The magnetocrystalline anisotropy energy of nickel

Physica 39 477-498

Franse, J. J. M. De Vries, G. 1968

THE MAGNETOCRYSTALLINE ANISOTROPY ENERGY OF NICKEL by J. J. M. FRANSE

and G. DE VRIES

Natuurkundig Laboratorium der Universiteit van Amsterdam, Nederland

synopsis Torque measurements on a nickel single crystal sphere show large contributions of higher order constants to the anisotropy energy at low temperatures; these contributions depend on purity. A description of the anisotropy energy with cubic harmonics is given and the first six constants in this description have been calculated at 296, 195, 77, 20 and 4.2 K.

1. Introduction. In this paper we describe measurements on the magnetocrystalline anisotropy energy of nickel at different temperatures between 1.5 K and 296 K. First of all we base our description on the usual expansion of this energy: EA

=

Ko

+

Kls

+

&p

+

K 3s2 +

K&

+ K5s3 + K6p2 + . . . . .

(1)

where s = CC$X~ + CL&; + CZ!~; and p = tz~c&~; al, a2 and a3 being the direction cosines of the magnetization with respect to the crystal axis and Kl, Kz etc. the anisotropy constants. These constants can be derived for instance from torque measurements by a least squares adjustment. At low temperatures, where many constants become important, the results for Kg are dependent on the number of constants used in the least squares adjustment. For this reason we present in section 6 our experimental results for the (100) and (110) planes in such a way that the contributions of higher order constants are clear, without an explicit calculation of the appropriate constants. The analysis of torque curves in the (111) plane is complicated at low temperatures, by large additional torques as well as by the complex combination of anisotropy constants which appears in the expression for the torque in this plane. Neglecting higher constants we can get, however, a good estimation of Ks and with the aid of this constant we make in section 7 an attempt to determine Kl, Kz and Ka from the combined torque curves in the (1 OO), (110) and (111) planes. Because it is not possible to give an accurate description of torque curves -

477 -

478

J. J. M. FRANSE

AND

G. DE VRIES

at low temperature with the anisotropy constants given in eq. (l), we shall discuss in section 9 an expansion of the anisotropy energy in cubic harmonics. From a theoretical point of view the use of cubic harmonics is sometimes preferable; it will be evident that the experimental results are better described by these functions, though even a description with six constants does not yield a satisfactory fit to the experimental torque curves at 4.2 I<. Some further experiments concerning the influence of purity and magnetic field on EA are given in section 10. 2. Experimental. Our starting material was Johnson & Matthey nickel (purity 99.999% Ni) from which a single crystal rod was grown by the Bridgman method. The surface of this rod showed regions with low angle boundaries parallel to the growing direction. X-ray exposures indicate that the overall spread in orientation is in the order of 0.5 degree. From this single crystal rod a sphere was machined; by repeatedly chemical etching and mechanical grinding the deformed surface layer was removed. Finally the sphere was polished with diamond powder (0.1 p) in a cone of copper, resulting in a diameter + = 13.37 + 0.002 mm. Fig. 1a

--outer tube

-inner

tube

(b)

Fig. 1. measurements

(a)

view of the apparatus

for torque

(b)

details

of the apparatus

of construction

down to 1.5 K.

and fixation

of the crystal.

