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The interpretation of magnetic susceptibilities
CHEMICAL
PHYSICS
THE
LETTERS
1 (1967)
143 - 144.
INTERPRETATION
NORTH -HOLLAND
PUBLISHING
OF MAGNETIC
A. D. BUCKINGHAM Scl~ooZof Chonistq,
. AMSTERDAM
SUSCEPTIBILITIES
and MISS S. M. MALM
thivm-siQReceived
COMPANS
of Bristd.
15 May
BristoL 8. UX.
1967
It is suggested that magnetic susceptibilities of transition metal complexes should be interpreted in terms of the magnetic moment and f’polarizabilityv’ of each of the populated eigen-states. In some cases, these constants can easily be obtained from the temperature-dependence of the susceptibility. Data for RU(h?H3)6Cl3 have been analysed and the ground state magnetic moment is in good agreement with that obtained from the E.S.R. spectrum.
Magnetic susceptibilities of transition metal complexes are normally interpreted in terms of an effective magnetic moment defined as ,ueff = dI3kTXIN[ 11. However, I-ceff varies with temperature if either a large temperature-independent paramagnetism exists or if T is such that more than a single electronic state is populated [2,3]. It would seem to be preferable to analyse the susceptibility in terms of the relevant constants of the quantum states of the complex - these are its permanent and induced magnetic moments. If T is sufficiently small for only the ground electronic state to tc appreciably populated, a plot of x versus ~-1 is linear, and its slope is related to the permanent moment and the intercept to the “magnetic polarizability”. The magnetic moment of a mole of magnetically dilute material in the direction of a unit field along the z-axis is
=&$[[ F; i
xzz
-I-
(pf))‘/(kT)]
of the molecule, or unit cell, 3 state @i in the direction of the magnetic field:
where the actual magnetic moment in state Qi, L\f M(= +.Ji, . . . , -Ji) is Wi,l~ I- I%% -I-2S,)( &,&; is the component of the electronic angular momentum in the direction of the field; L, and S, are components of the electronic orbital and spin an ular momenta; p is the Bohr magneton, and g(j ‘i is a principal component of the g-tensor for the zth state. The mean component of the induced magnetic moment of a molecule ;J1state *: _ in the direczon of the field is .!j$l where ,$
and
k xx+xyy+xtz)=
= F A+
+
(1)
and the mean susceptibility x=f
= @a)(i)
&dia)(i) -9
6) + (j~@))~/(3kT)]
=
(2)
where N
_
rJI=i + 1) exp t-E&‘)
i -F’U;
+ 1) exp (-E/kT)
’
(3)
(Wi + 1) is the degeneracy of the unperturbed ’ quantum state & with energy y and electronic angular momentum Ji (=S, 5, 2,. . . ). The permanent moment component ,@) in eq. (1) is the root-mean-square magnet% moment May 1467
is the molecular The ,u(i) in
eq. (2) is -g(i) 04 Ji(Jpl)
diamagnetic
susceptibility.
g(i) = J$(g$’
+ g&j” + g(t)2) is a function Of ;F/i
where
zz
only and can be interpreted in terms of parameters such as an “orbital reduction factorR ki which is a measure of eiectron delocalization [4]. However E(i) given by eqs. (2) and (5) is not a function 143
‘
A. D. BUCKINGHAM
144
of hialone and so other reduction factors (I$) are involved in its interpretation. At sufficiently low temperatures, only the ground state, which may be degenerate, is occupied; then a pl of x versus T-l is linear with a slope of &i(l~e’, o )2/3k and an intercept of X$(o) at r-1 = 0. The electron spin resonance spectrum of the solid at low temperature is also related to P(O). At higher temperatures, excited states, with different constants ~(3 and ((1) may be populated, leading to curvature in the graph. li here are two non-degenerate states Qo and 9 differing in the energy by an amount A, then J O) = p(l) = 6 and &“) =
and MISS S. M. MALM
ted by least squares to a linear equation in T-1 (T = 90-ZOOoK). This results in an intercept at T-1 = 0 of (655 f 25) x 10-6 emu/mole and a
slope of 0.377 f 0.009 emu OK/mole. The average g-value calculated from the slope taking Jo = 3 is 2.00 f 0.03 since the linearity of the plot indicates that only the ground state is appreciably populated. Single crystal electron spin resonances aves an average g-value of 1.95 f 0.02 at 200K [6]. The intercept might be interpreted as a diamagnetic contribution of -209 x 10-6 emu/mole fcstknated from Pascal constants) and a secondorder paramagnetism of 850 x 10-6 emu/mole if the origin is chosen at the Ru nucleus.
4 + {;
$1) = -5 + 6;
(7)
where .$ is the contribution to e(o) and 4(l) from the &agle states, I&I and I&~? respective1 ; 6; and ,$i are +he residual contributions to { vo) and t(1) from the higher excited states. If A <
REFERENCES [l] B. N. Figgis and J. Lewis.
(a
[Z]
and the susceptibility behaves as if the system were comprised of molecules in a state 4th a magnetic moment VW and a magnetic polarizahility $(
[3]
X =N[*(tA
+ C;i> +
[4] [S] [6]
Progress in Inorganic Chemistry 6 (1964) 37; B. N. Figgis, J. Lewis, F. E. Mabbs and G.A. Webb. J.Chem.Soc.A (1967) 442. J. H. van Vlcck. The Theory of Electric and Magnetic Susceptibilities (Oxford University Press, 1932). J.S.Griffith. Tile Theory of Transition Metal Ions (Cambridge University Press, 1961). K.W.H.Stevens, Proc.Roy.Soc. A219 (1953) 542. B. K. Figgis, .J. Lewis, F. E.Mabbs and G.A. Webb, J.Chcm.Soc._X (1966) 422. J. H. E.Griffiths, J.O!ven and I.M.Ward. Proc.Roy. Sot. A219 (1953) 526.
PHYSICS
THE
LETTERS
1 (1967)
143 - 144.
INTERPRETATION
NORTH -HOLLAND
PUBLISHING
OF MAGNETIC
A. D. BUCKINGHAM Scl~ooZof Chonistq,
. AMSTERDAM
SUSCEPTIBILITIES
and MISS S. M. MALM
thivm-siQReceived
COMPANS
of Bristd.
15 May
BristoL 8. UX.
1967
It is suggested that magnetic susceptibilities of transition metal complexes should be interpreted in terms of the magnetic moment and f’polarizabilityv’ of each of the populated eigen-states. In some cases, these constants can easily be obtained from the temperature-dependence of the susceptibility. Data for RU(h?H3)6Cl3 have been analysed and the ground state magnetic moment is in good agreement with that obtained from the E.S.R. spectrum.
Magnetic susceptibilities of transition metal complexes are normally interpreted in terms of an effective magnetic moment defined as ,ueff = dI3kTXIN[ 11. However, I-ceff varies with temperature if either a large temperature-independent paramagnetism exists or if T is such that more than a single electronic state is populated [2,3]. It would seem to be preferable to analyse the susceptibility in terms of the relevant constants of the quantum states of the complex - these are its permanent and induced magnetic moments. If T is sufficiently small for only the ground electronic state to tc appreciably populated, a plot of x versus ~-1 is linear, and its slope is related to the permanent moment and the intercept to the “magnetic polarizability”. The magnetic moment of a mole of magnetically dilute material in the direction of a unit field along the z-axis is
=&$[[ F; i
xzz
-I-
(pf))‘/(kT)]
of the molecule, or unit cell, 3 state @i in the direction of the magnetic field:
where the actual magnetic moment in state Qi, L\f M(= +.Ji, . . . , -Ji) is Wi,l~ I- I%% -I-2S,)( &,&; is the component of the electronic angular momentum in the direction of the field; L, and S, are components of the electronic orbital and spin an ular momenta; p is the Bohr magneton, and g(j ‘i is a principal component of the g-tensor for the zth state. The mean component of the induced magnetic moment of a molecule ;J1state *: _ in the direczon of the field is .!j$l where ,$
and
k xx+xyy+xtz)=
= F A+
+
(1)
and the mean susceptibility x=f
= @a)(i)
&dia)(i) -9
6) + (j~@))~/(3kT)]
=
(2)
where N
_
rJI=i + 1) exp t-E&‘)
i -F’U;
+ 1) exp (-E/kT)
’
(3)
(Wi + 1) is the degeneracy of the unperturbed ’ quantum state & with energy y and electronic angular momentum Ji (=S, 5, 2,. . . ). The permanent moment component ,@) in eq. (1) is the root-mean-square magnet% moment May 1467
is the molecular The ,u(i) in
eq. (2) is -g(i) 04 Ji(Jpl)
diamagnetic
susceptibility.
g(i) = J$(g$’
+ g&j” + g(t)2) is a function Of ;F/i
where
zz
only and can be interpreted in terms of parameters such as an “orbital reduction factorR ki which is a measure of eiectron delocalization [4]. However E(i) given by eqs. (2) and (5) is not a function 143
‘
A. D. BUCKINGHAM
144
of hialone and so other reduction factors (I$) are involved in its interpretation. At sufficiently low temperatures, only the ground state, which may be degenerate, is occupied; then a pl of x versus T-l is linear with a slope of &i(l~e’, o )2/3k and an intercept of X$(o) at r-1 = 0. The electron spin resonance spectrum of the solid at low temperature is also related to P(O). At higher temperatures, excited states, with different constants ~(3 and ((1) may be populated, leading to curvature in the graph. li here are two non-degenerate states Qo and 9 differing in the energy by an amount A, then J O) = p(l) = 6 and &“) =
and MISS S. M. MALM
ted by least squares to a linear equation in T-1 (T = 90-ZOOoK). This results in an intercept at T-1 = 0 of (655 f 25) x 10-6 emu/mole and a
slope of 0.377 f 0.009 emu OK/mole. The average g-value calculated from the slope taking Jo = 3 is 2.00 f 0.03 since the linearity of the plot indicates that only the ground state is appreciably populated. Single crystal electron spin resonances aves an average g-value of 1.95 f 0.02 at 200K [6]. The intercept might be interpreted as a diamagnetic contribution of -209 x 10-6 emu/mole fcstknated from Pascal constants) and a secondorder paramagnetism of 850 x 10-6 emu/mole if the origin is chosen at the Ru nucleus.
4 + {;
$1) = -5 + 6;
(7)
where .$ is the contribution to e(o) and 4(l) from the &agle states, I&I and I&~? respective1 ; 6; and ,$i are +he residual contributions to { vo) and t(1) from the higher excited states. If A <
REFERENCES [l] B. N. Figgis and J. Lewis.
(a
[Z]
and the susceptibility behaves as if the system were comprised of molecules in a state 4th a magnetic moment VW and a magnetic polarizahility $(
[3]
X =N[*(tA
+ C;i> +
[4] [S] [6]
Progress in Inorganic Chemistry 6 (1964) 37; B. N. Figgis, J. Lewis, F. E. Mabbs and G.A. Webb. J.Chem.Soc.A (1967) 442. J. H. van Vlcck. The Theory of Electric and Magnetic Susceptibilities (Oxford University Press, 1932). J.S.Griffith. Tile Theory of Transition Metal Ions (Cambridge University Press, 1961). K.W.H.Stevens, Proc.Roy.Soc. A219 (1953) 542. B. K. Figgis, .J. Lewis, F. E.Mabbs and G.A. Webb, J.Chcm.Soc._X (1966) 422. J. H. E.Griffiths, J.O!ven and I.M.Ward. Proc.Roy. Sot. A219 (1953) 526.