- Email: [email protected]
Interpretation of the magnetostriction of rare earth metals single crystals in the paramagnetic region
J. Phys, Chem. Solids, 1972,Vol.33. pp. 1005-1015. PergamonPress. Printedin Great Britain
INTERPRETATION OF THE MAGNETOSTRICTION OF RARE EARTH METALS SINGLE CRYSTALS IN THE PARAMAGNETIC REGION P. BOUTRON Laboratoire d'Electrostatique et de Physique du M6tal C.N.R.S., Cedex n° 166, 38-G renoble-G are, France (Received 21 J t m e 1971 )
A b s t r a c t - W e determine expressions for the magnetostriction of a magnetic crystal above the ordering temperature in terms of the exchange coefficients and of the crystalline potential parameters. By interpreting the basal plane magnetostriction results we deduce the corresponding crystalline potential parameter which is compared with the value calculated from the point charge model. We obtain the derivatives of exchange vs. the crystal parameters: the anisotropy of exchange interaction appears to be important. INTRODUCTION
RARE earths single crystals have an important thermal expansion anomaly in the ordered region and the crystal is distorted by a magnetic field; these properties can also be observed in the paramagnetic region. The magnetostriction of heavy rare earth single crystals has been measured in the ordered and paramagnetic regions by the strain gauges method[I-6] and with piezoelectric pickups [7.8]. The anomalous thermal expansion has been measured using X-ray diffraction [9-11 ]. The thermal variation of the magnetostriction constants is described approximately by the theoretical expression 15l~_( ~ - J (M/gj P.BJ) ), where 15/z is the normalised Bessel function and .L,¢'-~ the inverse Langevin function [ 12]. The total energy is the sum of the elastic, exchange and anisotropy energy and of the energy due to the external magnetic field. The crystal distorts in order to minimize this energy. Hence the magnetostriction can be expressed as a function of the derivatives of the exchange and crystalline field parameters with respect to the crystal parameters[13]. When the crystalline field is known the variation of exchange with the lattice parameters can be evaluated from the variation of the Nrel temperatures with stress. Knowing the
strain dependence of exchange and anisotropy energies it has been possible to give a quantitative interpretation of the anomalous thermal expansion of gadolinium, terbium and dysprosium [ 13]. By contrast magnetostriction measurements had not received an interpretation in the paramagnetic region. In a previous paper, we have established the expression for the paramagnetic susceptibility of an anisotropic crystal with a general symmetry[14, 15] when J or S is a good quantum number. Using this general expression, we will give (Section 1) the paramagnetic susceptibility of a distorted crystal. Since the derivatives of the susceptibility with respect to the strains are related to the values of the latter by the equilibrium conditions, the experimental values of the strain are functions of the derivatives of exchange and crystalline field with respect to the lattice parameters. In Section 2, we establish a relation between the parameter u~, and ( n y - n x ) which characterize the basal plane anisotropy energy of the distorted crystal, and we compare the experimental value oftt~Cz with its value in the point charge model. We calculate (Section 3) the derivatives of exchange with respect to the crystal parameters for the cases of isotropic and anisotropic exchange.
1005
JPCS Vol.33 No. 5--D
1006
P. B O U T R O N
In the first case the derivatives o f n x and n~ are distinct; this fact confirms that the exchange is anisotropic, in agreement with the study of the paramagnetic anisotropy in a high magnetic field [ 16].
where M is the magnetization[17]. In the paramagnetic region, in a weak magnetic field, the susceptibility is a constant independent of field strength, for a given shape of the crystal, and the magnetic free energy is
1. MAGNETOSTRICTION AND SUSCEPTIBILITY OF A DISTORTED CRYSTAL IN THE PARAMAGNETIC REGION
~,,, = --½xnH 2.
(a) Crystal equilibrium The spectroscopic free energy ~- of a magnetic crystal is the sum of the elastic and magnetic free energies, the latter containing a term due to strain, called the magnetoelastic energy. Let 0 be an atom located at the origin, and let c be the lattice parameter of a rare earth crystal in the direction of the crystalline 6-fold axis, which we call Oz. Let a be the distance between the atom at 0 and one of its nearest neighbours in the basal plane, and 0x the axis containing these two atoms. Let 0y be the direction perpendicular to 0x and 0z and such that the basis 0xyz is right handed in the hexagonal crystal, and let B be the nearest neighbour of 0 in the direction 0y. We call b the distance 0B, and we call 0y the direction of 0B in the strained crystal. When there is only a rhombohedric deformation, the directions 0x, 0y, 0z are orthogonal, and b = a~¢~-. The elastic free energy of a rare earth single crystal can be written:
(2)
The susceptibility Xn : M . H / H 2 depends on a, b, c, on the distortions of the crystal in a field, and on the direction of the field with respect to the deformed crystal. When the crystal is in a uniform magnetic field, the torque produced by appropriate shearing stresses maintains the crystal in equilibrium, and shearing stresses do not change the values of a, b or c. In the absence of other strains, the values of a, b, c corresponding to the equilibrium of the crystal at a given temperature and magnetic field are obtained by minimizing the free energy with resp= ct to these parameters:
c11eaa(H) + c12eob(H) + c,3ecc(H) 1 a~n(eij
O.O. ~ ) H z
(3a)
c,~_e.o (H ) + Cl,etjb (H ) + c13ecc (H ) 1 =-~OXn(eij=O.O.~o)H2
(3b)
ebb 2
2
~ e = ½cH ( e.~ + ebb ) + ½c33e~c+ clle.~ebb cv~ea, (H) + c13ebb (H) + c33ecc ( H )
+ c,3(eo~ec~+ ebbecc) + (CH -- C12)e~o + 2c44 (e~,c + e~,,).
(1)
I
-5OXn(e~j=O.O.~o)H2
(3c)
ecc
The co are the elastic moduli. The quantities e,,a, ebb, ecc are the relative variations of the parameters a, b, c of the crystal; eob, e0c, eoo are the distortions: xOy = (7r/2) --2eob, and c3. In the absence of stress and magnetic interactions, eoa = eb~ = ec~ = 0 and the basis Oxyz is orthogonal. When the shape and the temperature of the crystal are fixed, the variation of with the magnetic field H is given by d~- = - - M . dH for axes linked to the crystal, and
to first order in the strains; 0, ~o are the polar angles of M with respect to the axes of the unstrained crystal. (b) Susceptibility o f a strained cl~ystal If we make the assumption that the molecular field is given as a tensorial function of magnetization by the relation I-I,, = hM, the magnetic susceptibility at the temperature T
MAGNETOSTRICTION
OF
along one of the coordinate axis is, for a rhombohedral deformation [ 14.18]:
RARE
EARTH
1 = 1 [ T-n*(ab c)
1007
In the point charge model, UOz =
(2J -- 1 ) (2J + 3 ) tto ] 5k J (4)
METALS
~~2o~.1(r2 ) ~ 3 z i 2 - r i 2
u~ = - 3-~e-~ a , ( r ~) .
.
~
(8a)
~x?-y? ri 5
(8b)
i
to 0th order when regarded as an expansion in where Z is a coefficient of which the value is 3 1/T, the index i meaning x, y or z, and the n~ in the absence of screening, e the electronic being the components of the tensor n along the charge, (r z) the average value of the square of diagonal. C is the Curie constant, k the Boltz- the distance between a 4 f electron and the mann constant, J the angular momentum of centre of the ion, and where x~, y~ and z~ are the the ground state of the free ion. The crystal- coordinates of the ith ion and r~ its distance line potential V can be expanded in the opera- from the origin; a.~ is a coefficient tabulated in tor equivalents 0~'~~ of the tesseral harmonics Hutchings [ 19]. in the ith direction[19]. The term u°~ is the When the crystal is ordered ferromagneticoefficient of the operator 0°i(J)= 3J~--J cally, in the classical approximation the ( J + 1). The contribution of the other coeffi- exchange energy is E~x=--½M~M. In this cients to the susceptibility is zero in this order. approximation, when there is no external In the absence of strain, field, the magnetic energy of the crystal is the sum of exchange and anisotropy energies. The ugx=~.= , o --~U2z. [14] (5) magnetic free energy of the strained crystal is When the basal plane is strained, if the basis Oxyz remains orthogonal, to 2nd order in the
operator equivalents the crystalline potential is V=
[3j,2--J(J+
l ) ] u ° . + (J.~2--J~7)u2C.
