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Adiabatic rapid passage with radiation damping
Volume 4 1A, number 4
PHYSICS LETTERS
9 October 1972
ADIABATIC RAPID PASSAGE WITH RADIATION
DAMPING
M. ODEHNAL, V. PETlb,.I'(~EK
Institute of Nuclear Physics, [~eL Czechoslovakia Received 8 August 1972 The familiar Bloch equations are numerically solved for the case of strong radiation coupling during an adiabatic rapid passage. The behaviour of the magnetization appears to be more complicated than predicted by a previous solution of Bloom.
It is well-known from experiments on nuclear magnetic resonance (NMR), ferromagnetic resonance, on masers etc. that the resonant circuit enclosing a material with a large macroscopic magnetization Menhances the backward coupling of the nuclear (electronic) spin system (creating the 3'/) with itself. This coupling manifests itself in the radiation damping process [ 1 ]. A complete mathematical treatment of this coupling requires a solution of a set of coupled nonlinear differential equations. Bloembergen and Pound [ 1] set up this set of equations and gave a discussion o f some interesting cases encountered in resonance experiments. Bloom [2] simplified this set of five equations and left three Bloch equations only. He supplemented the externally applied rf magnetic field by a so called ringing field, i.e, a field created in the structure by the motion of the coherent M-vector. One of the cases analysed by Bloom was also an adiabatic rapid passage (ARP) through a NMR line. Bloom's solution, based on certain assumptions, is not valid for large coupling of the spin system and the resonant.circuit. This fact may have some important consequences for an ARP in solids, especially, for the ARP method of the reversal of the sign o f the polarization of nuclei in a polarized target. We analysed an ARP through a NMR line under the common conditions of an NMR experiment using the Bloom equations and solving them by a computer. These equations can be written as (for one coil NMR arrangement):
M0
+6(0
+ d
v
1
v
=-PMn2 , (1) uMz
where Mz, u and o are the components of the magnetization Min the rotating frame (rotating around the laboratory Z-axis with angular frequency w of the applied rf field) in the z-(z IIZ), x-,y-axes, respectively, 0 = 2zr~Q3"Mo/(3"H1),rl and Q the filling and quality factors of the NMR coil, respectively, 3' the gyromagnetic ration, ~ = 3'H0, T2 is the transversal relaxation time (TI~ oo). We supposed that our system (I = 1/2), placed in a large external field H 0 (11z) is subjected to an externally applied rf field Hi(X-axis of the laboratory frame X,Y,Z). A linear sweep of the frequency 60 of the rf generator through an interval A co (A60 is larger than the NMR linewidth) with a rate A60/At is assumed. The parameter 6(t) is given by 8 ( 0 = {6o0 - w(t)}/(3"H1) where 60(0 = 6o0 - A60(t0 - O / A t and t o is the instant when 60 = 600" M0 is the thermal equilibrium value of the magnetization and r = 3"Hlt. An ARP can be performed if tp < T2, TI, tp>> (7//1)-1 and d60/dt~3"2H21, where tp is the time of the passage through the line. Bloom solved a similar set of equations as in (1) under quasi-steady state conditions. He obtained a symmetrical solution for an ARP with respect to the starting value ofMz/M0 provided p < 1. If O > 1 an ARP is no more possible. In fig. 1 we have shown one o f our solutions of eqs. (1) for an ARP with the value o f o = 0.1, where one can see a strong asymmetry in the behaviour of 359
Volume 41A, number 4
/ ,/
PHYSICS LETTERS
N\
/\ /X
-O2 -0~
/
iI /2 /I/ X
II/''
-
/ /
-o8~/' ,I
_,.oiL/ -4
i/
/ -2
0
2
4o- u.~ co
Fig. 1 Variation of the z-component of the M-vector during an ARP for different initial values of (Mz/Mo) i. Curve 1 with (Mz/Mo) i = +1 and curve 2 with (Mz/Mo) i = 1 are numerical solution of eqs. (1) for O = 0.1. Curve 3 corresponds to the Bloom solution for p2 = 0.5 and (Mz/MOi= -1. the M-vector. For p ~< 10 - 2 we obtain practically the same results as Bloom. However, for larger values of p the behaviour of M is more complicated and approaches the transient two-level maser behaviour [3]. If we start the ARP from initial value (Mz/Mo) i = +1 the M-vector starts to follow the continuously varying effective field Heft in rotating frame (curve 1). When it reaches a certain critical value o f M z / M 0 (depending on p) there is a cooperative phenomenon: the spin system emits strongly its excess o f orientational energy, the vector M is decoupled from the H etf and the following motion is nonadiabatic. This radiative transient return of M on the curve 1 is not shown because
