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NMR dipolar echoes in solids containing spin-12 pairs
Volume 28, number.4.
1.5October 1974
CHEMICAL PHYSICS LETTERS
‘.
‘,
NMR DIPOLAR ECHOES
IN SOLTDS CONTAINING
SPIN-i PAIRS
N. BODEN, Y.K. LEVINE and R.T. SQUIRES Departmenfof Fhysical @emistry; 77re Universe 6~~ofLeeds. Leeds LS2 9JT. Received
20 June
UK. 1974
A model of loosely-coupled
spin-d pairs is used to explain the proton NhlR dipolar echo behaviour observed in previously predicted and observed in solids containing single spin-.; nuclei and is shown to originate in thedifferentiation of the intrapair from the interpair dipolar interactions.
crystdline hydrate% This behaviour is distinctly different from,that
,When a resonant 90”-~--90°g0~ rf p&se sequence is applied to a rigid dipolar coupled spin system, a spin echo occurs at time 2~ after the first pulse provided the pulse separation T is small compared to the spin-spin reiaxation time T2. This echo has its origin iri the dipolar interactions and vie therefore refer to it as a dipolar echo. The formation and shape of this echo have been investigated by a number of workers [14] for the special case of a solid, containing single identical spins; here the dipolar interactions are homogeneous. In most cases of practical importance the spins are not single but are,arranged in groups. In these c;Lses we expect distinctly different behaviour beczuse the intragroup spin-spin interactions can be differentiated from the intergroup ones,.i.e., the dipolar intenctions are no longer perfectly homogenkous. The simplest possible case of such spin groupings is phere the spins are arranged in equivalenf pairs, e.g., Hz0 in crystalline hydrates and >CH2 in methylene chains. We have recently demonstiated [5] how the, proton spin echo produced by a 90°-r-90e90~ sequence in powdered hydrates such as CaS(!14.2Hz0, ‘AlC1,.6H,O, and #,C20,.Hz0 has a different behaviour to that predicted and observed in solids containing sinde identical spiks. A result of corisidkable
practical interest is that the maximum echo amplitude E(r) has.i.7 dependence ,E(r) = E(0)
. . ..
:
=$My(inter)
,
(2)
where My(’ Inter) is the interpair van -fleck second moment.[6]. The behaviour expressed in eqs. (I) and (2) has been explained theoretiklly in terms of a model of loose!y coupled spin-i pairs [7,8]. Herein, we show’how the same.model explains the echo responses to the pulse sequences 90”-.r-~so~ (XY sequence) tid 90”-7-$-,0 (XX sequence) for the protons in powdered gypsum (CaS04.2H2P). The model of locselycoupled spin3 pairs assumes the interpair separation to be considerably larger than the intrapair diskce so ‘&at the spins within each pair may be cokiderkd as magnetically equivalent and XCd(intra) > J$(inter); I&(intra) and ZQ(bter) arc. tie intra- and inter-pair dipolar hamihonians, respectiiely. J&(inter) may therefore be treated as a srn2l-l perturbation of.the combined hzmiitonkn FfC=3C, + Xd(intra) where X, is the Zeeman harniltonianl It is only necessary tci consider the truncated dipolar ham.iltonian LKi such that [JCz, ZC@ntra) + JQ*(inter)] i?p(. inter ) is constructed -.
(1)
: [~~(intra),~.‘(intsr)]
= Cl.
’
-(3)
s-o as to satisfy the condition
=0
.,
.‘.
..’
:
:. _‘.’
M&inter)
from the interpair of the Frequency
1
(4)
:
‘.
:,.
,
expC-$%Iz(inter)& :
.. ‘. . .
,‘.
describe.d by
where Mz(inter) is the contribution interactions to the second moment spectrum. Significantly,
‘: ,‘.
.‘..
,:
: y . :
_‘. ,‘._’
523
(.
for 3re XY sequenc&pri$.nate in the interpair cl.ipolar interactions. In the absence.af these interactions no &Y sequence icho wih exist -&td the XY Sequence will o$y give?a singe compqnbnt echo with am$itude’pro.
