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Oscillatory free induction decay in angular distribution of nuclear radiation
Volume 106A, number 4
PHYSICS LETTERS
3 December 1984
OSCILLATORY FREE INDUCTION DECAY IN ANGULAR DISTRIBUTION OF NUCLEAR RADIATION L.N. SHAKHMURATOVA
Kazan State Pedagogical Institute, Kazan 420021, USSR Received 8 May 1984
A theoretical consideration of the free induction decay (FID) signal formation in the angular distribution of radiation, emitted by oriented nuclei after excitation by a long mdiofrequency resonance pulse tp • T2*,is carried out. The well-
known theorem on the oscillatory FID signal is expanded for the case of second rank tensors.
Recently the spin echo and free induction decay (FID) were detected by observation of the change in anisotropy in the angular distribution of radiation, emitted by oriented nuclei [ 1 ]. Theoretical consideration of the observation of spin echo and FID signals in the angular distribution of nuclear radiation (ADNR) is of special interest, in particular when the spin system by a long pulse (tp ~, T~2,where T~ is the inhomogeneous dephasing time) is being excited. The investigation of the oscillatory behaviour of the FID under condition of strong inhomogeneity for a two-level system, observed in NMR or in optical resonance experiments has been carried out in refs. [2,3] and as a result the theorem on this phenomenon has been formulated. In the present paper the multilevel nuclear spin system I and oscillatory FID signal in ADNR is theoretically considered, and the theorem is expanded for the case of higher rank tensors (~, > 1), observed in ADNR experiments. The possibility of an oscillatory FID signal in the case of axial geometry of ADNR experiment, when an extra rr/2-pulse is used to restore the initial axial symmetry of the nuclear spin system and to allow exploration of the spin system dynamics in the perpendicular plane [ 1], is investigated. The statistical tensors px, connected with the density matrix of the nuclei I by Clebsch-Gordon coefficients may be represented in the form [4] pq22(t2)= ~
ql,Kl
Gqlq2Q XlXat 2 _ t l .)P q l -(t l . ,
(1)
0.375-9601 •84•$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where Gqxlq2(t2 - t 1) are the perturbation factors q ks (PFs), defined by the interaction hamiltonian on the definite time interval t 2 - tl, here 0 ~< k ~< 21, -~, ~< q ~< ~,. We obtained the PF, for the spin system I, excited by a long pulse tp ~, ~2, when the initial orientation of nuclei is defined by kl(01, ~01) and the direction of nuclear radiation registration is k2(02, ~P2): r-I -1 ^-i(~l+¢z) -11 ei(~ol+~o2)) '-'2 2 ~ + G 22 = / { cos 2y(1 + x 2 ) 1/2 _ 4 cos y ( l + x2) 1/2 \\ (l +X2) 2 (1 +X2) 2
+3c°sy(I+x2)I/21 +x 2 - 3 (1 X-~x22)2) × cos bx cos(ab + a ) ) ,
(2a)
( r 2 -2 ^-2i(~o1+¢2) ~ r-22^2i(~al+~a2)\ u 2 2c ~-~ 22 ~ t
(1 + x 2 ) 2
31),)
+4(1+x2)2
cosy(1 + x 2 ) 1/2 (1 + x 2 ) 2
cos2bxcos2(ab+a
,
(2b)
wherey = ¢Oltp, b = COl(t - tp), a = WO/Wl,X = Awl/ 6Ol, o = COot" + 2A -- ~01 -- ~02; ~o0, col, A are frequency, amplitu~ and phase of the pulsed field, the bracket ( ) denotes an average over te inhomogeneous line 195
Volume 106A, number 4
PIIYSICS LETTERS
shape of gaussian form. By utilization of a Laplace transform [5,2] these expressions may be written in compact form, indicating the duration of the FID oscillations, for example,
3 I)eccmber 1984
-~ ~ . ~
(G21 - 12 e-i@l+~2) + G -1221 ei@l+~2)) =~rcos [a b
X
(4 j ( y \
?~ ~
l
~
. . . . . . .
~ v u-v v. v.V_~ .......
S)2Jo(x/4S 2 - b 2 ) u ( s - b/2) ds
0
- 2 f (y - s)ZJo(x/~ 0
b2)u(s - b) ds
- 3 ~ Jo(~s2 - b2)u(s - b) ds )],
(3)
IA" /y"
0 where
u(s-b)=O, S < b , =~1 , S = b , = 1, S > b .
