Lecture 2 Origin of the NMR Signal

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Lecture 2Origin of the NMR Signal

• Linda K. Nicholson
• Jan. 29, 2003

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Helpful Math Background

• 1. vectors (pp. 4 - 5, Farrar Becker, handout)
• 2. derivatives (pp. 53 - 56, Math Handbook,
  handout)
• 3. Taylor series (pp. 110 - 113, Math Handbook,
  handout)
• 4. linear algebra (e.g. pp. 176 - 237, G.
  Arfken, Mathematical Methods for Physicists)

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What gives rise to an NMR signal in a protein?

• Spectroscopy
• Nuclear Spin
• Nuclear spin interactions
• Proteins and spin-1/2 nuclei
• Proteins and quadrupolar nuclei

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Electromagnetic radiation

• Electromagnetic radiation is composed of magnetic
  and electronic waves
• From R.S. Macomber (1988) NMR spectroscopy
  Essential Theory and Practice
• The frequency is defined as n 1/to, where to is
  the peak-to-peak time.
• A wave travels l (distance) in to, so that the
  speed of the radiation (c, the speed of light,
  3x108 m/s) is defined as

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• From p. 24, Spin Dynamics1
• 1Spin Dynamics Basics of Nuclear Magnetic
  Resonance, by Malcolm H. Levitt, John Wiley, 2002

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• Radiofrequency energy (DE for nuclear spin state
  transitions)
• l 1011 to 3 x 107 nm
• n 106 to 1010 Hz
• By setting the frequency of electromagnetic
  radiation (n, or equivalently w) to the resonance
  condition, transitions between nuclear spin
  states can be induced
• (i.e. one can do NMR spectroscopy!).

allowed spin states
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What gives rise to an NMR signal in a protein?

• Spectroscopy
• Nuclear Spin
• Nuclear spin interactions
• Proteins and spin-1/2 nuclei
• Proteins and quadrupolar nuclei

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The Nuclear Magnet

• An individual nucleus
• A nucleus can be thought of classically as a
  tiny bar magnet that has a local magnetic field
  associated with it, expressed by the magnetic
  moment vector.

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Angular momentum

• A rotating object possesses angular momentum

Figure 1.1, p.6, Spin Dynamics1
right hand rule
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Angular momentum is quantized

• Example Rotational energy of a molecule
• At the level of atoms and molecules, only
  specific rotational states are allowed

Ltot J(J 1)1/2h h 1.054 x 10-34 Js
Figure 1.2, p.6, Spin Dynamics1
diatomic molecule
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rotational energy of a molecule

• Is proportional to the square root of the total
  angular momentum
• EJ BJ(J 1)
• B rotational constant for the molecule (size)
• The rotational motion of proteins is treated
  classically as an ordinary rigid body

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Different types of angular momentum

• Rotational angular momentum of atom or molecule
• Orbital angular momentum of electron
• Spin angular momentum of electron
• Spin angular momentum of nucleus

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Spin angular momentum

• An intrinsic property (not due to rotation)
• Is quantized S(S 1)1/2h
• Particles with spin S have 2S 1 sublevels
  (degenerate without B or E field)
• bosons particles with integer spin
• fermions particles with half-integer spin
• Arises from nothing, it just is

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Combining angular momenta
Total angular momentum of system with two sources
of angular momentum J1 and J2 is given by J3(J3
1)1/2h, where
Figure 1.3, p.6, Spin Dynamics1
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The Pauli principle

• Two fermions may not have identical quantum
  states
• Explains
• Periodic table
• Stability of chemical bond
• Conductivity of metals

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The fundamental particles

• leptons low mass, 6 varieties
• electron (electric charge of e, spin ½)
• quarks heavy, six varieties, all spin ½
• three with charge 2e/3
• three with electric charge e/3
• force particles mediate interactions
• photons (no mass, no electric charge, spin 1)
• gluons (strong nuclear force, holds nucleus
  together)
• vector bosons (weak nuclear force, radioactive
  b-decay)

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Neutrons and protons
3 quarks, stuck together by gluons
Figure 1.4, Spin Dynamics1
Figure 1.5, Spin Dynamics1
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Nuclear spin energy levels(in absence of
magnetic field)
Figure 1.7, Spin Dynamics1
Figure 1.6, Spin Dynamics1
Ground state nuclear spin empirical property of
each isotope
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Guidelines for determining spin of isotopes

• mass number atomic number (Z) S
  Detectable
• odd even or odd 1/2, 3/2,
  5/2 ... yes
• even even 0
  no
• even odd 1, 2, 3 ...
  yes
• Possible number of spin states 2S 1
• 1H S 1/2 2(1/2) 1 2 m 1/2
• 14N S 1 2(1) 1 3 m -1, 0, 1

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NMR-active nuclei in proteins
H
H
O

• Naturally abundant
• 1H, spin ½
• 31P, spin ½
• Enriched via bacterial expression
• 2H, spin 1
• 13C, spin ½
• 15N, spin ½

N
C
Ca
H
Cb
H
H
H
H
H
OH
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What is the relationship between the magnetic
moment vector and the angular momentum vector?

