CGS制单位转化
今天我发现网上有人问了关于SRIM那本书的问题，原来参数看起来不一样的原因是SRIM书上的单位是厘米-克-秒单位制(centimetre-gram-second system,CGS)，而不是现在比较常用的国际单位制(Système International d'Unités(French),SI)，那么就让我记一下有关的笔记吧。
| Quantity | Quantity symbol | CGS unit name | Unit symbol | Unit definition | In coherent SI units |
|---|---|---|---|---|---|
| length, position | \(L, x\) | centimetre | cm | 1/100 of metre | \(10^{−2}\) m |
| mass | \(m\) | gram | g | 1/1000 of kilogram | \(10^{−3}\) kg |
| time | \(t\) | second | s | 1 second | 1 s |
| velocity | \(v\) | centimetre per second | cm/s | cm/s | \(10^{−2}\) m/s |
| acceleration | \(a\) | gal | Gal | cm/\(\text{s}^2\) | \(10^{−2}\) m/\(\text{s}^2\) |
| force | \(F\) | dyne | dyn | g⋅cm/\(\text{s}^2\) | \(10^{−5}\) N |
| energy | \(E\) | erg | erg | g⋅\(\text{cm}^2\)/\(\text{s}^2\) | \(10^{−7}\) J |
| power | \(P\) | erg per second | erg/s | g⋅\(\text{cm}^2\)/\(\text{s}^{3}\) | \(10^{−7}\) W |
| pressure | \(p\) | barye | Ba | g/(cm⋅\(\text{s}^2\)) | \(10^{−1}\) Pa |
| dynamic viscosity | \(μ\) | poise | P | g/(cm⋅s) | \(10^{−1}\) Pa⋅s |
| kinematic viscosity | \(ν\) | stokes | St | \(\text{cm}^2\)/s | \(10^{−4}\) \(\text{m}^2\)/s |
| wavenumber | \(k\) | kayser (K) | \(\text{cm}^{-1}\) | \(\text{cm}^{-1}\) | 100 \(\text{m}^{-1}\) |
| Quantity | Symbol | SI unit | ESU unit | Gaussian unit | EMU unit |
|---|---|---|---|---|---|
| electric charge | \(q\) | 1 C | ≘ (\(10^{-1}\)\(c\)) statC (Franklin) | ≘ (\(10^{-1}\)\(c\)) statC (Franklin) | ≘ (\(10^{-1}\)) abC |
| electric flux | \(Φ_E\) | 1 V⋅m | ≘ (4π × \(10^{-1}\)\(c\)) statC (Franklin) | ≘ (4π × \(10^{-1}\)\(c\)) statC (Franklin) | ≘ (\(10^{-1}\)) abC |
| electric current | \(I\) | 1 A | ≘ (\(10^{-1}\)\(c\)) statA (Fr⋅\(\text{s}^{−1}\)) | ≘ (\(10^{-1}\)\(c\)) statA (Fr⋅\(\text{s}^{−1}\)) | ≘ (\(10^{-1}\)) Bi |
| electric potential / voltage | \(\Phi / V, U\) | 1 V | ≘ (\(10^8 c^{−1}\)) statV | ≘ (\(10^8 c^{−1}\)) statV | ≘ (\(10^8\)) abV |
| electric field | \(E\) | 1 V/m | ≘ (\(10^6 c^{−1}\)) statV/cm | ≘ (\(10^6 c^{−1}\)) statV/cm | ≘ (\(10^6\)) abV/cm |
