Lecture 7: Chemical shift, one-pulse experiment
General strategy
1. Define p at t = 0
2. Describe evolution of p using the relevant Hamiltonians usually several steps
3. Calculate the expectation value of the measured quantity (magnetization components in the x,y plane) as (M+) = (Mx + \My) =AfTr{p M+)  
The procedure requires knowledge of
1. relation(s) describing the initial state of the system (p(0))
2. all Hamiltonians (^f7)
3. the operator representing the measurable quantity (M_|_)
The procedure requires knowledge of
1. the operator representing the measurable quantity (M_|_)
2. all Hamiltonians (^f7)
3. relation(s) describing the initial state of the system (p(0))
The procedure requires knowledge of
1. the operator representing the measurable quantity (M_|_)
2. all Hamiltonians (jr)
3. relation(s) describing the initial state of the system (p(0))
Operator of measured quantity
M+ = MX + \My
M+ =Ni(tx + Uy) = A/V+
The procedure requires knowledge of
1. the operator representing the measurable quantity (M_|_)
2. all Hamiltonians (^f7)
3. relation(s) describing the initial state of the system (p(0))
Hamiltonian of static field Bq
Ad,lab — -1bqIz
Hamiltonian of the radio-frequency field Bi
Phase = 0 (x):
^l,rot =(—7^0 — ^rot)^ — jBiIx = Q.IZ + w\Ix
Phase = tt/2 (y):
#l,rot — (—7^0 — ^rot)^ — l^lly —       + u\Iy
Chemical shift Hamiltonian
Hg — —j(lxBe,x + IyBe,y + IzBe,z) — —7( Ix ly Iz )
	°xx	$xy	°xz		( B0,x ^	
7( Ix ly Tz )	fiyx		Syz		B0,y	= -7J
	y $zx	Szy	&zz j		(Bo,z)	
H — ^0,lab + Hs,\ + Hs,a + ^S,r
Chemical shift Hamiltonian
Isotropic component (independent of orientation)
Anisotropic (axially symmetric) component (depends on <p):
H$3 = — 7#0£a(3 sin # cos# cos (pfx+3 sin # cos# sin (ply+(3 cos2 l)fz)
Rhombic (asymmetric) component (depends on x):
H§r = —7^o^r(   (— cos 2\ sin # cos# cos ip + sin 2% sin # cos# sin ip)Tx +
(— cos 2x sin # cos # sin ip — sin 2% sin # cos79 cos c^)7y + ((cos2xsin2tf)^)
Secular approximation
Molecular motions do not resonate with the precession frequency —7^0
Be,xTx and Be,yTy oscillate rapidly with frequency close to —jBq Bq > Be     much faster oscillations than precession about Be,x, Be,y effectively average to zero on timescale longer than 1/(7^0) (~ ns)
Terms with Be,xTx and Be,yTy can be neglected on timescales > ns
y
\
Be,x + \Be,y -(B0 + Be,z)
\
J
Averaging in isotropic solvent
No orientation of the molecule is preferred
all values of x are equally probable and independent of # cos 2x = 0
ax = sin$cosc/? ay = sin t9sin <p az = costf
ax + av + az = 1 =^ ai + a? + ai = 1     3a| — 1 = (3 cos2 i? — 1) = 0
Secular approximation
Isotropic component:
Hs,\ = -7B0S\(TZ) Anisotropic (axially symmetric) component:
Hfta = —7^o^a(3 sin # cos# cos cpTx+3 sin # cos# sin (ply+(3 cos2 l)fz)
Rhombic (asymmetric) component:
H$r = —7^o^r(   (— cos 2x sin tfcostfcosc/? + sin 2% sin # cos# sin ip)Tx +
(— cos 2x sin # cos # sin ip — sin 2% sin # cos79 cos c^)7y + ((cos 2X sin2 79)^)
Averaging in isotropic solvent
Isotropic component:
Hs,\ = -7B0S\(TZ) Anisotropic (axially symmetric) component:
Hfta = — 7#o£a(3 sin # cos# cos cpTx+3 sin # cos# sin (ply+(3 cos2 tf—l)fz)
