Fundamentals of spectroscopy

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Spectral bands from the electromagnetic spectrum

Outline • Light-matter interaction

– Visible, outer electrons – X-rays, inner electrons – Infrared, molecular vibrations – Micro- and radiowaves, electron & nuclear spins

• Line widths • Detection modes

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Interactions between electromagnetic radiation and sample

• Electromagnetic radiation transfers energy • Sample is composed of atoms, molecules • By examining the resulting

electromagnetic radiation after it has intracted with the sample - conclusions can be drawn about the object under study

Forces in atoms and molecules • Forces in the universe

– Gravity and the electro-magnetic, weak and strong forces

• In atoms (and for most processes in our daily life) the forces have electric & magnetic character

• We have direct attractive and repulsive forces between the charged particles in the atoms, but we also have magnetic interactions

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Energy levels in an atom

Niels Bohr 1913

λ

Energy

Principal quantum number n determines the distance from the nucleus

Origin of stationary states

Schrödinger Equation Erwin

Schrödinger 1925

Hydrogen atom wave functions

http://sevencolors.org/post/hydrogen-atom-orbitals

Wavefunctions in QM: probability distribution of the electron, i.e. the electron cannot be seen as a localized particle

Quantum numbers: (n, l, ml) n - pricipal quantum number n = 1, 2, ... l - angular quantum number l = 0,1,2,...,n-1; s, p, d, f ml - magnetic quantum number ml = -l, ..., l

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Energy level diagram for the sodium atom

s, p, d, f corresponds to different orbital angular momentum (0, ℏ, 2ℏ, 3ℏ) of the electron

Why can’t we go from one state to any other state?

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The transition probability from one stationary state to another is proportional to

� 𝐹𝐹𝐹𝐹𝐹 𝑠𝑠𝐹𝑠𝑠 × 𝐹𝐹𝑠𝑠𝑖𝐹𝑖𝑠𝐹𝑖𝐹 𝑤𝐹𝑠𝑤 𝑠𝑤𝑠 𝑓𝐹𝑠𝐹𝑓 × 𝐹𝐹𝐹𝑠𝐹𝐹𝐹 𝑠𝑠𝐹𝑠𝑠𝑆𝑝𝑝𝑝𝑝

⇑ Odd parity

Wave functions with even angular momentum quantum numbers have even parity and

Wave functions with odd angular momentum quantum numbers have odd parity

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Energy level diagram for the sodium atom

The photon carries one unit of angular momentum and can therefore take a p-electron to s or d but not to f

Strongest line

Energy levels in Lithium

Spin

• Electrons have spin, s=1/2. In an atom with two outer electrons, these can have opposite or (if allowed by the Pauli principle) equal spin directions

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Energy levels in Helium

Singlets Total spin is zero

Triplets Total spin is one

Energy levels in Calcium

Spectral bands from the electromagnetic spectrum

Oscillations

• The oscillation frequency of a system depends on the mass(es) involved and the restoring force

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https://en.wikipedia.org/wiki/Oscillation

𝑓 =1

2𝜋𝑘𝑚

f = oscillation frequency k = spring constant m = mass

The spring exerts a force F = kx on the mass, m, where x is the displacement from the equilibrium position

How do we get from the visible to the X-ray region?

• Increase the ”spring constant” that is the restoring force on the electron

• In fact, the energy of the innermost electron increases as Z2

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Spectral bands from the electromagnetic spectrum

X-ray production

Page and/or figure references

In green: Sune Svanberg, Atomic and molecular spectroscopy, Springer Verlag In blue: Wolfgang Demtröder, Atoms, Molecules and Photons, Springer

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Bremsstrahlung X-rays can be produced by accelerating/deccelerating

charges. The radiated power is proportional to the

acceleration/deccelation squared

Section 7.5.1

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Collisions can excite inner shell electons to highly excited states. X-ray radiation is emitted when these electrons decay back to the inner shells.

Characteristic lines , Section 7.5.2

Sune Svanberg, Atomic and molecular spectroscopy, Springer Verlag, Fig 5.1

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Collisions can excite inner shell electons to highly excited states. X-ray radiation is emitted when these electrons or other bound electrons decay back to the inner shells. These characteristic lines are superposed on the continuous brehmsstrahlung background

Characteristic lines , Section 7.5.2

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Absorption of X-ray radiation as a function of X-ray wave length

Cu, Z = 29 Ag, Z = 47

page 276

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The water window Fig 10.25, page 271

The short wavelength offers very good resolution. Operating in the water window provides very good contrast between water and proteins in e.g., cells or tissue. Developing good microscopic techniques and sources in

this wavelength region is an active research field.

Spectral bands from the electromagnetic spectrum

Molecular spectra • For molecules we can in addition to

electronic transitions have vibrational and rotational transitions

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Molecular spectra • The proton/electron mass ratio is ~103

• The atomic nuclei in a molecule are ”glued” together by the outer electrons, ”force constant” should be similar as for outer electrons where the electronic transitions are a few electron volts

• Outer electron transitions in atoms are typically a few eV, thus vibrational energies are ~0.1 eV

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𝑓 =1

2𝜋𝑘𝑚

Energy separation in molecules

Fig 2.10, page 23

Spectral bands from the electromagnetic spectrum

Molecular energies

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Distance between nuclei is ~1Å

