Physics 202 - galileo.phys.virginia.edugalileo.phys.virginia.edu/~pqh/202_23n.pdfPhysics 202 – p....

Physics 202Professor P. Q. Hung

311B, Physics Building

Physics 202 – p. 1/41

Atomic Physics

The Hydrogen Atom: Spectrum

Pre-Bohr Observation: Emission of light byatoms in a gas. Let the light pass through anarrow-slit aperture ⇒ Discrete set of lines ofdifferent colors (or wavelengths).

For hydrogen, Balmer empirically fitted thespectrum withλ = (364.5 nm) n2

n2−4

n = 3, 4, 5, ....

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Atomic Physics


Physics 202 – p. 3/41

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Physics 202 – p. 4/41

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One can invert the previous equation to give:1λ

= R( 122 − 1

n2 )

where R = 10.97373 µm−1 is the Rydbergconstant.

More general empirical formula which alsoapplies to heavier atoms: The Rydberg-Ritzformula1λ

= R Z2( 1n2

2

− 1n2

1

)

where n1 > n2 and Z = 1 for hydrogen.

Physics 202 – p. 5/41

Atomic Physics

The Hydrogen Atom: Spectrum; Special cases

n2 = 2, n1 = n: Balmer series.This series covers the visible spectrum. Thelongest wavelength is when n = 3 ⇒λ = 656.3 nm, on the reddish side. Theshortest wavelength is when n = ∞ ⇒λ = 364.6 nm, on the ultraviolet side.

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Atomic Physics


n2 = 1, n1 = 2, 3, ...: Lyman series.λ = (91.13 nm) n2

n2−1

n = 2, 3, ...This is in the ultraviolet regime. The longestwavelength is when n = 2 ⇒ λ = 121.5 nm.The shortest wavelength is when n = ∞ ⇒λ = 91.13 nm.

Physics 202 – p. 7/41

Atomic Physics


n2 = 3, n1 = 4, 5, ...: Paschen series’λ = (820.14 nm) n2

n2−9

n = 4, 5, ....This is in the infrared region. The longestwavelength is when n = 4 ⇒ λ = 1874.6 nm.The shortest wavelength is when n = ∞ ⇒λ = 820.14 nm.

Physics 202 – p. 8/41

Atomic Physics


Physics 202 – p. 9/41

Atomic Physics

The Bohr Model

Two crucial elements: The Rutherford’spicture 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 thenucleus will lose energy through radiation andwill collapse into the nucleus.Something else: Bohr’s model in 1913.

Physics 202 – p. 10/41

Atomic Physics

The Bohr Model: Bohr’s 2 postulates

Electrons move in certainstationary, non-radiating, circular orbitsconsistent with Coulomb’s law and Newton’slaw and specified by the quantization ofangular momentumL = mvr = n~

n: an integer.

Radiation of frequencyf =

Ei−Ef

h

occurs when the electrom jumps from orbit iof energy Ei to orbit f of energy Ef .

Physics 202 – p. 11/41

Atomic Physics

The Bohr Model: Bohr’s 2 postulates

Electrons move in certainstationary, non-radiating, circular orbitsconsistent with Coulomb’s law and Newton’slaw and specified by the quantization ofangular momentumL = mvr = n~

n: an integer.

Radiation of frequencyf =

Ei−Ef

h

occurs when the electrom jumps from orbit iof energy Ei to orbit f of energy Ef . Physics 202 – p. 11/41

Atomic Physics

The Bohr Atom

Physics 202 – p. 12/41

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The Bohr Atom

Physics 202 – p. 13/41

Atomic Physics

The Bohr Model: Consequences of Bohr’s 2postulates

Circular orbit:Take an atom with a positively chargednucleus of charge Ze at the center and oneelectron going around it. The potential energyis U = −kZe2/r. The total energy isE = 1

2mv2 + U . For a circular orbit,kZe2/r2 = mv2/r ⇒ 1

2mv2 = kZe2/2r ⇒E = −kZe2/2r.But how do we know what r is for variousorbits? Physics 202 – p. 14/41

Atomic Physics


Orbital angular momentum is quantized:From kZe2/r2 = mv2/r, one findsrn = n2 ~

2

mkZe2 = n2 a0

Z(1)

a0 = ~2

mke2 = 0.0529 nmis the Bohr’s radius.

Physics 202 – p. 15/41

Atomic Physics


How did Bohr derive the Rydberg-Ritzformula?Using f =

Ei−Ef

hand E = −kZe2/2r, one finds

f = (kZe2

2hr)( 1

rf− 1

ri). Using (1) and f = c/λ,

one finds the Rydberg-Ritz formula withR = mk2e4

4πc~3

In very good agreement with the empiricaldetermination of the Rydberg constant.

Physics 202 – p. 16/41

Atomic Physics


How does one write the total energy of anelectron in an orbit rn?From E = −kZe2/2r and Eq. (1), one findsEn = −Z2 E0

n2 (2)whereE0 = k2e4m

2~2 ≈ 13.6 eV

is the ionization or binding energy of theelectron in the ground state of the hydrogenatom. It is the minimum energy to remove theelectron from the hydrogen atom.

Physics 202 – p. 17/41

Atomic Physics


For the hydrogen atom, one hasEn = −13.6 eV

n2

E1 = −13.6 eV is the ground state energy ofthe hydrogen atom.Caveat: Bohr’s model only applies to the casewhere there is one electron. It fails for atomshaving several electrons!

