fundamentals of spectroscopy - atomic physics · forces in atoms and molecules • forces in the...

Fundamentals of spectroscopy

1

Spectral bands from the electromagnetic

spectrum

Outline

• Interactions between electro-magnetic fields and

matter

– Visible, outer electrons

– X-rays, inner electrons

– Infrared, molecular vibrations

– Micro- and radiowaves, electron & nuclear spins

• Line widths

• Detection modes

3

Interactions between electromagnetic

radiation and sample

• Electro-magnetic radiation transfer energy

• Sample is composed of atoms, molecules

• By examining the resulting electro-

magnetic 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

5

Spectral bands from the electromagnetic

spectrum

Energy levels in an atom

λ

Energy

Principal quantum number n determines

the distance from the nucleus

How would the atmosphere look at ~120 nm?

What electron orbits are allowed?

An accelerating/deccelerating charge emits radiation

For example, a synchrotron sends out radiation in every bend

Why does an electron orbiting a nucleus then not send out

radiation, lose energy and finally collapse and hit the nucleus?

Fig 2.3

The de Broglie wavelength, l, is given by mv=h/l Eq. 2.2.

What electron orbits are allowed?

Schrödinger Equation

Erwin

Schrödinger

1925

The electron cloud is described by a wave-function, Y

Niels Bohr

1913

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

Generation of electro-magnetic fields

Figure from G Jönsson & E Nilsson

Våglära och Optik, Teach Support

11

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

14

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?

15

The transition probability from one stationary state to

another is proportional to

𝐹𝑖𝑛𝑎𝑙 𝑠𝑡𝑎𝑡𝑒 × 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑓𝑖𝑒𝑙𝑑 × 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑡𝑎𝑡𝑒

𝑆𝑝𝑎𝑐𝑒

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

16

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 for example not an

s-electron to f

Strongest line

Even Odd Even Odd

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

http://cwx.prenhall.com/bookbind/pubbooks/

hillchem3/medialib/media_portfolio/07.html 18

Energy levels in Helium

Singlets

Total spin is zero

Triplets

Total spin is one

Fig 2.6

Fig 2.7

Energy levels in Calcium

Which line for detection?

When to use

intercombination line?

Spectral bands from the electromagnetic

spectrum

Oscillations

• The oscillation frequency of a system

depends on the mass(es) involved and the

restoring force

22

https://en.wikipedia.

org/wiki/Oscillation

𝑓 =1

2𝜋

𝑘

𝑚 f = oscillation frequency

k = spring constant

m = mass

The spring exerts a force F = kx on a mass, m, where

x is the displacement from the equilibrium position

How could we change the electron ”spring constant” in an atom?

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

23

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

26

27

Bremsstrahlung

X-rays can be produced by

accelerating/deccelerating

charges. The radiated power

is proportional to the

acceleration/deccelation

squared

Section 7.5.1

28

Collisions can excite inner shell electrons 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

29

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

Fig 2.8

30

Absorption of X-ray radiation as a function

of X-ray wave length

Cu, Z = 29

Ag, Z = 47

page 276

Sune Svanberg, Atomic

and molecular

spectroscopy, Springer

Verlag, Fig 5.1

31

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

33

Fig 2.15

Page 25

How could we estimate the vibration frequency?

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

34

𝑓 =1

2𝜋

𝑘

𝑚

Energy separation in molecules

Fig 2.10, page 23

Spectral bands from the electromagnetic

spectrum

Molecular energies

37

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 = ℎ𝑣 𝐸(𝑒𝑉) =ℎ𝑣

𝑒

Compare Eq 2.1, page 16 & Fig 2.2, page 17

Vibration frequencies for

different molecular groups

39 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

41

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 m = IAân,

where ân is a unit vector normal to the

surface A.

m

42

Spin

• Electrons have spin, s=1/2.

http://cwx.prenhall.com/bookbind/pubbooks/

hillchem3/medialib/media_portfolio/07.html

43

Magnetic moments in atoms

• Orbital magnetic moment, 𝝁𝐿 = −𝜇𝐵L

• Spin magnetic moment, 𝝁𝑠 = −𝑔𝑠𝜇𝐵S

• 𝑔𝑠2,

• Nuclear magnetic moment, 𝝁𝐼 = 𝑔𝐼𝜇N I

• I is the nuclear spin

44

𝜇𝑁𝜇𝐵=𝑚𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑚𝑝𝑟𝑜𝑡𝑜𝑛

≈1

2000

Interaction between a magnetic

moment and a magnetic field

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

m

The energy, E, of a magnetic moment, m, in

a magnetic field B is given by the scalar

product E=-mB 45

Electron & nuclear spins in a

magnetic field

46

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

As E=-mB the energy difference between electron spin-up & spin-down for, e. g., B=1T is

consequently about 11.6*10-5 eV (~30 GHz, ~1 cm)

p

Spectral bands from the electromagnetic

spectrum

Outline

• Interactions between electro-magnetic fields and

matter

– Visible, outer electrons

– X-rays, inner electrons

– Infrared, molecular vibrations

– Micro- and radiowaves, electron & nuclear spins

• Line widths

• Detection modes

48

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

49

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

50

– 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.

51

Line widths of spectroscopic signals

Fig 2.22

Outline

• Interactions between electro-magnetic fields and

matter

– Visible, outer electrons

– X-rays, inner electrons

– Infrared, molecular vibrations

– Micro- and radiowaves, electron & nuclear spins

• Line widths

• Detection modes

52

Detection modes

• Fluorescence

• Absorption

• Scattering

• Reflection

53

Fig 2.16

Page 25

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

Ab

so

rpti

on

Flu

ore

sc

en

ce

Vibrational

relaxation

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 = s × N

s: 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, l, unchanged in

the scattering process)

– Rayleigh scattering, scattering on objects (atoms,

molecules, particles . . . etc.) much smaller than the

wavelength, scattering cross section ~l-4

– Mie scattering, scattering on larger particles

• Inelastic scattering (the wavelength, l, is changed

in the scattering process)

– Raman scattering

57

In Raman scattering molecules undergo transitions in which an

incident photon is absorbed and another scattered photon is emitted

at a different wavelength

Raman Scattering

1930

Fig 2.18

58

Chandrasekhara

Venkata Raman

Vibration frequencies for

different molecular groups

59 Page 161

Cross sections (s) (page 69)

• Resonant absorption s = 10-16 cm2

• Rayleigh scattering s = 10-26 cm2

• Raman scattering s = 10-29 cm2

• Mie scattering s = 10-26-10-8 cm2

• With 1015 photons/cm2 the probability for

resonant absorption equals 10% etc.

60

Outline

• Interactions between electro-magnetic fields and

matter

– Visible, outer electrons

– X-rays, inner electrons

– Infrared, molecular vibrations

– Micro- and radiowaves, electron & nuclear spins

• Line widths

• Detection modes

61

End

62