Pc Chapter 42
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Chapter 42 Atomic Physics
Importance of the Hydrogen Atom
The hydrogen atom is the only atomic system that can be solved exactly Much of what was learned in the twentieth century about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+
More Reasons the Hydrogen Atom is Important
The hydrogen atom proved to be an ideal system for performing precision tests of theory against experiment
Also for improving our understanding of atomic structure
The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms
This allows us to understand the periodic table
Final Reasons for the Importance of the Hydrogen Atom
The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids The full mathematical solution of the Schrödinger equation applied to the hydrogen atom gives a complete and beautiful description of the atom’s properties
Atomic Spectra
A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge Observation and analysis of these spectral lines is called emission spectroscopy The simplest line spectrum is that for atomic hydrogen
Uniqueness of Atomic Spectra
Other atoms exhibit completely different line spectra Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples
Emission Spectra Examples
Absorption Spectroscopy
An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source
Absorption Spectrum, Example
A practical example is the continuous spectrum emitted by the sun The radiation must pass through the cooler gases of the solar atmosphere and through the Earth’s atmosphere
Balmer Series
In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen
Hα is red, λ = 656.3 nm
Hβ is green, λ = 486.1 nm
Hγ is blue, λ = 434.1 nm
Hδ is violet, λ = 410.2 nm
Emission Spectrum of Hydrogen – Equation
The wavelengths of hydrogen’s spectral lines can be found from 1 1 1 RH 2 2 λ 2 n
RH is the Rydberg constant
RH = 1.097 373 2 x 107 m-1
n is an integer, n = 3, 4, 5,… The spectral lines correspond to different values of n
Other Hydrogen Series
Other series were also discovered and their wavelengths can be calculated 1 1 Lyman series: RH 1 2 n 2, 3 , 4 ,K λ
Paschen series: Brackett series:
n
1 1 1 RH 2 2 n 4 , 5 , 6 ,K λ 3 n 1 1 1 RH 2 2 n 5 , 6 , 7 ,K λ 4 n
Joseph John Thomson
1856 – 1940 Received Nobel Prize in 1906 Usually considered the discoverer of the electron Worked with the deflection of cathode rays in an electric field
Early Models of the Atom
J. J. Thomson established the charge to mass ratio for electrons His model of the atom
A volume of positive charge Electrons embedded throughout the volume
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
Rutherford’s Thin Foil Experiment
Experiments done in 1911 A beam of positively charged alpha particles hit and are scattered from a thin foil target Large deflections could not be explained by Thomson’s model
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
It 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
Niels Bohr
1885 – 1962 An active participant in the early development of quantum mechanics Headed the Institute for Advanced Studies in Copenhagen Awarded the 1922 Nobel Prize in physics
For structure of atoms and the radiation emanating from them
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 He applied Planck’s ideas of quantized energy levels to orbiting electrons
Bohr’s Theory, cont.
This model is now considered obsolete It has been replaced by a probabilistic quantum-mechanical theory The model can still be used to develop ideas of energy quantization and angular momentum quantization as applied to atomic-sized systems
Bohr’s Assumptions for Hydrogen, 1
The electron moves in circular orbits around the proton under the electric force of attraction
The Coulomb force produces the centripetal acceleration
Bohr’s Assumptions, 2
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 This representation claims the centripetally accelerated electron does not emit energy and eventually spirals into the nucleus
Bohr’s Assumptions, 3
Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit
The transition cannot be treated classically The frequency emitted in the transition is related to the change in the atom’s energy The frequency is independent of frequency of the electron’s orbital motion The frequency of the emitted radiation is given by Ei – Ef = hƒ
Bohr’s Assumptions, 4
The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of
Mathematics of Bohr’s Assumptions and Results
Electron’s orbital angular momentum mevr = nħ where n = 1, 2, 3,…
The total energy of the atom is 2 1 e E K U mev 2 ke 2 r
The total energy can also be expressed as k ee 2 E 2r
Note, the total energy is negative, indicating a bound electron-proton system
Bohr Radius
The radii of the Bohr orbits are quantized
n 2h2 rn n 1, 2, 3 ,K 2 me kee This shows that the radii of the allowed orbits have discrete values—they are quantized
When n = 1, the orbit has the smallest radius, called the Bohr radius, ao ao = 0.0529 nm
Radii and Energy of Orbits
A general expression for the radius of any orbit in a hydrogen atom is