• Any motion of a charged body has an associated
  magnetic field. The flow of negatively charged
  electrons through a loop of wire has a magnetic
  moment (µ) whose magnitude is equal to Ai, where
  A is the area enclosed by the loop and i is the
  current. The direction of the resulting magnetic
  moment is specified by the right hand rule.

m Ai
i
A
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• For an electron in a Bohr atom, the current
  (opposite the direction of the movement of the
  electron) is given by i -ew/2p, where e is the
  charge on the electron and w is the angular
  velocity vector for the electron. The angular
  momentum, P, is P mer2w, where me is the mass of
  the electron and r is the distance from the
  nucleus. The current is

Figure 2.3, Spin Dynamics1
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For an electron in a Bohr atom

• The magnitude of the magnetic moment (µ) from
  the definition above is Ai. Since the electrons
  are assumed to orbit in a circular path, the area
  is pr2, so
• The vector expression is written as
• (Note that for an electron, the angular momentum
  and magnetic moment vectors are antiparallel).
  The expression for could also have been derived
  from the classical expression , where
  is a unit vector pointing from the nucleus to the
  orbiting electron.

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• Defining µB (the Bohr magneton) as , the
  expression is rewritten as
• A proportionality factor (g-factor) is
  introduced to allow for generality between
  orbital and spin angular momentum. The g-factor
  is unity for orbital angular momentum, and is
  approximately 2 for spin angular momentum
•  

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The gyromagnetic ratio, g

• An analogous equation can be written for spin
  angular momentum of the nucleus, where gN is the
  nuclear g-factor and µN is the nuclear magneton
• We now define the gyromagnetic ratio (g) as the
  magnitude of the ratio of the magnetic moment to
  the spin angular momentum
• and the relationship between angular
  momentum and magnetic moment becomes
• Hence, the angular momentum and magnetic moment
  vectors associated with nuclear spin are pointed
  in the same direction and are related by a
  constant.

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Gyromagnetic ratio g
Figure 2.5, Spin Dynamics1
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Magnetic moment due to spin
Figure 2.4, Spin Dynamics1
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Magnetic susceptibility, c

• How easily a material develops a magnetic moment
  upon exposure to a magnetic field

minduced mo-1VcB
Figure 2.2, Spin Dynamics1
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Effect of a magnetic field

• No magnetic field 2I 1 spin states are
  degenerate (i.e. they all have the same energy).
• With magnetic field Spin states separate in
  energy (larger values of m have lower energy)
• The separation of energy levels in a magnetic
  field is called the nuclear Zeeman effect. The
  energy of a spin state is given by

E -m?Bo
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Emag -m?B

• Magnetic energy depends on the relative
  orientations of m and B

Figure 2.1, Spin Dynamics1
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• We know that the magnitude of the angular
  momentum vector is fixed by the value of the
  nuclear spin quantum number
• and that the z-component of the angular momentum
  vector is given by
• Iz hm
• where m is the magnetic quantum number
• m (-I, -I1, ..., I-1, I)
• Iz has 2I1 possible values

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• Thus, the discrete values of Iz are always
  smaller than I, and m can never be aligned with
  Bo. The minimum energy occurs when the
  projection of m onto B is the greatest. Hence,
  the energies of the m allowed spin states are
  proportional to their projection onto Bo
• where Em the energy of the state
• m magnetic quantum number
• Bo magnetic field strength
• g gyromagnetic ratio
• h Plancks constant/2p

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Without external magnetic field

• random orientations
• degenerate

Figure 2.6, Spin Dynamics1
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Alignment of spins in Bo
Figure 2.7, Spin Dynamics1

• Compass needle

Bo
Iz h/2
I(h/2)v3
Iz -h/2
Spin ½ magnetic moment
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precession
Figure 2.10, Spin Dynamics1
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The basis for precession

• Motion of Nuclear Magnets in Presence of Magnetic
  Fields

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