Unit 18 Notes – Atomic Physics and Quantum Effects

Chapter 30,31

Blackbody radiation – A blackbody absorbs all incident light rays. all bodies, no matter how hot or cold, emit electromagnetic waves. We can see the waves emitted by very hot objects because they are within the visible spectrum (light bulb filament; red-hot metal). At lower temps we can’t see the waves but they are still there. For example, the human body emits waves in the infrared range. This is why we can use infrared detecting devices to “see” in the dark.

The distribution of energy in blackbody radiation is independent of the material from which the blackbody is constructed – it depends only on the temperature.

The diagrams below show the intensity of various frequencies of EM radiation emitted by blackbodies of various temperatures. Note that as the temperature increases, the energy emitted (area under curve) increases and the peak in the radiation shifts to higher frequencies.

This is important because classical physics predicts a completely different curve that increases to infinite intensity in the ultraviolet region. (thus called the Ultraviolet Catastrophe). The only way to make sense of this finding is by saying energy is quantized (Planck’s quantum hypothesis)

For an in-depth discussion of the classical physics equation (Rayleigh-Jeans Law) that leads to the Ultraviolet Catastrophe and the subsequent development of Planck’s Law for blackbody radiation, go here:

Photons and the photoelectric effect

-Photoelectric effect experiment:

  • if light of a sufficiently high frequency shines on a metal plate, electrons are ejected from the plate. The ejected electrons move toward a positive electrode called the collector and cause a current to register on an ammeter.

-From this (the photoelectric effect), we see that light comes in energy packets called “photons” [this word comes from photoelectrons – electrons emitted with the help of light]

-energy of a photon:

E = hfwhere h = Planck’s constant and f = frequency

-Planck’s Constant, h = 6.626x10-34 Js

-this is what Einstein got his Nobel Prize for in 1921

  • Einstein says when light shines on a metal, a photon can give up its energy to an electron in the metal.
  • If the photon has enough energy to do the work of removing the electron from the metal, the electron can be ejected.
  • The work required depends on how strongly the electron is held
  • For the least strongly held electrons, the necessary work has a minimum value and is called the work function,, of the metal(your book calls it W0). If a photon has energy in excess of the work needed to remove an electron, the excess energy appears as kinetic energy of the ejected electron.
  • Thus, the least strongly held electrons are ejected with the maximum kinetic energy Kmax.
  • Applying the conservation of energy principle, Einstein proposed this relation to describe the photoelectric effect:

hf = Kmax + 

or

photon energy = max KE of ejected e- + min. work to eject e-

  • Note that the intensity of the light isn’t important. The frequency is.

Below the cutoff frequency, fo, electrons will not be emitted. Note that at that point, Kmax =0, so hf = ; thus fo = /h

If a potential is applied to oppose the current, it is the “stopping potential” (it stops the electrons).

Note the similarity between this graph and the kinetic energy-frequency graph. remember that K=eV

Importance of photoelectric effect: classical physics predicts that any frequency of light can eject electrons as long as the intensity is high enough. Experimental data shows there is a minimum (cutoff frequency) that the light must have. classical physics predicts that the kinetic energy of the ejected electrons should increase with the intensity of the light. again, experimental data shows this is not the case; increasing the intensity of the light only increases the number of electrons emitted, not their kinetic energy. THUS the photoelectric effect is strong evidence for the photon model of light.

Wave-Particle duality: a wave can exhibit particle-like characteristics and a particle can exhibit wave-like characteristics. EM radiation is an example.

Einstein theorized that light comes in quantized “packets” of energy called photons. Support for the photon model include: blackbody radiation, the photoelectric effect, Compton Effect.

The momentum of the photon

-Combining E=mc2 and p=mv, you get: p / E = v / c2

-The photon travels at the speed of light, so v = c and p / E = 1 / c

-Therefore the momentum, p, of the photon is p = E / c

-But we also know that E = hf and λ=c/f, so

momentum of a photon

The De Broglie wavelength and the wave nature of matter

-Just like waves can be particle-like, particles can be wave-like

-Wavelength of a particle is given by the same relation that applies to a photon:

-Where h is Planck’s constant, p is momentum; λ is known as the de Broglie wavelength of the particle

-X-ray diffraction provides support for this. the spacing of atoms in a crystal is on the order of the wavelength of x-rays, thus the crystal can act as a diffraction grating for the x-rays. Different crystal configurations produce distinct diffraction patterns. Diffraction patterns are evidence for wave-nature.

