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Magnetostriction in terms of hexagonal and cubic harmonics up to degree 6
Journal of Magnetism and Magnetic Materials 20 (1980) 258-264 o North-Holland Publishing Company
MAGNETOSTRICTION IN TERMS OF HEXAGONAL AND CUBIC HARMONICS UP TO DEGREE 6
R.J. POTTON and G.J. REELER Department of Pure and Applied Physics, Universityof Salford, Salford iUS 4WT, Lancashire, UK Received 27 November 1979
Tables are given which ahow the magnetostiiction constants jk~$e and magnetoelastic coupling coefficientsJ&& to be evaluated from the Fourier coefficients of Al/l in various crystallographic planes. The difference between anisotropy coefficients at zero strain and zero stress is given for hexagonal crystals for the fist time in terms of the ‘8. At all stages the indices which indicate the symmetry type of a function (n, f) are clearly distingukhed from those which do not (i, k, 0.
1. Introduction
There are clear advantages in expressing magnetostriction in terms of coefficients associated with the irreducible representations of the symmetry group of the particular crystal concerned. However, the different multiplicities of bases of an irreducible representation found among strain functions and symmetrized polynomials have not prevlously been distinguished clearly, and indeed some confusion has occurred through multiplicity being ascribed to representations rather than to their bases. In a previous paper [l] we developed a notation which separates those indices which indicate the symmetry type of a function k, i) from those which do not (j, k, 0 and in which separate indices j and k are used (if necessary) for strains and polynomials, respectively, to distinguish between independent bases of the same irreducible representation. We have now produced tables, in terms of this notation, which allow the magnetostriction coefficients ‘,Af (for I sf 6) and magnetoelastic coupling coefficients J>f to be evaluated from the Fourier coefficients of the dependence of linear strain Al/Zon magnetization direction in various crystallographic planes. The magnetoelastic energy also results in a difference between anisotropy coefficients at zero strain and zero stress, and the difference in terms of the hexagonal magnetoelastic coupling coefficients is given for the first time.
2. Magnetostriction In our notation, the magnetoelastic energy is written
where jefii are the irreducible strain functions and kKpi are polynomials of degree 1 in the direction cosines oi of the magnetization. The strains and polynomials transform according to the row i of the irreducible representation fi of the symmetry group of the crystal [2,3], and the prefures j and k distinguish linearly independent bases among the strains and polynomials, respectively. The magnetoelastic coupling coefficients J&‘ can be obtained experimentally from the magnetostriction coef258
R.J. Potton, G.J. Keeler f Magnetostriction in terms of harmonics
259
ficients $$I;( using the relations
where lJ d are elastic stiffness constants. In practice the ‘,Ar are not measured directly, but from the dilation Al/l in a particular crystallographic direction having cosines &. In ref. [l] we related (for selected values of /3 and I< 4) the ‘,A? to the Fourier components of the variation of Al/Z with magnetization angle 6 in various symmetry planes. Recent measurements by Mishima et al. [4] suggest that the expansion of AZ/l to degree 4 in a may be inadequate, and we have extended the above analysis to degree 6 for both cubic and hexagonal crystals.‘The results are given in tables 1 and 2. Table 1 Fourier components
of magnetostriction
for mirror symmetry planes in cubic materials
a) (100) plane a2 [OOl] -
0
Wll
8
0
0
0
0
0
0
0
4
23
0
a2
a2 [lo01 a4
10011
a4
10111
a4
[lo01
: &(4@
0
0
-4
i
0
-3
-3
0
-2
$
0
0
0
0
4
O
0
0
0
0
2
1
0
0
-1
0
0
0
-2
0
4
0
2
-2
0
0
a6 [ool]
0
0
0
0
0
‘:
0
0
06 [011]
0
0
0
0
0
0
0
i76 [ 1001
0
0
0
0
0
0
0
0 4
4 ?4
0 -4
-5: ‘l$
0 -11 m
0
3
0
3 75 s -3
y
0
0
Tl
-54
-Y
*I2
“I
-8 0
b) ( lie)plane a2
8
[OOl] .
