- Email: [email protected]
A study of the magnetic interactions in α-Fe2O3 through scattering of neutrons by spin waves
Samuelsen, E . J . 1967
Physica 34 241-245
A S T U D Y OF T H E MAGNETIC I N T E R A C T I O N S IN ~-Fe~O 3 T H R O U G H S C A T T E R I N G OF N E U T R O N S BY S P I N WAVES *) b y E. J. S A M U E L S E N I n s t i t u t t for Atomenergi, Kjeller, Norway Synopsis
The velocities of spin waves propagating in two crystallographic directions have been determined experimentally at room temperature. These velocities are theoretically expressed in terms of four nearest-neighbour exchange interaction constants. It is shown that the interactions involving Fe-O-Fe bonding angles close to 90° are of considerable strength. F r o m the rhombohedral s y m m e t r y of the a-Fe208 lattice one can deduce t h a t for small spin wave m o m e n t a the acoustic branch of the spin-wave spectrum will have a dispersion relation of the form
Eae(q) =
h2 2too
"C'q(1 +
A "cosZ ;~)~
(1)
where Z is the angle between the spin wave propagation direction and the rhombohedral [I 11] axis. q is the wave vector for the spin waves. The constants C a n d A are determined b y the magnetic interaction constants and the crystal structure parameters (lattice constant a, lattice angle ,,). F r o m spin wave theory, starting from the Heisenberg H a m i l t o n i a n and neglecting anisotropy field terms, one gets 1) C-----
2too h ~ .21J4JS-a(1
--cosa)~[(6+3p+r)(2+p--s)]~
(2)
and 1+2cos~ 1 + A - - - - 4(1 - - c o s ~ )
1
(2+ 41br-- s(3p + r) )
2+p--s
3p+r--3s
(3)
The interaction constant J a a n d the relative constants ~b, r, s are defined in table I. All of the four constants are believed to be of the same sign z). S -= ~ is the spin of the iron atoms. The constant h2/2mo is included for *) Work sponsored jointly b y the Reactor Centrum Nederland and I n s t i t u t t for Atomenergi. --
241
--
242
E.J. SAMUELSEN
convenience, m0 is the neutron mass. In this notation C and A are related to the velocities v± and v / / o f spin wave propagation respectively normal to and parallel to E111] b y h v j_ - -
h C
2m0
v//--
2m0
.C.(1 + A)~
(4)
We have determined C and A b y means of neutron scattering experiments b y the diffraction method 3) of two different natural specimen of a-Fez03 (Origin Krager~, Norway, and Elba). The diffraction method involves measuring the widths (F) of the intensity peaks due to inelastic scattering from spin waves as a function of the crystal setting (80) from suitable Bragg positions (0B) of the crystal. TABLE
1
A b s o l u t e ( n o t a t i o n of ref, 2) a n d r e l a t i v e i n t e r a c t i o n c o n s t a n t s to n e a r e s t n e i g h b o u r s in a - F e ~ O 8
@
--
4-
m
Interaction constants
Fe-O-Fe -angle
Spin p a i r
Absolute
133 ° 117 ° 94 ° 88 °
++ +-
Relative
Y4
1
J~
P
J2 J1
s
r
The cutoffs of the peaks are mainly determined by q vectors in the plane of scattering. When this plane is such that E111] is approximately normal to it, cos z Z of eq. (1) will be close to zero. Then it can be shown 1) from the requirements of conservation of energy and momentum in the scattering process that F k0 [At -
2
(5)
c
(1 -
provided that ko C ~
- -
1-
~'
ko
- -
2z~ 2
•
,
2 is the incident neutron wavelength, and A =8sin0Bcos
0B+--
sin~-
(6)
For spin waves connected to the (111)* reciprocal lattice point, [111] is of course in the plane of scattering, and cos e X corresponding to the cutoffs will vary with dO. In this case one gets 1) P
ko
IA]
--
2
c
(1 -
Ap
.(1 + A cos2(OB + 80)/(1 -- A))-½
(7)
SCATTERING OF NEUTRONS BY SPIN WAVES IN a-Fe203
243
We performed measurements at the (11i)* reciprocal lattice point, with such an orientation of the crystal that eq. (5) should be closely valid. The measured 1'/2 (for the Krager0 crystal, the Elba crystal gives similar results) versus JA]/(I --A)½ is reproduced in figure I, showing a straight line, the slope of which determined C = (95 -[- 2.5) x l0 s cm -1. This corresponds to v L = (3.0 -¢- 0.1) X 106 cm/s, which is a value somewhat lower than that obtained by G o e d k o o p and R i s t e 4) by means of a cruder version of the diffraction method. i
i
i
i
I
I
r
3.10-2
T "~ 2.TO-/2
V
?.
