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Spin-rotational relaxation of protons in some solid ammonium compounds
Volume 81A, number 2,3
PHYSICS LETTERS
12 January 1981
SPIN-ROTATIONAL RELAXATION OF PROTONS IN SOME SOLID AMMONIUM COMPOUNDS I K. SHIMOMURA Faculty of Integrated Arts and Sciences, Hiroshima Universityl Hiroshima, Japan and M. YOSHIDA, A. SANJOH and H. NEGITA Department of Chemistry, Faculty of Sciences, Hiroshima University, Hiroshima, Japan Received 4 March 1980 Revised manuscript received 25 September 1980
A large minimum of T1D of 0.5-0.75 s for protons in NH4C104, (NH4)2S2Os and (NH4)2Ce(NO3)6 has been observed. It is interpreted in terms of spin-rotational relaxation. Weak and strong collision approaches are presented and the spinrotation field is estimated to be about 10-: G, which is two orders of magnitude smaller than the usual dipolar field.
The spin-lattice relaxation times T 1 , T l p in the laboratory and rotating frames and the relaxation time of the dipolar energy TID for protons in polycrystalline NH4C104, (NH4)2S208 and (NH4)2Ce(NO3) 6 were measured at 18 MHz using a pulse spectrometer which has been described elsewhere [1] by applying three types of pulse sequences; 9 0 t - 9 0 , 90-long pulse (t) and 9 0 - T 2 - 4 5 - t - 4 5 , respectively, where the amplitude H I of the rf field in the long pulse was 2.5 G. Measurements were made over a temperature range between 380 K and the liquid nitrogen temperature 77 K. The proton line width was measured at 40 MHz using a JEOL BE-1 broad line spectrometer. The line shape and the second moments do not change appreciably in the temperature region between 133 K and 373 K for NH4C104: AHp_p 3.6 G, (AH 2) ~ 1.3 G 2. We found a remarkably large minimum in the T1D curve at or below room temperature, while the T 1 curve did not show any such minimum (figs. I, 2 and 3). An experiment on NH4C104 containing 15N in place of 14N showed the same results. At lower temperature, below 165 K, T1D in NH4C104 decreases with decreasing temperature, and the T 1 curve also shows the same tendency, which is due to the reorientation of the NH~" ions. The ratio of the relaxation
times T 1/T 1D increases from 2 to 3 with increasing temperature. This implies that the mode of the fluctuating field changes from no correlation to complete correlation [2] as the temperature increases. The value of the minimum of T1D is 0.5-0.75 s for the three compounds, and is four orders of magnitude larger than that of the reorientation of the NI-I~4ions when the dipolar energy relaxes through the modulation of the dipole interaction [ 3 ] . For the proton T 1 in solid NH4I, Sharp and Pinter [4] discussed the spin-rotational relaxation process which has been found in liquids [5,6]. Ikeda and McDowell [7] proposed the same mechanism for NH4C104 and interpreted T 1 above 200 K. In this model we expect T1D ~ T 1 on the high temperature side of the minimum of T 1 ; by using this model we cannot explain the minimum of T 1D. Nor can we ascribe it to the onset of translational diffusion of NH,~ ions since we do not find motional narrowing of protons due to diffusion for temperatures above the minimum of T 1D in NH4C104. This is consistent with the prediction that a larger minimum of T1D contributes less to the narrowing of the resonance line [8]. These compounds have large anions. The reorientation of large massive anions may trigger rotations of the nearly free NH~ ion around one of its symmetry axes,
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
189
Volume 81A, number 2,3
PHYSICS LETTERS
12 January 1981
(NHa)2Ce(NO3)6
NH4CIO4 T1
°°°
°°
o
T1
6>°°
"U" 10 Q.,
ocoOOOoo
o
oo o
"riD
ee
o
"3 ID
o
e •
oeO
I/) v
o
10
o o "[1D
•
%
o
o
2
a
I--.