THE

MAGNETOCRYSTALLINE

ANISOTROPY

ENERGY

OF NICKEL

479

gives a view of the apparatus for measuring torque curves down to 1.5 K ; torsion angles up to 600 seconds of arc are observed by autocollimation with an accuracy of better than 1”. The nickel sphere is fixed in a german silver tube, which is held under constant stress jnside another tube between two torsion wires, the upper wire being much more rigid than the lower one, see fig. lb. Calibration of the apparatus occurred at room temperature with known mechanical torques; at all measurements the upper torsion wire was kept at this temperature. The measurements were performed in an Oerlikon magnet (pole diameter 200 mm, pole distance 52 mm) at a field of 18100 Oe; the inhomogenity of the field was certainly less than 1 Oe/mm. With steps of 1 degree the magnet could be rotated around the apparatus over an angle of 180 degrees; this angle could be set within & of a degree. For the determination of the angle between the magnetic field and a reference direction in the crystal also the torsion angle was taken into account. 3. Contributions to the torque cuurves. When analysing torque curves we encounter torques of different origin arising from: a. the inner anisotropy of the crystal, b. non-perfect form or mounting of the crystal, c. magnetic inclusions in the apparatus. To avoid contributions of type-c beryllium bronze and german silver proved to be satisfactory for the essential parts; these torques were reduced to no more than 1 dynecm. Centering of the apparatus occurred within 1 mm, so we expect contributions of type b not to be larger than 200 dynecm/cc. Under influence of these torques the points of zero torque in the (100) and (110) planes are shifted at room temperature over about & degree, at lower temperatures no shift in these points could be observed. The anisotropy constants given in eq. (1) can be calculated from the torque sub a; for simple crystallographic planes, like the (100) and (110) planes we can derive directly from eq. (1) an expression for the torque: LA=__=

ah

ae

-KG'

-

K@'

+ .....

with for the (100) plane s’( 100) = sin 20 cos 28 and p’( 100) = 0 and for the (I 10) plane ~‘(110) = & sin 28 (1 + 3~0s 28) andp’(110) = ~‘(110) x +sin28, 8 being for both planes the angle between the direction of the magnetization and the [OOl] direction. Since X-ray exposures show a mosaic structure of our nickel crystal and the setting of the crystallographic planes is not perfect, we have to consider the influence of small deviations in orientation on torque curves.

J. J. M. FRANSE

480

AND G. DE VRIES

4. Influence of small deviations in orientation on torqzce curves. Suppose that n, the normal of the desired plane, does not coincide with the rotation axis n’ of the specimen in the magnetic field, but makes an angle 4 with it, see fig. 2.

Fig. 2. The vectors

Let p be an direction /3 is plane (n, n’); Between n, p,

n, p, q are transformed by the rotation $0 into n’, p’, q’.

unit vector in a reference direction of the desired plane. The determined by the intersection of the desired plane and the p and q are transformed by the rotation 4~ into p’ and q’. q and n’, p’, q’ the following relations exist:

n’=ncos+

+psin+cos/?

+ q sin + sin @,

p’=-nsin~cos~+p{l+(cos~-1)cos~~}+q(cos~-l)sin~cos~, q’=

-nsin$sinp+p(cos+-

When the magnetization for its orientation 01.

l)sinj3cosp

(3) +q{l+(cos~-

l)sin2j3}.

vector is lying in the plane (p’, q’) we can write a = p’ cos 0 + q’ sin 0

(4)

with 8 the angle between M and p’, and by substituting eq. (3) in eq. (4) we can determine a with respect to the crystallographic axis: a = n[-sin +

4 cos p cos 0 -

sin + sin /? sin 01 +

P[{l+ @OS+- 1)CO.+!?) cos 8 + (cos 4 - 1) sin p cos /3sin e] +

+q[{l

+(cos+-

1) sinsB}sinO+

(cos$-

l)sinjIcosBcose].

Making use of this expression for a, we shall calculate the torques for (p’, q’) planes, which have small deviations in orientation with simple crystallographic planes.

THE MAGNETOCRYSTALLINEANISOTROPYENERGY

481

OF NICKEL

(100) plane: The unit vectors n, p and Q are lying respectively [OlO] directions

in the [ 1001, [OOl] and

and we find for the direction cosines :

al = -sin

4 sin /I sin e -

sin 4 cos p cos 8,

as = (1 + (cos$ -

1) sir+)sinf3

~=(l

~)cos2~~c0se+(cos~-

+(cos$-

+ (cos+ -

For 4 is small this results in the following

1) sinpcos/3cose, l)sinpcos/Isin

e.

expression for the torque:

La = - Kr[sin 20 cos 20 + +s( -sin 20 cos 28 - cos 2/? sin 28 + + &sin2@cos28)

+ . . . . .] + . . . . .