~0~ -o:~
"°Zll/] t~ ^ .o4.t~/L/v "- ,~ /
It may be shown that expressions (2a) and (2b) are different from zero, when tp ~< t ~< 3tp and tp ~< t ~< 2tp, respectively. Our results do not contradict the theorem formulated in ref. [2]. Here G~21q2 describes the evolution of second rank tensors whereas the theorem concerns the first rank tensor observed in conventional NMR, so that the G~ltq2 oscillations, corresponding to the first rank tensors, are in accordance with the theorem. In the ADNR technique the higher rank ~. ~> 2 tensors are detected, consequently the FID signal's oscillations may exist at t > 2tp, but their duration definitely correlates with the pulse duration for all ql, q2 values. The computed results are in accordance with these conclusions. High-frequency oscillations are modulated by a slowly oscillating component, damping over a definite time interval (fig. 1). Also slow oscillations for the second rank tensors appear earlier than 6Olt p = 27r, whereas in the case of the first rank tensor oscillations appear only when the pulse area satisfies COltp/> 2n [2]. It often appears that in ADNR techniques the high frequency component is difficult to resolve. In this case observation of the slowly oscillating component may be realized by utilization of axial experimental geometry with short tp2 ,~ T~ and an extra ~r/2-pulse in accordance with the experiment [ 1]. Nuclear radiation is detected immediately after the extra pulse end, 196
Fig. I. Oscillations of (a)(G 1-12 exp[-i@l +~2)] + G-~12 × exp[i(~ol * ~2)]), (b)(G22-22exp[-2i(~oI +~02)] + G-2222 X exp[2i(~ol + ~02)])after the long pulse action ofy = wit p = lr/2, rr, 31r/2 area. Here b = tol(t - tp), ot = 0. then the anisotropy of ADNR is described by PFs of the form
(
= --~\ +.
(1+X2)2
cosyO+ b,2
(l+x2)------~
3 ) 4(1 +X2) 2 COS 2y(1 +X2) 1/2
+ 3[(
_ 1
4 k~2(1 + x 2 ) 2
(1 ~ +x2)
c o s y ( l +x2) 1/2
2x 2 + 1 +x2)1/2) + (1 +x2) 2 cos 2 y ( l cos
+
(
(1+
X2) 3/2 s i n y ( l + x 2 ) 1/2
× s i n 2y(1 + x 2 ) ' / 2 ) sin
2bx]),
2bx
(1 +x2) 312 (4)
Volume 106A, number 4
l.q
,
~
PHYSICS LETTERS
,
~
,
~,
3 December 1984
.
z)
,1 qq t
~
5
"t
t
~. 62
~
$
e
~.sg
1.5q
I.~
Fig• 2. Oscillations of complicated form for two pulse excitation, represented as a function of the pulse interval, depending upon long pulse a~cay = toltpt is (1)y = ~r[2; (2) y = lr; (3) y = 37r/2; (4) y = 21r; (5)y = 51r/2. The continuous line corresponds to R = to 1/o = 0.45 ; the dotted line to R = 0.1, where a is the inhomogeneous broadening. The decaying end for c > y is the extra short pulse effect•
c = tolr. The
where c = w I r, r is the time interval between the pulses, A 1 - A 2 -- n / 2 = 0, other designations are as before w h e n tp ~ t p r It should be noted, that due to the two pulses (long pulse and short one) different statistical tensors o f the second rank are mixed, so that superposition of a n u m b e r of low-frequency oscillations takes place and the resulting pattern can have a complicated form. By means o f Laplace-transform techniques expression (4) has been brought to a form to d e m o n s t r a t e that the oscillations last for tp~ (fig. 2). The amplitude o f these oscillations depend on T~2, so that it proves that these complicated oscillations c a n n o t be regarded as spin echo signals•
References
[1 ] H.R. Foster, P. Cooke, D.H. Chaplin, P. Lynam and G.V.H. Wilson, Phys. Rev. Lett. 38 (1977) 1546. [2] A. Schenzle, N.C. Wong and R.G. Brewer, Phys. Rev. A21 (1980) 887; A22 (1980)635. [3] M. Kunimoto, T. Endo, S. Nakaishi and T. Hashi, Phys. Lett. 80A (1980) 84. [4] E. Matthias, B. Olsen, D.A. Shirley, J.E. Templeton and R.M. Steffen, Phys. Rev. A4 (1971) 1626. [5] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables (Nat. Bureau of Standards, Washington, 1964) ch. 29.
