The Boltzmann Distribution Matrix

T h e quantum mechanical expectation value of an operator X at time t is [cf. equations ( 1 . 2 1 ) and ( 1 . 2 2 ) in Chapter 3 ]

<X>, = Ψ\ί)ΧΨ(ί) = 2 2* , ( ί Κ * ( 0 Χ * , (3.2)

3 k

440 9. M U L T I P L E QUANTUM TRANSITIONS

where Ψ(ί) is the state vector at time t and the Cj(t) its expansion coeffi-cients relative to a suitable initial basis. T h e state vector Ψ(ΐ) can be related to ¥^(0) by a unitary operator U(t)>

Ψ(ΐ) = U(t)W(0), (3.3) so that

<X>, = W>(0)U*(t)XU(t)W(0)

= Χ2,(0)^(0)[£Λ(0ΧΕ/(ί)]« .

^(3.4)

j k This equation can be written

<X>t = tr C(0)U*(t)XU(t), (3.5)

where

C,,(0) = o(0K*(0). (3.6) T h e statistical average of (3.5) is

&>t = trpU*(t)XU(t), (3.7)

where ρ denotes the statistical average of C(0). By assumption, the system is initially at thermal equilibrium, so that

Pa = 0, f or i Φ h

2« = ^ . = « ρ [ - * ( β<- ί 2 , ) / * Γ ]>

Pa J ⁿ

where iV^ is the occupation number of the ith eigenstate with energy fiQi. If the trace of ρ is normalized to the number of systems contained in a macroscopic unit of volume, ρ can be written in operator form as

i V0e x p t - « J R0/ * R ]

9 tr e x p [ - H^JkT] 9 { }

where 3#*Q is the stationary hamiltonian prevailing at t = 0. In the following discussion, the conditions | HQi | <^ kT> tr = 0, will be satisfied, so that (3.8) simplifies to

9 dim S V kT r K }

where dim S is the dimension of the spin space.

T o illustrate the use of (3.9), consider a single nucleus with spin I in a stationary ζ field H0 . In this case, ^ = -γΗ0Ι^ζ, and dim S = 2 7 + 1 ; hence

'-2ΓΪτ ('+4£4 <

·'°>

3. S P I N ECHO EXPERIMENTS 441

T h e statistical average of I^z at t = 0 is, by (3.7),

</;>„ = t r , / , = j t r /² + & t r l * \ N0yhH0I(I + 1)

T h e steady nuclear polarization is

in agreement with the result derived in Chapter 1. Similar calculations show that

<4 >o

^{= <}^T^V^{\ = 0.}

As a second application of (3.10), the magnetization in the xy plane following a single φ pulse will be calculated. T h e time-dependent hamiltonian is

Jf(t) = —{ω0Ιζ + γΗλ(Ιχ cos wt — I^y sin cot)}, (3.11) where ω⁰ = y i /⁰ . Introducing the transformation (2.6), one finds that

8Φ

i-âf= -{ΔΙ,+γΗ^Φ, (3.12) where Δ = ω⁰ — ω. T h e solution of (3.12) is

Φ(ί) = exp[i(f -^ί⁰^)(ΔΙ^ζ

+ ^{^4} ^)]Φ(0).

^(3.13)

If the rf pulse is applied from t⁰ = 0 to t = τ⁰ , and the rf frequency is so close to ω⁰ that Δτ⁰ <^ 1, r⁰ J/g can be dropped from the exponential argument, and (3.13) simplifies to

Φ(τ⁰) = ^'*Φ(0). (3.14)

For £ > τ⁰ , the solution of (3.12) is

0(t) = ei J i /20 ( r^o) , (3.15)

since ^ = 0 for / > τ⁰ . Combining (3.14), (3.15), and (2.6), one obtains

W(t) = ^ν^ν 'Φ ^Ο)

= e?*-o<V>W(0), (3.16)

442 9. M U L T I P L E Q U A N T U M T R A N S I T I O N S

since and N o w

so that

— I — I tr I e-^xl^e1*1* 2 / + 1 \ A 7 V^z

e-iœ0tIzJ±eiœ0tIz _ eTiu0tJ±

tr e-WxfreWx = tr J± = 0.

tr 7Z ε-¹*¹*!^*¹* — tr 7± £?*>V²é?-*>7*

= tr{(/v sin <p + 72 cos φ)/*}

= ± 5 / ( / + l ) ( 2 / + l) 8 i n ç > ,

M ± ( 0 = ±i'Af⁰ ****** sin φ, or

M^x(t) = M⁰ sin φ sin ω0ί, My(t) = M0 sin φ cos ω0£.