The points of zero torque for the (p’, q’) plane occur for 0 N (-$2/4) sin 2/I and 7d/4 + (+2/2) cos 28 instead of 0 = 0 and z/4 in the (100) plane; if I$ < 2” the difference is within our experimental accuracy. (110) plane: We write for the unit vectors in this case: n=-..--.- l J2

[1101, p = [OOl]

and

q=--...- ’ J2

[iio]

and find for the torque: sin 20( 1 + 3 cos 20) + +s{$ sin 28 cos 40 -

LA = -Kl[& -

9) sin48 -

($cos2/9+

3 sin 28 cos 20 +

sin28) + . . . . . .] + ,....

and also for this torque there is experimentally in the (110) plane for 4 < 2”.

no difference with the torque

(111) plane: The unit vectors are given by: n=--.---

1

J3

The expression

[ill];

‘=

-![i lo] J2

and

q = +

[112].

for the torque becomes for small 4:

LA = -AK2

sin 68 -

C[sin p(sin 28 -

COS~(COS 28 + 2 cos 4e)] +

.....

2 sin 48) (5)

with C N + JZ $(KI + &Ks) neglecting higher order constants. Contributions proportional to 4 are not negligible even for small values of 4; at room temperature C will be comparable with &Ks for 4 = 2”. These additional torques disturb the symmetry of the torque curves in the (111) plane considerably, see for instance a torque curve given by Sato and Chandrasekharr).

482

J. J. M. FRANSE

For a description

of torque

AND G. DE VRIES

curves

in this plane we can make an ex-

pansion of the torque in a Fourier series : LA = -

C

Si sin ie -

x

i=2,4,6

According

to eq. (5) the following

and

the angle I# can be calculated

3

(b=

(6)

relations between the St and the Ci exist :

2ss = -S4 Moreover,

Cz cos ie.

i=2,4

2(Kl+

QK2)

2cs = cq.

(7)

from these coefficients:

1/s:+ c: 2

(8)

*

5. Determinatiolz of the direction of the magnetization. In this section we assume that the orientation of the observed plane is perfect. The free energy of a ferromagnetic specimen in a magnetic field can be expressed by : E(&, 6) = &

+ &(E,

6) -

M*H

+ &(N di)

M

with E and 6 given by fig. 3.

Fig. 3. The

direction

of a is given by 0 and E, these equilibrium conditions.

angles can be calculated

Values for E and S can be derived from the equilibrium -

aE

a&

= 0

and

aE

_

ae

=

from

conditions:

0.

Since the orientation of the magnetic field with respect to the crystallographic axis is known in most cases, we know the direction of M. For symmetric

THE

MAGNETOCRYSTALLINE

ANISOTROPY

planes the equilibrium

conditions

sin 6 = LAIMH. For non-symmetric

planes

observed

ENERGY

OF NICKEL

483

lead in the case of a sphere to E = 0 and

the direction

of M is not restricted

to the

plane and we can write:

a = n sin E + p cos 0 cos E + q sin 8 cos E. Taking

for n, p and q respectively J3 -J-,111], JJ2

[I lo]

and

5

[ii21

we find for the torque in the (111) plane: LA = -p with p=&Kz+&K4+ is small. The equilibrium and 6 sin 6 N LaIMH

sin 60 -

qJ2 cos 38 sin E + . . . . .

. . . . . and qzKl+iKz+$KS+ conditions lead now to the relations and

sin E N -(q,/2

sin 30)/3[MH

(10) . . . . for for small

E E

+ I)

with I = -K1 Substitution

+ ;Kz(l

+ $icos 60) -

of the expression

&K3(5

+ 4 cos 68) + . . . . .

for sin E in eq. (10) gives: sin 68 + . . . . .

