en>ChrisGualtieri: Checkwiki 61 + General Fixes using AWB

2013-11-27T01:42:12Z

Checkwiki 61 + General Fixes using AWB

← Older revision		Revision as of 03:42, 27 November 2013
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	My name is ~~Erin but you can call me anything you such~~ as. ~~Administering databases has been his day job for~~ a ~~while~~. The ~~important things I adore most~~ to ~~play curling~~ and ~~I have actually been doing it for~~ a ~~long time~~. ~~She's always liked living~~ in ~~Indiana~~. ~~Inspect out his web site here: http:~~//~~mysocial~~.ws/~~index~~.~~php?do~~=/~~profile-704241~~/~~info~~/		In [[Rotational-vibrational coupling\|rotational-vibrational]] and electronic [[spectroscopy]] of [[diatomic molecule]]s, '''Hund's coupling cases''' are idealized cases where specific terms appearing in the molecular [[Hamiltonian (quantum mechanics)\|Hamiltonian]] and involving couplings between [[Angular momentum\|angular momenta]] are assumed to dominate over all other terms. There are five cases, traditionally notated with the letters (a) through (e). Most diatomic molecules are somewhere between the idealized cases (a) and (b).<ref name="brown">{{cite book\|first1=John M.\|last1=Brown\|first2=Alan\|last2=Carrington\|title=Rotational Spectroscopy of Diatomic Molecules\|publisher= Cambridge University Press\|year=2003\|isbn= 0521530784}}</ref>

			== Angular momenta ==
			To describe the Hund's coupling cases, we use the following angular momenta:
			*<math>\mathbf L</math>, the electronic orbital angular momentum
			*<math>\mathbf S</math>, the electronic spin angular momentum
			*<math>\mathbf J_a=\mathbf L+\mathbf S</math>, the total electronic angular momentum
			*<math>\mathbf J</math>, the total angular momentum of the system
			*<math>\mathbf N=\mathbf J-\mathbf S</math>, the total angular momentum minus the electron spin
			*<math>\mathbf R=\mathbf N-\mathbf L</math>, the rotational angular momentum of the nuclei

			== Choosing the applicable Hund's case ==
			Hund's coupling cases are idealizations. The appropriate case for a given situation can be found by comparing three strengths: the electrostatic coupling of <math>\mathbf L</math> to the internuclear axis, the spin-orbit coupling, and the rotational coupling of <math>\mathbf L</math> and <math>\mathbf S</math> to the total angular momentum <math>\mathbf J</math>.
			{\| class="wikitable"
			\|-
			! Hund's case !! Electrostatic !! Spin-orbit !! Rotational
			\|-
			\| (a) \|\| strong \|\| intermediate \|\| weak
			\|-
			\| (b) \|\| strong \|\| weak \|\| intermediate
			\|-
			\| (c) \|\| intermediate \|\| strong \|\| weak
			\|-
			\| (d) \|\| intermediate \|\| weak \|\| strong
			\|-
			\| rowspan="2" \| (e) \|\| rowspan="2" \| weak \|\| intermediate \|\| strong
			\|-
			\| strong \|\| intermediate
			\|}
			The last two rows are degenerate because they have the same [[good quantum number]]s.<ref>{{cite doi\|10.1080/00268979400100074}}</ref>

			== Case (a) ==
			In case (a), <math>\mathbf L</math> is electrostatically coupled to the internuclear axis, and <math>\mathbf S</math> is coupled to <math>\mathbf L</math> by [[spin-orbit coupling]]. Then both <math>\mathbf L</math> and <math>\mathbf S</math> have well-defined axial components <math>\Lambda</math> and <math>\Sigma</math>, respectively. <math>\boldsymbol\Omega</math> defines a vector of magnitude <math>\Omega=\Lambda+\Sigma</math> pointing along the internuclear axis. Combined with the rotational angular momentum of the nuclei <math>\mathbf R</math>, we have <math>\mathbf J=\boldsymbol\Omega+\mathbf R</math>. In this case, the procession of <math>\mathbf L</math> and <math>\mathbf S</math> around the nuclear axis is assumed to be much faster than the nutation of <math>\boldsymbol\Omega</math> and <math>\mathbf R</math> around <math>\mathbf J</math>.