Some orbitals are bonding and some are anti-bonding

Energy scale is in cm-1

Energy conversions

Unit nm Joule eV

Hz cm-1

1 nm 1 1.99∙10-16 1.24∙103 3.00∙1017 1.00∙109

1 Joule

1.99∙10-16 1 6.24∙1018 1.51∙1033 5.03∙1022

1 eV 1.24∙103 1.60∙10-19 1 2.42∙1014 8.07∙103

1 Hz 3.00∙1017 6.63∙10-34 4.14∙10-15 1 3.34∙10-11

1 cm-1 1.00∙109 1.99∙10-23 1.24∙10-4 3.00∙1010 1

Wavelength Energy Frequency Wavenumber

𝐸 = 𝑤𝑣 𝐸(𝑠𝑒) =𝑤𝑣𝑠

Vibration frequencies for different molecular groups

35 Page 161

Spectral bands from the electromagnetic spectrum

Forces in atoms and molecules • Forces in the universe

– Gravity and the electro-magnetic, weak and strong forces

• In atoms (and for most processes in our daily life) the forces have electric & magnetic character

• We have direct attractive and repulsive forces between the charged particles in the atoms, but we also have magnetic interactions

37

Magnetic moment

Figure adapted from page http://www.sr.bham.ac.uk/xmm/fmc2.html

A current, I, enclosing an area, A, generates a magnetic moment µ = IAân, where ân is a unit vector normal to the surface A.

µ

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Spin

• Electrons have spin, s=1/2.

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Magnetic moments in atoms

• Orbital magnetic moment, 𝝁𝐿 = −𝜇𝐵L • Spin magnetic moment, 𝝁𝑠 = −𝑔𝑠𝜇𝐵S

• 𝑔𝑠≈2,

• Nuclear magnetic moment, 𝝁𝐼 = 𝑔𝐼𝜇N I • I is the nuclear spin

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𝜇𝑁𝜇𝐵

=𝑚𝑝𝑒𝑝𝑝𝑒𝑒𝑒𝑒

𝑚𝑝𝑒𝑒𝑒𝑒𝑒≈

12000

Interaction between a magnetic moment and a magnetic field

Figure adapted from page http://www.sr.bham.ac.uk/xmm/fmc2.html

µ

The energy, E, of a magnetic moment, µ, in a magnetic field B is given by the scalar product E=-µ⋅B

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Electron & nuclear spins in a magnetic field

42 https://wiki.metropolia.fi/display/Physics/Nuclear+magnetic+resonance

For example, the energy difference between electron spin-up & spin-down for B=1T is consequently about 11.6*10-5 eV

Spectral bands from the electromagnetic spectrum

Outline • Light-matter interaction

– Visible, outer electrons – X-rays, inner electrons – Infrared, molecular vibrations – Micro- and radiowaves, electron & nucler spins

• Line widths • Detection modes

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Line widths of spectroscopic signals – Optical frequencies are close to 1015 Hz. – The frequency width of an atomic/molecular transition in

gas at low pressure is ~1 GHz due to Doppler broadening and 10-100 GHz due to collisions at atmospheric pressure

– Below, part of solar spectrum. Many spectral lines can be discerned within a narrow interval

45 nanometers Fig 6.87, page 178

Line widths of spectroscopic signals – In liquids and solid state materials atoms/molecules are

much closer. Outer electrons interact from different atoms/molecules interact strongly, lifetimes are short and lines are much broader

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– However, electrons in deeper shells are shielded by the outer electrons. Lines can then still be narrow also in liquids and solids. E.g. in rare earth doped materials.

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Line widths of spectroscopic signals

Fig 2.22

Outline • Light-matter interaction

– Visible, outer electrons – X-rays, inner electrons – Infrared, molecular vibrations – Micro- and radiowaves, electron & nucler spins

• Line widths • Detection modes

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Detection modes

• Fluorescence • Absorption • Scattering

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Fluorescence spectroscopy

0 1 2 3 4 5

S0

0 1 2 3 4 5

S1

0 1 2 3 4 5

S2

Ener

gy

Abso

rptio

n

Fluo

resc

ence

Vibrational relaxation

Tissue autofluorescence

Fluorescing marker

Solids & liquids typically have significant vibrational (and rotational) relaxation

Absorption spectroscopy

Beer-Lambert law

Absorption coefficient: μa [cm-1] ”probability for absorption event per unit length”

μa = σ × N σ: cross section [cm2] N: concentration [cm-3]

Absorption measurement, example

600 700 800 900

(nm)

10 0

10 1

10 2

a ( c

m - 1

)

Absorption coefficients

Hb HbO2

Muscle

Scattering

• Elastic scattering (wavelength, λ, unchanged in the scattering process) – Rayleigh scattering, scattering on objects (atoms,

molecules, particles . . . etc.) much smaller than the wavelength, scattering cross section ~λ-4

– Mie scattering, scattering on larger particles

• Inelastic scattering (the wavelength, λ, is changed in the scattering process) – Raman scattering

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Raman Scattering

1930 Fig 2.18 54

Chandrasekhara Venkata Raman

In Raman scattering molecules undergo transitions in which an incident photon is absorbed and another scattered photon is emitted

Cross sections (σ) (page 69)

• Resonant absorption σ = 10-16 cm2

• Rayleigh scattering σ = 10-26 cm2

• Raman scattering σ = 10-29 cm2

• Mie scattering σ = 10-26-10-8 cm2

• With 1015 photons/cm2 the probability for

resonant absorption equals 10% etc.

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Vibration frequencies for different molecular groups

56 Page 161

Outline • Light-matter interaction

– Visible, outer electrons – X-rays, inner electrons – Infrared, molecular vibrations – Micro- and radiowaves, electron & nucler spins

• Line widths • Detection modes

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End

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