(11)

The coefficient of sin 68 has a small dependence on 0; at the analysis of torque curves we shall neglect this dependence. A similar expression for the torque in the (111) plane has previously been given by Birss and Walliss). 6. Experimental results of the (100)and @IO) planes.We shall turn our attention now especially to the presence of higher order constants in eq. (1). If all constants except K1 are zero, we find from eq. (2) for the torque in TABLE Maxima

and minima

I in the torque

curve

of the (110) plane (103 erg/cc) T (in K) 296 195 77 20 4.2

1

LI

1

LII

( LIILII

33.7 145.5 496

14.4 - 61.0 -187

- 2.34 -2.38 - 2.66

670 682

-213 -214

-3.14 -3.19 *0.05

484

J. J. M. FRANSE

AND

G. DE VRIES

the (110) plane: LA = --k’l

‘a sin 2641 + 3 cos 28).

(12)

The function 2 sin 2f3(1 + cos 20) has two different extrema, LI and LII, respectively for 01 = 25.5” and 011 = 71.3”, with a ratio LI/LII= -2.67. In an experimental torque curve LA is given as a function of the angle

Fig. 4. The experimental

values for LA/S’ (1 lo), calculated in the (110) plane.

from torque measurements

THE

MAGNETOCRYSTALLINE

ANISOTROPY

between H and the [OOI] direction

ENERGY

and the positions

OF NICKEL

of the extrema

485 are

not the same as in eq. (12) ; the ratio of the extrema in this experimental curve remains, however, -2.67. Deviations in this ratio can easily be determined and give a direct indication for the presence of higher order constants in eq. (2). Our experimental results, given in table I, point out that the contributions of higher order constants to this ratio change sign at about 77 K. A similar result has been obtained by Joseph and Thorsena). Using eq. (2) we can write for the torque in the (1 IO) plane : LA = -s’(llO)[A1

+ Azsh2e

+ A3Sin48

f

. . . ..I

(13)

with between brackets a power series in sins 13 and the A$ being simple combinations of a limited number Ka : A1 = K1, A2

=

+K2

+

2&,

A3 = -$K3

+ $K4 + 3K5,

etc.

The e-dependence of the expression between brackets in eq. (13) is caused only by higher order constants. For each measuring point the quantity L~/s'(l 10) has been calculated and the results are given in fig. 4. Around the angles of zero torque the spreading in the points increases since every torque has been measured with the same absolute accuracy. It is possible to give a similar description as eq. (13) for the torque in the

Fig. 5. The experimental values for LA/S’ (loo), calculated from torque measurements in the (100) plane.

486

T. T. M. FRANSE AND G. DE VRIES

(100) plane : LA = --s’(100)[B~

+ &sin220

+ Basin428

+ . . . ..I

(14)

with Bi = Kr, Ba = *K3 etc. The results for this plane are given in fig. 5. At low temperatures the curves, given in figs. 4 and 5, are very complex; for the description of the (100) plane, for instance, at least 4 constants are needed at 4.2 K. The determination of these constants will be discussed in section 8. 7. Experimelztal results of the (III) plane. In interpreting the measurements of the (111) plane we have, as mentioned previously, to take into account for the following disturbing effects: a. errors in orientation giving rise to sin 20, sin 419, cos 28 and cos 40 terms; between the coefficients of these terms the relations (7) exist.

I O0

Fig. 6. Torque

I

v 11080

,,

x 14200

,,

0 18600

((

I

(111)

I

9o”

plane

K

20

I

I

g_

curves in the (111) plane at different magnetic of the [f lo] direction.

180’

fields for determination

THE

MAGNETOCRYSTALLINE

ANISOTROPY

ENERGY

OF NICKEL

487

b.

torques arising from asymmetric mounting are not negligible; they disturb the relations (7) especially at high temperature. c. the direction of the magnetization is not confined to the (111) plane; as a consequence the coefficient of sin 613is dependent on field strength. To make a least squares adjustment, for determining the coefficients in expression (6), we must know the position of the [i lo] direction. This direction cannot directly be derived from torque curves since the points of zero torque are shifted or have disappeared completely. The field dependence of the sin 60 coefficient, however, enables us to determine this direction within half a degree, by measuring torque curves at different field strengths preferentially at low temperature; these curves intersect in [I IO] and [T 121 directions, see fig. 6. At every temperature and every field strength a set of constants Sr and CZ has been determined. The coefficients Sz, Sq, Cs and Cq are independent of the magnetic field, change with temperature like KI, and have ratios in ac-

