:

 

 

Where is the number of atoms in the upper level (population of the upper level), it the transition probability for transitions from upper level to lower level , and is the photon energy (or the energy difference between the upper level and lower level). Although both and are well defined quantities for each line of a given atom, the population values depend on plasma conditions in a given light source, and they are this different for different sources.

 

Taking into account this issue, the following points should be kept in mind when using relative intensities:

 

  1. There is no common scale for relative intensities. The values from different databases or different publications use different scales. The relative intensities have meaning only within a given spectrum.
  2. The relative intensities are most useful in comparing strengths of spectral lines that are not separated widely. This results from the fact that most relative intensities are not corrected for spectral sensitivity of the measuring instruments.

 

  1. Relative intensities are source dependent (either laboratory or astrophysical detections)

 

2.1.3.14                                                          Line.observedFlux

Integrated intensity of the line profile over a given wavelength range

 

2.1.3.15                                                          Line.observedFluxWaveMin

Minimum wavelength for observedFlux integration.

 

2.1.3.16                                                          Line.observedFluxWaveMax

Maximum wavelength for observedFlux integration.

 

2.1.3.17                                                          Line.significanceOfDetection

The significance of line detection in an observed spectrum. It can be expressed in terms of signal-to-noise ratio, or detection probability (usually null hypothesis probability that a given observed line is due to a statistical background fluctuation).

 

 

2.1.3.18                                                          Line.transitionType

String indicating the first non zero term in the expansion of the operator  in the atomic transition probability integral:

 

 

Possible values correspond to, e.g., "electric dipole", "magnetic dipole", "electric quadrupole", etc., or their corresponding common abbreviations E1, M1, E2, etc.

2.1.3.19                                                          Line.strength

 

In theoretical works, the line strength S is widely used (Drake 1996):

 

 

Where  and  are the initial- and final-state wavefunction and  is the transition matrix element of the appropriate multipole operator . For example, the relationship between A, f, and S for electric dipole (E1 or allowed) transitions in S.I. units (A in s-1 ,n in s-1 , S in m2 C2, e0 in  C2.N-1.m-2, h in J.s) are:

 

 

2.1.3.20                                                          Line.observedBroadeningCoefficient

Width of the line profile (expressed as Full Width Half Maximum) induced by a process of “type=Broadening”.

 

2.1.3.21                                                          Line.observedShiftingCoefficient

Shift of the transition laboratory wavelength(/frequency/wavenumber) induced by a process of “type=Energy shift”. It is expressed by the difference between the peak intensity wavelength(frequency/wavenumber) in the observed profile and the laboratory wavelength(frequency/wavenumber).

 

 

2.1.4      Species

 

 This class is a placeholder for a future model, providing a full description of the physical and chemical property of the chemical element of compound where the transition originating the line occurs

 

 

 

2.1.4.1                                                             Species.name

Name of the chemical element or compound including ionisation status.  Examples of valid names are: CIV for Carbon three times ionised, N2H+ for the Dyazenylium molecule, etc (see Appendix A for standard chemical element names).

 

 

2.1.5      Level

 

 The scope of this class is to describe the quantum mechanics properties of each level, between which the transition originating the line occurs.

 

2.1.5.1                                                             Level.totalStatWeight

Statistical weight associated to the level including all degeneracies, expressed as the total number of terms pertaining to a given level.

 

2.1.5.2                                                             Level.nuclearStatWeight

The same as Level.totalStatWeight for nuclear spin states only

 

2.1.5.3                                                             Level.landeFactor

A dimensionless factor g that accounts for the splitting of normal energy levels into uniformly spaced sublevels in the presence of a magnetic field. The level of energy is split into levels of energy:

 

, ,…,

 

Where  is the magnetic field and is a proportionality constant.

 

In the case of the L-S coupling (see appendix C), the Lande factor  is specified as the combination of atomic quantum numbers, which enters in the definition of the total magnetic moment  in the fine structure interaction:

 

 

where  is the Bohr magneton, defined as:

 

 

Where

is the elementary charge

is the Planck constant

is the electron rest mass

 

In terms of pure quantum numbers:

 

 

2.1.5.4                                                             Level.lifeTime

Intrinsic lifetime of a level due to its radiative decay.

 

2.1.5.5                                                             Level.energy

The binding energy of an electron belonging to the level.

 

2.1.5.6                                                             Level.energyOrigin

Human readable string indicating the nature of the energy origin. Examples: “Ionization energy limit”, “Ground state energy” of an atom, “Dissociation limit” for a molecule, etc

 

2.1.5.7                                                             Level.quantumState

A representation of the level quantum state through its set of quantum numbers

 

2.1.5.8                                                             Level.nuclearSpinSymmetryType

A string indicating the type of nuclear spin symmetry. Possible values are: “para”, “ortho”, “meta”

 

2.1.5.9                                                             Level.parity

Eigenvalue of the parity operator. Values (+1,-1)

 

2.1.5.10                                                          Level.configuration

For atomic levels, the standard specification of the quantum numbers nPrincipal (n) and lElectronicOrbitalAngularMomentum (l) for the orbital of each electron in the level; an exponent is used to indicate the numbers of electrons sharing a given n and l. For example, 1s2,2s2,2p6,5f. The orbitals are conventionally listed according to increasing n, then by increasing l, that is, 1s, 2s, 2p, 3s, 3p, 3d, …..

Closed shell configurations may be omitted from the enumeration.

 

For molecular states, similar enumerations takes place involving appropriate representations.

 

 

2.1.6      QuantumState

 

 

 

2.1.6.1                                                             QuantumState.mixingCoefficient

A positive or negative number (double) giving the squared or the signed linear coefficient corresponding to the associated component in the expansion of the eigenstate (QuantumState in the DM). It varies from 0 to 1 (or -1 to 1)

 

2.1.6.2                                                             QuantumState.quantumNumber

In order to allow for a simple mechanism for quantum numbers coupling, the QuantumNumber object is reduced to the minimum set of needed attributes to identify a quantum number. Coupling is then implemented by specifying combinations of the different quantum numbers.

 

See Appendix C.

 

2.1.6.3                                                             QuantumState.termSymbol

The term (symbol) to which this quantum state belongs, if applicable.

 

For example, in the case of Spin-Orbit atomic interaction, a term describes a set of (2S+1)(2L+1) states belonging to a definite configuration and to a definite L and S. The notation for a term is for the LS coupling is, at follows:

 

 

where

 

For instance, 3P1 would describe a term in which L=1, S=1 and J=1. If J is not present, this term symbol represents the 3 different possible levels (J=0,1,2)

 

See appendix C for more examples of different couplings.

