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The Fermi surface of white tin
Volume 23, number 3
I ~ H Y S I C S LE T T E R S
THE
FERMI
SURFACE
OF
17 October 1966
WHITE
TIN
M. D. S T A F L E U and A. R. DE V R O O M E N Fysisch Laboratorium, Katholieke Universiteit, Nijmegen, The Netherlands Received 28 September 1966
The De Haas-Van Alphen effect is used to determine the F e r m i surface of white tin. The results affirm qualitatively a recent band calculation.
T h e De H a a s - V a n A l p h e n e f f e c t in w h i t e t i n h a s f o r the l a t e s t t i m e b e e n m e a s u r e d by G o l d and P r i e s t l e y [1], but we f e l t it n e c e s s a r y to do it a g a i n , u s i n g new t e c h n i q u e s that h a v e b e e n d e v e l o p e d s i n c e 1960, b e c a u s e we e x p e c t e d to find a d d i t i o n a l d e t a i l s . W e h a v e m e a s u r e d De H a a s - V a n A l p h e n f r e q u e n c i e s by the S h o e n b e r g S t i l e s m e t h o d [2], at a m o d u l a t i o n f r e q u e n c y of 10 k c / s , w i t h t h e h e l p of a l o c k - i n a m p l i f i e r , d e t e c t i n g at the s e c o n d h a r m o n i c . T h e m a g n e t i c f i e l d H (up to 35 kG) w a s s w e p t i n v e r s e l y p r o p o r t i o n a l to t i m e , a l l o w i n g the u s e of a l o w f r e q u e n c y s e l e c t i v e a m p l i f i e r to s e l e c t the d i f f e r e n t De H a a s - V a n A l p h e n f r e q u e n c i e s . O u r r e s u l t s a r e c o l l e c t e d in fig. 1. Our i n t e r p r e t a t i o n , g i v e n in t a b l e 1, u s e s the n e a r l y - f r e e e l e c t r o n m o d e l l a n g u a g e i n t r o d u c e d by Gold and
P r i e s t l e y , and a f f i r m s q u a l i t a t i v e l y the r e c e n t l y c a l c u l a t e d F e r m i s u r f a c e of W e i s z [3]. In his c o n s t r u c t i o n the f i r s t and s e c o n d B r i l l o u i n z o n e a r e c o m p l e t e l y f i l l e d , and the t h i r d z o n e c o n s i s t s of t h e " p i l l a r s " , c e n t e r e d in X. T h e 4(a) and 4(b) z o n e s , w h i c h a r e s e p a r a t e d s u r f a c e s in t h e n e a r l y - f r e e - e l e c t r o n m o d e l , a r e now c o n n e c t e d in the n e i g h b o u r h o o d of W, f o r m i n g a " n e c k " . The f i f t h z o n e i s e m p t y at F, and t h e " p e a r s " and "double p a n c a k e s " a r e m o r e c o n n e c t e d t h a n in t h e n e a r l y - f r e e - e l e c t r o n m o d e l , s u c h that the 5× o r b i t s do not e x i s t . T h e s i x t h z o n e i s e m p t y at V and K, but t h e r e i s a " p o c k e t " of e l e c t r o n s a r o u n d W; so, n e a r t h i s point the m o s t s t r i k i n g d i f f e r e n c e s b e t w e e n W e i s z ' c o n s t r u c t i o n and the nearly-free-electron model appear. L i k e Gold and P r i e s t l e y we i n t e r p r e t o u r
Table 1 Numerical values and interpretation of observed De Haas-Van Alphen frequencies at the principal axes (unit: 107 gauss; m b = magnetic breakdown; the nomenclature is that of Gold and Priestley, systematically extended). [001]
[110]
[100]
A
0.171 • 0.002
351
C1
0,440 ~ 0.005
6
C1
0.57 • 0.01
B
0.326 • 0.003
352
D1
0.88 ~ 0.02
2C 1
D1
1.16 • 0.03
2C 1
C2
0.445 ± 0.005
6
H1
2,08 • 0.02
5v2
H3
2.55 + 0.05
4E
E
3.50
* 0.04
47
K1
3.40 ~ 0.04
4: . e a r r i n g .