rn = n2ao
The energy of any orbit is k ee 2 1 En 2 n 1, 2,3, K 2ao n This becomes En = - 13.606 eV/ n2
Active Figure 42.8
(SLIDESHOW MODE ONLY)
Specific Energy Levels
Only energies satisfying the previous equation are allowed The lowest energy state is called the ground state
This corresponds to n = 1 with E = –13.606 eV
The ionization energy is the energy needed to completely remove the electron from the ground state in the atom
The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram
Quantum numbers are given on the left and energies on the right The uppermost level, E = 0, represents the state for which the electron is removed from the atom
Frequency of Emitted Photons
The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is E i Ef k e e 2 1 1 ƒ 2 2 h 2ao h nf ni
It is convenient to look at the wavelength instead
Wavelength of Emitted Photons
The wavelengths are found by 1 1 ƒ k ee 2 1 1 1 2 2 RH 2 2 λ c 2ao hc nf ni nf ni
The value of RH from Bohr’s analysis is in excellent agreement with the experimental value
Extension to Other Atoms
Bohr extended his model for hydrogen to other elements in which all but one electron had been removed ao rn n Z k ee 2 Z 2 En 2 n 1, 2, 3 ,K 2ao n 2
Z is the atomic number of the element
Difficulties with the Bohr Model
Improved spectroscopic techniques found that many of the spectral lines of hydrogen were not single lines
Each “line” was actually a group of lines spaced very close together
Certain single spectral lines split into three closely spaced lines when the atoms were placed in a magnetic field
Bohr’s Correspondence Principle
Bohr’s correspondence principle states that quantum physics agrees with classical physics when the differences between quantized levels become vanishingly small
Similar to having Newtonian mechanics be a special case of relativistic mechanics when v << c
The Quantum Model of the Hydrogen Atom
The potential energy function for the hydrogen atom is
e2 U (r ) k e r ke is the Coulomb constant r is the radial distance from the proton to the electron
The proton is situated at r = 0
Quantum Model, cont.
The formal procedure to solve the hydrogen atom is to substitute U(r) into the Schrödinger equation and find the appropriate solutions to the equations Because it is a three-dimensional problem, it is easier to solve if the rectangular coordinates are converted to spherical polar coordinates
Quantum Model, final
ψ(x, y, z) is converted to ψ(r, θ, φ) Then, the space variables can be separated: ψ(r, θ, φ) = R(r), ƒ(θ), g(φ) When the full set of boundary conditions are applied, we are led to three different quantum numbers for each allowed state
Quantum Numbers, General
The three different quantum numbers are restricted to integer values They correspond to three degrees of freedom
Three space dimensions
Principal Quantum Number
The first quantum number is associated with the radial function R(r)
It is called the principal quantum number It is symbolized by n
The potential energy function depends only on the radial coordinate r The energies of the allowed states in the hydrogen atom are the same En values found from the Bohr theory
Orbital and Orbital Magnetic Quantum Numbers
The orbital quantum number is symbolized by ℓ
It is associated with the orbital angular momentum of the electron It is an integer
The orbital magnetic quantum number is symbolized by mℓ
It is also associated with the angular orbital momentum of the electron and is an integer
Quantum Numbers, Summary of Allowed Values
The values of n can range from 1 to ∞ The values of ℓ can range from 0 to n - 1 The values of mℓ can range from –ℓ to ℓ Example:
If n = 1, then only ℓ = 0 and mℓ = 0 are permitted
If n = 2, then ℓ = 0 or 1
If ℓ = 0 then mℓ = 0
If ℓ = 1 then mℓ may be –1, 0, or 1
Quantum Numbers, Summary Table
Shells
Historically, all states having the same principle quantum number are said to form a shell
Shells are identified by letters K, L, M,…
All states having the same values of n and ℓ are said to form a subshell
The letters s, p, d, f, g, h, .. are used to designate the subshells for which ℓ = 0, 1, 2, 3,…
Shell and Subshell Notation, Summary Table
Wave Functions for Hydrogen
The simplest wave function for hydrogen is the one that describes the 1s state and is designated ψ1s(r) ψ1s (r )
1 3 o
πa
e r ao
As ψ1s(r) approaches zero, r approaches ∞ and is normalized as presented ψ1s(r) is also spherically symmetric
This symmetry exists for all s states
Probability Density
The probability density for the 1s state is
ψ1s
2
1 2r ao e 3 πao
The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr
Radial Probability Density
A spherical shell of radius r and thickness dr has a volume of 4πr2 dr The radial probability function is P(r) = 4πr2 |ψ|2
P(r) for 1s State of Hydrogen
The radial probability density function for the hydrogen atom in its ground state is 4r 2 2r ao P1s (r ) 3 e ao
The peak indicates the most probable location The peak occurs at the Bohr radius
P(r) for 1s State of Hydrogen, cont.
