HAL Id: jpa-00247537
https://hal.archives-ouvertes.fr/jpa-00247537
Submitted on 1 Jan 1991
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Demagnetizing field effects on high resolution NMR spectra in solutions with paramagnetic impurities
E. Belorizky, P. Fries, W. Gorecki, M. Jeannin
To cite this version:
E. Belorizky, P. Fries, W. Gorecki, M. Jeannin. Demagnetizing field effects on high resolution NMR spectra in solutions with paramagnetic impurities. Journal de Physique II, EDP Sciences, 1991, 1 (5), pp.527-541. �10.1051/jp2:1991187�. �jpa-00247537�
Classification Physics Abstracts
75 20 76.60J
Demagnetizing field effects on high resolution NMR spectra in
solutions with paramagnetic impurities
E. Belonzky (I), P. H. Fnes (2), W. Gorecki (I) and M. Jeann~n (I)
(I) Laboratoire de Spectromktne Physique (associk au CNRS), Umversitb Joseph Founer, BP. 87, 38402 Samt-Martin-d'Hdres Cedex, France
(2) CEN Grenoble, DRF Chimie (Equipe Chimie de Coordination), BP 85X, 38041 Grenoble Cedex, France
(Received 21 December 1990, accepted 4 February 1991)
Rksi~mk. On expnme le champ dbmagnbtisant comme la somme d'un tenure diamagnbtique, dfi
aux moments magnktiques klectromques des molkcules de la solution mduits par le champ
appliqub, et d'un tenure provenant des spins klectronJques des impuretbs paramagnbtiques Les effets de ces deux contnbutions, de signes opposbs, sur les raies de rbsonance nuclbaire sont
calculbs en fonction de la su~sceptibihtb dJarnagnbtique de la solution, de la concentration en
molbcules paramagnktiques et de la gbombtne de l'bchantillon par rapport I la direction du champ La thbone est illustrke par une btude de la RMN des protons de (CH~)( en solution dans
D~O et couplks I des ions Mn~+ de spin dlevb. Des dbplacements de raies et des klargissements
considbrables de plusieurs p p m sont observbs, lorsque la gbomktne du tube est modifibe et, sauf dans le cas d'un bchantillon sphdnque, les raies observbes sont dissymbtnques. La thbone et
l'expbnence sont en bon accord
Abstract. The demagnet1zlng field is expressed as the sum of a diamagnetic term due to the molecular electronic magnetic moments of the solution which are induced by the extemal field and of a term ansing from the electronic spins of the paramagnetic impunties. The effects of these two contributions of opposite signs on the NMR line are calculated as a function of the
diamagnetic susceptibility of the solution, of the concentration of the paramagnetic molecules and of the sample geometry with respect to the extemal field The theory is illustrated m
D~O solutions by a spectroscopic study of the (CH~)I Protons coupled with manganous
Mn~+ high spin paramagnetic ions. Considerable shifts and hnewidths of several p-p m are observed when the geometry of the vessel is modified and, except for sphencal samples, the
lJneshape is asymmetncal A good agreement with the theory is obtained.
1. Introduction.
It is well known that the introduction of paramagnetic impunties iq liquid solutions g~ves nse to relative frequency sh~fts « of the NMR l~nes of the solvent nuclei which depend on the
shape of the vessel containing the solution and on the field direction. Typically, for long and
perfectly cyhndncal coaxial vessels positioned parallel and perpendicular w~th respect to the field axis, it was shown [1, 2] that
«j «~ = 2 arx~~~~, (1)
528 JOURNAL DE PHYSIQUE II M 5
where x~~~~ is the volume susceptibility brought about by the presence of paramagnetic impurities m the solution Th~s property is used in the so called field axis method », first
described by Beconsall et al. [3] as a titration method and was developed by several authors
[4, 5] The method was improved by Delpuech et al. [2, 6] m order to measure the titration of dissolved oxygen m benzene and hexafluorobenzene solutions contained in sealed sample
tubes under pressure.