. [-2n~-- nx-- ny n= nML 1
•
2
1
+ aM2 -- tiM2 (nx -- n~,) + nxyaMflM 4
(6) + nuzflMYM + nzzYMaM] M 2
Let us identify the average values of V obtained from equation (6) and from the expressions deduced from (6) by cyclic permutation of indices x, y and z on the state vector IJ ) corresponding to the eigenvalue J of operator J~. We obtain the relations u% = - ½ u L + ~ 1[12C 2,
(7a)
ug,~
(7b)
2c - - lu° 2 2Z - - -~u 2 2Z
and the additional relations obtained by cyclic permutations of x, y, z. More general/y, if 0x, 0y and 0z are symmetry axes of a magnetic crystal, where either J or S is a good quantum number, the coefficients u°i and u~ corresponding to the symmetry axes are related by these relations.
2c 2 --~,~,2 ) +~#o, (TM~_~)+ V~(~, _~ V~OzaM~M --2s --ls -+ V2zflM~/M-1VlCz~tMOL M
(9)
to 2nd order in the spherical harmonics of the magnetization direction expressed here in terms of the direction cosines of the magnetization a,., flat, TM with n = (nx + n . + n.)/3. The magnetoelastic free energy can be written: -- Eme = Bl~'°e '~'l + Bea'°e a'2 + Bl"'2e "'l (TM2-½) !~ a,2na +,-,2 ~ ' 2 (TM2-~,) -f- BY'2elY½ (0~4 2 - - f~M 2) -4- BY'2e2YOtMflM
+ B ~'1el~fla~tyg+
B~'2e2CyMOtM
( 1 O)
P. B O U T R O N
1008
in terms of the various strain modes e ~z corresponding to the symmetry Cab. These modes are characterized by the indices a, y, e, which are defined in Ref. [20]. Identifying the magnetoelastic free energy with the variation of the magnetic free energy, we showed[13] that
when c/a = V ~ , i.e. for the perfect hexagonal close packed structure. From the equation (11) we have the result that u% is proportional to
[.Ou~( a, b = a~/3, c)
The coefficient u~ is zero for b = a~/3; u~ is proportional to
Oeaa
"]b,c
= [Ou°~(a, b = a~v/~, c).] Oebb
a,c
On (a, b_= a'v~, c)]
aeaa
c
a3\2a
2 b
(11)
(u% is proportional to I~o). When the exchange is anisotropic, we have established in the same way that
I
1(!%
and
[Ou~C~(a, b = oV/3, c).] 0 eaa
b,c
= -- [ Ou~ ( a' b = a~v/3' c
(16)
J~,c =I On(a'b=a'v/~'c)
..~
(12)
[On=(a, b = aV'3, c) aeaa lb.C
aebb
a,c
= [an"( o' b = eaaaN/3' c
(14)
and the relation analogous to equation (12) f o r ?/z-
Let u°z ( a, c) be given by u% ( a, b = a v e , c) then
Ou~ (a, c) = 2[ Ou~(a' b = a~v/3, c) ae,, ffe~, ]b,c" (15) We also have the analogous relation for n. The numerical calculation shows that, as in the face centered cubic lattice, u2° is zero
(c) F,'operties of the strained crystal If we know the derivatives of exchange and crystalline field parameters with respect to a, b and c, we can calculate the strain of the crystal as a function of the applied magnetic field by using the equations (3). We notice that the reciprocal susceptibilities and their derivatives are field independent. Hence when the field is applied along one of the axes 0x, 0y, 0z, the strain varies as the square of the corresponding magnetization in the molecular field approximation. This property could have been anticipated since the strain varies as 15/2(&a - l ( M / g J ~ J ))[l 2]. Experimentally, strain varies approximately as the square of magnetization above the ordering temperature [2]. The identities
a x x _ aX, Oea. aena
aXx
ax,
Oeoa Oeaa
axe_ Ox, Oecc
OX~ -- axz Oeaa OeOb
Oecc (17)
result from equations (11)-(14). For an unstrained crystal, the susceptibility Xn for the
MAGNETOSTRICTION OF RARE EARTH METALS
field in a general direction is related to the susceptibilities corresponding to a magnetic field parallel to 0x, 0y or 0z by [ 14] Xtl = OQI2Xx+ flnZX.~J+ yH2Xz
(18)
where an, fin, Tn are the direction cosines of the magnetic field. We deduce the angular variation of the strain to first order: e.. (H) + ebb(H) = (e..~ + e~oz) (1 --yn 2)
+ (e..~+eo~)yn z e . . ( H ) - - el~t~( H )
=
(e..~ -- e~)
(or." - -
(19)
flu2) (20)
ec~(H) = ec~(1 - - y n 2) + ecc.Y. 2
(21)
where the indices x, y and z are related to the strain corresponding to a field parallel to Or, 0y, or Oz. This relation between strain components is valid for a given field intensity. The strain components due to a field parallel to one of the coordinate axes are related by: eaax=
ebbv
1009
hi "'2 = e . . z - - k ( e o . ~ + eb~z)
(27)
kla'°=½(eaax+eaau+eaaz)
(28)
h2~'2 = e c ~ - ecc~
(29)
~k2ff,0 =
1 e ~ + 2eccx). ~(
(30)
It has been established experimentally that the sign of the quantity h("2--½h v'2= e ~ - - e ~ o ~ changes at 225 K [2]. This is not anomalous, since e..z varies as X~2H2 and e..~ as xx2H z and since X~ and Xx vary differently with temperature (equation (4)). 2. UNIAXIAL ANISOTROPY AND ANISOTROPY INDUCED BY A RHOMBHOEDRIC DISTORTION
Numerical calculation of the factors in equation (8) gives: x ' 2--y'2 ~ - l'36b/a~a3~/3
(31)
ri 5
l c_~ ~v~ 2 ~ 2a 2 b 3z,. ~ - r ; = - - 4 - 1 2 a3
(32)
enay=ebbx
(22) eaaz = ebbz
eccx=ecc,.
At low temperatures, the strain in an arbitrary direction is often expressed in terms of spherical harmonics of the magnetization direction [211. So eoo(H) + e~,t,(H) = 2h,"'°+ 2h,"'Z(TM"-½) (23) e.. (H) -- ebb (H) = h ~'' (OLM2 -- tim"~) e~c(H) = ,.2 x ~.0~x .,2(., --,.,_, ~ r i 2- - k ) ,
(24)
•, c _
(,2)
b/a-C
tg2z - - a2zZ. ( r2) F W
a3
uOz = aO~Z (i.2)r,,.
az
and
(25)
If in the paramagnetic region, we use the same expressions but with the direction cosines of the magnetic field, the coefficients of these polynomials are related to the coefficients of equations (19), (20) and (21) by: h ~'2 = e.. z..-- eb~
with an accuracy better than 3 per cent. We note that the value of the first coefficient is approximately one third of the second. Let us write
(26)
for an unstrained crystal, where (r2)vw is the value of (r'-') calculated by Freeman and Watson[22], 2c
b~=~ The quantities
and
c°z= agz_ c/a--a3
a2z,2C aO, b~ and c°z are given
1010
P. B O U T R O N
Table 1 for terbium, dysprosium, holmium and erbium; az*~and a% are of the same magnitude but opposite in sign. (a) P a r a m a g n e t i c region Using (22), we deduce from equation (3) and (4) that u~ and n~--n~ are related, in the paramagnetic region, to the strain induced by a field of intensity H by 5kC
u~-~ ( 2 J - - 1 ) ( 2 J + 3 ) ( n y - n ~ ) =
where the strain is not strictly proportional to the square of magnetization. Hence, we have used the measurements made at the highest temperatures, which are 280 K for Tb, 240 K for Dy, 141 K for Ho and 120 K for Er. The elastic moduli of gadolinium [23] and dysprosium [24] are known. We have obtained those of the other rare earths by assuming a linear variation with the atomic number. We have used for these calculations the values of the elasticity coefficients of Gd, and Dy at 298 K: their temperature variation is weak in the paramagnetic region. We have defined U2C -- ,,2~ +
2z---2z
[eaau(H) -- eaax(H) ] (cn -- c,2) X
Xx2
C
(2J-- 1 ) ( 2 J + 3 ) H2 10k
The deformation of the basal plane in the presence of a magnetic field has been measured in the paramagnetic region for the cases of terbium [ 1,21 ], dysprosium [2], holmium [3] and erbium [4]. The measurements have been performed at temperatures which are not much higher than the ordering temperature,
5kC
( 2 J - - 1 ) ( 2 J + 3 ) (n~
We have evaluated U ~ (Table 2). By analogy with the expression for u2z, 2c we have defined the coefficient z~ by U2~ = . 2 ~ 2 ~
(b/a -- "V'3) a3
z"2Zu2Z
(b) A b s o l u t e zero The magnetostriction can also be expressed
T a b l e 1.