The average value of r for the ground state of hydrogen is 3/2 ao
The graph shows asymmetry, with much more area to the right of the peak
According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory
Electron Clouds
The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud This shows the probability density as a function of position in the xy plane The darkest area, r = ao, corresponds to the most probable region
Wave Function of the 2s state
The next-simplest wave function for the hydrogen atom is for the 2s state
n = 2; ℓ = 0
The wave function is 1 ψ 2 s (r ) 4 2π
3
1 2 r r 2ao 2 e ao ao
ψ 2s depends only on r and is spherically symmetric
Comparison of 1s and 2s States
The plot of the radial probability density for the 2s state has two peaks The highest value of P corresponds to the most probable value
In this case, r ≈ 5ao
Active Figure 42.13
(SLIDESHOW MODE ONLY)
Physical Interpretation of ℓ
The magnitude of the angular momentum of an electron moving in a circle of radius r is L = mevr The direction of L is perpendicular to the plane of the circle In the Bohr model, the angular momentum of the electron is restricted to multiples of
Physical Interpretation of ℓ, cont.
According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum:
L l l 1 h l 0 , 1, 2,K n 1
L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system
Physical Interpretation of mℓ
The atom possesses an orbital angular momentum There is a sense of rotation of the electron around the nucleus, so that a magnetic moment is present due to this angular momentum There are distinct directions allowed for the magnetic moment vector µ with respect to the magnetic field vector B
Physical Interpretation of mℓ, 2
Because the magnetic moment µ of the atom can be related to the angular momentum vector, L, the discrete direction of µ translates into the fact that the direction of L is quantized Therefore, Lz, the projection of L along the z axis, can have only discrete values
Physical Interpretation of mℓ, 3
The orbital magnetic quantum number mℓ specifies the allowed values of the z component of orbital angular momentum Lz = mℓ The quantization of the possible orientations of L with respect to an external magnetic field is often referred to as space quantization
Physical Interpretation of mℓ, 4
L does not point in a specific direction
Even though its z-component is fixed Knowing all the components is inconsistent with the uncertainty principle
Imagine that L must lie anywhere on the surface of a cone that makes an angle θ with the z axis
Physical Interpretation of mℓ, final
θ is also quantized Its values are specified through Lz ml cos θ L l l 1 mℓ is never greater than ℓ, therefore θ can never be zero
Zeeman Effect
The Zeeman effect is the splitting of spectral lines in a strong magnetic field In this case the upper level, with ℓ = 1, splits into three different levels corresponding to the three different directions of µ
Spin Quantum Number ms
Electron spin does not come from the Schrödinger equation Additional quantum states can be explained by requiring a fourth quantum number for each state This fourth quantum number is the spin magnetic quantum number ms
Electron Spins
Only two directions exist for electron spins The electron can have spin up (a) or spin down (b) In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases
Electron Spins, cont.
The concept of a spinning electron is conceptually useful The electron is a point particle, without any spatial extent
Therefore the electron cannot be considered to be actually spinning
The experimental evidence supports the electron having some intrinsic angular momentum that can be described by ms Dirac showed this results from the relativistic properties of the electron
Spin Angular Momentum
The total angular momentum of a particular electron state contains both an orbital contribution L and a spin contribution S Electron spin can be described by a single quantum number s, whose value can only be s=½ The spin angular momentum of the electron never changes
Spin Angular Momentum, cont
The magnitude of the spin angular momentum is 3 S s(s 1)h h 2
The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number ms = ± ½
ms = + ½ corresponds to the spin up case ms = - ½ corresponds to the spin down case
Spin Angular Momentum, final
The z component of spin angular momentum is Sz = ms = ± ½
Spin angular moment S is quantized
Spin Magnetic Moment
The spin magnetic moment µspin is related to the spin angular momentum by e μspin
me
S
The z component of the spin magnetic moment can have values μspin , z
eh 2me
Wolfgang Pauli
1900 – 1958 Important review article on relativity
At age 21
Discovery of the exclusion principle Explanation of the connection between particle spin and statistics Relativistic quantum electrodynamics Neutrino hypothesis Hypotheses of nuclear spin
The Exclusion Principle
The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure The exclusion principle states that no two electrons can ever be in the same quantum state
Therefore, no two electrons in the same atom can have the same set of quantum numbers
Filling Subshells
Once a subshell is filled, the next electron goes into the lowest-energy vacant state
If the atom were not in the lowest-energy state available to it, it would radiate energy until it reached this state
Orbitals
An orbital is defined as the atomic state characterized by the quantum numbers n, ℓ and mℓ
From the exclusion principle, it can be seen that only two electrons can be present in any orbital
One electron will have spin up and one spin down
Allowed Quantum States, Example
In general, each shell can accommodate up to 2n2 electrons
Hund’s Rule
Hund’s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins
Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled
Periodic Table
Dmitri Mendeleev made an early attempt at finding some order among the chemical elements He arranged the elements according to their atomic masses and chemical similarities The first table contained many blank spaces and he stated that the gaps were there only because the elements had not yet been discovered
Periodic Table, cont.