~ ~ 1.10-2 ~ oa
0
/
.
9
A /. /
I~
o
'
3
7 /
I
I
I
1
I
o.I
0.2
0,3
0.4
o.s
I AI/(1-A)'~
I o.6
b
Fig. 1. E x p e r i m e n t a l l y d e t e r m i n e d p e a k half widths versus the crystal setting p a r a m e t r e IAI/(1 -- A)~ for spin w a v e s connected to the (l 1 i)* (©)and t h e (111)* (A for ~0 > 0, V for 60 < 0) reciprocal lattice points. P o i n t a represents t h e half-width of t h e B r a g g peak, indicating the i n s t r u m e n t a l resolution. N e u t r o n w a v e l e n g t h ~ = 1.22/~. Curve 1: S t r a i g h t line t h r o u g h t h e (111)* d a t a points, determines C in eq. 5. Curves 2 and 3 : Eq. 7 for 60 > 0 and 60 < 0, respectively, w i t h t h e d e t e r m i n e d C = 95 × 10 s cm -1, and z/ = --0.47.
In figure 1 are also shown the (lll)*-data. In this case no straight line can be drawn through the points and the origin, but eq. 7 can be fitted to the data quite well by zJ = --0.47 -4- 0.08. This corresponds to v//-~ (2.2 ~ 0 . 2 ) × 106cm/s. The uncertainties are estimated from the intensity statistics. In several publications3)5a, b) the authors have assumed 1 > p > s r ~ 0, mainly based on the early suggestions (6a) that superexchange interactions should become weaker as the cation-anion-cation angle approaches 90 °. Inserting s ~ r ----- 0 in eq. 3 our zl-value gives p ~ 2.7, which certainly indicates that the assumption is doubtful as this value of # does not indicate a n y monotonic variation of interaction strength with angle. Consequently we conclude that s and r should not be neglected. In fact recent theories on superexchange (6b) show that in particular for cations
244
E.J.
SAMUELSEN
with half filled 3d-shell like Fe ~+ the 90 ° superexchange m a y be of comparable strength to that of 180°. Thus H a Ip e r n 7), taking the oxygen s-orbitals as well as the p-orbitals into account in the superexchange mechanism, concludes t h a t in a-Fe203 p represents probably the weakest interaction. We have tried to combine our data with the information given by the N~el transition temperature of a-Fe2Oa. The N6el point (TN) can be related to the interaction constants by means of several models, of which the Green's function (G.f.) interpretation implicity involves a spin wave model, and so should be the one best fitted to combination with our data. As G.f. calculations do not exist for rhombohedral four-sublattice antiferromagnetics we used the molecular field (m,f.) expression s) 3p + r -- 3s + 6 =
3 kBTN 2S(S -{- 1) IJ41
(8)
but scaled the right hand side by 1.23 which is the ratio between the quantities kBTN/[Jaf[ given by the m.f. and the G.f. methods for b.c.c, structures s). Using the determined C and A in eq. (2) and eq. (3) and combining with eq. (8) we obtained one set of solutions of p and r as a function of s satisfying the stability conditions for the ~-FeaOa spin structure (s < 2, 3s < 3p + r, originating from the spin wave theory 1)). For 0 < s < 2 we obtained by this procedure (which of course is open to question) r > s >~ p, r varying between 0.1 and 3. JJ4[ varies slightly with s and is mainly determined by the scaled eq. (8) to about 3.3 meV. Without scaling eq. (8) no acceptable solution for p and r is found. One can obtain further information from the optical spin wave branch. The energy at q = 0 of this branch is 1): E°pt(0) = 2 [J4[ .S. (12(2 - s)(3p + r - 3s))½
