Lecture 10: J-coupling, spin echoes
Direct dipole-dipole coupling
J-coupling (indirect, through-bond)
J-coupling Hamiltonian
\
V
-^2,2?
B2,z;
2ir 7172
V
Jxx Jxy Jxz Jyx Jyy Jyz Jzx Jzy Jzz
	' V2,x N
•	V2,y
	( V2,z )
8 = —fii • B2 =
2tt 7172
Ml,x Pl,y fJ>l,z ) • J- •
/M2,xX
M2,y V M2,z )
í
y
\
Ij2,x
hu
J2,zj
I
2n{Tl,x h,y h
z
Jxx Ja
\
xy Jxz Jyx Jyy Jyz Jzx J zy Jzz
	(f \ l2,x
•	{2, y
	{ J2,z )
J-coupling Hamiltonian
H J — 27r(JxxIl,xl2,x + Jxyll,xl2,y + Jxzh,x^2
z
-\-JyxIi,yl2,x + Jyyll,yl2,y + Jyzli,yl2,z
-\-JzxI\,zl2,x + JzyIl,zl2,yJzzIl,zl2,z
Classical nucleus-electron interaction
Classical nucleus-electron interaction
Classical nucleus-electron interaction
Classical nucleus-electron interaction
Classical nucleus-electron interaction
Fermi interaction
J-coupling
Stationary state
of unperturbed a electrons
Favorable state of electron-coupled nuclei
Unavorable state of
electron-coupled nuclei
Scalar coupling
2
V
Jxx Jxy Jxz Jyx Jyy Jyz Jzx Jzy Jzz
	( f \
•	T2,y
	{ J2,z )
		Jxy	Jxz			(Jxx	0	0 )
	Jyx	Jyy	Jyz	—>		0	Jyy	0
	v Jzx	Jzy	Jzz )			( o	0	Jzz i
= 2ir
Jxx + Jyy + Jzz
(1	0	°1		(1	0	0)
0	1	0	= 2tt j	0	1	0
o	0	1		o	0	1
j = (JXx + Jyy + Jzz)/3 isotropic constant (scalar) (2Jzz — Jyy — Jxx)/6 = 0 no anisotropy (Jxx — Jyy)/2 = 0 no rhombicity
Hj — nJ (2IizI2z + 2IixI2x + 2Tiyl2y
J-coupling constants
• 1J(31P-1H) < 700 Hz
• 1 J(13C-1H) 140 Hz to 200 Hz
• 1J(15N-1H) -90 Hz
• 1 J(13C-13C) 30 Hz to 60 Hz
• 3J(1H-1H) < 15 Hz torsion angle
Secular approximation
• isotropic J: no ensemble averaging is needed
• 71 = 72 and S\ i « S\2 (strong coupling):
Hj = 7rJ {2I-]_ZI2Z + 2Iixl2x + 2Tlyf2y)
71 7^ 72 or l<5j,i — ^i,2l^o > 27r| JI (wea/c coupling)
Hj = 2irJIlzl2z = (2Ilzl2
z
Density matrix at thermal equilibrium
Diagonal elements:
PZ « \ + 71(1 + ^U)^L + 72(1 +2) ^
4     "------"afceT '  "------8kBT 16kBT
^^-71(1^11)^ + 7,(1+^^
^ - t - 7id + sLl)^L _ 72(1 + f 2) -Jfi
4     '           ,'xy8feBT                 '   8/cBT 16fcBT ttJ < 0.00001 7B0,
< 0.00002 (1H), |<5i>2| < 0.0002 (13C,15N)
~eq      i /     I              i            \ JjBoh
Density matrix evolution
Qx = -7l^o(l + ^2 = -72^o(l + ^i,2)
Weak coupling:     = q\.f\z + ^2^2z + ^(2^12^2^) ■^iz< ^izi ^iz^iz commute =>•
[>j, J^] = jj^ p = CJ?j ->   CJj cos(oot) + CJ\ COS(wt)
	
	
	/
	
	
	P
3x for     = <0\z,tPiziltPxz^iz in any order
A
	'Slz —	y Qiŕ
		Six
	/	
Sly'/		s,
Slx		
1~S
lz
	>slz
	
	y -2SXxJ2z
	/
	
	xS2z
\-Slz	
B
-2J*i
J2J^2z ^
_       7T J
«^1
lx
C
D
	VSlzJ?2z
	_       7T J
	/
	/ '-->-
2-^1,-/2, i
	'-2./1:../2:.	i-2./1:./2-,	
	>Slz		^SXzJ2z
f	/" /		s.
Sly/	A/		2Slyy2z
Six		Six	
\	[Slz	\	1-2SlzI2z l2Slzjr2z
1 1 1 y	/ /	1 1 1	\OJl \*-2Slxy2z
-J-
ly
lx\
2S^S2z
^-lSlzl2z
Density matrix evolution
p after a 90° pulse: p(b) = \jt + h^(—^iy — ^2y)
y
y
x
c\ = cos(Qit) c2 = cos(Q2*)
Cj = COS(7T Jt)
+n^ix —>
-cicj
+C1SJ + S1CJ
C2CJ
+S2CJ +S2«J
^lx^2z Six
2^1y^2z
^2y
^2x^lz ^2x
2^2y^l
*i = sinCQ-^) «2 = sin(ft2*)
sj = s\n(7rJt)
Spectrum
M+ = M (7l(/lx + \hy) + 72(/2x + \hy))
(M_|_) = Tr{p(t)M+} oc
_ ^e--R2,ií ^ei(Qi-7rJ)i _|_ ei(Qi+7rJ)^ _|_ e-R2,2t f^e\(n2-TTJ)t _|_ ei(Q2+^)ť
&{y(a;)} =
NYhzB0
16fcBT
«2,1
«2.1
V
ä| ! + (w - Qi + ttJ)2    R2, 1 + (w - Qi - ttJ)2
«2.2
+
«2.2
\
Ä| 2 + (w - Q2 + Tr J)2    R2, 2 + (w - Q2 - ^-02
Homo- and heteronuclear pairs
Homonuclear: 7^^=72, J^i-p^j, Heteronuclear: 71 7^72, ^7,^7, e
Homo- and heteronuclear pairs
uj
uj
UJ
UJ
HOMEWORK:
Spin echoes Sections 10.5-10.8
Spin echoes A ,HI-
13C or15N-
B ,HLJ
13C or15N-
C     'h'-j
13C or15N-*
D ,HI_I
13C or15N-"