1. Chapter 28
Atomic Physics

2. Importance of Hydrogen Atom
Hydrogen is the simplest atom
The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms
This enables us to understand the periodic table

3. More Reasons the Hydrogen Atom is so Important
The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment
Also for improving our understanding of atomic structure
Much of what we know about the hydrogen atom can be extended to other single-electron ions
For example, He+ and Li2+

4. Early Models of the Atom
J.J. Thomson’s model of the atom
A volume of positive charge
Electrons embedded throughout the volume
A change from Newton’s model of the atom as a tiny, hard, indestructible sphere

5. Early Models of the Atom, 2
Rutherford
Planetary model
Based on results of thin foil experiments
Positive charge is concentrated in the center of the atom, called the nucleus
Electrons orbit the nucleus like planets orbit the sun

6. Difficulties with the Rutherford Model
Atoms emit certain discrete characteristic frequencies of electromagnetic radiation
The Rutherford model is unable to explain this phenomena
Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency
The radius should steadily decrease as this radiation is given off
The electron should eventually spiral into the nucleus
It doesn’t

7. Emission Spectra A gas at low pressure has a voltage applied to it
A gas emits light characteristic of the gas
When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed
Each line has a different wavelength and color
This series of lines is called an emission spectrum

8. Examples of Spectra

9. Emission Spectrum of Hydrogen – Equation
The wavelengths of hydrogen’s spectral lines can be found from
RH is the Rydberg constant
RH = x 107 m-1
n is an integer, n = 1, 2, 3, …
The spectral lines correspond to different values of n

10. Spectral Lines of Hydrogen
The Balmer Series has lines whose wavelengths are given by the preceding equation
Examples of spectral lines
n = 3, λ = nm
n = 4, λ = nm

11. Absorption Spectra
An element can also absorb light at specific wavelengths
An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas
The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum
The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum

12. Applications of Absorption Spectrum
The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere
The various absorption lines can be used to identify elements in the solar atmosphere
Led to the discovery of helium

13. The Bohr Theory of Hydrogen
In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory
His model includes both classical and non-classical ideas
His model included an attempt to explain why the atom was stable

14. Bohr’s Assumptions for Hydrogen
The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction
The Coulomb force produces the centripetal acceleration

15. Bohr’s Assumptions, cont
Only certain electron orbits are stable
These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation
Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion
Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state
The “jump” cannot be treated classically

16. Bohr’s Assumptions, final
The electron’s “jump,” continued
The frequency emitted in the “jump” is related to the change in the atom’s energy
It is generally not the same as 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

17. Mathematics of Bohr’s Assumptions and Results
Electron’s orbital angular momentum
me v r = n ħ where n = 1, 2, 3, …
The total energy of the atom
The energy can also be expressed as

18. Bohr Radius The radii of the Bohr orbits are quantized
This shows that the electron can only exist in certain allowed orbits determined by the integer n
When n = 1, the orbit has the smallest radius, called the Bohr radius, ao
ao = nm

19. Radii and Energy of Orbits
A general expression for the radius of any orbit in a hydrogen atom is
rn = n2 ao
The energy of any orbit is
En = eV/ n2

20. Specific Energy Levels
The lowest energy state is called the ground state
This corresponds to n = 1
Energy is –13.6 eV
The next energy level has an energy of –3.40 eV
The energies can be compiled in an energy level diagram
The ionization energy is the energy needed to completely remove the electron from the atom
The ionization energy for hydrogen is 13.6 eV

21. Energy Level Diagram
The value of RH from Bohr’s analysis is in excellent agreement with the experimental value
A more generalized equation can be used to find the wavelengths of any spectral lines

22. Generalized Equation
For the Balmer series, nf = 2
For the Lyman series, nf = 1
Whenever an transition occurs between a state, ni to another state, nf (where ni > nf), a photon is emitted
The photon has a frequency f = (Ei – Ef)/h and wavelength λ

23. Bohr’s Correspondence Principle
Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small
Similar to having Newtonian Mechanics be a special case of relativistic mechanics when v << c

