Azimuthal quantum number: Difference between revisions

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The term "azimuthal quantum number" was introduced by [[Arnold Sommerfeld]] in 1915<ref>{{Cite book |last=Whittaker |first=Edmund Taylor |title=A history of the theories of aether and electricity |date=1989 |publisher=Dover |isbn=978-0-486-26126-3 |series=Dover classics of science and mathematics |location=New York}}</ref>{{rp|II:132}} as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, {{mvar|ℓ}}, arises from quantization of orbital angular momentum. Some textbooks<ref name=Schiff>{{Cite book |last=Schiff |first= Leonard  |title=Quantum mechanics |publisher= McGraw-Hill |year= 1949}}</ref>{{rp|199}} and the ISO standard 80000-10:2019<ref name="iso">{{Cite web |title=ISO Online Browsing Platform |url=https://www.iso.org/obp/ui/en/#iso:std:iso:80000:-10:ed-2:v1:en |access-date=2024-02-20 |at=10-13.3}}</ref> call {{mvar|ℓ}} the '''orbital angular momentum quantum number'''.
The term "azimuthal quantum number" was introduced by [[Arnold Sommerfeld]] in 1915<ref>{{Cite book |last=Whittaker |first=Edmund Taylor |title=A history of the theories of aether and electricity |date=1989 |publisher=Dover |isbn=978-0-486-26126-3 |series=Dover classics of science and mathematics |location=New York}}</ref>{{rp|II:132}} as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, {{mvar|ℓ}}, arises from quantization of orbital angular momentum. Some textbooks<ref name=Schiff>{{Cite book |last=Schiff |first= Leonard  |title=Quantum mechanics |publisher= McGraw-Hill |year= 1949}}</ref>{{rp|199}} and the ISO standard 80000-10:2019<ref name="iso">{{Cite web |title=ISO Online Browsing Platform |url=https://www.iso.org/obp/ui/en/#iso:std:iso:80000:-10:ed-2:v1:en |access-date=2024-02-20 |at=10-13.3}}</ref> call {{mvar|ℓ}} the '''orbital angular momentum quantum number'''.


The energy levels of an atom in an external magnetic field depend upon the {{mvar|''m''{{sub|ℓ}}}} value so it is sometimes called the  magnetic quantum number.<ref>{{Cite book |last=Eisberg |first=Robert M. |title=Quantum physics of atoms, molecules, solids, nuclei, and particles |last2=Resnick |first2=Robert |date=2009 |publisher=Wiley |isbn=978-0-471-87373-0 |edition=2. ed., 37. [Nachdr.] |location=New York}}</ref>{{rp|240}}
The energy levels of an atom in an external magnetic field depend upon the {{mvar|''m''{{sub|ℓ}}}} value so it is sometimes called the  magnetic quantum number.<ref>{{Cite book |last1=Eisberg |first1=Robert M. |title=Quantum physics of atoms, molecules, solids, nuclei, and particles |last2=Resnick |first2=Robert |date=2009 |publisher=Wiley |isbn=978-0-471-87373-0 |edition=2. ed., 37. [Nachdr.] |location=New York}}</ref>{{rp|240}}


The lowercase letter {{mvar|ℓ}}, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter {{mvar|L}} is used.<ref name="iso"/>
The lowercase letter {{mvar|ℓ}}, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter {{mvar|L}} is used.<ref name="iso"/>


