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K. JELŠOVSKÁ et al.: NUCLEAR MAGNETIC RESONANCE SPECTRAL FUNCTION AND MOMENTS FOR ...
( i)
d
B - the induction of the magnetic field formed by the
paramagnetic ions Me2+ (Me = Mn, Ni), including
demagnetising effects which operate in these substances;
( n)
B d - the induction of the magnetic field in the area of a
single nuclei formed by another nucleus of the quasiisolated
pair H-H.
The inductions B d and B d can be expressed according
to papers [1, 3] in the form:
292
( i)
( n)
( )
( ) (
2
i µ 0 ⋅µ
i Br
2 2
Bd = A1 cos ϑ + A2
sin ϑcos2 ϕ
4π ⋅3k T −θ
and
B
( n) 0 p
d
3
2 4π
rp
where:
+ B sin ϑsin 2ϕ + B sin 2ϑsin ϕ
3
2
1 2
+ B3 sin 2ϑ cosϕ
+ C
2 ⎟⎠
(2)
3 µ µ
= ± −
2 ( 3cos ϑ 1 ) ,
⎞
⎟
(3)
µ 0 - the permeability of vacuum,
µ i - the magnetic moment of paramagnetic ions,
k - Boltzmann’s constant,
T - the temperature,
θ - Curie-Weiss constant,
µ p - the proton’s magnetic moment,
r p - the proton - proton distance in the crystalline water.
The angles ϕ and ϑ characterize the orientation of
external magnetic field r B in the reference frame used.
The parameters A , A , B , B , B and C depend on the
1 2 1 2 3
configuration of paramagnetic ions and resonating nuclei
and are expressed as [1, 3]:
A = 3∑ r P cos β ,
1
−3
l 2 l
l
1
A = ∑ r P cos β cos2 α ,
−3
2
2
2 l
l 2 l l
1
B = ∑ r P cos β sin 2 α ,
−3
2
1
2 l
l 2 l l
1
B = ∑ r P cos β sin α ,
−3
l
2
2 l
l 2 l l
1
B = ∑ r P cos β cos α ,
−3
l
3
2 l
l 2 l l
1
C = − A1,
3
(4)
where:
rl
- the vector joining the reference nucleus with the l
- th paramagnetic ion,
α , β - the angles characterizing the orientation of r l l l and
m
P are Legendre polynomials.
2
According to equations (2) and (3) the local magne-
( i) ( n)
tic field Bloc = Bd + Bd
may be thought as a quadratic
form relative to the components of the unit vector
er ( sin ϑcos ϕ,sin ϑsin ϕ,cos ϑ)
parallel to Br. This quadratic
form may be diagonalized and the roots of the secular
equation determine the invariant parameters of the local
magnetic field (denoted as B x ,
+ B y ,
+ B z ,
+ Bx ,
− By ,
− z
B− ), through
which the spatial function may be expressed [1].
According to [1, 3] the second moment of NMR spectrum
may be expressed in the form:
M
2
ABr
=
( T − θ)
2 2 ,
where
2 4
0 ⎟ i
⎟ 2
(5)
⎛ µ ⎞ µ 4 2 2 2 2 2
A = ⎜
⎡A1 3 ( A2 B1 B2 B3
) ⎤
⎜ ⋅ ⋅ + + + + .
⎜⎝ 4π ⎟⎠
9k 45 ⎢⎣ ⎥⎦
(6)
The equation (5) expresses the temperature and the
field dependence of the second moment in paramagnetic.
For isolated proton pairs the second moment M 20 may be
expressed in the form [1, 4]:
M
0 p
20 3
5 4πrp
2
9 ⎛µ µ ⎞
⎟
=
⎜ ⎟
⎜ ⎟ .
⎜⎝ ⎠ ⎟
(7)
For the spectral function f 0 (x) of the isolated pairs of
the nuclei, we used the following form [1, 3, 7]:
+ −
F0 ( x) = f0 ( x) + f0 ( x) , x = B− Br
(8)
for (ε) = + or –
ε ε
⎧⎪ 0,
Bz < x <
Bx
⎪ K( k)
( ε) ( ε)
⎪
, Bx < x < By
,
⎪ ( ε) ( ε) ( ε)
⎪ ( B ) ( )
( )
z − x By − B
ε
x
f0 ( x)
= ⎪
⎨ (9)
⎪ ⎛1 ⎞
⎪ K ⎜ ⎟
⎪ ⎜ ⎟
⎪ ⎜⎝ k ⎟
( ε) ( ε
⎪
⎠
)
⎪
, By < x < Bz
,
⎪ ( ε) ( ε) ( ε)
⎪ ( x− Bx ) ( Bz − By
)
⎪⎩
where:
( ) ( ) ,
METALURGIJA 45 (2006) 4, 291-297