Download Atomic Physics

Physics 202
Professor P. Q. Hung
311B, Physics Building
Physics 202 – p. 1/4
Atomic Physics
The Hydrogen Atom: Spectrum
Pre-Bohr Observation: Emission of light by
atoms in a gas. Let the light pass through a
narrow-slit aperture ⇒ Discrete set of lines of
different colors (or wavelengths).
Physics 202 – p. 2/4
Atomic Physics
The Hydrogen Atom: Spectrum
Pre-Bohr Observation: Emission of light by
atoms in a gas. Let the light pass through a
narrow-slit aperture ⇒ Discrete set of lines of
different colors (or wavelengths).
For hydrogen, Balmer empirically fitted the
spectrum with
n2
λ = (364.5 nm) n2 −4
n = 3, 4, 5, ....
Physics 202 – p. 2/4
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 3/4
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 4/4
Atomic Physics
The Hydrogen Atom: Spectrum
One can invert the previous equation to give:
1
1
1
=
R(
−
λ
22
n2 )
where R = 10.97373 µ m−1 is the Rydberg
constant.
Physics 202 – p. 5/4
Atomic Physics
The Hydrogen Atom: Spectrum
One can invert the previous equation to give:
1
1
1
=
R(
−
λ
22
n2 )
where R = 10.97373 µ m−1 is the Rydberg
constant.
More general empirical formula which also
applies to heavier atoms: The Rydberg-Ritz
formula
1
1
2 1
=
R
Z
(
−
)
λ
n22
n21
where n1 > n2 and Z = 1 for hydrogen.
Physics 202 – p. 5/4
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 2, n1 = n: Balmer series.
This series covers the visible spectrum. The
longest wavelength is when n = 3 ⇒
λ = 656.3 nm, on the reddish side. The
shortest wavelength is when n = ∞ ⇒
λ = 364.6 nm, on the ultraviolet side.
Physics 202 – p. 6/4
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 1, n1 = 2, 3, ...: Lyman series.
n2
λ = (91.13 nm) n2 −1
n = 2, 3, ...
This is in the ultraviolet regime. The longest
wavelength is when n = 2 ⇒ λ = 121.5 nm.
The shortest wavelength is when n = ∞ ⇒
λ = 91.13 nm.
Physics 202 – p. 7/4
Atomic Physics
The Hydrogen Atom: Spectrum; Special cases
n2 = 3, n1 = 4, 5, ...: Paschen series’
n2
λ = (820.14 nm) n2 −9
n = 4, 5, ....
This is in the infrared region. The longest
wavelength is when n = 4 ⇒ λ = 1874.6 nm.
The shortest wavelength is when n = ∞ ⇒
λ = 820.14 nm.
Physics 202 – p. 8/4
Atomic Physics
The Hydrogen Atom: Spectrum
Physics 202 – p. 9/4
Atomic Physics
The Bohr Model
Two crucial elements: The Rutherford’s
picture of the atom and the Balmer formula
(or more generally the Rydberg-Ritz formula).
Physics 202 – p. 10/4
Atomic Physics
The Bohr Model
Two crucial elements: The Rutherford’s
picture of the atom and the Balmer formula
(or more generally the Rydberg-Ritz formula).
Defect with the Rutherford’s picture:
Classically, the electron in orbit around the
nucleus will lose energy through radiation and
will collapse into the nucleus.
Something else: Bohr’s model in 1913.
Physics 202 – p. 10/4
Atomic Physics
The Bohr Model: Bohr’s 2 postulates
Electrons move in certain
stationary, non-radiating, circular orbits
consistent with Coulomb’s law and Newton’s
law and specified by the quantization of
angular momentum
L = mvr = n~
n: an integer.
Physics 202 – p. 11/4
Atomic Physics
The Bohr Model: Bohr’s 2 postulates
Electrons move in certain
stationary, non-radiating, circular orbits
consistent with Coulomb’s law and Newton’s
law and specified by the quantization of
angular momentum
L = mvr = n~
n: an integer.
Radiation of frequency
Ei −Ef
f= h
occurs when the electrom jumps from orbit i
of energy Ei to orbit f of energy Ef .
Physics 202 – p. 11/4
Atomic Physics
The Bohr Atom
Physics 202 – p. 12/4
Atomic Physics
The Bohr Atom
Physics 202 – p. 13/4
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2
postulates
Circular orbit:
Take an atom with a positively charged
nucleus of charge Ze at the center and one
electron going around it. The potential energy
is U = −kZe2 /r. The total energy is
E = 12 mv 2 + U . For a circular orbit,
kZe2 /r2 = mv 2 /r ⇒ 12 mv 2 = kZe2 /2r ⇒
E = −kZe2 /2r.