I

532

l

5.3

’

‘6

v

C*

I

0

I

5

I

10 (H-K,/M,)“(1650e1)

Fig. 7. The constants, describing a torque curve in the (111) plane, plotted (H - Kl/M)--1.

cordance to (7), see fig. 7. The coefficient .Sa has to be proportional to (H + r/M)-1 and by extrapolation to H --t co we can find Sr(= --T]‘8Kz . . . . .). These extrapolations are given in figs. 7 and 8 by taking -AK4 + --KI for Y. The slopes of the extrapolation lines must be equal to q2/3M; for 195 K and 77 K the experimental and calculated slopes are in good agreement, at lower temperatures the contributions of higher constants to r and especially to 4 become larger and the slopes can not exactly be calculated.

488

J. J. M.

FRANSE

J

AND

G. DE VRIES

”

J?0.15_

- 0.15

: k

* 0

o- - o-

_o-

- _---

_--o-

VP 0.10 _

- 0.10

/

5

0

10

4_

I I I /

/

0 I

20

K

-lo

I

t

I

( H- K,/MS)-’ ( 10’0 k;

Fig. 8. Determination

of ST in the (111) plane by graphical extrapolation; Se are plotted against (H - Kl/IM)-1.

values for

Except those at 20 K these measurements have all been performed at the same value of p and 4; using eq. (8) we found for the latter # = 1.3”. For some temperatures the measurements have been repeated on another (111) plane of the same sphere, 4 being 0.7” in this case; both values of 4 give the same results for Sg and Sr, see table II. Somewhere between 300 K

THE MAGNETOCRYSTALLINE

ANISOTROPY TABLE

ENERGY

489

OF NICKEL

II

Values for ST (in 10s erg/cc) at different deviations in orientation 4 1.3” 0.7”

I

296 K

195K

1

1.3 + 0.1

77 K

1

5.0 f 0.2

-4.4 -4.8

4.2 K

20 K

& 1.0 l 1.0

-23.0

-23.0 -22.0

& 2.0

& 2.0 It 2.0

and 80 K ST reaches a maximum value and at about 80 K Sr changes sign. A maximum for Sy in the same temperature region can also be deduced from measurements given by Hofmanna), but has not been found by Birss and Wallis 2). 8. Determination of anisotro$y constants from torque curves in the (IOO), @anes. We shall first discuss now two methods to calculate the anisotropy constants, given in eq. (1) from torque curves in the (100) and (110) planes. In the first place we can make a least squares adjustment to determine the quantities A ( and & in eqs. (13) and (14), from which the Kc can easily be calculated. We have not succeeded to find unique values for the At and Bi, since these values are strongly dependent on the number of constants used in the least squares adjustment. The difficulties we are finding to calculate these constants are caused by the small differences between the functions ~‘(110) sinsV3 in the (110) plane, see fig. 9. At a graphical extrapolation near the [OOI] direction, the higher (ITO) and (III)

TABLE

III

Values of constants (in 10s erg/cc) by graphical extrapolation of torque curves near the [OOl] direction calculated ( 10 0) plane

T (in K)

(110) plane

Bl 57f1 - 248 f - 845 & -1162 * -1217 f

296 195 77 20 4.2

2 10 10 10

from ST and Bs

Bs

A1

As

As

0 f 0.5 5&l - 8255 -155 -& 10 - 170 + 30

58kl - 248 & 2 -839 + 10 -1174 & 10 -1210 + 10

- 13&l - 57f5 -290 & 20 -390 & 50 -550 + 50

- 11 - 65 -286 -455 -475

TABLE

IV

The first three anisotropy constants (in 10s erg/cc) calculated in different ways -