These results are exactly what would be predicted on the basis of classical arguments. T h e ψ pulse turns the ζ magnetization through an angle ψ about the χ axis as in Fig. 9.18(b). T h e y magnetization immediately following the pulse is M⁰ sin φ, and, after the pulse is turned off, this component precesses in the negative sense about H⁰e^z. C. Modulation of the Echo Envelope

T h e modulation of the echo amplitude following a pair of 90° pulses will be calculated for the AB system (25, 24). T o simplify the discussion, nuclear magnetic relaxation, molecular diffusion, and the inhomogeneity of H⁰ will be neglected (23, 24).

T h e stationary hamiltonian for the system will be written

= ~ { έ ( " Α + ω^Β)Ι^ζ + £ 8 ( / A , - I^Bz) + Jl^A · IB} , (3.17) where δ = ωΑ — ω^Β. Since \fiS\<^ kT> \fij\<^ kT, and dim S = 4, (3.9) becomes

JVQ ) Y , ά(ω^Α + ω^Β)^T I

P = -4-\l+ 2kT~^h\'^{( 3 1 8}⁾

which is of the form (3.3), with U = Λ ^ . From (3.7) and (3.16), the χ and y components of the nuclear magnetization are given by

yh(ïïy^t = M^œ± iM^y

2fïfT

^tr

I (

⁷

+

^JT^Iz) e-^e-^hl^é^hé^

3. S P I N ECHO EXPERIMENTS 443

T h e time-dependent hamiltonian during a pulse is

JT(t) = - { O ^ A * + ωΒΙΒζ + JIA · I^B + γΗχ(Ιχ cos œt - Iy sin œt)}. (3.19) Introducing the transformation (2.6), one finds that the Schrödinger equation in the rotating coordinate system is

8Φ

i^R = * & (3.20) where

*τ = - { ( ωΑ - œ)IAz + (ωΒ - œ)IBz + JLA · IB + yH.Q. (3.21) It will now be assumed that ω^Α , ω^Β , ω, / , and τ⁰ satisfy the conditions

I ω^Α — œ i τ⁰ , I α>^Β — ω | τ⁰ , | / | τ⁰ <^ 1. (3.22) Under these conditions, only YHJ.X need be retained in (3.21) during

a pulse. Therefore,

Φ(τ⁰) = ^ Φ ( Ο ) , Φ(τ + 2τ0) = β^Φ(τ + τ0).

When / i j = 0, the hamiltonian in the rotating system is + ωΙ^ζ = J^⁰\ so that

* ( τ + τ0) = e x p( - T^ ' ) * K ) .

Φ(ί) = exp[-i(f - τ - 2τ0)^0']Φ(2τ0 + τ) (ί > τ + 2τ0).

Combining the preceding equations, and returning to the laboratory coordinate system, one obtains

W(t) = exp(iœtl^z) exp[—i(t — r ) ^ ' ] exp(i(pl^x) exp(—ζτ^') εχρ(ΐφΙ^χ)Ψ(0), (3.23) where use has been made of the fact that 2 τ⁰ <<ζ τ.

T h e nuclear magnetization in the XY plane is equal to the statistical average of YFT^I+}T or YFI(J~^)T . From (3.23),

= ^^ωΨ⁺( 0 ) ί /⁺( ί ) /⁺ί/ ( 0^ψ( 0 ) , (3.24) where

U = exp[-i(t - r)Ji?0'] exp(iV4) β χ ρ ( - ί τ ^ ' ) ^χρ(ιΨΙ^χ). (3.25) T h e statistical average of (3.24) is, therefore,

< F > , = N* W A+ ? > ^ {« I.U*WU(t)}. (3.26)

444 9. M U L T I P L E Q U A N T U M T R A N S I T I O N S

where

1 — · - Q Q = -Ar^y R = (P + s*)^.