== Relation to atomic orbitals ==
== Relation to atomic orbitals ==
[[File:Vector model of orbital angular momentum.svg|right|thumb|Azimuthal quantum number: Illustrating (five) alternative ''orbital angular momentum'' shapes  as "cones"{{mdash}}here portraying (five) alternative values in terms of the reduced Planck constant, {{math|''ħ''}}.]]
There are four quantum numbers{{mdash}}''n'', ''ℓ'', ''m''<sub>''ℓ''</sub>, ''m''<sub>''s''</sub>{{mdash}} connected with the energy states of an isolated atom's electrons. These four numbers specify the unique and complete quantum state of any single [[electron]] in the [[atom]], and they combine to compose the electron's [[wavefunction]], or ''[[atomic orbital#Electron properties|orbital]]''.
There are four quantum numbers{{mdash}}''n'', ''ℓ'', ''m''<sub>''ℓ''</sub>, ''m''<sub>''s''</sub>{{mdash}} connected with the energy states of an isolated atom's electrons. These four numbers specify the unique and complete quantum state of any single [[electron]] in the [[atom]], and they combine to compose the electron's [[wavefunction]], or ''[[atomic orbital#Electron properties|orbital]]''.


When solving to obtain the wave function, the [[Schrödinger equation]] resolves into three equations that lead to the first three quantum numbers, meaning that the three equations are interrelated. The ''azimuthal quantum number'' arises in solving the polar part of the wave equation{{mdash}}relying on the [[spherical coordinate system]], which generally works best with models having sufficient aspects of [[spherical symmetry]].
When solving to obtain the wave function, the [[Schrödinger equation]] resolves into three equations that lead to the first three quantum numbers, meaning that the three equations are interrelated. The ''azimuthal quantum number'' arises in solving the polar part of the wave equation{{mdash}}relying on the [[spherical coordinate system]], which generally works best with models having sufficient aspects of [[spherical symmetry]].


[[File:Vector model of orbital angular momentum.svg|left|thumb|Azimuthal quantum number: Illustrating (five) alternative ''orbital angular momentum'' shapes  as "cones"{{mdash}}here portraying (five) alternative values for the reduced Planck constant, {{math|''ħ''}}.]]
An electron's angular momentum, {{math|''L''}}, is related to its quantum number {{math|''ℓ''}} by the following equation:


An electron's angular momentum, {{math|''L''}}, is related to its quantum number {{math|''ℓ''}} by the following equation:
<math display="block">\mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi,</math>
<math display="block">\mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi,</math>
where {{math|''ħ''}} is the [[reduced Planck constant]], {{math|'''L'''}} is the orbital ''angular momentum operator'' and <math>\Psi</math> is the wavefunction of the electron. The quantum number {{math|''ℓ''}} is always a non-negative integer: 0, 1, 2, 3, etc. (Notably, {{math|'''L'''}} has no real meaning except in its use as the angular momentum operator; thus, it is standard practice to use the quantum number {{math|''ℓ''}} when referring to angular momentum).
where {{math|''ħ''}} is the [[reduced Planck constant]], {{math|'''L'''}} is the orbital ''angular momentum operator'' and <math>\Psi</math> is the wavefunction of the electron. The quantum number {{math|''ℓ''}} is always a non-negative integer: 0, 1, 2, 3, etc. (Notably, {{math|'''L'''}} has no real meaning except in its use as the angular momentum operator; thus, it is standard practice to use the quantum number {{math|''ℓ''}} when referring to angular momentum).


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The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The {{math|''ℓ m''}} quantum number corresponds to specific [[spherical harmonics]] and are commonly used to describe features observed in spectroscopic methods such as [[X-ray photoelectron spectroscopy]]<ref>{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}}</ref> and [[electron energy loss spectroscopy]].<ref>{{Cite book |last=Egerton |first=R.F. |url=https://link.springer.com/10.1007/978-1-4419-9583-4 |title=Electron Energy-Loss Spectroscopy in the Electron Microscope |date=2011 |publisher=Springer US |isbn=978-1-4419-9582-7 |location=Boston, MA |language=en |doi=10.1007/978-1-4419-9583-4}}</ref> (The notation is slightly different, with [[X-ray notation]] where K, L, M are used for excitations out of electron states with <math>n=0, 1, 2</math>.)
The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The {{math|''ℓ m''}} quantum number corresponds to specific [[spherical harmonics]] and are commonly used to describe features observed in spectroscopic methods such as [[X-ray photoelectron spectroscopy]]<ref>{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}}</ref> and [[electron energy loss spectroscopy]].<ref>{{Cite book |last=Egerton |first=R.F. |url=https://link.springer.com/10.1007/978-1-4419-9583-4 |title=Electron Energy-Loss Spectroscopy in the Electron Microscope |date=2011 |publisher=Springer US |isbn=978-1-4419-9582-7 |location=Boston, MA |language=en |doi=10.1007/978-1-4419-9583-4}}</ref> (The notation is slightly different, with [[X-ray notation]] where K, L, M are used for excitations out of electron states with <math>n=0, 1, 2</math>.)