T&K)

from AI,BI and SF

1 h-1

195 77 20 296 4.2

1

- 248 - 842 -1168 57 -1214

Ks -

1

90 a3 - 410 23 410

fromXq......Xfa Ks

(

- 10 -164 -310 0 -340

1

Kl

Ks

Ks

- 248 - a43 -1173 57 -1233

26 - 80 90 - 390 530

0 - 11 -160 - 350 -230

I 1

I 1

490

J. J. M. FRANSE

AND

G. DE VRIES

0.:

0.5

02

0.4

0.:

0.3

0.:

0.2

0.1

0.1

0.c

0.0

-0.1

0.1

-0.2

0.2

o-

ao-

DO

I/

Yu-

Fig. 9. Graphical representation of the functions, which determine the constants AI, A 2 etc. in the (110) plane.

THE

constants

MAGNETOCRYSTALLINE

ANISOTROPY

ENERGY

491

OF NICKEL

have in principle no influence on the determination

of the lower

ones. This second method has two disadvantages: a. in the neighbourhood of 8 = 0 there is a relatively large uncertainty the measuring points. b. we use information about only a limited part of the torque curves.

in

I

122_

do

o. __-**--

_---

0

_-o--O

_- ~_0.0-0-0

ooo

0

0 0 0

0

0 120_

t

0

0

0.0 2

0.04

Fig. 10. Graphical

determination

of the anisotropy

4.2 K from torque

measurements

in the (110) plane;

of the dashed line and the vertical

I

I

I

0.0 0

constants

sin’J

0.06

for the (110) plane

I at

A1 is given by the intersection axis for 6 = 0 and AZ by the slope of the dashed line.

By measuring the torque curve of the (110) plane around 8 = 0 in steps of 0.1 degree, we have made an examination for the usefulness of this method at 4.2 K, see fig. 10. From fig. 10 only the first two constants Ai and As can be determined with a fair accuracy; values for AI and AZ for other temperatures and values for B1 and Bs have been determined in the same manner from the measuring points given in figs. 4 and 5, see table III. Neglecting K4 and higher constants in Sr we can calculate values for A2 from Bs and Sy and in table II we compare them with the experimental values for A2 from the (110) plane. The agreement between both values permits us to give in table IV the first three anisotropy constants in the expression (1) ; K1 is the mean of A1 and B1, K2 = - 18s: and Ka = = 2B2. The description of the torque curves with three constants is at low temperature very unsatisfactory as may be clear from fig. 11, and for a better description we have to use another development of El. 9. Descriptiort of the magnetocrystalline anisotropy energy with cztbic harmonics. Sometimes it is convenient to give an expression of EA in terms

1. 1. M. FRANSE AND G. DE VRIES

492

_200..-. ewyirnent~,,,_$

x

>

900

Fig. 11. Description of LA/S’ (110) at 4.2 K with values for Kl, Kz and K3, given in table IV.

of spherical harmonics. Under special conditions relations between the constants in this expression and the magnetization are given by Zeners) and by Van Vleck6) : J/p+ I)/2 x1-s . (15) The results of magnetic anisotropy measurements are usually not given by the 3-1 and for that reason relation (15) is applied for cubic crystals to combinations of the K$: Kr + &Ks

and

+ . . . . . N 1M,1o

Ks + . . . . . N Mil.

(16)

For nickel it is known that the first relation (16) does not hold since the change with temperature of K1 is much too large in comparison with that of the magnetization. In this section we shall give a description of our experimental results with spherical harmonics, making use of the orthonormal set of cubic harmonics given by Mueller and Priestley 7) : EA =

Hl, the cubic harmonics, harmonics :

where

F x s:H;(6

(17)

4)

1

are symmetric

combinations

and

CEO= Yf.

with

of spherical

THE

MAGNETOCRYSTALLINE

ANISOTROPY

ENERGY

l

l-4

l

1=6

OF NICKEL

493

5

x 1=8

0

4

3

2

1

0

.l

.2

,3

4

30’

60’

y’

90’

Fig. 12. The functions dHi(e, n/4)/30, determining the anisotropy constants e (110) plane.