Recently, we have shown [7] that in purely diamagnetic solutions or liquids, there are also important frequency sh~fts « of the NMR l~nes of the liquid nuclei together with considerable
broaden~ng of the l~nes which are sample shape dependent This is due to magnetic dipolar
interactions of the studied nuclei with the electronic diamagnetic moments induced by the extemal applied magnetic field Then, we have a classical mhomogeneous demagnetizing field effect which was investigated m h~gh resolution NMR of l~quid samples. The theory of this
mhomogeneous broadening was performed and successfully compared w~th expenmental
results conceming the NMR of protons of tetramethylammonium ions in heavy water solution and of pure benzene [7]
In the present paper we extend the previous study to solutions containing paramagnetic impurities. Here the inhomogeneous demagnetizing field is the sum of two opposite contributions : one arising from the field induced diamagnetic moments of the molecules of the solution and the other from the field induced paramagnetic moments of the impurities
It should be noticed that usually the paramagnetic spins of the impurities interact with nuclear spins of the diamagnetic molecules of the solution, not only through the d~polar
interactions responsible for the demagnet~zing field, but also through short range scalar hyperfine interactions. It is very useful to separate these two effects which play an important
role in the relaxation mechanisms of these systems In this work wa have chosen systems in
which there is a coulombic repulsion between the ions carrying the studied nuclei and the paramagnetic species, in order to avoid any hyperfine interaction and to master the dipolar
effects. But, we will show that our technique is a powerful method for separating the dipolar and scalar interactions when the latter are present Th~s separation is of fundamental
importance in order to study the dynam~c behaviour of an ion pair like (CH~)~P+ /free radical
in D~O solution (electrolytic solution). Indeed, it was not possible to interpret the
longitudinal relaxation rates of ~H, ~~C and ~~P nuclei by only considenng the dipolar coupling
between the investigated nuclei and the free radical electronic spins [8]. Furthermore, the observed chemical shifts and hnewidths were somewhat erratic because no care was taken of the sample geometry
In section 2 the theory of the inhomogeneous broadening is presented The experimental
results concern~ng the NMR of protons of (CH~)~NCI in presence of Mn~+
ions m heavy
water solution are discussed m section 3.
2. Theory.
We are interested m the local field H, at sites of equJvalent nuclei of spin I and of gyromagnetic ratio yj m a diamagnetic solution, with volume diamagnetic susceptib~l~ty
xd,a, contain~ng paramagnetic impunties w~th volume susceptibility xpara The local field
H, is
H, = Ho(I ha + H~,~ (2)
f§ is the applied field. ha is the usual chem~cal shift which is independent of the
susceptibility It takes into account the field contribution from the molecule m which the proton is located and is independent of the sample shape. lid,~ is the dipolar field given by
~dtP
~ ~~~~l'~~~~
)
~
~ ~~~~) ~~~~
)
~~~
where ~~ is the electronic diamagnetic moment of a g~ven molecule m the liquid
(N number density of these molecules) and r~~ is the position of the diamagnet~c moment
~~ w~th respect to the reference nuclear spin I. Similarly, ~s
= g~MB S is the electronic paramagnetic moment of the impunty with spin S (gs is the Landk factor) and r~s is the relative position of th~s spin w~th respect to the nucleus I. As discussed m reference [7]
~~ and ~s are approximated as point dipoles
In order to calculate lIa~ we use the Lorentz method. The volume of the hqu~d sample is div~ded in two pans : (I) a sphere centered at the reference nucleus, of radius 1lo much larger than the minimum distance of approach b of two molecular centers, but much smaller than the macroscopic sample size ; (ii) the volume outside the sphere. The field H~~~ is the sum of two terms Hj,~ + H][~. The first, II~~~ is the field produced by the induced electronic moments
(both diamagnetic and paramagnetic) inside the sphere 1lo and the second Hj[~ by those outside the sphere. According to the rotational invanance of the local structure of the hquJd around the studied nuclei, it is clear that Hj~~ = 0 Notice that in the case of a sphencal solid sample with an uniform distnbution of the electron~c moments, Hj~~ = 0. Here, the relative
equihbnum distribution of the interacting molecules w~th~n the sphere 1lo is not uniform ~pair
correlation effects), but it remains isotropic because of the random diffusing motions of the molecules.
In order to calculate ll~[~ we use the classical method [9] considenng a continuous uniform distnbution of the electronic moments. We define
M
" (Xd>a + Xpara)1i0 (4)
as the total magnet~zation per unit volume We have
N~g) »( S(S + I)
~P~"
3 kT ' ~~~
where N~ is the number density of paramagnetic species. Denoting by z the d~rection of the
applied field l§, we have in th~s direct~on according to equation (3)
H][~ = M- ~~~ ~
dV, (6)
v r~
r
where the integration is taken over the volume V between the spherical surface
6~ and the intemal vessel S. Equation (6) can be rewntten [10]
H$,~ = M; div dV
=
~( ~
~~° ~'~~ l, (7)
v r ~ r S r
where z
= zk ~k unit vector parallel to Oz) and dS, dSJ are onented outside the surface
S of the vessel and the sphere So of radius llo. Finally, we obtain,
H][~ = M, ~ "
N~,) (8a)
w~th
N,,
=
l~
(8b)S r~
530 JOURNAL DE PHYSIQUE II M 5
Hence, the component of the local field in the direction of the applied field depends on its location and on the sample shape (external surface 5~ through the demagnetization coefficient N,~. For a spherical sample it is well known that N~~ is uniform N~~
=
~"
and 3
H][~ = 0. But, for a cyhndncal sample, H~~ is not uniform and th~s leads to an average shift and to a broadening of the resonance line.