The Compton Effect:

-when an X-ray photon strikes an electron in a piece of graphite, the X-ray scatters in one direction, and the electron recoils in another direction after the collision (like two billiard balls colliding on a pool table)

-the scattered photon has a frequency f ’ that is smaller than the frequency f of the incident photon, thus the photon loses energy in the collision

-the difference between the two frequencies depends on the angle at which the scattered photon leaves the collision

-Similar to the analysis for the kinetic energy and work for photoelectric effect, we find:

  • The electron is assumed to be initially at rest and essentially free (not bound to the atoms of the material)
  • According to principle of conservation of energy:

hf = hf ’ + K

or

energy of incident photon = energy of scattered photon + KE of e-

-For an initially stationary electron, conservation of total linear momentum requires that:

  • Momentum of incident photon = momentum of scattered photon + momentum of electron

From this point, these equations are combined with the relativistic equations for energy and momentum to derive the equation for Compton Scattering. an in-depth explanation of the derivation can be found

-The difference between the wavelength λ’ of the scattered photon and the wavelength λ of the incident photon is related to the scattering angle θ by:

  • m is the mass of the electron. h/mc is the “Compton wavelength of the electron” and is h/mc = 2.43x10-12 m.

It is interesting to note that according to Einstein’s theory of relativity, the rest mass of a photon is zero. However, it is never at rest, it is always moving (at the speed of light!) so it does have a finite momentum (even though p=mv doesn’t work)

The Davisson-Germer experiment demonstrated the wave nature of the electron, confirming the earlier hypothesis of deBroglie. Davisson and Germer measured the energies of electrons scattering from a metal surface. Electrons from a heated filament were accelerated by a voltage and allowed to strike the surface of nickel metal, which could be rotated to observe angular dependence of the scattered electrons. They found that at certain angles there was a peak in the intensity of the scattered electron beam.In fact, the electron beam was scattered by the surface atoms on the nickel at the exact angles predicted for the diffraction of x-rays according to Bragg's formula nλ=2dsinθ, with a wavelength given by the de Broglie equation, λ=h/p. X-rays are accepted to be wavelike, thus this is evidence for wavelike behavior of electron.

This is Davisson-Germer’s data relating the intensity of scattered electrons as a function of accelerating voltage for a particular angle.

X-Rays

-can be produced when electrons, accelerated through a large potential difference, collide with a metal target (made from molybdenum or platinum for example)contained within an evacuated glass tube

-a plot of X-ray intensity per unit wavelength versus wavelength consists of sharp peaks or lines superimposed on a broad continuous spectrum

X-ray spectrum shown here is produced when a molybdenum target is bombarded with electrons that have been accelerated through a potential difference

-when the energetic electrons impact the target metal, they undergo a rapid deceleration (braking). as the electrons suddenly come to rest they give off high-energy radiation in the form of X-rays over a wide range of wavelengths. This is referred to as “Bremsstrahlung continuum” (bremsstrahlung is German for “braking radiation”) This is the base for the peaks seen in the graph

-the sharp peaks are called characteristic lines or characteristic X-rays because they are characteristic of the target material.

-the characteristic lines are marked Kα and Kβ because they involve the n=1 or K shell of a metal atom. (K shell is the innermost electron shell)

-if an electron with enough energy strikes the target, one of the K-shell electrons can be knocked entirely out of a target atom

-an electron in one of the outer shells can then fall into the K shell, and an X-ray photon is emitted in the process. Kα is a change from n=2 to n=1; Kβ is a change from n=3 to n=1

-there is a cutoff wavelength (as seen in the diagram). an impinging electron cannot give up any more than all of its KE, thus an emitted X-ray photon can have an energy no more than the KE of the impinging electron. the wavelength that corresponds to this is the cutoff wavelength (max frequency-min wavelength).

K=eV, E=hf thus eV=hf. f=c/λ, so…

V is the potential difference applied across the X-ray tube; e is the charge of an electron

Line Spectra

-for a solid object the radiation emitted has a continuous range of wavelengths, some of which are in the visible region of the spectrum.