| electric displacement field | \(D\) | 1 C/\(\text{m}^2\) | ≘ (\(10^{−5}c\)) statC/\(\text{cm}^2\) (Fr/\(\text{cm}^2\)) | ≘ (\(10^{−5}c\)) statC/\(\text{cm}^2\) (Fr/\(\text{cm}^2\)) | ≘ (\(10^{−5}\)) abC/\(\text{cm}^2\) |
| electric dipole moment | \(p\) | 1 C⋅m | ≘ (10\(c\)) statC⋅cm | ≘ (10\(c\)) statC⋅cm | ≘ (10) abC⋅cm |
| magnetic dipole moment | \(μ\) | 1 A⋅\(\text{m}^2\) | ≘ (\(10^3c\)) statC⋅\(\text{cm}^2\) | ≘ (\(10^3\)) Bi⋅\(\text{cm}^2\) = (\(10^3\)) erg/G | ≘ (\(10^3\)) Bi⋅\(\text{cm}^2\) = (\(10^3\)) erg/G |
| magnetic B field | \(B\) | 1 T | ≘ (\(10^4 c^{−1}\)) statT | ≘ (\(10^4\)) G | ≘ (\(10^4\)) G |
| magnetic H field | \(H\) | 1 A/m | ≘ (4π × \(10^{−3}c\)) statA/cm | ≘ (4π × \(10^{−3}\)) Oe | ≘ (4π × \(10^{−3}\)) Oe |
| magnetic flux | \(Φ_m\) | 1 Wb | ≘ (\(10^8 c^{−1}\)) statWb | ≘ (\(10^8\)) Mx | ≘ (\(10^8\)) Mx |
| resistance | \(R\) | 1 Ω | ≘ (\(10^9 c^{−2}\)) s/cm | ≘ (\(10^9 c^{−2}\)) s/cm | ≘ (\(10^9\)) abΩ |
| resistivity | \(ρ\) | 1 Ω⋅m | ≘ (\(10^{11} c^{−2}\)) s | ≘ (\(10^{11} c^{−2}\)) s | ≘ (\(10^{11}\)) abΩ⋅cm |
| capacitance | \(C\) | 1 F | ≘ (\(10^{−9} c^2\)) cm | ≘ (\(10^{−9} c^2\)) cm | ≘ (\(10^{−9}\)) abF |
| inductance | \(L\) | 1 H | ≘ (\(10^9\) \(c^{−2}\)) \(\text{cm}^{−1}⋅\text{s}^2\) | ≘ (\(10^9\) \(c^{−2}\)) \(\text{cm}^{−1}⋅\text{s}^2\) | ≘ (\(10^9\)) abH |
对于公式 \[ \begin{aligned} \varepsilon &=\frac{a E_{c}}{Z_{1} Z_{2} e^{2}} \\ E_{c} &=\frac{E_{0} M_{2}}{M_{1}+M_{2}} \\ a &=\frac{0.8853 a_{0}}{Z_{1}^{0.23}+Z_{2}^{0.23}} \end{aligned} \] 其中，玻尔半径\(a_0=0.529\)Å,\(E_0\)单位为keV，要得到： \[ \varepsilon=\frac{32.53 M_{2} E_{0}}{Z_{1} Z_{2}\left(M_{1}+M_{2}\right)\left(Z_{1}^{0.23}+Z_{2}^{0.23}\right)} \] 即在CGS单位制下有： \[ \frac{0.8853a_0}{e^2}=32.53 \] 该问题下，有人给出了： \[ 32.53 \approx 0.8853 \cdot 5.29 \cdot 10^{-9} \cdot 1000 \cdot 4.8 \cdot 10^{-10} \cdot \frac{1}{300} /\left(4.8 \cdot 10^{-10}\right)^{2} \] 唔，看起来和国际单位制的变换还是有些不一样的，让我们一步步地看看：
\[ \begin{aligned} &a_0=0.529\text{Å}=0.529*10^{-8}\text{cm}=5.29*10^{-9}\text{cm}\\ &e=4.803 204 27*10^{-10}\text{Fr} \end{aligned} \] 由于CSG（或者说CSG中，ESU）单位制中电压单位(statV)关于SI单位制的电压单位(V)有： \[ 1\text{V}=10^{8}c^{-1}\text{statV}\approx \frac{1}{300}\text{statV} \] 由于\(E_0\)给的是千电子伏特(keV)单位，所以要再添上比例系数： \[ 1000*\frac{1}{300}*e \] 将上面的全部代入式子即得结果。