PA1140 Waves and Quanta Unit 4: Revision Dr Matt Burleigh (S4)

1. PA1140 Waves and Quanta Unit 4: Revision Dr Matt Burleigh (S4) http://www.star.le.ac.uk/~mbu/lectures.html

2. PA1140 Waves and Quanta Previous Lecture Slides for Unit 4: http://www.star.le.ac.uk/~mbu/lectures.html • Bohr theory • Atomic size and shape • Mass and binding energy • Radioactivity, fission and fusion

3. Ch. 37 Atomic Spectra Rydberg-Ritz empirical formula: the wavelengths of lines in a spectrum of H are given by: Where n1 and n2 are integers and R is the Rydberg constant

5. Bohr Model of Atom Know Bohr’s postulates Derive frequency/wavelength of lines: Where the Rydberg constant, R, is: Derive energy of Bohr orbits: Understand energy level diagrams

6. Ch. 40 Nuclear Physics Radioactivity Derive number of nuclei N remaining after time t: where l is the decay constant and N0 is the number of nuclei at t=0 Derive decay rate R: where R0= lN0= rate of decay at t=0 Average lifetime: Derive half life:

7. Ch. 40 Nuclear size and Shape • Atomic number (Z) and mass number (A) • Radius of nucleus: • Mass and binding energy • Volume is proportional to A, so density constant • Nucleus looks like a liquid drop • For light nuclei N~Z • For heavier nuclei the number of neutrons increases • The extra uncharged neutrons act to stabilize heavy nuclei from repulsive electrostatic forces

8. Nuclear reactions b-Decay g-Decay a-Decay Q value, exothermic & endothermic Understand fission & fusion

10. 1 1 1 1 1 1 1 1 1 1

12. Substitute into given Rydberg formula for ni = 3, nf = 2 Hal = 656.3 nm ni = 4, nf = 2 Hbl = 486.1 nm l = 121.6 nm E = hc/l=10.2 eV Maximum energy which can be absorbed is equal to the electron energy, 12.9 eV Energy states for Hydrogen are En=-E1/n2=-13.6/n2 eV (from memory or use given formula for transition energies ) So energy states available are n=1 -13.6eV, n=2 -3.4eV, n=3 -1.51eV, n=4 -0.85eV, n=5 -0.544eV Transition energies then are (1->2) 10.2eV, (1->3) 12.1eV, (1->4) 12.75eV, (1->5) 13.056eV So we can reach n=4. Longest wavelength will then correspond to the smallest transition from this state, n=4 -> n=3, Back to Rydberg formula l = 1875 nm

13. B6. Describe what is meant by the decay constant of a radioactive nucleus. [2] Describe what is meant by the half-life of a radioactive source. [2] Write down an equation relating the half-life to the decay constant. [2] A radioactive nucleus with decay constant is produced in a nuclear reactor at a rate R0nuclei per second. Assuming that the number of nuclei initially present is zero, show that the number of nuclei N after time t is given by the expression: [8] The rate of production of 22Na in a reactor is 1015 nuclei s–1. Production continues for a period of one year. What is the decay rate of the 22Na sample a further one year after the completion of the irradiation? [6] The half-life of 22Na is 2.6 years. There are 3.15  107 seconds in a year.

14. Describe what is meant by the decay constant of a radioactive nucleus: If radioactive decay is a random process, we expect the number of nuclei that decay after time dt to be proportional to N and t. The constant of proportionality l is called the decay constant. Describe what is meant by the decay constant of a radioactive nucleus The half life t1/2 is defined as the time it takes the number of nuclei and the decay rate to decrease by half 2 Write down an equation relating the half-life to the decay constant 2 2

16. Bohr’s postulates – Bookwork! • Bohr proposed that certain “magical” circular orbits existed, called “stationary states”, which did not radiate, and that electrons could only exist in these states, with radiation occurring when they made the transition from one to the other. • (2) He also postulated that the frequency of the radiation from spectral lines was determined by energy conservation during transitions from one stationary state to the other. i.e. From E=hf, where h is Planck’s constant (3). Trial and error led Bohr to his third postulate, that angular momentum is quantized, specifically that 6 n is the quantum number of the state

17. 2 So angular momentum quantization IS given by standing wave condition 2 4 2 Substituting in n, n-1 for large n 1 In the radial direction then there is no uncertainty in r. momentum is this direction is zero, and also has no uncertainty. So the Bohr model clearly violates the uncertainty principle. 1 2