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Home , Spin (physics)

Lecture #8 Nuclear Hamiltonian* • Topics – Introduction – External Interactions – Internal Interactions • Handouts and Reading assignments – van de Ven: Chapters 2.1-2.2 – Biographies: Ramsey, Zeeman – Levitt, Chapters 7 (optional)

* many figures in this lecture from Levitt. 1 The Spin Density • Spin density operator: σˆ (t) – Completely describes the state of a spin system. Isnt the trace a matrix • The expectation of any A ˆ : rather than operator € operation? ˆ ˆ ˆ pure state: A = Tr{σ ψ A} inner product in Liouville space € statistical mixture of states: Aˆ = Tr{σˆ Aˆ } € • Example: x, y, and z : ˆ ˆ My = γ!Tr{σ I y} ˆ ˆ ˆ Mx = γ! I x = γ!Tr{€σ I x} similarly M = γ!Tr σˆ Iˆ z { z} 2 € € € € Liouville-von Neuman Equation • of σ ˆ ( t ) : ∂ ˆ σˆ = −i[Hˆ ,σˆ ] = −iHˆ σ ˆ ∂t • Hamiltonian Hˆ - operator corresponding to energy of the system € • If H ˆ time independent:€ Hˆ (t) = Hˆ ˆ € I z σˆ ˆ I y −iHˆ t iHˆ t ˆ σˆ (t) = e σˆ (0)e = e−iHt σˆ (0) σˆ rotates around in operator space € Superoperator€ notation textbook notation ˆ € I x € € € • Key: find the Hamiltonian!

€ 3 The Spin Hamiltonian Revisited • In general, H ˆ is the sum of different terms representing different physical interactions. ˆ ˆ ˆ ˆ H = H1 + H 2 + H 3 +! Examples: 1) interaction of spin with B0 € 2) interaction with of other nuclei 3) spin-spin • Life€ is easier if: € ˆ – H i are time independent. Terms that depend on spatial orientation may average to zero with rapid molecular tumbling (as if often seen in vivo). ˆ ˆ € – H i, H j terms commute, ˆ then e − i H tσ ˆ can be computed in any order.

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ −i( H 1 + H 2 )t −iH 1t −iH 2t −iH 2t −iH 1t [H i,H j ] = 0 e = e e = e e € 4 €

€ € Electromagnetic Interactions • Think of an as a lumpy magnet with non- uniform positive • Nuclear spin Hamiltonian contains terms which describe the orientation dependence of the nuclear energy

Nuclear interacts with magnetic fields

Nuclear electric charge interacts with electric fields

ˆ ˆ elec ˆ mag H = H + H 5

€ Electromagnetic Interactions quadrapole • Electric Interactions monopole dipole Nuclear electric charge distributions can be expressed as a sum of ! (0) ! (1) ! (2) ! multipole components. C (r ) = C (r ) + C (r ) + C (r ) +" Based on symmetry arguments relating the shape of a nucleus and its spin: C(n)=0 for n>2I, and, within experimental error, odd terms disappear (not obvious) Hˆ elec = 0 (for spin I =1/2) Hence, for spin-1/2 nuclei there are no electrical € energy terms that depend on orientation or internal nuclear structure, i.e. Spin-1/2 nuclei behaves exactly like point charges! • Magnetic Interactions (weve seen this before) € Nuclear magnetic moment is primarily just the dipole moment.

magnetic moment ! ! ! ! ˆ mag ˆ H = −µˆ ⋅ B = −γ"I ⋅ B local 6

€ Spin Hamiltonian: Overview

Relative magnitudes Is tissue a solid or liquid? 7 External Magnetic Fields

• Static Field B0 ! ! B B e Hˆ static B Iˆ Iˆ = 0 z = −γ 0 z = −ω 0 z unit vector in z direction

€ • With RF €excitation (assuming B1 along the x axis) ! ! ! B = B cosωt e − sinωt e RF 1( x y ) Hˆ ext Hˆ static Hˆ RF ˆ ˆ ˆ = + = −γB0I z −γB1(I x cosωt − I y sinωt ) € In the rotating frame, near resonance (ω ≈ ω 0 ) € ˆ ext ˆ € H ≈ −ω1I x 8 € € Internal Interactions • In general, the form of the internal nuclear spin Hamiltonian is quite complicated.

• Well make two simplifying approximations. Secular approx. Molecular ˆ ˆ 0 ˆ 0 H int H int H int

1. Secular approximation: large B0 field dominate some of the internal spin interactions. € 2. Motional averaging: € with rapid molecular tumbling,€ some interaction terms fluctutate with time and can be replaced by their motionally averaged values. Terms with zero time-average are dropped. • Discarded internal spin Hamiltonian terms are responsible for spin relaxation. 9 Motional Averaging • Molecular motion

• Molecular orientation depends on time, hence secular Hamiltonian terms can be ˆ 0 written as H int ( Θ ( t ) ). These terms can be replaced by their time averages: τ ˆ 0 1 ˆ 0 ergodicity ˆ 0 ˆ 0 H int = τ ∫ H int (Θ(t))dt H int = H int p(Θ)dΘ 0 ∫ € p(Q)=probability density for molecule having orientation Q Isotropic materials: Hˆ isotropic 1 Hˆ 0 d int = N ∫ int (Θ) Θ € € normalization 10

€ Motional Averaging • Diffusion On the timescale of an NMR experiment, molecules in a liquid largely diffuse within a small spherical volume a few tens of microns in diameter (known as a diffusion sphere).

