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On the calculation of dipole fields in nonellipsoidal crystals
1. Phys. Chem. solids. 1975,
Vol. 36. pp. 1307-1309.
Pergamon Press.
Printed in Great Britain
ON THE CALCULATION OF DIPOLE FIELDS IN NONELLIPSOIDAL CRYSTALS P. J. BECKER and A. KASTEN Physikalisches Institut der Universitat Karlsruhe, Germany (Received 3 January 1975;accepted 21 April 1975) Abstract-A method for calculating the local magnetic dipole field inside a crystal of arbitrary shape is given in the case of magnetic saturation. As an example the dependence of the dipole field on the position inside the crystal has been calculated for a dysprosium vanadate (DyVO,) sample. 1. INTRODUCTION
Zeeman effect measurements in optical absorption spectroscopy show the dependence of the energetic positions of the absorption lines on the external magnetic field. The
field which is responsible for the observed shifts or splittings, the local field IS,,, is the sum of external field Hext, the internal magnetic dipole field Hdd, and the exchange field I&. The dipole part is shape dependent, but for ellipsoidal samples it is uniform and can be calculated in the well known method sketched briefly in Section 2.1. In many cases it is impossible to get ellipsoidal crystals and hence it is desirable to have a method to calculate the dipole field inside a sample of arbitrary shape. In Section 2.2 we shall describe our method of calculating those fields in the case of magnetic saturation which has already been used in preceeding works[l, 21. The described method is particularly suited for Ising like systems where the direction of the magnetic moments is determined by the crystal field. 2. CALCULATION OF THE DIPOLE SUMS
2.1 Ellipsoidal samples The dipole field at the site of a central ion inside a paramagnetic or magnetically ordered crystal of ellipsoidal shape is usually calculated as follows. Inside a spherical region (Lorentz sphere) one has to sum the contributions of each magnetic moment individually. The rest of the sample is regarded to be magnetised homogeneously. This method yields a dipole field of [3] (Fig. la): Hdd=H,+HQ+H,= + j-/j-
($%;)d’r-1
c; (yri Sphere
field of a sphere (Lorentz field). For samples of ellipsoidal shapes with the magnetisation along a principal axis, say a, HZ can be expressed in terms of the corresponding demagnetising factor A$ [4]. eqn (1) can thus be written:
(14 This local field is uniform over the whole volume[5]. If the sample is not ellipsoidal, the dipole field is not uniform even in the case of paramagnetic saturation or complete ferromagnetic ordering. At different sites inside the crystal one gets different values of the dipole field and it is impossible to define a demagnetising factor which has the same value for all sites inside the crystal. 2.2 Nonellipsoidal samples For circular cylinders and rectangular prisms Joseph and Schliimann[6] have shown that in a iirst approximation the dipole fields can be calculated following eqn (la) if one defines a demagnetising tensor the components of which depend on the position inside the sample. We want to describe another method to calculate the dipole fields in
-7) j- j- (Fr-;)d’r
mi is the magnetic moment of the i-th ion inside the Lorentz sphere, ri the vector of the distance from the central ion. m is the magnetisation inside the volume element d3r, being uniform for ellipsoidal samples, r is the distance vector from the central ion to this volume element. For a given direction of the magnetisation, HI only depends on the microscopic structure of the crystal, II2 only on the macroscopic shape. H, is the demagnetising
Fig. 1. (a) Schematic diagram of the contributions to the local magnetic field inside ellipsoidal samples following eqn (1). (b) Contributions to the local magnetic field inside samples of arbitrary shape following eqn (3). Note that H, and H, cancel out.
1307
1308
P. J. BECKER and A. KASTEN
the case of magnetic saturation which will be more appropriate if only the fields at some few points inside the crystal are needed as in spectroscopic work. For crystals of arbitrary shape the dipole field at the site of a central ion is:
where the sum runs over all magnetic moments of the entire volume V of the sample. As we will show, it is however not necessary to perform the summation over the whole crystal, since the summation over a volume V’ which is geometrically equivalent to the macroscopic shape yields the same result. For simplicity we shall discuss in the following the case of a rectangular prism, but the results are directly applicable to samples of arbitrary shapes. As in Section 2.1 the sample of volume V = xo . yO. zois regarded to be magnetised homogeneously outside a certain region V’ = (xdk) . (yolk). (zolk). In this case V’ is not spherical (see Fig. lb). One gets a dipole field of:
The integration limits have to be chosen as sketched two dimensionally in Fig. 2. Both integrals in eqn (3) have the same value, i.e. their difference vanishes as can easily be seen by making the transformations x’ = (x/k), y’ = (y/k), and z’ = (z/k). Hence the dipole field of eqn (3) reduces to:
(34 As in the case of the Lorentz sphere in Section 2.1, V’ must be sufficiently large to ensure that the summation converges. The position of the central ion is arbitrary and hence the dipole field can be calculated at each site inside (or outside) the sample. Since each volume can be approximated by a sum of rectangular prisms, eqn (3a) holds for an arbitrary shape of V’. 3. CALCULATIONS ONWO, At room temperature DyVO, has zircon structure (space group 14,/amd) with the point symmetry iim2 at the site of a paramagnetic Dy’+ ion[7]. The lattice parameters are a0 = 7.136 A and co = 6.307 A[8]. Figure 3 shows a unit cell of DyVO,. The axes x, y, and z point tBelow 14K DyVOI undergoes a crystallographic distortion to orthorhombic symmetry (space group Imma) and the a- and b-axes are no longer equivalent.