.,
_; .. ‘.:‘,,where Em is the mth eigenvalue of jc,O.The matrix ‘cJements.of R, are constructed from the Wigner rota: tidn matrices a$& (-til2; &n/2) for the XY skquencl? and.Q$,(O, /3,0) for the XX sequence. I3q. (6) shows _, &thatan echo,occurs when : : : (Ejj&‘,&(r+)=‘(E,_-E&; . .. ,_i.e.,.at t = 27. ‘Jbe.an&lar dependence of the matimum ~ch~~p~,tudes E(j), normalized to.E(90’)iy ._ r =‘l, are giGen 5y . . ..’
XY se&nce
E@)&,
.XX &&enc&’
E(j3)xx =‘-&n2Pcasj3
.: : :
0
30*-T - SO*,,. sequence
.
= sin2p + $in2P cOs2p , .
Expanding ecj..(5) in a power series in time gives in the laboratory frame, to second ordkr in time IE(ii&,
:.
G&t
=’ ,’ %
-
27)
&?fi
+
F&t)
cos2@
:
:[Gi(t - 2~) t: E;(t)} sin’p co&
,
:
-&r&m at t.=.27 and F(r) is’a~tran$&t sQm$ I~ .’
..,. . 2,: .:
.’ ...’
., ‘, : ., : . I’ :
: :
‘, ‘.
‘.’
‘_:. .:; :
.,-
,“..’
,;. :
maxi muin echo amplitude‘s E(s) as 3 r%nction. ii tons-k powdered CaSCi0.2~~0 at 298 K. .-. .;: ‘. , ‘_ : : ..,_ ‘, ‘_ ; ), -_:, ; ,‘: :. _ : ,: .,: ; i. ‘., ,;. .’ 1 ,_ ..:; -,. .. ‘;. :
T* for pro..‘;, ‘. :
._ :.
_.
:
:
CREhUCAL.PHYS1C.SLETTERS 15 October 1974 ..,” portional to.sin*& The angular dependence of the Furthermore, we also find that E(T) for the ,- echo amplitudes predicted by these equations is ob90°~~45"9,~sequence has, et long times, a gaussian kit-red for.the protons ii-r powdered gypsum. dependence on r2 and the decay constant. is identical '.Eqs.(10) and (11) also give, to second order in. with that for t@ 904--T-54”44’uo sequence. Thus, time, the 7 dependence of the maximum echo amplithe model of loosely-coupled spin-$ pairs provides a ‘cudes’E(r); these are good explanation for the general doubIe-pulse spin echo behaviour observed in STsum. Ebb fi sin2pC1 -$ +f2(intei) 72 + .._} Volume 28, number 4
+$iti2p
cos2pil
-&f2(inter)72
+ . ..I
(12)
Referekes
and E(.r)xx~~sin2~cos~1
-f$j~2(inler)52+...}
. (13)
I&g. 1 shows that the E(T) for both 90”-r-90”,,~ and 90”-r-54°44’00 sequences have a gaussian dependence on 72 over at least 97% of their decay. The ratie of the two decay constants, 1.7, compares favourably with the ratio predicted by eqs. (12) and (13), which is 2.0.
[ 11 ,IG. Powles and J.H. Straqe, Ptoc Phyi Sac (London) 82 (1963) 6. [2] P. Mansfield, Phys. Rev. 137 (1965) A961. [3] R. Hausscr and G. Sieglc, Phys Letters 19 (1965) 356. [4] G. Sieglc, Z Naturfoxch. 232 (1968) 91. [5j N. Boden and M. Mortimer, (l973) 53%.
Chem.
Phyr
Letters
21
[6] J.H. VA Fleck, Phys Rev. 74 (1948) 1168. Y.K. Levine, bf. Mortimer and R.T. Squires, Phys. Letters 46A (i974) 329.
[7] N. +den,
[8] N. Boden
and Y.K. Levine,
Mol. Phyr,
to be published.
