PY3P05

Lecture 1-2: Introduction to Atomic SpectroscopyLecture 1-2: Introduction to Atomic Spectroscopy

o Line spectra

o Emission spectra

o Absorption spectra

o Hydrogen spectrum

o Balmer Formula

o Bohr’s Model

Bulb

Sun

Na

H

Hg

Cs

Chlorophyll

Diethylthiacarbocyaniodid

Diethylthiadicarbocyaniodid

Absorption spectra

Emission spectra

PY3P05

Types of SpectraTypes of Spectra

o Continuous spectrum: Produced by solids, liquids & dense gases produce - no “gaps” in wavelength of light produced:

o Emission spectrum: Produced by rarefied gases – emission only in narrow wavelength regions:

o Absorption spectrum: Gas atoms absorb the same wavelengths as they usually emit and results in an absorption line spectrum:

PY3P05

Emission and Absorption SpectroscopyEmission and Absorption Spectroscopy

Gas cloud

12

3

PY3P05

Line SpectraLine Spectra

o Electron transition between energy levels result in emission or absorption lines.

o Different elements produce different spectra due to differing atomic structure.

H

He

C

PY3P05

Emission/Absorption of Radiation by AtomsEmission/Absorption of Radiation by Atoms

o Emission/absorption lines are due to radiative transitions:

1. Radiative (or Stimulated) absorption:

Photon with energy (E = h = E2 - E1) excites electron from lower energy level.

Can only occur if E = h = E2 - E1

1. Radiative recombination/emission:

Electron makes transition to lower energy level and emits photon with energy

h’ = E2 - E1.

E2

E1

E2

E1

E =h

PY3P05

Emission/Absorption of Radiation by AtomsEmission/Absorption of Radiation by Atoms

o Radiative recombination can be either:

a) Spontaneous emission: Electron minimizes its total energy by emitting photon and making transition from E2 to E1.

Emitted photon has energy E’ = h’ = E2 - E1

a) Stimulated emission: If photon is strongly coupled with electron, cause electron to decay to lower energy level, releasing a photon of the same energy.

Can only occur if E = h = E2 - E1 Also, h’ = h

E2

E1

E2

E1

E2

E1

E2

E1

E’ =h’

E =hE

’ =h’

E =h

PY3P05

Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen

o In ~1850’s, optical spectrum of hydrogen was found to contain strong lines at 6563, 4861 and 4340 Å.

o Lines found to fall closer and closer as wavelength decreases.

o Line separation converges at a particular wavelength, called the series limit.

o Balmer found that the wavelength of lines could be written

where n is an integer >2, and RH is the Rydberg constant.

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

€

1/λ = RH

1

22−

1

n2

⎛

⎝ ⎜

⎞

⎠ ⎟

6563 4861 4340

H H H

(Å)

PY3P05

Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen

o If n =3, =>

o Called H - first line of Balmer series.

o Other lines in Balmer series:

o Balmer Series limit occurs when n .

€

1/λ = RH

1

22−

1

32

⎛

⎝ ⎜

⎞

⎠ ⎟=> λ = 6563 Å

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Highway 6563 in New Mexico

€

1/λ∞ = RH

1

22−

1

∞

⎛

⎝ ⎜

⎞

⎠ ⎟=> λ∞ ~ 3646 Å

Name Transitions Wavelength (Å)

H 3 - 2 6562.8

H 4 - 2 4861.3

H 5 - 2 4340.5

PY3P05

Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen

Lyman UV nf = 1, ni2

Balmer Visible/UV nf = 2, ni3

Paschen IR nf = 3, ni4

Brackett IR nf = 4, ni5

Pfund IR nf = 5, ni6

Series Limits

o Other series of hydrogen:

o Rydberg showed that all series above could bereproduced using

o Series limit occurs when ni = ∞, nf = 1, 2, …

€

1/λ = RH

1

n f2

−1

ni2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

PY3P05

Simplest Atomic Spectrum: HydrogenSimplest Atomic Spectrum: Hydrogen

o Term or Grotrian diagram for hydrogen.

o Spectral lines can be considered as transition between terms.