(Xl
.Yl)
I
( x2*Y1)
(x2-x,) ‘X0
(Y2-Y,l=Y,
Fig. 2. Two dimensionalvisualisationof the integrationlimits of eqn (3).
z
/JY
x
Fig. 3. Unit cell of DyVO,.The arrowsindicatethe alignmentof the magneticmomentsin the antiferromagneticstate. along the crystallographic a, 6, and c axes which are parallel to the crystal growth faces. With these lattice parameters and the g-factors of Dy” in DyV04 at low temperaturest g, = 19.1, g, = g, ~0, the dipole sum of eqn (3a) has been calculated for the antiferromagnetic configuration shown in Fig. 3 as well as for paramagnetic saturation parallel x. Each paramagnetic ion belongs to one of four sublattices (sites 14 in Fig. 3) and the dipole sums were computed for each sublattice separately. Figure 4 shows the dependence of the antiferromagnetic dipole field on the volume of the summation area taken to be either a sphere or a rectangular prism consisting of complete unit cells. In the latter case a minimum summation over three unit cells in each direction gave a dipole field which was only 0.03 kOe higher than the value for extremely large summation areas (697 kOe). In the case of the ferromagnetic dipole sums the minimum summation volume can increase considerably if one of the extensions of the sample is very small, since from the central ion the extension in each direction must exceed three unit cells to obtain a comparable accuracy.
On the calculation of dipole fieldsin nonellipsoidalcrystals
vo~umcLA?
Fig. 4. Dipole sums of DyV04 in the antiferromagnetic state versus the volume of the summation area. iV is the number of p~rna~tic ions included. The summation area was a sphere (circles) of radius r or a rect~g~~ prism consisting of complete unit cells (crosses). Figure 5 shows one example for the dependence of the ferromagnetic (or paramagnetic saturated) dipole field on the position of the central ion inside the crystal (sample II of Ref.[2]) with Ifdd = -5.39 kOe at the centre of the crystal.
1309
Fig. 5. Dependenceof the paramagneticsaturated (parallel x) dipolefield on the position inside a DyVO, sample (rectangular prism). Tbe circles are calculated with eqn (3a), the solid lines with eqn (la) using the position dependent demagnetising factors of
Ref.[4k of the dipole field may yield “curved” spectral lines which we have sometimes observed. Acknowledgements-We are indebted to Prof. Dr. H. G. Kahle for many valuable discussions and to G. Domann for his help in computing the dipole sums. The computer calculations were performed on the Univac 1108 of the Rechenzentrum der Universitiit Karlsrube. This work was supported by the Deutsche Forschungsgemeinschaft.
4. CONCLUSION
Our calculations show that the dipole field at any site inside a nonellipsoidal antife~oma~etic, ferromagnetic, or p~ma~etic saturated crystal can be obtained by summing over the contributions of the magnetic moments inside a region around the central ion which is geometrically equivalent to the macroscopic crystal shape. The convergence is much smoother when the summation is performed over complete unit cells, rather than over spheres with the same volume. The dependence of the dipole field on the position causes inhomogeneously broadened lines. When the spectra are taken photographically where the sample is imaged onto the slit the variation
REFERENCES
Kasten A., Diploma work, Phys. Inst. der Universitiit Karlsruhe (1971),unpublished. 2. Kasten A. and Becker P. J., L Phys. C: Solid Stare Physics 7, 3120 (1974). 3. See for example: Hellwege, K. H., Ej~~~hru~g in die Fesfk~~~~hysjk ff. Springer Verlag, Heidelberg (1970). 4. Osborn J. A., Phys. Rev. 67, 351 (1945). 5. Maxwell J. C., A Treatise on Electricify and Magnetism 3rd Edit. Dover Publications, Inc. New York (1954). 6. Joseph R. I. and Schliimann E., J. Appl. Phys. 36,1579 (1965). 7. Wckofl, R. W. CL Crystal Structures 2nd Edit. Vol. 3. Interscience Publishers, New York (1%5). 8. Gijbel H. and Will G., Phys. Stat. So/id. (b) 50, 147 (1972). 1.