-.
: -.:
:-:
j25
1.5October 1974
CHEMICAL PHYSICS LETTERS
‘.
‘,
NMR DIPOLAR ECHOES
IN SOLTDS CONTAINING
SPIN-i PAIRS
N. BODEN, Y.K. LEVINE and R.T. SQUIRES Departmenfof Fhysical @emistry; 77re Universe 6~~ofLeeds. Leeds LS2 9JT. Received
20 June
UK. 1974
A model of loosely-coupled
spin-d pairs is used to explain the proton NhlR dipolar echo behaviour observed in previously predicted and observed in solids containing single spin-.; nuclei and is shown to originate in thedifferentiation of the intrapair from the interpair dipolar interactions.
crystdline hydrate% This behaviour is distinctly different from,that
,When a resonant 90”-~--90°g0~ rf p&se sequence is applied to a rigid dipolar coupled spin system, a spin echo occurs at time 2~ after the first pulse provided the pulse separation T is small compared to the spin-spin reiaxation time T2. This echo has its origin iri the dipolar interactions and vie therefore refer to it as a dipolar echo. The formation and shape of this echo have been investigated by a number of workers [14] for the special case of a solid, containing single identical spins; here the dipolar interactions are homogeneous. In most cases of practical importance the spins are not single but are,arranged in groups. In these c;Lses we expect distinctly different behaviour beczuse the intragroup spin-spin interactions can be differentiated from the intergroup ones,.i.e., the dipolar intenctions are no longer perfectly homogenkous. The simplest possible case of such spin groupings is phere the spins are arranged in equivalenf pairs, e.g., Hz0 in crystalline hydrates and >CH2 in methylene chains. We have recently demonstiated [5] how the, proton spin echo produced by a 90°-r-90e90~ sequence in powdered hydrates such as CaS(!14.2Hz0, ‘AlC1,.6H,O, and #,C20,.Hz0 has a different behaviour to that predicted and observed in solids containing sinde identical spiks. A result of corisidkable
practical interest is that the maximum echo amplitude E(r) has.i.7 dependence ,E(r) = E(0)
. . ..
:
=$My(inter)
,
(2)
where My(’ Inter) is the interpair van -fleck second moment.[6]. The behaviour expressed in eqs. (I) and (2) has been explained theoretiklly in terms of a model of loose!y coupled spin-i pairs [7,8]. Herein, we show’how the same.model explains the echo responses to the pulse sequences 90”-.r-~so~ (XY sequence) tid 90”-7-$-,0 (XX sequence) for the protons in powdered gypsum (CaS04.2H2P). The model of locselycoupled spin3 pairs assumes the interpair separation to be considerably larger than the intrapair diskce so ‘&at the spins within each pair may be cokiderkd as magnetically equivalent and XCd(intra) > J$(inter); I&(intra) and ZQ(bter) arc. tie intra- and inter-pair dipolar hamihonians, respectiiely. J&(inter) may therefore be treated as a srn2l-l perturbation of.the combined hzmiitonkn FfC=3C, + Xd(intra) where X, is the Zeeman harniltonianl It is only necessary tci consider the truncated dipolar ham.iltonian LKi such that [JCz, ZC@ntra) + JQ*(inter)] i?p(. inter ) is constructed -.
(1)
: [~~(intra),~.‘(intsr)]
= Cl.
’
-(3)
s-o as to satisfy the condition
=0
.,
.‘.
..’
:
:. _‘.’
M&inter)
from the interpair of the Frequency
1
(4)
:
‘.
:,.
,
expC-$%Iz(inter)& :
.. ‘. . .
,‘.
describe.d by
where Mz(inter) is the contribution interactions to the second moment spectrum. Significantly,
‘: ,‘.
.‘..
,:
: y . :
_‘. ,‘._’
523
(.
for 3re XY sequenc&pri$.nate in the interpair cl.ipolar interactions. In the absence.af these interactions no &Y sequence icho wih exist -&td the XY Sequence will o$y give?a singe compqnbnt echo with am$itude’pro.