Intermolecular interactions Long-range interactions dont average to within the diffusion sphere zero, but are very small. average out to zero.

11 B0- Interactions When a material is placed in a magnetic field it is magnetized to some degree and modifies the field…

Electrons in an circulate about B0, Shielding: generating a magnetic moment opposing the applied magnetic field.

• Global effects: magnetic susceptibility s B0 = (1− χ)B0 field inside sample bulk magnetic susceptibility applied field

Hereafter well use B0 to refer to the internal field (to be revisited when we talk€ about field inhomogeneities and shimming). • Local effect: shielding constant (Dont confuse with the Different experience spin density operator!) different electron cloud densities. B = B0(1−σ) 12

€ The Zeeman Hamiltonian ! • The interaction energy! between! the magnetic field, B , and the magnetic moment, µ = γ I , is given by the Zeeman Hamiltonian. Whos this ! ! ! !ˆ Zeeman guy? Classical: E = −γB ⋅ I QM: Hˆ = −γB ⋅ I zeeman€ • The formal€ correction for chemical shielding is: !ˆ ! € Hˆ I (1 )B € 3 3 shielding zeeman = −γ −σ where σ = × • In vivo, rapid molecular tumbling averages out the non-isotropic components.

€ σ = σ iso = Tr(σ /3) (anisotropic€ components do contribute to spin relaxation) ! ˆ ˆ • Hence for B = [0,0,B0 ]: H Zeeman = −γ(1−σ)B0I z = −ωIˆ € z 13

€ € € Chemical Shift • Energy diagram: water fat

H2O ...(CH2)nCH3 e- ... E e-

is electrophilic

B = 0 B = B0 Frequency (Hz)

ν = γB0(1−σ)/2π water 220 Hz @ 1.5 T € € fat First complete theory of chemical shift€ was developed by Norman Ramsey in 1950. Usually plotted as 4.7 1.3 0.9 ppm relative frequency shielding 6 δ =10 (ν −ν ref )/ν ref frequency 14

€ Magnetic • Nuclei with spin ≠ 0 act like tiny magnetic dipoles.

# µ0 &# µ & Bµx = % (% (( 3sinθ cosθ) $ 4π '$ r3 '

permeability of free space Bµy = 0 # µ &# µ & B = % 0 (% ( 3cos2 θ −1 € µz $ 4π '$ r3 '( ) falls off as r3 Dipole at origin € Magnetic Field in y=0 plane

€

Lines of Bµz Bµx 15 Dipolar Coupling • Dipole fields from nearby spins interact (i.e. are coupled).

• Rapid fall off with distance causes this to be primarily a intramolecular effect.

Spins remain aligned with B0

Water with molecule tumbling in a magnetic field

Interaction is

time variant! 16 The Dipole Hamiltonian • Mathematically speaking, the general expression is: µ γ γ & "ˆ "ˆ 3 "ˆ " "ˆ " ) ! ˆ 0 I S where vector from H dipole = − 3 !( I ⋅ S − 2 (I ⋅ r )(S ⋅ r )+ r 2πr ' r * spin I to spin S • Secular approximation:

$ !ˆ !ˆ ' µ γ γ 2 Hˆ d& 3Iˆ Sˆ I S ) where d €0 I S 3cos 1 dipole = z z − ⋅ = − 3 "( ΘIS − ) € % ( 4πr dipole coupling angle between B0 constant and vector from spins I and S € ˆ - with isotropic tumbling (e.g. liquid): H dipole = 0 What is line splitting? - without tumbling (e.g. solid): line splitting

Note: big effects on relaxation for both cases.€ 17 J-Coupling • The most obvious interactions between neighboring nuclei is their mutual dipole coupling. • However this anisotropic interaction averages out to zero for freely tumbling molecules. • Consider the following NMR spectrum: Line splitting ?! Ethanol

CH3 OH CH2

Chemical Shift (ppm)

18 J-Coupling: Mechanism • At very small distances (comparable to the nuclear radius), the dipolar interaction between an electron and is replaced by an isotropic interaction called Fermi contract interaction. Energy Diagram A simple model of J-coupling less stable

I e e S I spin senses more stable of S spin I e e S !ˆ !ˆ independent of molecular orientation • Interaction energy ∝ −γ eγ n I ⋅ S Scalar coupling • What’s in a name? J-coupling Spin-spin coupling

€ Through-bond (vs through- space) interaction Indirect interaction 19 J-Coupling: Energy Diagram populations Zeeman Splitting J-Coupling Single quantum coherences ?? Energy