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Atomic Physics

The Bohr Atom: Consequences

Physics 202 – p. 19/41

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The Bohr Atom: Example 1The Lithium atom Li has Z = 3. Since it isneutral, it also has 3 electrons. Bohr’s model isnot applicable. If two electrons are stripped away,one ends up with the ion Li2+. What is theionization energy of Li2+?Solution:Since one has only one electron left, the Bohr’smodel can be used. The ionization energy isfound by putting n = 1 and Z = 3 in Eq. (2).E1 = −32(13.6 eV ) = −122 eV .So the minimum ionization energy needed toremove the remaining electron is 122 eV .

Physics 202 – p. 20/41

Atomic Physics

The Bohr Atom: Example 2Use the Bohr’s model to calculate the quatumnumber n of the Earth in its orbit about the Sun.Solution:1) Replace ke2 by GMEMS in Eq. (1) with Z andusing rn = 6 × 106 m.2) rn = n2 ~

2

GM2

EMS⇒

3) n = ME

√rnGMS

~= 2 × 1072. A large number!

Lesson: For a very large quantum number n, weget back the results of classical physics!

Physics 202 – p. 21/41

Atomic Physics

Origin of angular momentum quantization

Crucial ingredient of Bohr’s model: thequantization of angular momentum,L = mvr = n~. Origin?De Broglie’s wave-particle duality

Electrons behave like waves. On an orbit,standing wave phenomena ⇒ condition forstanding wave:2πr = nλλ = h

p

p = mv⇒ L = mvr = n h

2π .⇒ Wave nature of particles is crucial inatomic physics.

Physics 202 – p. 22/41

Atomic Physics


Physics 202 – p. 23/41

Atomic Physics

Origin of angular momentum quantizationShortcomings of Bohr’s model:

No prediction for relative intensities ofspectral lines.

Cannot describe complex atoms.

Lack of theoretical foundation.

Physics 202 – p. 24/41

Atomic Physics


Better picture: wave mechanics or quantummechanics:

Propagation of matter waves is governed bythe Schroedinger equation.

Matter wave is described by a wave functionwhose absolute square represents theprobability to find the particle within a rangeof position (roughly speaking)

Physics 202 – p. 25/41

Atomic Physics

Quantum MechanicsIn Bohr’s model, only one quantum number: nIn quantum mechanics:

Principal quantum number n.

Orbital quantum number l.l = 0, 1, 2, .., (n − 1).Magnitude of angular momentum:L =

√

l(l + 1)~.

Physics 202 – p. 26/41

Atomic Physics

Quantum Mechanics

Magnetic quantum number ml.ml = −l, ..,−1, 0, 1, .., l.Lz = ml~.

Spin quantum number ms. ms = ±12.

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Quantum mechanics: Hydrogen atom

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Pauli Exclusion Principle

Pauli Exclusion principle:No two electrons can have the same set ofvalues for n, l, ml, ms.

An atom is characterized by energy levels,e.g. n = 1, l = 0 is the lowest energy level orground state.

Ground state of an atom: the electrons are inthe lowest available energy levels consistentwith the Pauli exclusion principle.

Physics 202 – p. 29/41

Atomic Physics


Shell: levels with n. K,L,M,N,O,...

Subshell: For a given shell, levels withdifferent l, ml, ms. For l, one usually call: s, p,d, f, g, h,...

The Pauli Exclusion principle allows only amaximum number of electrons that can fit intoan energy level or subshell.

Physics 202 – p. 30/41

Atomic Physics


For each l, there can be 2(2l + 1) electronsthat can fit. (This is because of the number ofcombinations 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. 31/41

Atomic Physics


For each l, there can be 2(2l + 1) electronsthat can fit. (This is because of the number ofcombinations 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. 31/41

Atomic Physics


Physics 202 – p. 32/41

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Physics 202 – p. 35/41

Atomic Physics

Atomic RadiationTwo types of X-rays: bremsstrahlung andcharacteristic X-rays

bremsstrahlung: radiation from energeticelectrons which decelerate upon hitting atarget. Continuous spectrum.

characteristic X-rays: An incoming electronknocks off say a K-shell electron. That void isfilled immediately by electrons from outershells ⇒ X-rays. To do that the incomingelectron must have at leastEK = −13.6 eV (Z−1)2

12 .

Physics 202 – p. 36/41

Atomic Physics

Atomic RadiationTwo types of X-rays: bremsstrahlung andcharacteristic X-rays

bremsstrahlung: radiation from energeticelectrons which decelerate upon hitting atarget. Continuous spectrum.

characteristic X-rays: An incoming electronknocks off say a K-shell electron. That void isfilled immediately by electrons from outershells ⇒ X-rays. To do that the incomingelectron must have at leastEK = −13.6 eV (Z−1)2

12 .Physics 202 – p. 36/41

Atomic Physics

Atomic Radiation

Notice that the energy of the electron in theother shells is En = −13.6 eV (Z−1)2

n2 .

Characteristic peaks, e.g. Kα,Kβ.Read the book concerning lasers, etc...

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Atomic Physics

Atomic Radiation

Physics 202 – p. 38/41

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Atomic Radiation

Physics 202 – p. 39/41

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Atomic Radiation

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Atomic Radiation

Physics 202 – p. 41/41

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