Chapter 28:Atomic Physics

1. Homework : Read and understand the lecture note. Chapter 28:Atomic Physics • Emission and absorption spectrum • Atoms emit and absorb light of specific wavelengths. Atoms from • different elements have different wavelengths of emitted and absorbed • light. These specific wave- • lengths are the same for • both emitted and absorbed • light. • In 1885, Balmer found a • formula that described • these wavelengths for • hydrogen atom. AtomicSpectra (n=3,l=656.3 nm) etc.

2. Bohr Theory of Hydrogen (1913) • Basic assumptions of Bohr theory • The electron moves in circular orbits • about the proton under the influence • of the Coulomb force of attraction. • Only certain electron orbits are stable. • These are orbits in which the hydrogen • atom does not emit energy in form of • EM radiation. Hence the total energy • of the atom is constant and classical • mechanics can be used to describe • the electron’s motion. • Radiation is emitted by the hydrogen atom when electron jumps • from a more energetic initial state to a less energetic state. • The frequency f of the radiation emitted in the jump is related to • the change in the atom’s energy and is independent of the • frequency of the electron’s orbital motion: • The size of the allowed electron orbits is determined by a condition • imposed on the electron’s orbital angular momentum: centripetal force

3. Bohr Theory of Hydrogen (1913) • Consequences of Bohr theory • The electrical potential energy of the atom: • The total energy of the atom assuming that the nucleus is at rest: • From Newton’s 2nd law applied to the electron: The negative sign indicates the ele- ctron is bound to the proton • From the 4th assumption and this : radii of allowed electron orbits Bohr’s radius • Energy levels of electron :

4. Bohr Theory of Hydrogen (1913) • Consequences of Bohr theory (cont’d) • Ionization energy : The upper most level corresponds to E=0 and n= and the energy needed to remove the electron completely from the atom (ionization energy = E1). • Ground state : The lowest energy state (n=1) is called the ground state. • Emitted/absorbed photon : From the 3rd postulate, in the transition of the electron from an orbit with principal quantum number ni to another with nf, it emits a photon of frequency f given by:

5. Bohr Theory of Hydrogen (1913) • Consequences of Bohr theory (cont’d) • Wavelengths of emitted/absorbed photons Since lf=c, • Named transitions : - The Lyman series nf=1, ni=2,3,4,… - The Balmer series nf=2, ni=3,4,5,… - The Paschen series nf=3, ni=4,5,6,… • Bohr’s correspondence principle • Quantum mechanics is in agreement with classical physics when • the energy differences between quantized levels are very small.

6. Modification of Bohr Theory • Success of Bohr theory • It explains the Balmer series and other series. • It predicts correctly a value for the Rydberg constant. • It gives an expression for the radius of the atom. • It predicts the energy levels of hydrogen. This theory gives a model of what the atom looks like and how it behaves. With some refinements and modifications, it can be used as a model for other atoms than hydrogen atom. • Extension of Bohr theory to hydrogen-like atoms • A hydrogen-like atom contains only one electron : He-, Li2-, Be3- etc. • To extend the theory to hydrogen-like atoms, replace e2 with Ze2 • where Z is the atomic number of the element.

7. Modification of Bohr Theory • Sommerfeld’s extention of Bohr theory • Sommerfeld extended the Bohr theory to include elliptical orbits. • Sommerfeld model introduced, in addition to the principal quantum • number n, a new quantum number called orbital quantum number l , • where the value of l ranges from 0 to n-1 in integer step. For a given n, l =0,1,…,n-1 : n=2->l =0,1…. • An electron in any one of the allowed energy states of a hydrogen • atom may move in any one of a number oforbits corresponding to • different l. • All states with the same principal quantum • number n are said to form a shell, which is • identified by the letter K,L,M,…corresponding • to n = 1,2,3,… • The letters s, p, d, f, g,… are used to • designate the states with l = 0,1,2,3,4,…

8. Modification of Bohr Theory • Orbital magnetic quantum number ml • Another quantum number was introduced when it was discovered • that the spectral lines of a gas are split into several closely spaced • lines when the gas is placed in a strong magnetic field (Zeeman effect). Orbital magnetic quantum number ml : -l =< ml =< l (2l+1 states) • Spin magnetic quantum number ms • Yet another quantum number was introduced • when it was discovered that the spectral lines • of a gas are actually split into two closely • spaced lines (fine structure) even without a strong magnetic field due to • spinning of electrons. Spin magnetic quantum number ms: ms =-1/2,+1/2 • Number of allowed states with n and l Example : p subshell has 2(2x1+1)=6 possible states 2(2l+1)

9. De Broglie Waves and Hydrogen Atom • Quantization of angular momentum and de Broglie waves • De Broglie found an interpretation of the Bohr’s angular momentum • quantization in terms of his wave theory. • An electron orbit would be stable (allowed) only if it contained an • integral number of electron wavelengths. The de Broglie wavelength of an electron is: This argument strengthened the wave theory of matter.