But how do we know what r is for various
orbits?
Physics 202 – p. 14/4
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2
postulates
Orbital angular momentum is quantized:
From kZe2 /r2 = mv 2 /r, one finds
2 ~2
rn = n mkZe2 = n2 aZ0 (1)
~2
a0 = mke2 = 0.0529 nm
is the Bohr’s radius.
Physics 202 – p. 15/4
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2
postulates
How did Bohr derive the Rydberg-Ritz
formula?
Ei −Ef
Using f = h and E = −kZe2 /2r, one finds
kZe2
f = ( 2hr )( r1f − r1i ). Using (1) and f = c/λ,
one finds the Rydberg-Ritz formula with
mk 2 e4
R = 4πc~3
In very good agreement with the empirical
determination of the Rydberg constant.
Physics 202 – p. 16/4
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2
postulates
How does one write the total energy of an
electron in an orbit rn ?
From E = −kZe2 /2r and Eq. (1), one finds
En = −Z 2 En20 (2)
where
k 2 e4 m
E0 = 2~2 ≈ 13.6 eV
is the ionization or binding energy of the
electron in the ground state of the hydrogen
atom. It is the minimum energy to remove the
electron from the hydrogen atom.
Physics 202 – p. 17/4
Atomic Physics
The Bohr Model: Consequences of Bohr’s 2
postulates
For the hydrogen atom, one has
En = − 13.6n2eV
E1 = −13.6 eV is the ground state energy of
the hydrogen atom.
Caveat: Bohr’s model only applies to the case
where there is one electron. It fails for atoms
having several electrons!
Physics 202 – p. 18/4
Atomic Physics
The Bohr Atom: Consequences
Physics 202 – p. 19/4
Atomic Physics
The Bohr Atom: Example 1
The Lithium atom Li has Z = 3. Since it is
neutral, it also has 3 electrons. Bohr’s model is
not applicable. If two electrons are stripped away,
one ends up with the ion Li2+ . What is the
ionization energy of Li2+ ?
Solution:
Since one has only one electron left, the Bohr’s
model can be used. The ionization energy is
found by putting n = 1 and Z = 3 in Eq. (2).
E1 = −32 (13.6 eV ) = −122 eV .
So the minimum ionization energy needed to
remove the remaining electron is 122 eV .
Physics 202 – p. 20/4
Atomic Physics
The Bohr Atom: Example 2
Use the Bohr’s model to calculate the quatum
number n of the Earth in its orbit about the Sun.
Solution:
1) Replace ke2 by GME MS in Eq. (1) with Z and
using rn = 6 × 106 m.
~2
2
2) rn = n GM 2 MS ⇒
√
E
ME rn GMS
~
= 2 × 1072 . A large number!
3) n =
Lesson: For a very large quantum number n, we
get back the results of classical physics!
Physics 202 – p. 21/4
Atomic Physics
Origin of angular momentum quantization
Crucial ingredient of Bohr’s model: the
quantization of angular momentum,
L = mvr = n~. Origin?
De Broglie’s wave-particle duality
Physics 202 – p. 22/4
Atomic Physics
Origin of angular momentum quantization
Crucial ingredient of Bohr’s model: the
quantization of angular momentum,
L = mvr = n~. Origin?
De Broglie’s wave-particle duality
Electrons behave like waves. On an orbit,
standing wave phenomena ⇒ condition for
standing wave:
2πr = nλ
λ = hp
p = mv
h
⇒ L = mvr = n 2π
.
Physics 202 – p. 22/4
Atomic Physics
Origin of angular momentum quantization
Physics 202 – p. 23/4
Atomic Physics
Origin of angular momentum quantization
Shortcomings of Bohr’s model:
No prediction for relative intensities of
spectral lines.
Physics 202 – p. 24/4
Atomic Physics
Origin of angular momentum quantization
Shortcomings of Bohr’s model:
No prediction for relative intensities of
spectral lines.
Cannot describe complex atoms.
Physics 202 – p. 24/4
Atomic Physics
Origin of angular momentum quantization
Shortcomings of Bohr’s model:
No prediction for relative intensities of
spectral lines.
Cannot describe complex atoms.
Lack of theoretical foundation.
Physics 202 – p. 24/4
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantum
mechanics:
Physics 202 – p. 25/4
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantum
mechanics:
Propagation of matter waves is governed by
the Schroedinger equation.
Physics 202 – p. 25/4
Atomic Physics
Origin of angular momentum quantization
Better picture: wave mechanics or quantum
mechanics:
Propagation of matter waves is governed by
the Schroedinger equation.