Vol. 36. pp. 1307-1309.
Pergamon Press.
Printed in Great Britain
ON THE CALCULATION OF DIPOLE FIELDS IN NONELLIPSOIDAL CRYSTALS P. J. BECKER and A. KASTEN Physikalisches Institut der Universitat Karlsruhe, Germany (Received 3 January 1975;accepted 21 April 1975) Abstract-A method for calculating the local magnetic dipole field inside a crystal of arbitrary shape is given in the case of magnetic saturation. As an example the dependence of the dipole field on the position inside the crystal has been calculated for a dysprosium vanadate (DyVO,) sample. 1. INTRODUCTION
Zeeman effect measurements in optical absorption spectroscopy show the dependence of the energetic positions of the absorption lines on the external magnetic field. The
field which is responsible for the observed shifts or splittings, the local field IS,,, is the sum of external field Hext, the internal magnetic dipole field Hdd, and the exchange field I&. The dipole part is shape dependent, but for ellipsoidal samples it is uniform and can be calculated in the well known method sketched briefly in Section 2.1. In many cases it is impossible to get ellipsoidal crystals and hence it is desirable to have a method to calculate the dipole field inside a sample of arbitrary shape. In Section 2.2 we shall describe our method of calculating those fields in the case of magnetic saturation which has already been used in preceeding works[l, 21. The described method is particularly suited for Ising like systems where the direction of the magnetic moments is determined by the crystal field. 2. CALCULATION OF THE DIPOLE SUMS
2.1 Ellipsoidal samples The dipole field at the site of a central ion inside a paramagnetic or magnetically ordered crystal of ellipsoidal shape is usually calculated as follows. Inside a spherical region (Lorentz sphere) one has to sum the contributions of each magnetic moment individually. The rest of the sample is regarded to be magnetised homogeneously. This method yields a dipole field of [3] (Fig. la): Hdd=H,+HQ+H,= + j-/j-
($%;)d’r-1
c; (yri Sphere
field of a sphere (Lorentz field). For samples of ellipsoidal shapes with the magnetisation along a principal axis, say a, HZ can be expressed in terms of the corresponding demagnetising factor A$ [4]. eqn (1) can thus be written:
(14 This local field is uniform over the whole volume[5]. If the sample is not ellipsoidal, the dipole field is not uniform even in the case of paramagnetic saturation or complete ferromagnetic ordering. At different sites inside the crystal one gets different values of the dipole field and it is impossible to define a demagnetising factor which has the same value for all sites inside the crystal. 2.2 Nonellipsoidal samples For circular cylinders and rectangular prisms Joseph and Schliimann[6] have shown that in a iirst approximation the dipole fields can be calculated following eqn (la) if one defines a demagnetising tensor the components of which depend on the position inside the sample. We want to describe another method to calculate the dipole fields in
-7) j- j- (Fr-;)d’r
mi is the magnetic moment of the i-th ion inside the Lorentz sphere, ri the vector of the distance from the central ion. m is the magnetisation inside the volume element d3r, being uniform for ellipsoidal samples, r is the distance vector from the central ion to this volume element. For a given direction of the magnetisation, HI only depends on the microscopic structure of the crystal, II2 only on the macroscopic shape. H, is the demagnetising
Fig. 1. (a) Schematic diagram of the contributions to the local magnetic field inside ellipsoidal samples following eqn (1). (b) Contributions to the local magnetic field inside samples of arbitrary shape following eqn (3). Note that H, and H, cancel out.
1307
1308
P. J. BECKER and A. KASTEN
the case of magnetic saturation which will be more appropriate if only the fields at some few points inside the crystal are needed as in spectroscopic work. For crystals of arbitrary shape the dipole field at the site of a central ion is:
where the sum runs over all magnetic moments of the entire volume V of the sample. As we will show, it is however not necessary to perform the summation over the whole crystal, since the summation over a volume V’ which is geometrically equivalent to the macroscopic shape yields the same result. For simplicity we shall discuss in the following the case of a rectangular prism, but the results are directly applicable to samples of arbitrary shapes. As in Section 2.1 the sample of volume V = xo . yO. zois regarded to be magnetised homogeneously outside a certain region V’ = (xdk) . (yolk). (zolk). In this case V’ is not spherical (see Fig. lb). One gets a dipole field of:
The integration limits have to be chosen as sketched two dimensionally in Fig. 2. Both integrals in eqn (3) have the same value, i.e. their difference vanishes as can easily be seen by making the transformations x’ = (x/k), y’ = (y/k), and z’ = (z/k). Hence the dipole field of eqn (3) reduces to:
(34 As in the case of the Lorentz sphere in Section 2.1, V’ must be sufficiently large to ensure that the summation converges. The position of the central ion is arbitrary and hence the dipole field can be calculated at each site inside (or outside) the sample. Since each volume can be approximated by a sum of rectangular prisms, eqn (3a) holds for an arbitrary shape of V’. 3. CALCULATIONS ONWO, At room temperature DyVO, has zircon structure (space group 14,/amd) with the point symmetry iim2 at the site of a paramagnetic Dy’+ ion[7]. The lattice parameters are a0 = 7.136 A and co = 6.307 A[8]. Figure 3 shows a unit cell of DyVO,. The axes x, y, and z point tBelow 14K DyVOI undergoes a crystallographic distortion to orthorhombic symmetry (space group Imma) and the a- and b-axes are no longer equivalent.
