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LSU Historical Dissertations and Theses
Graduate School
1968
The Magnetoacoustic Effect in Mercury.
Tommy Earl Bogle
Louisiana State University and Agricultural & Mechanical College
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69-4453
BOGLE, Tommy Earl, 1940THE MAGNETOACOUSTIC EFFECT IN MERCURY.
Louisiana State U niversity and Agricultural and
Mechanical College, Ph.D., 1968
P hysics, solid state
U niversity M icrofilm s, Inc., A nn A rbor, M ichigan
THE MAGNETOACOUSTIC E FFECT IN MERCURY
A Dissertation
Submitted to the Graduate Faculty of the
Louisiana State U n i v e r s i t y and
A g r i c u l t u r a l and Mechanical College
in p a r t i a l f u l f i l l m e n t of the
requirements f o r the degree of
Doctor of Philosophy
in
The Department of Physics and Astronomy
B.S.,
by
Tommy Earl Bogle
Louisiana Polytechnic I n s t i t u t e ,
August, 1968
1962
ACKNOWLEDGMENTS
F i r s t and foremost the author wishes t o thank Dr.
G ren ie r and Dr.
J u l i a n B.
Coon f o r t h e i r guidance and as s is t a n c e
throughout the course of t h i s work.
to Dr.
Claude G.
The author
is e s p e c i a l l y indebted
Coon f o r h is c o n t r i b u t i o n t o the p s e u do po t en t ia l s e c t i o n .
The author wishes to thank the e n t i r e
as s is t a n c e
low temperature group f o r
in va ri ous phases o f the experiment.
Thanks ar e due to
the t e c h n i c a l s t a f f f o r a s s is t a n c e and e s p e c i a l l y to Mrs.
Boothby who prepared the drawings and Mrs.
Barbara
Pat M i l l s who typed and
proofread t h i s manuscript.
The aut hor a l s o wishes to acknowledge the Atomic Energy
Commission f o r f i n a n c i a l support o f the experiment.
as s is t a n c e rece ive d from the Dr.
of the L.S.U.
Charles E.
The f i n a n c i a l
Coates Memorial Fund
Foundation donated by George H. Coates f o r p r ep a ra ­
t i o n o f t h i s manuscript
is g r a t e f u l l y acknowledged.
TABLE OF CONTENTS
Page
INTRODUCTION
2
II.
STRUCTURE
4
11.
EXPERIMENT
8
IV.
THEORY AND DATA ANALYSIS
15
EXPERIMENTAL RESULTS AND DISCUSSION
19
1.
V.
A.
1.
(n o )
19
(110)
3-
Results f o r q p a r a l l e l to the
di r e c t i on
(112)
4.
Results f o r q p a r a l l e l to the (100),
(111), and [110] d i r e c t i o n s
28
5-
Summary of e l e c t r o n data
29
6.
Pseudopotential C o e f f i c i e n t s
30
22
Hole Surface
1.
2.
3VI.
■—
>
Results f o r q p a r a l l e l to the
di r e c t i on
Results f o r q p a r a l l e l to the
di r e c t i on
2.
B.
19
Electr on Surface
26
3^
Results f o r q p a r a l l e l to the
d i re c t i on
(110)
Results f o r q p a r a l l e l to the
di r e c t i on
(112)
Summary o f hole r e s u l t s
3^
42
^3
45
CONCLUSION
iii
TABLE OF CONTENTS
Page
APPENDIX A, ADiscussion of
the k PWC a l c u l a t i o n
^7
APPENDIX B, C a lc u la t i o n of the A t t e n u a t i o n C o e f f i c i e n t
in Transverse Magnetic Fi el ds
55
REFERENCES
73
VITA
75
Iv
LIS T OF TABLES
Table
I.
Physical constants and c r y s t a l l o g r a p h i c
data f o r mercury.
II.
Comparison of c a l i p e r s and areas of lens
obtained by d i f f e r e n t methods in (110)
plane.
111.
Comparison of r a d i i
o f openings in f i r s t
zone hole surface.
IV.