360
9 October 1972
it starts out of the scale of the figure at ( 6 % - co)/TH1 = 7. If we start the ARP with (Mz/Mo) = --1 the nonadiabatic emission of the energy starts very early in the beginning of the passage (curve 2, dashed double curve represents an envelope of oscillatory transition of Mz/Mo). After this cooperative action, related 'also to the superradiant emission [4], the vector M c a n be locked again to the varying Heft, follows it adiabatically for some time and, in dependence on the value o f O, finishes the passage normally or is overthrown again to the position of the minimum o f its orientational energy. Curve 3 represents one of the Bloom solutions of ARP. The difference o f our and his solution is clear. One can note that even for so large value o f p there are no transient phenomena in his curve. It is well known that the Bloch equations are not valid for solids and one can apply these results to solids qualitatively only. However, the radiation damping manifests itself in solid samples very often (see for example the broadening or sharpening o f NMR lines in the experiments with highly polarized targets) and we can expect that this study could help understanding the peculiar asymmetry in the ARP observed on proton polarized target at Dubna [5]. A detailed discussion of different solution of eqs. (1) will be published elsewhere.
References [i] [2] [3] [4] [5]
N. Bloembergen, R.V. Pound, Phys. Rev. 95 (1954) 8. S. Bloom, J. Appl. Phys. 28 (1957) 800. I.M. Firth, Physica 29 (1963) 857. R.H. Dicke, Phys. Rev. 93 (1954) 99. L.B. Parfenov, B.S. Neganov, Polarized proton target, Published by Joint Institute for Nuclear Research, Dubna, USSR, 1968, No. 13-4143, (in Russian).
PHYSICS LETTERS
9 October 1972
ADIABATIC RAPID PASSAGE WITH RADIATION
DAMPING
M. ODEHNAL, V. PETlb,.I'(~EK
Institute of Nuclear Physics, [~eL Czechoslovakia Received 8 August 1972 The familiar Bloch equations are numerically solved for the case of strong radiation coupling during an adiabatic rapid passage. The behaviour of the magnetization appears to be more complicated than predicted by a previous solution of Bloom.
It is well-known from experiments on nuclear magnetic resonance (NMR), ferromagnetic resonance, on masers etc. that the resonant circuit enclosing a material with a large macroscopic magnetization Menhances the backward coupling of the nuclear (electronic) spin system (creating the 3'/) with itself. This coupling manifests itself in the radiation damping process [ 1 ]. A complete mathematical treatment of this coupling requires a solution of a set of coupled nonlinear differential equations. Bloembergen and Pound [ 1] set up this set of equations and gave a discussion o f some interesting cases encountered in resonance experiments. Bloom [2] simplified this set of five equations and left three Bloch equations only. He supplemented the externally applied rf magnetic field by a so called ringing field, i.e, a field created in the structure by the motion of the coherent M-vector. One of the cases analysed by Bloom was also an adiabatic rapid passage (ARP) through a NMR line. Bloom's solution, based on certain assumptions, is not valid for large coupling of the spin system and the resonant.circuit. This fact may have some important consequences for an ARP in solids, especially, for the ARP method of the reversal of the sign o f the polarization of nuclei in a polarized target. We analysed an ARP through a NMR line under the common conditions of an NMR experiment using the Bloom equations and solving them by a computer. These equations can be written as (for one coil NMR arrangement):
M0
+6(0
+ d
v
1
v
=-PMn2 , (1) uMz