+4
aZ(110) a4 Wll
: &(4n)-Y
0
-& z 1
a4(110)
0
0
3
a6 [ool]
0
0
0
0
0’9’q
0
0
0
oy
a6(110)
0
/
-;
f33 is
o--Is 4
0
0
11 T
-1 4
The terms in the left hanQectors denote the particular Fourier (cosine) components, at, and the measuring direction. A constant numerical factor of 16 has been extracted from each matrix, and awkward numerical factors associated with a particulv Af have been collected into the right hand vectors (for 1= 2 and 4 these correspond to the factors absorbed intQ the constants k+fi defined in ref. [ 11. We a minus sign omitted from the second row of 1 in [ 11). The reference direction is to or for to the [ 1101 and liO] measuring directions can conveniently be as ( llO)), with f signs representing, respectively, these two directions.
260
R.J. Potton, G.J. Keel& /Magnetostriction
Table 2 Fourier components of magnetostriction
in terms of harmonics
for mirror symmetry planes in hexagonal materials
a) Basal plane
\ a2
[cl
-0
b/b1
i8
0 8
0
0
0’ 0
0
0-
0
0
0
0
0
0
0
0
0
0
0
~16
0
0
0
* 13
a6 [cl
0
0
0
16
0
0
0
a6 [a/b1
0
0
0
0
16
0
0
18
0
-8
0
0
-4
Y
-8
4 i7
r8
-;
Y
a2
a4 [cl
addbl
= 16(4n)-1/2
3
0
*y
11
b) b plane
-8
i4
0
0
14
0
0
0
0
14
rl
r2
0
0
0
0
0
0
0
0
0
0
3
-; -1
TA
3
0
3
*f 0
3 2
o-
0
f
*U
4
*& 0 i# 0 *i_
c) b plane, measuring direction at an angle p to the c direction (in plane)
The total magnetostriction
is given by
Sh2p +bzW1sin28 + b4[#] sin (~),=(~).cos%+($lu
4.e + b6[p]
sh~ 60
In the above table, the reference direction is taken to be the II direction throughout, and for measurements at an angle fi to the c direction, it is assumed that the magnetization is rotated from the (I direction towards the measuring direction (we note a minus’ sign omitted from the last row of table 2 in ref. [ 11). The a_and b measuring directions are given top,“ther, wits tit * signs representing respectively these two d$ec$rns. The 1Ar always appear in the combination (& + 2a @, ) for measurements in the c direction, and as (& - 4 &$) for measurements in the basal plane, and for convenience these ‘combinations have been called :A? and :A:, respectively. For the b plane data, c’a@F has been written to mean that combination appropriate to the particular measuring direction. The above data for the b plane also applies to the (I plane, with the fast sign again representing the in-plane measurement, prcvided the sign of every coefficient fork = 2 is reversed.
R.J. Potton, C.J. Keeler /Magnetostriction
in terms of harmonics
261
The above authors used an ad hoc expression for Al/l, and in fact for hexagonal crystals the dilation has never been properly expressed in terms of ‘,Af to degree 6. We have chosen the (un-normalized) basis functions of the 7 and E representations for hexagonal symmetry as (*p;” cos m@, p;” sin m#), the minus sign being necessary for m = 4 (7 representation) and m = 5 (E representation). Our choice of associating the minus sign, when it occurs, with the even function determines a choice of sign for *Al (in conformity with previous work [ 1,5]) and for ,A: and z Ai. The expressions for the dilation in both cubic and hexagonal crystals are given in the appendix.
3. The free energy at zero stress The magnetoelastic contribution to the free energy results in the anisotropy energy at zero stress being different to that at zero strain by terms such as Zi k#Kf’ k,,K,!A’ which are invariants of the symmetry group of the crystal. These appear as modifications to the zero strain anisotropy coefficients. Using our notation it can be shown
Table 3 The coefficients ,,,,,dF,.,(l,
p 1’ I” k”
k) for cubic crystals
d(4)
d(6)
d(8)
I” 1’ k’ 1
0 4 0
1
6
OO8
1
4
4
2x !@Ti 11 x 13
2
2
24
4x5 -v11
2 13
2 4
3x5 11
2 6
9
2
-22
1
7
4
4
2
2 _-2X3X5 llJ? 5 2x11
4
qezz 2x 13
11 v- 13
e261
4
2x3 13
11 v 5
2 4
2 6
r
2
2X3x5 11 3x13 v- 2
YVLZ 21 3 -ii v ?
