Chapter 28 Atomic Physics. Plum Pudding Model of the Atom J. J. Thomson’s “Plum Pudding” model...
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Transcript of Chapter 28 Atomic Physics. Plum Pudding Model of the Atom J. J. Thomson’s “Plum Pudding” model...
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Chapter 28
Atomic Physics
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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 Thomson1856 – 1940
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Scattering Experiments
• 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
Ernest Rutherford1871 – 1937
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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
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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
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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
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Emission Spectrum of Hydrogen• The wavelengths of hydrogen’s spectral lines can be
found from
• RH = 1.097 373 2 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 = 656.3 nm
• n = 4, λ = 486.1 nm
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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
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• 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
The Bohr Theory of Hydrogen
Niels Henrik David Bohr1885 – 1962
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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 atomremains constant and classicalmechanics can be used to describethe electron’s motion
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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
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Radii and Energies of Orbits
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Radii and Energies of Orbits
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Radii and Energies of Orbits
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• The radii of the Bohr orbits are quantized
• When n = 1, the orbit has the smallest radius, called the Bohr radius, ao = 0.0529 nm
• A general expression for the radius of any orbit in a hydrogen atom is rn = n2 ao
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Radii and Energies of Orbits
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• 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 = - 13.6 eV / n2
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Energy Level Diagram2
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Energy Level Diagram
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• 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 beused to find the wavelengths of anyspectral lines
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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 hydrogenis 13.6 eV
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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 quantumnumber are said to form a shell, whereas thestates with given values of n and ℓ are saidto form a subshell Arnold Johannes
Wilhelm Sommerfeld1868 – 1951
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Modifications of the Bohr Theory – Elliptical Orbits
Arnold Johannes Wilhelm Sommerfeld
1868 – 1951
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Pieter Zeeman1865 – 1943
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
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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
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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
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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
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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
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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
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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
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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
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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
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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 Pauli1900 – 1958
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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
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Filling Shells
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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 Mendeleyev1834 – 1907
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The Periodic Table
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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
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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
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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
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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
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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
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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 littleenergy for electrons to jump from the partially filled to one of the nearby empty states
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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