where Mz, u and o are the components of the magnetization Min the rotating frame (rotating around the laboratory Z-axis with angular frequency w of the applied rf field) in the z-(z IIZ), x-,y-axes, respectively, 0 = 2zr~Q3"Mo/(3"H1),rl and Q the filling and quality factors of the NMR coil, respectively, 3' the gyromagnetic ration, ~ = 3'H0, T2 is the transversal relaxation time (TI~ oo). We supposed that our system (I = 1/2), placed in a large external field H 0 (11z) is subjected to an externally applied rf field Hi(X-axis of the laboratory frame X,Y,Z). A linear sweep of the frequency 60 of the rf generator through an interval A co (A60 is larger than the NMR linewidth) with a rate A60/At is assumed. The parameter 6(t) is given by 8 ( 0 = {6o0 - w(t)}/(3"H1) where 60(0 = 6o0 - A60(t0 - O / A t and t o is the instant when 60 = 600" M0 is the thermal equilibrium value of the magnetization and r = 3"Hlt. An ARP can be performed if tp < T2, TI, tp>> (7//1)-1 and d60/dt~3"2H21, where tp is the time of the passage through the line. Bloom solved a similar set of equations as in (1) under quasi-steady state conditions. He obtained a symmetrical solution for an ARP with respect to the starting value ofMz/M0 provided p < 1. If O > 1 an ARP is no more possible. In fig. 1 we have shown one o f our solutions of eqs. (1) for an ARP with the value o f o = 0.1, where one can see a strong asymmetry in the behaviour of 359
Volume 41A, number 4
/ ,/
PHYSICS LETTERS
N\
/\ /X
-O2 -0~
/
iI /2 /I/ X
II/''
-
/ /
-o8~/' ,I
_,.oiL/ -4
i/
/ -2
0
2
4o- u.~ co
Fig. 1 Variation of the z-component of the M-vector during an ARP for different initial values of (Mz/Mo) i. Curve 1 with (Mz/Mo) i = +1 and curve 2 with (Mz/Mo) i = 1 are numerical solution of eqs. (1) for O = 0.1. Curve 3 corresponds to the Bloom solution for p2 = 0.5 and (Mz/MOi= -1. the M-vector. For p ~< 10 - 2 we obtain practically the same results as Bloom. However, for larger values of p the behaviour of M is more complicated and approaches the transient two-level maser behaviour [3]. If we start the ARP from initial value (Mz/Mo) i = +1 the M-vector starts to follow the continuously varying effective field Heft in rotating frame (curve 1). When it reaches a certain critical value o f M z / M 0 (depending on p) there is a cooperative phenomenon: the spin system emits strongly its excess o f orientational energy, the vector M is decoupled from the H etf and the following motion is nonadiabatic. This radiative transient return of M on the curve 1 is not shown because
360
9 October 1972
it starts out of the scale of the figure at ( 6 % - co)/TH1 = 7. If we start the ARP with (Mz/Mo) = --1 the nonadiabatic emission of the energy starts very early in the beginning of the passage (curve 2, dashed double curve represents an envelope of oscillatory transition of Mz/Mo). After this cooperative action, related 'also to the superradiant emission [4], the vector M c a n be locked again to the varying Heft, follows it adiabatically for some time and, in dependence on the value o f O, finishes the passage normally or is overthrown again to the position of the minimum o f its orientational energy. Curve 3 represents one of the Bloom solutions of ARP. The difference o f our and his solution is clear. One can note that even for so large value o f p there are no transient phenomena in his curve. It is well known that the Bloch equations are not valid for solids and one can apply these results to solids qualitatively only. However, the radiation damping manifests itself in solid samples very often (see for example the broadening or sharpening o f NMR lines in the experiments with highly polarized targets) and we can expect that this study could help understanding the peculiar asymmetry in the ARP observed on proton polarized target at Dubna [5]. A detailed discussion of different solution of eqs. (1) will be published elsewhere.
References [i] [2] [3] [4] [5]
N. Bloembergen, R.V. Pound, Phys. Rev. 95 (1954) 8. S. Bloom, J. Appl. Phys. 28 (1957) 800. I.M. Firth, Physica 29 (1963) 857. R.H. Dicke, Phys. Rev. 93 (1954) 99. L.B. Parfenov, B.S. Neganov, Polarized proton target, Published by Joint Institute for Nuclear Research, Dubna, USSR, 1968, No. 13-4143, (in Russian).