K3
4 . 3 5 ~: 0.05
mb: H 3 + N 1
F1
5.4
• 0.1
5k
K2
3.85
mb~K 1 + C 1
L3
6.85 • 0.1
5~
F2
6.85
• 0.1
5ff
L1
5.55 :~ 0.1
mb
N1
1.65 * 0.03
G1
11.0
• 0.2
4~
L2
6.9
± 0.2
2K 1
G4
10.5 + 0.2
4(~
M1
9.1
~ 0.2
M2
11,8 ± 0.4
4//
N1
1.55
3~
±0.05
±0.05
6
E
rob: K 1 + L 1
Other branches: C 3 (6); D 2 (2C2); D3 (.9); F3 (.9); G2 (4?/.9); G3 (4~.9); H 2 (5Vl); K 4, K 5 (mb: H 3 + N1):
L4 (58); N 2, N 3 (3c) 179
Volume 23, n u m b e r 3
PHYSICS
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17 October 1966
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A 106
I
I 20
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(100) I I 40
i 60
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I 00 °
I
(001) I 20
(11o) I
I 40 °
i 80
I
I 60
I
I 40
I
I 20 •
Fig. 1. The m e a s u r e d De Haas-Van Alphen frequencies (F) with H in the (100), (001), and (110) planes, respectively. The open c i r c l e s indicate relatively weak signals. A - o s c i l l a t i o n s a s t h e c e n t r a l 351 o r b i t s . F r o m t h e a n g u l a r d e p e n d e n c e i n (100) a n d (110) i t m a y b e d e r i v e d t h a t t h e e x t r e m a l c r o s s s e c t i o n ±[001] is very nearly circular. The cross section is a m i n i m u m up t o 70 ° i n b o t h p l a n e s , i m p l y i n g a n o n - c e n t r a l m a x i m u m o r b i t 352; we i n t e r p r e t o u r B - o s c i l l a t i o n s i n t h i s w a y , w h e r e a s N1, N 2 a n d N 3 a r e i n t e r p r e t e d a s 3E o r b i t s . T h e " p i l l a r s " , w h i c h h a v e , a c c o r d i n g to o u r m e a s u r e ments, a "dumb-bell'shape, are rather nearly s u r f a c e s of r e v o l u t i o n a r o u n d X t ). T h e e x t r e m a l a r e a ±[100] i s o n l y 6% l e s s t h a n t h a t ± [ 1 1 0 ] , where we found only one 3c-like De Haas-Van A l p h e n f r e q u e n c y . I n o u r B - o s c i l l a t i o n s we h a v e f o u n d b e a t s i n (110), f r o m w h i c h w e d e r i v e t h a t t h e e x t r e m a l a r e a s d i f f e r 7% w h e n H i s o r i e n t a t e d a t 50 ° f r o m [001], i n t h e F X P a n d L X P 180
planes, respectively. In t h e f o u r t h z o n e we i n t e r p r e t K 1 a s t h e " e a r r i n g " o r b i t [3], t h a t a r i s e s f r o m t h e 4(a) a n d 4(b) z o n e s b e i n g c o n n e c t e d . K 1 i s s h a r p l y c u t off a t 72 ° i n (100), f r o m w h i c h f o l l o w s a n e s t i m a t e of t h e n e c k - s i z e . We h a v e not f o u n d De H a a s - V a n Alphen frequencies with H//[001], .~hat c a n b e a s c r i b e d to t h e n e c k i t s e l f , n o r f r e q u e n c i e s f o r H / / [ 1 1 0 ] t h a t c o r r e s p o n d to c o u n t e r p a r t o r b i t s of t h e " e a r r i n g s " . T h e s e o r b i t s , h o w e v e r , m a y b e not v e r y f a v o u r a b l e f o r a m e a s u r a b l e D e H a a s V a n A l p h e n e f f e c t . M 2 i s a s c r i b e d t o 477 o r b i t s , w h i c h a r e n o n - c e n t r a l f o r H n e a r [100]. L i k e G o l d a n d t ) r i e s t l e y we a s c r i b e G 1 to t h e c e n t r a l 4~ o r b i t s o n t h e " c y l i n d e r s " . A s t h e y a r e m a x i m u m o r b i t s , a l s o n o n - c e n t r a l (4a) o r b i t s m a y b e e x p e c t e d . A l t h o u g h t h e a c c u r a c y of o u r m e a s u r e -
Volume 23, number 3
PHYSICS LETTERS