o A consequence of atomic energy levels, is that transitions can only occur between certain terms. Called a selection rule. Selection rule for hydrogen: n = 1, 2, 3, …

PY3P05

Bohr Model for HydrogenBohr Model for Hydrogen

o Simplest atomic system, consisting of single electron-proton pair.

o First model put forward by Bohr in 1913. He postulated that:

1. Electron moves in circular orbit about proton under Coulomb attraction.

2. Only possible for electron to orbits for which angular momentum is quantised, ie., n = 1, 2, 3, …

3. Total energy (KE + V) of electron in orbit remains constant.

4. Quantized radiation is emitted/absorbed if an electron moves changes its orbit.

€

L = mvr = nh

PY3P05

Bohr Model for HydrogenBohr Model for Hydrogen

o Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron on charge -e and mass m. Assume M>>m so nucleus remains at fixed position in space.

o As Coulomb force is a centripetal, can write (1)

o As angular momentum is quantised (2nd postulate): n = 1, 2, 3, …

o Solving for v and substituting into Eqn. 1 => (2)

o The total mechanical energy is:

o Therefore, quantization of AM leads to quantisation of total energy.

€

1

4πε0

Ze2

r2= m

v 2

r

€

mvr = nh

€

E =1/2mv 2 + V

∴ En = −mZ 2e4

4πε0( )22h2

1

n2 n = 1, 2, 3, …

€

r = 4πε0

n2h2

mZe2

=> v =nh

mr=

1

4πε0

Ze2

nh

(3)

PY3P05

Bohr Model for HydrogenBohr Model for Hydrogen

o Substituting in for constants, Eqn. 3 can be written eV

and Eqn. 2 can be written where a0 = 0.529 Å = “Bohr radius”.

o Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1):

o For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV

€

En =−13.6Z 2

n2

€

r =n2a0

Z

PY3P05

Bohr Model for HydrogenBohr Model for Hydrogen

o The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate):

or (4)

where

o Theoretical derivation of Rydberg formula.

o Essential predictions of Bohr model

are contained in Eqns. 3 and 4.

€

1/λ =E i − E f

hc=

1

4πε0

⎛

⎝ ⎜

⎞

⎠ ⎟

2me4

4πh3cZ 2 1

n f2

−1

ni2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

1/λ = R∞Z 2 1

n f2

−1

ni2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

R∞ =1

4πε0

⎛

⎝ ⎜

⎞

⎠ ⎟

2me4

4πh3c

PY3P05

Correction for Motion of the NucleusCorrection for Motion of the Nucleus

o Spectroscopically measured RH does not agree exactly with theoretically derived R∞.

o But, we assumed that M>>m => nucleus fixed. In reality, electron and proton move about common centre of mass. Must use electron’s reduced mass ():

o As m only appears in R∞, must replace by:

o It is found spectroscopically that RM = RH to within three parts in 100,000.

o Therefore, different isotopes of same element have slightly different spectral lines.

€

=mM

m + M

€

RM =M

m + MR∞ =

μ

mR∞

PY3P05

Correction for Motion of the NucleusCorrection for Motion of the Nucleus

o Consider 1H (hydrogen) and 2H (deuterium):

o Using Eqn. 4, the wavelength difference is therefore:

o Called an isotope shift.

o H and D are separated by about 1Å.

o Intensity of D line is proportional to fraction of D in the sample.

€

RH = R∞

1

1+ m / M H

=109677.584

RD = R∞

1

1+ m / MD

=109707.419

cm-1

cm-1

Balmer line of H and D

€

=H − λ D = λ H (1− λ D /λ H )

= λ H (1− RH /RD )

PY3P05

Spectra of Hydrogen-like AtomsSpectra of Hydrogen-like Atoms

o Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc).

o Spectrum of 4He+ is almost identical to H, but just offset by a factor of four (Z2).

o For He+, Fowler found the following

in stellar spectra:

o See Fig. 8.7 in Haken & Wolf.€

1/λ = 4RHe

1

32−

1

n2

⎛

⎝ ⎜

⎞

⎠ ⎟

0

20

40

60

80

100

120

Energy(eV)

1

n2

1

23

Z=1H

Z=2He+

Z=3Li2+

n n

1

2

34

13.6 eV

54.4 eV

122.5 eV

PY3P05

Spectra of Hydrogen-like AtomsSpectra of Hydrogen-like Atoms

o Hydrogenic or hydrogen-like ions:

o He+ (Z=2)

o Li2+ (Z=3)

o Be3+ (Z=4)

o …

o From Bohr model, the ionization energy is:

E1 = -13.59 Z2 eV

o Ionization potential therefore increases rapidly with Z.

