KEY CONCEPTS ATOMIC PHYSICS & QUANTUM EFFECTS

1. PHOTONS & THE PHOTOELECTRIC EFFECT Max Planck explained blackbody radiation with his quantum hypothesis, which states that the energy of a thermal oscillator, Eosc, is not continuous, but instead is a discrete quantity given by the equation:

Eosc = nhf n = 1, 2, 3,... where f is the frequency and h is a constant now known as Planck’s constant. Albert Einstein extended the idea by adding that all emitted radiation is quantized. He suggested that light is composed of discrete quanta, rather than of waves. According to his theory, each particle of light, known as a photon, has an energy E given by: E = hf Einstein’s theory helped him explain a phenomenon known as the photoelectric effect, in which a photon of light strikes a photosensitive material and causes an electron to be ejected from the material. A photocell constructed from photosensitive material can produce an electrical current when light shines on it.

The kinetic energy, K, of a photoelectron displaced by a photon of energy, hf, is given by: K = hf - φ where the work function, φ, is the minimum energy needed to free the electron from the photosensitive material. No photoemission occurs if the frequency of the incident light falls below a certain cutoff frequency – or threshold frequency – given by: φ f0 = h Einstein's theory explained several aspects of the photoelectric effect that could not be explained by classical theory: • The kinetic energy of photoelectrons is dependent on the light’s frequency. • No photoemission occurs for light below a certain threshold frequency. • Photocurrent flows immediately when the light’s frequency is greater than the threshold frequency.

Arthur Compton provided further evidence supporting the quantization of light theory with his x-ray scattering experiments. In the Compton effect scattered radiation,

λ, always has a slightly longer wavelength than incident radiation, λ0 :

λ0 < λ KEY CONCEPTS ATOMIC PHYSICS & QUANTUM EFFECTS The change in the wavelength depends on the scattering angle, θ, and a constant known as the Compton wavelength, λc , but not on the scattering material: Δλ = λ 1− cosθ c( )

2. ATOMIC ENERGY LEVELS The appreciable backscattering of alpha particles by gold foil observed by Ernest Rutherford in 1911 seemed to indicate that most of the mass of an atom is concentrated in a minute nucleus with an associated positive charge. In 1913 Niels Bohr proposed a new theory of the atom, in which electrons move in circular orbits around the nucleus. In order to explain the emission lines (e.g., the Balmer series) observed in hydrogen, he postulated that the angular momentum, L, of the hydrogen electron is quantized and can have only certain discrete values given by: nh L = mvr = n =1, 2, 3,... 2π where h is Planck’s constant and n is the principle quantum number. The ionization energy necessary to completely remove an electron from a hydrogen atom is –13.6 eV. Therefore, the possible energy levels for a hydrogen electron are given by: −13.6 En = 2 eV n = 1, 2, 3,... n The lowest energy level (n = 1) is known as the ground state, and the higher levels are known as excited states. An electron must absorb just the right amount of energy from a photon in order to move from one level to a higher level. Conversely, when an electron moves from a higher level to a lower level, a photon with a particular energy is emitted. Fluorescence is the phenomenon in which an atom absorbs a photon and immediately thereafter emits of one or more photons, each having a lower energy than the original, absorbed photon. If the emission is delayed for a measurable length of time, the phenomenon is known as phosphorescence.

3. WAVE-PARTICLE DUALITY In 1924 Louis de Broglie suggested that, if light can act like a particle, then maybe a particle, such as an electron, can behave as a wave. The de Broglie wavelength, λ, of such a particle wave is related to Planck’s constant, h, and the momentum, p, of the particle: h h λ = = mv p The de Broglie theory was used to show that the angular momentum of the hydrogen electron is quantized, as previously proposed by Bohr. Proof of de Broglie’s concept of wave-particle duality came in 1927, when Davisson and Germer studied the diffraction of electrons by a nickel crystal. They

2 KEY CONCEPTS ATOMIC PHYSICS & QUANTUM EFFECTS showed that the wavelength of a diffracted beam of electrons calculated from standard wave theory, corresponds to the de Broglie wavelength: λ = d sinθ = 0.165 nm (wave theory) h λ = = 0.167 nm (de Broglie wavelength) p George Thomson also showed that, when a beam of electrons is scattered by a material, it forms a diffraction pattern similar to that observed for x-rays.

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