1
•
i
A
i
2
TIj~
~,:
**
%
(H1:2.5 G) L
,
i
,
i
i
,
~
10 103/T (K -~)
4
6
° e o
i
12
8
Fig. 1. The temperature dependence of T1, Tip and T 1D in NH4C104. and the spin-rotational interaction appears. Since the rotation axis may change at each collision, we assume that the correlation time r c of the rotation is equal to that of the reorientation of anions which is supposed to be much longer than that of the NHz~ ions. The spin-rotational hamiltonian can be expressed in terms of the proton spin operator I, the rotational angular momentum J and the spin-rotational coupling tensor C which is most conveniently given with respect to the molecule-fixed coordinate frame, as [9]
%r(t) = z.
c-J(t).
Using the semiclassical approach [5,10] in which we consider the coupling to be a scalar, 9~sr can be replaced with
(NH4)2S2Os T1
oooo o o
I0
oo o
o o
'~D
•
2
3 4 IO~/T
5 6 (K-1)
Fig. 3. The temperature dependence of T 1 and T 1D in (NH4)2Ce(NO3)6.
~sr(t) = - - ~ # ( t ) ' Z (1)
= - 7 ~ [ h x ( t ) l x + h y ( t ) l y + hz(t)lz] ,
where H ( h x , hy, hz) is the magnetic field produced at the proton by the rotation of the NH~ ion and we define the laboratory frame by x, y, z, with the z-axis parallel to the external field H 0. The components h x , hy and h z are linear functions of the components of the angular momentum of the ions. At low temperatures, collisions are so infrequent that r c is longer than the spin-spin relaxation time T 2 and strong collision theory may be applied, while at high temperatures r c becomes so short that weak collision theory may hold. 1. Weak collision case. The relaxation times T 1 and T 1 are written in terms of the Zeeman terms or ~gl ~o=- T h H l l x ) as well as the fluctuation term ~sr(t) in the hamiltonian. Using eq. (2) of ref. [11] and eq. (2.27) of ref. [12], we find T 1 and Tlp as
o
%
•
= v2 (hx2 + h2)
o
T1
i..=
"c
(2)
1 + 002 7 c2 '
I
1 - 3'2 h'~ 2
3
4
5
6
IOyT ¢K-') Fig. 2. The temperature dependence of T1 and T1D in (NI-I4)2 $20s. 190
rc
2 2
Tlo
1 + co 1 r c
II
+ 1_,,,2t~2" 2 t ,,y
Te -COl) r c
TC
+ 1 +(co o
]
•,
(3)
Volume 81A, number 2,3
PHYSICS LETTERS
where 601 = 7H1 and an overbar denotes ensemble average. Since 600 >> 6°1, we have = ,,/2h--~
rc -7c l + c o 2 r 2 +72hy2 1+-- 6002rc2"
TXÜ
can use the sudden approximation, and the energy change can be written as A ( ~ s r ) = Tr p(~0rk, --C~Ork),
(4)
From eqs. (2) and (4), it is expected that the relaxation time Tlp will be very much shorter than T 1 when 600rc >> 1, while Tlp will have an col-dependence , when COlt c >> 1. This has been observed experimentally.
2. Strong collision case. Following the SlichterAilion approach [13], in which the period r c between collisions is assumed to be longer than T 2, a common spin temperature is established and we can describe the spin state in terms of the density matrix in the rotating frame: p
=
(73
in terms Of~s0rk and 9~s0rk,, the spin-rotational hamiltonians before and after a collision, respectively. By evaluating the traces in the high temperature limit, one finds
(dO-1/dO = - [Tr(~Ork)2/Tr(~T(o) 2 rcl 0 -1 .
(8)
Then, using the following formula:
~Ork = -7l~hzklz ,
(9)
one obtains the relaxation time Tlp of the system in the rotating frame as,
r~p 1 = [h2z/(H? + H 2 + h2)] r c 1 ,
(10)
where
exp(-C~o/kO)/Z ,
where ~ o = ~ 1 + ~ d 0 +~Or, 0 is the spin temperature, Z is the partition function, ~Tgd is the dipolar part of the hamiltonian and the superscript zero denotes the secular part. Since the total energy in the rotating frame is
=- raxI4
12 January 1981
+
(5)
we can find the rate of the energy change, neglecting the lattice coupling of the dipolar and Zeeman systems, as
dE]dt = A(~Or)/r c .
