Transcript 投影片 1 - Academia Sinica

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 |n> are the solution of the time-independent Schroedinger equation. The “bra, <n|
and “ket, |n>” , and the angular momentum operator can be written in the matrix form as:
IXA =
Thus:
The N2
Where
terms can be put in the matrix form as follow:
= d*mn, i.e. D is a Hermitian matrix
Thus,
The angular momentum operators for spin ½ systems are:
For spin 1:
For a coupled A(½ )X(½) system
= NoA
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
For an isolated spin ½ system:
For A(½)X(½) system:
Density matrix description of the 2D heteronuclear correlated spectroscopy
For a coupled two spin ½ system, AX there are four energy
states (Fig. I.1); (1) |++>; (2) |-+>; (3) |+->, and (4) |-->.
The resonance frequencies for observable single quantum
transitions (flip or flop) among these states are:
1QA: 12 (|++>  |-+>)= A + J/2;
24 (|-+>  |->)= X - J/2; 1Qx: 13 (|++>  |+->)= X + J/2;
34
(|+->  |-->)= A - J/2;
Other unobservable transitions are:
Double quantum transition 2QAX (Flip-flip): |++>  |-->
(flop-flop): |-->  |++>
Zero quantum transitions (flip-flop):
ZQAX: |+->  |-+> or |-+>  |+->
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.
(uncoupled)
(1)
(2)
(3)
(4)
(coupled)
1. Equilibrium populations:
At 4.7 T:
For a CH system, A = 13C and X = 1H and x  4C  q  4p
Thus,
Therefore,
Hence:
where
Unitary matrix
4
2. The first pulse:
where
The pulse created 1QX (proton) (non-vanishing d13 and d24)
3. Evolution from t(1) to t(2):
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.
A and X, respectively, and TrH
are the rotating frame resonance frequencies of spin
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
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
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 d12+d34 we can’t 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,
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.