Photoelectric Effect

• work function: $$\mathrm{W}=h v_{0}=\frac{\mathrm{hc}}{\lambda_{0}}$$

• Work Function is Minimum For Cesium: $(1.9 \mathrm{eV})$

• Photoelectric current is directly proportional to intensity of incident radiation. $(v-$ constant $)$

• Photoelectrons ejected from metal have kinetic energies ranging from 0 to $\mathrm{KE}_{\max }$.

Here, $$KE_{max}=e V_s \quad V_s - \text{stopping potential}$$

• Stopping potential Stopping potential is independent of intensity of light used ( $v$-constant)

• Intensity in the terms of electric field is $$ \mathrm{I}=\frac{1}{2} \in_{0} \mathrm{E}^{2} \cdot \mathrm{c} $$

• Momentum of one photon is $$p = \frac{h}{\lambda}$$

• Einstein equation for photoelectric effect is

$$hv=w_0+k_{max}\Rightarrow \hspace{2mm}\frac{hc}{\lambda}=\frac{hc}{\lambda_0}+eV_s$$

• Energy:

$$\Delta \mathrm{E}=\frac{12400 \mathrm{eV}}{\lambda\left(\mathrm{A}^{0}\right)} $$

Angular momentum

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

Force due to radiation (Photon) (no transmission)

(i) When light is incident perpendicularly

(a) $$\quad a=1 \quad r=0$$

$$ \mathrm{F}=\frac{\mathrm{IA}}{\mathrm{C}}, \quad \text { Pressure }=\frac{\mathrm{I}}{\mathrm{C}} $$

(b) $$\quad r=1, \quad a=0$$

$$ F=\frac{2 I A}{c}, \quad P=\frac{2 I}{c} $$

(c) when $$0<r<1 \quad\text{and}\quad a+r=1$$

$$ F=\frac{I A}{c}(1+r), P=\frac{I}{c}(1+r) $$

When light is incident at an angle $\theta$ with vertical.

(a) $$ a=1,\quad r=0$$

$$ F=\frac{I A \cos \theta}{C}, \quad P=\frac{F \cos \theta}{A}=\frac{I}{C} \cos 2 \theta $$

(b) $$ r=1,\quad a=0$$

$$ F=\frac{2 I A \cos ^{2} \theta}{c}, \quad P=\frac{2 I \cos ^{2} \theta}{c} $$

(c) $$0<r<1, \quad a+r=1$$

$$ P=\frac{I \cos ^{2} \theta}{C}(1+r) $$

De Broglie wavelength:

$$ \lambda=\frac{\mathrm{h}}{\mathrm{mv}}=\frac{\mathrm{h}}{\mathrm{P}}=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mKE}}} $$

Radius And Speed Of Electron In Hydrogen Like Atoms:

$$r_n=\frac{n^2}{Z}a_0, \quad a_0=0.529 \stackrel{\circ}{A}$$

$$V_n=\frac{Z}{n}V_0, \quad V_0=2.19\times10^6 m/s$$

Energy In nth Orbit:

$$ E_{n}=E_{1} \cdot \frac{Z^{2}}{n^{2}} \quad E_{1}=-13.6 \mathrm{eV} $$

Wavelength Corresponding To Spectral Lines:

$$ \frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right] $$

• Lyman series: $\mathrm{n}_1=1,$ $\mathrm{n}_2=2,3,4$

• Balmer series: $\mathrm{n}_1=2,$ $\mathrm{n}_2=3,4,5$.

• Paschen series: $\mathrm{n}_1=3,$ $\mathrm{n}_2=4,5,6$

• The lyman series is an ultraviolet and Paschen, Brackett and Pfund series are in the infrared region.

• Total number of possible transitions, is $\frac{\mathrm{n}(\mathrm{n}-1)}{2}$, (from nth state)

• If effect of nucleus motion is considered,

$$ r_{n} = \left(0.529 , \text{\AA}\right) \frac{n^{2}}{Z} \cdot \frac{m}{\mu} $$

$$E_{n}=(-13.6 \mathrm{eV}) \frac{Z^{2}}{n^{2}} \cdot \frac{\mu}{m}$$

Here $\mu$ is reduced mass: $$\mu=\frac{M m}{(M+m)}, M-\text { mass of nucleus }$$

Minimum wavelength for $x$-rays:

$$\lambda_{min}=\frac{hc}{eV_0}=\frac{12400}{V_0(volt)}\stackrel{\circ}{A}$$

Moseley’s Law:

$$\sqrt{v}=a(z-b)$$

where: $a$ and $b$ are positive constants for one type of $x$-rays (independent of $z$ )

Average Radius Of Nucleus:

$$R=R_{0} A^{1 / 3}, \quad R_{0}=1.1 \times 10^{-15} M$$

$$A \text { - mass number }$$

Binding energy of nucleus of mass $M$:

$$B=\left(Z M_{p}+N M_{N}-M\right) C^{2}$$

Alpha - Decay Process:

$$^A_ZX\rightarrow\frac{A-4}{Z-2}Y+^4_2He$$

$$Q=\left[m(^A_ZX)-m\left(\frac{A-4}{z-2}Y\right)-m(^4_2He)\right]C^2$$

Beta- minus decay

$$\begin{gathered} { } _Z^A X \rightarrow{ } _{Z+1}^A Y+\beta^{-}+v^{-} \ \text {Q-value }=\left[m\left({ } _z^A X\right)-m\left({ } _{Z+1}^A Y\right)\right] c^2 \end{gathered}$$

Beta plus-decay

$$\begin{aligned} & { } _z^A X \longrightarrow{ } _{Z-1}^A Y+\beta+ v^{+} \ & Q-\text { value }=\left[m\left({ } _z^A X\right)-m\left({ } _{Z-1}^A Y\right)-2 m e\right] c^2 \end{aligned}$$

Electron capture : when atomic electron is captured, $x$-rays are emitted.

$$^A_zX+e\rightarrow\frac{A}{Z-1}Y+v$$

Radioactive Decay:

In radioactive decay, number of nuclei at instant $t$ is given by $$N=N_{0} e^{-\lambda t}$$

where: $\lambda$ is decay constant.

• Activity of sample : $$\quad A=A_{0} e^{-\lambda t}$$

• Activity per unit mass is called specific activity.

• Half life : $$T_{1 / 2}=\frac{0.693}{\lambda}$$

• Average life : $$T_{av}=\frac{T_{1/2}}{0.693}$$

• A radioactive nucleus can decay by two different processes having half lives $t_{1}$ and $t_{2}$ respectively.

Effective half-life of nucleus is given by $$\frac{1}{t}=\frac{1}{t_{1}}+\frac{1}{t_{2}}$$