(i + ρ

)ν2 > (i

⁺

ρ2)ΐ/2 » * g

⁺

#

Relative to this basis, the matrices for I+ and Iy are

(

0 c + s c — s 0 0 0 0 c + s 0 0 0 c - s 0 0 0 0

(

^{0 — (c}

+ 5) — (c —

^{s) 0}

£T +

S 0 0 ~ ( c + S)

c-s 0 0 -(c-i)l'

0 C

+ ί C — 5

⁰

T h e matrices for e x p ( — z V ^ ' ^ e x p ^ V J ^ ' ) and exp[i(J — r)Jf⁰']I⁺ exp[-i(i - T)J^Q] may now be computed by equation (1.27) of Chapter 3.

T h e matrix for ei(7Tl2)Ix is, by Table 4.1 and equation (3.17) of Chapter 2,

(

1 i(c + s) i(c — s) —1

i(c + s) 1 - 2cs -(c2 - s²) i(c + s) i{c - s) -(c2 -s2) 1 + 2cs i(c - s)

- 1 i(c + s) i(c — s) 1

T h e contribution from the identity operator of (3.18) vanishes, since tr Uf(t)I+U(t) = 0.

In the special case where φ = π/2, the trace in (3.26) simplifies to

tr

^I^z^U^f^(t)I+U(t)

= tr exp(—ιτ^')/,, exp(zYJ^') exp (—i^Z^

X exp[i(f - ^t ^)JÎJ']/+ exp[-i(f - t )JT

⁰

'] exp /,). (3.27)

T h e remaining trace is most conveniently evaluated relative to the basis that diagonalizes «#^\ The eigenvectors and eigenvalues of J ^ ' are

«ι = l + +>, "2 =

| + _>+,|-+>,

% = -*!+-> + Ί-+>, «4 = I—>,

ßi ^{= - Î W}

^{+ ω}^Β^{- 2ω}

+ i / } ,

ί2² = fâj - R),

3. S P I N ECHO EXPERIMENTS 445 T h e trace calculation is straightforward, but very tedious. T h e echo amplitude at t — 2r is obtained by collecting the coefficients of exp[—(ij2)(œA + ωΒ — 2œ)(t — 2τ)]. One finds that the absolute value of the echo amplitude is given by

δ2

2R2 1 + 2δ² 2 sin2 \]τ sin2 \ R T (3.28) F °r I /1 ^ I δ I, this reduces to

1 — 2 sin2 \]τ sin² \ δτ | (3.29) Figure 9.19 shows the observed echo envelope for the protons in 2-bromo-5-chlorothiophene. T h e modulation of the envelope is in good agreement with (3.28), and the values of / and δ are in good agreement with the results obtained from steady-state experiments (cf. Chapter 6).

T h e method of calculation used for two 90° pulses may also be used to calculate the signals observed with more complicated pulse sequences (27).

τ in seconds

FIG. 9.19. O b s e r v e d e c h o envelope for t h e p r o t o n s in 2 - b r o m o - 5 - c h l o r o t h i o p h e n e . T h e lower plot of t h e o b s e r v e d e n v e l o p e is n o r m a l i z e d t o u n i t y in t h e u p p e r p l o t to correct for t h e d a m p i n g effects of n u c l e a r relaxation a n d diffusion. [ H a h n a n d M a x w e l l (23).]

446 9. M U L T I P L E Q U A N T U M T R A N S I T I O N S R E F E R E N C E S

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4. S. Yatsiv, Phys. Rev. 1 1 3 , 1522 (1959).

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In document and Multiple Quantum Transitions, Double (Pldal 34-41)

= Χ2*,(0)^*(0)[£Λ(0ΧΕ/(ί)]« .