197
PHYSICS LETTERS
3 December 1984
OSCILLATORY FREE INDUCTION DECAY IN ANGULAR DISTRIBUTION OF NUCLEAR RADIATION L.N. SHAKHMURATOVA
Kazan State Pedagogical Institute, Kazan 420021, USSR Received 8 May 1984
A theoretical consideration of the free induction decay (FID) signal formation in the angular distribution of radiation, emitted by oriented nuclei after excitation by a long mdiofrequency resonance pulse tp • T2*,is carried out. The well-
known theorem on the oscillatory FID signal is expanded for the case of second rank tensors.
Recently the spin echo and free induction decay (FID) were detected by observation of the change in anisotropy in the angular distribution of radiation, emitted by oriented nuclei [ 1 ]. Theoretical consideration of the observation of spin echo and FID signals in the angular distribution of nuclear radiation (ADNR) is of special interest, in particular when the spin system by a long pulse (tp ~, T~2,where T~ is the inhomogeneous dephasing time) is being excited. The investigation of the oscillatory behaviour of the FID under condition of strong inhomogeneity for a two-level system, observed in NMR or in optical resonance experiments has been carried out in refs. [2,3] and as a result the theorem on this phenomenon has been formulated. In the present paper the multilevel nuclear spin system I and oscillatory FID signal in ADNR is theoretically considered, and the theorem is expanded for the case of higher rank tensors (~, > 1), observed in ADNR experiments. The possibility of an oscillatory FID signal in the case of axial geometry of ADNR experiment, when an extra rr/2-pulse is used to restore the initial axial symmetry of the nuclear spin system and to allow exploration of the spin system dynamics in the perpendicular plane [ 1], is investigated. The statistical tensors px, connected with the density matrix of the nuclei I by Clebsch-Gordon coefficients may be represented in the form [4] pq22(t2)= ~
ql,Kl
Gqlq2Q XlXat 2 _ t l .)P q l -(t l . ,
(1)
0.375-9601 •84•$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where Gqxlq2(t2 - t 1) are the perturbation factors q ks (PFs), defined by the interaction hamiltonian on the definite time interval t 2 - tl, here 0 ~< k ~< 21, -~, ~< q ~< ~,. We obtained the PF, for the spin system I, excited by a long pulse tp ~, ~2, when the initial orientation of nuclei is defined by kl(01, ~01) and the direction of nuclear radiation registration is k2(02, ~P2): r-I -1 ^-i(~l+¢z) -11 ei(~ol+~o2)) '-'2 2 ~ + G 22 = / { cos 2y(1 + x 2 ) 1/2 _ 4 cos y ( l + x2) 1/2 \\ (l +X2) 2 (1 +X2) 2
+3c°sy(I+x2)I/21 +x 2 - 3 (1 X-~x22)2) × cos bx cos(ab + a ) ) ,
(2a)
( r 2 -2 ^-2i(~o1+¢2) ~ r-22^2i(~al+~a2)\ u 2 2c ~-~ 22 ~ t
(1 + x 2 ) 2
31),)
+4(1+x2)2
cosy(1 + x 2 ) 1/2 (1 + x 2 ) 2
cos2bxcos2(ab+a
,
(2b)
wherey = ¢Oltp, b = COl(t - tp), a = WO/Wl,X = Awl/ 6Ol, o = COot" + 2A -- ~01 -- ~02; ~o0, col, A are frequency, amplitu~ and phase of the pulsed field, the bracket ( ) denotes an average over te inhomogeneous line 195
Volume 106A, number 4
PIIYSICS LETTERS
shape of gaussian form. By utilization of a Laplace transform [5,2] these expressions may be written in compact form, indicating the duration of the FID oscillations, for example,
3 I)eccmber 1984
-~ ~ . ~
(G21 - 12 e-i@l+~2) + G -1221 ei@l+~2)) =~rcos [a b
X
(4 j ( y \
?~ ~
l
~
. . . . . . .
~ v u-v v. v.V_~ .......
S)2Jo(x/4S 2 - b 2 ) u ( s - b/2) ds
0
- 2 f (y - s)ZJo(x/~ 0
b2)u(s - b) ds
- 3 ~ Jo(~s2 - b2)u(s - b) ds )],
(3)
IA" /y"
0 where
u(s-b)=O, S < b , =~1 , S = b , = 1, S > b .