merits i s not v e r y good at high De H a a s - V a n Alphen f r e q u e n c i e s , we b e l i e v e that we can d i s ti n g u i s h the G4 points at t h e s e 4a o r b i t s . It is e a s i l y s e e n that H 3 should be i n t e r p r e t e d as 4~ o r b i ts . F r o m the topology of the fourth zone it f o l lo w s that at e v e r y o r i e n t a t i o n of H one should always m e a s u r e ~, 77, o r e o r b i t s . So we b e l i e v e that G 2 and G 3 should be a s c r i b e d to (partly n o n - c e n t r a l ) 7/ and ~ o r b i t s . F i n a l l y , E a g r e e s v e r y well with the T o r b i t s . In the fifth zone L 3 is e a s i l y r e c o g n i z e d as 5~ o r b i t s . H 1 and H 2 a g r e e with 5v o r b i t s around the " c o n n e c t i o n s " [3] b e tw e e n the " p e a r s " ; the c e n t r a l 5 v1 o rb i t m u s t have a m i n i m u m at about 70 ° in (100), which points to H 2. The H 1 points p r o b a b l y denote n o n - c e n t r a l 5v 2 o r b i t s . With the m a g n e t i c field n e a r [100] in (100), n o n - c e n t r a l (58) o r b i t s around a " p e a r " and a " c o n n e c t i o n " may be ex p ect ed , and they a g r e e with L 4. With H n e a r [001], two t y p es of o r b i t s a r e expected: 5a (around the " p e a r " top) a g r e e s with F2, 5~ (between four " p e a r s " and four " c o n n e c t i o n s " ) a g r e e s with F 1. C1, C2, and C 3 can be a s c r i b e d to a c l o s e d s u r f a c e . It a g r e e s v e r y well with the s i x t h - z o n e " p o c k e t " at W. The i n t e r p r e t a t i o n of the C - m e a s u r e m e n t s around 35 ° in (100) is u n c l e a r , however. Magnetic breakdown is to be expected between the t h i r d and fourth zone in the LXP plane. K3, K4, and K 5 a g r e e v e r y w e l l with s e l f - i n t e r s e c t i n g o r b i t s of d i f f e r e n t types. F r o m the c a l c u l a t i o n of W e i s z it follows that the e n e r g y l e v e l s of the fourth, fifth, and sixth z o n e s n e a r W a r e v e r y c l o s e t o g e t h e r . So we think it p o s s i b l e b reak d o w n to o c c u r t h e r e too. L 1 a g r e e s with an orbit, c e n t e r e d in F, between
17 October 1966
two " e a r r i n g s " of the fourth zone, and two " p e a r s " of the fifth zone, m o r e o r l e s s like the } o r b i t of the n e a r l y - f r e e - e l e c t r o n model. M 1 would a g r e e with a s e l f - i n t e r s e c t i n g orbit, c o n s i s t i n g of an L1 and a K 1 type orbit, and K 2 could be a s c r i b e d to breakdown b e t w e e n a K 1 and a C 1 type orbit. Magnetic b r e a k d o w n b ei n g dependent on the spacing of the e n e r g y l e v e l s , this s u p p l i e s a s t r o n g t e s t f o r any band calculation. F i n a l l y , D1, D2, and L 2 a r e i n t e r p r e t e d as second h a r m o n i c s of C1, C2, and K1, r e s p e c tively. We have no explanation of D 3 and F 3. Our m e a s u r e m e n t s a g r e e q u a l i t a t i v e l y with the c o n s t r u c t i o n of W e i s z , but q u a n t i t a t i v e l y t h e r e a r e n o t i c e a b l e d i f f e r e n c e s . We only m e n tion the shape of the t h i r d - z o n e " p i l l a r s " , which a r e , in W e i s z ' calculation, f ar f r o m s u r f a c e s of revolution. His c a l c u l a t i o n is b a s e d on Gantm a k h e r ' s [4] s i z e effect m e a s u r e m e n t s , which m ay be in e r r o r up to 10%. A p r e l i m i n a r y OPW c a l c u l a t i o n , b ased on our m e a s u r e m e n t s , points to a b e t t e r a g r e e m e n t b et w een e x p e r i m e n t a l r e sults and the " m o d e l p o t e n t i a l " of A n i m al u and Heine [5], than follows f r o m W e i s z ' calculation. We thank Dr. W e i s z f o r sending us a p r e p r i n t of his a r t i c l e . 1. A.V. Gold and M. G. Priestley, Phil. Mag. 5 (1960) 1089. 2. D. Shoenberg and P. J. Stiles, Proc. Roy. Soc. A281 (1964) 62. 3. G. Weisz, Phys. Rev., to be published. 4. V. F. Gantmakher, JETP 44 (1963) 811 (Soviet Phys. JETP 17 (1963) 549); JETP 46 (1964) 2028 (Soviet Phys. JETP 19 (1964) 1366). 5. A.O.E.Animalu and V. Heine, Phil. Mag. 12 (1965) 1249.
