R.G. Griffin Winter School 2020 Page 1 Notes on Dynamic Nuclear Polarization 1. Overhauser Effect The following simple derivation shows that the enhancement in a electron nuclear DNP experiment, Overhauser is (gS/gI)~660. 1.1 Energy levels and transitions of a model four level system Consider the Hamiltonian with electron and nuclear Zeeman terms and a hyperfine term of magnitude A. The hyperfine term couples two spin-1/2 particles I=1/2 and S=1/2. We assume that and therefore with mS=±1/2 and mI=±1/2 we obtain resonance frequencies and the transitions given below. NMR transtions EPR transitions In the absence of applied alternating fields the populations are in thermal equilbrium where is the transition rate from state à and ’s are the populations of the two states. This can be rewritten in terms of Boltzmann factors as H = γ S !B 0 S Z − γ I !B 0 I Z + A ! I i ! S γ S !B 0 > A and γ S >> γ I H = γ S !B 0 S Z − γ I !B 0 I Z + AI Z S Z ω S = γ S B 0 + A ! m I and ω I = γ I B 0 − A ! m S W εη , ε ' η' P εη = W ε ' η', εη P ε ' η' W εη , ε ' η' εη ε ' η' P ij
Transcript
R.G. Griffin Winter School 2020 Page 1
Notes on Dynamic Nuclear Polarization
1. Overhauser Effect The following simple derivation shows that the enhancement in a electron nuclear DNP
experiment, Overhauser is (gS/gI)~660.
1.1 Energy levels and transitions of a model four level system Consider the Hamiltonian with electron and nuclear Zeeman terms and a hyperfine term of magnitude A.
The hyperfine term couples two spin-1/2 particles I=1/2 and S=1/2. We assume that
and therefore
with mS=±1/2 and mI=±1/2 we obtain resonance frequencies and the transitions given below.
NMR transtions EPR transitions
In the absence of applied alternating fields the populations are in thermal equilbrium
where is the transition rate from state à and ’s are the populations of
the two states. This can be rewritten in terms of Boltzmann factors as
H = γ S!B0SZ − γ I!B0IZ + A!I i!S
γ S!B0 > A and γ S >> γ I
H = γ S!B0SZ − γ I!B0IZ + AIZSZ
ωS = γ SB0 +A!mI and ω I = γ IB0 −
A!mS
Wεη,ε 'η'Pεη =Wε 'η',εηPε 'η'
Wεη ,ε 'η' εη ε 'η' Pi j
R.G. Griffin Winter School 2020 Page 2
1.2 Overhauser effect (A. Overhauser, 1953 and Carver and Slichter, 1953) Now consider the same system subject to irradiation of one of the electron transitions, we, as illustrated below.
Under irradiation we can write rate equations for the populations
(1)
(2)
(3)
(4)
The probabilities sum to unity . We solve the equations in the steady state when the
driving field is strong and saturates the transition. Therefore,
and (4) yields
Wεη,ε 'η'Wε 'η',εη
=Pε 'η'Pεη
= e Eεη−Eεη⎛⎝⎜
⎞⎠⎟ / kT
dP1dt = P2W21−P1W12 + (P1−P2 )ωe
dP2dt = P1W12 −P2W21+P3W32 −P2W23 + (P1−P2 )ωe
dP3dt = P2W23 −P3W32 +P4W43 −P3W34
dP4dt = P3W34 −P4W43
Pi1
4
∑ =1
ωe >>W12,W21 1 → 2P1=P2
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which is thermal equilibrium for and . Equation (3) yields
Since and and and are in equilibrium. Therefore, and are in equilibrium. We define
2. Solid effect -- DNP with forbidden transitions (Jefferies and Abragam)
Similar arguments can be extended to the Solid Effect that is based on forbidden transitions.
The solid effect utilizes zero and double quantum transitions that require operators of the
form
However, in the DNP experiment we excite these nomally DQ and ZQ transitions with a microwave field. This is represented be . In a separate note we show how this is accomplished. Assume that we have admixtures of states which permit excitation of these transitions. We saturate the transition with microwaves . Thus, . Proceeding as before