 

For molecular quantum states, it is a shorthand expression of the group irreductible representation and angular momenta that characterize the state of a molecule, i.e its electronic quantum state. A complete description of the molecularTermSymbol can be found in « Notations and Conventions in Molecular Spectroscopy: Part 2. Symmetry notation » (IUPAC Recommendations 1997), C.J.H. Schutte et at, Pure & Appl. Chem., Vol. 69, no. 8, pp. 1633-1639, 1997. The molecular term symbol contains the irreductible representation for the molecular point groups with right subscripts and superscripts, and a left superscript indicating the electron spin multiciplicity, Additionaly it starts with an symbol ~X (i.e., ~ on X) (ground state), Ã, ~B (i.e. ~ on B), ... indicating excited states of the same multiplicity than the ground state X or ã, ~b (~ on b), ... for excited states of different  multiplicity.

 

2.1.7      QuantumNumber

 

 The scope of this class is to describe the set of quantum numbers describing each level.

 

2.1.7.1                                                             QuantumNumber.label

The name of the quantum number. It is a string like “F”, “J”, “I1”, etc., or whatever human readable string that identifies the quantum number

 

2.1.7.2                                                             QuantumNumber.type

A string describing the quantum number. Recommended values are (see Appendix B for a description):

 

totalNuclearSpinI

totalMagneticQuantumNumberI

totalMolecularProjectionI

nuclearSpin

parity

serialQuantumNumber

nPrincipal

lElectronicOrbitalAngularMomentum

sAngularMomentum

jTotalAngularMomentum

fTotalAngularMomentum

lMagneticQuantumNumber

sMagneticQuantumNumber

jMagneticQuantumNumber

fMagneticQuantumNumber

asymmetricTAU

asymmetricKA

asymmetricKC

totalSpinMomentumS

totalMagneticQuantumNumberS

totalMolecularProjectionS

totalElectronicOrbitalMomentumL

totalMagneticQuantumNumberL

totalMolecularProjectionL

totalAngularMomentumN

totalMagneticQuantumNumberN

totalMolecularProjectionN

totalAngularMomentumJ

totalMagneticQuantumNumberJ

totalMolecularProjectionJ

intermediateAngularMomentumF

totalAngularMomentumF

totalMagneticQuantumNumberF

vibrationNu

vibrationLNu

totalVibrationL

vibronicAngularMomentumK

vibronicAngularMomentumP

hinderedK1

hinderedK2

 

2.1.7.3                                                             QuantumNumber.numeratorValue

The numerator of the quantum number value

 

2.1.7.4                                                             QuantumNumber.denominatorValue

The denominator of the quantum number value. If not explicitly specified, it is defaulted to “1” (meaning that the corresponding quantum number value is a multiple integer)

2.1.7.5                                                             QuantumNumber.description

A human readable string, describing the nature of the quantum number. Standard descriptions are given at the Appendix B for those quantum numbers whose names are given above.  For a quantum number not appearing above, the descritpion shall be given here.

 

 

2.1.8      Process

The scope of this class is to describe the physical process responsible for the generation of the line, or for the modification of its physical properties with respect to those measured in the laboratory. The complete description of the process is relegated to specific placeholder called “model” which will describe specific physical models for each process.

 

 

2.1.8.1                                                             Process.type

String identifying the type of process. Possible values are: "Matter-radiation interaction", "Matter-matter interaction", "Energy shift", "Broadening".

 

2.1.8.2                                                             Process.name

String describing the process: Example values (corresponding to the values of "type" listed above) are: "Photoionization", "Collisional excitation", "Gravitational redshift", "Natural broadening".

 

2.1.8.3                                                             Process.model

A theoretical model by which a specific process might be described.

 

 

2.1.9      Environment

The scope of this class is describing the physical properties of the ambient gas, plasma, dust or stellar atmosphere where the line is generated.

 

 

2.1.9.1                                                             Environment.temperature

The temperature in the line-producing plasma.

 

2.1.9.2                                                             Environment.opticalDepth

The optical depth in the line-producing plasma for the transition described by "initialLevel" and "finalLevel".

 

2.1.9.3                                                             Environment.particledensity

The particle density in the line-producing plasma.

 

2.1.9.4                                                             Environment.massdensity

The mass density in the line-producing plasma.

 

2.1.9.5                                                             Environment.pressure

The pressure in the line-producing plasma.

 

2.1.9.6                                                             Environment.entropy

The entropy of the line-producing plasma.

 

2.1.9.7                                                             Environment.mass

The total mass of the line-producing gas/dust cloud or star.

 

2.1.9.8                                                             Environment.metallicity

As customary in astronomy, the metallicity of an element is expressed as

the logarithmic ratio between the element and the Hydrogen abundance,

normalized to the solar value. If the metallicity of a celestial object

or plasma is expressed through a single number, this refers to the iron

abundance.

 

2.1.9.9                                                             Environment.extinctionCoefficient

A quantitative observable k, which expresses the suppression of the emission line intensity due to the presence of optically thick matter along the line-of-sight. It is a measure of the intervening gas density through one of the following equations:

                                               

 

where  is the particle density,  is the integrated cross section,  is the integrated opacity and  the matter density.

 

2.1.9.10                                                          Model

Placeholder for future detailed theoretical models of the environment plasma where the line appears.

 

2.1.10 Source

This class gives a basic characterization of the celestial source, where an astronomical line has been observed

2.1.10.1                                                         Source.IAUname

The IAUname of the source

 

2.1.10.2                                                         Source.name

An alternative or conventional name of the source

 

2.1.10.3                                                         Source.coordinates

Coordinates of the source. Link to IVOA Space Time Coordinates data model

 

3         UCDs

 

The following is a list of the UCDs that should accompany any of the object attributes in their different serializations.

 

They are based in The UCD1+ controlled vocabulary Version 1.23(IVOA Recommendation, 2 Apr 2007).

 

 

There is one table per each of the objects in the Data Model. The left column indicates the object attribute, and the right column the UCD. Items appearing in (bold) correspond to other objects in the model.