* * * * *
ANTIFERROMAGNETISM
OF
K2MnF 4
D. J. B R E E D Natuurkundig L a b o r a t o r i u m d e r Universiteit van A m s t e r d a m , The Netherlands
Received 12 September 1966 In this note we report magnetic measurements of a K2MnF 4 single crystal. K2MnF 4 single c r y s t a l s w e r e p r e p a r e d in an a r g o n a t m o s p h e r e , s t a r t i n g f r o m a m i x t u r e of about 0.2 m o l e KMnF 3 ( p r e c i p i t a t e d f r o m aqueous solutions of KF and MnCl 2 [1]) and 0.6 mole
KHF 2 in a c o n i c a l platinum c r u c i b l e . The KF : MnF 2 r a t i o is c h o s e n on the e x p e c t e d a n a l ogy with the K F - M g F 2 s y s t e m [2]. A f t e r d r y i n g in the f u r n a c e , the m i x t u r e was 181
I ~ H Y S I C S LE T T E R S
THE
FERMI
SURFACE
OF
17 October 1966
WHITE
TIN
M. D. S T A F L E U and A. R. DE V R O O M E N Fysisch Laboratorium, Katholieke Universiteit, Nijmegen, The Netherlands Received 28 September 1966
The De Haas-Van Alphen effect is used to determine the F e r m i surface of white tin. The results affirm qualitatively a recent band calculation.
T h e De H a a s - V a n A l p h e n e f f e c t in w h i t e t i n h a s f o r the l a t e s t t i m e b e e n m e a s u r e d by G o l d and P r i e s t l e y [1], but we f e l t it n e c e s s a r y to do it a g a i n , u s i n g new t e c h n i q u e s that h a v e b e e n d e v e l o p e d s i n c e 1960, b e c a u s e we e x p e c t e d to find a d d i t i o n a l d e t a i l s . W e h a v e m e a s u r e d De H a a s - V a n A l p h e n f r e q u e n c i e s by the S h o e n b e r g S t i l e s m e t h o d [2], at a m o d u l a t i o n f r e q u e n c y of 10 k c / s , w i t h t h e h e l p of a l o c k - i n a m p l i f i e r , d e t e c t i n g at the s e c o n d h a r m o n i c . T h e m a g n e t i c f i e l d H (up to 35 kG) w a s s w e p t i n v e r s e l y p r o p o r t i o n a l to t i m e , a l l o w i n g the u s e of a l o w f r e q u e n c y s e l e c t i v e a m p l i f i e r to s e l e c t the d i f f e r e n t De H a a s - V a n A l p h e n f r e q u e n c i e s . O u r r e s u l t s a r e c o l l e c t e d in fig. 1. Our i n t e r p r e t a t i o n , g i v e n in t a b l e 1, u s e s the n e a r l y - f r e e e l e c t r o n m o d e l l a n g u a g e i n t r o d u c e d by Gold and