Rhombic (asymmetric) component:
H$r = — 7#o^r(   (— cos 2x sin tfcostfcosc/? + sin 2\ sin 79 cos# sin c/?)fx +
(— cos2xsin # cos # sin <p — sin 2xsin coscos(p)Iy + ((cos 2X sin2 &)TZ)
Secular approximation and averaging
Isotropic component:
Anisotropic (axially symmetric) component:
H$3 = — 7#0£a(3 sin # cos# cos (pfx+3 sin # cos# sin (ply+(3 cos2 l)fz)
Rhombic (asymmetric) component:
H$r = — 7#o^r(   (— cos 2x sin tfcostfcosc/? + sin 2\ sin 79 cos# sin c/?)fx +
(— cos 2x sin # cos # sin <p — sin 2\ sin 79 cos# cos ip)Ty + ((cos2xsin2tf)i;)
Hamiltonian without radio waves (B0, S\)
#0,lab = -7^o(l + 5\)Tz = uolz ^O,rot — (^0 — ^rot)^ — ^-Tz
Hamiltonian with radio waves (Bq, 5\, B{) Phase = 0 (x)\
#l,rot — (—7^0 — urox)Iz — lB\Ix = Q.IZ + u\Ix
Phase = tt/2 (y):
#l,rot — (—7^0 — ^rot)^ — l^lly = &Iz + uily
Hamiltonian without radio waves (B0, S\)
#0,lab = -7^o(l + 5\)Tz = uolz ^O,rot — (^0 — ^rot)^ — ^-Tz
Hamiltonian with radio waves (Bq, 5\, B{)
Phase = 0 (x), close to resonance Q <c qui.
^l,rOt =QIz + W CJl/a;
Phase = 7r/2 (y), close to resonance Q <c u;i:
#l,rot =QIz + ^l^r ~ t^i-Ty
The procedure requires knowledge of
1. the operator representing the measurable quantity (M_|_)
2. all Hamiltonians (^f7)
3. relation(s) describing the initial state of the system (p(0))
Thermal equilibrium as the initial state
Classically, Sj = —jlj ■ Bq = — /iBqCOS'Sj = —^zjBq
Boltzmann: P(#) = e-£W)/k&T « l - S($)/kBT for        <c fcBT
Quantum mechanically, 5 is eigenvalue of H = —jBq(1 + 5\)Tz ~ —jBqTz
2 "r 4feBT
0
V
0
1 7^0ft
2 4fcBT y
'l 0^
V0 1;
+
4fcBT
/
V
1 0 0 -1
J
= J?t + kJz
Mixed state: The two-dimensional density matrix does not imply that all magnetic moments are in one of two eigenstates!
Relaxation due to chemical shift anisotropy
Bloch - Wangsness - Redfield semiclassical theory
(spin magnetic moments classically, molecular environment classically)
Rl = \b2 (|J(u;0) + ^J(-wo)) ~ \b2J(u0) R2 = b2 (|j(0) + |j(u>0)) « R0 +
Same equations as derived classically
One pulse experiment
HOMEWORK: Section 7.8
Conclusions
Density matrix evolves as
p(t) oc (^xcos(Qt + 0) + ^sin(Qt + 0) + terms orthogonal to Magnetization rotates during signal acquisition as
(M+) = |M+|e_jR2teiQt = |M+|e_jR2tei(^ (cos(Qt) + i sin(Qt))
unimportant phase shift which is empirically corrected
Fourier transform gives a complex signal proportional to
M-f2h2B0
R2
\
— I
4kBT   \R% + (uo- Q)2      R\ + (u - Q)2;
Signal
p(t) oc (^xcos(Qt + 0) + ^sin(Qt + 0) + terms orthogonal to
cosine modulation of Jx = real component of signal sine modulation of Jy = imaginary component of signal
• i-H • i-H
M+|e_jR2te^(cos(Qt) + i sin(Qt))
Spectrum
After Fourier transformation:
R2
AkBT   [R2, + (w - Q)
	A
	i 0
^3
0^
Signal-to-noise ratio
Signal/noise = K
h2N\j\5/2B%/2
,3/2 3/2 A B ^sample
1 _ 9—^2^2,max
o ,1/2 il2 62,max
Relaxation
Relaxation: ~ l/i?2 f°r lon9 acquisition time t2,max l/i?2 ~ 6Drot/62 for large rigid spherical molecules
6Drot = 3fcBT 47rr3r/(T)
1/&2 = 7_2jBq 25^2 for chemical shift anisotropy,
but chemical shift anisotropy is usually not dominant
High field/high 7 usually advantageous (exception: 13C=)