 

 

Line

initialLevel

(Level)

finalLevel

(Level)

initialElement

(ChemicalElement)

finalElement

(ChemicalElement)

wavelength

em.wl

wavenumber

em.wn

frequency

em.freq

airWavelength

em.wl

einsteinA

phys.at.transProb

oscillatorStrength

phys.at.oscStrength

weightedOscillStrength

phys.at.WOscStrength

intensity

spect.line.intensity

observedFlux

phot.flux

observedFluxWaveMin

em.wl

observedFluxWaveMax

em.wl

significanceOfDetection

stat.snr

process

(Process)

lineTitle

meta.title

transitionType

meta.title

strength

spect.line.strength

observedBroadeningCoefficient

spect.line.broad

observedShiftingCoefficient

phys.atmol.lineShift

 

 

Species

name

meta.title

 

 

Level

type

meta.title

totalStatWeight

phys.atmol.sweight

nuclearStatWeight

phys.atmol.nucweigth

lifeTime

phys.atmol.lifetime

energy

phys.energy

quantumState

(QuantumState)

energyOrigin

phys.energy

landeFactor

phys.at.lande

nuclearSpinSymmetryType

phys.atmol.symmetrytype

parity

phys.atmol.parity

energyOrigin

phys.energy

configuration

phys.atmol.configuration

 

 

QuantumState

normalizedProbability

stat.normalProb

quantumNumber

phys.atmol.qn

termSymbol

phys.atmol.termSymbol

 

 

QuantumNumber

label

meta.title

type

meta.title

numeratorValue

meta.number

denominatorValue

meta.number

description

meta.note

 

 

 

 

 

Process

model

(Model)

name

meta.title

 

 

Environment

temperature

phys.temperature

opticalDepth

phys.absorption.opticalDepth

density

phys.density

pressure

phys.pressure

extinctionCoefficient

phys.absorption

entropy

phys.entropy

mass

phys.mass

metallicity

phys.abund.Z

model

(Model)


4         Working examples

 

4.1           The Hyperfine Structure of N2H+

 

This example refers to the measurement of the hyperfine structure of the J=10 transition in diazenlyium (N2H+) at 93 Ghz (Caselli et al. 1995) toward the cold (kinetic temperature TK~10 K) dense core of the interstellar cloud L1512. Due to the closed-shell 1Σ configuration of this molecule, the dominant hyperfine interactions are those between the molecular electric field gradient and the electric quadrupole moments of the two nitrogen nuclei. Together they produce a splitting of the J=10 in seven components. The astronomical measurements are much more accurate than those obtainable on the Earth, due to the excellent spectral resolution (~0.18 km s-1 FWHM), which correspond to the thermal width at ~20K, much a lower temperature than achievable in the laboratory.

 

 

Table 1 – Observed  properties of the N2H+ hyperfine structure components

J F1 F → J'F'1F'

 (MHz)

 (MHz)

1 0 1 → 0 1 2

93176.2650

0.0011

1 2 1 → 0 1 1

93173.9666

0.0012

1 2 3 → 0 1 2

93173.7767

0.0012

1 2 2 → 0 1 1

93173.4796

0.0012

1 1 1→ 0 1 0

93172.0533

0.0012

1 1 2 → 0 1 2

93171.9168

0.0012

1 1 0 → 0 1 1

93171.6210

0.0013

 

 

whereis the transition frequency – as derived assuming the same Local Standard Rest velocity for all observed spectral lines – and   its relative uncertainty.

 

 Estimates of the N2H+ optical depth, excitation temperature and intrinsic line width were made by fitting the hyperfine splitting complex. They yielded:

 

·          

·          

·            Δv = 183±1 m s-1

 

 However, the same paper reports evidence for deviations from a single temperature excitation  in the following transitions: (F1,F) = (1,2) → (1,2) and (1,0) → (1,1)

 

 We show below an example of instantiation of the current Line Data Model for one of the components of the N2H+ hyperfine transition (e.g. the transition in the first row of Tab.1).

In what follows, SI units are assumed whenever pertinent and PhysicalQuantity.error indicates the statistical uncertainty on a measured quantity.

 

4.1.1      The values in the model

In what follows we give the values attached to each of the model items pertinent for the case. For sake of simplicity, we report here the transition in the first row of Table 1 only. Likewise, the class attributes have been given values in pseudo-code way.

 

Initial Level (one QuantumState defined by three QuantumNumber(s)):

 

·         Line.initialLevel.quantumState.quantumNumber.label := “J”

·         Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumJ

·         Line.initialLevel.quantumState.quantumNumber.description := “Pure quantum number”

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue := 1

·         Line.initialLevel.quantumState.quantumNumber.denominatorValue :=1

 

·         Line.initialLevel.quantumState.quantumNumber. label:= “F1

·         Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF

·         Line.initialLevel.quantumState.quantumNumber.description:= “Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I1

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue := 0

 

 

·         Line.initialLevel.quantumState.quantumNumber. label:= “F”

·         Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF

·         Line.initialLevel.quantumState.quantumNumber.description := “Resulting total angular momentum; coupling of I2 and F1

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue := 1

·         Line.initialLevel.quantumState.quantumNumber.denominatorValue :=1

 

 

 

Final Level (one QuantumState defined by three QuantumNumber(s)):

 

 

·         Line.finalLevel.quantumState.quantumNumber. label:= “J”

·         Line.finalLevel.quantumState.quantumNumber.type := “jtotalAngularMomentum”

·         Line.finalLevel.quantumState.quantumNumber.description := “Total angular momentum excluding nuclear spins. Pure quantum number”

·         Line.finalLevel.quantumState.quantumNumber.numeratorValue := 1

·         Line.initialLevel.quantumState.quantumNumber.denominatorValue :=1

 

 

 

 

·         Line.initialLevel.quantumState.quantumNumber. label:= “F1

·         Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF

·         Line.initialLevel.quantumState.quantumNumber.description:= “Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I1

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue := 0

 

 

·         Line.initialLevel.quantumState.quantumNumber. label:= “F”

·         Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF

·         Line.initialLevel.quantumState.quantumNumber.description := “Resulting total angular momentum; coupling of I and J”

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue := 2

·         Line.initialLevel.quantumState.quantumNumber.denominatorValue := 1

 

Line specific attributes:

·         Line.airWavelength.value:= 3.21755760x10-3

·         Line.airWavelength.unit.expression:= “m”

·         Line.airWavelength.Unit.scaleSI:= 1

·         Line.airWavelength.Unit.dimEquation:= “L”

 

 

Process specific attributes (Broadening):

 

 

·         Line.process.type := “Broadening”

·         Line.process.name := “Intrinsic line width”

·         Line.observedBroadeningCoefficient.value := 183

·         Line.observedBroadeningCoefficient.unit.expression := “m/s”

·         Line.observedBroadeningCoefficient.Unit.scaleSI := 1

·         Line.observedBroadeningCoefficient.Unit.dimEquation := “LT-1”

 

 

Environment specific attributes:

 

 