~0~ -o:~
"°Zll/] t~ ^ .o4.t~/L/v "- ,~ /
It may be shown that expressions (2a) and (2b) are different from zero, when tp ~< t ~< 3tp and tp ~< t ~< 2tp, respectively. Our results do not contradict the theorem formulated in ref. [2]. Here G~21q2 describes the evolution of second rank tensors whereas the theorem concerns the first rank tensor observed in conventional NMR, so that the G~ltq2 oscillations, corresponding to the first rank tensors, are in accordance with the theorem. In the ADNR technique the higher rank ~. ~> 2 tensors are detected, consequently the FID signal's oscillations may exist at t > 2tp, but their duration definitely correlates with the pulse duration for all ql, q2 values. The computed results are in accordance with these conclusions. High-frequency oscillations are modulated by a slowly oscillating component, damping over a definite time interval (fig. 1). Also slow oscillations for the second rank tensors appear earlier than 6Olt p = 27r, whereas in the case of the first rank tensor oscillations appear only when the pulse area satisfies COltp/> 2n [2]. It often appears that in ADNR techniques the high frequency component is difficult to resolve. In this case observation of the slowly oscillating component may be realized by utilization of axial experimental geometry with short tp2 ,~ T~ and an extra ~r/2-pulse in accordance with the experiment [ 1]. Nuclear radiation is detected immediately after the extra pulse end, 196
Fig. I. Oscillations of (a)(G 1-12 exp[-i@l +~2)] + G-~12 × exp[i(~ol * ~2)]), (b)(G22-22exp[-2i(~oI +~02)] + G-2222 X exp[2i(~ol + ~02)])after the long pulse action ofy = wit p = lr/2, rr, 31r/2 area. Here b = tol(t - tp), ot = 0. then the anisotropy of ADNR is described by PFs of the form
(
= --~\ +.
(1+X2)2
cosyO+ b,2
(l+x2)------~
3 ) 4(1 +X2) 2 COS 2y(1 +X2) 1/2
+ 3[(
_ 1
4 k~2(1 + x 2 ) 2
(1 ~ +x2)
c o s y ( l +x2) 1/2
2x 2 + 1 +x2)1/2) + (1 +x2) 2 cos 2 y ( l cos
+
(
(1+
X2) 3/2 s i n y ( l + x 2 ) 1/2
× s i n 2y(1 + x 2 ) ' / 2 ) sin
2bx]),
2bx
(1 +x2) 312 (4)
Volume 106A, number 4
l.q
,
~
PHYSICS LETTERS
,
~
,
~,
3 December 1984
.
z)
,1 qq t
~
5
"t
t
~. 62
~
$
e
~.sg
1.5q
I.~
Fig• 2. Oscillations of complicated form for two pulse excitation, represented as a function of the pulse interval, depending upon long pulse a~cay = toltpt is (1)y = ~r[2; (2) y = lr; (3) y = 37r/2; (4) y = 21r; (5)y = 51r/2. The continuous line corresponds to R = to 1/o = 0.45 ; the dotted line to R = 0.1, where a is the inhomogeneous broadening. The decaying end for c > y is the extra short pulse effect•
c = tolr. The
where c = w I r, r is the time interval between the pulses, A 1 - A 2 -- n / 2 = 0, other designations are as before w h e n tp ~ t p r It should be noted, that due to the two pulses (long pulse and short one) different statistical tensors o f the second rank are mixed, so that superposition of a n u m b e r of low-frequency oscillations takes place and the resulting pattern can have a complicated form. By means o f Laplace-transform techniques expression (4) has been brought to a form to d e m o n s t r a t e that the oscillations last for tp~ (fig. 2). The amplitude o f these oscillations depend on T~2, so that it proves that these complicated oscillations c a n n o t be regarded as spin echo signals•
References
[1 ] H.R. Foster, P. Cooke, D.H. Chaplin, P. Lynam and G.V.H. Wilson, Phys. Rev. Lett. 38 (1977) 1546. [2] A. Schenzle, N.C. Wong and R.G. Brewer, Phys. Rev. A21 (1980) 887; A22 (1980)635. [3] M. Kunimoto, T. Endo, S. Nakaishi and T. Hashi, Phys. Lett. 80A (1980) 84. [4] E. Matthias, B. Olsen, D.A. Shirley, J.E. Templeton and R.M. Steffen, Phys. Rev. A4 (1971) 1626. [5] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables (Nat. Bureau of Standards, Washington, 1964) ch. 29.
197