(Xl
.Yl)
I
( x2*Y1)
(x2-x,) ‘X0
(Y2-Y,l=Y,
Fig. 2. Two dimensionalvisualisationof the integrationlimits of eqn (3).
z
/JY
x
Fig. 3. Unit cell of DyVO,.The arrowsindicatethe alignmentof the magneticmomentsin the antiferromagneticstate. along the crystallographic a, 6, and c axes which are parallel to the crystal growth faces. With these lattice parameters and the g-factors of Dy” in DyV04 at low temperaturest g, = 19.1, g, = g, ~0, the dipole sum of eqn (3a) has been calculated for the antiferromagnetic configuration shown in Fig. 3 as well as for paramagnetic saturation parallel x. Each paramagnetic ion belongs to one of four sublattices (sites 14 in Fig. 3) and the dipole sums were computed for each sublattice separately. Figure 4 shows the dependence of the antiferromagnetic dipole field on the volume of the summation area taken to be either a sphere or a rectangular prism consisting of complete unit cells. In the latter case a minimum summation over three unit cells in each direction gave a dipole field which was only 0.03 kOe higher than the value for extremely large summation areas (697 kOe). In the case of the ferromagnetic dipole sums the minimum summation volume can increase considerably if one of the extensions of the sample is very small, since from the central ion the extension in each direction must exceed three unit cells to obtain a comparable accuracy.
On the calculation of dipole fieldsin nonellipsoidalcrystals
vo~umcLA?
Fig. 4. Dipole sums of DyV04 in the antiferromagnetic state versus the volume of the summation area. iV is the number of p~rna~tic ions included. The summation area was a sphere (circles) of radius r or a rect~g~~ prism consisting of complete unit cells (crosses). Figure 5 shows one example for the dependence of the ferromagnetic (or paramagnetic saturated) dipole field on the position of the central ion inside the crystal (sample II of Ref.[2]) with Ifdd = -5.39 kOe at the centre of the crystal.
1309
Fig. 5. Dependenceof the paramagneticsaturated (parallel x) dipolefield on the position inside a DyVO, sample (rectangular prism). Tbe circles are calculated with eqn (3a), the solid lines with eqn (la) using the position dependent demagnetising factors of
Ref.[4k of the dipole field may yield “curved” spectral lines which we have sometimes observed. Acknowledgements-We are indebted to Prof. Dr. H. G. Kahle for many valuable discussions and to G. Domann for his help in computing the dipole sums. The computer calculations were performed on the Univac 1108 of the Rechenzentrum der Universitiit Karlsrube. This work was supported by the Deutsche Forschungsgemeinschaft.
4. CONCLUSION
Our calculations show that the dipole field at any site inside a nonellipsoidal antife~oma~etic, ferromagnetic, or p~ma~etic saturated crystal can be obtained by summing over the contributions of the magnetic moments inside a region around the central ion which is geometrically equivalent to the macroscopic crystal shape. The convergence is much smoother when the summation is performed over complete unit cells, rather than over spheres with the same volume. The dependence of the dipole field on the position causes inhomogeneously broadened lines. When the spectra are taken photographically where the sample is imaged onto the slit the variation
REFERENCES
Kasten A., Diploma work, Phys. Inst. der Universitiit Karlsruhe (1971),unpublished. 2. Kasten A. and Becker P. J., L Phys. C: Solid Stare Physics 7, 3120 (1974). 3. See for example: Hellwege, K. H., Ej~~~hru~g in die Fesfk~~~~hysjk ff. Springer Verlag, Heidelberg (1970). 4. Osborn J. A., Phys. Rev. 67, 351 (1945). 5. Maxwell J. C., A Treatise on Electricify and Magnetism 3rd Edit. Dover Publications, Inc. New York (1954). 6. Joseph R. I. and Schliimann E., J. Appl. Phys. 36,1579 (1965). 7. Wckofl, R. W. CL Crystal Structures 2nd Edit. Vol. 3. Interscience Publishers, New York (1%5). 8. Gijbel H. and Will G., Phys. Stat. So/id. (b) 50, 147 (1972). 1.