·         Line.process.model.excitationTemperature.value := 4.9

·         Line.process.model.excitationTemperature.error := 0.1

·         Line.process.model.excitationTemperature.unit.expression := “K”

·         Line.process.model.excitationTemperature.Unit.scaleSI := 1

·         Line.process.model.excitationTemperature.Unit.dimEquation := “K”

·         Line.process.model.opticalDepth.value := 7.9

·         Line.process.model.opticalDepth.unit.expression := “”

·         Line.process.model.opticalDepth.Unit.scaleSI := 1

·         Line.process.model.opticalDepth.Unit.dimEquation := “”

·         Line.process.model.opticalDepth.error := 0.3

 

 

Initial and final (identical) Specie(s):

 

·         Line.initialElement.name := “N2H+

·         Line.finalElement.name := “N2H+

 

Source specific attributes:

 

·         Line.source.name := “L1512”

 

 

 

4.1.2      JSON representation

 

{

            "Line": {

 

                        "source": {

                                    "name": "L1512"

                        }

 

                        "initialElement": {

                                    "name": "N2H+"

                        }

 

                        "finalElement": {

                                    "name": "N2H+"

                        }

 

                        "initialLevel": {

                                    "quantumNumber": {

                                                "label": "J"

                                                "type": "totalAngularMomentumJ"

                                                "description": "Pure quantum number"

                                                "numeratorValue": "1"

                                                "denominatorValue": "1"

                                    }

                                    "quantumNumber": {

                                                "label": "F1"

                                                "type": "totalAngularMomentumF”

                                                "description": "Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I1"

                                                "numeratorValue": "0"

                                    }

                                    "quantumNumber": {

                                                "label": "F"

                                                "type": "totalAngularMomentumF"

                                                "description": 'Resulting total angular momentum; coupling of I2 and F1"

                                                "numeratorValue": "1"

                                                "denominatorValue": "1"

                                    }

                        }

 

                        "finalLevel": {

                                    "quantumNumber": {

                                                "label": "J"

                                                "type": "jtotalAngularMomentum"

                                                "description":  "Total angular momentum excluding nuclear spins. Pure quantum number"

                                                "numeratorValue": "1"

                                                "denominatorValue": "1"

                                    }

                                    "quantumNumber": {

                                                "label": "F1"

                                                "type": "totalAngularMomentumF"

                                                "description": "Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I1"

                                                "numeratorValue": "0"

                                    }

                                    "quantumNumber": {

                                                "label": "F"

                                                "type": "totalAngularMomentumF"

                                                "description": "Resulting total angular momentum; coupling of I and J"

                                                "numeratorValue": "2"

                                                "denominatorValue": "1"

                                    }

                        }

 

                        "airWavelength": {

                                    "value":  "3.21755760x10-3"

                                    "unit": {

                                                "expression": "m"

                                                "scaleSI": "1"

                                                "dimEquation": "L"

                                    }          

                        }

                        "process": {

                                    "type": "Broadening"

                                    "name": "Intrinsic line width"

 

                                    "model": {

                                                "excitationTemperature": {

                                                            "value": "4.9"

                                                            "error": "0.1"

                                                            "unit": {

                                                                        "expression": "K"

                                                                        "scaleSI": "1"

                                                                        "dimEquation": "K"

                                                            }

                                                }

                                                "opticalDepth": {

                                                            "value": "7.9"

                                                            "error": "0.3"

                                                            "unit": {

                                                                        "expression": ""

                                                                        "scaleSI": "1"

                                                                        "dimEquation": ""

                                                            }          

                                                }

                                    }

                        }

                       

                        "observedBroadeningCoefficient": {

                                    "value": "183"

                                    "unit" : {

                                                "expression": "m/s"

                                                "scaleSI": "1"

                                                "dimEquation": "LT-1"

                                    }          

                        }

            }

}

 


4.1.3      UML instantiation diagram

 

 

.

 

 

Please note that some physical quantities (marked with an asterisk) have not been fully instanced to simplify the graphics.
 

4.2      Radiative Recombination Continua: a diagnostic tool for X-Ray spectra of AGN

The advent of a new generation of large X-ray observatories is allowing us to obtain spectra of unprecedented quality and resolution on a sizeable number of Active Galactic Nuclei (AGN). This has revived the need for diagnostic tools, which can properly characterize the properties of astrophysical plasmas encompassing  the nuclear region, where the gas energy budget is most likely dominated by the high-energy AGN output.

 

 Among these spectra diagnostics, Radiative Recombination Continua (RRC) play a key role, as they unambiguously identify photoionized plasmas, and provide unique information on their physical properties. The first quantitative studies which recognized the importance of RRC in X-ray spectra date back to the early '90, using Einstein (Liedahl et al. 1991; Kahn & Liedahl 1991) and ASCA (Angelini et al. 1995) observations. The pioneer application of the RRC diagnostic to AGN is due to Kinkhabwala et al. (2002; K02), who analysed a long XMM-Newton/RGS observation of the nearby Seyfert 2 galaxy NGC1068 (z=0.003793, corresponding to a recession velocity of 1137 km s-1). We will refer to the results reported in their paper hereafter.

 

 K02 report the detection of RRC from 6 different ionic species. Their observational properties are shown in Tab.3. The RRC temperature kTe is

 

Tab.3 – Properties of the RRC features in the XMM-Newton/RGS spectrum of NGC1068

Ion

kTe (eV)

Flux

(10-4 ph cm-2 s-1)

I (eV)

CV

2.5±0.5

4.3±0.4

392.1

CVI

4.0±1.0

2.8±0.3

490.0

NVI

3.5±2.0

2.1±0.2

552.1

NVII

5.0±3.0

1.1±0.1

667.1

OVII

4.0±1.3

2.4±0.2

739.3

OVIII

7.0±3.5

1.2±0.1

871.4

 

 

derived from the RRC profile fit, as the width of the RRC profile ΔEkTe. The average RRC photon energy is EI+kTe, where I is the ionization potential of the recombined state. If the plasma is highly over ionized (kT«I) – as expected in X-ray photoionized nebulae (Kallman & McCray 1982) – then  ΔE/EkTe/I. Therefore, the specification of kTe (extracted from Tab.2 in K02) and I (extracted from table of photo ionization potentials) is enough to know the energy of the feature.

