
International Virtual Observatory Alliance 
Simple Spectral Lines Data Model
Version 1.0
Working Draft 13 July 2009
This version:
WDSSLDM1.020090713
Latest version:
http://www.ivoa.net/Documents/SSLDM
Previous versions:
Editors: Pedro Osuna, Jesus Salgado
Authors:
Pedro Osuna
Matteo Guainazzi
Jesus Salgado
MarieLise 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
^{4.1} The Hyperfine Structure of N2H^{+}
4.1.3 UML instantiation diagram
4.2 Radiative Recombination Continua: a diagnostic tool for XRay spectra of AGN
4.2.3 UML Instantiation diagram
5 Appendix A: List of Atomic Elements
6 Appendix B: List of quantum numbers
6.1.2 totalMagneticQuantumNumberI
6.1.3 totalMolecularProjectionI
6.2 Quantum numbers for hydrogenoids
6.2.2 lElectronicOrbitalAngularMomentum
6.3 Pure rotational quantum numbers
6.4 Quantum numbers for n electron systems (atoms and molecules)
6.4.2 totalMagneticQuantumNumberS
6.4.3 totalMolecularProjectionS
6.4.4 totalElectronicOrbitalMomentumL
6.4.5 totalMagneticQuantumNumberL
6.4.6 totalMolecularProjectionL
6.4.8 totalMagneticQuantumNumberN
6.4.9 totalMolecularProjectionN
6.4.11 totalMagneticQuantumNumberJ
6.4.12 totalMolecularProjectionJ
6.4.13 intermediateAngularMomemtunF
6.4.15 totalMagneticQuantumNumberF
6.5 Vibrational and (ro)vibronic quantum numbers
6.5.4 vibronicAngularMomentumK
6.5.5 vibronicAngularMomentumP
6.5.6 rovibronicAngularMomentumP
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 VOtools, which can extract emission/absorption line information from observed spectra or narrowband 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 boundbound and freebound transitions, but not freefree 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 lineemitting/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 nonelectromagnetic transitions.
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
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.
Value of the measure. General Number format.
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.
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
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:
String representation of the unit
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
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.E10 m 1.E10 L
Unit.expression= Angstrom Unit.scaleSI=1.E10 Unit.dimEquation=L
1 erg/cm^2/s/Angstrom = 1.E7 Kg/m/s^3 1.E7 ML1T3
Unit.expression= erg/cm^2/s/Angstrom Unit.scaleSI= 1.E7 Unit.dimEquation= ML1T3

See, e.g., IVOA SSAP for more examples
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.
A small description title identifying the line. This is useful when identification is not secure or not yet established.
A full description of the initial level of the transition, originating the line.
A full description of the final level of the transition, originating the line
A full description of the initial state of the atom (including its ionization state) or molecule, where the line transition occurs.
A full description of the final state of the atom (including its ionization state) or molecule, where the line transition occurs. For boundbound atomic transitions, it follows: "initialElement"="finalElement".
Wavelength in the vacuum of the transition originating the line.
Frequency in the vacuum of the transition originating the line.
Wavenumber in the vacuum of the transition originating the line.
Wavelength in the air of the transition originating the line.
Einstein A coefficient, defined as the probability per unit time for spontaneous emission in a boundbound transition from "initialLevel" to "finalLevel".
If positive ("absorption oscillator strength"): the quantity defined by the relation:
where is the Einstein Acoefficient for spontaneous emission between "initialLevel" and "finalLevel"  characterized by the energy difference , ^{ }, , , and h are the usual symbols for the finestructure constant, electron mass, speed of light and Planck constant, respectively; is the statistical weight of the ith level. The subscripts "i" and "f" refer to the "initialLevel" and "finalLevel", respectively. As usual throughout this document, units are S.I. with ^{ }e_{0} expressed in .
If negative ("emission oscillator strength") the quantity (f_{if}) is defined by:
whereis the weighted oscillator strength.
The product between "oscillatorStrength" and the statistical weight of the "initialLevel"
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:
Integrated intensity of the line profile over a given wavelength range
Minimum wavelength for observedFlux integration.
Maximum wavelength for observedFlux integration.
