Quantum Physical Phenomena in Life (and Medical) Sciences

Péter Maróti Professor of Biophysics, University of Szeged, Hungary

Principles of spectroscopy

Suggested texts:

P. Maróti, G. Laczkó and L. Szalay: Medical Physics I-II, University of Szeged, 1988.

S. Damjanovich, J. Fidy and J. Szőlősi: Medical Biophysics, Semmelweis, Budapest 2006

P.W. Atkins: Physical Chemistry, Oxford University Press, Oxford, 5th Edition, 1994

Definition of the spectrumSpectrum: distribution of a physical quantity according to energy.

Physical quantities can be e.g. emitted or absorbed light quanta (photons)

Energy can be represented by wavelength, frequency, wave number etc.

1/ 21/ 2002 Vision 11

http:/ / insight.med.utah.edu/ Webvision/ imageswv/ spectra.jpeg

Example:

the absorption spectrum.

Selective absorption of the light by visual pigments of the cones and rod in the human eye. This is necessary to the color vision.

Electromagnetic spectrum

wavelength

radio microwave infrared ultraviolet X-ray

low energy high energyvisible light

frequency

energy

Principal types of spectroscopiesSpectroscopy Wavelength Observed energy,

Quantum transition

Gamma (γ)

Emission < 0.1 nm Gamma radiation coupled to transitions in nucleus

Mössbauer 0.01 – 10 nm Resonance-reabsorption of gamma radiation

Röntgen (X-ray) 0.01 – 10 nm X-rays from reactions in internal shells of the atom

Optical

Ultraviolet (UV) 200 – 300 nm

Electron transitionsVisible 400 – 800 nm

Infrared (IR)

0.8 – 200 μm Molecular vibrationsRaman

Magnetic and dielectric

Microwave 0.01 – 10 cm Molecular rotations

EPR

10 – 105 cm

Electron spins

NMR Nuclear spins

Alternative current (AC) 1 cm - ∞

Molecular conformationsDirect current (DC) ∞

Magnetic Resonance Spectroscopy

Electron Spin Resonance (ESR)or

Electron Paramagnetic Resonance (EPR)

and

Nuclear Magnetic Resonance (NMR)

Energy of transition

=E/h

(E = h

Energy ()

magnetic field strength (B0)0

Energy ()

magnetic field strength (B0)0

2

0hBhE

mI = ±½

mI = -½

mI = +½

ms = ±½

ms = -½

ms = +½

0BghE

EPR ~9.5GHz @ 0.34 T

NMR ~10-200 MHz @ 4.7 T

RT= 0.593 kcal/mol at 298oK

- frequency Hz

- wavenumber cm-1

- wavelength nm

~

c

1~

RT=0.002 kcal/mol at 10oK

Radiowave

Microwave

EPR and NMR spectroscopy

Magnetic induction (gauss, 100 μT)

singlet tripletquartet

Chemical shift

1) EPR spectrum: absorbed electromagnetic radiation as a function of the magnetic induction.

2) The magnitude of the applied magnetic induction is smaller than in EPR but electromagnetic radiation of larger frequency (microwave) is needed.

3) Spin-marker: compounds with stable and unpaired electrons.

4) Kinetic measurements: to track translation and rotation in the ms time domain.

1) NMR spectrum: absorbed electromagnetic radiation as a function of the magnetic induction. The area under the line (band) of the spectrum is proportional to the number of the absorbing nuclea.

2) The position of the lines depends on the interactions („chemical shift”): the structure of the electron clouds modifies the local magnetic field (induction) and therefore the condition for resonance absorption is shifted. Excellent tool for determination of the chemical structure.

The protons in three positions offer three different groups of lines.

OHCH2 CH3

Magnetic Resonance Spectroscopy

1. Electrons have “spin”, - rotation of the charge about its axis generates a magnetic field at each electron.

2. Electrons in orbitals with two electrons are spin-paired, - one with ms +½ (, ), one with ms -½ (, ), - so that the spins and magnetic fields cancel ( ).