Hydrogenic isoelectronic

sequences

0

20

40

60

80

100

120

Energy(eV)

1

n2

1

23

Z=1H

Z=2He+

Z=3Li2+

n n

1

2

34

13.6 eV

54.4 eV

122.5 eV

PY3P05

Implications of Bohr ModelImplications of Bohr Model

o We also find that the orbital radius and velocity are quantised:

and

o Bohr radius (a0) and fine structure constant () are fundamental constants:

and

o Constants are related by

o With Rydberg constant, define gross atomic characteristics of the atom.

€

rn =n2

Z

m

μa0

€

a0 =4πε0h

2

me2

Rydberg energy RH 13.6 eV

Bohr radius a0 5.26x10-11 m

Fine structure constant 1/137.04

€

vn = αZ

nc

€

=e2

4πε0hc

€

a0 =h

mcα

PY3P05

Exotic AtomsExotic Atoms

o Positroniumo electron (e-) and positron (e+) enter a short-lived bound state, before they

annihilate each other with the emission of two -rays (discovered in 1949).o Parapositronium (S=0) has a lifetime of ~1.25 x 10-10 s. Orthopositronium

(S=1) has lifetime of ~1.4 x 10-7 s.o Energy levels proportional to reduced mass => energy levels half of hydrogen.

o Muonium:o Replace proton in H atom with a meson (a “muon”). o Bound state has a lifetime of ~2.2 x 10-6 s.o According to Bohr’s theory (Eqn. 3), the binding energy is 13.5 eV.o From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV.

o Antihydrogen:o Consists of a positron bound to an antiproton - first observed in 1996 at CERN!o Antimatter should behave like ordinary matter according to QM.o Have not been investigated spectroscopically … yet.

PY3P05

Failures of Bohr ModelFailures of Bohr Model

o Bohr model was a major step toward understanding the quantum theory of the atom - not in fact a correct description of the nature of electron orbits.

o Some of the shortcomings of the model are:

1. Fails describe why certain spectral lines are brighter than others => no mechanism for calculating transition probabilities.

2. Violates the uncertainty principal which dictates that position and momentum cannot be simultaneously determined.

o Bohr model gives a basic conceptual model of electrons orbits and energies. The precise details can only be solved using the Schrödinger equation.

PY3P05

Failures of Bohr ModelFailures of Bohr Model

o From the Bohr model, the linear momentum of the electron is

o However, know from Hiesenberg Uncertainty Principle, that

o Comparing the two Eqns. above => p ~ np

o This shows that the magnitude of p is undefined except when n is large.

o Bohr model only valid when we approach the classical limit at large n.

o Must therefore use full quantum mechanical treatment to model electron in H atom.

€

p = mv = mZe2

4πε0nh

⎛

⎝ ⎜

⎞

⎠ ⎟=

nh

r

€

p ~h

Δx~

h

r

PY3P05

Hydrogen SpectrumHydrogen Spectrum

o Transitions actually depend on more than a single quantum number (i.e., more than n).

o Quantum mechanics leads to introduction on four quntum numbers.

o Principal quantum number: n

o Azimuthal quantum number: l

o Magnetic quantum number: ml

o Spin quantum number: s

o Selection rules must also be modified.

PY3P05

Atomic Energy ScalesAtomic Energy Scales

Energy scale Energy (eV) Effects

Gross structure 1-10 electron-nuclear attraction

Electron kinetic energy

Electron-electron repulsion

Fine structure 0.001 - 0.01 Spin-orbit interaction

Relativistic corrections

Hyperfine structure 10-6 - 10-5 Nuclear interactions