By noting the columns in which some missing elements should be located, he was able to make rough predictions about their chemical properties Within 20 years of the predictions, most of the elements were discovered The elements in the periodic table are arranged so that all those in a column have similar chemical properties
Periodic Table, Explained
The chemical behavior of an element depends on the outermost shell that contains electrons For example, the inert gases (last column) have filled subshells and a wide energy gap occurs between the filled shell and the next available shell
Hydrogen Energy Level Diagram Revisited
The allowed values of ℓ are separated Transitions in which ℓ does not change are very unlikely to occur and are called forbidden transitions
Such transitions actually can occur, but their probability is very low compared to allowed transitions
Selection Rules
The selection rules for allowed transitions are
Δℓ = ±1 Δmℓ = 0, ±1
The angular momentum of the atom-photon system must be conserved Therefore, the photon involved in the process must carry angular momentum
The photon has angular momentum equivalent to that of a particle with spin 1 A photon has energy, linear momentum and angular momentum
Multielectron Atoms
For multielectron atoms, the positive nuclear charge Ze is largely shielded by the negative charge of the inner shell electrons The outer electrons interact with a net charge that is smaller than the nuclear charge 2 13.6 Zeff Allowed energies are En eV 2 n
X-Ray Spectra
These x-rays are a result of the slowing down of high energy electrons as they strike a metal target The kinetic energy lost can be anywhere from 0 to all of the kinetic energy of the electron The continuous spectrum is called bremsstrahlung, the German word for “braking radiation”
X-Ray Spectra, cont.
The discrete lines are called characteristic x-rays These are created when
A bombarding electron collides with a target atom The electron removes an inner-shell electron from orbit An electron from a higher orbit drops down to fill the vacancy
X-Ray Spectra, final
The photon emitted during this transition has an energy equal to the energy difference between the levels Typically, the energy is greater than 1000 eV The emitted photons have wavelengths in the range of 0.01 nm to 1 nm
Moseley Plot
Henry G. J. Moseley plotted the values of atoms as shown λ is the wavelength of the Kα line of each element
The Kα line refers to the photon emitted when an electron falls from the L to the K shell
From this plot, Moseley developed a periodic table in agreement with the one based on chemical properties
Stimulated Absorption
When a photon has energy hƒ equal to the difference in energy levels, it can be absorbed by the atom This is called stimulated absorption because the photon stimulates the atom to make the upward transition
Active Figure 42.24
(SLIDESHOW MODE ONLY)
Spontaneous Emission
Once an atom is in an excited state, the excited atom can make a transition to a lower energy level Because this process happens naturally, it is known as spontaneous emission
Stimulated Emission
In addition to spontaneous emission, stimulated emission may also occur Stimulated emission may occur when the excited state is a metastable state
Stimulated Emission, cont.
A metastable state is a state whose lifetime is much longer than the typical 10-8 s An incident photon can cause the atom to return to the ground state without being absorbed Therefore, you have two photons with identical energy, the emitted photon and the incident photon
They both are in phase and travel in the same direction
Lasers – Properties of Laser Light
Laser light is coherent
The individual rays in a laser beam maintain a fixed phase relationship with each other There is no destructive interference
Laser light is monochromatic
The light has a very narrow range of wavelengths
Lasers – Properties of Laser Light, cont.
Laser light has a small angle of divergence
The beam spreads out very little, even over long distances
Lasers – Operation
It is equally probable that an incident photon would cause atomic transitions upward or downward
Stimulated absorption or stimulated emission
If a situation can be caused where there are more electrons in excited states than in the ground state, a net emission of photons can result
This condition is called population inversion
Lasers – Operation, cont.
The photons can stimulate other atoms to emit photons in a chain of similar processes The many photons produced in this manner are the source of the intense, coherent light in a laser
Conditions for Build-Up of Photons
The system must be in a state of population inversion The excited state of the system must be a metastable state
In this case, the population inversion can be established and stimulated emission is likely to occur before spontaneous emission
The emitted photons must be confined in the system long enough to enable them to stimulate further emission
This is achieved by using reflecting mirrors
Laser Design – Schematic
The tube contains the atoms that are the active medium An external source of energy pumps the atoms to excited states The mirrors confine the photons to the tube
Mirror 2 is only partially reflective
Energy-Level Diagram for Neon in a Helium-Neon Laser
The atoms emit 632.8-nm photons through stimulated emission The transition is E3* to E2
* indicates a metastable state
Laser Applications
Applications include:
Medical and surgical procedures Precision surveying and length measurements Precision cutting of metals and other materials Telephone communications
Importance of the Hydrogen Atom
The hydrogen atom is the only atomic system that can be solved exactly Much of what was learned in the twentieth century about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+
More Reasons the Hydrogen Atom is Important
The hydrogen atom proved to be an ideal system for performing precision tests of theory against experiment
Also for improving our understanding of atomic structure
The quantum numbers that are used to characterize the allowed states of hydrogen can also be used to investigate more complex atoms
This allows us to understand the periodic table
Final Reasons for the Importance of the Hydrogen Atom
The basic ideas about atomic structure must be well understood before we attempt to deal with the complexities of molecular structures and the electronic structure of solids The full mathematical solution of the Schrödinger equation applied to the hydrogen atom gives a complete and beautiful description of the atom’s properties