 

4.2.1       The values in the model

 

Initial Level description:

·         Line.initialLevel.quantumState.quantumNumber.label:= “n”

·         Line.initialLevel.quantumState.quantumNumber.type:= “nPrincipal

·         Line.initialLevel.quantumState.quantumNumber.numeratorValue:=1

·         Line.initialLevel.quantumState.quantumNumber.denominatorValue:=1

 

 

Final Level:

·         Line.finalLevel.quantumState.quantumNumber.label:= “n”

·         Line.finalLevel.quantumState.quantumNumber.type:= “nPrincipal

·         Line.finalLevel.quantumState.quantumNumber.numeratorValue:= 1

·         Line.finalLevel.quantumState.quantumNumber.denominatorValue:= 1

 

 

 

Initial Element:

·         Line.initialElement.species.name := “CVI”

 

Final Element:

·         Line.finalElement.species.name := “CV”

 

 

(Observed) Line specific attributes

 

·         Line.wavelength.value := 394.6

·         Line.wavelength.unit.expression := “eV”

·         Line.wavelength.Unit.scaleSI := 1.6E-19

·         Line.wavelength.Unit.dimEquation := “ML2T-2”

 

·         Line.observedFlux.value := 2.8E-4

·         Line.observedFlux.error := 0.3E-4

·         Line.observedFlux.unit.expression := “photons*cm-2*s-1”

·         Line.observedFlux.unit.scaleSI = 1.E4

·         Line.observedFlux.unit.dimEquation := “L-2T-1”

 

·         Line.transitionType := “Radiative Recombination Continuum”

 

Process specific attributes

·         Line.Process.type := “Energy shift”

·         Line.Process.name := “Cosmological redshift”

 

·         Line.observedShiftingCoefficient.value := 1137

·         Line.observedShiftingCoefficient.unit.expression := “km/s”

·         Line.observedShiftingCoefficient.unit.scaleSI := 1.E3

·         Line.observedShiftingCoefficient.unit.dimEquation := “MT-1”

 

 

Environment specific attributes:

 

·         Line.environment.temperature.value := 1.9E5

·         Line.environment.temperature.unit.expression := “K”

·         Line.environment.temperature.unit.scaleSI := 1

·         Line.environment.temperature.unit.dimEquation := “K”

 

 

 

Source specific attributes:

 

 

4.2.2      JSON representation

{

            "Line": {

                        "transitionType":  “Radiative Recombination Continuum”

                        "wavelength": {

                                    "value":  "394.6"

                                    "unit": {

                                                "expression": eV"

                                                "scaleSI": "1.6E-19"

                                                "dimEquation": "ML2T-2"

                                    }          

                        }

                        "observedFlux": {

                                    "value":  "2.8E-4"

                                    "error": "0.3E-4"

                                    "unit": {

                                                "expression": photons*cm-2*s-1"

                                                "scaleSI": "1.E4"

                                                "dimEquation": "L-2T-1"

                                    }          

                        }

                        "source": {

                                    "name": "NGC1068"

                        }

                        "initialElement": {

                                    "name": "CVI"

                        }

                        "finalElement": {

                                    "name": "CV"

                        }

                        "initialLevel": {

                                    "quantumNumber": {

                                                "label": "n"

                                                "type": "nPrincipal"

                                                "numeratorValue": "1"

                                                "denominatorValue": "1"

                                    }

                        }

                        "finalLevel": {

                                    "quantumNumber": {

                                                "label": "n"

                                                "type": "nPrincipal"

                                                "numeratorValue": "1"

                                                "denominatorValue": "1"

                                    }

                        }

                        "process": {

                                    "type": "Energy shift"

                                    "name": "Cosmological redshift"

                        }

                        "observedShiftingCoefficient": {

                                    "value": "1137"

                                    "unit" : {

                                                "expression": "km/s"

                                                "scaleSI": "1.E3"

                                                "dimEquation": "MT-1"

                                    }          

                        }

                        "environment": {

                                    "temperature": {

                                                "value": "1.9E5"

                                                "unit": {

                                                            "expression": "K"

                                                            "scaleSI": "1"

                                                            "dimEquation": "K"

                                                }          

                                    }

                        }

            }

}

4.2.3      UML Instantiation diagram

 


 

4.3      References

 

 

[1] [Angelini L. et al]

Astrophysical Journal, 1995, 449, L41 (1995)

 

[2] [Condon E.U./Shortley G.H.]

The Theory of Atomic Spectra,

Cambridge University Press

ISBN 0521092094 (1985)

 

[3] [Shore B.W./Menzel D.H.]

Principles of Atomic Spectra

Wiley Series in Pure and Applied Spectroscopy,

John Wiley & Sons Inc

ISBN 047178835X (1968)

 

[4] [Caselli P. et al]

Astrophysical Journal, 455, L77 (1995)

 

[5] [Drake G.W.F.]

Atomic, Molecular and Optical Physics Handbook, Chap.21

(AIP Woodbury:NY) (1996)

 

[6] [Kahn S.M./Liedahl D.A.]

In “Iron Line Diagnostic in X-ray Sources”, (Berlin:Springer), (1991)

 

[7] [Kallman T.R./McCray R.]

Astrophysical Journal Supplement, 50, 263 (1982)

 

[8] [Kinkhabwala A. et al]

Astrophysical Journal, 575,732 (2002)

 

[9] [Liedahl D. et al.]

1991, AIP Conf. Proc. 257, 181 (1991)

 

[10] [Liedahl D./Paerels F.]

Astrophysical Journal, 468, L33 (1996)

 

[11] [Martin W.C./Wiese W.L.]

Atomic Spectroscopy. A compendium of Basic Ideas, Notation, Data and Formulas

http://physics.nist.gov/Pubs/AtSpec/index.html

 

[12] [Rybicki C.B./Lightman A.P.]

Radiative Processes in Astrophysics

Wiley Interscience, John Wiley & Sons

ISBN 0-471-82759-2 (1979)

 

[13] [Salgado J./Osuna P./Guainazzi M./Barbarisi I./Dubernet ML./Tody D.] IVOA Simple Line Access Protocol v0.9 http://www.ivoa.net/internal/IVOA/SpectralLineListsDocs/SLAP_v0.9.pdf

 

[14] [Derriere S./Gray N./Mann R./Preite A./McDowell J./ Mc Glynn T./ Ochsenbein F./Osuna P./Rixon G./Williams R.]

An IVOA Standard for Unified Content Descriptors v1.10 http://www.ivoa.net/Documents/latest/UCD.html

 

[15] [Tody D./Dolensky M./McDowell J./Bonnarel F./Budavari T./Busko I./ Micol A./Osuna P./Salgado J./Skoda P./Thompson R./Valdes F.]

IVOA Simple Spectral Access Protocol v1.4

http://www.ivoa.net/Documents/latest/SSA.html

 

 

 

 

 


 

5         Appendix A: List of Atomic Elements

List of Elements extracted from the IUPAC Commission on Atomic Weights and Isotopic Abundances. (http://www.chem.qmul.ac.uk/iupac/)

 

 


List of Elements in Atomic Number Order.