The significance of line detection in an observed spectrum. It can be expressed in terms of signaltonoise ratio, or detection probability (usually null hypothesis probability that a given observed line is due to a statistical background fluctuation).
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.
In theoretical works, the line strength S is widely used (Drake 1996):
_{ }
Where and _{ }are the initial and finalstate 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 m^{2} C^{2}, e_{0} in C^{2}.N^{1}.m^{2}, h in J.s) are:
Width of the line profile (expressed as Full Width Half Maximum) induced by a process of “type=Broadening”.
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).
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
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).
The scope of this class is to describe the quantum mechanics properties of each level, between which the transition originating the line occurs.
Statistical weight associated to the level including all degeneracies, expressed as the total number of terms pertaining to a given level.
The same as Level.totalStatWeight for nuclear spin states only
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 LS 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:
Intrinsic lifetime of a level due to its radiative decay.
The binding energy of an electron belonging to the level.
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
A representation of the level quantum state through its set of quantum numbers
A string indicating the type of nuclear spin symmetry. Possible values are: “para”, “ortho”, “meta”
Eigenvalue of the parity operator. Values (+1,1)
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, 1s^{2},2s^{2},2p^{6},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.
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)
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.
The term (symbol) to which this quantum state belongs, if applicable.
For example, in the case of SpinOrbit 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, ^{3}P_{1} 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. 16331639, 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.
The scope of this class is to describe the set of quantum numbers describing each level.
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
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
The numerator of the quantum number value
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)
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.
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.
String identifying the type of process. Possible values are: "Matterradiation interaction", "Mattermatter interaction", "Energy shift", "Broadening".
String describing the process: Example values (corresponding to the values of "type" listed above) are: "Photoionization", "Collisional excitation", "Gravitational redshift", "Natural broadening".
A theoretical model by which a specific process might be described.
The scope of this class is describing the physical properties of the ambient gas, plasma, dust or stellar atmosphere where the line is generated.
The temperature in the lineproducing plasma.
The optical depth in the lineproducing plasma for the transition described by "initialLevel" and "finalLevel".
The particle density in the lineproducing plasma.
The mass density in the lineproducing plasma.
The pressure in the lineproducing plasma.
The entropy of the lineproducing plasma.
The total mass of the lineproducing gas/dust cloud or star.
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.
A quantitative observable k, which expresses the suppression of the emission line intensity due to the presence of optically thick matter along the lineofsight. 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.
Placeholder for future detailed theoretical models of the environment plasma where the line appears.
This class gives a basic characterization of the celestial source, where an astronomical line has been observed
The IAUname of the source
An alternative or conventional name of the source
Coordinates of the source. Link to IVOA Space Time Coordinates data model
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) 
This example refers to the measurement of the hyperfine structure of the J=1→0 transition in diazenlyium (N_{2}H^{+}) at 93 Ghz (Caselli et al. 1995) toward the cold (kinetic temperature T_{K}~10 K) dense core of the interstellar cloud L1512. Due to the closedshell ^{1}Σ 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=1→0 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 N_{2}H^{+ }hyperfine structure components
J F_{1 }F → J'F'_{1}F' 
(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 N_{2}H^{+ }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: (F_{1,}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 N_{2}H^{+ }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.
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 pseudocode 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:= “F_{1}”
· Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF”
· Line.initialLevel.quantumState.quantumNumber.description:= “Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I_{1}”
· 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_{2} and F_{1}”
· 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:= “F_{1}”
· Line.initialLevel.quantumState.quantumNumber.type := “totalAngularMomentumF”
· Line.initialLevel.quantumState.quantumNumber.description:= “Resulting angular momentum including nuclear spin for one nucleus; coupling of J and I_{1}”
· 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 := “LT1”
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 := “N_{2}H^{+}”
· Line.finalElement.name := “N_{2}H^{+}”
Source specific attributes:
· Line.source.name := “L1512”
{
"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": "3.21755760x103"
"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": "LT1"
}
}
}
}
.
Please note that some physical quantities (marked with an
asterisk) have not been fully instanced to simplify the graphics.