Atomic Spectra
A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge Observation and analysis of these spectral lines is called emission spectroscopy The simplest line spectrum is that for atomic hydrogen
Uniqueness of Atomic Spectra
Other atoms exhibit completely different line spectra Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples
Emission Spectra Examples
Absorption Spectroscopy
An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source
Absorption Spectrum, Example
A practical example is the continuous spectrum emitted by the sun The radiation must pass through the cooler gases of the solar atmosphere and through the Earth’s atmosphere
Balmer Series
In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen
Hα is red, λ = 656.3 nm
Hβ is green, λ = 486.1 nm
Hγ is blue, λ = 434.1 nm
Hδ is violet, λ = 410.2 nm
Emission Spectrum of Hydrogen – Equation
The wavelengths of hydrogen’s spectral lines can be found from 1 1 1 RH 2 2 λ 2 n
RH is the Rydberg constant
RH = 1.097 373 2 x 107 m-1
n is an integer, n = 3, 4, 5,… The spectral lines correspond to different values of n
Other Hydrogen Series
Other series were also discovered and their wavelengths can be calculated 1 1 Lyman series: RH 1 2 n 2, 3 , 4 ,K λ
Paschen series: Brackett series:
n
1 1 1 RH 2 2 n 4 , 5 , 6 ,K λ 3 n 1 1 1 RH 2 2 n 5 , 6 , 7 ,K λ 4 n
Joseph John Thomson
1856 – 1940 Received Nobel Prize in 1906 Usually considered the discoverer of the electron Worked with the deflection of cathode rays in an electric field
Early Models of the Atom
J. J. Thomson established the charge to mass ratio for electrons His model of the atom
A volume of positive charge Electrons embedded throughout the volume
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
Rutherford’s Thin Foil Experiment
Experiments done in 1911 A beam of positively charged alpha particles hit and are scattered from a thin foil target Large deflections could not be explained by Thomson’s model
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
It 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
Niels Bohr
1885 – 1962 An active participant in the early development of quantum mechanics Headed the Institute for Advanced Studies in Copenhagen Awarded the 1922 Nobel Prize in physics
For structure of atoms and the radiation emanating from them
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 He applied Planck’s ideas of quantized energy levels to orbiting electrons
Bohr’s Theory, cont.
This model is now considered obsolete It has been replaced by a probabilistic quantum-mechanical theory The model can still be used to develop ideas of energy quantization and angular momentum quantization as applied to atomic-sized systems
Bohr’s Assumptions for Hydrogen, 1
The electron moves in circular orbits around the proton under the electric force of attraction
The Coulomb force produces the centripetal acceleration
Bohr’s Assumptions, 2
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 This representation claims the centripetally accelerated electron does not emit energy and eventually spirals into the nucleus
Bohr’s Assumptions, 3
Radiation is emitted by the atom when the electron makes a transition from a more energetic initial state to a lower-energy orbit
The transition cannot be treated classically The frequency emitted in the transition is related to the change in the atom’s energy The frequency is independent of frequency of the electron’s orbital motion The frequency of the emitted radiation is given by Ei – Ef = hƒ
Bohr’s Assumptions, 4
The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of
Mathematics of Bohr’s Assumptions and Results
Electron’s orbital angular momentum mevr = nħ where n = 1, 2, 3,…
The total energy of the atom is 2 1 e E K U mev 2 ke 2 r
The total energy can also be expressed as k ee 2 E 2r
Note, the total energy is negative, indicating a bound electron-proton system
Bohr Radius
The radii of the Bohr orbits are quantized
n 2h2 rn n 1, 2, 3 ,K 2 me kee This shows that the radii of the allowed orbits have discrete values—they are quantized
When n = 1, the orbit has the smallest radius, called the Bohr radius, ao ao = 0.0529 nm
Radii and Energy of Orbits
A general expression for the radius of any orbit in a hydrogen atom is
rn = n2ao
The energy of any orbit is k ee 2 1 En 2 n 1, 2,3, K 2ao n This becomes En = - 13.606 eV/ n2
Active Figure 42.8
(SLIDESHOW MODE ONLY)
Specific Energy Levels
Only energies satisfying the previous equation are allowed The lowest energy state is called the ground state
This corresponds to n = 1 with E = –13.606 eV
The ionization energy is the energy needed to completely remove the electron from the ground state in the atom
The ionization energy for hydrogen is 13.6 eV
Energy Level Diagram
Quantum numbers are given on the left and energies on the right The uppermost level, E = 0, represents the state for which the electron is removed from the atom
Frequency of Emitted Photons
The frequency of the photon emitted when the electron makes a transition from an outer orbit to an inner orbit is E i Ef k e e 2 1 1 ƒ 2 2 h 2ao h nf ni
It is convenient to look at the wavelength instead
Wavelength of Emitted Photons
The wavelengths are found by 1 1 ƒ k ee 2 1 1 1 2 2 RH 2 2 λ c 2ao hc nf ni nf ni
The value of RH from Bohr’s analysis is in excellent agreement with the experimental value
Extension to Other Atoms
Bohr extended his model for hydrogen to other elements in which all but one electron had been removed ao rn n Z k ee 2 Z 2 En 2 n 1, 2, 3 ,K 2ao n 2
Z is the atomic number of the element
Difficulties with the Bohr Model
Improved spectroscopic techniques found that many of the spectral lines of hydrogen were not single lines
Each “line” was actually a group of lines spaced very close together
Certain single spectral lines split into three closely spaced lines when the atoms were placed in a magnetic field
Bohr’s Correspondence Principle
Bohr’s correspondence principle states that quantum physics agrees with classical physics when the differences between quantized levels become vanishingly small
Similar to having Newtonian mechanics be a special case of relativistic mechanics when v << c
The Quantum Model of the Hydrogen Atom
The potential energy function for the hydrogen atom is
e2 U (r ) k e r ke is the Coulomb constant r is the radial distance from the proton to the electron
The proton is situated at r = 0
Quantum Model, cont.
