Landé g-factor

The factor comes about during the calculation of the 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,{{Cite web|url = http://hyperphysics.phy-astr.gsu.edu/HBASE/quantum/Lande.html|title = Magnetic Interactions and the Lande' g-Factor|date = 25 January 1999|access-date = 14 October 2014|website = HyperPhysics|publisher = Georgia State University|last = Nave|first = C. R.}}

:$g\_J=\; g\_L\backslash frac\{J(J+1)-S(S+1)+L(L+1)\}\{2J(J+1)\}+g\_S\backslash frac\{J(J+1)+S(S+1)-L(L+1)\}\{2J(J+1)\}.$

The orbital $g\_L$ is equal to 1, and under the approximation $g\_S\; =\; 2$, the above expression simplifies to

:$g\_J\; =\; 1+\backslash frac\{J(J+1)+S(S+1)-L(L+1)\}\{2J(J+1)\}.$

Here, J is the 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_L and g_S are other g-factors of an electron. For an $S=0$ atom, $g\_J=1$ and for an $L=0$ atom, $g\_J=2$.

If we wish to know the g-factor for an atom with total atomic angular momentum $\backslash vec\{F\}=\backslash vec\{I\}+\backslash vec\{J\}$ (nucleus + electrons), such that the total atomic angular momentum quantum number can take values of $F=J+I,\; J+I-1,\; \backslash dots,|J-I|$, giving

:$\backslash begin\{align\}$

g_F &= g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}+g_I\frac{\mu_\text{N}}{\mu_\text{B}}\frac{F(F+1)+I(I+1)-J(J+1)}{2F(F+1)} \\

&\approx g_J\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}

\end{align}

Here $\backslash mu\_\backslash text\{B\}$ is the Bohr magneton and $\backslash mu\_\backslash text\{N\}$ is the nuclear magneton. This last approximation is justified because $\backslash mu\_N$ is smaller than $\backslash mu\_B$ by the ratio of the electron mass to the proton mass.

The following working is a common derivation.{{Cite book|title = Solid state physics|last = Ashcroft|first = Neil W.|publisher = Saunders College|year = 1976|isbn = 9780030493461|url = https://books.google.com/books?id=FRZRAAAAMAAJ|last2 = Mermin|first2 = N. David}}{{Cite book|title = Modern Atomic and Nuclear Physics|last = Yang|first = Fujia|publisher = World Scientific|year = 2009|isbn = 9789814277167|pages = 132|edition = Revised|last2 = Hamilton|first2 = Joseph H.|url = https://books.google.com/books?id=LXv8Xh3GE6oC&pg=PA132}}

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

:$\backslash vec\; \backslash mu\_L=\; -\backslash vec\; L\; g\_L\; \backslash mu\_\{\backslash rm\; B\}/\backslash hbar$

:$\backslash vec\; \backslash mu\_S=\; -\backslash vec\; S\; g\_S\; \backslash mu\_\{\backslash rm\; B\}/\backslash hbar$

:$\backslash vec\; \backslash mu\_J=\; \backslash vec\; \backslash mu\_L\; +\; \backslash vec\; \backslash mu\_S$

where

:$g\_L\; =\; 1$

:$g\_S\; \backslash approx\; 2$

Note that negative signs in the above expressions are because an electron carries negative charge, and the value of $g\_S$ can be derived naturally from Dirac's equation. The total magnetic moment $\backslash vec\; \backslash mu\_J$, as a vector operator, does not lie on the direction of total angular momentum $\backslash vec\; J\; =\; \backslash vec\; L+\backslash vec\; S$, because the g-factors for orbital and spin part are different. However, due to Wigner-Eckart theorem, its expectation value does effectively lie on the direction of $\backslash vec\; J$ which can be employed in the determination of the g-factor according to the rules of angular momentum coupling. In particular, the g-factor is defined as a consequence of the theorem itself

:$\backslash langle\; J,J\_z|\backslash vec\; \backslash mu\_J|J,J\text{'}\_z\backslash rangle\; =\; -g\_J\backslash mu\_\{\backslash rm\; B\}\backslash langle\; J,J\_z|\backslash vec\; J|J,J\text{'}\_z\backslash rangle$

Therefore,

:$\backslash langle\; J,J\_z|\backslash vec\; \backslash mu\_J|J,J\text{'}\_z\backslash rangle\backslash cdot\backslash langle\; J,J\text{'}\_z|\backslash vec\; J|J,J\_z\backslash rangle\; =\; -g\_J\backslash mu\_\{\backslash rm\; B\}\backslash langle\; J,J\_z|\backslash vec\; J|J,J\text{'}\_z\backslash rangle\backslash cdot\backslash langle\; J,J\text{'}\_z|\backslash vec\; J|J,J\_z\backslash rangle$

:$\backslash sum\_\{J\text{'}\_z\}\backslash langle\; J,J\_z|\backslash vec\; \backslash mu\_J|J,J\text{'}\_z\backslash rangle\backslash cdot\backslash langle\; J,J\text{'}\_z|\backslash vec\; J|J,J\_z\backslash rangle\; =\; -\backslash sum\_\{J\text{'}\_z\}g\_J\backslash mu\_\{\backslash rm\; B\}\backslash langle\; J,J\_z|\backslash vec\; J|J,J\text{'}\_z\backslash rangle\; \backslash cdot\backslash langle\; J,J\text{'}\_z|\backslash vec\; J|J,J\_z\backslash rangle$

:$\backslash langle\; J,J\_z|\backslash vec\; \backslash mu\_J\backslash cdot\; \backslash vec\; J|J,J\_z\backslash rangle\; =\; -g\_J\backslash mu\_\{\backslash rm\; B\}\backslash langle\; J,J\_z|\backslash vec\; J\backslash cdot\backslash vec\; J|J,J\_z\backslash rangle\; =\; -g\_J\backslash mu\_\{\backslash rm\; B\}\; \backslash quad\; \backslash hbar^2\; J(J+1)$

One gets

:$\backslash begin\{align\}$

g_J\langle J,J_z|\vec J\cdot\vec J|J,J_z \rangle &= \langle J,J_z|g_L {{\vec L}\cdot {\vec J}}+g_S {{\vec S} \cdot {\vec J}}| J,J_z\rangle \\

&= \langle J,J_z|g_L {\left[\vec L^2+\frac{1}{2}\left(\vec J^2-\vec L^2-\vec S^2\right)\right]}+g_S {\left[\vec S^2+\frac{1}{2}\left(\vec J^2-\vec L^2-\vec S^2\right)\right]}|J,J_z\rangle \\

&= \frac{g_L \hbar^2}{2}[ J(J+1)+L(L+1)-S(S+1)]+ \frac{g_S \hbar^2}{2}[ J(J+1)-L(L+1)+S(S+1)]\\

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)}}

\end{align}

The following table gives the calculated Lande g-factors for some common term symbols in the approximation $g\_S=2$.

• Einstein–de Haas effect,
• Zeeman effect,
• g-factor (physics).

{{DEFAULTSORT:Lande G-Factor}}

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Category:Nuclear physics