3. Most molecular bonds are formed by coalescence of atomic orbitals so as to satisfy the lower energy state arising from spin-pairing. Most molecules therefore have all orbitals occupied by magnetically “silent” electron pairs.

4. If an electron is added to or subtracted from such a molecule (by reduction or oxidation) an unpaired electron is introduced which is not spin coupled, and therefore acts as a magnet. Such molecules are said to be paramagnetic.

5. Some bonds have unpaired electrons, giving them an inherent paramagnetic property.

6. In EPR spectroscopy, we take advantage of this paramagnetic property to measure properties of the the paramagnetic center, and its interaction with other local magnets. These interactions provide information about local structure, including local environment, distances, angles, polarity, etc.

Electron Paramagnetic Resonance (EPR) spectroscopy

Electron Paramagnetic Resonance

Energy ()

magnetic field strength (B0)0

ms = ±½

ms = -½

ms = +½

0BghE

The photon energy for resonance depends on the applied field. The field aligns the spins, to give a splitting of the electron energy levels, with a population in the ms +½ state that is greater than that in the ms -½ state. Absorption of a photon at the resonance energy flips the spins. The net absorbance is due to transitions from ms +½ to ms -½ levels.

No field External field applied

Absorption of microwaves changes the energy level so that the small fraction of spins in the lower energy state are flipped into the opposite orientation. Saturation occurs when flips in both directions occur with equal probability. The power needed to saturate depends on the relaxation time of the spin transition.

Relaxation

The general principles of EPR and NMR are essentially the same, but with the differences in energies.

However, the nomenclature is different, mainly for historical reasons. The resonance condition for EPR is described by the equations:

Here, g, the dimensionless Landé g-value, describes the resonance energy. Values around 2.003 are found for simple free-radical systems. The term (written also in form of μB) is the Bohr magneton:

= eh/(4 me) = 927.4 x 10-26 JT-1 in SI units, or

= /h = e/(4 me) = 13.996 x 109 Hz T-1 in frequency units, or

eV = /(e/C) = 5.7884 x 10-5 eV T-1 in electron volt units, where (e/C) is the electron charge, 1.602 x 10-19 C.

02

1

2

1 BgEEhE

smBgE 0

The nomenclature for EPR

Nuclear Magnetic Resonance (NMR) spectroscopy

1. The atomic nucleus is made up of protons (+ charge) and neutrons (neutral).

2. Like electrons, protons and neutrons (or nucleons) are quantum mechanical entities, and their energetic properties can be described by operators, wave functions, and quantum numbers.

3. Both protons and neutrons have a spin ½, with spin quantum number, mI, which can have values of (up, +½, or ), or (down, -½, or ).

4. Total nuclear energy levels are lower if the constituent protons and neutrons are spin-paired, so that for most nuclei, the spins (and magnetic fields) cancel. Protons are spin-paired with protons, neutrons with neutrons.

5. Isotopes that have an odd number of either protons or neutrons have a net spin, according to the Table below:

Number of protons Number of neutrons Nuclear spin, (I)

Even Even 0

Even Odd 1/2 or 3/2 or 5/2 …

Odd Even 1/2 or 3/2 or 5/2 …

Odd Odd 1 or 2 or 3 …

Energy ()

magnetic field strength (B0)0

2

0hBhE

mI = ±½

mI = -½

mI = +½

Nuclear Magnetic Resonance

In the absence of an external magnetic field, the spins of the nuclei are arranged more or less at random. When a magnetic field is applied, the spins align with the field, just like bar magnets. However, because the energy involved is so small, they can flip direction relatively easily.

The consequence is that the energy level of the nucleus splits in a magnetic field, as shown on the right. The population of spins in the mI +½ () state is (very slightly) higher than in the mI -½ () state, because the energy is lower.

