1. Chapter 28
Atomic Physics

2. Plum Pudding Model of the Atom
J. J. Thomson’s “Plum Pudding” model of the atom:
Electrons embedded throughout the a volume of positive charge
A change from Newton’s model of the atom as a tiny, hard, indestructible sphere
Sir Joseph John Thomson
1856 – 1940

3. Scattering Experiments
Ernest Rutherford
1871 – 1937
The source was a naturally radioactive material that produced alpha particles
Most of the alpha particles passed though the foil
A few deflected from their original paths
Some even reversed their direction of travel

4. Planetary Model of the Atom
Based on results of thin foil scattering experiments, Rutherford’s Planetary model of the atom:
Positive charge is concentrated in the center of the atom, called the nucleus
Electrons orbit the nucleus like planets orbit the sun

5. Difficulties with the Rutherford Model
Atoms emit certain discrete characteristic frequencies of electromagnetic radiation but 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, but it doesn’t

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

7. Emission Spectrum of Hydrogen
The wavelengths of hydrogen’s spectral lines can be found from
RH = x 107 m-1 is the Rydberg constant and n is an integer, n = 3, 4, 5, …
The spectral lines correspond to different values of n
n = 3, λ = nm
n = 4, λ = nm
Johannes Robert Rydberg
1854 – 1919

8. 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
Such 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

9. 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 was an attempt to explain why the atom was stable and included both classical and non-classical ideas
Niels Henrik David Bohr
1885 – 1962

10. The Bohr Theory of Hydrogen
The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction, which produces the centripetal acceleration
Only certain electron orbits are stable
In these orbits electrons do 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

11. The Bohr Theory of Hydrogen
Radiation is emitted when the electrons “jump” (not in a classical sense) from a more energetic initial state to a lower state
The frequency emitted in the “jump” is related to the change in the atom’s energy: Ei – Ef = h ƒ
The size of the allowed electron orbits is determined by a quantization condition imposed on the electron’s orbital angular momentum:
me v r = n ħ where n = 1, 2, 3, …; ħ = h / 2 π

12. Radii and Energies of Orbits

13. Radii and Energies of Orbits

14. Radii and Energies of Orbits
The radii of the Bohr orbits are quantized
When n = 1, the orbit has the smallest radius, called the Bohr radius, ao = nm
A general expression for the radius of any orbit in a hydrogen atom is rn = n2 ao

15. Radii and Energies of Orbits
The lowest energy state (n = 1) is called the ground state, with energy of –13.6 eV
The next energy level (n = 2) has an energy of –3.40 eV
The energies can be compiled in an energy level diagram with the energy of any orbit of En = eV / n2

16. Energy Level Diagram

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

18. Energy Level Diagram
The uppermost level corresponds to E = 0 and n  
The ionization energy: energy needed to completely remove the electron from the atom
The ionization energy for hydrogen
is 13.6 eV

19. Modifications of the Bohr Theory – Elliptical Orbits
Sommerfeld extended the results to include elliptical orbits
Retained the principal quantum number, n, which determines the energy of the allowed states
Added the orbital quantum number, ℓ, ranging from 0 to n-1 in integer steps
All states with the same principal quantum
number are said to form a shell, whereas the
states with given values of n and ℓ are said
to form a subshell
Arnold Johannes Wilhelm Sommerfeld
1868 – 1951

20. Modifications of the Bohr Theory – Elliptical Orbits
Arnold Johannes Wilhelm Sommerfeld
1868 – 1951

21. Modifications of the Bohr Theory – Zeeman Effect
Another modification was needed to account for the Zeeman effect: splitting of spectral lines in a strong magnetic field, indicating 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
Pieter Zeeman
1865 – 1943

22. 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

23. 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

24. 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 doublet in some lines
A classical description of electron spin is incorrect: the electron cannot be located precisely in space, thus it cannot be considered to be a spinning solid object

25. de Broglie Waves in the Hydrogen Atom
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

26. de Broglie Waves in the Hydrogen Atom
This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems
By applying wave theory to the electrons in an atom, de Broglie was able to explain the appearance of integers in Bohr’s equations as a natural consequence of standing wave patterns

27. Quantum Mechanics and the Hydrogen Atom
Schrödinger’s wave equation was subsequently applied to hydrogen and other atomic systems - one of the first great achievements of quantum mechanics
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

28. 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

29. Electron Clouds
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

30. The Pauli Exclusion Principle
No two electrons in an atom or in the same location can ever have the same set of values of the quantum numbers 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
Wolfgang Ernst Pauli
1900 – 1958

31. Filling Shells
As a general rule, the order that electrons fill an atom’s subshell is:
1) Once one subshell is filled, the next electron goes into the vacant subshell that is lowest in energy
2) Otherwise, the electron would radiate energy until it reached the subshell with the lowest energy
3) A subshell is filled when it holds 2(2ℓ+1) electrons

32. Filling Shells

33. Dmitriy Ivanovich Mendeleyev
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 is explained by quantum numbers and Pauli’s Exclusion Principle
Dmitriy Ivanovich Mendeleyev
1834 – 1907

35. The Periodic Table

36. 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

37. Explanation of Characteristic X-Rays
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

38. 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

39. Energy Bands in Solids 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, and white represents energy gaps
Electrons can have any energy within the allowed bands and cannot have energies in the gaps

40. 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

41. 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

42. 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

43. 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 light-color area in the valence band represents holes – empty states in the valence band created by electrons that have jumped to the conduction band

44. Semiconductors
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

45. 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

46. 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

47. 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

48. 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

49. 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, and a depletion region
Mobile donor electrons from the n side nearest the junction diffuse to the p side, leaving behind immobile positive ions

50. A p-n Junction
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

51. 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

52. Applications of Semiconductor Devices
Rectifiers: change AC voltage to DC voltage
Transistors: may be used to amplify small signals
Integrated circuits: a collection of interconnected transistors, diodes, resistors and capacitors fabricated on a single piece of silicon