The advent of a new generation of large Xray 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 highenergy 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 Xray 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 XMMNewton/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 kT_{e}_{ }is
Tab.3 – Properties of the RRC features in the XMMNewton/RGS spectrum of NGC1068
Ion 
kT_{e} (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 ΔE≈kT_{e}. The average RRC photon energy is E≈I+kT_{e}, where I is the ionization potential of the recombined state. If the plasma is highly over ionized (kT«I) – as expected in Xray photoionized nebulae (Kallman & McCray 1982) – then ΔE/E≈kT_{e}/I. Therefore, the specification of kT_{e} (extracted from Tab.2 in K02) and I (extracted from table of photo ionization potentials) is enough to know the energy of the feature.
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 := 1.6E19
· Line.wavelength.Unit.dimEquation := “ML2T2”
· Line.observedFlux.value := 2.8E4
· Line.observedFlux.error := 0.3E4
· Line.observedFlux.unit.expression := “photons*cm2*s1”
· Line.observedFlux.unit.scaleSI = 1.E4
· Line.observedFlux.unit.dimEquation := “L2T1”
· 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 := “MT1”
^{ }
^{ }
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:
{
"Line": {
"transitionType": “Radiative Recombination Continuum”
"wavelength": {
"value": "394.6"
"unit": {
"expression": eV"
"scaleSI": "1.6E19"
"dimEquation": "ML2T2"
}
}
"observedFlux": {
"value": "2.8E4"
"error": "0.3E4"
"unit": {
"expression": photons*cm2*s1"
"scaleSI": "1.E4"
"dimEquation": "L2T1"
}
}
"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": "MT1"
}
}
"environment": {
"temperature": {
"value": "1.9E5"
"unit": {
"expression": "K"
"scaleSI": "1"
"dimEquation": "K"
}
}
}
}
}
[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 Xray 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 0471827592 (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
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 longestlived 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 112118 are under review. The temporary system recommended by J Chatt, Pure Appl. Chem., 51, 381384 (1979) is used above. The names of elements 101109 were agreed in 1997 (See Pure Appl. Chem., 1997, 69, 24712473),for element 110 in 2003 (see Pure Appl. Chem., 2003, 75, 16131615) and for element 111 in 2004 (see Pure Appl. Chem., 2004, 76, 21012103).
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 M_{A} >, can be simultaneous eigenfunctions of three types of operators : the magnitude A^{2}, the component of A onto the internuclear axis A_{z}, and the component of A on the laboratory quantization axis A_{Z}. The basis function labels A, a and M_{A} correspond to eigenvalues of A^{2}, A_{z}, and A_{Z}, respectively ħ^{2}A(A+1), ħ a and ħ M_{A}.
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)
total nuclear spin of one atom or a molecule, I
total magnetic quantum number, where is the eigenvalue of the operator
total nuclear spin projection quantum number
where is the eigenvalue of the operator
nuclear spin of individual nucleus which composes a molecule, noted;
eigenvalue of the parity operator applied to the total wavefunction. It takes the value “0” for even parity and “1” for odd parity
A serial quantum number that labels states to which no good or nearly good quantum numbers can be assigned to.
principal quantum number n
orbital angular momentum of an electron where
is the eigenvalue of the operator (called as well azimuthal quantum number).
spin angular momentum of an electron, only where is the eigenvalue of the operator)
total angular momentum of one electron, and . is the eigenvalue of the operator, where
total angular momentum , including nuclear spin . is the eigenvalue of the operator, where
orbital magnetic quantum number, where is the eigenvalue of the operator
spin magnetic quantum number, where is the eigenvalue of the operator
orbital magnetic quantum number, where is the eigenvalue of the operator
orbital magnetic quantum number, where is the eigenvalue of the operator
Index t labelling asymmetric rotational energy levels for a given rotational quantum number N.
Note: The solution of the Schrödinger equation for an asymmetrictop 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 (N_{t}). This index t goes from N for the lowest energy of the set to +N for the highest energy, and is equal to (K_{a} – K_{c}).