When photons of the “right” energy are absorbed, the spin of the nucleus flips between the two states. If we look at the diagram, we can see that the energy gap between the two states is dependent on the strength of the applied magnetic field (magnetic induction, B0). As a consequence, photons are absorbed which have a particular energy at a particular field. This is what is meant by resonance.

Here, is the gyromagnetic ratio for the nucleus, B0 is the magnetic induction, and h is the Planck’s constant.

When we come to discuss pulsed NMR, we will need to refer to circular motion. Here the preferred frame of reference is that of the circular motion of precession. We define the precession or Larmor frequency:

2

0hBhE

00 2 B

Resonance conditions at NMR

Some consequences of the energy scale.

The transitions leading to NMR absorption have energies in the radio frequency range, depending on nucleus () and the strength of the magnetic field generated by the magnet.

NMR machines are rated by the frequency at which the proton is in NMR resonance for the magnet they are built around, so we have 200 MHz, 500 MHz, 750 MHz and even 1 GHz NMR spectrometers. To achieve the higher fields, high electrical currents are needed, which can be achieved using superconducting coils, - these are generally called superconducting magnets (costing $M).

Since = 2=B0, for a 200 MHz machine we need a magnet generating 4.7 Tesla; for a 500 MHz machine we need 11.74 Tesla, etc.

As we increase the energy gap (increase frequency), the small differences of energy for transitions of nuclear magnets in different environments are also increased, and our NMR spectrum will be better resolved. In addition, we also increase the population difference for the two spin states as we increase the energy (E/RT is increased), as discussed in the next slide.

These factors make a big difference in the amount of time needed to generate a data set, - for example in solution of the structure of a protein.

As we have seen, the energy of electromagnetic waves is generally expressed in frequency, but energy scales are all related through Planck’s constant and the speed of light, so we can express these energies in J/mol, eV, or any other energy units.

For a particular temperature, any transition can be described in terms of an equilibrium between two states. For example, when we flip the energy level of a proton in a magnetic field, we have two states, or , and or , separated by E, calculated as above. Let us use N and N to represent the relative populations in the two states. We can represent the equilibration between these two states by N NThen the ratio of these populations is given by the Boltzman distribution:

N / N = e E / RT

Remembering that the energies of the transitions associated with NMR (and those of EPR) are both very much less than RT, we can see that only a very small excess of states will be in the lower energy level. For the proton at 500 MHz, at ~25oC, since E = h, and Planck and Avagadro constants are 6.626.10-34 J.s and 6.02.1023 mol-1 respectively, energies in the RF range (~500 MHz) are 0.2 J/mol.

N / N = 1.00008

It is these one in ten thousand spins that are available to absorb a photon to provide an NMR signal.

Molecular vibrationsVibrational spectroscopy

Frederick William Herschel(~1800)

He discovered the non-visible (infrared) part of the spectrum of the sun.

Elektromagnetic spectrum

X-RAY0.2 nm

ULTRA-VIOLET2 nm

VISIBLE400-800 nm

INFRARED MICROWAVE3 mm-20 cm

RADIO10 m-30 km

FAR IRMid IRNEAR IR

wavelength (cm) 7.8·10-5 to 3·10-4 3·10-4 to 3·10-3 3·10-3 to 3·10-2

wavenumber (cm-1) 12820 to 4000 4000 to 400 400 to 33

Basic (normal) vibrations

Overtones and combinations of the normal vibrations

Rotations

Information about the molecules

3. Particles (atoms and molecules) in parabolic potential well

02 xL

220 v

2

1

2

1mxkE

The energy profile of a harmonic oscillator is parabolic. The particle is freely moving inside the parabolic potential well.

Semiclassic treatment

The total energy of the particle is

which is equal to the kinetic energy between the walls.

Those energy levels are allowed which can be covered with standing waves

Ln 2

n

Possible velocities of the particlen0n

n 244v

E

k

m

hn

mx

hn

m

h

Possible energies of the particle

,...3,2,1,0where4

28v

2

1

n

n

222nn

nhnE

E

k

m

hnmE

Energy levels of the harmonic oscillator

Solution from the Schrödinger equation:

En = h·ν(n + ½), where n = 0, 1, 2, ...