24. Successes of the Bohr Theory
Explained several features of the hydrogen spectrum
Accounts for Balmer and other series
Predicts a value for RH that agrees with the experimental value
Gives an expression for the radius of the atom
Predicts energy levels of hydrogen
Gives a model of what the atom looks like and how it behaves
Can be extended to “hydrogen-like” atoms
Those with one electron
Ze2 needs to be substituted for e2 in equations
Z is the atomic number of the element

25. Modifications of the Bohr Theory – Elliptical Orbits
Sommerfeld extended the results to include elliptical orbits
Retained the principle quantum number, n
Added the orbital quantum number, ℓ
ℓ ranges from 0 to n-1 in integer steps
All states with the same principle quantum number are said to form a shell
The states with given values of n and ℓ are said to form a subshell

26. Modifications of the Bohr Theory – Zeeman Effect
Another modification was needed to account for the Zeeman effect
The Zeeman effect is the splitting of spectral lines in a strong magnetic field
This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field
A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced
m ℓ can vary from - ℓ to + ℓ in integer steps

27. Modifications of the Bohr Theory – Fine Structure
High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field
This splitting is called fine structure
Another quantum number, ms, called the spin magnetic quantum number, was introduced to explain the fine structure

28. de Broglie Waves
One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation why the restriction occurred
de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths

29. de Broglie Waves in the Hydrogen Atom
In this example, three complete wavelengths are contained in the circumference of the orbit
In general, the circumference must equal some integer number of wavelengths
2  r = n λ n = 1, 2, …

30. de Broglie Waves in the Hydrogen Atom, cont
The expression for the de Broglie wavelength can be included in the circumference calculation
me v r = n 
This is the same quantization of angular momentum that Bohr imposed in his original theory
This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems
Schrödinger’s wave equation was subsequently applied to atomic systems

31. QUICK QUIZ 28.1
In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.

32. QUICK QUIZ 28.1 ANSWER
(b). The circumference of the orbit is n times the de Broglie wavelength (2πr = nλ), so there are three times as many wavelengths in the n = 3 level as in the n = 1 level. Also, by combining Equations 28.4, 28.6 and the defining equation for the de Broglie wavelength (λ = h/mv), one can show that the wavelength in the n = 3 level is three times as long.

33. Quantum Mechanics and the Hydrogen Atom
One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atom
The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments

34. Quantum Number Summary
The values of n can range from 1 to  in integer steps
The values of ℓ can range from 0 to n-1 in integer steps
The values of m ℓ can range from -ℓ to ℓ in integer steps
Also see Table 28.2

35. How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?
QUICK QUIZ 28.2

36. QUICK QUIZ 28.2 ANSWER
The quantum numbers associated with orbital states are n, , and m . For a specified value of n, the allowed values of  range from 0 to n – 1. For each value of , there are (2  + 1) possible values of m. (a) If n = 3, then  = 0, 1, or 2. The number of possible orbital states is then [2(0) + 1] + [2(1) + 1] + [2(2) + 1] = = 9. (b) If n = 4, one additional value of  is allowed ( = 3) so the number of possible orbital states is now 9 + [2(3) + 1] = = 16

37. When the principal quantum number is n = 5, how many different values of (a)  and (b) m are possible?
QUICK QUIZ 28.3

38. QUICK QUIZ 28.3 ANSWER
(a) For n = 5, there are 5 allowed values of , namely  = 0, 1, 2, 3, and (b) Since m ranges from – to + in integer steps, the largest allowed value of  ( = 4 in this case) permits the greatest range of values for m. For n = 5, there are 9 possible values for m: -4, -3, -2, –1, 0, +1, +2, +3, and +4.