Comparison of ou ts id e dimensions of the
T - s e c t i o n as measured from L, T, and X.
LIS T OF FIGURES
F ig u re
Page
1.
The B r i l l o u i n zone f o r mercury.
6
2.
Sample mold.
9
3.
Block diagram of expe rim ent al apparatus.
k.
A p l o t of a l l C - c a l i p e r s obtained f o r q p a r a l l e l to
the (110) d i r e c t i o n .
11
Dimensions taken from Brandt
and Rayne's dHvA- 3 PW c a l c u l a t i o n are shown f o r
comparison as are the dimensions due to Loucks.
A l l data were taken a t 165 MHz.
5.
20
A p o la r p l o t of data w i t h q In the (110) d i r e c t i o n .
A p r oj ec t ed view of the B r i l l o u i n zone in the
direction
( 110)
is shown along w i t h a sketch o f the f r e e
e l e c t r o n lenses.
The data were taken a t a
frequency of 165 MHz.
6.
21
A p o la r p l o t of data w it h q in the
(110) d i r e c t i o n .
A p r oj ec t ed view of the B r i l l o u i n zone in the
( 110)
d i r e c t i o n is shown along w i t h a sketch of the
f r e e e l e c t r o n lenses.
The data were taken a t a
frequency of I 35 MHz.
2k
vi
Page
Fi gure
7-
A p o l a r p l o t o f data w fth q 5 ° from the (112)
direction.
A p r o j e c t e d view of the B r i l l o u i n
zone in the (112) d i r e c t i o n is shown along w it h
a sketch of the f r e e e l e c t r o n lenses.
The data
were taken a t a frequency of 165 MHz.
8.
27
A p l o t of the
ex pe rim ent al data in the (110)
cross se c ti o n
of the Fermi su rface compared
w it h the f r e e
e l e c t r o n surface and the r e s u l t s
of a 4 PW c a l c u l a t i o n .
The t h i c k lin e s
the f r e e e l e c t r o n su rf ac e .
i n di c at ed by the s o l i d
i n d ic a t e
The T - s e c t i o n
l i n e and centered on U
is the r e s u l t o f the present c a l c u l a t i o n as is
the sec tio n in d ic a t e d by the t h i n l i n e o v e r l y i n g
the data centered on L.
9-
31
A r e p r e s e n t a t i o n of the f i r s t zone hole surface^
in mercury in the extended zone scheme.
The
e x p e r i m e n t a l l y observed o r b i t s ar e i n di c at ed
w it h Greek l e t t e r s .
10.
35
A r e p r e s e n t a t i o n of the mercury Fermi su rface
cross s e c tio n by the (110) plane w i t h various
caliper
11.
in d ic a t e d .
38
A r e p r e s e n t a t i o n of the t 1 hole o r b i t w it h
corresponding n o t a t io n
i n d ic a t e d .
41
Figure
12.
Page
Pseudopotential f o r mercury computed by Animalu
and Heine w it h the values c a l c u l a t e d
in t h i s
work as w e l l as those of Brandt and Rayne
shown f o r comparison.
A reduced value of kp
was used which accounts f o r the f a c t t h a t the
present values ar e displaced s l i g h t l y t o the
r i g h t of Brandt
13.
and Rayne's values .
52
A p l o t of nine d i f f e r e n t cross sections of the
T - s e c t i o n centered on U in the
(110) plane w it h
the corresponding values of
V jqq.> and V^lO
i n d ic a t e d .
Ik .
5^
The f i e l d - d e p e n d e n t f a c t o r
in the a t t e n u a t i o n
of a l o n g i t u d i n a l sound wave moving pe r pe ndi cul ar
to the magnetic
15.
field .
66
The f i e l d - d e p e n d e n t f a c t o r
in the a t t e n u a t i o n
of a t ra nsv er se sound wave p o l a r i z e d perpen­
d i c u l a r t o the magnetic f i e l d and propagating
a l s o in a d i r e c t i o n p er pe nd ic ul a r to the f i e l d .
16.