From equations (2) and (8) the resonance frequency of a g~ven nucleus is :
2 " ~ " YI H> " YI H0ii A" (Xd<a + Xpan) Nz= ~) (9)
Denoting by vo the resonance frequency of the nucleus m the field Ho, and by Si the form factor
5~=N;~-~", (10)
we obtain
v = vo(I tr) (lla)
with the relative frequency sh~ft
" " A" + (Xd>a + Xpara) St ii b)
For a sphencal sample we simply have «
= ha
Now, we consider a cylinder sample w~th heigh h and diameter d. First, the direct field
l§ is assumed to be parallel to the cylinder axis It is easy to show that at the center of the
cylinder
~2 -1/2
N~~ = 4 ar I +
~ ,
(12)
h
while at a distance c from the center, on the axis of the cylinder, 0 « c « h/2,
~"~~"'~ (~
h ~2c j~~ ~ ~~~ i~ h~2c j~~ ~ ~~l' ~~~~
For an arbitrary point inside the cylinder N~; must be nunlencally calculated from equation (8b). Th~s has been done for a large number of nuclear positions m a rectangular sect~on of the
cylinder containing the central axis for various values of the ratio hid. As an example we report m figure I the obtained values of N== for hid
= I
Then, we have performed a numencal mtegrat~on m order to obtain the normalized distribution f(Si) for vanous values of hid The numencal technique is explained in the
appendix
z
Ho 3,68 2,24
6,95
axe
Fig. I Values ofN== m a rectangular section of a cylinder with hid
= I The field Ho is parallel to the axis of the cylinder
f S~ h/d=lO.O
2.
I.
1.
0.
0.
-8.-6.-~.-2. 0. 2. ~. 6. 8.
(a S~
O.
f S~ h/d=I.OO f S~ h/d=O. 3~
0.
0.
0.
0.
o. o.
o.
-8.-6.-~.-2. 0. 2. ~. 6. 8. -8.-6.-~.-2. 0. 2. ~. 6. 8.
(b S~ (c S~
Fig 2 Theoretical dipolar shifts and line shapes f(Si) for a cyhndncal sample for various values of
the ratio hid, when the field llj is parallel to the axis of the cylinder
532 JOURNAL DE PHYSIQUE II lK° 5
We represent m figure 2 the d~stnbution f(Si) for hid = 10 (thin cylinder), hid
= I, and
hid
=
0 34 (flat cylinder). The san~e work has been done when the field llo is perpendicular to the axis of the cylinder. In th~s case we have :
N~~=
1@
(14a)S r
and
Si
= N~~ ~ " (14b)
We report m figure 3 the corresponding distribution function f(Sr) for hid
= 1.
1.
f S~ ) h/d=I.OO
0.
0.
0.
0.
-8.-6.-~.-2. 0. 2. ~. 6. 8.
S~
Fig 3 Normalized dJstnbution f(Si) for a cyhndncal sample when the field Hois perpendicular to the
ams of the cylinder
For a g~ven species of nuclei, neglecting the homogeneous broadening ansing from
relaxation, the normalized mhomogeneous line shape F(v is, according to equation (9) and
(I1)
f(Sf) 2 1Tf (Sf)
~~ ~
V0(Xd>a + Xpara) ~ YJH0(Xd>a + Xpara)1
~~~~
In th~s equation, v = v~ Au, where v~ is the resonant frequency for a sphencal sample
yi Ho
v~ = (l ha (16)
and
From equation (15), we see that F( v is proportional to f(Si) Furthermore, the relative sh~ft is gJven by
"d "
@
"
~~
" Sf(Xdia + Xpara) (18)
s ~0
We consider the situation where the applied field I§ is parallel to the axis of the cylinder. For low concentrations of paramagnet~c impunties we have (x~~ + x~~~) < 0. It can be seen from figure 2a that the theoretical line f(Sr) for a thin cylinder (hid
= 10) is displaced towards the low Si values with respect to the ong~n (sphencal sample)~ i e towards the low frequencies v(hv » 0) The line is dissymmetncal w~th a slower decrease towards the h~gh frequencies
For hid
= I, the line is dissymmetncal and centered around Si= 0 but is considerably
broadened (Fig 2b) Finally, for a flat cylinder (hid
= 0.34 ) the line is very dissymmetncal
and displaced towards the h~gh values of Si, i.e towards the h~gh frequencies The broadening
is also larger than in the prev~ous cases (Fig 2c). For higher concentrations of paramagnetic impurities, when (xd,a + xpara) »0 we must have shifts m the opposite direction to that
observed m the previous case and the asymmetry of the line shape must be inverted. We also
predict from equation (17) the existence of a cntical concentration of paramagnetic impurities
for which (xdia + x para) =
0 leading to the absence of any sh~ft and broadening of the line For th~s concentration the line is posit~oned at the same frequency for all sample shapes. The
resonance line for a cyhndncal sample must be very narrow as for a sphencal sample.