(r2)F,,(~, 2)
Tb
Dy
Ho
Er
8.2115
0"203
0.195
0' 186 22 3~ x 52 x'7
1
aj a~°en 10-14 cgs/at/A 3 a22c en 10-14 cgs/at/,~ 3
9xll
nx).
1
32x5x7
1
--2x32x52
--5.06
--3.06
- 1.03
1.12
5-01
3-03
1.02
-- 1- I 1
a (Ik at 273 K)
3.607
3"594
3.582
3.562
c (,~ at 273 K)
5.70
5.652
5.626
5"598
bz2C(K) k
7.75
4-75
1.61
-- 2.45
-~(K)
0.414
0"288
0-102
--0.1 ] 1
- ~ (Al6onard in K, at 273 K)
1.08
0-695
0.181
-- 0.348
Z ~ corr. to u2° £r 1Fw para.
2.60
2-42
1.77
3.14
MAGNETOSTRICT1ON OF RARE EARTH METALS
1011
Table 2.
1
u'.-',~
-ib__v~(K)
Tb
Dy
Ho
Er
6.8
5-1
1"5
--1 "54
0'88
1.07
0-93
t3
-2C g,2z
h y'-°(0 K)
8.5X 10-3(a) 8.5x I0 -:~ 8.7 x 10-:~(b)
1 u.',y(1)
2.5x 10-3
--5-4x 10-3 -3.66
7.9
5'2
1.4
1.02
1 I ~""
0.87
--11 X 10-3*
10 X 10-3
---(K)
t_b_v~
0"86
a -t2c
zz(l)
( e . . - et,b) (0 K) (RX)
a.u" - -
fl.u z
--1
i U'.,_F(H) --(K)
I
10.0
kb_v~
2-06
6-1
a t2C zz~ (II)
1.29
1"28
(a) Du Plessis, (b) Rhyne and Legvold. *Value obtained by extrapolation.
in terms of u~g and n ~ - n~ at absolute zero [13]. In the ordered state, the anisotropy energy can be considered as a perturbation compared to the exchange energy. From equation (6), we deduce the classical expression of crystalline potential for the same type of deformation V ---- aI2° /. 2 _ ½ ) + O2~r_ 2 -2 2 Z t. y M • 2z k~M
/3,u2)
1
(
b/a -- V 3 =
1
u~ 4
nu--nxga21xBzJ) 4 J-- ½
c H - - c v , kT.2(0K)
2,,/-3 g (g-½) (34)
1 Cl~--c,., ( e a , - - e o b ) ( M , OK) 2V3 J (J -- ½) (aM 2 -- tiM-)
with lP.oz(OK) , o _ = 2J(J--~)U2z
(35)
-- "C l 2C V'.,~(OK) = J (J--~)u2z.
(36)
and
By minimizing the free energy we obtain[ 13] ( e,,.
The parameters u.~g and (n.--nx) are related by:
--
et,i,) (M) = --
2 (aM" -- fl,,fl) C;m - - C12
X [ OI2'~c4- M " 0 (t&--_nx)] LOe,,, 4 Oeaa Jb,c
(37)
(38)
We note that the relative contribution to anisotropy and magnetostriction of n.,j--nx compared to u~g is 10J ( J + 1 ) / 3 ( 2 J + 3 ) times greater in the paramagnetic region than at absolute zero. Hence the thermal variation of exchange and one ion anisotropy are different in the ferromagnetic region. The parameter X~'2 has been determined at low temperatures from strain gauge measurements. The deformation was measured for crystals in which the magnetic moments are made parallel to each other and the basal plane with
1012
P. B O U T R O N
a magnetic field. The value of k ~'~ has been extrapolated to absolute zero[I-4]. We have set UI2C
~,2C -a- n ~ l - -
•,~ = - 2 z - -
4
nz
J ga~IZn~J-- ½
and Ut2c ~ 2Z
~t2C~2C "<, 2 z U 2 z
b/a - N/3
z~ and z'~ are nearly the same. The single ion contribution dominates, and the coefficient Z ( ( r 2 ) / ( r 2 ) ~ . w ) is included between 1 and 2. It is interesting to compare the anisotropy due to a distorsion and the uniaxial anisotropy. The latter is characterized in the absence of a distorsion by the two parameters u°~ and n,--n~. The anisotropy of the paramagnetic susceptibility equation (4) can be written:
a 3
1 From the measurements of k r'2 we have obtained a value of U'~g(I) (Table 2). The values of tl,~c 2c are nearly equal for terbium, 2~ and U 2z dysprosium and holmium. This could have been anticipated, since for these three rare earths, X~.2 varies as J~12(.L,~-~(M/gjtxnJ)). For erbium, U.~gand U'~g do not agree; and it is not surprising, since h ~." does not vary as I512(Z-~(M/gj~nJ))[4]. The reason may be that the two ion contribution dominates. But there is some uncertainty regarding the magnetic structure of erbium in the magnetic fields corresponding to the measurements of Rhyne and Legvold. The structure may be complex and so we cannot draw any certain conclusion. Terbium and dysprosium are ferromagnetic at low temperatures. In the ferromagnetic region, e~, -- ebb has been measured by X-Rays on small single crystals[9, 10], in. zero magnetic field. It can be shown that in these small crystals Weiss domains are all parallel or antiparallel to each other and e ~ - eob is related to U'~g by [38]. We have given Table 2 the second value U'~,~ (II), obtained by X-rays, of U'~g for terbium and dysprosium. These values obtained by the two methods are nearly the same.
X~
1 _3(2J--I)(2.I+3) iOkC
Xx
UO" --
(39)
with UO = .o z
10kC 3(2,I-1)(2J+3)
(n~-n~).
(40)
From the measurement of the anisotropy of the paramagnetic susceptibility[15], we find, in the case of dysprosium U.°,_:/k = 0.695 K. The paramagnetic susceptibility of dysprosium has been measured in a high magnetic field[16]. We have deduced the ratio n J n x from these measurements in the tensorial molecular field model, taking into account the parasitic effect of magnetostriction[16]. The error introduced by using the molecular field approximation decreases with temperature. At 280 K, the highest temperature where measurements have been performed, the ratio n J n x is !.35. The values of n~ and n~ are also related by I o (~n~+~n.~.)C = '0 , : ,~ - ".~v~,x "
(41)
where O,,z and 01,x are the paramagnetic Curie temperatures in the 0z and 0x directions respectively. Let us assume that the relative error introduced by using the molecular field approximation is the same for n~ and n.,., and (c) D i s c u s s i o n thus the uncertainty on the ratio nz/n.,, is U~g and U '~g are different functions of u~g smaller. The correct values of nz and nx are and n~j- n~: we could evaluate u.~~ , and n, - n~ in the ratio 1-35 and are solutions of equation by a system of two equations with two un- (41). They are nz = 12-55 and n x = 9 - 3 in qc knowns. However, U ~ and U '~gare not known cgs e.m.u, units/at.gr., so n z - n ~ = 3.25, with sufficient precision. We note that in the and u°z/k = 1.3 K for dysprosium, which gives case of terbium and dysprosium the values of Z((r'-')/(r-~)~.,w) = 4.5. We notice that the ex-
MAGNETOSTRICTION OF RARE EARTH METALS change anisotropy has a higher contribution to uniaxial anisotropy than to basal plane anisotropy. The value of Z((r")/(r2)r.,) is much higher for u°, than for lt;_z. "~ If we regard the ratio (r2)/r.. as a constant the screening coefficient Z/3 is much bigger for tt°z than tl.2z,2cthough they are both second order parameters. This fact could perhaps be explained in a model more accurate than the point charge model, i.e. the conduction electrons are not completely free and can contribute to the anisotropy. Other possible sources of the disagreement could be: contributions from higher order terms in the basal plane anisotropy of the strained crystal at low temperatures, biquadratic exchange or single ion anisotropy arising from anisotropic exchange.