Were now Alternative energy considering pairs of diagram spins

• The J-Coupling Hamiltonian J-coupling constant (Hz) !ˆ !ˆ !ˆ !ˆ ˆ 2 JI S 2 J(Iˆ Sˆ Iˆ Sˆ Iˆ Sˆ ) H J = I JS = π ⋅ = π x x + y y + z z

product operators 20

€ € € Summary: Nuclear Spin Hamiltonian

• Free ignore for now: effect small or time average=0 vanishes for spin=1/2 ˆ ˆ ˆ ˆ ˆ H = H Zeeman + H dipole + H J + H quadrupole WI and WS are resonant frequencies in rotating frame: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Ω = γ(1−σ )B −ω = ω −ω H = −ΩI I z + −ΩS S z + 2πJ(I xS x + I yS y + I zS z ) I I 0 rf I ΩS = γ(1−σ S )B0 −ω rf = ω S −ω

€ • With RF excitation (assuming B1 along the€ x axis) € € ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ I ˆ S ˆ H = −ΩI I z − ΩS S z + 2πJ(I xS x + I yS y + I zS z ) −ω1 I x −ω1 S x

%ω I >> Ω and ω S >> Ω ˆ I ˆ S ˆ 1 I 1 S (secular approx.) H ≈ −ω1 I x −ω1 S x for & I S 'ω 1 , ω1 >> 2πJ € Above equations hold for both homonuclear and heteronuclear cases.

€ 21 € Next lecture: , Populations, and Coherences

22 Biography: Norman Ramsey

Norman F. Ramsey was born in Washington, D.C. and 'was educated in the United States and England; he earned five degrees in physics including the Ph.D. (Columbia 1940) and the D.Sc. (Cambridge, 1964). Ramsey's scientific research focused on the properties of molecules, atoms, nuclei and elementary particles and includes key contributions to the knowledge of magnetic moments, the structural shape of nuclear particles, the of nuclear , the thermodynamics of energized populations of atoms and molecules (e.g. those in masers and ) and . Ramsey not only contributed basic advances in the theoretical understanding of the physics involved in his research, he also made pioneering advances in the methods of investigation; in particular, he contributed many refinements of the molecular beam method for the study of atomic and molecular properties, he invented the separated oscillatory field method of exciting resonances and, with the collaboration of his students, he was the principal inventor of the atomic maser. The separated oscillatory field method provides extremely high resolution in atomic and molecular spectroscopy and it is the practical for the most precise atomic clocks; likewise the atomic hydrogen maser made even higher levels of spectroscopic resolution possible and it also functions as the basis for atomic clocks having the highest levels of stability for periods extending to several hours.' 'During World War II his involvement with MIT's Radiation Laboratory led to the development of 3 cm radar and later he was leader of the Delivery Group of the Manhattan Project in Los Alamos. In 1947 he moved to Harvard University where he continues research and writing as the Higgins Professor of Physics, Emeritus. His research has ranged from atomic beams to particle physics. Ramsey participated in the founding of Brookhaven National Laboratory and served as the first Chairman of its Physics Department. He was Chairman of the Atomic Energy Commission's High Energy Physics Advisory Panel in 1963 when the recommendation was made to build a 200 GeV accelerator. Ramsey was then instrumental in the creation of Fermilab as Founding President of the Universities Research Association (URA), the Laboratory's management organization, from 1966 until 1981. He smoothly oversaw the operation of the Laboratory from his URA offices in Washington when he was not personally visiting the site to be involved with Fermilab's successful development. … Ramsey has also served as Chairman of the General Advisory Committee of the Atomic Energy Commission … . He has won many awards, including the Rabi Prize, the Rumford Premium of the American Academy of Arts and Sciences, and the 1988 National Medal of Science. In 1989 he received the for Physics for his research leading to the development of the hydrogen maser and the cesium .' 23 Biography: Pieter Zeeman

Born at Zonnemair in the Netherlands, Zeeman studied at Leiden University and received a doctorate in 1893. This was for his work on the Kerr effect, which concerns the effect of a magnetic field on light. In 1896 he discovered another magnetooptical effect, which now bears his name – he observed that the spectral lines of certain elements are split into three lines when the sample is in a strong magnetic field perpendicular to the light path; if the field is parallel to the light path the lines split into two. This work was done before the development of quantum , and the effect was explained at the time using classical theory by Hendrik Antoon Lorentz, who assumed that the light was emitted by oscillating .

This effect (splitting into three or two lines) is called the normal and it can be explained using 's theory of the atom. In general, most substances show an anomalous Zeeman effect, in which the splitting is into several closely spaced lines – a phenomenon that can be explained using and the concept of electron spin.

Zeeman was a meticulous experimenter and he applied his precision in measurement to the determination of the in dense media, confirming Lorentz's prediction that this was related to wavelength. Also, in 1918, he established the equality of gravitational and inertial mass thus reconfirming Einstein's equivalence principle, which lies at the core of general relativity theory.

Zeeman and Lorentz shared the 1902 Nobel Prize for physics for their work on magnetooptical effects. 24