Supplementary information for:
Theory for cross effect dynamic nuclear polarization under magic-angle spinning in solid state
nuclear magnetic resonance: the importance of level crossings
I.
Equations for dependence of interactions on MAS
We will be using the Euler angles according to the zyz convention used in Mehring, Appendix
1
B.
For an orientation of the biradical with electron dipole-dipole and hyperfine couplings, we use 10
angles:
The orientation of each electrons’ g-tensor: α1, β1, γ1; α2, β2, γ2
The orientation of the dipole-dipole coupling: βee, γee
The orientation of the hyperfine coupling: βen, γen
Electron g-factor anisotropy:
For the electron g-factor anisotropy, we need the three Euler angles, α, β, and γ, the nitrogen nuclear
spin (Nz = +1, 0, or -1), and the frequency, f, appropriate for an electron with a g-factor of 2 in the
magnetic field.
First, we define the frequencies for the three principal axes of the g-anisotropy, based on experimental
values for a Tempo nitroxide.2
f x  f 2.0061 / 2   18.76  106 N z
f y  f 2.0021/ 2  92.4 106 N z
(S1)
f z  f 2.0094 / 2  18.2  106 N z
We also need to use elements of the rotation matrix, R (Mehring, pg 219).1 We refer to the matrix
elements by Rnn, where the first number is the row, and the second the column.
 cos  cos  cos   sin  sin 
R   cos  cos  sin   sin  cos 

cos  sin 

sin  cos  cos   cos  sin 
 sin  cos  sin   cos  cos 
sin  sin 
 sin  cos  
sin  sin  

cos  
(S2)
The final result is composed of 5 terms:
e  2 c0  c1 cos(r t )  c2 sin( r t )  c3 cos(2r t )  c4 sin( 2r t )
c0  13  f x  f y  f z 
c1  2 3 2  f x  R11  R31  f y  R12  R32  f z  R13  R33
c2  2 3 2  f x  R21  R31  f y  R22  R32  f z  R23  R33
c3  13  f x  R112  R212   f y  R122  R222   f z  R132  R232 
c4 
2
3
f
x
 R11  R 21  f y  R 22  R12  f z  R13  R 23
1
(S3)
Electron dipole-dipole coupling:
We write it in the high field limit, which means only the part that commutes with S1z + S2z.
d  d max

2 sin(  ee ) cos(  ee ) cos(r t   ee )  12 sin 2 (  ee ) cos(2(r t   ee ))

(S4)
Electron-nucleus hyperfine coupling:
We write it in the high field limit with respect to the electron, which means only the part that
commutes with S1z.
hzz  hzz,max
hxz  hzz,max
 2 sin(  ) cos(  ) cos( t   ) 
 3cos ( )  1  sin(  ) cos( 
en

2
2
4
h yz  hzz, max 
3
2
en
en
r
1
2
en
en
sin(  en ) cos(  en ) sin( r t   en ) 
6
4
1
2
en
sin 2 (  en ) cos( 2(r t   en ))
) cos(r t   en ) 
2
4

sin 2 (  en ) cos( 2(r t   en ))
sin 2 (  en ) sin( 2(r t   en ))


(S5)
II. Solid effect
The solid effect under MAS can be modeled as consisting of an adiabatic level crossing. In this
case, the microwaves cross the resonance condition of ωm ~ ωe ± ωn, and the microwaves and the
hyperfine coupling provide the second order coupling between the direct product states. In this case,
we can again use a 4X4 Hamiltonian, for one nucleus and one electron, since the second electron is not
part of the process. So, spin product states 1, 3, 5, and 7 can be used as our starting states for the solid
effect Hamiltonian. Removing the second electron energy and electron dipole-dipole coupling results
in the Hamiltonian HSE.
1
H SE
1
7

3
5





1
2
(e  n  hzz )
0
1
 e im t
2 1
1
(hxz  ih yz )
2
7
0
1
( e  n  hzz )
2
1
( hxz  ih yz )
2
1
 e im t
2 1
3
1e imt
1
( hxz  ih yz )
2
1
( e  n  hzz )
2
0
1
2
5
( hxz  ih yz )
1
 e imt
2 1
0
1
(e  n  hzz )
2
1
2