The formal procedure to solve the hydrogen atom is to substitute U(r) into the Schrödinger equation and find the appropriate solutions to the equations Because it is a three-dimensional problem, it is easier to solve if the rectangular coordinates are converted to spherical polar coordinates
Quantum Model, final
ψ(x, y, z) is converted to ψ(r, θ, φ) Then, the space variables can be separated: ψ(r, θ, φ) = R(r), ƒ(θ), g(φ) When the full set of boundary conditions are applied, we are led to three different quantum numbers for each allowed state
Quantum Numbers, General
The three different quantum numbers are restricted to integer values They correspond to three degrees of freedom
Three space dimensions
Principal Quantum Number
The first quantum number is associated with the radial function R(r)
It is called the principal quantum number It is symbolized by n
The potential energy function depends only on the radial coordinate r The energies of the allowed states in the hydrogen atom are the same En values found from the Bohr theory
Orbital and Orbital Magnetic Quantum Numbers
The orbital quantum number is symbolized by ℓ
It is associated with the orbital angular momentum of the electron It is an integer
The orbital magnetic quantum number is symbolized by mℓ
It is also associated with the angular orbital momentum of the electron and is an integer
Quantum Numbers, Summary of Allowed Values
The values of n can range from 1 to ∞ The values of ℓ can range from 0 to n - 1 The values of mℓ can range from –ℓ to ℓ Example:
If n = 1, then only ℓ = 0 and mℓ = 0 are permitted
If n = 2, then ℓ = 0 or 1
If ℓ = 0 then mℓ = 0
If ℓ = 1 then mℓ may be –1, 0, or 1
Quantum Numbers, Summary Table
Shells
Historically, all states having the same principle quantum number are said to form a shell
Shells are identified by letters K, L, M,…
All states having the same values of n and ℓ are said to form a subshell
The letters s, p, d, f, g, h, .. are used to designate the subshells for which ℓ = 0, 1, 2, 3,…
Shell and Subshell Notation, Summary Table
Wave Functions for Hydrogen
The simplest wave function for hydrogen is the one that describes the 1s state and is designated ψ1s(r) ψ1s (r )
1 3 o
πa
e r ao
As ψ1s(r) approaches zero, r approaches ∞ and is normalized as presented ψ1s(r) is also spherically symmetric
This symmetry exists for all s states
Probability Density
The probability density for the 1s state is
ψ1s
2
1 2r ao e 3 πao
The radial probability density function P(r) is the probability per unit radial length of finding the electron in a spherical shell of radius r and thickness dr
Radial Probability Density
A spherical shell of radius r and thickness dr has a volume of 4πr2 dr The radial probability function is P(r) = 4πr2 |ψ|2
P(r) for 1s State of Hydrogen
The radial probability density function for the hydrogen atom in its ground state is 4r 2 2r ao P1s (r ) 3 e ao
The peak indicates the most probable location The peak occurs at the Bohr radius
P(r) for 1s State of Hydrogen, cont.
The average value of r for the ground state of hydrogen is 3/2 ao
The graph shows asymmetry, with much more area to the right of the peak
According to quantum mechanics, the atom has no sharply defined boundary as suggested by the Bohr theory
Electron Clouds
The charge of the electron is extended throughout a diffuse region of space, commonly called an electron cloud This shows the probability density as a function of position in the xy plane The darkest area, r = ao, corresponds to the most probable region
Wave Function of the 2s state
The next-simplest wave function for the hydrogen atom is for the 2s state
n = 2; ℓ = 0
The wave function is 1 ψ 2 s (r ) 4 2π
3
1 2 r r 2ao 2 e ao ao
ψ 2s depends only on r and is spherically symmetric
Comparison of 1s and 2s States
The plot of the radial probability density for the 2s state has two peaks The highest value of P corresponds to the most probable value
In this case, r ≈ 5ao
Active Figure 42.13
(SLIDESHOW MODE ONLY)
Physical Interpretation of ℓ
The magnitude of the angular momentum of an electron moving in a circle of radius r is L = mevr The direction of L is perpendicular to the plane of the circle In the Bohr model, the angular momentum of the electron is restricted to multiples of
Physical Interpretation of ℓ, cont.