10. Hydrogen Atom • Quantum numbers Spin Magnetic Quantum Number

11. Electron Clouds • Wave function and electron clouds • The solution of the wave equation yields a wave function Y that depends • on the quantum number n, l, and ml. • The quantity |Y|2 DVP gives a probability of • finding the electron in a small volume DVP • around a point P. • The maximum of the probability coincides • with the Bohr radius. Quantum theory predicts that the electron is not located at a fixed point. Probability per unit length of finding the electron at a distance r from the nucleus for 1s state of hydrogen atom

12. Exclusion Principle and Periodic Table • Pauli exclusion principle (1925) No two electrons in an atom can ever have the same set of values for the set of quantum numbers n, l, ml, and ms. • The Pauli exclusion principle explains the electronic structure of • complex atoms as a succession of filled levels with different quantum • numbers increasing in energy, where the outermost electrons are • primarily responsible for the chemical properties of the element. • General rule for the order that electrons fill a subshell • Once one subshell is filled, the next electron goes into the vacant • subshell that is the lowest in energy. • If the atomwere not in the lowest energy state available, it would • radiate energy until it reached that state. • A subshell is filled when it contains 2(2l+1) electrons. For hydrogen atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2) or (1,0,0,-1/2) 1 electron 1s1

13. Exclusion Principle and Periodic Table • General rule for the order that electrons fill a subshell (cont’d) For helium atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2) and (1,0,0,-1/2) 2 electrons 1s2 For lithium atom at the ground state : two electrons in 1s subshell one electron in 2s subshell 2 electrons 1s2 1s22s1 1 electron 2s1 ( 2p subshell has higher energy)

14. Exclusion Principle and Periodic Table • Periodic table (Mendeleev 1871)

15. Characteristic X-Rays • Characteristic x-rays • The discrete lines in an x-ray spectrum • are called characteristic x-rays. • When an electron beam bombards • a metal to produce x-rays, knocked-off • electrons leave vacancies at the energy • levels at which they were located. • Then these vacancies are filled in by • electrons that were at higher energy • states. In such a transition, a photon • is emitted with an energy corresponding • to the difference in energy between the • initial and the final energy of the electron • that has filled a vacated state.

16. Atomic Transitions • Stimulated absorption process • When an atom is hit by a photon • that carries an energy equal to • the energy difference between • two states, an electron at the • lower energy state can move up • to the higher energy state by • absorbing the incident photon. • This process is called stimulated • absorption process and the higher • energy state is called an excited • state. • When this happens, there is a • probability that the electron that • has moved up to the excited state • goes back to a lower energy state • by emitting a photon (spontaneous • emission).

17. Atomic Transitions • Stimulated absorption process (cont’d) • When an atom is in the excited • state E2 and a photon with energy • hf=E2-E1 is incident on it, the • incoming photon increases the • probability that the excited atom • will return to the ground state by • emitting a second photon with the • same energy (stimulated emission). The emitted photon is exactly in phase with the incident photon.

18. Laser • Population inversion • When an incident photon causes atomic transitions, stimulated • absorption and stimulated emission happen with the equal • probability. • When light is incident on a system of atoms, there is usually a net • absorption. This is because when the system is in thermal equilibrium • there are many more atoms in the ground state than in excited states. • If there are more atoms in excited states than in the ground state • (population inversion), a net emission can result. • A mechanism exists to realize the population inversion: laser.

19. Laser • Laser (light amplification by simulated emission of radiation) • There are three conditions to achieve • laser action: • The system must be in a state of • population inversion. • The excited state of the system must • be metastable state (longer lifetime • than otherwise short liftime). Then • stimulated emission occurs before • spontaneous emission. • The emitted photons must be confined • within the system long enough to allow • them to stimulate further emission from • other excited atoms.