At No

Symbol

Name

Notes

1

H

Hydrogen

1, 2, 3

2

He

Helium

1, 2

3

Li

Lithium

1, 2, 3, 4

4

Be

Beryllium

 

5

B

Boron

1, 2, 3

6

C

Carbon

1, 2

7

N

Nitrogen

1, 2

8

O

Oxygen

1, 2

9

F

Fluorine

 

10

Ne

Neon

1, 3

11

Na

Sodium

 

12

Mg

Magnesium

 

13

Al

Aluminium

 

14

Si

Silicon

2

15

P

Phosphorus

 

16

S

Sulfur

1, 2

17

Cl

Chlorine

3

18

Ar

Argon

1, 2

19

K

Potassium

1

20

Ca

Calcium

1

21

Sc

Scandium

 

22

Ti

Titanium

 

23

V

Vanadium

 

24

Cr

Chromium

 

25

Mn

Manganese

 

26

Fe

Iron

 

27

Co

Cobalt

 

28

Ni

Nickel

 

29

Cu

Copper

2

30

Zn

Zinc

 

31

Ga

Gallium

 

32

Ge

Germanium

 

33

As

Arsenic

 

34

Se

Selenium

 

35

Br

Bromine

 

36

Kr

Krypton

1, 3

37

Rb

Rubidium

1

38

Sr

Strontium

1, 2

39

Y

Yttrium

 

40

Zr

Zirconium

1

41

Nb

Niobium

 

42

Mo

Molybdenum

1

43

Tc

Technetium

5

44

Ru

Ruthenium

1

45

Rh

Rhodium

 

46

Pd

Palladium

1

47

Ag

Silver

1

48

Cd

Cadmium

1

49

In

Indium

 

50

Sn

Tin

1

51

Sb

Antimony

1

52

Te

Tellurium

1

53

I

Iodine

 

54

Xe

Xenon

1, 3

55

Cs

Caesium

 

56

Ba

Barium

 

57

La

Lanthanum

1

58

Ce

Cerium

1

59

Pr

Praseodymium

 

60

Nd

Neodymium

1

61

Pm

Promethium

5

62

Sm

Samarium

1

63

Eu

Europium

1

64

Gd

Gadolinium

1

65

Tb

Terbium

 

66

Dy

Dysprosium

1

67

Ho

Holmium

 

68

Er

Erbium

1

69

Tm

Thulium

 

70

Yb

Ytterbium

1

71

Lu

Lutetium

1

72

Hf

Hafnium

 

73

Ta

Tantalum

 

74

W

Tungsten

 

75

Re

Rhenium

 

76

Os

Osmium

1

77

Ir

Iridium

 

78

Pt

Platinum

 

79

Au

Gold

 

80

Hg

Mercury

 

81

Tl

Thallium

 

82

Pb

Lead

1, 2

83

Bi

Bismuth

 

84

Po

Polonium

5

85

At

Astatine

5

86

Rn

Radon

5

87

Fr

Francium

5

88

Ra

Radium

5

89

Ac

Actinium

5

90

Th

Thorium

1, 5

91

Pa

Protactinium

5

92

U

Uranium

1, 3, 5

93

Np

Neptunium

5

94

Pu

Plutonium

5

95

Am

Americium

5

96

Cm

Curium

5

97

Bk

Berkelium

5

98

Cf

Californium

5

99

Es

Einsteinium

5

100

Fm

Fermium

5

101

Md

Mendelevium

5

102

No

Nobelium

5

103

Lr

Lawrencium

5

104

Rf

Rutherfordium

5, 6

105

Db

Dubnium

5, 6

106

Sg

Seaborgium

5, 6

107

Bh

Bohrium

5, 6

108

Hs

Hassium

5, 6

109

Mt

Meitnerium

5, 6

110

Ds

Darmstadtium

5, 6

111

Rg

Roentgenium

5, 6

112

Uub

Ununbium

5, 6

114

Uuq

Ununquadium

5, 6

116

Uuh

Ununhexium

see Note above

118

Uuo

Ununoctium

see Note above

1.      Geological specimens are known in which the element has an isotopic composition outside the limits for normal material. The difference between the atomic weight of the element in such specimens and that given in the Table may exceed the stated uncertainty.

2.      Range in isotopic composition of normal terrestrial material prevents a more precise value being given; the tabulated value should be applicable to any normal material.

3.      Modified isotopic compositions may be found in commercially available material because it has been subject to an undisclosed or inadvertant isotopic fractionation. Substantial deviations in atomic weight of the element from that given in the Table can occur.

4.      Commercially available Li materials have atomic weights that range between 6.939 and 6.996; if a more accurate value is required, it must be determined for the specific material [range quoted for 1995 table 6.94 and 6.99].

5.      Element has no stable nuclides. The value enclosed in brackets, e.g. [209], indicates the mass number of the longest-lived isotope of the element. However three such elements (Th, Pa, and U) do have a characteristic terrestrial isotopic composition, and for these an atomic weight is tabulated.

The names and symbols for elements 112-118 are under review. The temporary system recommended by J Chatt, Pure Appl. Chem., 51, 381-384 (1979) is used above. The names of elements 101-109 were agreed in 1997 (See Pure Appl. Chem., 1997, 69, 2471-2473),for element 110 in 2003 (see Pure Appl. Chem., 2003, 75, 1613-1615) and for element 111 in 2004 (see Pure Appl. Chem., 2004, 76, 2101-2103).

 

6         Appendix B: List of quantum numbers

 

The list contains the most usual quantum numbers in atomic and molecular spectroscopy. The list is not exhaustive and is opened to new entries.

 

Note for molecules:  Angular momemtum basis functions, |A a MA >, can be simultaneous eigenfunctions of three types of operators : the magnitude A2, the component of A onto the internuclear axis Az, and the component of A on the laboratory quantization axis AZ. The basis function labels A, a and MA correspond to eigenvalues of  A2, Az, and AZ, respectively ħ2A(A+1), ħ a and ħ MA

 

Note for intermediate coupling: Intermediate coupling occurs in both atomic and molecular physics. The document below gives some explanations about intermediate coupling in atomic physics, these explanations can be transposed to molecular physics (as for intermediate coupling between different Hund's cases). As described below, levels can be labelled by the least objectionable coupling case, by linear combinaison of pure coupling basis functions (the linear coefficients can be determined in a theoretical approach: this is planned for in the data model), or simply by a sort of serial number (see serialQuantumNumber  below)

6.1      Various Quantum numbers

6.1.1      totalNuclearSpinI

 total nuclear spin of one atom or a molecule, I

 

6.1.2      totalMagneticQuantumNumberI

total magnetic quantum number,  where  is the eigenvalue of the  operator

 

6.1.3      totalMolecularProjectionI

total nuclear spin projection quantum number  

where  is the eigenvalue of the  operator

 

6.1.4      nuclearSpin

nuclear spin of individual nucleus  which composes a molecule, noted;

 

6.1.5      parity

eigenvalue of the parity operator applied to the total wavefunction. It takes the value “0” for even parity and “1” for odd parity

 

6.1.6      serialQuantumNumber

A serial quantum number that labels states to which no good or nearly good quantum numbers can be assigned to.