P r i e s t l e y , and a f f i r m s q u a l i t a t i v e l y the r e c e n t l y c a l c u l a t e d F e r m i s u r f a c e of W e i s z [3]. In his c o n s t r u c t i o n the f i r s t and s e c o n d B r i l l o u i n z o n e a r e c o m p l e t e l y f i l l e d , and the t h i r d z o n e c o n s i s t s of t h e " p i l l a r s " , c e n t e r e d in X. T h e 4(a) and 4(b) z o n e s , w h i c h a r e s e p a r a t e d s u r f a c e s in t h e n e a r l y - f r e e - e l e c t r o n m o d e l , a r e now c o n n e c t e d in the n e i g h b o u r h o o d of W, f o r m i n g a " n e c k " . The f i f t h z o n e i s e m p t y at F, and t h e " p e a r s " and "double p a n c a k e s " a r e m o r e c o n n e c t e d t h a n in t h e n e a r l y - f r e e - e l e c t r o n m o d e l , s u c h that the 5× o r b i t s do not e x i s t . T h e s i x t h z o n e i s e m p t y at V and K, but t h e r e i s a " p o c k e t " of e l e c t r o n s a r o u n d W; so, n e a r t h i s point the m o s t s t r i k i n g d i f f e r e n c e s b e t w e e n W e i s z ' c o n s t r u c t i o n and the nearly-free-electron model appear. L i k e Gold and P r i e s t l e y we i n t e r p r e t o u r
Table 1 Numerical values and interpretation of observed De Haas-Van Alphen frequencies at the principal axes (unit: 107 gauss; m b = magnetic breakdown; the nomenclature is that of Gold and Priestley, systematically extended). [001]
[110]
[100]
A
0.171 • 0.002
351
C1
0,440 ~ 0.005
6
C1
0.57 • 0.01
B
0.326 • 0.003
352
D1
0.88 ~ 0.02
2C 1
D1
1.16 • 0.03
2C 1
C2
0.445 ± 0.005
6
H1
2,08 • 0.02
5v2
H3
2.55 + 0.05
4E
E
3.50
* 0.04
47
K1
3.40 ~ 0.04
4: . e a r r i n g .
K3
4 . 3 5 ~: 0.05
mb: H 3 + N 1
F1
5.4
• 0.1
5k
K2
3.85
mb~K 1 + C 1
L3
6.85 • 0.1
5~
F2
6.85
• 0.1
5ff
L1
5.55 :~ 0.1
mb
N1
1.65 * 0.03
G1
11.0
• 0.2
4~
L2
6.9
± 0.2
2K 1
G4
10.5 + 0.2
4(~
M1
9.1
~ 0.2
M2
11,8 ± 0.4
4//
N1
1.55
3~
±0.05
±0.05
6
E
rob: K 1 + L 1
Other branches: C 3 (6); D 2 (2C2); D3 (.9); F3 (.9); G2 (4?/.9); G3 (4~.9); H 2 (5Vl); K 4, K 5 (mb: H 3 + N1):
L4 (58); N 2, N 3 (3c) 179
Volume 23, n u m b e r 3
PHYSICS
[OOll c~
1'100~
G,
o• |
:1:.:.:..;" ..+°p
10 e
LETTERS
2 •._~ room°|
. . . . . . . . . . .
.'.'..,
'
!l
• ....~...,..,+."
"
.'~
LI
.," e ° "
. OoO
oedP,,,eol
oeedplOo o o o
LIP
•
"
I
l K2 ~q(°••o.
o~ lUJe
/
. . " % , .~_o,.to. . . . . .
• ...,..,,
, ,,
.m ..'"~
• 08 °
F2 I<,~....,-
)'
K~ ~o
KS
J~n'o~ +~pllodp 11,• d~$" K3 °
"qb l oib tJq~e I I ° 8 ~))eqlnl
e•
Ir
oo°• *8oo °eoo
• I
",,,,,,,_,
ooe.~
...