(6)
We evaluate the total energy by using the relation: E = Tr(pg(o). Since the actual duration of a collision is short, one
H 2 =¼72I~2I(1 + 1 ) ~
(1 - 3 cos20/)2/Rt,
1
R~ is the distance of the fth intermolecular spin pair, Oj is the angle between R~ ahd H O. Similarly, by putt i n g H 1 = 0 in eq. (10), we can get a formula for T1D:
=
2 + hl)]
(11)
Consider the case in which the temperature is low enough, so that the correlation time r e due to collisions is longer than the time A (>> T2) during which the spin-rotational interaction is active following a collision [5]. Assuming that a collision occurs at t = 0 and 9~0r = 0 for t < 0, it is easily shown that the energy does not change at t --- 0; on the other hand before t = A a new spin temperature is established, therefore we obtain
Table 1 Minimum of TID, activation energy and spin-rotation field. NH4C104
(NH4)2 $2 08
(NH4)2Ce(NO3)6
E a (keal/mol) r E (s) (anion reorientation)
4.5 0.6 5 6.6 X 10-l°
3.5 0.75 5
3.3 0.5 7.5
7.7 × 10 - 9
E a (keal/mol)(NH~ reorientation)
0.65
1.6
(hz2)l/2(G)
9.2 X 10-3
8.5 X 10 -3
T1D minimum (10a/T) (s)
~ 1 0 -10
1.6 ~10 -2
191
Volume 81A, number 2,3 E = Tr p •
=
-(ZkO) -1 T r ( ~ o ) 2 ,
PHYSICS LETTERS (12)
where ~ o = ~ 1 + ~ 0 +~Ork" Since Ug0rk, = 0 at t = A+, the energy change at t = A is
a(C~Or) = Tr p(c~Osrk, --C~Ork) = -(ZkO)- 1 Tr (~;~Ork) 2 .
(13)
In this case one obtains the same formulae as eqs. (10) and (I 1). We estimated 72h2 as 105 s - 2 from eq. (11), which was four orders of magnitude smaller than the usual dipolar fluctuation, and evaluated the activation energy as 5 - 7 . 5 kcal/mol which was associated with the reorientation o f the anions. The results are shown in table 1 together with the value of the minimum of T1 D, its temperature, and the activation energy o f ion reorientation. The minimum o f T 1 according to eq. (2) is co0[co 1 times larger than that o f eq. (4) and is hidden b y other relaxations. For a single crystal of NH 4 C10 4, we found an anisotropy of T l p , which was ascribed to the anisotropy o f T r ( ~ 0 ) Z a n d Tr (~0r) 2. The analysis is in progress.
192
12 January 1981
References [1] K. Shimomura, R. Sugimoto and H. Negita, J. Sci. Hiroshima Univ. Ser. A38 (1974) 71. [2] M. Goldman, Spin-temperature (Clarendon, Oxford, 1970). [3] M. Punkkinen, J. Nonmetals 2 (1974) 79; K. Morimoto, K. Shimomura and M. Yoshida, J. Phys. Soc. Japan 45 (1978) 1965. [4] A.R. Sharp and M.M. Pinter, J. Chem. Phys. 53 (1970) 2428. [5] R.J.C. Brown, H.S. Gutowsky and K. Shimomura, J. Chem. Phys. 38 (1963) 76. [6] P.S. Hubbard, Phys. Rev. 131 (1963) 1155. [7] R. Ikeda and C.A. McDowell, Chem. Phys. Lett. 14 (1972) 389. [8] D. Wolf, Spin-temperature and nuclear-spin relaxation in matter (Clarendon, Oxford, 1979). [9] N.F. Ramsey, Molecular beams (Oxford U.P., London, 1956). [10] D.K. Green and J.G. Powles, Proc. Phys. Soc. 85 (1965) 87; R.G. Gordon, J. Chem. Phys. 44 (1966) 1184; R.H. Faulk and M. Eisner, J. Chem. Phys. 44 (1966) 2926; H. Boehme and M. Eisner, J. Chem.Phys. 46 (1967) 4242; C.H. Wang, J. Magn. Reson. 9 (1973) 75. [11] R.V. Steenwinkel, Z. Naturforsch. 24 (1969) 15 26. [12] D. Woff and P. Jung, Phys. Rev. B12 (1975) 3596. [13] C.P. Slichter and D.C. Ailion, Phys. Rev. 135 (1964) A1099.