 

6.2      Quantum numbers for hydrogenoids

 

6.2.1      nPrincipal

principal quantum number n

 

6.2.2      lElectronicOrbitalAngularMomentum

orbital angular momentum of an electron  where

is the eigenvalue of the operator (called as well azimuthal quantum number).

 

6.2.3      sAngularMomentum

spin angular momentum of an electron,  only where  is the eigenvalue of the  operator)

 

6.2.4      jTotalAngularMomentum

total angular momentum of one electron,  and .  is the eigenvalue of the  operator, where

 

6.2.5      fTotalAngularMomentum

total angular momentum , including nuclear spin .  is the eigenvalue of the  operator, where

 

6.2.6      lMagneticQuantumNumber

orbital magnetic quantum number,  where  is the eigenvalue of the  operator

 

6.2.7      sMagneticQuantumNumber

spin magnetic quantum number,  where  is the eigenvalue of the  operator

 

6.2.8      jMagneticQuantumNumber

  orbital magnetic quantum number,   where  is the eigenvalue of the  operator

 

6.2.9      fMagneticQuantumNumber

  orbital magnetic quantum number,   where  is the eigenvalue of the  operator

6.3      Pure rotational quantum numbers

 

6.3.1      asymmetricTAU

Index t labelling asymmetric rotational energy levels for a given rotational quantum number N.

 

Note: The solution of the Schrödinger equation for an asymmetric-top molecule gives for each value of N, (2N+1) eigenfunctions with its own energy. It is customary to keep track of them by adding the subscript t to the N value (Nt). This index t goes from -N for the lowest energy of the set to +N for the highest energy, and is equal to (Ka – Kc).

 

6.3.2      asymmetricKA

For a given N, energy levels may be specified by Ka Kc (or K-1 K1, or K- K+ are alternative notations), where Ka is the K quantum number for the limiting prolate (B=C) and Kc for the limiting oblate (B=A). In the notation (K-1 K1) the subscripts “1” and “-1” correspond to values of the asymmetry parameter

where A, B, C are rotational constants of the asymmetric molecule (by definition A>B>C)

 

6.3.3      asymmetricKc

 see asymmetricKA

 

6.4      Quantum numbers for n electron systems (atoms and molecules)

 

6.4.1      totalSpinMomentumS

it is the total spin quantum number S, S can be integral or half-integral. is the eigenvalue of the operator, where

 

6.4.2      totalMagneticQuantumNumberS

total spin magnetic quantum number,  where  is the eigenvalue of the  operator.

 

6.4.3      totalMolecularProjectionS

total spin projection quantum number , , where  is the eigenvalue of the operator.

 

6.4.4      totalElectronicOrbitalMomentumL

it is the total orbital angular momentum ,  is integral.  is the eigenvalue of the  operator, where

 

6.4.5      totalMagneticQuantumNumberL

total orbital magnetic quantum number, , where  is the eigenvalue of the  operator.

 

6.4.6      totalMolecularProjectionL

total orbital projection quantum number, where is the absolute value of the eigenvalue of the  operator (Hund’s cases (a) and (b) in the case of a diatomic)

6.4.7      totalAngularMomentumN

is the total angular momentum N exclusive of nuclear and electronic spin, N is integral. For a molecule in a close-shell state totalAngularMomemtumN is the pure rotational angular momentum.

 

6.4.8      totalMagneticQuantumNumberN

total orbital magnetic quantum number, where  is the eigenvalue of the  operator

 

 

6.4.9      totalMolecularProjectionN

absolute value of the component of the angular momentum along the axis of a symmetric (or quasi-symmetric) rotor, usually noted .is the eigenvalue of the operator, with values

For open shell diatomic molecules, it corresponds to “totalMolecularProjectionL” (), so we advise to preferentially use “totalMolecularProjectionL”

 

Note: The symbol  is also used in spectroscopy to describe the component of the vibronic angular momentum (excluding spin) along the axis for linear polyatomic molecules. In this model, we prefer to uniquely identify this specific case by a different type of Quantum Number: “vibronicAngularMomentumK”, defined thereafter.

 

6.4.10 totalAngularMomentumJ

 

is the total angular momentum J exclusive of  nuclear spin, J can be integral or half-integral.

For atoms:

 

For molecules:

 

6.4.11 totalMagneticQuantumNumberJ

total magnetic quantum number,  where

is the eigenvalue of the  operator.

 

6.4.12 totalMolecularProjectionJ

absolute value of the component of the angular momentum  along the molecular axis, noted  where  is the eigenvalue of the  operator.

For linear molecules with no nuclear spin (or no nuclear spin coupled to the molecular axis), it is the absolute value of the component of the total electronic angular momentum on the molecular axis (Hund’s cases (a) and (c)). When and are defined (Hund’s case (a)):

 

For linear molecules with a nuclear spin coupled to the molecular axis, it includes as well the component of the nuclear spin on the molecular axis.