- J eoo
_~0
E
°O e4
• H1
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17 October 1966
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Fig. 1. The m e a s u r e d De Haas-Van Alphen frequencies (F) with H in the (100), (001), and (110) planes, respectively. The open c i r c l e s indicate relatively weak signals. A - o s c i l l a t i o n s a s t h e c e n t r a l 351 o r b i t s . F r o m t h e a n g u l a r d e p e n d e n c e i n (100) a n d (110) i t m a y b e d e r i v e d t h a t t h e e x t r e m a l c r o s s s e c t i o n ±[001] is very nearly circular. The cross section is a m i n i m u m up t o 70 ° i n b o t h p l a n e s , i m p l y i n g a n o n - c e n t r a l m a x i m u m o r b i t 352; we i n t e r p r e t o u r B - o s c i l l a t i o n s i n t h i s w a y , w h e r e a s N1, N 2 a n d N 3 a r e i n t e r p r e t e d a s 3E o r b i t s . T h e " p i l l a r s " , w h i c h h a v e , a c c o r d i n g to o u r m e a s u r e ments, a "dumb-bell'shape, are rather nearly s u r f a c e s of r e v o l u t i o n a r o u n d X t ). T h e e x t r e m a l a r e a ±[100] i s o n l y 6% l e s s t h a n t h a t ± [ 1 1 0 ] , where we found only one 3c-like De Haas-Van A l p h e n f r e q u e n c y . I n o u r B - o s c i l l a t i o n s we h a v e f o u n d b e a t s i n (110), f r o m w h i c h w e d e r i v e t h a t t h e e x t r e m a l a r e a s d i f f e r 7% w h e n H i s o r i e n t a t e d a t 50 ° f r o m [001], i n t h e F X P a n d L X P 180
planes, respectively. In t h e f o u r t h z o n e we i n t e r p r e t K 1 a s t h e " e a r r i n g " o r b i t [3], t h a t a r i s e s f r o m t h e 4(a) a n d 4(b) z o n e s b e i n g c o n n e c t e d . K 1 i s s h a r p l y c u t off a t 72 ° i n (100), f r o m w h i c h f o l l o w s a n e s t i m a t e of t h e n e c k - s i z e . We h a v e not f o u n d De H a a s - V a n Alphen frequencies with H//[001], .~hat c a n b e a s c r i b e d to t h e n e c k i t s e l f , n o r f r e q u e n c i e s f o r H / / [ 1 1 0 ] t h a t c o r r e s p o n d to c o u n t e r p a r t o r b i t s of t h e " e a r r i n g s " . T h e s e o r b i t s , h o w e v e r , m a y b e not v e r y f a v o u r a b l e f o r a m e a s u r a b l e D e H a a s V a n A l p h e n e f f e c t . M 2 i s a s c r i b e d t o 477 o r b i t s , w h i c h a r e n o n - c e n t r a l f o r H n e a r [100]. L i k e G o l d a n d t ) r i e s t l e y we a s c r i b e G 1 to t h e c e n t r a l 4~ o r b i t s o n t h e " c y l i n d e r s " . A s t h e y a r e m a x i m u m o r b i t s , a l s o n o n - c e n t r a l (4a) o r b i t s m a y b e e x p e c t e d . A l t h o u g h t h e a c c u r a c y of o u r m e a s u r e -
Volume 23, number 3
PHYSICS LETTERS
merits i s not v e r y good at high De H a a s - V a n Alphen f r e q u e n c i e s , we b e l i e v e that we can d i s ti n g u i s h the G4 points at t h e s e 4a o r b i t s . It is e a s i l y s e e n that H 3 should be i n t e r p r e t e d as 4~ o r b i ts . F r o m the topology of the fourth zone it f o l lo w s that at e v e r y o r i e n t a t i o n of H one should always m e a s u r e ~, 77, o r e o r b i t s . So we b e l i e v e that G 2 and G 3 should be a s c r i b e d to (partly n o n - c e n t r a l ) 7/ and ~ o r b i t s . F i n a l l y , E a g r e e s v e r y well with the T o r b i t s . In the fifth zone L 3 is e a s i l y r e c o g n i z e d as 5~ o r b i t s . H 1 and H 2 a g r e e with 5v o r b i t s around the " c o n n e c t i o n s " [3] b e tw e e n the " p e a r s " ; the c e n t r a l 5 v1 o rb i t m u s t have a m i n i m u m at about 70 ° in (100), which points to H 2. The H 1 points p r o b a b l y denote n o n - c e n t r a l 5v 2 o r b i t s . With the m a g n e t i c field n e a r [100] in (100), n o n - c e n t r a l (58) o r b i t s