en>Yobot: Removed invisible unicode characters + other fixes, removed: using AWB (10691)

2015-01-06T11:59:22Z

Removed invisible unicode characters + other fixes, removed: using AWB (10691)

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	~~{{Lead too long\|date=September 2013}}~~		Jena Vella is title people use to call me though I don't really like being called like where. One of his favorite hobbies is to ready flowers but now he is wanting to earn money with it. Illinois is where me and my husband live. Dispatching is what she does and it's something she enjoy. Go to my how do people find out more: http://thoughtsindixie.blogspot.com/

	~~In [[physics]], the '''Landé ''g''-factor'''~~ is ~~a particular example of a [[g-factor (physics)\|~~'~~'g''-factor]], namely for an [[electron]] with both [[Spin (physics)\|spin]] and [[orbit]]al [[angular momentum\|angular momenta]]~~. ~~It is named after [[Alfred Landé]], who first described it in 1921.~~

	~~In [[atomic physics]], it is a multiplicative term appearing in the expression for the energy levels of an [[atom]] in a weak [[magnetic field]]. The [[quantum state]]s~~ of ~~[[electron]]s in [[atomic orbital]]s are normally [[degenerate energy level\|degenerate in energy]], with the degenerate states all sharing the same angular momentum. When the atom~~ is ~~placed in a weak magnetic field, however, the degeneracy is lifted.~~

	The factor comes about during the calculation of the [[Perturbation theory (quantum mechanics)\|first-order perturbation]] in the energy of an atom when a weak uniform magnetic field (that is, weak in comparison to ~~the system's internal magnetic field)~~ is ~~applied~~ to ~~the system. Formally we can write the factor as,<ref>http://hyperphysics.phy-astr.gsu.edu/HBASE/quantum/Lande.html Hyperphysics: Magnetic Interactions and the Landé ''g''-Factor</ref>~~

	~~:<math>g_J= g_L\frac{J(J+1)-S(S+1)+L(L+1)}{2J(J+1)}+g_S\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}~~.~~</math>~~

	~~The orbital ''g''-factor~~ is ~~equal to 1,~~ and ~~under the approximation <math>g_S = 2 </math>, the above expression simplifies to~~

	~~:<math>g_J \approx \frac{3}{2}+\frac{S(S+1)-L(L+1)}{2J(J+1)}~~.~~</math>~~

	~~Here, ''J''~~ is ~~the [[Total angular momentum quantum number\|total electronic angular momentum]], ''L'' is the orbital angular momentum,~~ and ''S'' is the [[spin angular momentum]]. Because ''S''=1/2 for electrons, one often sees this formula written with 3/4 in place of ''S''(''S''+1). The quantities ''g<sub>L</sub>'' and ''g<sub>S</sub>'' are other [[g-factor (physics)\|''g''-factors]] of an electron.

	~~If we wish~~ to ~~know the ''g''-factor for an atom with total atomic angular momentum F=I+J,~~

	:~~<math>g_F= g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}+g_I\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)}</math>~~

	~~:<math>\approx g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)} </math>~~

	~~This last approximation is justified because <math>g_I</math> is smaller than <math>g_J</math> by the ratio of the electron mass to the proton mass.~~

	~~==A derivation==~~
	The following derivation basically follows the line of thought in <ref>http://books.google.com.br/books?id=FRZRAAAAMAAJ&q=ashcroft+solid+state+physics&dq=ashcroft+solid+state+physics&hl=en&sa=X&ei=ci3OUOqENYKG8QS6mYDQDQ&ved=0CC8Q6AEwAA Solid State Physics By Neil W. Ashcroft and N. David Mermin</ref> and.<ref>http://~~books~~.~~google~~.com.br/books?id=LXv8Xh3GE6oC&pg=PA132&dq=lande's+g+factor&hl=en&sa=X&ei=R9CuUMToDoWy8QTDzoHYCA&ved=0CDMQ6AEwAQ#v=onepage&q=lande's%20g%20factor&f=false Modern Atomic and Nuclear Physics: Revised Edition By Fujia Yang, Joseph H. Hamilton</ref>

