Lecture Density Matrix Formalism

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Lecture Density
Matrix Formalism A tool used to describe the
state of a spin ensemble, as well as its
evolution in time.
The expectation value X-component of the magnetic
moment of nucleus A Where ? is the wave
function and is a linear combination of the
eigenstates of the form Where ngt are the
solution of the time-independent Schroedinger
equation. The bra, ltn and ket, ngt , and the
angular momentum operator can be written in the
matrix form as

IXA Thu
s
2
The N2 terms can be put in the
matrix form as follow Where
dmn, i.e. D is a Hermitian matrix Thus,


No?A The angular momentum operators for
spin ½ systems are For spin 1 For a
coupled A(½ )X(½) system
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Using the expression
-(4/p)MoA(d11/2 d22/2 d33/2
d44/2) Where And
Remember
and Similarly

and In modern NMR spectrometers we
normally do quadrature detection, i.e. For
nucleus A we have Similarly, for nucleus
X The density matrix at thermal equilibrium


Thus,
if n ? m and Evolution of the
density matrix can be obtained by solving the
Schoedinger equation to give Effect of
radiofrequency pulse Where R is the rotation
matrix

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For an isolated spin ½ system For A(½)X(½)
system
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Density matrix description of the 2D
heteronuclear correlated spectroscopy
(uncoupled)
(coupled)
For a coupled two spin ½ system, AX there are
four energy states (Fig. I.1) (1) gt (2)
-gt (3) -gt, and (4) --gt. The resonance
frequencies for observable single quantum
transitions (flip or flop) among these states
are 1QA ?12 (gt ? -gt) ?A J/2 ?24
(-gt ? --gt) ?X - J/2 1Qx ?13 (gt ?
-gt) ?X J/2 ?34 (-gt ? --gt) ?A -
J/2 Other unobservable transitions are Double
quantum transition 2QAX (Flip-flip) gt ?
--gt
(flop-flop) --gt ? gt Zero quantum
transitions (flip-flop)
ZQAX -gt ? -gt or -gt ?
-gt Density matrix of the coupled spin system
is shown on Table I.1. The diagonal elements are
the populations of the states. The off-diagonal
elements represent the probabilities of the
corresponding transitions.
(1)
(2)
(3)
(4)
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1. Equilibrium populations
At 4.7 T
For a CH system, A 13C and X 1H and ?x ? 4?C
? q ? 4p Thus,
Hence
where ? 4
Unitary matrix
Therefore,
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2. The first pulse
where

• The pulse created 1QX (proton) (non-vanishing d13
  and d24)
• 3. Evolution from t(1) to t(2)

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To calculate D(2) we need to calculate the
evolution of only the non-vanishing elements,
i.e. d13 and d24 in the rotating frame.

are the rotating frame
resonance frequencies of spin A and X,
respectively, and ?TrH is the transmitter (or
reference) frequency. Hence
where B and C are the complex
conjugates of B and C, respectively. 4. The
second pulse (rotation w.r.t. 13C) D(3)
R180XCD(2)R-1180XC 5. Evolution from t(3) to
t(4) (? is lab frame and ? is rotating frame
resonance frequency) Substituting B and C
into the equations we get
? J
is absent ? Decoupled due to spin echo
sequence

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6. The role of ?1 (Evolution with coupling)

and Let ? 1/2J and We have Let
Thus, 7. The third and fourth pulses
Combine the two rotations into one

and
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• D(7) R180XCD(2)R-1180XC D(5)
• Proton magnetization, d13 and d24 has been
  transferred to the carbon magnetization,
• d12 and d34 with
  and

8. The role of ?2 Signal is proportional to
d12d34 we cant detect signal at this time
otherwise s will cancel out. The effect of ?2 is
as follow (Only non-vanishing elements, d12 and
d13 need to be considered

• Hence
• For ?2 0 the terms containing s cancel.
• For ?2 1/2J we have

9. Detection During this time proton is
decoupled and only 13C evolve. Thus,
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• As described in Appendix B, in a quasrature
  detectin mode the total magnetization MTC is
• 
  
  if we reintroduce the p/4 factor.
• Thus,
• This is the final signal to be detected. The 13C
  signal evolve during detection time, td at a
  freqquency
• ?H and is amplitude modulated by proton
  evolution ?The. Fourier transform with respect
  to td and te
• results in a 2D HETCOR spectrum as shown on
  Fig. I.3a. The peak at -?H is due to
  transformation of
• sine function due to
  
• The negative peak can be removed by careful
  placing the reference frequency and the spectral
  width
• or by phase cycling.
• If there is no 180o pulse during te we will see
  spectrum I.3.b
• If there is also no 1H decoupling is during
  detection we will get spectrum I.3.c.

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