2x3 v- 13
3 v- 11 x 17
Table 4 The ,jk.d&,(l, P
a
k) for hexagonal crystals
l’k’ I”k”
I”k” I’k’
d(2)
0
2
1
0
4
0
61
0
62
0
81
0
82
2
2
$ss
617
2
4
617
4x54 7x 11
2
61
2
62
4
4
4x54 7x 11
2x 243 7 x 11 x 13
2
2
-4J5 7
217
2
41
&/m
-4x3xsJ3 7x 11
2
42
2
61
2
62
41
41
41
42
42
42
2
2
2
4
44)
462)
4
1
1
3x5 11
2X5 11
2 v- 13
‘2x7 v- 13
Js
4x7 x 13 x 17
-2 8
4x5 iQE
v- 5x 7
17
2X5X49 11 x 13Ji7
-4&
2
11
2x7 v- 13 x 17
2 v- 2 5x11 16JJ 7x 11
-2x 27 7x 13
8x49 11 x 13fl 4
4 x 27 11 x 13
2J5 7
-8 ll@
2x7 11 x 13Jn
-S/7
24-5 7
2X5,/i% 7x 11
-I6
VZ
-2
62 4
d@, 2)
1
61 2
48,1)
1
7
~2
46,U
-4
4 2x 7x
174 11
2x 243 7 x 11 x 13
-2 llJr3
-16x 49 11 x 13fl
7 v- 5x
17
R.J. Potton, G.J. Keeler lhfagnetostriction
263
in terms of harmonics
that the corrections are (kKdo=o
-
(k“de=o
=-
k,k,,d;,,,
I,
(I, k) c
ii
ji’sp l&Bj! ,$B;,
.
This equation is a generalization of eq. (21b) of Callen and Callen [2]; the latter remains valid whenj is not redundant only if the stiffness constant is replaced by the compliance ji’sp. Tables 3 and 4 give the k,k,, dp,,,,,(1, k) for 1’ t I” < 8 for crystals of cubic and hexagonal symmetry. For cubic crystals (in which j is redundant) the k’k”dFt”(r) up t0 I= 6 have been giVen in table Iv Of ref. [2]. .Wenote, however, that &(4) and d&,(6) should read 40/(1 lfi) and (30/l 1) 40, respectively, instead of zero.
Acknowledgement We acknowledge the interest shown by Professor R.R. Birss in this paper an abbreviated version of which was presented in September 1979 at the Munich International Conference on Magnetism.
The form of AZ/Zfor cubic crystals is A+!
= 3‘4$ +fiA~(S[cY~fi~] + ZjfiA;(s[o#] + ;flA;(llcY:cY;o; t ;~~A;(lls[cYpp~] + &m + ie
- 5) +fiA:
L?S[Crjoj&flj]t zflA$(S[(Y4]
- $[cr~] - ;s[&3;]
t -“1)+ ?JfiAO
2(S[oiojozfli@j] - JS[cUiOj&/3j])
t @[L$] - E) - &S[c$] - 15S[ol4@] +5s[op]
IA: 2(1 lS[otoio$&&] *AZ 2(8S[oioi(o:
t 5S[&i’]
- 6S[oio,o$@rfli] + fS[oioiprfli])
+ oT)fiiflj] - ~S[CU~O~(O~ + of)2flifli]),
where S[ ] denotes cyclic summation. The form of AZ/Zfor hexagonal crystals is &Al/l
= 5 ‘A; + fi
“A#
- 5)
- 5) t w{i
‘A; t & ‘A;(& - $>(o;
- 4)
- 5)
264
R.J. Potton, G.J. Keeler / Magnetostriction in terms of harmonics
References [l] [2] [3] [4]
R.R. Birss, G.J. Keeler, P. Pearson and R.J. Potton, J. Phys. El1 (1978) 928. E.R. Callen and H.B. Callen, Phys. Rev. 129 (1963) 578. E.R. Callen and H.B. Callen, Phys. Rev. Al39 (1965)‘455. A. Mkhuma, H. Fujii and T. Okamoto, J. Phys Sot. Japan 40 (1976) 962. [ 51W.P. Mason, Phys. Rev. 96 (1954) 302.