1013
Table 3. Tb I Oxx xJ Oe,, I ax~ x~--"oe,,,, t Ox~. x~.~0e,.~
Dy
16
--19
--16
--52
83
73
1 OX=
x..'-'Oe,,, I Ox~ x:~Oe,.,.
Ho
Er.
--55
-1.5
124
42
The above valuesare in cgs units/at.gr.
OXZoe- Xz"-C[0 (nzC)Oe (2J-1 3)5k ) (2J +
(44)
3. V A R I A T I O N OF E X C H A N G E W I T H L A T T I C E PARAMETER FOR DYSPROSIUM
The magnetostriction of dysprosium has been measured in the paramagnetic region in fields both parallel and perpendicular to the c-axis[2]. Above the ordering temperature measurements have been performed only for a field parallel to the b-axis of terbium[l] and parallel to the c-axis of erbium[4]. For holmium the experimental data are insufficient for analysis [3]. Using equation (3), we have obtained the derivatives of the susceptibility with respect to strain (Table 3). Knowing these derivatives, we have obtained the intrinsic variation of the susceptibility with field and the exchange anisotropy [16]. From equation (4) giving the susceptibility of a strained crystal, using equations ( l l )-(15), we obtain the relations
1/axx
Ox,.',
~ t0e,,, + ~ )
"[1 a(n.,.+n.,,)(a.b,c) = ~-~ -2 C Oe ....
-~ ( 2 J - - l ) ( 2 J + - ox... x,,- FO,,,,c, ae,.,, C L ~e,.,.
3) au°z ae ....
-I ( 2 J - - 1 ) ( 2 J + 3 )
I Ok
(42)
o.~/.°.1 Oec,. J
(43)
Ott°~]Oe J
where e is e,,, or e,.,.. We know the exchange anisotropy n~--nx and the crystalline field parameter u~Jk for dysprosium. When b = aN/3, u% varies as (l/a :3) ( c / a - x / - ~ ) , hence we can determine the derivatives of u°~. Thus we obtain the derivatives of exchange using equations (42), (43). (44):
l CO(n.,.+n,,) ( a . b . c ~ = 2 3e,,,, '
884
C On,.. (a. b. c) = + 1896
Oeee
(45)
anz C~ (a. b. c) = - - 12 01lz C ~e,.,. ( a . b . c )
=+28
in K/atom. We have also evaluated the derivatives of exchange assuming isotropic exchange. If the results obtained for the derivatives of n~. and n~ are very different, this difference is a supplementary evidence of anisotropic exchange. For, if the derivatives of nx and n,
1014
P. B O U T R O N
were very different it would be very improbable that n~ and nz would have the same values in the unstrained crystal. Using an isotropic exchange model, the derivatives of n~ and nz are
1 c O ( n ~ + n . ) (a, b, c) = - - 7 0 6 2 Oeaa C an~ (a, b, c) = 1492 Oecc
(46)
C Onz (a, b, c) ----- 368 C 0n~ (a, b. c) = 836 Oecc in K/atom and are very different. We interpret this as strong evidence for anisotropic exchange. We note that the derivatives of nx have also been obtained from the variation of Nrel temperature with pressure assuming isotropic exchange [ 13]. The values obtained are:
1 C 3 ( naea, x + n ,-) ( a , b , c ) = - 2 6 9
results. Paramagnetic magnetostriction alone gives the derivatives of nz and nz. Indeed the interpretation of the variation of the Nrel temperature with stress gives only the derivatives of exchange corresponding to the easy magnetization direction. In the ordered state, the forces required to restrain the sample when a magnetic field is applied perpendicular to an easy axis can lead to large stresses and the magnetostriction can be difficult to interpret. We have found that the ratio of the derivatives of nx or nz with respect to ecc to the derivatives with respect to e,, are near --2. This is in agreement with the theoretical calculations of exchange anisotropy in the free electron model. Finally, the above method of interpretation of paramagnetic magnetostriction is very general and can be applied to all substances where J or S is a good quantum number. In the case of dysprosium, more precise experimental results would be desirable permitting a more exact comparison with theory. For the other rare earths more complete results will be necessary for their interpretation. Acknowledgernents-The author is indebted to D. Bloch, J. L. F r r o n and Prof. G. E. Everett for a critical reading of the manuscript and for helpful discussions.
and 012x
C~
(a, b, c) = 1488
(n is related to the quantity 1 of [13] by n = (252/gT ~,~J2)l). 4. CONCLUSION
The interpretation of the paramagnetic magnetostriction of rare earth single crystals has given supplementary information regarding the parameters characterizing exchange and the crystalline field. We have obtained the dysprosium exchange derivatives for the magnetization both parallel and perpendicular to the c-axis. By performing the calculations assuming isotropic exchange, we have shown that it is in fact anisotropic, in agreement with the high f i e l d paramagnetic susceptibility
REFERENCES 1. R H Y N E J. J. and L E G V O L D S., Phys. Rev. 138, A507 (1965). 2. C L A R K A. E., D E S A V A G E B. F. and B O Z O R T H R. M., Phys. R ev. 138, A 216 (1965). 3. R H Y N E J. J., L E G V O L D S. and R O D I N E E. T., Phys. Rev. 154, 266 (1967). 4. R H Y N E J. J. and L E G V O L D S., Phys. Rev. 140, A 2143 (1965). 5. L E G V O L D S., A L S T A D J. and R H Y N E J. J., Phys. Rev., Lett. 10, 509 (1963). 6. B O Z O R T H R. M. and W A K I Y A M A T., J. phys. Soc.Japan 18, 97 (1963). 7. B E L O V K. P., L E V I T I N R. Z. and P O N O M A R E V B. K.,J. appl. Phys. 39, 3285 (1968). 8. BELOV K. P., L E V I T I N R. Z. and P O N O M A R E V B. K., Soviet Phys. JETP 24, I 101 (1967). 9. D A R N E L L F.J.,Phys. Rev. 132, 1098 (1963). 10. D A R N E L L F. J. and M O O R E E. P.,J. appl. Phys. 34, 1337 (1963). 11. D A R N E L L F. J., Phys. Rev. 130, 1825 (1963). 12. C A L L E N E. R. and C A L L E N H. B., Phys. Rev. 129, 578 (1963).
M A G N E T O S T R I C T I O N OF RARE EARTH METALS 13. BARTHOL1N H., BEILLE J., BLOCH D., BOUTRON P. and FI~RON J. L., 42, 1679 (1971). 14. BOUTRON P.,J. de Phys. 30,413 (1969). 15. ALI~ONARD R., BOUTRON P. and BLOCH D., J. Phys. Chem. Solids 30, 2277 (1969). 16. BOUTRON P., FI~RON J. L., H U G G. and MORIN P., J. de Phys. 32, suppl~.ment au No. 2-3, C1-368 (1971). 17. BOUTRON P.,Ann. Phys. 3, 359 (1968). 18. BOUTRON P., Colloque Intern. du C.N.R.S. sur les Elements des Terres Rares p. 197 (1969).