39. Spin Magnetic Quantum Number
It is convenient to think of the electron as spinning on its axis
The electron is not physically spinning
There are two directions for the spin
Spin up, ms = ½
Spin down, ms = -½
There is a slight energy difference between the two spins and this accounts for the Zeeman effect

40. Electron Clouds
The graph shows the solution to the wave equation for hydrogen in the ground state
The curve peaks at the Bohr radius
The electron is not confined to a particular orbital distance from the nucleus
The probability of finding the electron at the Bohr radius is a maximum

41. Electron Clouds, cont
The wave function for hydrogen in the ground state is symmetric
The electron can be found in a spherical region surrounding the nucleus
The result is interpreted by viewing the electron as a cloud surrounding the nucleus
The densest regions of the cloud represent the highest probability for finding the electron

42. The Pauli Exclusion Principle
No two electrons in an atom can ever be in the same quantum state
In other words, no two electrons in the same atom can have exactly the same values for n, ℓ, m ℓ, and ms
This explains the electronic structure of complex atoms as a succession of filled energy levels with different quantum numbers

43. The Periodic Table
The outermost electrons are primarily responsible for the chemical properties of the atom
Mendeleev arranged the elements according to their atomic masses and chemical similarities
The electronic configuration of the elements explained by quantum numbers and Pauli’s Exclusion Principle explains the configuration

44. Krypton (atomic number 36) has how many electrons in its next to outer shell (n = 3)? (a) 2 (b) 4 (c) 8 (d) 18
QUICK QUIZ 28.4

45. QUICK QUIZ 28.4 ANSWER
(d). Krypton has a closed configuration consisting of filled n=1, n=2, and n=3 shells as well as filled 4s and 4p subshells. The filled n=3 shell (the next to outer shell in Krypton) has a total of 18 electrons, 2 in the 3s subshell, 6 in the 3p subshell and 10 in the 3d subshell.

46. Characteristic X-Rays
When a metal target is bombarded by high-energy electrons, x-rays are emitted
The x-ray spectrum typically consists of a broad continuous spectrum and a series of sharp lines
The lines are dependent on the metal
The lines are called characteristic x-rays

47. Explanation of Characteristic X-Rays
The details of atomic structure can be used to explain characteristic x-rays
A bombarding electron collides with an electron in the target metal that is in an inner shell
If there is sufficient energy, the electron is removed from the target atom
The vacancy created by the lost electron is filled by an electron falling to the vacancy from a higher energy level
The transition is accompanied by the emission of a photon whose energy is equal to the difference between the two levels

48. Moseley Plot λ is the wavelength of the K line
K is the line that is produced by an electron falling from the L shell to the K shell
From this plot, Moseley was able to determine the Z values of other elements and produce a periodic chart in excellent agreement with the known chemical properties of the elements

49. Atomic Transitions – Energy Levels
An atom may have many possible energy levels
At ordinary temperatures, most of the atoms in a sample are in the ground state
Only photons with energies corresponding to differences between energy levels can be absorbed

50. Atomic Transitions – Stimulated Absorption
The blue dots represent electrons
When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level
These higher levels are called excited states
ΔE = hƒ = E2 – E1
In general, ΔE can be the difference between any two energy levels

51. Atomic Transitions – Spontaneous Emission
Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon
This process is called spontaneous emission

52. Atomic Transitions – Stimulated Emission
An atom is in an excited stated and a photon is incident on it
The incoming photon increases the probability that the excited atom will return to the ground state
There are two emitted photons, the incident one and the emitted one
The emitted photon is in exactly in phase with the incident photon

53. Population Inversion
When light is incident on a system of atoms, both stimulated absorption and stimulated emission are equally probable
Generally, a net absorption occurs since most atoms are in the ground state
If you can cause more atoms to be in excited states, a net emission of photons can result
This situation is called a population inversion

54. Lasers To achieve laser action, three conditions must be met
The system must be in a state of population inversion
The excited state of the system must be a metastable state
Its lifetime must be long compared to the normal lifetime of an excited state
The emitted photons must be confined in the system long enough to allow them to stimulate further emission from other excited atoms
This is achieved by using reflecting mirrors

55. Production of a Laser Beam

56. Laser Beam – He Ne Example
The energy level diagram for Ne
The mixture of helium and neon is confined to a glass tube sealed at the ends by mirrors
A high voltage applied causes electrons to sweep through the tube, producing excited states
When the electron falls to E2 in Ne, a nm photon is emitted