MAGNETOSTRICTION IN TERMS OF HEXAGONAL AND CUBIC HARMONICS UP TO DEGREE 6
R.J. POTTON and G.J. REELER Department of Pure and Applied Physics, Universityof Salford, Salford iUS 4WT, Lancashire, UK Received 27 November 1979
Tables are given which ahow the magnetostiiction constants jk~$e and magnetoelastic coupling coefficientsJ&& to be evaluated from the Fourier coefficients of Al/l in various crystallographic planes. The difference between anisotropy coefficients at zero strain and zero stress is given for hexagonal crystals for the fist time in terms of the ‘8. At all stages the indices which indicate the symmetry type of a function (n, f) are clearly distingukhed from those which do not (i, k, 0.
1. Introduction
There are clear advantages in expressing magnetostriction in terms of coefficients associated with the irreducible representations of the symmetry group of the particular crystal concerned. However, the different multiplicities of bases of an irreducible representation found among strain functions and symmetrized polynomials have not prevlously been distinguished clearly, and indeed some confusion has occurred through multiplicity being ascribed to representations rather than to their bases. In a previous paper [l] we developed a notation which separates those indices which indicate the symmetry type of a function k, i) from those which do not (j, k, 0 and in which separate indices j and k are used (if necessary) for strains and polynomials, respectively, to distinguish between independent bases of the same irreducible representation. We have now produced tables, in terms of this notation, which allow the magnetostriction coefficients ‘,Af (for I sf 6) and magnetoelastic coupling coefficients J>f to be evaluated from the Fourier coefficients of the dependence of linear strain Al/Zon magnetization direction in various crystallographic planes. The magnetoelastic energy also results in a difference between anisotropy coefficients at zero strain and zero stress, and the difference in terms of the hexagonal magnetoelastic coupling coefficients is given for the first time.
2. Magnetostriction In our notation, the magnetoelastic energy is written
where jefii are the irreducible strain functions and kKpi are polynomials of degree 1 in the direction cosines oi of the magnetization. The strains and polynomials transform according to the row i of the irreducible representation fi of the symmetry group of the crystal [2,3], and the prefures j and k distinguish linearly independent bases among the strains and polynomials, respectively. The magnetoelastic coupling coefficients J&‘ can be obtained experimentally from the magnetostriction coef258
R.J. Potton, G.J. Keeler f Magnetostriction in terms of harmonics
259
ficients $$I;( using the relations
where lJ d are elastic stiffness constants. In practice the ‘,Ar are not measured directly, but from the dilation Al/l in a particular crystallographic direction having cosines &. In ref. [l] we related (for selected values of /3 and I< 4) the ‘,A? to the Fourier components of the variation of Al/Z with magnetization angle 6 in various symmetry planes. Recent measurements by Mishima et al. [4] suggest that the expansion of AZ/l to degree 4 in a may be inadequate, and we have extended the above analysis to degree 6 for both cubic and hexagonal crystals.‘The results are given in tables 1 and 2. Table 1 Fourier components
of magnetostriction
for mirror symmetry planes in cubic materials
a) (100) plane a2 [OOl] -
0
Wll
8
0
0
0
0
0
0
0
4
23
0
a2
a2 [lo01 a4
10011
a4
10111
a4
[lo01
: &(4@
0
0
-4
i
0
-3
-3
0
-2
$
0
0
0
0
4
O
0
0
0
0
2
1
0
0
-1
0
0
0
-2
0
4
0
2
-2
0
0
a6 [ool]
0
0
0
0
0
‘:
0
0
06 [011]
0
0
0
0
0
0
0
i76 [ 1001
0
0
0
0
0
0
0
0 4
4 ?4
0 -4
-5: ‘l$
0 -11 m
0
3
0
3 75 s -3
y
0
0
Tl
-54
-Y
*I2
“I
-8 0
b) ( lie)plane a2
8
[OOl] .