Conclusions:

(a) The energy levels of the oscillator are equidistant: ΔE = h·ν.

(b) The selection rule: Δn = ± 1.

Transitions are allowed between adjacent states only (in first order). Transitions among not neigboring levels are forbidden.

(c) The minimum energy of the oscillator (n = 0) differs from zero: E0 = ½ hν. The zero-point energy is not zero, because of the uncertainty principle: the particle is confined, its position is not completely uncertain, and therefore its momentum, and hence its kinetic energy, cannot be exactly zero.

For typical molecular oscillator, the zero-point energy is about

E0 = 3·10-20 J = 100 meV

The Boltzmann-energy at room temperature is ½ kBT = 25 meV

Uniform ladder of spacing h·ν

Infrared (IR) spectroscopyThe energy levels are associated with vibrations of bonds in molecules.

Each bond type will absorb IR light at one wavelength only, but there are different modes of vibrations:

(a)valence vibrations and

(b)straching vibrations.

Bond Wavelength (μm)

C – N (aliphatic) 3.3 – 3.7

C – H (aromatic) 3.2 – 3.3

C – O 9.55 – 10.0

C – O (aldehydes) 5.75 – 5.80

S – H 3.85 – 3.90

Example: normal modes of vibrations of CO2

All (arbitrarily complex) vibrations of a molecule can be decomposed into the sum of normal modes.

Normal vibrations

R H out of plane bending

R

R

H

H

R

R

H

H

R

R

H

H

R

R

H

H

R

R

H

H

RH

antiszimmetric szimmetric

rocking

in plane bending

stretch

scissoring

wavenumber, 1/λ

Measuring methodsMeasuring methods

IR Reflexion IR Transmission

Detector

Detector

IR TransmissionIR

Reflexion

IRAbsorption

IR lightIR light

Advantages of the IR spectroscopy

Non-invasive determination of the sucrose content of the blood

IR light source

Detector

Advantages of the IR spectroscopy

Remote sensing and measurement of hazardous materials

GratingSample

Fourier transform (FT) techniques in the infrared (IR) spectrometers (FTIR)

xkkIxI 2cos1)()(

The heart of the FTIR spectrometer is a Michelson interferometer which works by splitting the beam from the sample into two and introducing a varying path difference x into one of the two arms. When the two components recombine, there is a phase difference between them, and they interfere either constructively or destructively depending on the extra path that one has taken. The detected signal oscillates as the two components alternately come into and out of phase as the path difference is changed. If the radiation has wavenumber k (=1/λ), the detected signal varies with x as

The interferometer converts the presence of a particular component in the signal into a variation in intensity of the radiation reaching the detector. An actual signal consists of radiation spanning a large number of wavenumbers, and the total intensity at the detector is the sum of all their oscillating intensities

0

d)2cos(1)()( kxkkIxI

The problem is to find I(k), the variation of intensity with wavenumber, which is the spectrum we require, from the record of values I(x). The step is a standard technique of mathematics: Fourier transformation. known unknown

Raman spectroscopyAbout 1 in 107 of the incident laser photons collide with the molecules in the sample, give up some of their energy to the molecular vibrations, and emerge with a lower energy (Stokes radiation).

Other incident photons may collect energy from the molecular vibrations, and emerge as higher-frequency (anti-Stokes) radiation.

Basic properties:

-The shifts in frequency are small (good monochromator is needed) and

-- the intensities of the lines are low (high laser energy is needed).

An advantage of Raman spectroscopy over infrared spectroscopy is that the radiation can be entirely in the visible range, so the complications arising from needing to select a range of infrared-transparent sample cells are avoided.

Comparison of Raman and IR spectroscopies:

advantages and disadvantages

The two methods utilize complementer properties of the same molecule.