			The good quantum numbers in case (a) are <math>\Lambda</math>, <math>S</math>, <math>\Sigma</math>, <math>J</math> and <math>\Omega</math>. We express the rotational energy operator as <math>H_{rot}=B\mathbf R^2=B(\mathbf J-\mathbf L-\mathbf S)^2</math>, where <math>B</math> is a rotational constant. There are, ideally, <math>2S+1</math> fine-structure states, each with rotational levels having relative energies <math>BJ(J+1)</math> starting with <math>J=\Omega</math>.<ref name="brown" />

			== Case (b) ==
			In case (b), the spin-orbit coupling is weak or non-existent (in the case <math>\Lambda=0</math>). In this case, we take <math>\mathbf N=\boldsymbol\lambda+\mathbf R</math> and <math>\mathbf J=\mathbf N+\mathbf S</math> and assume <math>\mathbf L</math> precesses quickly around the internuclear axis.

			The good quantum numbers in case (b) are <math>\Lambda</math>, <math>N</math>, <math>S</math>, and <math>J</math>. We express the rotational energy operator as <math>H_{rot}=B\mathbf R^2=B(\mathbf N-\mathbf L)^2</math>, where <math>B</math> is a rotational constant. The rotational levels therefore have relative energies <math>BN(N+1)</math> starting with <math>N=\Lambda</math>.<ref name="brown" />

			== Case (c) ==
			In case (c), the spin-orbit coupling is stronger than the coupling to the internuclear axis, and <math>\lambda</math> and <math>\sigma</math> from case (a) cannot be defined. Instead<math>\mathbf L</math> and <math>\mathbf S</math> combine to form <math>\mathbf J_a</math>, which has a projection along the internuclear axis of magnitude <math>\Omega</math>. Then <math>\mathbf J=\boldsymbol\Omega+\mathbf R</math>, as in case (a).

			The good quantum numbers in case (c) are <math>J_a</math>, <math>J</math>, and <math>\Omega</math>.<ref name="brown" />

			== Case (d) ==
			In case (d), the rotational coupling between <math>\mathbf L</math> and <math>\mathbf R</math> is much stronger than the electrostatic coupling of <math>\mathbf L</math> to the internuclear axis. Thus we form <math>\mathbf N</math> by coupling <math>\mathbf L</math> and <math>\mathbf R</math> and the form <math>\mathbf J</math> by coupling <math>\mathbf N</math> and <math>\mathbf S</math>.

			The good quantum numbers in case (d) are <math>L</math>, <math>R</math>, <math>N</math>, <math>S</math>, and <math>J</math>. Because <math>R</math> is a good quantum number, the rotational energy is simply <math>H_{rot}=B\mathbf R^2=BR(R+1)</math>.<ref name="brown" />

			== Case (e) ==
			In case (e), we first form <math>\mathbf J_a</math> and then form <math>\mathbf J</math> by coupling <math>\mathbf J_a</math> and <math>\mathbf R</math>. This case is rare but has been observed.<ref>{{cite doi\|10.1063/1.472999}}</ref>

			The good quantum numbers in case (e) are <math>J_a</math>, <math>R</math>, and <math>J</math>. Because <math>R</math> is once again a good quantum number, the rotational energy is <math>H_{rot}=B\mathbf R^2=BR(R+1)</math>.<ref name="brown" />

			== References ==
			<references />

			[[Category:Spectroscopy]]

en>Melcombe: change tag to nofootnotes

2012-07-09T13:08:33Z

change tag to nofootnotes

New page

My name is Erin but you can call me anything you such as. Administering databases has been his day job for a while. The important things I adore most to play curling and I have actually been doing it for a long time. She's always liked living in Indiana. Inspect out his web site here: http://mysocial.ws/index.php?do=/profile-704241/info/

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en>ChrisGualtieri: Checkwiki 61 + General Fixes using AWB

en>Melcombe: change tag to nofootnotes