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Influence of crystallographic displacement and g-factors on the groundstate of cerous magnesium nitrate
Solid State Communications, Vol. 18, PP. 473 474, 1976.
Pergamon Press.
Printed in Great Britain
INFLUENCE OF CRYSTALLOGRAPHIC DISPLACEMENT AND g-FACTORS ON THE GROUNDSTATE OF CEROUS MAGNESIUM NITRATE* P.H.E. Meijer Physics Dept. Catholic University, Washington, D.C. 20064., U.S.A. and National Bureau of Standards, Washington, D.C. 20234., U.S.A. R.G. Lockhart Catholic University, Washington, D.C. 20064., U.S.A. and Th. Niemeijer Technical University, Deift, The Netherlands (Received 7 September 1975) Since the thermodynamic properties of cerous magnesium nitrate play such an important role in low temperature physics we computed the influence of all variations still possible under the given set of known data: Small variation in the lattice structure, in particular the displacement of the center plane, small variations of the g-factor and the combination thereof. COMPUTATIONS for the Luttinger—Tisza ground state for CMN have been reported by Daniels and Felsteiner1 and by Meijer and Niemeijer.2 The main goal of the last publication was to investigate whether the assumption of larger super lattices could lead to lower ground state energies. In this paper the low temperature lattice constants of Schifferl were used which were not yet available when Daniels and Felsteiner’s work was published, In neither paper, however, was the small “displacement” of the center atom, already reported by the X-ray work
Table 1 the values indicated are given in mK and correspond to a point in k.space given by 011. The use of the Miller indices was explained in reference.2 Notice that, in case ~ = 0, 011 relates to the basis vectors A, B, C as in the same way as the indices 100 related to a, b, c. This is important in order to establish the relation with the previous work. The reader is reminded that 011 is degenerated with 101 and 110 and in the same way the indices 100 are degenerate with 010 and 001. The table shows that small changes in the g 11 -factor have no influence, but that the displacement of the middle layer does have an influence on the energy. The effect of the displacement of every other layer gives a lowering of the Luttinger Tisza (L—T) ground state, which can be approximately expressed by: L~.R’E= 0 09Th
of Zalkin, completely taken into account. By displacement we mean that; the lattice has actually a periodicity in the c-direction twice the distance between the successive triangular a b planes, because every other plane is slightly displaced with respect to the position it would have if all planes were equidistant. The primitive translation vectors in this model are related to the vectors used previously (compare ref. 3) by A =inbthe + unit c, B =cella + atom byc, C = a + b and the additional u
=
v
=
w
—
/
~(1 + ~c’/c)
Table 1. Lowest state CMN1n mK according to the
where the parameter C’ is expressed with the help of the relative displacement: c’ = c(1 6). If6 = 0 we recuperate the lattice previously used. c is the distance between the triangular planes. The results for the lowest energy level are given in
L.T. method
~,
—
*
-
where ~ is the relative displacement of5the middle~layer. available = According to the crystallographic data 0.045.
—
g 0.032 1.9467 1.9552 —
.c’ c1
Supported by NSF Grant No. 37916.
= =
c c(1
—
0.045)
—
g 0 1.9467 1.9552 —
Why is the L T ground state so insensitive to slight 473
474
INFLUENCE OF g-FACTORS ON CEROUS MAGNESIUM NITRATE
variations in the g-factor and why is the ground state lowered rather than raised by the displacement? The first question can be understood if one considers the complete set of 6 by 6 matrices. Take one of the three that contains the lowest state. This matrix can be “almost” factorized into a 4 by 4 and a 2 by 2 matrix. By almost, we mean that the elements that prevent complete factorization are extremely small. It turns out that the 4 by 4 provides the ground state and that that section of the matrix is completely determined by the g 1 factors, hence the insensitivity to small changes in g11 To answer the second question. The system we are considering Inside can be each looked upon asthe twolattice interpenetration sublattices. sublattice sums remain
Vol. 18, No. 4
the same, but the interlattice sums vary when the lattices are displaced with respect to each other. Since the interlattice sums represent a set of off-diagonal elements in the matrices, and since these off-diagonal elements are minimal when the lattice has the highest symmetry, we fmd that the enlargement of these elements leads to a further spreading of the eigenvalues and hence to a lowering of the ground state. In all four cases the strong condition4 is fulfilled. The eigenvectors are the same whether 6 is zero or not. Ifg11 = 0 they lie along the a axis, the b axis or along the third side of the triangle. If g11 0 they are slightly 2 lifted out of the a b plane. This result is not new.
REFERENCES 1. 2.
DANIELS J.M. & FELSTEINER J., Can. J. Phys. 42, 1469 (1964). MEIJER P.H.E. & NIEMEIJER Th., Phys. Rev. B7, 1984 (1973).
3.
MEIJER P.H.E. & O’KEEFE D.J., Phys. Rev. B!, 3786 (1970). (The lattice vectors a, b, c are called a
4.
in this paper.) LUTTINGER J.M. & TISZA L., Phys. Rev 70, 954 (1946)
1, a2, a3
5.
ZALKIN A., FORRESTER J.D. & TEMPLETON D. H., J. Chem. Phys. 39, 7881(1963); SCHIFERL D., J. Chem. Phys. 52, 3234 (1970).