Differences between IR and Raman methods

  Raman IR

1It is due to the scattering of light by the vibrating molecules.

It is the result of absorption of light by vibrating molecules.

2The vibration is Raman active if it causes a change in polarisability.

The vibration is IR active if there is a change in dipole moment during the vibration.

3The molecule need not possess a permanent dipole moment.

The vibration concerned should have a change in dipole moment due to that vibration.

4 Water can be used as a solvent.Water cannot be used due to its intense absorption.

5Sample preparation is not very elaborate sample can be almost in any state.

Sample preparation is elaborateGaseous samples can rarely be used.

6Gives an indication of covalent character in the molecule.

Gives an indication of ionic character in the molecule.

7 Cost of instrumentation is very highComparatively inexpensive.

Complexity and complementarity of the infrared (IR) and the Raman spectra

wavenum

ber, 1/cm

streching of the methyl group

rocking of the methyl group

deformation of the methyl group

Hoop: out-of-plane motion of H atoms

Double bond vibrations

Single bond vibrations

Electron Transitions

to rebrush your mindsee the previous chapters on

„Electron in hyperbolic and rectangular shape of potential wells”

and

for additional information see the next slide-show on

„Visible absorption and fluorescence spectroscopy: Luminescence”

Problems for Seminar

1. An electron is confined to a molecule of length 1.0 nm. What is (a) its minimum energy and (b) the minimum excitation energy from that state?

2. Estimate a typical nuclear excitation energy by calculating the first excitation energy of a proton (mass of m = 1.673·10-27 kg, the Planck’s constant is h = 6.626·10-34 J·s) confined to a region roughly equal to the diameter of a nucleus, 1·10-14 m.

3. Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33·10-26 kg and force constant 155 N/m.

4. For a harmonic oscillator consisting of a particle of mass 1.33·10-25 kg, the difference in adjacent energy levels is 4.82·10-21 J. Calculate the force constant of the oscillator.

5. Calculate the minimum excitation energies of (a) a pendulum of length 1.0 m on the surface of the Earth, (b) the 33 Hz quartz crystal of a watch and (c) the bond between two oxygen atoms in the O2 molecule, for which the force constant is k = 1177 N/m.

Problems for Seminar

6. Determine the wavefunction and the probability distribution of the harmonic oscillator in the first excited state. Compare these functions with those of the ground state.

7. Calculate the longest and the shortest wavelength transitions in the Balmer series of atomic hydrogen.

8. Do the same with the Lyman and Paschen series of the H atom.

9. Calculate the saparation between the two lowest levels for an oxygen molecule in a one-dimensional container of length 5.0 cm. At what value of n does the energy of the molecule reach 1/2·kBT at 300 K, and what is the separation of this level from the one immediately below?

10. To a crude first approximation, a π-electron in a linear polyene may be considered to be a particle in a one-dimensional box. The polyene β-carotene contains 22 conjugated C atoms, and the average internuclear distance is 140 pm. Each state up to n = 11 is occupied by two electrons. Calculate the separation in energy between the ground state and the first excited state in which one electron occupies the state with n = 12. What is the wavelength of the radiation required to produce a transition between these two states?

Problems for Seminar

H35Cl H81Br HI CO NO

2990 2650 2310 2170 1904

Calculate the force constants of the bonds and arrange them in order of increasing stiffness.

12. An Ar atom rotates in a circle around a fixed axis (z) with orbital angular momentum quantum number ml = 2. Its energy of rotation is Erot = 2.47·10-23 J. Calculate the radius (r) of the rotation. (The angular momentum about the axis (Jz = r·p) is quantized: Jz = ml ·(h/2π).)

11. The inverse value of the reduced (effective) mass of a diatomic molecule is the sum of the inverse values of the individual masses of the atoms mA and mB: 1/μ = 1/mA + 1/mB. The table contains the infrared absorption wavenumbers (in cm-1) of the following diatomic molecules: