A Pulse EPR Primer
FIDs and Echoes
Applications
ESEEM
Relaxation Time
Measurement
2 + 1, DEER, ELDOR
Structural Elucidation
Dynamics, Distances
EXSY
Measurement of Slow
Inter & Intra-molecular
Chemical Exchange
and Molecular Motions
Measurement of Long
Distances
Topics
The Rotating Frame
The Effect of B1
FIDs (Free Induction Decays)
FT (Fourier Transform) Theory
Spin Echoes
Relaxation Times
Rotating Frame
The Axis System
Magnet
x
B1
z
y
B0
Rotating Frame
The Larmor Frequency
M0
B0
z
y
x
wL = -g B0
Rotating Frame
Linearly and Circularly
Polarized Light
Rotating Frame
The Rotating Frame
Rotating Frame
B1 in both Frames
M0
B0
z
M0
z
w0
w0
y
x
w0
Lab
Frame
y
x
B1
Rotating
Frame
w0
Rotating Frame
Tip Angles
z
z
z
M0
M0
z
a
a
M
y
x
M
x
B1
y
y
y
x
B1
MW
ON
MW
OFF
a = -g |B1| tp
x
Rotating Frame
Pulse Phases
z
z
z
z
M
B1
M
y
x
B1
M
B1
x
+x
y
y
B1
y
x
x
+y
M
-x
-y
Rotating Frame
Transverse Magnetization
in Both Frames
z
z
M
M
y
y
w0
x
x
Rotating
Frame
Lab
Frame
Rotating Frame
Generation of
Microwaves
S
N
wL
M
Rotating Frame
Off-resonance Effects
z
z
M
M
y
y
w
x
x
Rotating Frame
The Effective Field
Beff = B12 + B02
Rotating Frame
M
Sin(x)/x Behavior
10
8
6
4
2
0
2
4
6
8
10
ww
M-y = M0
sin( 1 + (w/w1)2

1 + (w/w1)2
)
Rotating Frame
Excitation Bandwidth
16 ns
32 ns
48 ns
64 ns
-65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0
[MHz]
5
10
15
20
25
30
35
40
45
50
55
60
Relaxation Times
Spin Temperature and
Populations
E
nantiparallel
kT
=
e
n
parallel
z
y
x
z
z
M0
y
M
y
x
x
M
Thermal
Equilibrium
/2 Pulse
 Pulse
Relaxation Times
Longitudinal Magnetization
Recovery
/2 Pulse
/ Pulse
+1
M /M
z
0
0
2
4
6
8
10
t/T
2
4
6
8
10
t/T
1
1
-1
Mz(t) = M0 1- e

-t/T1
Mz(t) = M0 1- 2 e

-t/T
1
Relaxation Times
Effect of Excessive
Repetition Times
0
2000
4000
6000
8000
10000
12000
Time [ns]
14000
Mz(SRT) = M0 1- e
16000
18000
-SRT/T
1
20000
22000
Relaxation Times
Homogeneous &
Inhomogeneous Broadening
Homogeneous Broadening
The lineshape is determined by the relaxation
time. The spectrum is the sum of a large
number of lines each having the same
Larmor frequency and linewidth.
Lorentzian Lineshapes
Inhomogeneous Broadening
The lineshape is determined by the
unresolved couplings. The spectrum is the
sum of a large number of narrower
homogeneously broadened lines each having
the different Larmor frequencies.
Gaussian Lineshapes
Relaxation Times
A FID
(Free Induction Decay)
-t/T
2
M-y(t) = M e
My
1
1
1
1
0
0
0
0
1
1
1
1
Fourier Theory
Fourier transforms convert
time domain signals into
frequency domain signals
and vice versa.
Fourier Theory
Time Behavior of
Magnetization
M-y(t) = M cos(wt)
z
Cos(wt)
M
y
w
x
Mx(t) = M sin(wt)
Sin(wt)
Fourier Theory
The Complex Axis
System
-iwt
Mt(t) = M e
Im
i
e = cos() + i sin()
M

Re
Fourier Theory
The Fourier Transform
+
-iwt
F(w) = 
f(t)
e
dt

-
+
1 
iwt
f(t) = 2  F(w) e
dw
-
Fourier Theory
Some Fourier Facts
Even functions (f(-t) = f(t) or symmetric) have purely real
Fourier transforms.
Odd functions (f(-t) = -f(t) or anti-symmetric) have purely
imaginary Fourier transforms.
Fourier Theory
Some Fourier Facts
An exponential decay in the time domain is a lorentzian in
the frequency domain.
A gaussian decay in the time domain is a gaussian in the
frequency domain.
Fourier Theory
Some Fourier Facts
Quickly decaying signals in the time domain are broad in the
frequency domain.
Slowly decaying signals in the time domain are narrow in
the frequency domain.
Fourier Theory
A Simple Fourier Transform
Fourier Theory
Fourier Theory
F(w)
f (t)
Re
a)
0
t
0
t
sin(w0t)
c)
0
t
e-t / T2
d)
0
t
-t2 / 2

0 
(t+) - (t-)
w0
t
w
0
0
w0
0
w0
0
0
(w+w0) - (ww0)
0
0
1/T2
w
(1/T2)2 + w2
(1/T2)2 + w2
0
0


e-w / 2 
e
e)
0
(w+w0) + (ww0)
cos(w0t)
b)
w0
Im
w
w
w


e-w / 2  erf(w/22)
0
0
Sin(w)/w
0
w
Fourier Theory
Addition Properties
f(t) + g(t) = F(w) + G(w)
F(w)
f (t)
0
t
w0
+
0
w0
w
+
t
w0
=
0
0
0
w0
w
=
t
w0
0
w0
w
Fourier Theory
Shift Properties
-iwt
iwt
f(t - t)  F(w) e
f(t) e
F(w)
f (t)
Re
t
0
0
t
t
 F(w-w)
Im
w0
0
w0
0
w0
0
w0
0
w
w
Fourier Theory
Convolution Properties
+
 f() g(t-) d
f(t) * g(t) = 
-
*
=
Fourier Theory
Convolution Theorem
f(t) * g(t)  F(w)  G(w)
F(w) * G(w)  f(t)  g(t)
Fourier Theory
A Practical Example
Re
Im
-A
0
+A
Fourier Theory
A Practical Example
Use Convolution
*
=
Fourier Theory
A Practical Example
Use Addition
w
+
=
t
+
=
cos(At)
1
1+ cos(At)
Fourier Theory
A Practical Example
Use the Convolution Theorem
X
=
Fourier Theory
Linewidth Effects
F(w)
f (t)
Fourier Theory
Splitting Effects
F(w)
f (t)
Fourier Theory
Field Effects
F(w)
w
f (t)
Fourier Theory
Field vs Frequency
Frequency
Field Sweep
Fourier Theory
Field vs Frequency
w0 + w
w0
B0
w0 - w
Echoes
Spin Echoes

2

Echoes
Spin Echoes
Echoes
Spin Echoes with
Inhomogeneous
Broadening
Echoes
Phase Memory Time, TM
-2/TM
Echo Height()  e
Echoes
Spectral Diffusion
Echoes
Spin Lattice Relaxation
Echoes
ESEEM
0
0
1
2000
2
3
4
4000
6000
8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000
[ns]
5
7
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
[MHz]
6
8
Echoes
Stimulated
Echo
Hahn
Echo
t1

2
0
2
2
2
Stimulated
Hahn
Echo
Refocused Echo
Echo
Hahn
Echo
2t1+ 
2t1+ 2
t1+2
2t1
Echoes
Effect of Pulse Lengths
with Two Equal
Pulses