-the continuous range of wavelengths is characteristic of the entire collection of atoms that make up the solid.

-in contrast, individual atoms, free of the strong interactions that are present in a solid, emit only certain specific wavelengths, rather than a continuous range. these wavelengths are characteristic of the atom.

  • a low-pressure gas in a sealed tube can be made to emit EM waves by applying a sufficiently large potential difference between two electrodes located within the tube
  • with a grating spectroscope the individual wavelengths emitted by the gas can be separated and identified as a series of bright fringes or lines. these series of lines are called the LINE SPECTRA.

Bohr Model of Hydrogen atom

-Bohr tried to derive the formula that describes the line spectra that was developed by Balmer using trial and error. he used Rutherford’s model of the atom, the quantum ideas of Planck and Einstein, and the traditional description of a particle in uniform circular motion.

-assumptions of Bohr model:

  • electron in H moves in circular motion
  • only orbits where the angular momentum of the electron is equal to an integer times Planck’s constant divided by 2π are allowed
  • electrons in allowed orbits do not radiate EM waves. Thus the orbits are stable. (if it emitted radiation, it would lose energy and spiral into the nucleus)
  • EM radiation is given off or absorbed only when an electron changes from one allowed orbit to another. ΔE = hf (ΔE is energy difference between orbits and f is frequency of radiation emitted or absorbed)

-Bohr theorized that a photon is emitted only when the electron changes orbits from a larger one with a higher energy to a smaller one with a lower energy.

Bohr Energy levels in electron volts

-the centripetal force that provides the circular orbit is electrostatic force (Coulomb’s Law). setting this equal to centripetal force we get:

or

-note that electrostatic force between an electron and a nucleus with Z protons is ke(Ze)/r2 = kZe2/r2. (Z = # of protons = atomic number)

-angular momentum is L=rmv . the second assumption listed above states the angular momentum of a given orbit, Ln, equals an integar, n, times h/2π; or Ln=nh/2π. setting these equal and solving for v gives:

-combining these two equations gives:

(note that n is the orbital number)

-solving for rn gives us

Bohr orbital radius

-energy of electron is sum of kinetic and potential energies:

E = K + U = ½ mv2 + U

-the electrostatic potential energy is U = -kZe2/r and using the first equation in this section we get:

-substituting in the Bohr orbital radius, rn, from above:

-grouping all of the constants yields:

-plugging in values for the constants gives us:

Z= atomic number; n = energy level (1,2,3,…)

-when an electron in an initial orbit with a larger energy Ei, drops to a lower orbit with energy Ef, the emitted photon has an energy of Ei – Ef , consistent with the law of conservation of energy. combining this with Einstein’s E = hf, we get

Ei – Ef = hf

-To find the wavelengths in hydrogen’s line spectrum (Z=1), we apply that to the equation above:

-given that E = hc/λ :

-the value in the first parenthesis is only constants. calculating this value gives the Rydberg Constant (1.097x107) – the exact value that Balmer found

de Broglie Waves and Bohr Model

-the angular momentum assumption in Bohr’s model is there because it produces results in agreement with experiment.

-however, de Broglie matter-wave relationship explains the significance:

-think of matter (electron) waves as analogous to a wave on a string – except that the string is a circle representing the electrons orbit. the standing wave must fit an integral number of wavelengths into the circumference of the orbit: nλ=2πr. combine this with p=h/λ:

-rearranging and multiplying both sides of the equation by r gives us angular momentum

-Thus this condition of the Bohr model is a reflection of the wave nature of matter

Energy Levels

Energy of the photon

Energy level Diagrams

Example: An electron releases energy as it moves back to its ground state position. As a result, photons are emitted. Calculate the POSSIBLE wavelengths of the emitted photons.

Line Spectra of Hydrogen Atom

-Lyman series occurs when electrons make transition s from higher energy levels with ni = 2, 3, 4, … to the first energy level where nf = 1.

-notice when an electron transitions from n=2 to n=1, the longest wavelength photon in the Lyman series is emitted, since the energy change is the smallest possible. when the electron transitions from highest to lowest, the shortest wavelength is emitted.

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