1015
19. HUTCH1NGS M. T., Solid State Phys. 16, 227 (1964). 20. COOPER B. R., Solid State Phys. 21, 393 (1968). 21. Du PLESSIS P. de V., Phil Mag. 18, 145 (1963). 22. FREEMAN A. J. and WATSON R. E., Phys. Rev. 127, 2058 (1962). 23. FISHER E. S. and DEVER D., Proc. 6th Rare Earth Conf., Gatlinburg, p. 522 (1967). 24. FISHER E. S. and DEVER D., Trans. Metall. Soc. A IME 239, 48 (1967), and private communication.
INTERPRETATION OF THE MAGNETOSTRICTION OF RARE EARTH METALS SINGLE CRYSTALS IN THE PARAMAGNETIC REGION P. BOUTRON Laboratoire d'Electrostatique et de Physique du M6tal C.N.R.S., Cedex n° 166, 38-G renoble-G are, France (Received 21 J t m e 1971 )
A b s t r a c t - W e determine expressions for the magnetostriction of a magnetic crystal above the ordering temperature in terms of the exchange coefficients and of the crystalline potential parameters. By interpreting the basal plane magnetostriction results we deduce the corresponding crystalline potential parameter which is compared with the value calculated from the point charge model. We obtain the derivatives of exchange vs. the crystal parameters: the anisotropy of exchange interaction appears to be important. INTRODUCTION
RARE earths single crystals have an important thermal expansion anomaly in the ordered region and the crystal is distorted by a magnetic field; these properties can also be observed in the paramagnetic region. The magnetostriction of heavy rare earth single crystals has been measured in the ordered and paramagnetic regions by the strain gauges method[I-6] and with piezoelectric pickups [7.8]. The anomalous thermal expansion has been measured using X-ray diffraction [9-11 ]. The thermal variation of the magnetostriction constants is described approximately by the theoretical expression 15l~_( ~ - J (M/gj P.BJ) ), where 15/z is the normalised Bessel function and .L,¢'-~ the inverse Langevin function [ 12]. The total energy is the sum of the elastic, exchange and anisotropy energy and of the energy due to the external magnetic field. The crystal distorts in order to minimize this energy. Hence the magnetostriction can be expressed as a function of the derivatives of the exchange and crystalline field parameters with respect to the crystal parameters[13]. When the crystalline field is known the variation of exchange with the lattice parameters can be evaluated from the variation of the Nrel temperatures with stress. Knowing the
strain dependence of exchange and anisotropy energies it has been possible to give a quantitative interpretation of the anomalous thermal expansion of gadolinium, terbium and dysprosium [ 13]. By contrast magnetostriction measurements had not received an interpretation in the paramagnetic region. In a previous paper, we have established the expression for the paramagnetic susceptibility of an anisotropic crystal with a general symmetry[14, 15] when J or S is a good quantum number. Using this general expression, we will give (Section 1) the paramagnetic susceptibility of a distorted crystal. Since the derivatives of the susceptibility with respect to the strains are related to the values of the latter by the equilibrium conditions, the experimental values of the strain are functions of the derivatives of exchange and crystalline field with respect to the lattice parameters. In Section 2, we establish a relation between the parameter u~, and ( n y - n x ) which characterize the basal plane anisotropy energy of the distorted crystal, and we compare the experimental value oftt~Cz with its value in the point charge model. We calculate (Section 3) the derivatives of exchange with respect to the crystal parameters for the cases of isotropic and anisotropic exchange.
1005
JPCS Vol.33 No. 5--D
1006
P. B O U T R O N
In the first case the derivatives o f n x and n~ are distinct; this fact confirms that the exchange is anisotropic, in agreement with the study of the paramagnetic anisotropy in a high magnetic field [ 16].
where M is the magnetization[17]. In the paramagnetic region, in a weak magnetic field, the susceptibility is a constant independent of field strength, for a given shape of the crystal, and the magnetic free energy is
1. MAGNETOSTRICTION AND SUSCEPTIBILITY OF A DISTORTED CRYSTAL IN THE PARAMAGNETIC REGION
~,,, = --½xnH 2.
(a) Crystal equilibrium The spectroscopic free energy ~- of a magnetic crystal is the sum of the elastic and magnetic free energies, the latter containing a term due to strain, called the magnetoelastic energy. Let 0 be an atom located at the origin, and let c be the lattice parameter of a rare earth crystal in the direction of the crystalline 6-fold axis, which we call Oz. Let a be the distance between the atom at 0 and one of its nearest neighbours in the basal plane, and 0x the axis containing these two atoms. Let 0y be the direction perpendicular to 0x and 0z and such that the basis 0xyz is right handed in the hexagonal crystal, and let B be the nearest neighbour of 0 in the direction 0y. We call b the distance 0B, and we call 0y the direction of 0B in the strained crystal. When there is only a rhombohedric deformation, the directions 0x, 0y, 0z are orthogonal, and b = a~¢~-. The elastic free energy of a rare earth single crystal can be written:
(2)
The susceptibility Xn : M . H / H 2 depends on a, b, c, on the distortions of the crystal in a field, and on the direction of the field with respect to the deformed crystal. When the crystal is in a uniform magnetic field, the torque produced by appropriate shearing stresses maintains the crystal in equilibrium, and shearing stresses do not change the values of a, b or c. In the absence of other strains, the values of a, b, c corresponding to the equilibrium of the crystal at a given temperature and magnetic field are obtained by minimizing the free energy with resp= ct to these parameters:
c11eaa(H) + c12eob(H) + c,3ecc(H) 1 a~n(eij
O.O. ~ ) H z
(3a)
c,~_e.o (H ) + Cl,etjb (H ) + c13ecc (H ) 1 =-~OXn(eij=O.O.~o)H2
(3b)
ebb 2
2
~ e = ½cH ( e.~ + ebb ) + ½c33e~c+ clle.~ebb cv~ea, (H) + c13ebb (H) + c33ecc ( H )
+ c,3(eo~ec~+ ebbecc) + (CH -- C12)e~o + 2c44 (e~,c + e~,,).
(1)
I
-5OXn(e~j=O.O.~o)H2
(3c)
ecc
The co are the elastic moduli. The quantities e,,a, ebb, ecc are the relative variations of the parameters a, b, c of the crystal; eob, e0c, eoo are the distortions: xOy = (7r/2) --2eob, and c3. In the absence of stress and magnetic interactions, eoa = eb~ = ec~ = 0 and the basis Oxyz is orthogonal. When the shape and the temperature of the crystal are fixed, the variation of with the magnetic field H is given by d~- = - - M . dH for axes linked to the crystal, and
to first order in the strains; 0, ~o are the polar angles of M with respect to the axes of the unstrained crystal. (b) Susceptibility o f a strained cl~ystal If we make the assumption that the molecular field is given as a tensorial function of magnetization by the relation I-I,, = hM, the magnetic susceptibility at the temperature T
MAGNETOSTRICTION
OF
along one of the coordinate axis is, for a rhombohedral deformation [ 14.18]:
RARE
EARTH
1 = 1 [ T-n*(ab c)
1007
In the point charge model, UOz =
(2J -- 1 ) (2J + 3 ) tto ] 5k J (4)
METALS
~~2o~.1(r2 ) ~ 3 z i 2 - r i 2
u~ = - 3-~e-~ a , ( r ~) .
.
~
(8a)
~x?-y? ri 5
(8b)
i
to 0th order when regarded as an expansion in where Z is a coefficient of which the value is 3 1/T, the index i meaning x, y or z, and the n~ in the absence of screening, e the electronic being the components of the tensor n along the charge, (r z) the average value of the square of diagonal. C is the Curie constant, k the Boltz- the distance between a 4 f electron and the mann constant, J the angular momentum of centre of the ion, and where x~, y~ and z~ are the the ground state of the free ion. The crystal- coordinates of the ith ion and r~ its distance line potential V can be expanded in the opera- from the origin; a.~ is a coefficient tabulated in tor equivalents 0~'~~ of the tesseral harmonics Hutchings [ 19]. in the ith direction[19]. The term u°~ is the When the crystal is ordered ferromagneticoefficient of the operator 0°i(J)= 3J~--J cally, in the classical approximation the ( J + 1). The contribution of the other coeffi- exchange energy is E~x=--½M~M. In this cients to the susceptibility is zero in this order. approximation, when there is no external In the absence of strain, field, the magnetic energy of the crystal is the sum of exchange and anisotropy energies. The ugx=~.= , o --~U2z. [14] (5) magnetic free energy of the strained crystal is When the basal plane is strained, if the basis Oxyz remains orthogonal, to 2nd order in the
operator equivalents the crystalline potential is V=
[3j,2--J(J+
l ) ] u ° . + (J.~2--J~7)u2C.