68
A r e p r e s e n t a t i o n of an e l e c t r o n o r b i t on the
Fermi
surface f o r two d i f f e r e n t values of the
magnetic f i e l d H.
The magnetic f i e l d
is
d i r e c t e d out o f the page and a sound wave is
propagated in the X d i r e c t i o n .
orbit
The corresponding
in r e c i p r o c a l space is shown a t the r i g h t .
vi i i
JO
ABSTRACT
Geometric resonances in the u l t r a s o n i c a t t e n u a t i o n have been
observed in high p u r i t y mercury s i n g l e c r y s t a l s w i t h l o n g i t u d i n a l
sound waves propagated along f i v e c r y s t a 1lographic d i r e c t i o n s a t
frequencies up t o 165 MHz.
Of the f i v e ,
onl y data f o r the ( 11 0) ,
( 11 0) , and (112) d i r e c t i o n s are rep ort ed.
The dominant resonance
branches have been assigned to c a l i p e r s of the second band e l e c t r o n
lens-shaped surfaces w i t h three major symmetry c a l i p e r s being
obtained.
The remainder of the resonance branches have been assigned
to o r b i t s on the f i r s t band hole sur face.
Various breakthrough
dimensions of the hole surface were determined from these o r b i t s .
The pse udopotential c o e f f i c i e n t s corresponding to the planes bounding
the f i r s t B r i l l o u i n zone in mercury have been estimated by comparing
the geometric resonance data w it h the r e s u l t s o f a fo ur pseudowave
c a l c u l a t i o n n e g le c ti n g spin o r b i t coupling.
I.
INTRODUCTION
The Fermi su rface o f c r y s t a l l i n e mercury has been s tu di ed both
e x p e r i m e n t a l l y and t h e o r e t i c a l l y by a number of i n v e s t i g a t o r s
few years.
in the past
The de Haas van Alphen (dHvA) e f f e c t has y i e l d e d s e v e r a l
extremal cross s e c t i o n a l ar ea s* ;
extremal c a l i p e r s of a p o r t i o n of the
2
surface have been determined from p r e l i m i n a r y magnetoacoustic data ;
e f f e c t i v e masses on the Fermi su r f a c e have been determined from an
Azbel-Kaner c y c lo t r o n resonance experiment , and i n fo rm a t i o n about the
topology has been obt ain ed from magnetoresistance measurements.
firs t
k 5
The
t h e o r e t i c a l d e t e r m in a ti o n o f the band s t r u c t u r e was a three plane
wave pse udopotential c a l c u l a t i o n
fo ur parameter model
R
1
based on the dHvA d at a.
An e m p i r i c a l
based on magnetoresistance data was l a t e r presented
and used as a s t a r t i n g p o i n t , along w i t h the dHvA d at a,
plane wave pse ud op ot en ti al c a l c u l a t i o n
f o r an e i g h t
in cl u d in g spin o r b i t coupling and
5
co n st ra in ed to s a t i s f y the requirement o f compensation.
A detailed
g
r e l a t i v i s t i c augmented plane wave (RAPW) c a l c u l a t i o n
has been performed
w i t h the r e s u l t s y i e l d i n g a su r f a c e t o p o l o g i c a l l y e q u i v a l e n t t o t h a t
obtained from the plane wave c a l c u l a t i o n s .
A l l of these c a l c u l a t i o n s
y i e l d a model in reasonable agreement w i t h the above experiments w i t h
the f o l l o w i n g ex c ep tio n s.
The c a l c u l a t i o n s p r e d i c t cross s e c t i o n a l areas
belonging t o the second band e l e c t r o n su r f a c e much too la r g e t o be in
agreement w i t h the e x p e r i m e n t a l l y observed values.
d o p o t e n t i a l and RAPW c a l c u l a t i o n s f a i l
Also both pseu-
t o p r e d i c t c o r r e c t l y the ang ula r
e x t e n t of the e x p e r i m e n t a l l y observed open o r b i t s as seen from the
2
3
magnetoresistance data, whereas the four parameter model uniquely
ex p la i n s t h i s angular range.
