Dual Nature of Matter and Radiation

Wave Theory of Light, Einstein’s Photoelectric Equation and Particle Nature of Light

  • ELNSTEIN’S PHOTO ELECTRIC EQUATION : \tt h\nu=h\nu_{0}\ +\ \frac{1}{2}\ mV_{max}^2
  • \tt \frac{1}{2}\ mV_{max}^{2}\ =\ h\left(\nu-\nu_0\right)=hc\left(\frac{1}{\lambda}-\frac{1}{\lambda_{0}}\right)
  • \tt V_{max}=\sqrt{\frac{2h\left(\nu-\nu_{0}\right)}{m}}=\sqrt{\frac{2\ hc}{m}\frac{\left[\lambda_{0}-\lambda\right]}{\lambda\ \lambda_{0}}}
  • \tt \frac{\left[K\ E_{max}\right]_1}{\left[K\ E_{max}\right]_2}=\frac{h\nu_{1}-W}{h\nu_{2}-W}=\frac{\nu_{1}-\nu_{0}}{\nu_{2}-\nu_{0}}
  • \tt \frac{\left[K\ E_{max}\right]_1}{\left[K\ E_{max}\right]_2}=\frac{hc\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{0}}\right)}{{hc\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{0}}\right)}}=\frac{\left(\frac{1}{\lambda_{1}}-\frac{1}{\lambda_{0}}\right)}{{\left(\frac{1}{\lambda_{2}}-\frac{1}{\lambda_{0}}\right)}}
  • Energy of photon in eV = \tt E=\frac{12400}{\lambda\left(in\ Angstrom\right)} in eV
  • Work function = \tt W= \frac{12400}{\lambda_{0}} in e.V
  • The graph between maximum kinetic Energy and maximum velocity increases linearly with the frequency of the incident radiation.

  • The photoelectric effect is an instantaneous phenomenon. The time lag between incidence of radiation and emissions of photo electron is very small.
  • The graph between stopping potential, Vs and incident frequency (ν) is a straight line with slope \tt \frac{h}{e}

  • The graph between K.Emax and incident frequency, ν is a straight line with slope h

  • In photoelectric effect all the emitted photoelectrons do not have same kinetic energy. The K.E ranges from 0 to K.Emax (or) 0 to h (ν – ν0)
  • Intensity of the radiation obeys inverse square law \tt I\propto\frac{1}{d^{2}}\ and\ i\propto I\Rightarrow\ i\propto\ \frac{1}{d^{2}}
  • Photo electric Cells convert light energy to electrical energy.
  • Vacuum type of photo emissive cells has instantaneous response.
  • In gas filled photo emissive cells, photo current will not be proportional to intensity of the incident radiation.
  • In the perspective of De.Broglie, matter behaves both as particles and waves.
  • de-BROGLIE wavelength associated with the matter wave of that particle is \tt \lambda=\frac{h}{mv}=\frac{h}{p}=\frac{h}{\sqrt{2mE}}
  • de-BROGLIE wave equation: \tt \lambda\propto\frac{1}{\nu} & if ν = 0, λ = ∞
  • \tt \lambda\propto\frac{1}{m} {λ = de-Broglie wavelength m = mass}
  • \tt \lambda\propto\frac{1}{p} {λ = de-Broglie wavelength P = momentum}
  • \tt \lambda\propto\frac{h}{\sqrt{2\ mqv}}{ν = potential difference h = plancks constant q = charge}
  • For an electron \tt \lambda\approx\frac{12.27}{\sqrt{v}}\ A^{o}
  • For a proton, \tt \lambda=\frac{0.286}{\sqrt{v}}\ A^{o}
  • For deuteron, \tt \lambda=\frac{0.202}{\sqrt{E}}\ A^{o}
  • For α particle, \tt \lambda=\frac{0.101}{\sqrt{v}}\ A^{o}
  • For neutron, \tt \lambda=\frac{0.286}{\sqrt{E}}\ A^{o}
  • If the de Broglie waves are associated with electron, then these should be diffracted like x-rays. we can determine the wave length of these waves using the Bragg’s formula, 2d sin θ = n λ
  • \tt \theta=\left[\frac{180-\phi}{2}\right] = Bragg’s angle
  • de Broglie wave length can also be verified using the formula \tt \lambda=\frac{12.27}{\sqrt{v}}=\frac{12.27}{\sqrt{54}}=1.67\ A^{o}
  • X-rays are produced by the deceleration of high energy electrons bombarding a hard metal target.
  • The target used to produce x-rays should have a) high atomic weight b) high melting point c) high thermal conductivity.
  • X-rays are electromagnetic waves of very short wavelength i.e order of wave length 0.1 Å – 100Å, order of frequency 1016 Hz – 1019 Hz, order of energy 124eV – 124 KeV
  • Intensity of x-rays depends on the number of electron striking the target
  • High frequency x-rays are called HARD X-RAYS.
  • Low frequency X-rays are called SOFT X-RAYS.
  • CONTINUOUS X-RAY SPECTROM is produced when high speed electron are suddenly stopped by a metal target.
  • Continuous X-ray spectrum contains all wave lengths above a minimum wavelength λm
  • \tt \lambda_{min}=\frac{hc}{eV}=\frac{12400}{v}\ A^{o}
  • \tt \nu_{max}=\frac{eV}{h}, maximum frequency of emitted X-ray photon.
  • Efficiency of X-ray tube, \tt \eta=\frac{output\ power}{input\ power}\times100
  • CHARACTERISTIC X-RAY SPECTRUM is produced due to transition of electrons from higher energy level to lower energy level in target atoms.
  • Wavelengths of the X-rays of characteristic X-ray spectrum depend only on atomic number of the target element and independent of target potential.
  • In characteristic X-rays, K-series of lines are obtained when transition takes place from higher levels to K-shell.
  • E < E < E, E > E
  • I > I > I
  • \tt \nu_{k\beta} = \nu_{k\alpha} + \nu_{L\alpha} \Rightarrow \frac{1}{\lambda_{k\beta}} =\frac{1}{\lambda_{k\alpha}} + \frac{1}{\lambda_{L\alpha}}
  • \tt E_{k} - E_{L} = h\nu_{k\alpha} = \frac{hc}{\lambda_{k\alpha}}

View the Topic in this video From 00:19 To 28:33

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1. The balance of the photon energy is used up in giving the electron a kinetic energy of \frac{1}{2}mv^{2}

hv = \phi_{0} + \frac{1}{2}mv^{2}

2. Kinetic energy of photoelectrons is ΔKE = hv − hv0 = h(v − v0)

= hc \left[\frac{1}{\lambda} - \frac{1}{\lambda_{0}}\right] = 12400 \left[\frac{1}{\lambda (A^0)} - \frac{1}{\lambda_{0}(A^0)}\right]eV