 

6.4.13 intermediateAngularMomemtunF

is associated to the intermediate quantum number  where (or ) + any other vector

6.4.14 totalAngularMomentumF

is the total angular momentum  including nuclear spin, can be integral or half-integral.  is the eigenvalue of the operator, where for atoms:

 

 

and for molecules with m nuclear spins:

6.4.15 totalMagneticQuantumNumberF

total magnetic quantum number,  where  is the eigenvalue of the operator

 

6.5      Vibrational and (ro-)vibronic quantum numbers

6.5.1      vibrationNu

vibrational modes  (following Mulliken conventions). By default the vibrational mode is a normal mode. If the vibrational mode is fairly localised, the bond description will be included in the attribute “description” of “QuantumNumber”

6.5.2      vibrationLNu

angular momentum associated to degenerate vibrations,

or

6.5.3      totalVibrationL

total vibrational angular momentum is the sum of all angular momenta associated to degenerate vibrations:

6.5.4      vibronicAngularMomentumK

is the sum of the total vibrational angular momentum  and of the electronic orbital momentum about the internuclear axis

here  (and ) are unsigned quantities. This is used for linear polyatomic molecules. (see p.25 Volume III pf Herzberg, and REC. (recommendation) 17 of Muliken, 1955)

 

6.5.5      vibronicAngularMomentumP

is the sum of the total vibrational angular momentum  and of the total electronic orbital momentum about the internuclear axis  

here  (and ) are unsigned quantities. This is used for linear polyatomic molecules. (see p.26 Volume III pf Herzberg, and REC. 18 of Muliken, 1955)

 

 

6.5.6      rovibronicAngularMomentumP

total resultant axial angular momentum quantum number including electron spin: . (REC. 26 of Mulliken, 1955)

6.5.7      hinderedK1, hinderedK2

for internal free rotation of 2 parts of a molecule, (see p.492, Volume II of Herzberg), 2 additional projection quantum numbers are necessary:  and , such that total rotational energy is given by:

 

 

Where  is totalAngularMomentumN andis totalMolecularProjectionN (see p.492, Volume II of Herzberg (1964))

 

 

 

 

7         APPENDIX C:Description of couplings for atomic Physics

7.1      LS coupling

Usually the strongest interactions among the electrons of an atom are their mutual Coulomb repulsions. These repulsions affect only the orbital angular momenta, and not the spins. It is thus most appropriate to first couple together all the orbital angular momenta to give eigenfunctions

of L2  and LZ , with L the total orbital angular momentum of the atom. Similarly all spins are coupled together to give the eigenfunctions of  S2  and SZ ,  with S the total spin angular momentum; then and S  are coupled together to give eigenfunctions of  J2  and JZ  , where J=L+S.

 

When the coupling conditions within an atom correspond closely to pure LS-coupling conditions, then the quantum states of an atom can be accuratly described in terms of LS-coupling quantum numbers:

 

Giving values of L and S specifies a term, or more precisely a ``LS term'', because on may also refer to ``terms'' of a different sort when discussing other coupling schemes (In order to completely specify a term it is necessary to give not only values of L and S, but also values of all lower-order quantum numbers, such as nili.

·                     Giving values of L, S, J specifies a level

·                     Giving values of L, S, J, MJ specifies a state

·                     The value of (2S+1) is called the multiplicity of the term

 

For LS-coupled functions, the notation introduced by Russel and Saunders is universally used : 2S+1LJ, where numerical values are to be substituted for (2S+1) and J, and the appropriate letter symbol is used for L (S, P, ..); except when discussing the Stark or Zeeman effect, there is usually no need to specify the value of MJ .

 

7.2       jj coupling

 

With increasing Z, the spin-orbit interactions become increasingly more important; in the limit in which these interactions become much stronger than the Coulom terms, the coupling conditions approach pure jj coupling.

 

In the jj-coupling scheme, basis functions are formed by first coupling the spin of each electron to its own orbital angular momentum, and then coupling together the various resultants ji in some arbitrary order to obtain the total angular momentum J.

For two-electron configurations, the coupling scheme may be described by the condensed notation [(l1 s1)j1 , (l2, s2)j2]JMJ with the usual jj-notation for energy levels (j1, j2)J [analogous to the Russel Saunders notation 2S+1LJ].

 

7.3      jK coupling

For configurations containing only two electron outside of closed shells, the common limiting type of pair coupling (energy levels tend to appear in pairs), jK coupling, occurs when the strongest interaction is the spin-orbit interaction of the more tightly bound electron, and the next strongest  interaction is the spin-independent (direct) portion of the Coulomb interaction between the 2 electrons.

The corresponding angular-momemtum coupling scheme is l1 + s1 =  j1 , j1 + l2 = K, K + s2 = J, or notation {[(l1 s1)j1 , l2]K,s2}JM with the standard energy level notation j1[K]J.

 

7.4      LK coupling

The other limiting form of pair coupling is called LK (or Ls) coupling. In two-electron configurations, it corresponds to the case in which the direct Coulomb interaction is greater then the spin-orbit interaction of either electron, and the spin-orbit interaction of the inner electron is next most important. The coupling scheme is l1 + l2 = LL + s1 = K, K + s2 = J, or notation {[(l1 l2)L, s1]K,s2}JM with the standard energy level notation L[K]J.

 

7.5      Intermediate coupling

Frequently the coupling conditions do not lie particularly close even to one of these four cases; such situation is referred to as intermediate coupling. The energy levels can only be labelled in terms of the least objectionable of the four pure-coupling schemes (with the understanding that these labels may give a poor description of the true angular-momentum properties of the corresponding quantum states). In many cases, however, the coupling conditions are so hopelessly far from any pure-coupling scheme that it is meaningless to do anything more than label the energy levels and quantum states by means of serial numbers or some similar arbitrary device, or to list the values of the largest few eigenvector components (or the squares thereof) in the expansion of the total wavefunction.

 

  “The wavefunctions of levels are often expressed as eigenvectors that are linear combinations of basis states in one of the standard coupling schemes. Thus, the wave function Y(aJ) for a level labeled  aJ might be expressed in terms of normalized LS coupling basis states F(gSLJ): Y(aJ)=gSL c(gSLJ)F(SLJ)The c(gSLJ) are expansion coefficients , and gSL |c(SLJ|2 = 1(Martin & Wiese)

 

The expansion coefficients are called “mixingCoefficient” in this document.

 

The squared expansion coefficients for the various gSL terms in the composition of the aJ level are conveniently expressed as percentages, whose sum is 100%. The notation for RS basis states has been used only for concreteness; the eigenvectors may be expressed in any coupling scheme, and the coupling schemes may be different for different configurations included in a single calculation (with configuration interaction). « Intermediate coupling » conditions for a configuration are such that calculations in both LS and jj coupling yield some eigenvectors representing significant mixtures of basis states.

The largest percentage in the composition of a level is called the purity of the level in that coupling scheme. The coupling scheme (or combination of coupling schemes if more than one configuration is involved) that results in the largest average purity for all the levels in a calculation is usually best for naming the levels. With regard to any particular calculation, one does well to remember that, as with other calculated quantities, the resulting eigenvectors depend on a specific theoretical model and are subject to the inaccuracies of whatever approximations the model involves.

Theoretical calculations of experimental energy level structures have yielded many eigenvectors having significantly less than 50% purity in any coupling scheme. Since many of the corresponding levels have nevertheless been assigned names by spectroscopists, some caution is advisable in the acceptance of level designations found in the literature. »