The angular momentum quantum numbers are also used when the electron states are described in methods such as [[Kohn–Sham equations|Kohn–Sham density functional theory]]<ref>{{Cite journal |last=Kohn |first=W. |last2=Sham |first2=L. J. |date=1965 |title=Self-Consistent Equations Including Exchange and Correlation Effects |url=https://link.aps.org/doi/10.1103/PhysRev.140.A1133 |journal=Physical Review |language=en |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |issn=0031-899X}}</ref> or with [[gaussian orbital]]s.<ref>{{Citation |last=Gill |first=Peter M.W. |title=Molecular integrals Over Gaussian Basis Functions |date=1994 |work=Advances in Quantum Chemistry |volume=25 |pages=141–205 |url=https://linkinghub.elsevier.com/retrieve/pii/S0065327608600192 |access-date=2024-02-20 |publisher=Elsevier |language=en |doi=10.1016/s0065-3276(08)60019-2 |isbn=978-0-12-034825-1|url-access=subscription }}</ref> For instance, in [[silicon]] the electronic properties used in [[semiconductor device]] are due to the p-like states with <math>l=1</math> centered at each atom, while many properties of [[transition metal]]s depend upon the d-like states with <math>l=2</math>.<ref>{{Cite book |last=Pettifor |first=D. G. |title=Bonding and structure of molecules and solids |date=1996 |publisher=Clarendon Press |isbn=978-0-19-851786-3 |edition=Reprint with corr |series=Oxford science publications |location=Oxford}}</ref>
The angular momentum quantum numbers are also used when the electron states are described in methods such as [[Kohn–Sham equations|Kohn–Sham density functional theory]]<ref>{{Cite journal |last1=Kohn |first1=W. |last2=Sham |first2=L. J. |date=1965 |title=Self-Consistent Equations Including Exchange and Correlation Effects |url=https://link.aps.org/doi/10.1103/PhysRev.140.A1133 |journal=Physical Review |language=en |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |bibcode=1965PhRv..140.1133K |issn=0031-899X}}</ref> or with [[gaussian orbital]]s.<ref>{{Citation |last=Gill |first=Peter M.W. |title=Molecular integrals Over Gaussian Basis Functions |date=1994 |series=Advances in Quantum Chemistry |volume=25 |pages=141–205 |url=https://linkinghub.elsevier.com/retrieve/pii/S0065327608600192 |access-date=2024-02-20 |publisher=Elsevier |language=en |doi=10.1016/s0065-3276(08)60019-2 |isbn=978-0-12-034825-1|url-access=subscription }}</ref> For instance, in [[silicon]] the electronic properties used in [[semiconductor device]] are due to the p-like states with <math>l=1</math> centered at each atom, while many properties of [[transition metal]]s depend upon the d-like states with <math>l=2</math>.<ref>{{Cite book |last=Pettifor |first=D. G. |title=Bonding and structure of molecules and solids |date=1996 |publisher=Clarendon Press |isbn=978-0-19-851786-3 |edition=Reprint with corr |series=Oxford science publications |location=Oxford}}</ref>


== History ==
== History ==

Latest revision as of 13:42, 27 December 2025

Template:Short description

File:HAtomOrbitals.png
The atomic orbital wavefunctions of a hydrogen atom: The azimuthal quantum number (Template:Mvar) is denoted by letter at the top of each column. The principal quantum number (Template:Mvar) is shown at the right of each row.