Pergamon Press.
Printed in Great Britain
INFLUENCE OF CRYSTALLOGRAPHIC DISPLACEMENT AND g-FACTORS ON THE GROUNDSTATE OF CEROUS MAGNESIUM NITRATE* P.H.E. Meijer Physics Dept. Catholic University, Washington, D.C. 20064., U.S.A. and National Bureau of Standards, Washington, D.C. 20234., U.S.A. R.G. Lockhart Catholic University, Washington, D.C. 20064., U.S.A. and Th. Niemeijer Technical University, Deift, The Netherlands (Received 7 September 1975) Since the thermodynamic properties of cerous magnesium nitrate play such an important role in low temperature physics we computed the influence of all variations still possible under the given set of known data: Small variation in the lattice structure, in particular the displacement of the center plane, small variations of the g-factor and the combination thereof. COMPUTATIONS for the Luttinger—Tisza ground state for CMN have been reported by Daniels and Felsteiner1 and by Meijer and Niemeijer.2 The main goal of the last publication was to investigate whether the assumption of larger super lattices could lead to lower ground state energies. In this paper the low temperature lattice constants of Schifferl were used which were not yet available when Daniels and Felsteiner’s work was published, In neither paper, however, was the small “displacement” of the center atom, already reported by the X-ray work
Table 1 the values indicated are given in mK and correspond to a point in k.space given by 011. The use of the Miller indices was explained in reference.2 Notice that, in case ~ = 0, 011 relates to the basis vectors A, B, C as in the same way as the indices 100 related to a, b, c. This is important in order to establish the relation with the previous work. The reader is reminded that 011 is degenerated with 101 and 110 and in the same way the indices 100 are degenerate with 010 and 001. The table shows that small changes in the g 11 -factor have no influence, but that the displacement of the middle layer does have an influence on the energy. The effect of the displacement of every other layer gives a lowering of the Luttinger Tisza (L—T) ground state, which can be approximately expressed by: L~.R’E= 0 09Th
of Zalkin, completely taken into account. By displacement we mean that; the lattice has actually a periodicity in the c-direction twice the distance between the successive triangular a b planes, because every other plane is slightly displaced with respect to the position it would have if all planes were equidistant. The primitive translation vectors in this model are related to the vectors used previously (compare ref. 3) by A =inbthe + unit c, B =cella + atom byc, C = a + b and the additional u
=
v
=
w
—
/
~(1 + ~c’/c)
Table 1. Lowest state CMN1n mK according to the
where the parameter C’ is expressed with the help of the relative displacement: c’ = c(1 6). If6 = 0 we recuperate the lattice previously used. c is the distance between the triangular planes. The results for the lowest energy level are given in
L.T. method
~,
—
*
-
where ~ is the relative displacement of5the middle~layer. available = According to the crystallographic data 0.045.
—
g 0.032 1.9467 1.9552 —
.c’ c1
Supported by NSF Grant No. 37916.
= =
c c(1
—
0.045)
—
g 0 1.9467 1.9552 —
Why is the L T ground state so insensitive to slight 473
474
INFLUENCE OF g-FACTORS ON CEROUS MAGNESIUM NITRATE
variations in the g-factor and why is the ground state lowered rather than raised by the displacement? The first question can be understood if one considers the complete set of 6 by 6 matrices. Take one of the three that contains the lowest state. This matrix can be “almost” factorized into a 4 by 4 and a 2 by 2 matrix. By almost, we mean that the elements that prevent complete factorization are extremely small. It turns out that the 4 by 4 provides the ground state and that that section of the matrix is completely determined by the g 1 factors, hence the insensitivity to small changes in g11 To answer the second question. The system we are considering Inside can be each looked upon asthe twolattice interpenetration sublattices. sublattice sums remain
Vol. 18, No. 4
the same, but the interlattice sums vary when the lattices are displaced with respect to each other. Since the interlattice sums represent a set of off-diagonal elements in the matrices, and since these off-diagonal elements are minimal when the lattice has the highest symmetry, we fmd that the enlargement of these elements leads to a further spreading of the eigenvalues and hence to a lowering of the ground state. In all four cases the strong condition4 is fulfilled. The eigenvectors are the same whether 6 is zero or not. Ifg11 = 0 they lie along the a axis, the b axis or along the third side of the triangle. If g11 0 they are slightly 2 lifted out of the a b plane. This result is not new.
REFERENCES 1. 2.
DANIELS J.M. & FELSTEINER J., Can. J. Phys. 42, 1469 (1964). MEIJER P.H.E. & NIEMEIJER Th., Phys. Rev. B7, 1984 (1973).
3.
MEIJER P.H.E. & O’KEEFE D.J., Phys. Rev. B!, 3786 (1970). (The lattice vectors a, b, c are called a
4.
in this paper.) LUTTINGER J.M. & TISZA L., Phys. Rev 70, 954 (1946)
1, a2, a3
5.
ZALKIN A., FORRESTER J.D. & TEMPLETON D. H., J. Chem. Phys. 39, 7881(1963); SCHIFERL D., J. Chem. Phys. 52, 3234 (1970).