Metabolic engineering

2014-01-08T20:47:21Z

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{{Lead too long|date=September 2013}}

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]]

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