+4
aZ(110) a4 Wll
: &(4n)-Y
0
-& z 1
a4(110)
0
0
3
a6 [ool]
0
0
0
0
0’9’q
0
0
0
oy
a6(110)
0
/
-;
f33 is
o--Is 4
0
0
11 T
-1 4
The terms in the left hanQectors denote the particular Fourier (cosine) components, at, and the measuring direction. A constant numerical factor of 16 has been extracted from each matrix, and awkward numerical factors associated with a particulv Af have been collected into the right hand vectors (for 1= 2 and 4 these correspond to the factors absorbed intQ the constants k+fi defined in ref. [ 11. We a minus sign omitted from the second row of 1 in [ 11). The reference direction is to or for to the [ 1101 and liO] measuring directions can conveniently be as ( llO)), with f signs representing, respectively, these two directions.
260
R.J. Potton, G.J. Keel& /Magnetostriction
Table 2 Fourier components of magnetostriction
in terms of harmonics
for mirror symmetry planes in hexagonal materials
a) Basal plane
\ a2
[cl
-0
b/b1
i8
0 8
0
0
0’ 0
0
0-
0
0
0
0
0
0
0
0
0
0
0
~16
0
0
0
* 13
a6 [cl
0
0
0
16
0
0
0
a6 [a/b1
0
0
0
0
16
0
0
18
0
-8
0
0
-4
Y
-8
4 i7
r8
-;
Y
a2
a4 [cl
addbl
= 16(4n)-1/2
3
0
*y
11
b) b plane
-8
i4
0
0
14
0
0
0
0
14
rl
r2
0
0
0
0
0
0
0
0
0
0
3
-; -1
TA
3
0
3
*f 0
3 2
o-
0
f
*U
4
*& 0 i# 0 *i_
c) b plane, measuring direction at an angle p to the c direction (in plane)
The total magnetostriction
is given by
Sh2p +bzW1sin28 + b4[#] sin (~),=(~).cos%+($lu
4.e + b6[p]
sh~ 60
In the above table, the reference direction is taken to be the II direction throughout, and for measurements at an angle fi to the c direction, it is assumed that the magnetization is rotated from the (I direction towards the measuring direction (we note a minus’ sign omitted from the last row of table 2 in ref. [ 11). The a_and b measuring directions are given top,“ther, wits tit * signs representing respectively these two d$ec$rns. The 1Ar always appear in the combination (& + 2a @, ) for measurements in the c direction, and as (& - 4 &$) for measurements in the basal plane, and for convenience these ‘combinations have been called :A? and :A:, respectively. For the b plane data, c’a@F has been written to mean that combination appropriate to the particular measuring direction. The above data for the b plane also applies to the (I plane, with the fast sign again representing the in-plane measurement, prcvided the sign of every coefficient fork = 2 is reversed.
R.J. Potton, C.J. Keeler /Magnetostriction
in terms of harmonics
261
The above authors used an ad hoc expression for Al/l, and in fact for hexagonal crystals the dilation has never been properly expressed in terms of ‘,Af to degree 6. We have chosen the (un-normalized) basis functions of the 7 and E representations for hexagonal symmetry as (*p;” cos m@, p;” sin m#), the minus sign being necessary for m = 4 (7 representation) and m = 5 (E representation). Our choice of associating the minus sign, when it occurs, with the even function determines a choice of sign for *Al (in conformity with previous work [ 1,5]) and for ,A: and z Ai. The expressions for the dilation in both cubic and hexagonal crystals are given in the appendix.
3. The free energy at zero stress The magnetoelastic contribution to the free energy results in the anisotropy energy at zero stress being different to that at zero strain by terms such as Zi k#Kf’ k,,K,!A’ which are invariants of the symmetry group of the crystal. These appear as modifications to the zero strain anisotropy coefficients. Using our notation it can be shown
Table 3 The coefficients ,,,,,dF,.,(l,
p 1’ I” k”
k) for cubic crystals
d(4)
d(6)
d(8)
I” 1’ k’ 1
0 4 0
1
6
OO8
1
4
4
2x !@Ti 11 x 13
2
2
24
4x5 -v11
2 13
2 4
3x5 11
2 6
9
2
-22
1
7
4
4
2
2 _-2X3X5 llJ? 5 2x11
4
qezz 2x 13
11 v- 13
e261
4
2x3 13
11 v 5
2 4
2 6
r
2
2X3x5 11 3x13 v- 2
YVLZ 21 3 -ii v ?
2x3 v- 13
3 v- 11 x 17
Table 4 The ,jk.d&,(l, P
a
k) for hexagonal crystals
l’k’ I”k”
I”k” I’k’
d(2)
0
2
1
0
4
0
61
0
62
0
81
0
82
2
2
$ss
617
2
4
617
4x54 7x 11
2
61
2
62
4
4
4x54 7x 11
2x 243 7 x 11 x 13
2
2
-4J5 7
217
2
41
&/m
-4x3xsJ3 7x 11
2
42
2
61
2
62
41
41
41
42
42
42
2
2
2
4
44)
462)
4
1
1
3x5 11
2X5 11
2 v- 13
‘2x7 v- 13
Js
4x7 x 13 x 17
-2 8
4x5 iQE
v- 5x 7
17
2X5X49 11 x 13Ji7
-4&
2
11
2x7 v- 13 x 17
2 v- 2 5x11 16JJ 7x 11
-2x 27 7x 13
8x49 11 x 13fl 4
4 x 27 11 x 13
2J5 7
-8 ll@
2x7 11 x 13Jn
-S/7
24-5 7
2X5,/i% 7x 11
-I6
VZ
-2
62 4
d@, 2)
1
61 2
48,1)
1
7
~2
46,U
-4
4 2x 7x
174 11
2x 243 7 x 11 x 13
-2 llJr3
-16x 49 11 x 13fl
7 v- 5x
17
R.J. Potton, G.J. Keeler lhfagnetostriction
263
in terms of harmonics
that the corrections are (kKdo=o
-
(k“de=o
=-
k,k,,d;,,,
I,
(I, k) c
ii
ji’sp l&Bj! ,$B;,
.
This equation is a generalization of eq. (21b) of Callen and Callen [2]; the latter remains valid whenj is not redundant only if the stiffness constant is replaced by the compliance ji’sp. Tables 3 and 4 give the k,k,, dp,,,,,(1, k) for 1’ t I” < 8 for crystals of cubic and hexagonal symmetry. For cubic crystals (in which j is redundant) the k’k”dFt”(r) up t0 I= 6 have been giVen in table Iv Of ref. [2]. .Wenote, however, that &(4) and d&,(6) should read 40/(1 lfi) and (30/l 1) 40, respectively, instead of zero.
Acknowledgement We acknowledge the interest shown by Professor R.R. Birss in this paper an abbreviated version of which was presented in September 1979 at the Munich International Conference on Magnetism.
The form of AZ/Zfor cubic crystals is A+!
= 3‘4$ +fiA~(S[cY~fi~] + ZjfiA;(s[o#] + ;flA;(llcY:cY;o; t ;~~A;(lls[cYpp~] + &m + ie
- 5) +fiA:
L?S[Crjoj&flj]t zflA$(S[(Y4]
- $[cr~] - ;s[&3;]
t -“1)+ ?JfiAO
2(S[oiojozfli@j] - JS[cUiOj&/3j])
t @[L$] - E) - &S[c$] - 15S[ol4@] +5s[op]
IA: 2(1 lS[otoio$&&] *AZ 2(8S[oioi(o:
t 5S[&i’]
- 6S[oio,o$@rfli] + fS[oioiprfli])
+ oT)fiiflj] - ~S[CU~O~(O~ + of)2flifli]),
where S[ ] denotes cyclic summation. The form of AZ/Zfor hexagonal crystals is &Al/l
= 5 ‘A; + fi
“A#
- 5)
- 5) t w{i
‘A; t & ‘A;(& - $>(o;
- 4)
- 5)
264
R.J. Potton, G.J. Keeler / Magnetostriction in terms of harmonics
References [l] [2] [3] [4]
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