PHYSICS LETTERS
12 January 1981
SPIN-ROTATIONAL RELAXATION OF PROTONS IN SOME SOLID AMMONIUM COMPOUNDS I K. SHIMOMURA Faculty of Integrated Arts and Sciences, Hiroshima Universityl Hiroshima, Japan and M. YOSHIDA, A. SANJOH and H. NEGITA Department of Chemistry, Faculty of Sciences, Hiroshima University, Hiroshima, Japan Received 4 March 1980 Revised manuscript received 25 September 1980
A large minimum of T1D of 0.5-0.75 s for protons in NH4C104, (NH4)2S2Os and (NH4)2Ce(NO3)6 has been observed. It is interpreted in terms of spin-rotational relaxation. Weak and strong collision approaches are presented and the spinrotation field is estimated to be about 10-: G, which is two orders of magnitude smaller than the usual dipolar field.
The spin-lattice relaxation times T 1 , T l p in the laboratory and rotating frames and the relaxation time of the dipolar energy TID for protons in polycrystalline NH4C104, (NH4)2S208 and (NH4)2Ce(NO3) 6 were measured at 18 MHz using a pulse spectrometer which has been described elsewhere [1] by applying three types of pulse sequences; 9 0 t - 9 0 , 90-long pulse (t) and 9 0 - T 2 - 4 5 - t - 4 5 , respectively, where the amplitude H I of the rf field in the long pulse was 2.5 G. Measurements were made over a temperature range between 380 K and the liquid nitrogen temperature 77 K. The proton line width was measured at 40 MHz using a JEOL BE-1 broad line spectrometer. The line shape and the second moments do not change appreciably in the temperature region between 133 K and 373 K for NH4C104: AHp_p 3.6 G, (AH 2) ~ 1.3 G 2. We found a remarkably large minimum in the T1D curve at or below room temperature, while the T 1 curve did not show any such minimum (figs. I, 2 and 3). An experiment on NH4C104 containing 15N in place of 14N showed the same results. At lower temperature, below 165 K, T1D in NH4C104 decreases with decreasing temperature, and the T 1 curve also shows the same tendency, which is due to the reorientation of the NH~" ions. The ratio of the relaxation
times T 1/T 1D increases from 2 to 3 with increasing temperature. This implies that the mode of the fluctuating field changes from no correlation to complete correlation [2] as the temperature increases. The value of the minimum of T1D is 0.5-0.75 s for the three compounds, and is four orders of magnitude larger than that of the reorientation of the NI-I~4ions when the dipolar energy relaxes through the modulation of the dipole interaction [ 3 ] . For the proton T 1 in solid NH4I, Sharp and Pinter [4] discussed the spin-rotational relaxation process which has been found in liquids [5,6]. Ikeda and McDowell [7] proposed the same mechanism for NH4C104 and interpreted T 1 above 200 K. In this model we expect T1D ~ T 1 on the high temperature side of the minimum of T 1 ; by using this model we cannot explain the minimum of T 1D. Nor can we ascribe it to the onset of translational diffusion of NH,~ ions since we do not find motional narrowing of protons due to diffusion for temperatures above the minimum of T 1D in NH4C104. This is consistent with the prediction that a larger minimum of T1D contributes less to the narrowing of the resonance line [8]. These compounds have large anions. The reorientation of large massive anions may trigger rotations of the nearly free NH~ ion around one of its symmetry axes,
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
189
Volume 81A, number 2,3
PHYSICS LETTERS
12 January 1981
(NHa)2Ce(NO3)6
NH4CIO4 T1
°°°
°°
o
T1
6>°°
"U" 10 Q.,
ocoOOOoo
o
oo o
"riD
ee
o
"3 ID
o
e •
oeO
I/) v
o
10
o o "[1D
•
%
o
o
2
a
I--.