In quantum mechanics, the azimuthal quantum number Template:Mvar is a quantum number for an atomic orbital that determines its orbital angular momentum and describes aspects of the angular shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron (the others being the principal quantum number Template:Mvar, the magnetic quantum number Template:Mvar, and the spin quantum number Template:Mvar).

For a given value of the principal quantum number Template:Mvar (electron shell), the possible values of Template:Mvar are the integers from 0 to n − 1Script error: No such module "Check for unknown parameters".. For instance, the n = 1Script error: No such module "Check for unknown parameters". shell has only orbitals with ℓ=0, and the n = 2Script error: No such module "Check for unknown parameters". shell has only orbitals with ℓ=0, and ℓ=1.

For a given value of the azimuthal quantum number Template:Mvar, the possible values of the magnetic quantum number Template:Mvar are the integers from mℓ=−ℓScript error: No such module "Check for unknown parameters". to mℓ=+ℓScript error: No such module "Check for unknown parameters"., including 0. In addition, the spin quantum number Template:Mvar can take two distinct values. The set of orbitals associated with a particular value of Template:Mvar are sometimes collectively called a subshell.

While originally used just for isolated atoms, atomic-like orbitals play a key role in the configuration of electrons in compounds including gases, liquids and solids. The quantum number Template:Mvar plays an important role here via the connection to the angular dependence of the spherical harmonics for the different orbitals around each atom.

Nomenclature

The term "azimuthal quantum number" was introduced by Arnold Sommerfeld in 1915[1]Template:Rp as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, Template:Mvar, arises from quantization of orbital angular momentum. Some textbooks[2]Template:Rp and the ISO standard 80000-10:2019[3] call Template:Mvar the orbital angular momentum quantum number.

The energy levels of an atom in an external magnetic field depend upon the Template:Mvar value so it is sometimes called the magnetic quantum number.[4]Template:Rp

The lowercase letter Template:Mvar, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter Template:Mvar is used.[3]

Relation to atomic orbitals

File:Vector model of orbital angular momentum.svg
Azimuthal quantum number: Illustrating (five) alternative orbital angular momentum shapes as "cones"Template:Mdashhere portraying (five) alternative values in terms of the reduced Planck constant, ħScript error: No such module "Check for unknown parameters"..

There are four quantum numbersTemplate:Mdashn, ℓ, mℓ, msTemplate:Mdash connected with the energy states of an isolated atom's electrons. These four numbers specify the unique and complete quantum state of any single electron in the atom, and they combine to compose the electron's wavefunction, or orbital.

When solving to obtain the wave function, the Schrödinger equation resolves into three equations that lead to the first three quantum numbers, meaning that the three equations are interrelated. The azimuthal quantum number arises in solving the polar part of the wave equationTemplate:Mdashrelying on the spherical coordinate system, which generally works best with models having sufficient aspects of spherical symmetry.

An electron's angular momentum, LScript error: No such module "Check for unknown parameters"., is related to its quantum number ℓScript error: No such module "Check for unknown parameters". by the following equation:

𝐋2Ψ=ℏ2ℓ(ℓ+1)Ψ,

where ħScript error: No such module "Check for unknown parameters". is the reduced Planck constant, LScript error: No such module "Check for unknown parameters". is the orbital angular momentum operator and Ψ is the wavefunction of the electron. The quantum number ℓScript error: No such module "Check for unknown parameters". is always a non-negative integer: 0, 1, 2, 3, etc. (Notably, LScript error: No such module "Check for unknown parameters". has no real meaning except in its use as the angular momentum operator; thus, it is standard practice to use the quantum number ℓScript error: No such module "Check for unknown parameters". when referring to angular momentum).