According to quantum mechanics, an atom in a state whose principle quantum number is n can take on the following discrete values of the magnitude of the orbital angular momentum:
L l l 1 h l 0 , 1, 2,K n 1
L can equal zero, which causes great difficulty when attempting to apply classical mechanics to this system
Physical Interpretation of mℓ
The atom possesses an orbital angular momentum There is a sense of rotation of the electron around the nucleus, so that a magnetic moment is present due to this angular momentum There are distinct directions allowed for the magnetic moment vector µ with respect to the magnetic field vector B
Physical Interpretation of mℓ, 2
Because the magnetic moment µ of the atom can be related to the angular momentum vector, L, the discrete direction of µ translates into the fact that the direction of L is quantized Therefore, Lz, the projection of L along the z axis, can have only discrete values
Physical Interpretation of mℓ, 3
The orbital magnetic quantum number mℓ specifies the allowed values of the z component of orbital angular momentum Lz = mℓ The quantization of the possible orientations of L with respect to an external magnetic field is often referred to as space quantization
Physical Interpretation of mℓ, 4
L does not point in a specific direction
Even though its z-component is fixed Knowing all the components is inconsistent with the uncertainty principle
Imagine that L must lie anywhere on the surface of a cone that makes an angle θ with the z axis
Physical Interpretation of mℓ, final
θ is also quantized Its values are specified through Lz ml cos θ L l l 1 mℓ is never greater than ℓ, therefore θ can never be zero
Zeeman Effect
The Zeeman effect is the splitting of spectral lines in a strong magnetic field In this case the upper level, with ℓ = 1, splits into three different levels corresponding to the three different directions of µ
Spin Quantum Number ms
Electron spin does not come from the Schrödinger equation Additional quantum states can be explained by requiring a fourth quantum number for each state This fourth quantum number is the spin magnetic quantum number ms
Electron Spins
Only two directions exist for electron spins The electron can have spin up (a) or spin down (b) In the presence of a magnetic field, the energy of the electron is slightly different for the two spin directions and this produces doublets in spectra of certain gases
Electron Spins, cont.
The concept of a spinning electron is conceptually useful The electron is a point particle, without any spatial extent
Therefore the electron cannot be considered to be actually spinning
The experimental evidence supports the electron having some intrinsic angular momentum that can be described by ms Dirac showed this results from the relativistic properties of the electron
Spin Angular Momentum
The total angular momentum of a particular electron state contains both an orbital contribution L and a spin contribution S Electron spin can be described by a single quantum number s, whose value can only be s=½ The spin angular momentum of the electron never changes
Spin Angular Momentum, cont
The magnitude of the spin angular momentum is 3 S s(s 1)h h 2
The spin angular momentum can have two orientations relative to a z axis, specified by the spin quantum number ms = ± ½
ms = + ½ corresponds to the spin up case ms = - ½ corresponds to the spin down case
Spin Angular Momentum, final
The z component of spin angular momentum is Sz = ms = ± ½
Spin angular moment S is quantized
Spin Magnetic Moment
The spin magnetic moment µspin is related to the spin angular momentum by e μspin
me
S
The z component of the spin magnetic moment can have values μspin , z
eh 2me
Wolfgang Pauli
1900 – 1958 Important review article on relativity
At age 21
Discovery of the exclusion principle Explanation of the connection between particle spin and statistics Relativistic quantum electrodynamics Neutrino hypothesis Hypotheses of nuclear spin
The Exclusion Principle
The four quantum numbers discussed so far can be used to describe all the electronic states of an atom regardless of the number of electrons in its structure The exclusion principle states that no two electrons can ever be in the same quantum state
Therefore, no two electrons in the same atom can have the same set of quantum numbers
Filling Subshells
Once a subshell is filled, the next electron goes into the lowest-energy vacant state
If the atom were not in the lowest-energy state available to it, it would radiate energy until it reached this state
Orbitals
An orbital is defined as the atomic state characterized by the quantum numbers n, ℓ and mℓ
From the exclusion principle, it can be seen that only two electrons can be present in any orbital
One electron will have spin up and one spin down
Allowed Quantum States, Example
In general, each shell can accommodate up to 2n2 electrons
Hund’s Rule
Hund’s Rule states that when an atom has orbitals of equal energy, the order in which they are filled by electrons is such that a maximum number of electrons have unpaired spins
Some exceptions to the rule occur in elements having subshells that are close to being filled or half-filled
Periodic Table
Dmitri Mendeleev made an early attempt at finding some order among the chemical elements He arranged the elements according to their atomic masses and chemical similarities The first table contained many blank spaces and he stated that the gaps were there only because the elements had not yet been discovered
Periodic Table, cont.