around a " p e a r " and a " c o n n e c t i o n " may be ex p ect ed , and they a g r e e with L 4. With H n e a r [001], two t y p es of o r b i t s a r e expected: 5a (around the " p e a r " top) a g r e e s with F2, 5~ (between four " p e a r s " and four " c o n n e c t i o n s " ) a g r e e s with F 1. C1, C2, and C 3 can be a s c r i b e d to a c l o s e d s u r f a c e . It a g r e e s v e r y well with the s i x t h - z o n e " p o c k e t " at W. The i n t e r p r e t a t i o n of the C - m e a s u r e m e n t s around 35 ° in (100) is u n c l e a r , however. Magnetic breakdown is to be expected between the t h i r d and fourth zone in the LXP plane. K3, K4, and K 5 a g r e e v e r y w e l l with s e l f - i n t e r s e c t i n g o r b i t s of d i f f e r e n t types. F r o m the c a l c u l a t i o n of W e i s z it follows that the e n e r g y l e v e l s of the fourth, fifth, and sixth z o n e s n e a r W a r e v e r y c l o s e t o g e t h e r . So we think it p o s s i b l e b reak d o w n to o c c u r t h e r e too. L 1 a g r e e s with an orbit, c e n t e r e d in F, between
17 October 1966
two " e a r r i n g s " of the fourth zone, and two " p e a r s " of the fifth zone, m o r e o r l e s s like the } o r b i t of the n e a r l y - f r e e - e l e c t r o n model. M 1 would a g r e e with a s e l f - i n t e r s e c t i n g orbit, c o n s i s t i n g of an L1 and a K 1 type orbit, and K 2 could be a s c r i b e d to breakdown b e t w e e n a K 1 and a C 1 type orbit. Magnetic b r e a k d o w n b ei n g dependent on the spacing of the e n e r g y l e v e l s , this s u p p l i e s a s t r o n g t e s t f o r any band calculation. F i n a l l y , D1, D2, and L 2 a r e i n t e r p r e t e d as second h a r m o n i c s of C1, C2, and K1, r e s p e c tively. We have no explanation of D 3 and F 3. Our m e a s u r e m e n t s a g r e e q u a l i t a t i v e l y with the c o n s t r u c t i o n of W e i s z , but q u a n t i t a t i v e l y t h e r e a r e n o t i c e a b l e d i f f e r e n c e s . We only m e n tion the shape of the t h i r d - z o n e " p i l l a r s " , which a r e , in W e i s z ' calculation, f ar f r o m s u r f a c e s of revolution. His c a l c u l a t i o n is b a s e d on Gantm a k h e r ' s [4] s i z e effect m e a s u r e m e n t s , which m ay be in e r r o r up to 10%. A p r e l i m i n a r y OPW c a l c u l a t i o n , b ased on our m e a s u r e m e n t s , points to a b e t t e r a g r e e m e n t b et w een e x p e r i m e n t a l r e sults and the " m o d e l p o t e n t i a l " of A n i m al u and Heine [5], than follows f r o m W e i s z ' calculation. We thank Dr. W e i s z f o r sending us a p r e p r i n t of his a r t i c l e . 1. A.V. Gold and M. G. Priestley, Phil. Mag. 5 (1960) 1089. 2. D. Shoenberg and P. J. Stiles, Proc. Roy. Soc. A281 (1964) 62. 3. G. Weisz, Phys. Rev., to be published. 4. V. F. Gantmakher, JETP 44 (1963) 811 (Soviet Phys. JETP 17 (1963) 549); JETP 46 (1964) 2028 (Soviet Phys. JETP 19 (1964) 1366). 5. A.O.E.Animalu and V. Heine, Phil. Mag. 12 (1965) 1249.
* * * * *
ANTIFERROMAGNETISM
OF
K2MnF 4
D. J. B R E E D Natuurkundig L a b o r a t o r i u m d e r Universiteit van A m s t e r d a m , The Netherlands
Received 12 September 1966 In this note we report magnetic measurements of a K2MnF 4 single crystal. K2MnF 4 single c r y s t a l s w e r e p r e p a r e d in an a r g o n a t m o s p h e r e , s t a r t i n g f r o m a m i x t u r e of about 0.2 m o l e KMnF 3 ( p r e c i p i t a t e d f r o m aqueous solutions of KF and MnCl 2 [1]) and 0.6 mole
KHF 2 in a c o n i c a l platinum c r u c i b l e . The KF : MnF 2 r a t i o is c h o s e n on the e x p e c t e d a n a l ogy with the K F - M g F 2 s y s t e m [2]. A f t e r d r y i n g in the f u r n a c e , the m i x t u r e was 181