Planck s s Radiation Law Planck made two modifications to the classical theory The oscillators (of electromagnetic origin) can only have certain discrete energies determined by E n = n h ν with n is an integer, ν is the frequency and h = 6.6261 x 10-34 J s is called Planck s constant The oscillators must absorb or emit energy in discrete multiples of the fundamental quantum of energy given by ΔE = h ν Blackbody Radiation The amount of radiation energy emitted by a blackbody of T and surface area A per unit time having a wavelength between λ and λ + dλ into a solid angle dω = sin θ dθ dφ is given by B ( T ) dλ da cos θ dω = B ( T ) dλ da cos θ sin θ dθ dφ 1
Blackbody Radiation Considering a star as a spherical blackbody of radius R and temperature T with each surface area da emitting radiation isotropically The energy per second emitted by this star with wavelength between λ and λ + dλ is the monochromatic luminosity L dλ = 2π π/2 φ =0 θ =0 B dλ da cos θ sin θ dθ dφ A = 4 π 2 R 2 B dλ = 8 π 2 R 2 h c 2 dλ λ 5 (e hc / λkt 1 ) Temperature and Color The intensity of light emitted by three hypothetical stars is plotted against wavelength Where the peak of a star s intensity curve lies relative to the visible light band determines the apparent color of its visible light The insets show stars of about these surface temperatures 2
The color of a star can be determined by using filters of narrow wavelength bands The apparent magnitude is measured through three filters in the standard UBV system Ultraviolet U band 365 nm ± 34 nm Blue B band 440 nm ± 49 nm Visual V band 550 nm ± 45 nm U - B color index = M U - M B B - V color index = M B - M V A star with a smaller B - V color index is bluer than a star with a larger B - V color index Color Indices Color-Color Diagram Relation between the U - B and B - V color indices for mainsequence stars Stars differ from ideal blackbodies as some light gets absorbed as it travels through the star s atmosphere 3
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Electromagnetic radiation interacts with electrons within metals and gives the electrons increased kinetic energy Light can give electrons enough extra kinetic energy to allow them to escape The ejected electrons are called photoelectrons The minimum extra kinetic energy which allows escape is called the work function φ of the material Minimum voltage for which I = 0 is called stopping potential V 0 2
The kinetic energies of the photoelectrons are independent of the light intensity The maximum kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light The smaller the work function φ of the emitter material, the smaller is the threshold frequency of the light that can eject photoelectrons When the photoelectrons are produced, however, their number is proportional to the intensity of light The photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light 3
Einstein suggested that the electromagnetic radiation field is quantized into particles called photons Each photon has the energy quantum Albert Einstein (1879-1955) E photon = h ν = h c λ where ν is the photon frequency and h is Planck s constant The photon travels at the speed of light in a vacuum, and its wavelength is given by c λ = ν Conservation of energy yields h ν = φ + K electron Explicitly, the energy of the most energetic electrons is K max = h ν φ The retarding potentials measured in the photoelectric effect are the opposing potentials needed to stop the most energetic electrons e V 0 = K max 4
Einstein in 1905 predicted that the stopping potential was linearly proportional to the light frequency, with a slope h, the same constant found by Planck K max = e V 0 = h ν φ Arthur Compton (1892-1962) E photon = h ν = h c λ = p c 5
Δ λ = λ f λ h I = ( 1 cos θ ) m e c = λ C ( 1 cos θ ) with Compton wavelength λ C = 0.00243 nm 6
Chemical elements were observed to produce unique wavelengths of light when burned or excited in an electrical discharge Emitted light is passed through a diffraction grating with thousands of lines per cm and diffracted according to its wavelength λ by the equation d sinθ = n λ with n = 0, 1, 2, where d is the distance between grating lines 1 λ = R 1 H 4 1 n 2 Johann Balmer (1825-1898) where n = 3,4,5 and R H = 1.097 10 7 m -1 Rydberg constant 7
Louis de Broglie (1892-1987) Let s consider atoms as a planetary model The force of attraction on the electron by the nucleus and Newton s second law give 1 e 2 υ = µ 2 r 2 r 4 π ε 0 where υ is the tangential velocity of the electron and the reduced mass m µ = e m p m e + m p Can estimate υ, taking r = 5 10-11 m υ 2 10 6 m/s < 0.01 c 8
The total energy of the atom can be written as the sum of kinetic and potential energy E = K + U = 1 e 2 1 8 π ε 0 r 4 π ε 0 e 2 r = 1 8 π ε 0 e 2 r 9
Niels Bohr (1885-1962) L = n h 2 π n ћ n = principle quantum number 2 π r = n λ = n h p L = r p = n h 2 π = n ћ 10
From h L = n 2 π n ћ and K = 8 π ε = 0 r e 2 µ υ 2 2 4 π ε r n = 0 n 2 ћ 2 = n 2 a 0 µ e 2 a 0 = 4 π ε 0 ћ 2 µ e 2 = 0. 53 x 10 quantized orbit (position) 10 m e 2 e 2 E E n = = = 0 8 π ε 0 r n 8 π ε a 0 0 n2 n 2 Emission of light occurs when the atom in an excited state decays to a lower energy state (n u n l ) h ν = E h E l where ν is the frequency of the photon 1 λ ν E h E l 1 1 = = = R H n 2 c h c l n h 2 and R H the Rydberg constant 11