(9)
Unfortunately the neutron flux at our present reactor is too low for us to investigate the optical branch. We shall, however, return to such experiments, after which a more extended account of the work on a-Fe2Oa will be published. The author is indebted to Dr. T. R i s t e for suggesting this problem and for all help and support during the work. Received 12-8-66
REFERENCES 1) S a m u e l s e n , E. J., to be published. 2) B e r t a u t , E. F., Proc. of the Int. Conf. on Magnetism, NottiI~gham, 1964, p. 516. 3) D i m i t r i j e v i d , Z., R z a n y , H., T o d o r o v i 6 , J. a n d W a n i c , A., Proc. of Inelastic Scattering of Neutrons, 1964, Vol. I, I.A.E.A., (Vienna, 1964) p. 443.
SCATTERING OF N E U T R O N S BY SPIN W A V E S IN
a-Fe203
245
4) G o e d k o o p , J. A. and R i s t e , T., Nature 185 (1960) 450. 5) a) Li, Y. Y., Phys. Rev. 102 (1956) 1015 b) I i d a , S., J. Phys. Soe. J a p a n 11 (1956) 1300. 6) a) A n d e r s o n , P. W., Phys. Rev. 79 (1950) 350. b) A n d e r s o n , P. W., Magnetism, Vol. I by R a d o , G. T. and S u h l , H. (Academic Press, 1964) p. 25. 7) H a l p e r n , V., Proe. Roy. Soc. 291A (1966) 113. 8) S m a r t , J. S., Advanced Course on Magnetic Interactions, Kjeller Report 93 (1964) 26.
Physica 34 241-245
A S T U D Y OF T H E MAGNETIC I N T E R A C T I O N S IN ~-Fe~O 3 T H R O U G H S C A T T E R I N G OF N E U T R O N S BY S P I N WAVES *) b y E. J. S A M U E L S E N I n s t i t u t t for Atomenergi, Kjeller, Norway Synopsis
The velocities of spin waves propagating in two crystallographic directions have been determined experimentally at room temperature. These velocities are theoretically expressed in terms of four nearest-neighbour exchange interaction constants. It is shown that the interactions involving Fe-O-Fe bonding angles close to 90° are of considerable strength. F r o m the rhombohedral s y m m e t r y of the a-Fe208 lattice one can deduce t h a t for small spin wave m o m e n t a the acoustic branch of the spin-wave spectrum will have a dispersion relation of the form
Eae(q) =
h2 2too
"C'q(1 +
A "cosZ ;~)~
(1)
where Z is the angle between the spin wave propagation direction and the rhombohedral [I 11] axis. q is the wave vector for the spin waves. The constants C a n d A are determined b y the magnetic interaction constants and the crystal structure parameters (lattice constant a, lattice angle ,,). F r o m spin wave theory, starting from the Heisenberg H a m i l t o n i a n and neglecting anisotropy field terms, one gets 1) C-----
2too h ~ .21J4JS-a(1
--cosa)~[(6+3p+r)(2+p--s)]~
(2)
and 1+2cos~ 1 + A - - - - 4(1 - - c o s ~ )
1
(2+ 41br-- s(3p + r) )
2+p--s
3p+r--3s
(3)
The interaction constant J a a n d the relative constants ~b, r, s are defined in table I. All of the four constants are believed to be of the same sign z). S -= ~ is the spin of the iron atoms. The constant h2/2mo is included for *) Work sponsored jointly b y the Reactor Centrum Nederland and I n s t i t u t t for Atomenergi. --