	~~Both orbital angular momentum and [[spin angular momentum]] of electron contribute to the magnetic moment. In particular, each of them alone contributes to the magnetic moment by the following form~~
	~~:<math>\vec \mu_L= \vec L g_L \mu_B</math>~~
	~~:<math>\vec \mu_S= \vec S g_S \mu_B</math>~~
	~~:<math>\vec \mu_J= \vec \mu_L + \vec \mu_S</math>~~
	~~where~~
	~~:<math>g_L = -1</math>~~
	~~:<math>g_S = -2</math>~~
	Note that negative signs in the above expressions are due to the fact that an electron carries negative charge, and the value of <math>g_S</math> can be derived naturally from [[Dirac's equation]]. The total magnetic moment <math>\vec \mu_J</math>, as a vector operator, does not lie on the direction of total angular momentum <math>\vec J = \vec L+\vec S</math>. However, due to [[Wigner-Eckart theorem]], its expectation value does effectively lie on the direction of <math>\vec J</math> which can be employed in the determination of ''g''-factor according to the rules of [[angular momentum coupling]]. In particular, ''g''-factor is defined as a consequence of the theorem itself
	~~:<math>\langle J,J_z\|\vec \mu_J\|J,J_{{z'}}\rangle = g_J\mu_B\langle J,J_z\|\vec J\|J,J_{z'}\rangle</math>~~
	~~Therefore,~~
	~~:<math>\langle J,J_z\|\vec \mu_J\|J,J_{z'}\rangle\cdot\langle J,J_{z'}\|\vec J\|J,J_z\rangle = g_J\mu_B\langle J,J_z\|\vec J\|J,J_{z'}\rangle\cdot\langle J,J_{z'}\|\vec J\|J,J_z\rangle</math>~~
	:<math>\sum_{J_{z'}}\langle J,J_z\|\vec \mu_J\|J,J_{z'}\rangle\cdot\langle J,J_{z'}\|\vec J\|J,J_z\rangle = \sum_{J_{z'}}g_J\mu_B\langle J,J_z\|\vec J\|J,J_{z'}\rangle \cdot\langle J,J_{z'}\|\vec J\|J,J_z\rangle</math>
	~~:<math>\langle J,J_z\|\vec \mu_J\cdot \vec J\|J,J_z\rangle = g_J\mu_B\langle J,J_z\|\vec J\cdot\vec J\|J,J_z\rangle</math>~~
	~~One gets~~
	~~:<math>g_J\langle J,J_z\|\vec J\cdot\vec J\|J,J_z \rangle = g_L {{\vec L}\cdot {\vec J}}+g_S {{\vec S} \cdot {\vec J}} </math>~~
	~~:<math>= g_L {(\vec L^2+\frac{1}{2}(\vec J^2-\vec L^2-\vec S^2))}+g_S {(\vec S^2+\frac{1}{2}(\vec J^2-\vec L^2-\vec S^2))} </math>~~
	~~:<math>g_J = g_L \frac{J(J+1)+L(L+1)-S(S+1)}{{2J(J+1)}}+g_S \frac{J(J+1)-L(L+1)+S(S+1)}{{2J(J+1)}} </math>~~

	~~==List of Landé ''g''-factors ==~~

	~~{\| class="wikitable"~~
	\|-
	~~! align="center" \| Element~~
	~~! align="center" \| Landé ''g''-factor~~
	\|-
	~~\| align="center" \| Gadolinium~~
	~~\| align="center" \|2.67~~
	\|}


	~~==See also==~~
	* [[Einstein-de Haas effect]]
	* [[Zeeman effect]]

	~~==References==~~
	~~<references~~ />

	~~{{DEFAULTSORT:Lande G-Factor}}~~
	~~[[Category:Atomic physics]]~~
	~~[[Category:Nuclear physics]]~~

213.113.146.186 at 20:47, 8 January 2014

2014-01-08T20:47:21Z

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2012-05-18T22:09:07Z

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en>Yobot: Removed invisible unicode characters + other fixes, removed: ­ using AWB (10691)

213.113.146.186 at 20:47, 8 January 2014

en>LittleWink: Disambiguated: diesel → diesel fuel

en>Yobot: Removed invisible unicode characters + other fixes, removed: using AWB (10691)