. [-2n~-- nx-- ny n= nML 1
•
2
1
+ aM2 -- tiM2 (nx -- n~,) + nxyaMflM 4
(6) + nuzflMYM + nzzYMaM] M 2
Let us identify the average values of V obtained from equation (6) and from the expressions deduced from (6) by cyclic permutation of indices x, y and z on the state vector IJ ) corresponding to the eigenvalue J of operator J~. We obtain the relations u% = - ½ u L + ~ 1[12C 2,
(7a)
ug,~
(7b)
2c - - lu° 2 2Z - - -~u 2 2Z
and the additional relations obtained by cyclic permutations of x, y, z. More general/y, if 0x, 0y and 0z are symmetry axes of a magnetic crystal, where either J or S is a good quantum number, the coefficients u°i and u~ corresponding to the symmetry axes are related by these relations.
2c 2 --~,~,2 ) +~#o, (TM~_~)+ V~(~, _~ V~OzaM~M --2s --ls -+ V2zflM~/M-1VlCz~tMOL M
(9)
to 2nd order in the spherical harmonics of the magnetization direction expressed here in terms of the direction cosines of the magnetization a,., flat, TM with n = (nx + n . + n.)/3. The magnetoelastic free energy can be written: -- Eme = Bl~'°e '~'l + Bea'°e a'2 + Bl"'2e "'l (TM2-½) !~ a,2na +,-,2 ~ ' 2 (TM2-~,) -f- BY'2elY½ (0~4 2 - - f~M 2) -4- BY'2e2YOtMflM
+ B ~'1el~fla~tyg+
B~'2e2CyMOtM
( 1 O)
P. B O U T R O N
1008
in terms of the various strain modes e ~z corresponding to the symmetry Cab. These modes are characterized by the indices a, y, e, which are defined in Ref. [20]. Identifying the magnetoelastic free energy with the variation of the magnetic free energy, we showed[13] that
when c/a = V ~ , i.e. for the perfect hexagonal close packed structure. From the equation (11) we have the result that u% is proportional to
[.Ou~( a, b = a~/3, c)
The coefficient u~ is zero for b = a~/3; u~ is proportional to
Oeaa
"]b,c
= [Ou°~(a, b = a~v/~, c).] Oebb
a,c
On (a, b_= a'v~, c)]
aeaa
c
a3\2a
2 b
(11)
(u% is proportional to I~o). When the exchange is anisotropic, we have established in the same way that
I
1(!%
and
[Ou~C~(a, b = oV/3, c).] 0 eaa
b,c
= -- [ Ou~ ( a' b = a~v/3' c
(16)
J~,c =I On(a'b=a'v/~'c)
..~
(12)
[On=(a, b = aV'3, c) aeaa lb.C
aebb
a,c
= [an"( o' b = eaaaN/3' c
(14)
and the relation analogous to equation (12) f o r ?/z-
Let u°z ( a, c) be given by u% ( a, b = a v e , c) then
Ou~ (a, c) = 2[ Ou~(a' b = a~v/3, c) ae,, ffe~, ]b,c" (15) We also have the analogous relation for n. The numerical calculation shows that, as in the face centered cubic lattice, u2° is zero
(c) F,'operties of the strained crystal If we know the derivatives of exchange and crystalline field parameters with respect to a, b and c, we can calculate the strain of the crystal as a function of the applied magnetic field by using the equations (3). We notice that the reciprocal susceptibilities and their derivatives are field independent. Hence when the field is applied along one of the axes 0x, 0y, 0z, the strain varies as the square of the corresponding magnetization in the molecular field approximation. This property could have been anticipated since the strain varies as 15/2(&a - l ( M / g J ~ J ))[l 2]. Experimentally, strain varies approximately as the square of magnetization above the ordering temperature [2]. The identities
a x x _ aX, Oea. aena
aXx
ax,
Oeoa Oeaa
axe_ Ox, Oecc
OX~ -- axz Oeaa OeOb
Oecc (17)
result from equations (11)-(14). For an unstrained crystal, the susceptibility Xn for the
MAGNETOSTRICTION OF RARE EARTH METALS
field in a general direction is related to the susceptibilities corresponding to a magnetic field parallel to 0x, 0y or 0z by [ 14] Xtl = OQI2Xx+ flnZX.~J+ yH2Xz
(18)
where an, fin, Tn are the direction cosines of the magnetic field. We deduce the angular variation of the strain to first order: e.. (H) + ebb(H) = (e..~ + e~oz) (1 --yn 2)
+ (e..~+eo~)yn z e . . ( H ) - - el~t~( H )
=
(e..~ -- e~)
(or." - -
(19)
flu2) (20)
ec~(H) = ec~(1 - - y n 2) + ecc.Y. 2
(21)
where the indices x, y and z are related to the strain corresponding to a field parallel to Or, 0y, or Oz. This relation between strain components is valid for a given field intensity. The strain components due to a field parallel to one of the coordinate axes are related by: eaax=
ebbv
1009
hi "'2 = e . . z - - k ( e o . ~ + eb~z)
(27)
kla'°=½(eaax+eaau+eaaz)
(28)
h2~'2 = e c ~ - ecc~
(29)
~k2ff,0 =
1 e ~ + 2eccx). ~(
(30)
It has been established experimentally that the sign of the quantity h("2--½h v'2= e ~ - - e ~ o ~ changes at 225 K [2]. This is not anomalous, since e..z varies as X~2H2 and e..~ as xx2H z and since X~ and Xx vary differently with temperature (equation (4)). 2. UNIAXIAL ANISOTROPY AND ANISOTROPY INDUCED BY A RHOMBHOEDRIC DISTORTION
Numerical calculation of the factors in equation (8) gives: x ' 2--y'2 ~ - l'36b/a~a3~/3
(31)
ri 5
l c_~ ~v~ 2 ~ 2a 2 b 3z,. ~ - r ; = - - 4 - 1 2 a3
(32)
enay=ebbx
(22) eaaz = ebbz
eccx=ecc,.
At low temperatures, the strain in an arbitrary direction is often expressed in terms of spherical harmonics of the magnetization direction [211. So eoo(H) + e~,t,(H) = 2h,"'°+ 2h,"'Z(TM"-½) (23) e.. (H) -- ebb (H) = h ~'' (OLM2 -- tim"~) e~c(H) = ,.2 x ~.0~x .,2(., --,.,_, ~ r i 2- - k ) ,
(24)
•, c _
(,2)
b/a-C
tg2z - - a2zZ. ( r2) F W
a3
uOz = aO~Z (i.2)r,,.
az
and
(25)
If in the paramagnetic region, we use the same expressions but with the direction cosines of the magnetic field, the coefficients of these polynomials are related to the coefficients of equations (19), (20) and (21) by: h ~'2 = e.. z..-- eb~
with an accuracy better than 3 per cent. We note that the value of the first coefficient is approximately one third of the second. Let us write
(26)
for an unstrained crystal, where (r2)vw is the value of (r'-') calculated by Freeman and Watson[22], 2c
b~=~ The quantities
and
c°z= agz_ c/a--a3
a2z,2C aO, b~ and c°z are given
1010
P. B O U T R O N
Table 1 for terbium, dysprosium, holmium and erbium; az*~and a% are of the same magnitude but opposite in sign. (a) P a r a m a g n e t i c region Using (22), we deduce from equation (3) and (4) that u~ and n~--n~ are related, in the paramagnetic region, to the strain induced by a field of intensity H by 5kC
u~-~ ( 2 J - - 1 ) ( 2 J + 3 ) ( n y - n ~ ) =
where the strain is not strictly proportional to the square of magnetization. Hence, we have used the measurements made at the highest temperatures, which are 280 K for Tb, 240 K for Dy, 141 K for Ho and 120 K for Er. The elastic moduli of gadolinium [23] and dysprosium [24] are known. We have obtained those of the other rare earths by assuming a linear variation with the atomic number. We have used for these calculations the values of the elasticity coefficients of Gd, and Dy at 298 K: their temperature variation is weak in the paramagnetic region. We have defined U2C -- ,,2~ +
2z---2z
[eaau(H) -- eaax(H) ] (cn -- c,2) X
Xx2
C
(2J-- 1 ) ( 2 J + 3 ) H2 10k
The deformation of the basal plane in the presence of a magnetic field has been measured in the paramagnetic region for the cases of terbium [ 1,21 ], dysprosium [2], holmium [3] and erbium [4]. The measurements have been performed at temperatures which are not much higher than the ordering temperature,
5kC
( 2 J - - 1 ) ( 2 J + 3 ) (n~
We have evaluated U ~ (Table 2). By analogy with the expression for u2z, 2c we have defined the coefficient z~ by U2~ = . 2 ~ 2 ~
(b/a -- "V'3) a3
z"2Zu2Z
(b) A b s o l u t e zero The magnetostriction can also be expressed
T a b l e 1.