57. Holography
Holography is the production of three-dimensional images of an object
Light from a laser is split at B
One beam reflects off the object and onto a photographic plate
The other beam is diverged by Lens 2 and reflected by the mirrors before striking the film

58. Holography, cont
The two beams form a complex interference pattern on the photographic film
It can be produced only if the phase relationship of the two waves remains constant
This is accomplished by using a laser
The hologram records the intensity of the light and the phase difference between the reference beam and the scattered beam
The image formed has a three-dimensional perspective

59. Energy Bands in Solids
In solids, the discrete energy levels of isolated atoms broaden into allowed energy bands separated by forbidden gaps
The separation and the electron population of the highest bands determine whether the solid is a conductor, an insulator, or a semiconductor

60. Energy Bands, Detail Sodium example
Blue represents energy bands occupied by the sodium electrons when the atoms are in their ground states
Gold represents energy bands that are empty
White represents energy gaps
Electrons can have any energy within the allowed bands
Electrons cannot have energies in the gaps

61. Energy Level Definitions
The valence band is the highest filled band
The conduction band is the next higher empty band
The energy gap has an energy, Eg, equal to the difference in energy between the top of the valence band and the bottom of the conduction band

62. Conductors
When a voltage is applied to a conductor, the electrons accelerate and gain energy
In quantum terms, electron energies increase if there are a high number of unoccupied energy levels for the electron to jump to
For example, it takes very little energy for electrons to jump from the partially filled to one of the nearby empty states

63. Insulators The valence band is completely full of electrons
A large band gap separates the valence and conduction bands
A large amount of energy is needed for an electron to be able to jump from the valence to the conduction band
The minimum required energy is Eg

64. Semiconductors A semiconductor has a small energy gap
Thermally excited electrons have enough energy to cross the band gap
The resistivity of semiconductors decreases with increases in temperature
The white area in the valence band represents holes

65. Semiconductors, cont
Holes are empty states in the valence band created by electrons that have jumped to the conduction band
Some electrons in the valence band move to fill the holes and therefore also carry current
The valence electrons that fill the holes leave behind other holes
It is common to view the conduction process in the valence band as a flow of positive holes toward the negative electrode applied to the semiconductor

66. Current Process in Semiconductors
An external voltage is supplied
Electrons move toward the positive electrode
Holes move toward the negative electrode
There is a symmetrical current process in a semiconductor

67. Doping in Semiconductors
Doping is the adding of impurities to a semiconductor
Generally about 1 impurity atom per 107 semiconductor atoms
Doping results in both the band structure and the resistivity being changed

68. n-type Semiconductors
Donor atoms are doping materials that contain one more electron than the semiconductor material
This creates an essentially free electron with an energy level in the energy gap, just below the conduction band
Only a small amount of thermal energy is needed to cause this electron to move into the conduction band

69. p-type Semiconductors
Acceptor atoms are doping materials that contain one less electron than the semiconductor material
A hole is left where the missing electron would be
The energy level of the hole lies in the energy gap, just above the valence band
An electron from the valence band has enough thermal energy to fill this impurity level, leaving behind a hole in the valence band

70. A p-n Junction
A p-n junction is formed when a p-type semiconductor is joined to an n-type
Three distinct regions exist
A p region
An n region
A depletion region

71. The Depletion Region
Mobile donor electrons from the n side nearest the junction diffuse to the p side, leaving behind immobile positive ions
At the same time, holes from the p side nearest the junction diffuse to the n side and leave behind a region of fixed negative ions
The resulting depletion region is depleted of mobile charge carriers
There is also an electric field in this region that sweeps out mobile charge carriers to keep the region truly depleted

72. Diode Action
The p-n junction has the ability to pass current in only one direction
When the p-side is connected to a positive terminal, the device is forward biased and current flows
When the n-side is connected to the positive terminal, the device is reverse biased and a very small reverse current results

73. Applications of Semiconductor Diodes
Rectifiers
Change AC voltage to DC voltage
A half-wave rectifier allows current to flow during half the AC cycle
A full-wave rectifier rectifies both halves of the AC cycle
Transistors
May be used to amplify small signals
Integrated circuit
A collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon