4.14.1: example magnetic moment data and their interpretation

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Summary of applicable formulae

1) Spin-Only magnetic moment
μ_s.o. = √{4S(S+1)} B.M.

2) For A and E ground terms
μ_eff = μ_s.o. (1-α λ /Δ) B.M.
Do not expect Temperature dependence.

3) For T ground terms with orbital angular momentum contribution
μ_S_+L = √{4S(S+1) + L(L+1)} B.M.
T terms generally show marked Temperature dependence.

The examples that follow are arranged showing the experimentally observed values, the theoretical "spin-only" value and possible variations expected.

A number of the examples involve "alums" where the central Transition Metal ion can be considered to be octahedrally coordinated by water molecules.

d¹

VCl₄
V(IV) tetrahedral

      80K       300K             μ_s.o. /B.M.
      1.6       1.6                   1.73

²E ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For less than a half-filled d shell, the sign of λ is positive so the effect on μ should be that μ_eff < μ_s.o.

VCl₆^2-
V(IV) octahedral

      80K       300K             μ_s.o. /B.M.
      1.4       1.8                   1.73

²T_2g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S_+L = 3 B.M. which is clearly much higher than the 1.8 B.M. found at 300K. So, μ_s.o. < μ_obs < μ_S_+L
showing that the magnetic moment is partially quenched.

d²

V³⁺ in (NH₄)V(SO₄)₂.12H₂O (an alum)
V(III) octahedral

      80K       300K             μ_s.o. /B.M.
      2.7       2.7                   2.83

³T_1g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S_+L = √(20) = 4.47 B.M. which is clearly much higher than the 2.7 B.M. found at 300K.
So, μ_obs < μ_s.o. < μ_S_+L
showing that the magnetic moment is significantly quenched.
In this case, there is no observed Temperature variation between 80 and 300K and it may require much lower temperatures to see the effect?

d³

Cr³⁺ in KCr(SO₄)₂.12H₂O (an alum)
Cr(III) octahedral

      80K       300K             μ_s.o. /B.M.
      3.8       3.8                   3.87

⁴A_2g ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For less than a half-filled d shell the sign of λ is positive so the effect on μ should be that μ_eff < μ_s.o.

d⁴

CrSO₄.6H₂O
Cr(II) octahedral

      80K       300K             μ_s.o. /B.M.
      4.8       4.8                   4.9

⁵E_g ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For less than a half-filled d shell the sign of λ is positive so the effect on μ should be that μ_eff < μ_s.o.

K₃Mn(CN)₆
Mn(III) low-spin octahedral

      80K       300K             μ_s.o. /B.M.
      3.1       3.2                   2.83

³T_1g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S_+L = √(20) = 4.47 B.M. which is clearly much higher than the 3.2 B.M. found at 300K.
So, μ_s.o. < μ_obs < μ_S_+L
showing that the magnetic moment is partially quenched.
In this case, there is a small Temperature variation observed between 80 and 300K.

d⁵

K₂Mn(SO₄)₂.6H₂O (an alum)
Mn(II) high-spin octahedral

      80K       300K             μ_s.o. /B.M.
      5.9       5.9                   5.92

⁶A_1g ground term - hence do not expect Temperature dependence and L=0 so no orbital contribution possible.
Expect μ_eff = μ_s.o.

K₃Fe(CN)₆
Fe(III) low-spin octahedral

      80K       300K             μ_s.o. /B.M.
      2.2       2.4                   1.73

²T_2g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S+L = √(9) = 3 B.M. which is clearly much higher than the 2.4 B.M. found at 300K.
So, μ_s.o. < μ_obs < μ_S_+L
showing that the magnetic moment is partially quenched.

d⁶

Fe²⁺ in (NH₄)₂Fe(SO₄)₂.6H₂O (an alum)
Fe(II) high-spin octahedral

      80K       300K             μ_s.o. /B.M.
      5.4       5.5                   4.9

⁵T_2g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S+L = √(30) = 5.48 B.M. which is close to the 5.5 B.M. found at 300K.
So, μ_s.o. < μ_obs ~ μ_S+L
showing that the magnetic moment is not showing much quenching.

d⁷

Cs₂CoCl₄
Co(II) tetrahedral

      80K       300K             μ_s.o. /B.M.
      4.5       4.6                   3.87

⁴A₂ ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For more than a half-filled d shell the sign of λ is negative so the effect on μ should be that μ_eff > μ_s.o.
The observed values are somewhat bigger than expected for the small (0.2 B.M.) variation due to equation 2) so other factors must be affecting the magnetic moment. These effects will not be covered in this course!

Co²⁺ in (NH₄)₂Co(SO₄)₂.6H₂O (an alum)
Co(II) high-spin octahedral

      80K       300K             μ_s.o. /B.M.
      4.6       5.1                   3.88

⁴T_1g ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S+L = √(27) = 5.2 B.M. which is close to the 5.1 B.M. found at 300K.
So, μ_s.o. < μ_obs ~ μ_S+L
showing that the magnetic moment is not showing much quenching.

d⁸

Ni²⁺ in (NH₄)₂Ni(SO₄)₂.6H₂O (an alum)
Ni(II) octahedral

      80K       300K             μ_s.o. /B.M.
      3.3       3.3                   2.83

³A_2g ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For more than a half-filled d shell the sign of λ is negative so the effect on μ should be that μ_eff > μ_s.o.
The observed values are somewhat bigger than expected for the small (0.2 B.M.) variation due to equation 2) so other factors must be affecting the magnetic moment. These effects will not be covered in this course!

(Et₄N)₂NiCl₄
Ni(II) tetrahedral

      80K       300K             μ_s.o. /B.M.
      3.2       3.8                   2.83

³T₂ ground term - hence do expect Temperature dependence and large variation from spin-only value may be observed at low temperatures.
Since there is a direct orbital angular momentum contribution we should calculate μ_S_+L from equation 3) above.
For a full S+L contribution this would give μ_S+L = √(20) = 4.47 B.M. which is higher than the 3.8 B.M. found at 300K.
So, μ_s.o. < μ_obs < μ_S_+L
showing that the magnetic moment is partially quenched.

d⁹

Cu²⁺ in (NH₄)₂Cu(SO₄)₂.6H₂O (an alum)
Cu(II) octahedral

      80K       300K             μ_s.o. /B.M.
      1.9       1.9                   1.73

²E_g ground term - hence don't expect Temperature dependence and small variation from spin-only value can be accounted for by equation 2) above. For more than a half-filled d shell the sign of λ is negative so the effect on μ should be that μ_eff > μ_s.o.

Contributors and Attributions

Prof. Robert J. Lancashire (The Department of Chemistry, University of the West Indies)

Summary of applicable formulae

d1

d2

d3

d4

d5

d6

d7

d8

d9

Contributors and Attributions

d¹

d²

d³

d⁴

d⁵

d⁶

d⁷

d⁸

d⁹