(r2)F,,(~, 2)
Tb
Dy
Ho
Er
8.2115
0"203
0.195
0' 186 22 3~ x 52 x'7
1
aj a~°en 10-14 cgs/at/A 3 a22c en 10-14 cgs/at/,~ 3
9xll
nx).
1
32x5x7
1
--2x32x52
--5.06
--3.06
- 1.03
1.12
5-01
3-03
1.02
-- 1- I 1
a (Ik at 273 K)
3.607
3"594
3.582
3.562
c (,~ at 273 K)
5.70
5.652
5.626
5"598
bz2C(K) k
7.75
4-75
1.61
-- 2.45
-~(K)
0.414
0"288
0-102
--0.1 ] 1
- ~ (Al6onard in K, at 273 K)
1.08
0-695
0.181
-- 0.348
Z ~ corr. to u2° £r 1Fw para.
2.60
2-42
1.77
3.14
MAGNETOSTRICT1ON OF RARE EARTH METALS
1011
Table 2.
1
u'.-',~
-ib__v~(K)
Tb
Dy
Ho
Er
6.8
5-1
1"5
--1 "54
0'88
1.07
0-93
t3
-2C g,2z
h y'-°(0 K)
8.5X 10-3(a) 8.5x I0 -:~ 8.7 x 10-:~(b)
1 u.',y(1)
2.5x 10-3
--5-4x 10-3 -3.66
7.9
5'2
1.4
1.02
1 I ~""
0.87
--11 X 10-3*
10 X 10-3
---(K)
t_b_v~
0"86
a -t2c
zz(l)
( e . . - et,b) (0 K) (RX)
a.u" - -
fl.u z
--1
i U'.,_F(H) --(K)
I
10.0
kb_v~
2-06
6-1
a t2C zz~ (II)
1.29
1"28
(a) Du Plessis, (b) Rhyne and Legvold. *Value obtained by extrapolation.
in terms of u~g and n ~ - n~ at absolute zero [13]. In the ordered state, the anisotropy energy can be considered as a perturbation compared to the exchange energy. From equation (6), we deduce the classical expression of crystalline potential for the same type of deformation V ---- aI2° /. 2 _ ½ ) + O2~r_ 2 -2 2 Z t. y M • 2z k~M
/3,u2)
1
(
b/a -- V 3 =
1
u~ 4
nu--nxga21xBzJ) 4 J-- ½
c H - - c v , kT.2(0K)
2,,/-3 g (g-½) (34)
1 Cl~--c,., ( e a , - - e o b ) ( M , OK) 2V3 J (J -- ½) (aM 2 -- tiM-)
with lP.oz(OK) , o _ = 2J(J--~)U2z
(35)
-- "C l 2C V'.,~(OK) = J (J--~)u2z.
(36)
and
By minimizing the free energy we obtain[ 13] ( e,,.
The parameters u.~g and (n.--nx) are related by:
--
et,i,) (M) = --
2 (aM" -- fl,,fl) C;m - - C12
X [ OI2'~c4- M " 0 (t&--_nx)] LOe,,, 4 Oeaa Jb,c
(37)
(38)
We note that the relative contribution to anisotropy and magnetostriction of n.,j--nx compared to u~g is 10J ( J + 1 ) / 3 ( 2 J + 3 ) times greater in the paramagnetic region than at absolute zero. Hence the thermal variation of exchange and one ion anisotropy are different in the ferromagnetic region. The parameter X~'2 has been determined at low temperatures from strain gauge measurements. The deformation was measured for crystals in which the magnetic moments are made parallel to each other and the basal plane with
1012
P. B O U T R O N
a magnetic field. The value of k ~'~ has been extrapolated to absolute zero[I-4]. We have set UI2C
~,2C -a- n ~ l - -
•,~ = - 2 z - -
4
nz
J ga~IZn~J-- ½
and Ut2c ~ 2Z
~t2C~2C "<, 2 z U 2 z
b/a - N/3
z~ and z'~ are nearly the same. The single ion contribution dominates, and the coefficient Z ( ( r 2 ) / ( r 2 ) ~ . w ) is included between 1 and 2. It is interesting to compare the anisotropy due to a distorsion and the uniaxial anisotropy. The latter is characterized in the absence of a distorsion by the two parameters u°~ and n,--n~. The anisotropy of the paramagnetic susceptibility equation (4) can be written:
a 3
1 From the measurements of k r'2 we have obtained a value of U'~g(I) (Table 2). The values of tl,~c 2c are nearly equal for terbium, 2~ and U 2z dysprosium and holmium. This could have been anticipated, since for these three rare earths, X~.2 varies as J~12(.L,~-~(M/gjtxnJ)). For erbium, U.~gand U'~g do not agree; and it is not surprising, since h ~." does not vary as I512(Z-~(M/gj~nJ))[4]. The reason may be that the two ion contribution dominates. But there is some uncertainty regarding the magnetic structure of erbium in the magnetic fields corresponding to the measurements of Rhyne and Legvold. The structure may be complex and so we cannot draw any certain conclusion. Terbium and dysprosium are ferromagnetic at low temperatures. In the ferromagnetic region, e~, -- ebb has been measured by X-Rays on small single crystals[9, 10], in. zero magnetic field. It can be shown that in these small crystals Weiss domains are all parallel or antiparallel to each other and e ~ - eob is related to U'~g by [38]. We have given Table 2 the second value U'~,~ (II), obtained by X-rays, of U'~g for terbium and dysprosium. These values obtained by the two methods are nearly the same.
X~
1 _3(2J--I)(2.I+3) iOkC
Xx
UO" --
(39)
with UO = .o z
10kC 3(2,I-1)(2J+3)
(n~-n~).
(40)
From the measurement of the anisotropy of the paramagnetic susceptibility[15], we find, in the case of dysprosium U.°,_:/k = 0.695 K. The paramagnetic susceptibility of dysprosium has been measured in a high magnetic field[16]. We have deduced the ratio n J n x from these measurements in the tensorial molecular field model, taking into account the parasitic effect of magnetostriction[16]. The error introduced by using the molecular field approximation decreases with temperature. At 280 K, the highest temperature where measurements have been performed, the ratio n J n x is !.35. The values of n~ and n~ are also related by I o (~n~+~n.~.)C = '0 , : ,~ - ".~v~,x "
(41)
where O,,z and 01,x are the paramagnetic Curie temperatures in the 0z and 0x directions respectively. Let us assume that the relative error introduced by using the molecular field approximation is the same for n~ and n.,., and (c) D i s c u s s i o n thus the uncertainty on the ratio nz/n.,, is U~g and U '~g are different functions of u~g smaller. The correct values of nz and nx are and n~j- n~: we could evaluate u.~~ , and n, - n~ in the ratio 1-35 and are solutions of equation by a system of two equations with two un- (41). They are nz = 12-55 and n x = 9 - 3 in qc knowns. However, U ~ and U '~gare not known cgs e.m.u, units/at.gr., so n z - n ~ = 3.25, with sufficient precision. We note that in the and u°z/k = 1.3 K for dysprosium, which gives case of terbium and dysprosium the values of Z((r'-')/(r-~)~.,w) = 4.5. We notice that the ex-
MAGNETOSTRICTION OF RARE EARTH METALS change anisotropy has a higher contribution to uniaxial anisotropy than to basal plane anisotropy. The value of Z((r")/(r2)r.,) is much higher for u°, than for lt;_z. "~ If we regard the ratio (r2)/
1013
Table 3. Tb I Oxx xJ Oe,, I ax~ x~--"oe,,,, t Ox~. x~.~0e,.~
Dy
16
--19
--16
--52
83
73
1 OX=
x..'-'Oe,,, I Ox~ x:~Oe,.,.
