原子物理 ~ 原子・原子核・素粒子

$$\begin{eqnarray*} \lambda &=& 3.646 \times 10^{-7} \ \mbox{m} \times \frac{n^2}{n^2 – 2^2} \ (n=3,4,5,6) \\ \\ \frac{1}{\lambda} &=& R \left( \frac{1}{n’^2} – \frac{1}{n^2} \right) \end{eqnarray*}$$

\lambda &=& 3.646 \times 10^{-7} \ \mbox{m} \times \frac{n^2}{n^2 – 2^2} \ (n=3,4,5,6)

\frac{1}{\lambda} &=& R \left( \frac{1}{n’^2} – \frac{1}{n^2} \right)

$$mvr = n \frac{h}{2\pi}$$

mvr = n \frac{h}{2\pi}

$$h \nu = E – E’$$

h \nu = E – E’

$$\begin{eqnarray*} m \frac{v_n^2}{r_n} &=& k_0 \frac{e^2}{r_n^2} \ (n=1,2,3 \cdots ) \\ \\ r_n &=& \left( \frac{h}{2 \pi} \right) ^2 \frac{n^2}{k_0 me^2} \ (n=1,2,3 \cdots ) \end{eqnarray*}$$

$$\begin{eqnarray*} U &=& – k_0 \frac{e^2}{r_n} \\ \\ E_n &=& K + U = \frac{1}{2} m v_n^2 – k_0 \frac{e^2}{r_n} = – k_0 \frac{e^2}{2r_n} \\ \\ E_n &=& – \frac{2 \pi ^2 k_0^2 me^4}{h^2} \cdot \frac{1}{n^2} \ (n=1,2,3 \cdots ) \\ \\ E_n &=& – \frac{2.18 \times 10^{-18}}{n^2} \ \mbox{J} = – \frac{13.6}{n^2} \ \mbox{eV} \ (n=1,2,3 \cdots ) \end{eqnarray*}$$

U &=& – k_0 \frac{e^2}{r_n}

E_n &=& K + U = \frac{1}{2} m v_n^2 – k_0 \frac{e^2}{r_n} = – k_0 \frac{e^2}{2r_n}

E_n &=& – \frac{2 \pi ^2 k_0^2 me^4}{h^2} \cdot \frac{1}{n^2} \ (n=1,2,3 \cdots )

E_n &=& – \frac{2.18 \times 10^{-18}}{n^2} \ \mbox{J} = – \frac{13.6}{n^2} \ \mbox{eV} \ (n=1,2,3 \cdots )

$$\begin{eqnarray*} h \nu &=& \frac{hc}{\lambda} E_n – E_{n’} \\ \\ \frac{1}{\lambda} &=& \frac{E_n – E_{n’}}{hc} = \frac{2 \pi ^2 k_0^2 me^4}{ch^3} \left( \frac{1}{n’^2} – \frac{1}{n^2} \right) \\ \\ \\ R &=& \frac{2 \pi ^2 k_0^2 me^4}{ch^3} \end{eqnarray*}$$

h \nu &=& \frac{hc}{\lambda} E_n – E_{n’}

\frac{1}{\lambda} &=& \frac{E_n – E_{n’}}{hc} = \frac{2 \pi ^2 k_0^2 me^4}{ch^3} \left( \frac{1}{n’^2} – \frac{1}{n^2} \right)

R &=& \frac{2 \pi ^2 k_0^2 me^4}{ch^3}

$$\begin{eqnarray*} _{88}^{226} \text{Ra} & \to & _{86}^{222} \text{Rn} + _{2}^{4} \text{He} \\ \\ _{81}^{206} \text{Ti} & \to & _{82}^{206} \text{Pb} + \text{e}^- \\ \end{eqnarray*}$$

_{88}^{226} \text{Ra} & \to & _{86}^{222} \text{Rn} + _{2}^{4} \text{He}

_{81}^{206} \text{Ti} & \to & _{82}^{206} \text{Pb} + \text{e}^-

$$N = N_0 \left( \frac{1}{2} \right) ^{\frac{t}{T}}$$

N = N_0 \left( \frac{1}{2} \right) ^{\frac{t}{T}}

$$\text{n} + ^{14}\text{N} \to \text{p} + ^{14}\text{C}$$

\text{n} + ^{14}\text{N} \to \text{p} + ^{14}\text{C}

$$\begin{eqnarray*} E &=& mc^2 \\ \\ m’ &=& \frac{m}{\sqrt{1- \frac{v^2}{c^2}}} \\ \\ E &=& m’ c^2 = \frac{mc^2}{\sqrt{1- \frac{v^2}{c^2}}} \\ \\ p &=& m’ v = \frac{mv}{\sqrt{1 – \frac{v^2}{c^2}}} \\ \\ E &=& \frac{mc^2}{\sqrt{1 – \frac{v^2}{c^2}}} = mc^2 \left( 1 – \frac{v^2}{c^2} \right) ^{-\frac{1}{2}} \simeq mc^2 + \frac{1}{2} mv^2 \end{eqnarray*}$$

E &=& mc^2

m’ &=& \frac{m}{\sqrt{1- \frac{v^2}{c^2}}}

E &=& m’ c^2 = \frac{mc^2}{\sqrt{1- \frac{v^2}{c^2}}}

p &=& m’ v = \frac{mv}{\sqrt{1 – \frac{v^2}{c^2}}}

E &=& \frac{mc^2}{\sqrt{1 – \frac{v^2}{c^2}}} = mc^2 \left( 1 – \frac{v^2}{c^2} \right) ^{-\frac{1}{2}} \simeq mc^2 + \frac{1}{2} mv^2

$$\begin{eqnarray*} \Delta m &=& Zm_\text{p} + Nm_\text{n} – M \\ \\ \Delta m \cdot c^2 &=& Zm_\text{p} c^2 + Nm_\text{n} c^2 – Mc^2 \end{eqnarray*}$$

\Delta m &=& Zm_\text{p} + Nm_\text{n} – M

\Delta m \cdot c^2 &=& Zm_\text{p} c^2 + Nm_\text{n} c^2 – Mc^2

$$_{1}^{2} \text{H} + _{1}^{3} \text{H} \to _{0}^{1} \text{n} + _{2}^{4} \text{He}$$

_{1}^{2} \text{H} + _{1}^{3} \text{H} \to _{0}^{1} \text{n} + _{2}^{4} \text{He}

$$_{92}^{235} \text{U} + _{0}^{1} \text{n} \to _{56}^{144} \text{Ba} + _{36}^{89} \text{Kr} + 3 _{0}^{1} \text{n}$$

_{92}^{235} \text{U} + _{0}^{1} \text{n} \to _{56}^{144} \text{Ba} + _{36}^{89} \text{Kr} + 3 _{0}^{1} \text{n}