1
•
i
A
i
2
TIj~
~,:
**
%
(H1:2.5 G) L
,
i
,
i
i
,
~
10 103/T (K -~)
4
6
° e o
i
12
8
Fig. 1. The temperature dependence of T1, Tip and T 1D in NH4C104. and the spin-rotational interaction appears. Since the rotation axis may change at each collision, we assume that the correlation time r c of the rotation is equal to that of the reorientation of anions which is supposed to be much longer than that of the NHz~ ions. The spin-rotational hamiltonian can be expressed in terms of the proton spin operator I, the rotational angular momentum J and the spin-rotational coupling tensor C which is most conveniently given with respect to the molecule-fixed coordinate frame, as [9]
%r(t) = z.
c-J(t).
Using the semiclassical approach [5,10] in which we consider the coupling to be a scalar, 9~sr can be replaced with
(NH4)2S2Os T1
oooo o o
I0
oo o
o o
'~D
•
2
3 4 IO~/T
5 6 (K-1)
Fig. 3. The temperature dependence of T 1 and T 1D in (NH4)2Ce(NO3)6.
~sr(t) = - - ~ # ( t ) ' Z (1)
= - 7 ~ [ h x ( t ) l x + h y ( t ) l y + hz(t)lz] ,
where H ( h x , hy, hz) is the magnetic field produced at the proton by the rotation of the NH~ ion and we define the laboratory frame by x, y, z, with the z-axis parallel to the external field H 0. The components h x , hy and h z are linear functions of the components of the angular momentum of the ions. At low temperatures, collisions are so infrequent that r c is longer than the spin-spin relaxation time T 2 and strong collision theory may be applied, while at high temperatures r c becomes so short that weak collision theory may hold. 1. Weak collision case. The relaxation times T 1 and T 1 are written in terms of the Zeeman terms or ~gl ~o=- T h H l l x ) as well as the fluctuation term ~sr(t) in the hamiltonian. Using eq. (2) of ref. [11] and eq. (2.27) of ref. [12], we find T 1 and Tlp as
o
%
•
= v2 (hx2 + h2)
o
T1
i..=
"c
(2)
1 + 002 7 c2 '
I
1 - 3'2 h'~ 2
3
4
5
6
IOyT ¢K-') Fig. 2. The temperature dependence of T1 and T1D in (NI-I4)2 $20s. 190
rc
2 2
Tlo
1 + co 1 r c
II
+ 1_,,,2t~2" 2 t ,,y
Te -COl) r c
TC
+ 1 +(co o
]
•,
(3)
Volume 81A, number 2,3
PHYSICS LETTERS
where 601 = 7H1 and an overbar denotes ensemble average. Since 600 >> 6°1, we have = ,,/2h--~
rc -7c l + c o 2 r 2 +72hy2 1+-- 6002rc2"
TXÜ
can use the sudden approximation, and the energy change can be written as A ( ~ s r ) = Tr p(~0rk, --C~Ork),
(4)
From eqs. (2) and (4), it is expected that the relaxation time Tlp will be very much shorter than T 1 when 600rc >> 1, while Tlp will have an col-dependence , when COlt c >> 1. This has been observed experimentally.
2. Strong collision case. Following the SlichterAilion approach [13], in which the period r c between collisions is assumed to be longer than T 2, a common spin temperature is established and we can describe the spin state in terms of the density matrix in the rotating frame: p
=
(73
in terms Of~s0rk and 9~s0rk,, the spin-rotational hamiltonians before and after a collision, respectively. By evaluating the traces in the high temperature limit, one finds
(dO-1/dO = - [Tr(~Ork)2/Tr(~T(o) 2 rcl 0 -1 .
(8)
Then, using the following formula:
~Ork = -7l~hzklz ,
(9)
one obtains the relaxation time Tlp of the system in the rotating frame as,
r~p 1 = [h2z/(H? + H 2 + h2)] r c 1 ,
(10)
where
exp(-C~o/kO)/Z ,
where ~ o = ~ 1 + ~ d 0 +~Or, 0 is the spin temperature, Z is the partition function, ~Tgd is the dipolar part of the hamiltonian and the superscript zero denotes the secular part. Since the total energy in the rotating frame is
=- raxI4
12 January 1981
+
(5)
we can find the rate of the energy change, neglecting the lattice coupling of the dipolar and Zeeman systems, as
dE]dt = A(~Or)/r c .