Atomic orbitals have distinctive shapes, (see top graphic) in which letters, s, p, d, f, etc., (employing a convention originating in spectroscopy) denote the shape of the atomic orbital. The wavefunctions of these orbitals take the form of spherical harmonics, and so are described by Legendre polynomials. The several orbitals relating to the different (integer) values of ℓ are sometimes called sub-shellsTemplate:Mdashreferred to by lowercase Latin letters chosen for historical reasonsTemplate:Mdashas shown in the table "Quantum subshells for the azimuthal quantum number".

Quantum subshells for the azimuthal quantum number
Azimuthal
quantum
number (ℓ) 		Historical
letter 		Historical
name[5]Template:Rp 		Maximum
electrons 		Shape
0 s sharp 2 Spherical (see this picture of spherical harmonics, top row).
1 p principal 6 Three dumbbell-shaped polar-aligned orbitals; one lobe on each pole of the x, y, and z axes (on both + and − axes).
2 d diffuse 10 Nine dumbbells and one doughnut, or "Unique shape #1" (see this picture of spherical harmonics, third row center).
3 f fundamental 14 "Unique shape #2" (see this picture of spherical harmonics, bottom row center).
4 g 18
5 h 22
6 i 26
The letters after the g sub-shell follow in alphabetical orderTemplate:Mdashexcepting letter j and those already used.

Each of the different angular momentum states can take 2(2ℓ + 1) electrons. This is because the third quantum number mℓ (which can be thought of loosely as the quantized projection of the angular momentum vector on the z-axis) runs from −ℓ to ℓ in integer units, and so there are 2ℓ + 1 possible states. Each distinct n, ℓ, mℓ orbital can be occupied by two electrons with opposing spins (given by the quantum number ms = ±Template:1/2), giving 2(2ℓ + 1) electrons overall. Orbitals with higher ℓ than given in the table are perfectly permissible, but these values cover all atoms so far discovered.

For a given value of the principal quantum number n, the possible values of ℓ range from 0 to n − 1Script error: No such module "Check for unknown parameters".; therefore, the n = 1Script error: No such module "Check for unknown parameters". shell only possesses an s subshell and can only take 2 electrons, the n = 2Script error: No such module "Check for unknown parameters". shell possesses an s and a p subshell and can take 8 electrons overall, the n = 3Script error: No such module "Check for unknown parameters". shell possesses s, p, and d subshells and has a maximum of 18 electrons, and so on.

A simplistic one-electron model results in energy levels depending on the principal number alone. In more complex atoms these energy levels split for all n > 1Script error: No such module "Check for unknown parameters"., placing states of higher ℓ above states of lower ℓ. For example, the energy of 2p is higher than of 2s, 3d occurs higher than 3p, which in turn is above 3s, etc. This effect eventually forms the block structure of the periodic table. No known atom possesses an electron having ℓ higher than three (f) in its ground state.

The angular momentum quantum number, ℓ and the corresponding spherical harmonic govern the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and trough, which has zero magnitudes. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number ℓ takes the value of 0. In a p orbital, one node traverses the nucleus and therefore ℓ has the value of 1. L has the value 2ℏ.

Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The wavelengths listed are for a hydrogen atom: Template:Unbulleted list

Addition of quantized angular momenta

Script error: No such module "labelled list hatnote". Given a quantized total angular momentum 𝐣 that is the sum of two individual quantized angular momenta ℓ1 and ℓ2, 𝐣=ℓ1+ℓ2 the quantum number j associated with its magnitude can range from |ℓ1−ℓ2| to ℓ1+ℓ2 in integer steps where ℓ1 and ℓ2 are quantum numbers corresponding to the magnitudes of the individual angular momenta.

Total angular momentum of an electron in the atom

File:LS coupling.svg
"Vector cones" of total angular momentum J (purple), orbital L (blue), and spin S (green). The cones arise due to quantum uncertainty between measuring angular momentum component.