3. Experiment and discussion.
We have performed expenments using a h~gh resolution Brucker WM200 Spectrometer working for the proton at a frequency of 200 MHz, with a direct field homogeneity of about 10 Hz, i e 0.05 ppm The mtemal diameter of the r f coil is 25 mm and its height is 40 mm We used three kinds of.pyrex vessels a sphencal one w~th internal diameter d
= 21mm, a
thin tube with internal diameter d
= 4 mm and height h
= 40 mm (hid
=
10), and a partially filled tube with d
=
21 mm, and h varying between 5.5 and 7 5 mm (0 26 « hid « 0 36 ). We have analysed the hneshift and line broadening of the NMR spectra for the twelve equivalent
protons of (CH~)~N+ ions of (CH~)~NCI m D~O solution m the presence of Mn~+
(S
= 512) paramagnetic impurities obtained from the dissociation of Mncl~.
The concentration of (CH~)~NCI m our solution was 0.I mole l~ ~, and that of MnC12 was
varied from 0 (diamagnetic solution) to 0.12 mole l~
Usually, the chemical shifts ha are studied by locking the rf frequency on a reference nucleus (deutenum of D~O). Because the locking frequency is affected, m the same way as
the studied nuclei by the sample shape effect, ha remains constant, independently of the
sample shape, although one observes different hneshapes and broaden~ng according to the geometry Here, m order to observe the sh~fts and the broadening of the l~nes due to the
dipolar inhomogeneous fields, it is necessary to lock the rf field on the deutenum nucleus for a sphencal sample and to keep this locking frequency constant for all geometnes.
First, we consider the diamagnetic solution The observed spectra are represented m figure 4 For a sphencal sample we observe a narrow symmetncal, line (Fig 4a) w~th a half
height hnewidth of 0.07 ppm. The satellite line displaced by 1.6 ppm with respect to the main line is due to residual HDO m D~O. For the thin cylinder (hid
=
10) the line is displaced by
2.95 ppm w~th a slight asymmetry towards high frequencies as expected (Figs 4b and 2a)
For a flat cylinder (hid
= 0.34 the line is very broad (about 2.5 ppm at half height) and the
maximum peak is displaced towards h~gh frequencies by 3 4 ppm (Fig 4c) Theoretically, if
we take the volume susceptibil~ty of D~O to be x~~~ =
0.72 x 10~ ~ (cgs) [I I], we obtain a
JOURNAL DE PHYSIQUE II -T " 5 MM >WI ,7
534 JOURNAL DE PHYSIQUE II M 5
a b
I
25 20 is lo S 25 20 is lo 5
PPm PPm
c
25 20 is lo 5
PPm
Fig 4 Observed resonance lines of protons of (CH~)~NCI m D~O at 200 MHz : (a) sphencal sample,
~b) thin cylinder (hid = 10)
,
(c) flat cylinder (hid = 0 34)
l~ne displaced by 2.92 ppm for the thin cylinder. For the flat cylinder the displacement of the maxbnum peak is predicted to be 3 0 ppm w~th a linewidth at half height of 1.5 ppm. The
larger expenmental linewidth value is probably due to the interference with the HDO peak.
Now, we introduce the paramagnetic impurities with a concentration C~ (mole l~~) of
Mn~+ ions. From equation (5) we obta~n for S=512, g~=2 and a temperature
T
=
293 K :
xpara = 15.0 x 10~~ C~(cgs) (19)
Then, the relative sh~fts are g~ven by
«~ =
~~
= Si(- 0.72 + 15 C~) 10~~. (20)
Us
The cntical concentration for which we have cancellation of the diamagnetic and paramagnetic
susceptibilit~es is then Cl
= 0.048 mole l~ ~. For C~ < Cl we expect the same kind of spectra than for the diamagnetic solution, but w~th decreasing shifts and a narrowing of the l~nes as
C~ increases. For C~
=
Cl the position of the lines must be independent of the sample shape
a b
25 20 15 lo 5 25 20 15 lo S
ppm ppm
c
I
25 20 15 lo 5
ppm
Fig 5. Observed lJneshapes of proton resonance lines of (CH3)4NCl m D20 m presence of paramagnetic impunties Mn~+ with vanous concentrations C~ (mole l'~) in a sphencal sample (a) C~ = 0, (b) C~
= 0.04, (c) C~ = 0.12