241
--
242
E.J. SAMUELSEN
convenience, m0 is the neutron mass. In this notation C and A are related to the velocities v± and v / / o f spin wave propagation respectively normal to and parallel to E111] b y h v j_ - -
h C
2m0
v//--
2m0
.C.(1 + A)~
(4)
We have determined C and A b y means of neutron scattering experiments b y the diffraction method 3) of two different natural specimen of a-Fez03 (Origin Krager~, Norway, and Elba). The diffraction method involves measuring the widths (F) of the intensity peaks due to inelastic scattering from spin waves as a function of the crystal setting (80) from suitable Bragg positions (0B) of the crystal. TABLE
1
A b s o l u t e ( n o t a t i o n of ref, 2) a n d r e l a t i v e i n t e r a c t i o n c o n s t a n t s to n e a r e s t n e i g h b o u r s in a - F e ~ O 8
@
--
4-
m
Interaction constants
Fe-O-Fe -angle
Spin p a i r
Absolute
133 ° 117 ° 94 ° 88 °
++ +-
Relative
Y4
1
J~
P
J2 J1
s
r
The cutoffs of the peaks are mainly determined by q vectors in the plane of scattering. When this plane is such that E111] is approximately normal to it, cos z Z of eq. (1) will be close to zero. Then it can be shown 1) from the requirements of conservation of energy and momentum in the scattering process that F k0 [At -
2
(5)
c
(1 -
provided that ko C ~
- -
1-
~'
ko
- -
2z~ 2
•
,
2 is the incident neutron wavelength, and A =8sin0Bcos
0B+--
sin~-
(6)
For spin waves connected to the (111)* reciprocal lattice point, [111] is of course in the plane of scattering, and cos e X corresponding to the cutoffs will vary with dO. In this case one gets 1) P
ko
IA]
--
2
c
(1 -
Ap
.(1 + A cos2(OB + 80)/(1 -- A))-½
(7)
SCATTERING OF NEUTRONS BY SPIN WAVES IN a-Fe203
243
We performed measurements at the (11i)* reciprocal lattice point, with such an orientation of the crystal that eq. (5) should be closely valid. The measured 1'/2 (for the Krager0 crystal, the Elba crystal gives similar results) versus JA]/(I --A)½ is reproduced in figure I, showing a straight line, the slope of which determined C = (95 -[- 2.5) x l0 s cm -1. This corresponds to v L = (3.0 -¢- 0.1) X 106 cm/s, which is a value somewhat lower than that obtained by G o e d k o o p and R i s t e 4) by means of a cruder version of the diffraction method. i
i
i
i
I
I
r
3.10-2
T "~ 2.TO-/2
V
?.
~ ~ 1.10-2 ~ oa
0
/
.
9
A /. /
I~
o
'
3
7 /
I
I
I
1
I
o.I
0.2
0,3
0.4
o.s
I AI/(1-A)'~
I o.6
b
Fig. 1. E x p e r i m e n t a l l y d e t e r m i n e d p e a k half widths versus the crystal setting p a r a m e t r e IAI/(1 -- A)~ for spin w a v e s connected to the (l 1 i)* (©)and t h e (111)* (A for ~0 > 0, V for 60 < 0) reciprocal lattice points. P o i n t a represents t h e half-width of t h e B r a g g peak, indicating the i n s t r u m e n t a l resolution. N e u t r o n w a v e l e n g t h ~ = 1.22/~. Curve 1: S t r a i g h t line t h r o u g h t h e (111)* d a t a points, determines C in eq. 5. Curves 2 and 3 : Eq. 7 for 60 > 0 and 60 < 0, respectively, w i t h t h e d e t e r m i n e d C = 95 × 10 s cm -1, and z/ = --0.47.