Due to the spin–orbit interaction in an atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin. These therefore change over time. However the total angular momentum JScript error: No such module "Check for unknown parameters". does commute with the one-electron Hamiltonian and so is constant. JScript error: No such module "Check for unknown parameters". is defined as 𝐉=𝐋+𝐒 LScript error: No such module "Check for unknown parameters". being the orbital angular momentum and SScript error: No such module "Check for unknown parameters". the spin. The total angular momentum satisfies the same commutation relations as orbital angular momentum, namely [Ji,Jj]=iℏεijkJk from which it follows that [Ji,J2]=0 where JiScript error: No such module "Check for unknown parameters". stand for JxScript error: No such module "Check for unknown parameters"., JyScript error: No such module "Check for unknown parameters"., and JzScript error: No such module "Check for unknown parameters"..

The quantum numbers describing the system, which are constant over time, are now jScript error: No such module "Check for unknown parameters". and mjScript error: No such module "Check for unknown parameters"., defined through the action of JScript error: No such module "Check for unknown parameters". on the wavefunction Ψ 𝐉2Ψ=ℏ2j(j+1)Ψ𝐉zΨ=ℏmjΨ

So that jScript error: No such module "Check for unknown parameters". is related to the norm of the total angular momentum and mjScript error: No such module "Check for unknown parameters". to its projection along a specified axis. The j number has a particular importance for relativistic quantum chemistry, often featuring in subscript in for deeper states near to the core for which spin-orbit coupling is important.

As with any angular momentum in quantum mechanics, the projection of JScript error: No such module "Check for unknown parameters". along other axes cannot be co-defined with JzScript error: No such module "Check for unknown parameters"., because they do not commute. The eigenvectors of jScript error: No such module "Check for unknown parameters"., sScript error: No such module "Check for unknown parameters"., mjScript error: No such module "Check for unknown parameters". and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of ℓScript error: No such module "Check for unknown parameters"., sScript error: No such module "Check for unknown parameters"., mℓScript error: No such module "Check for unknown parameters". and msScript error: No such module "Check for unknown parameters"..

Beyond isolated atoms

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File:Electron energy loss spectroscopy coreloss lsmo.svg
Example of inner shell ionization edge (core loss) EELS data from La0.7Sr0.3MnO3, acquired in a scanning transmission electron microscope.

The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The ℓ mScript error: No such module "Check for unknown parameters". quantum number corresponds to specific spherical harmonics and are commonly used to describe features observed in spectroscopic methods such as X-ray photoelectron spectroscopy[6] and electron energy loss spectroscopy.[7] (The notation is slightly different, with X-ray notation where K, L, M are used for excitations out of electron states with n=0,1,2.)

The angular momentum quantum numbers are also used when the electron states are described in methods such as Kohn–Sham density functional theory[8] or with gaussian orbitals.[9] For instance, in silicon the electronic properties used in semiconductor device are due to the p-like states with l=1 centered at each atom, while many properties of transition metals depend upon the d-like states with l=2.[10]

History

The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld.[11] The Bohr model was derived from spectroscopic analysis of atoms in combination with the Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. Orbits with zero angular momentum were considered as oscillating charges in one dimension and so described as "pendulum" orbits, but were not found in nature.[12] In three-dimensions the orbits become spherical without any nodes crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.

See also

References

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  1. ↑ Script error: No such module "citation/CS1".
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  5. ↑ Script error: No such module "citation/CS1".
  6. ↑ Script error: No such module "citation/CS1".
  7. ↑ Script error: No such module "citation/CS1".
  8. ↑ Script error: No such module "Citation/CS1".
  9. ↑ Script error: No such module "citation/CS1".
  10. ↑ Script error: No such module "citation/CS1".
  11. ↑ Script error: No such module "citation/CS1".
  12. ↑ Script error: No such module "Citation/CS1".

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External links

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