in the

494

J. J. M. FRANSE

AND

G. DE

VRIES

The coefficients u;~ are tabulated and the Hj form an orthonormal set over the unit sphere if the C lnz do so; summation over i is over degenerate sets like that which occurs for I = 12. The (100) and (110) planes are respectively given by I# = 0 and + = 7~14and for the torque in these planes we can write : aHi(f3, 0 or z/4)

-xzc;

L*=

i

ae

2

(18)

1

The differences between the functions aHi(O, n/4)/&3 for different 1 turn out to be large, see fig. 12, and the determination of the constants in eq. (18) is by consequence not strongly dependent on the number of constants used in a least squares adjustment. In fact we can now find the first five constants in eq. (18) from measurements in the (110) plane only, and it is possible to TABLE The first five anisotropy from measurements

constants

V

(in 103 erg/cc)

in a description

planes together, x4

( 110)

plane

(110) and (100) planes

I

The first six constants

X6

195 77 20 4.2

/

x4

1

18.5 & 0.2 80.2 f 275.4 f 364.5 371.3

1 3

f 3 & 3

-2.1 -0.3

& 0.05 + 0.1 f 0.3

9.7 & 1 11.5 * 1

x8

12.7

1.73

371.4

11.6

1.90

(in 108 erg/cc)

X6

-0.63

I

372.7

measurements

1

(

y-x,

I

42 -0.88

- 0.56

- 0.82

VI

in a description

with cubic harmonics

from

on the (1 LO) and (100) planes

X8

1

0.00 & 0.05 -0.18 -3.1

on the (110) and (100)

at 4.2 K

TABLE

7. 296

of EA with cubic harmonics

on the (110) plane only and from measurements

* f

0.05 0.2

-0.2 l 0.3 2.0 * 0.3

Xl0 0.00 & 0.05

x:2 0.00 5 0.05

x?2 0.00 + 0.05

0.00 * 0.05 -0.04 * 0.1

0.00 + 0.05 0.03 & 0.1

0.00 * 0.03 f

-1.0 -0.6

* 0.2 f 0.3

-0.8 -0.8

& 0.2 * 0.2

0.05 0.1

0.04 j, 0.1 0.15 * 0.1

compare them with the results of a least squares adjustment to the measuring points of the (100) and (110) planes together; both results are in fairly good agreement as is demonstrated for measurements at 4.2 K in table V. Table VI gives the first six constants, calculated by a least squares adjustment on the measuring points of the (100) and (110) planes together. The errors in these constants are a measure for the differences between the results of the (110) plane and of the (110) and (100) planes together. It is clear that in this case also for the higher constants Zg and &-s the relation (15) is not satisfied since both constants show a change in sign. Fig. 13 shows the description of the torque curve of the (110) plane at 4.2 K with an increasing number of constants; the sharp change in the torque curve for 8 - 18” can not be described even with six constants.

THE MAGNETOCRYSTALLINE

495

ANISOTROPY ENERGY OF NICKEL

Fig. 13. Description of the experimental results for the (110) plane near 0 = 0 with an increasing number of constants.

Between

Kr= Ks= KS= Kg=

the Kg and 37: the following

-3.232Y4-3.775.%-s-

15.03%-s-

relation

exists:

14.63&o-38.26.%-;,

+ .....

41 .52~s-43.43Xs+260.3~n,-259.1.%;,+529.2~;,+.

....

54.29%-8+49.73%-10+357.6~~~-66.14Z;,+. -944.8Zio-

1048 St,--2937

Kg=

-892.6X:,+317.5%&+.

Ks=

1339 z&+9353

.... .%;%+.

.... ....

%-;s+.

.... (19)

Using the relations (19), we can calculate the first three constants Kg, and for comparison these values are also given in table IV. The results for KS, given in this table, are in agreement with values for this constant obtained by Aubert 8). A change in sign for Ks at low temperature has never been found before. 10. Some additional ex$eriments. Some further experiments, concerning the dependence of the anisotropy energy on temperature below 4.2 K, on magnetic field and on purity were performed.