Matter wave is described by a wave function
whose absolute square represents the
probability to find the particle within a range
of position (roughly speaking)
Physics 202 – p. 25/4
Atomic Physics
Quantum Mechanics
In Bohr’s model, only one quantum number: n
In quantum mechanics:
Principal quantum number n.
Physics 202 – p. 26/4
Atomic Physics
Quantum Mechanics
In Bohr’s model, only one quantum number: n
In quantum mechanics:
Principal quantum number n.
Orbital quantum number l.
l = 0, 1, 2, .., (n − 1).
Magnitude
of
angular
momentum:
p
L = l(l + 1)~.
Physics 202 – p. 26/4
Atomic Physics
Quantum Mechanics
Magnetic quantum number ml .
ml = −l, .., −1, 0, 1, .., l.
Lz = ml ~.
Physics 202 – p. 27/4
Atomic Physics
Quantum Mechanics
Magnetic quantum number ml .
ml = −l, .., −1, 0, 1, .., l.
Lz = ml ~.
Spin quantum number ms . ms = ± 12 .
Physics 202 – p. 27/4
Atomic Physics
Quantum mechanics: Hydrogen atom
Physics 202 – p. 28/4
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:
No two electrons can have the same set of
values for n, l, ml , ms .
Physics 202 – p. 29/4
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:
No two electrons can have the same set of
values for n, l, ml , ms .
An atom is characterized by energy levels,
e.g. n = 1, l = 0 is the lowest energy level or
ground state.
Physics 202 – p. 29/4
Atomic Physics
Pauli Exclusion Principle
Pauli Exclusion principle:
No two electrons can have the same set of
values for n, l, ml , ms .
An atom is characterized by energy levels,
e.g. n = 1, l = 0 is the lowest energy level or
ground state.
Ground state of an atom: the electrons are in
the lowest available energy levels consistent
with the Pauli exclusion principle.
Physics 202 – p. 29/4
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Physics 202 – p. 30/4
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Subshell: For a given shell, levels with
different l, ml , ms . For l, one usually call: s, p,
d, f, g, h,...
Physics 202 – p. 30/4
Atomic Physics
Pauli Exclusion Principle
Shell: levels with n. K,L,M,N,O,...
Subshell: For a given shell, levels with
different l, ml , ms . For l, one usually call: s, p,
d, f, g, h,...
The Pauli Exclusion principle allows only a
maximum number of electrons that can fit into
an energy level or subshell.
Physics 202 – p. 30/4
Atomic Physics
Pauli Exclusion Principle
For each l, there can be 2(2l + 1) electrons
that can fit. (This is because of the number of
combinations of ml and ms .)
Physics 202 – p. 31/4
Atomic Physics
Pauli Exclusion Principle
For each l, there can be 2(2l + 1) electrons
that can fit. (This is because of the number of
combinations of ml and ms .)
Notation for the ground state of an atom:
n lnumber of electrons .
Hydrogen(H): 1 electron,1 s1 .
Helium (He): 2 electrons, 1 s2 .
Lithium (Li): 3 electrons, 1 s2 2 s1 .
Beryllium (Be): 4 electrons, 1 s2 2 s2
etc..
Physics 202 – p. 31/4
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 32/4
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 33/4
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 34/4
Atomic Physics
Pauli Exclusion Principle
Physics 202 – p. 35/4
Atomic Physics
Atomic Radiation
Two types of X-rays: bremsstrahlung and
characteristic X-rays
bremsstrahlung: radiation from energetic
electrons which decelerate upon hitting a
target. Continuous spectrum.
Physics 202 – p. 36/4
Atomic Physics
Atomic Radiation
Two types of X-rays: bremsstrahlung and
characteristic X-rays
bremsstrahlung: radiation from energetic
electrons which decelerate upon hitting a
target. Continuous spectrum.
characteristic X-rays: An incoming electron
knocks off say a K-shell electron. That void is
filled immediately by electrons from outer
shells ⇒ X-rays. To do that the incoming
electron must have at
least
(Z−1)2
EK = −13.6 eV 12 .
Physics 202 – p. 36/4
Atomic Physics
Atomic Radiation
Notice that the energy of the electron in the
(Z−1)2
other shells is En = −13.6 eV n2 .
Physics 202 – p. 37/4
Atomic Physics
Atomic Radiation
Notice that the energy of the electron in the
(Z−1)2
other shells is En = −13.6 eV n2 .
Characteristic peaks, e.g. Kα , Kβ .
Read the book concerning lasers, etc...
Physics 202 – p. 37/4
Atomic Physics
Atomic Radiation
Physics 202 – p. 38/4
Atomic Physics
Atomic Radiation
Physics 202 – p. 39/4
Atomic Physics
Atomic Radiation
Physics 202 – p. 40/4
Atomic Physics
Atomic Radiation
Physics 202 – p. 41/4