Ho
Er.
--55
-1.5
124
42
The above valuesare in cgs units/at.gr.
OXZoe- Xz"-C[0 (nzC)Oe (2J-1 3)5k ) (2J +
(44)
3. V A R I A T I O N OF E X C H A N G E W I T H L A T T I C E PARAMETER FOR DYSPROSIUM
The magnetostriction of dysprosium has been measured in the paramagnetic region in fields both parallel and perpendicular to the c-axis[2]. Above the ordering temperature measurements have been performed only for a field parallel to the b-axis of terbium[l] and parallel to the c-axis of erbium[4]. For holmium the experimental data are insufficient for analysis [3]. Using equation (3), we have obtained the derivatives of the susceptibility with respect to strain (Table 3). Knowing these derivatives, we have obtained the intrinsic variation of the susceptibility with field and the exchange anisotropy [16]. From equation (4) giving the susceptibility of a strained crystal, using equations ( l l )-(15), we obtain the relations
1/axx
Ox,.',
~ t0e,,, + ~ )
"[1 a(n.,.+n.,,)(a.b,c) = ~-~ -2 C Oe ....
-~ ( 2 J - - l ) ( 2 J + - ox... x,,- FO,,,,c, ae,.,, C L ~e,.,.
3) au°z ae ....
-I ( 2 J - - 1 ) ( 2 J + 3 )
I Ok
(42)
o.~/.°.1 Oec,. J
(43)
Ott°~]Oe J
where e is e,,, or e,.,.. We know the exchange anisotropy n~--nx and the crystalline field parameter u~Jk for dysprosium. When b = aN/3, u% varies as (l/a :3) ( c / a - x / - ~ ) , hence we can determine the derivatives of u°~. Thus we obtain the derivatives of exchange using equations (42), (43). (44):
l CO(n.,.+n,,) ( a . b . c ~ = 2 3e,,,, '
884
C On,.. (a. b. c) = + 1896
Oeee
(45)
anz C~ (a. b. c) = - - 12 01lz C ~e,.,. ( a . b . c )
=+28
in K/atom. We have also evaluated the derivatives of exchange assuming isotropic exchange. If the results obtained for the derivatives of n~. and n~ are very different, this difference is a supplementary evidence of anisotropic exchange. For, if the derivatives of nx and n,
1014
P. B O U T R O N
were very different it would be very improbable that n~ and nz would have the same values in the unstrained crystal. Using an isotropic exchange model, the derivatives of n~ and nz are
1 c O ( n ~ + n . ) (a, b, c) = - - 7 0 6 2 Oeaa C an~ (a, b, c) = 1492 Oecc
(46)
C Onz (a, b, c) ----- 368 C 0n~ (a, b. c) = 836 Oecc in K/atom and are very different. We interpret this as strong evidence for anisotropic exchange. We note that the derivatives of nx have also been obtained from the variation of Nrel temperature with pressure assuming isotropic exchange [ 13]. The values obtained are:
1 C 3 ( naea, x + n ,-) ( a , b , c ) = - 2 6 9
results. Paramagnetic magnetostriction alone gives the derivatives of nz and nz. Indeed the interpretation of the variation of the Nrel temperature with stress gives only the derivatives of exchange corresponding to the easy magnetization direction. In the ordered state, the forces required to restrain the sample when a magnetic field is applied perpendicular to an easy axis can lead to large stresses and the magnetostriction can be difficult to interpret. We have found that the ratio of the derivatives of nx or nz with respect to ecc to the derivatives with respect to e,, are near --2. This is in agreement with the theoretical calculations of exchange anisotropy in the free electron model. Finally, the above method of interpretation of paramagnetic magnetostriction is very general and can be applied to all substances where J or S is a good quantum number. In the case of dysprosium, more precise experimental results would be desirable permitting a more exact comparison with theory. For the other rare earths more complete results will be necessary for their interpretation. Acknowledgernents-The author is indebted to D. Bloch, J. L. F r r o n and Prof. G. E. Everett for a critical reading of the manuscript and for helpful discussions.
and 012x
C~
(a, b, c) = 1488
(n is related to the quantity 1 of [13] by n = (252/gT ~,~J2)l). 4. CONCLUSION
The interpretation of the paramagnetic magnetostriction of rare earth single crystals has given supplementary information regarding the parameters characterizing exchange and the crystalline field. We have obtained the dysprosium exchange derivatives for the magnetization both parallel and perpendicular to the c-axis. By performing the calculations assuming isotropic exchange, we have shown that it is in fact anisotropic, in agreement with the high f i e l d paramagnetic susceptibility
REFERENCES 1. R H Y N E J. J. and L E G V O L D S., Phys. Rev. 138, A507 (1965). 2. C L A R K A. E., D E S A V A G E B. F. and B O Z O R T H R. M., Phys. R ev. 138, A 216 (1965). 3. R H Y N E J. J., L E G V O L D S. and R O D I N E E. T., Phys. Rev. 154, 266 (1967). 4. R H Y N E J. J. and L E G V O L D S., Phys. Rev. 140, A 2143 (1965). 5. L E G V O L D S., A L S T A D J. and R H Y N E J. J., Phys. Rev., Lett. 10, 509 (1963). 6. B O Z O R T H R. M. and W A K I Y A M A T., J. phys. Soc.Japan 18, 97 (1963). 7. B E L O V K. P., L E V I T I N R. Z. and P O N O M A R E V B. K.,J. appl. Phys. 39, 3285 (1968). 8. BELOV K. P., L E V I T I N R. Z. and P O N O M A R E V B. K., Soviet Phys. JETP 24, I 101 (1967). 9. D A R N E L L F.J.,Phys. Rev. 132, 1098 (1963). 10. D A R N E L L F. J. and M O O R E E. P.,J. appl. Phys. 34, 1337 (1963). 11. D A R N E L L F. J., Phys. Rev. 130, 1825 (1963). 12. C A L L E N E. R. and C A L L E N H. B., Phys. Rev. 129, 578 (1963).
M A G N E T O S T R I C T I O N OF RARE EARTH METALS 13. BARTHOL1N H., BEILLE J., BLOCH D., BOUTRON P. and FI~RON J. L., 42, 1679 (1971). 14. BOUTRON P.,J. de Phys. 30,413 (1969). 15. ALI~ONARD R., BOUTRON P. and BLOCH D., J. Phys. Chem. Solids 30, 2277 (1969). 16. BOUTRON P., FI~RON J. L., H U G G. and MORIN P., J. de Phys. 32, suppl~.ment au No. 2-3, C1-368 (1971). 17. BOUTRON P.,Ann. Phys. 3, 359 (1968). 18. BOUTRON P., Colloque Intern. du C.N.R.S. sur les Elements des Terres Rares p. 197 (1969).
1015
19. HUTCH1NGS M. T., Solid State Phys. 16, 227 (1964). 20. COOPER B. R., Solid State Phys. 21, 393 (1968). 21. Du PLESSIS P. de V., Phil Mag. 18, 145 (1963). 22. FREEMAN A. J. and WATSON R. E., Phys. Rev. 127, 2058 (1962). 23. FISHER E. S. and DEVER D., Proc. 6th Rare Earth Conf., Gatlinburg, p. 522 (1967). 24. FISHER E. S. and DEVER D., Trans. Metall. Soc. A IME 239, 48 (1967), and private communication.