496

J. J. M. FRANSE

AND

G. DE VRIES

Lowering the temperature from 4.2 K to 1.5 K has no observable influence on the torque curves for the (1 IO) and (100) planes. For each temperature the measurements have been repeated at a field of 8200 Oe; the shape of the curves given in figs. 4 and 5 remains the same at the lower magnetic field, there is only a small shift in the whole curve. By measuring have determined

the increase of maximum torque with magnetic field, we this shift for the (100) plane at 296 K, 77 K and 4.2 K, see

fig. 14. /AA’

/ 9’

/

/

/

/

/-

/

-

_’ .

o 296

V’

K

A

77K

v

4.2 K

‘V __y-

0’

_o-

I H (IO’Oe)

20 torque

-V’

/V’

I

I

10

Fig. 14. Change of the maximum

/A’

/A-

I’

-0’

/

A’

‘A

C

/

A’

/

with magnetic

30

field, for the torque

curve of

the (100) plane.

L&‘(l

10) -

K1 are compared

for pure and

THE

-z

MAGNETOCRYSTALLINE

(iOO)plane

ANISOTROPY

77K

ENERGY

OF NICKEL

1~_(100)plan,y,-, -z Y hi

/ I I

mk

\

/ /

4.2 K

i;0T -z 3 so_

1

\ \ \

,

0 c

497

\ \

: , -pure

\

f

L

0

L----h0'

SO0

(1lO)plane

77K

z

Y Ei

mk 1

j-O

_z

J

!I

90’

(110) p_!ane 4.2 K /’ ’ ‘\ 1’ ‘\

-pure

--- less pure

3 -too_

Fig. 15. Comparison of the contributions of higher constants to the torque curves in the (100) and (110) planes for pure and less pure nickel at 77 K and 4.2 K.

less pure nickel;

the curves for the less pure nickel are simpler and can be

described with a smaller number of constants. The results of measurements on the disk parallel to the (111) plane are in accordance with those of the sphere for 77 K and 4.2 K. The purity of the nickel samples has been investigated in different ways. X-ray fluorescense exposures did not give indications for the presence of impurities in one of the three kinds of nickel. A chemical analysis on the nickel of the sphere and that of the three disks resulted in 20 p,p.m. Fe for the sphere and 150 p.p.m. Fe for the three small disks; Co, Cu, Al and Mn were not detectable or present in only a few p.p.m. Resistance measurements resulted in a ratio RN~/R~.z = 610 for the nickel of the sphere and 230 for that of the three small disks. 11. Discmsion. It seems improbable that the difference in shape between the samples of high and lower purity should lead to the observed differences in anisotropy energy.

498

THE MAGNETOCRYSTALLINE

ANISOTROPY

ENERGY

OF NICKEL

The importance of the higher order terms, the strong temperature dependence of the other terms and the effect of purity seem to suggest that the free path length of the electrons plays an important role. Possibly fine details of the energy band structure are important that become blurred if the mean free path of the electrons is too short. We hope to extend our investigations to the influences of different impurities in order to get some more clarity on these speculations. Acknowledgements. The authors wish to express their gratitude to Dr. J. Veerman for the many valuable discussions about the interpretation of torque curves and to Mr. H. F. van Aalst and Mr. N. Buis for their assistance at the experiments. Received 13-5-68

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9)

Sato, H. and Chandrasekhar, B. S., J. Phys. Chem. Solids l(l957) 228. Birss, R. R. and Wallis, P. M., Proc. Int. Conf. Magnetism Nottingham (1964). Joseph, A. S. and Thorsen, A. C., Proc. Int. Conf. Magnetism Nottingham (1964). Hofmann, U., Phys. Stat. Sol. 7 (1964) K145. Zener, C., Phys. Rev. 96 (1954) 1335. Van Vleck, J. H., J. Phys. Radium 20 (1959) 124. Mueller, F. M. and Priestley, M. G., Phys. Rev. 148 (1966) 638. Aubert, G., J. appl. Phys. 39 (1968) 504. Puzei, I. M., &west. Akad. Nauk USSR, ser. fir. 27 (1963) 1469.