In figure 1 are also shown the (lll)*-data. In this case no straight line can be drawn through the points and the origin, but eq. 7 can be fitted to the data quite well by zJ = --0.47 -4- 0.08. This corresponds to v//-~ (2.2 ~ 0 . 2 ) × 106cm/s. The uncertainties are estimated from the intensity statistics. In several publications3)5a, b) the authors have assumed 1 > p > s r ~ 0, mainly based on the early suggestions (6a) that superexchange interactions should become weaker as the cation-anion-cation angle approaches 90 °. Inserting s ~ r ----- 0 in eq. 3 our zl-value gives p ~ 2.7, which certainly indicates that the assumption is doubtful as this value of # does not indicate a n y monotonic variation of interaction strength with angle. Consequently we conclude that s and r should not be neglected. In fact recent theories on superexchange (6b) show that in particular for cations
244
E.J.
SAMUELSEN
with half filled 3d-shell like Fe ~+ the 90 ° superexchange m a y be of comparable strength to that of 180°. Thus H a Ip e r n 7), taking the oxygen s-orbitals as well as the p-orbitals into account in the superexchange mechanism, concludes t h a t in a-Fe203 p represents probably the weakest interaction. We have tried to combine our data with the information given by the N~el transition temperature of a-Fe2Oa. The N6el point (TN) can be related to the interaction constants by means of several models, of which the Green's function (G.f.) interpretation implicity involves a spin wave model, and so should be the one best fitted to combination with our data. As G.f. calculations do not exist for rhombohedral four-sublattice antiferromagnetics we used the molecular field (m,f.) expression s) 3p + r -- 3s + 6 =
3 kBTN 2S(S -{- 1) IJ41
(8)
but scaled the right hand side by 1.23 which is the ratio between the quantities kBTN/[Jaf[ given by the m.f. and the G.f. methods for b.c.c, structures s). Using the determined C and A in eq. (2) and eq. (3) and combining with eq. (8) we obtained one set of solutions of p and r as a function of s satisfying the stability conditions for the ~-FeaOa spin structure (s < 2, 3s < 3p + r, originating from the spin wave theory 1)). For 0 < s < 2 we obtained by this procedure (which of course is open to question) r > s >~ p, r varying between 0.1 and 3. JJ4[ varies slightly with s and is mainly determined by the scaled eq. (8) to about 3.3 meV. Without scaling eq. (8) no acceptable solution for p and r is found. One can obtain further information from the optical spin wave branch. The energy at q = 0 of this branch is 1): E°pt(0) = 2 [J4[ .S. (12(2 - s)(3p + r - 3s))½
(9)
Unfortunately the neutron flux at our present reactor is too low for us to investigate the optical branch. We shall, however, return to such experiments, after which a more extended account of the work on a-Fe2Oa will be published. The author is indebted to Dr. T. R i s t e for suggesting this problem and for all help and support during the work. Received 12-8-66
REFERENCES 1) S a m u e l s e n , E. J., to be published. 2) B e r t a u t , E. F., Proc. of the Int. Conf. on Magnetism, NottiI~gham, 1964, p. 516. 3) D i m i t r i j e v i d , Z., R z a n y , H., T o d o r o v i 6 , J. a n d W a n i c , A., Proc. of Inelastic Scattering of Neutrons, 1964, Vol. I, I.A.E.A., (Vienna, 1964) p. 443.
SCATTERING OF N E U T R O N S BY SPIN W A V E S IN
a-Fe203
245
4) G o e d k o o p , J. A. and R i s t e , T., Nature 185 (1960) 450. 5) a) Li, Y. Y., Phys. Rev. 102 (1956) 1015 b) I i d a , S., J. Phys. Soe. J a p a n 11 (1956) 1300. 6) a) A n d e r s o n , P. W., Phys. Rev. 79 (1950) 350. b) A n d e r s o n , P. W., Magnetism, Vol. I by R a d o , G. T. and S u h l , H. (Academic Press, 1964) p. 25. 7) H a l p e r n , V., Proe. Roy. Soc. 291A (1966) 113. 8) S m a r t , J. S., Advanced Course on Magnetic Interactions, Kjeller Report 93 (1964) 26.