(6)
We evaluate the total energy by using the relation: E = Tr(pg(o). Since the actual duration of a collision is short, one
H 2 =¼72I~2I(1 + 1 ) ~
(1 - 3 cos20/)2/Rt,
1
R~ is the distance of the fth intermolecular spin pair, Oj is the angle between R~ ahd H O. Similarly, by putt i n g H 1 = 0 in eq. (10), we can get a formula for T1D:
=
2 + hl)]
(11)
Consider the case in which the temperature is low enough, so that the correlation time r e due to collisions is longer than the time A (>> T2) during which the spin-rotational interaction is active following a collision [5]. Assuming that a collision occurs at t = 0 and 9~0r = 0 for t < 0, it is easily shown that the energy does not change at t --- 0; on the other hand before t = A a new spin temperature is established, therefore we obtain
Table 1 Minimum of TID, activation energy and spin-rotation field. NH4C104
(NH4)2 $2 08
(NH4)2Ce(NO3)6
E a (keal/mol) r E (s) (anion reorientation)
4.5 0.6 5 6.6 X 10-l°
3.5 0.75 5
3.3 0.5 7.5
7.7 × 10 - 9
E a (keal/mol)(NH~ reorientation)
0.65
1.6
(hz2)l/2(G)
9.2 X 10-3
8.5 X 10 -3
T1D minimum (10a/T) (s)
~ 1 0 -10
1.6 ~10 -2
191
Volume 81A, number 2,3 E = Tr p •
=
-(ZkO) -1 T r ( ~ o ) 2 ,
PHYSICS LETTERS (12)
where ~ o = ~ 1 + ~ 0 +~Ork" Since Ug0rk, = 0 at t = A+, the energy change at t = A is
a(C~Or) = Tr p(c~Osrk, --C~Ork) = -(ZkO)- 1 Tr (~;~Ork) 2 .
(13)
In this case one obtains the same formulae as eqs. (10) and (I 1). We estimated 72h2 as 105 s - 2 from eq. (11), which was four orders of magnitude smaller than the usual dipolar fluctuation, and evaluated the activation energy as 5 - 7 . 5 kcal/mol which was associated with the reorientation o f the anions. The results are shown in table 1 together with the value of the minimum of T1 D, its temperature, and the activation energy o f ion reorientation. The minimum o f T 1 according to eq. (2) is co0[co 1 times larger than that o f eq. (4) and is hidden b y other relaxations. For a single crystal of NH 4 C10 4, we found an anisotropy of T l p , which was ascribed to the anisotropy o f T r ( ~ 0 ) Z a n d Tr (~0r) 2. The analysis is in progress.
192
12 January 1981
References [1] K. Shimomura, R. Sugimoto and H. Negita, J. Sci. Hiroshima Univ. Ser. A38 (1974) 71. [2] M. Goldman, Spin-temperature (Clarendon, Oxford, 1970). [3] M. Punkkinen, J. Nonmetals 2 (1974) 79; K. Morimoto, K. Shimomura and M. Yoshida, J. Phys. Soc. Japan 45 (1978) 1965. [4] A.R. Sharp and M.M. Pinter, J. Chem. Phys. 53 (1970) 2428. [5] R.J.C. Brown, H.S. Gutowsky and K. Shimomura, J. Chem. Phys. 38 (1963) 76. [6] P.S. Hubbard, Phys. Rev. 131 (1963) 1155. [7] R. Ikeda and C.A. McDowell, Chem. Phys. Lett. 14 (1972) 389. [8] D. Wolf, Spin-temperature and nuclear-spin relaxation in matter (Clarendon, Oxford, 1979). [9] N.F. Ramsey, Molecular beams (Oxford U.P., London, 1956). [10] D.K. Green and J.G. Powles, Proc. Phys. Soc. 85 (1965) 87; R.G. Gordon, J. Chem. Phys. 44 (1966) 1184; R.H. Faulk and M. Eisner, J. Chem. Phys. 44 (1966) 2926; H. Boehme and M. Eisner, J. Chem.Phys. 46 (1967) 4242; C.H. Wang, J. Magn. Reson. 9 (1973) 75. [11] R.V. Steenwinkel, Z. Naturforsch. 24 (1969) 15 26. [12] D. Woff and P. Jung, Phys. Rev. B12 (1975) 3596. [13] C.P. Slichter and D.C. Ailion, Phys. Rev. 135 (1964) A1099.