For a given N, energy levels may be specified by K_{a} K_{c }(or K_{1} K_{1}, or K_{} K_{+} are alternative notations), where K_{a} is the K quantum number for the limiting prolate (B=C) and K_{c} for the limiting oblate (B=A). In the notation (K_{1} K_{1}) 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)
see asymmetricKA
it is the total spin quantum number S, S can be integral or halfintegral. is the eigenvalue of the operator, where
total spin magnetic quantum number, where is the eigenvalue of the operator.
total spin projection quantum number , , where is the eigenvalue of the operator.
it is the total orbital angular momentum , is integral. is the eigenvalue of the operator, where
total orbital magnetic quantum number, , where is the eigenvalue of the operator.
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)
is the total angular momentum N exclusive of nuclear and electronic spin, N is integral. For a molecule in a closeshell state totalAngularMomemtumN is the pure rotational angular momentum.
total orbital magnetic quantum number, where is the eigenvalue of the operator
absolute value of the component of the angular momentum along the axis of a symmetric (or quasisymmetric) 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.
is the total angular momentum J exclusive
of nuclear spin, J can be integral or halfintegral.
For atoms:
For molecules:
total magnetic quantum number, where
is the eigenvalue of the operator.
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.
is associated to the intermediate quantum number where (or ) + any other vector
is the total angular momentum including nuclear spin, can be integral or halfintegral. is the eigenvalue of the operator, where for atoms:
and for molecules with m nuclear spins:
total magnetic quantum number, where is the eigenvalue of the operator
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”
angular momentum associated to degenerate vibrations,
or
total vibrational angular momentum is the sum of all angular momenta associated to degenerate vibrations:
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)
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)
total resultant axial angular momentum quantum number including electron spin: . (REC. 26 of Mulliken, 1955)
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))
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 L^{2 } and L_{Z} , with L the total orbital angular momentum of the atom. Similarly all spins are coupled together to give the eigenfunctions of S^{2 } and S_{Z} , with S the total spin angular momentum; then L and S are coupled together to give eigenfunctions of J^{2 } and J_{Z} , where J=L+S.
When the coupling conditions within an atom correspond closely to pure LScoupling conditions, then the quantum states of an atom can be accuratly described in terms of LScoupling 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 lowerorder quantum numbers, such as n_{i}l_{i}.
· Giving values of L, S, J specifies a level
· Giving values of L, S, J, M_{J} specifies a state
· The value of (2S+1) is called the multiplicity of the term
For LScoupled functions, the notation introduced by Russel and Saunders is universally used : ^{2S+1}L_{J}, 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 M_{J} .
With increasing Z, the spinorbit 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 jjcoupling 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 j_{i} in some arbitrary order to obtain the total angular momentum J.
For twoelectron configurations, the coupling scheme may be described by the condensed notation [(l_{1} s_{1})j_{1} , (l_{2}, s_{2})j_{2}]JM_{J} with the usual jjnotation for energy levels (j_{1}, j_{2})_{J} [analogous to the Russel Saunders notation ^{2S+1}L_{J}].
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 spinorbit interaction of the more tightly bound electron, and the next strongest interaction is the spinindependent (direct) portion of the Coulomb interaction between the 2 electrons.
The corresponding angularmomemtum coupling scheme is l_{1} + s_{1} = _{ }j_{1} , j_{1 }+ l_{2} = K, K + s_{2} = J, or notation {[(l_{1} s_{1})j_{1} , l_{2}]K,s_{2}}JM with the standard energy level notation j_{1}[K]_{J}.
The other limiting form of pair coupling is called LK (or Ls) coupling. In twoelectron configurations, it corresponds to the case in which the direct Coulomb interaction is greater then the spinorbit interaction of either electron, and the spinorbit interaction of the inner electron is next most important. The coupling scheme is l_{1} + l_{2} = L, L + s_{1} = K, K + s_{2} = J, or notation {[(l_{1} l_{2})L, s_{1}]K,s_{2}}JM with the standard energy level notation L[K]_{J}.
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 purecoupling schemes (with the understanding that these labels may give a poor description of the true angularmomentum properties of the corresponding quantum states). In many cases, however, the coupling conditions are so hopelessly far from any purecoupling 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)=∑_{g}_{SL }c(gSLJ)F(SLJ)The c(gSLJ) are expansion coefficients , and ∑_{g}_{SL} 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. »