.,
_; .. ‘.:‘,,where Em is the mth eigenvalue of jc,O.The matrix ‘cJements.of R, are constructed from the Wigner rota: tidn matrices a$& (-til2; &n/2) for the XY skquencl? and.Q$,(O, /3,0) for the XX sequence. I3q. (6) shows _, &thatan echo,occurs when : : : (Ejj&‘,&(r+)=‘(E,_-E&; . .. ,_i.e.,.at t = 27. ‘Jbe.an&lar dependence of the matimum ~ch~~p~,tudes E(j), normalized to.E(90’)iy ._ r =‘l, are giGen 5y . . ..’
XY se&nce
E@)&,
.XX &&enc&’
E(j3)xx =‘-&n2Pcasj3
.: : :
0
30*-T - SO*,,. sequence
.
= sin2p + $in2P cOs2p , .
Expanding ecj..(5) in a power series in time gives in the laboratory frame, to second ordkr in time IE(ii&,
:.
G&t
=’ ,’ %
-
27)
&?fi
+
F&t)
cos2@
:
:[Gi(t - 2~) t: E;(t)} sin’p co&
,
:
-&r&m at t.=.27 and F(r) is’a~tran$&t sQm$ I~ .’
..,. . 2,: .:
.’ ...’
., ‘, : ., : . I’ :
: :
‘, ‘.
‘.’
‘_:. .:; :
.,-
,“..’
,;. :
maxi muin echo amplitude‘s E(s) as 3 r%nction. ii tons-k powdered CaSCi0.2~~0 at 298 K. .-. .;: ‘. , ‘_ : : ..,_ ‘, ‘_ ; ), -_:, ; ,‘: :. _ : ,: .,: ; i. ‘., ,;. .’ 1 ,_ ..:; -,. .. ‘;. :
T* for pro..‘;, ‘. :
._ :.
_.
:
:
CREhUCAL.PHYS1C.SLETTERS 15 October 1974 ..,” portional to.sin*& The angular dependence of the Furthermore, we also find that E(T) for the ,- echo amplitudes predicted by these equations is ob90°~~45"9,~sequence has, et long times, a gaussian kit-red for.the protons ii-r powdered gypsum. dependence on r2 and the decay constant. is identical '.Eqs.(10) and (11) also give, to second order in. with that for t@ 904--T-54”44’uo sequence. Thus, time, the 7 dependence of the maximum echo amplithe model of loosely-coupled spin-$ pairs provides a ‘cudes’E(r); these are good explanation for the general doubIe-pulse spin echo behaviour observed in STsum. Ebb fi sin2pC1 -$ +f2(intei) 72 + .._} Volume 28, number 4
+$iti2p
cos2pil
-&f2(inter)72
+ . ..I
(12)
Referekes
and E(.r)xx~~sin2~cos~1
-f$j~2(inler)52+...}
. (13)
I&g. 1 shows that the E(T) for both 90”-r-90”,,~ and 90”-r-54°44’00 sequences have a gaussian dependence on 72 over at least 97% of their decay. The ratie of the two decay constants, 1.7, compares favourably with the ratio predicted by eqs. (12) and (13), which is 2.0.
[ 11 ,IG. Powles and J.H. Straqe, Ptoc Phyi Sac (London) 82 (1963) 6. [2] P. Mansfield, Phys. Rev. 137 (1965) A961. [3] R. Hausscr and G. Sieglc, Phys Letters 19 (1965) 356. [4] G. Sieglc, Z Naturfoxch. 232 (1968) 91. [5j N. Boden and M. Mortimer, (l973) 53%.
Chem.
Phyr
Letters
21
[6] J.H. VA Fleck, Phys Rev. 74 (1948) 1168. Y.K. Levine, bf. Mortimer and R.T. Squires, Phys. Letters 46A (i974) 329.
[7] N. +den,
[8] N. Boden
and Y.K. Levine,
Mol. Phyr,
to be published.
-.
: -.:
:-:
j25