A Pulse EPR Primer

A Pulse EPR PrimerA Pulse EPR PrimerFIDs and Echoes

ApplicationsApplications

ESEEM Relaxation Time Measurement2 + 1, DEER, ELDOR

EXSY

Structural ElucidationDynamics, Distances

Measurement of Long Distances

Measurement of Slow Inter & Intra-molecular Chemical Exchange and Molecular Motions

TopicsTopics

The Rotating Frame

The Effect of B1

FIDs (Free Induction Decays)

FT (Fourier Transform) Theory

Spin Echoes

Relaxation Times

Rotating FrameRotating Frame

The Axis System

x

y

z

B1

B0

M agnet


The Larmor Frequency

L = - B0

B0

M0

x

y

z


Linearly and Circularly Polarized Light


The Rotating Frame


B1 in both Frames

M0

x

y

z

B1

0

0

B0

M0

x

y

z

0

0

LabFram e

RotatingFram e


Tip Angles

= - |B1| tp

M0

x

y

z

M

x

z

y

M0

x

y

z

B1

M

x

y

z

B1

M WO N

M WO FF


Pulse Phases

M

x

y

z

B1

+x

Mx

y

z

B1

+y

M

x

y

z

B1

-y

M

x

y

z

B1

-x


Transverse Magnetization in Both Frames

M

x

z

y

M

x

z

y

0

LabFram e

R otatingFram e


Generation of Microwaves

N S

L

M


Off-resonance Effects

M

x

z

y

M

x

z

y


The Effective Field

B eff 02 = B 1

2 + B


Sin(x)/x Behavior

10 8 6 4 2 0 2 4 6 8 10

M

M M-y 0 =

1 + ( / )12

sin 1( + (/ ) 12 )


Excitation Bandwidth

-65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 [M H z]

16 ns

32 ns

48 ns

64 ns

Relaxation TimesRelaxation Times

Spin Temperature and Populations

M0

x x

y y

z z

M

M

x

z

y

Therm alEquilib rium

/2 Pulse Pu lse

= enantipara lle l

nparalle l

EkT


Longitudinal Magnetization Recovery

2 4 6 8 1 0 2 4 6 8 1 0

t / T 1 t / T 1

M / Mz 0 0

-1

+ 1

/2 Pulse / Pulse

M (t) z = M 0 1- e -t/T1 M (t) z = M 0 1- 2 e -t/T1


Effect of Excessive Repetition Times

M (SRT) z = M 0 1- e -SRT/T1

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000Tim e [ns]


Homogeneous & Inhomogeneous Broadening

Homogeneous BroadeningThe 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 BroadeningThe 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


A FID (Free Induction Decay)

M (t) -y = M e -t/T2

1

0

1

M y

1

0

1

1

0

1

1

0

1

Fourier TheoryFourier Theory

Fourier transforms convert time domain signals into frequency domain signals and vice versa.


Time Behavior of Magnetization

M

x

z

y

Cos( t)

Sin( t)

M (t) -y = M cos( t)

M (t) x = M sin( t)


The Complex Axis System

M (t) t = M e -i t

e i = cos( ) + i s in( )

Im

Re

M


The Fourier Transform

F( ) = f(t) e -i t d t

+

-

f(t) = 1

F( ) 2 e i t d

+

-


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.

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.

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.

Some Fourier Facts


A Simple Fourier Transform



0 0

0

0

0 0

0 0

0 0

e - / 2 e - / 2

erf( / 2 ) 2

( + ) + 0 ( )0

( + ) - 0 ( )0

0

0

0 S in( )/

(1 /T2) + 2 2

(1 /T 2) + 2 2

1/T2

0t

0t

0t

0t

0t

e -t / T2

e - t / 2 2

(t+ ) - (t- )

cos( t) 0

s in ( t) 0

f (t) F ( ) Re Im

0 0

0

0

a)

d)

b)

c)

e)


Addition Properties

f(t) + g(t) = F( ) + G ( )

f (t) F ( )

0

0t

0 0

0

0t

0 0

0

0t

0 0

+

=

+

=


Shift Properties

0 0

0t

f (t) F( ) R e Im

0 0

0 0

0t

0 0

t

f(t - t) F ( ) e -i t f(t) F ( - ) e i t


Convolution Properties

f(t) * g(t) = f( ) g(t- ) d

+

-

* =


Convolution Theorem

f(t) * g(t) F ( ) G ( )

F( ) * G ( ) f(t) g(t)


A Practical Example

-A +A

R e

Im

0


A Practical Example

* =

Use Convolution


A Practical Example

Use Addition

cos(A t) 1+ cos(A t)1

+ =

+ =

t


A Practical Example

Use the Convolution Theorem

X =


Linewidth Effects

f (t) F ( )


Splitting Effects f (t) F( )


Field Effects f (t) F ( )


Field vs Frequency

Fie ld Sw eep

Frequency


Field vs Frequency

B 0

0 - 00 +

EchoesEchoes

Spin Echoes

2

EchoesEchoes

Spin Echoes

EchoesEchoes

Spin Echoes with Inhomogeneous

Broadening

EchoesEchoes

Phase Memory Time, TM

Echo H eight( ) e -2 /T M

EchoesEchoes

Spectral Diffusion

EchoesEchoes

Spin Lattice Relaxation

EchoesEchoes

ESEEM

0 1 2 3 4 5 6 7 8 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 [M Hz]

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 28000 30000 32000[ns]

EchoesEchoes

Stimulated Echo

2 2 2

t1

2 t1 t1 + 2 2 t1+ 2 t1+ 2 2 0

HahnEcho

HahnEcho

H ahnEcho

Stim ulatedEcho

R efocusedEcho

EchoesEchoes

Effect of Pulse Lengths with Two Equal

Pulses

Documents

A Pulse EPR Primer