(S6)
This Hamiltonian matrix has a form very similar to the one for the three-spin crossing (Eq. 8), with the
difference that the rotating microwaves have replaced the (non-rotating) electron-electron dipole
coupling. Again, we do not have a direct off-diagonal matrix element between the dominant spin
product states at the energy level crossing (3 and 5, or 1 and 7). But the combination of the hyperfine
coupling and the microwaves will create a second-order off-diagonal matrix element between those
states. So, we want to block diagonalize this Hamiltonian to calculate these second-order terms. First,
we will shift the Hamiltonian to the microwave rotating frame:
2
1
1
(e  n  hzz

  )




 (h

1
2
7
H SE,rot 
3
5
1
2
0
1
2
xz
1
1
2
0
m
7
3
1
2
( e  n  hzz
1
2
 m )
1
2
1
2
( hxz  ih yz )
 ih yz )
1
2
5
1
1
2
( hxz  ih yz )
1
2
( e  n  hzz
0
1
0
 m )
1








( hxz  ih yz )
1
2
(e  n  hzz
 m )
(S7)
Next, we want to derive the eigenstates of this Hamiltonian, to first order in the off-diagonal elements.
1
1


h
2 1
1SE  N1SE  1 
3  2 xz 5  ,
e  hzz  m
n  hzz 

1
1


h
2 1
3SE  N 3SE  3 
1  2 xz 7  ,
e  hzz  m
n  hzz 

1
1


h
2 1
5SE  N 5SE  5 
7  2 xz 1  ,
e  hzz  m
n  hzz 

1
1


h
2 1
7SE  N 7SE  7 
5  2 xz 3 
e  hzz  m
n  hzz 

(S8)
where NiSE is the normalization constant for that state. Then, we express the approximate Hamiltonian
in the basis of these states, writing the lowest order term in ω1, hxz, and hzz, and ignoring any terms of
third order and above.
1SE
 
H SE
7SE
3SE
5SE








1
2
1SE
(e  n  hzz
 m )
 e  n  m 
1
4 1hxz 

 e  m n 
0
0
7SE



e  n  m
1
4 1hxz 

 e  m n 
1
2 ( e  n  hzz
3SE
0
0
1
2
( e  n  hzz
 m )
   n  m 
 14 1hxz  e

 e  m n 



0

      
h 

      
(    h

 )

0
0
 m )
0
5SE
 14
e
n
(S9)
m
1 xz
e
1
2
e
m
n
n
zz
m
We can simplify the off-diagonal element further, by calculating its approximate value at the
appropriate level crossing. At the level crossing between states 1SE and 7SE, ωm ~ ωe + ωn, and the offdiagonal matrix element between these two states is ~ ω1 hxz / 2ωn. For the level crossing between 3SE
and 5SE, ωm ~ ωe - ωn, and the lowest-order term of the off-diagonal element between the states just has
the opposite sign (-ω1 hxz / 2ωn). Thus, we can see that the solid effect with MAS is mathematically
3
very similar to the three-spin crossing of the cross effect, but with the electron-electron dipole coupling
replaced by the microwave strength in the rotating frame.
III. Electron and nuclear polarization for an example radical orientation
Figure S1:
Electron polarizations (red and blue) and nuclear polarization (black) for the rotor orientation shown in
Figure 2. The polarizations are shown at the beginning of each rotor cycle, including the first 10 ms
when the hyperfine coupling is not present. When the hyperfine coupling is added at t=0, the nuclear
polarization increases, and the momentary decrease of the electron polarization difference is visible.
Also shown in this graph is the equality at equilibrium of the electron polarization difference (~0.3)
and the nuclear polarization (~200 = 0.3 x 660). 100 sets of random T2e fields were averaged.
4
IV. Electron polarization as a function of electron frequency across lineshape
Figure S2:
Average electron polarization across the electron lineshape for several MAS speeds, red circles 1010
Hz; green squares, 7000 Hz; black diamonds, 50010 Hz. Microwave frequency is 264 GHz.
V. Microwave frequency dependence of DNP as a function of nuclear Larmor frequency
Figure S3:
Equilibrium nuclear polarization, npol,av, as a function of microwave frequency, normalized at the
maximum positive nuclear polarization (at 264.0 GHz), for ωn = 400.9 MHz (●), 100 MHz (), and 50
MHz (+). The maximum nuclear polarization is 111, 311, and 548 times thermal polarization for ωn =
400.9 MHz, 100 MHz, and 50 MHz, respectively. Lines are drawn to guide the eye.
5
Supplementary References
1
2
M. Mehring, High Resolution NMR Spectroscopy in Solids. (Springer-Verlag, New York,
1976).
W. Snipes, J. Cupp, G. Cohn, and A. Keith, Biophysical Journal 14 (1), 20 (1974).
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