(p.416) Appendix A Atomic units
(p.416) Appendix A Atomic units
A.1 Atomic units in vacuum
In this book, we have consistently employed Hartree atomic units (a.u.). This convenient system of units is obtained by assigning a value of 1 to the free-electron mass and charge and to the reduced Planck's constant, i.e., m = e = ħ = 1.
Table A.1 lists various physical quantities, their fundamental atomic units, and the associated numerical values. Note that all references to the first Bohr orbit and the ionization energy of the hydrogen atom assume an infinite mass of the proton.
The numerical values in Table A.1 are expressed in the familiar SI units. Conversion factors to Gaussian units and more significant figures of the fundamental constants can be found in Drake (2006).
In SI units, the speed of light is given by
Together with the Bohr radius a _{0} = 1 a.u., this implies that in a.u. we have
The atomic unit of the magnetic field can be defined in several different ways. Since we work in SI units, we choose here the definition in which one unit of magnetic field, B _{0}, causes a unit Lorentz force on an electron moving with unit velocity. The value of B _{0} is then 2.35052 × 10^{5} T.
An often-used alternative definition of the magnetic-field unit posits that the electric and magnetic fields in a plane wave have the same magnitude. This implies that Maxwell's equations in atomic units and in Gaussian units have the same appearance, and it causes the magnetic-field unit to be (1/c)B _{0} = 1.715 × 10^{3} T. (p.417)
Table A.1 Fundamental atomic units.
Quantity |
Physical meaning |
Unit |
Value in SI units |
---|---|---|---|
Mass |
Free-electron mass |
m |
9.10938 × 10^{–31} kg |
Charge |
Absolute value of free-electron charge |
e |
1.60218 × 10^{–19} C |
Angular momentum |
Reduced Planck's constant |
ħ |
1.05457 × 10^{–34} Js |
Length |
Bohr radius of H atom |
${a}_{0}=\frac{4\pi {\in}_{0}{\hslash}^{2}}{m{e}^{2}}$ |
5.29177 × 10^{–11} m |
Velocity |
Electron speed in the first Bohr orbit |
${\upsilon}_{0}=\frac{{e}^{2}}{4\pi {\in}_{0}\hslash}$ |
2.18769 × 10^{6} m/s |
Time |
Time it takes an electron in the first Bohr orbit to travel one Bohr radius |
${\tau}_{0}=\frac{{a}_{0}}{{\upsilon}_{0}}$ |
2.41888 × 10^{–17} s |
Energy |
Twice the ionization energy of the H atom |
${E}_{H}=\frac{{e}^{2}}{4\pi {\in}_{0}{a}_{0}}$ |
4.35974 × 10^{–18} J |
Electric field |
Electric field in the first Bohr orbit |
${\epsilon}_{0}=\frac{e}{4\pi {\in}_{0}{a}_{0}^{2}}$ |
5.14221 × 10^{11} V/m |
Intensity |
Intensity of plane electromagnetic wave with electric-field amplitude ξ _{0} |
${\mathcal{I}}_{0}=\frac{1}{2}\sqrt{\frac{{\in}_{0}}{{\mu}_{0}}}{\epsilon}_{0}^{2}$ |
3.50945 × 10^{20} W/m^{2} |
Magnetic field |
Causes unit Lorentz force on electron with speed υ _{0} |
${B}_{0}=\frac{\hslash}{e{a}_{0}^{2}}$ |
2.35052 × 10^{5} T |
A.2 Atomic units in the effective-mass approximation
Next, we consider the case of charge carriers in semiconductors, which are treated in the effective-mass approximation (see Appendix K). Working with rescaled atomic units is convenient for describing the electronic structure and excitations in semiconductor nanostructures.
In the effective-mass approximation, the free-electron mass is replaced by an effective mass
The magnetic permeability μ _{r} of typical semiconductor materials, on the other hand, remains close to 1 and will be ignored in the following.
Table A.2 Atomic units for a semiconductor with effective mass m* and effective charge e*. The GaAs results use m_{r} = 0.067 and ∈_{r} = 12.4.
Quantity |
Unit |
a.u.*/a.u. for GaAs |
---|---|---|
Mass |
m* = m_{r} m |
0.067 |
Charge |
${e}^{*}=e/\sqrt{{\in}_{r}}$ |
0.284 |
Angular momentum |
ħ* = ħ |
1 |
Length |
${a}_{0}^{*}=\frac{4\pi {\in}_{0}{\hslash}^{2}}{{m}^{*}{e}^{*2}}=\frac{{\in}_{r}}{{m}_{r}}{a}_{0}$ |
185 |
Velocity |
${\upsilon}_{0}^{*}=\frac{{e}^{*2}}{4\pi {\in}_{0}\hslash}=\frac{{\upsilon}_{0}}{{\in}_{r}}$ |
0.0833 |
Time |
${\tau}_{0}^{*}=\frac{{a}_{0}^{*}}{{\upsilon}_{0}^{*}}=\frac{{\in}_{r}^{2}}{{m}_{r}}{\tau}_{0}$ |
2.29 × 10^{3} |
Energy |
${E}_{\text{H}}^{*}=\frac{{e}^{*2}}{4\pi {\in}_{0}{a}_{0}^{*}}=\frac{{m}_{r}}{{\in}_{r}^{2}}{E}_{\text{H}}$ |
4.36 × 10^{–4} |
Electric field |
${\epsilon}_{0}^{*}=\frac{{e}^{*}}{4\pi {\in}_{0}\sqrt{{\in}_{r}}{a}_{0}^{*2}}=\frac{{m}_{r}^{2}}{{\in}_{r}^{3}}{\epsilon}_{0}$ |
2.35 × 10^{–6} |
Intensity |
${\mathcal{I}}_{0}^{*}=\frac{1}{2}\sqrt{\frac{{\in}_{r}{\in}_{0}}{{\mu}_{0}}}{\epsilon}_{0}^{*2}=\frac{{m}_{r}^{4}}{{\in}_{r}^{11/2}}{I}_{0}$ |
19.5 × 10^{–12} |
Magnetic field |
${B}_{0}^{*}=\frac{\hslash}{e*\sqrt{{\in}_{r}}{a}_{0}^{*2}}=\frac{{m}_{r}^{2}}{{\in}_{r}^{2}}{B}_{0}$ |
1.03 × 10^{–4} |
The reader may wonder why we use the static dielectric constant of the material in the definitions of effective atomic units. This seems appropriate if only static phenom-ena are to be studied. However, the situation is less clear if one wants to study carrier dynamics and excitations, which would require the use of the frequency-dependent di-electric function. To avoid inconsistencies, the regime of validity of the effective-mass approximation is restricted to frequencies much below the band gap energy, which is typically much less than 1 eV. The use of this approximation is therefore limited to studying dynamical phenomena on low-frequency scales where it is justified to work with the static dielectric constant.