By noting the columns in which some missing elements should be located, he was able to make rough predictions about their chemical properties Within 20 years of the predictions, most of the elements were discovered The elements in the periodic table are arranged so that all those in a column have similar chemical properties
Periodic Table, Explained
The chemical behavior of an element depends on the outermost shell that contains electrons For example, the inert gases (last column) have filled subshells and a wide energy gap occurs between the filled shell and the next available shell
Hydrogen Energy Level Diagram Revisited
The allowed values of ℓ are separated Transitions in which ℓ does not change are very unlikely to occur and are called forbidden transitions
Such transitions actually can occur, but their probability is very low compared to allowed transitions
Selection Rules
The selection rules for allowed transitions are
Δℓ = ±1 Δmℓ = 0, ±1
The angular momentum of the atom-photon system must be conserved Therefore, the photon involved in the process must carry angular momentum
The photon has angular momentum equivalent to that of a particle with spin 1 A photon has energy, linear momentum and angular momentum
Multielectron Atoms
For multielectron atoms, the positive nuclear charge Ze is largely shielded by the negative charge of the inner shell electrons The outer electrons interact with a net charge that is smaller than the nuclear charge 2 13.6 Zeff Allowed energies are En eV 2 n
X-Ray Spectra
These x-rays are a result of the slowing down of high energy electrons as they strike a metal target The kinetic energy lost can be anywhere from 0 to all of the kinetic energy of the electron The continuous spectrum is called bremsstrahlung, the German word for “braking radiation”
X-Ray Spectra, cont.
The discrete lines are called characteristic x-rays These are created when
A bombarding electron collides with a target atom The electron removes an inner-shell electron from orbit An electron from a higher orbit drops down to fill the vacancy
X-Ray Spectra, final
The photon emitted during this transition has an energy equal to the energy difference between the levels Typically, the energy is greater than 1000 eV The emitted photons have wavelengths in the range of 0.01 nm to 1 nm
Moseley Plot
Henry G. J. Moseley plotted the values of atoms as shown λ is the wavelength of the Kα line of each element
The Kα line refers to the photon emitted when an electron falls from the L to the K shell
From this plot, Moseley developed a periodic table in agreement with the one based on chemical properties
Stimulated Absorption
When a photon has energy hƒ equal to the difference in energy levels, it can be absorbed by the atom This is called stimulated absorption because the photon stimulates the atom to make the upward transition
Active Figure 42.24
(SLIDESHOW MODE ONLY)
Spontaneous Emission
Once an atom is in an excited state, the excited atom can make a transition to a lower energy level Because this process happens naturally, it is known as spontaneous emission
Stimulated Emission
In addition to spontaneous emission, stimulated emission may also occur Stimulated emission may occur when the excited state is a metastable state
Stimulated Emission, cont.
A metastable state is a state whose lifetime is much longer than the typical 10-8 s An incident photon can cause the atom to return to the ground state without being absorbed Therefore, you have two photons with identical energy, the emitted photon and the incident photon
They both are in phase and travel in the same direction
Lasers – Properties of Laser Light
Laser light is coherent
The individual rays in a laser beam maintain a fixed phase relationship with each other There is no destructive interference
Laser light is monochromatic
The light has a very narrow range of wavelengths
Lasers – Properties of Laser Light, cont.
Laser light has a small angle of divergence
The beam spreads out very little, even over long distances
Lasers – Operation
It is equally probable that an incident photon would cause atomic transitions upward or downward
Stimulated absorption or stimulated emission
If a situation can be caused where there are more electrons in excited states than in the ground state, a net emission of photons can result
This condition is called population inversion
Lasers – Operation, cont.
The photons can stimulate other atoms to emit photons in a chain of similar processes The many photons produced in this manner are the source of the intense, coherent light in a laser
Conditions for Build-Up of Photons
The system must be in a state of population inversion The excited state of the system must be a metastable state
In this case, the population inversion can be established and stimulated emission is likely to occur before spontaneous emission
The emitted photons must be confined in the system long enough to enable them to stimulate further emission
This is achieved by using reflecting mirrors
Laser Design – Schematic
The tube contains the atoms that are the active medium An external source of energy pumps the atoms to excited states The mirrors confine the photons to the tube
Mirror 2 is only partially reflective
Energy-Level Diagram for Neon in a Helium-Neon Laser
The atoms emit 632.8-nm photons through stimulated emission The transition is E3* to E2
* indicates a metastable state
Laser Applications
Applications include:
Medical and surgical procedures Precision surveying and length measurements Precision cutting of metals and other materials Telephone communications
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