In the f o l l o w i n g sections the r e s u l t s o f a d e t a i l e d magnetoacoustic
experiment ar e reported in an e f f o r t to gain a more p r e c i s e knowledge
of the Fermi surface of mercury.
The d e t a i l s of a fo ur pseudowave
c a l c u l a t i o n o f sev er al cross sections o f the Fermi
a r e presented in Appendix A.
The a t t e n u a t i o n c o e f f i c i e n t of a sound
wave in the presence of a tra nsverse magnetic f i e l d
in Appendix B.
surface of mercury
Results are obtained both f o r
is c a l c u l a t e d a l s o
l o n g i t u d i n a l waves and
waves p o l a r i z e d perpe ndi cul ar t o the d i r e c t i o n o f propagation.
The
o s c i l l a t o r y behavior of the a t t e n u a t i o n c o e f f i c i e n t corresponding to
geometric resonances is discussed.
c a l i p e r dimension,
the o s c i l l a t i o n s .
C, of the Fermi
A relation
is obtained f o r the
surface in terms of the period of
I I.
STRUCTURE
The s t r u c t u r e o f c r y s t a l l i n e mercury has been discussed e x t e n s i v e l y
by a number of authors
and w i l l only be summarized here along w it h
the basic t o p o l o g i c a l f e a t u r e s of i t s Fermi
surface.
Mercury is a d i v a l e n t metal which c r y s t a l l i z e s a t approximately 223°K
i n t o a simple rhombohedral s t r u c t u r e w i t h one atom per u n i t c e l l .
The
rhombohedral s t r u c t u r e can be considered as a d i s t o r t e d v er s io n of the
more common face centered cubic s t r u c t u r e where the d i s t o r t i o n con sists
of s t r e t c h i n g along the t r i g o n a l a x i s .
A d e t a i l e d discussion of the
rhombohedral s t r u c t u r e in general can be found in sev er al textbooks.
*7 8
The l a t t i c e parameters of mercury as w e l l as other info rm at io n
p e r t a i n i n g to i t s c r y s t a l s t r u c t u r e are l i s t e d
in Table i .
The f i r s t
B r i l l o u i n zone w it h symmetry points and p r i n c i p a l symmetry axes labeled
is shown in Fig.
1.
The f i r s t zone is composed of th ree d i s t i n c t ,
i n e q u i v a l e n t faces centered on X, T, and L, and per pe nd ic ul a r to the
CllO),
{ill},
and £ 100} d i r e c t i o n s ,
respectively.
the l a r g e s t and ar e c l o s e s t to the zone ce n te r T.
{ill)
The £ 100} faces are
The hexagonal shaped
faces are about 1/ 2$ f u r t h e r from T than the re c t a n g u la r £ 110} faces.
Unli ke most systems,
the d i r e c t i o n s
in the rhombohedral system are
not g e n e r a l l y normal to planes having the same index.
to t h i s are the ( i l l ) ,
and <112) d i r e c t i o n s
( 1 1 0 ) , and <112) d i r e c t i o n s .
lie
in the (111) plane.
The exceptions
A l l three <110)
The [110] d i r e c t i o n
is
p a r a l l e l to the l i n e j o i n i n g T and W so t h a t the (110) plane b i s e c t s
the B r i l l o u i n zone along a l i n e X-U -T-U -L.
k
This plane and i t s
two other
5
TABLE I .
P hysical constants and c ry s ta llo g ra p h lc data fo r mercury a t 5°K>
Value
Symbol
a
a0
D efi ni tio n
2.9863A
o_ 1
2 - 3002A
R e cip ro cal space l a t t i c e v e c to r
a
70°
Real space rhombohedral angle
ea
10i*° 21. 7 1
R e cip ro cal space rhombohedral angle
a
22. 996a“3
Volume o f u n it c e l l in r e a l space
°-l
1.3707A
Free e le c tro n Fermi radius
0.5261 Ry
Free e le c tro n Fermi energy
14.^8 g/cm3
Dens i ty
%
kF
ef
p
x-u
6‘
°-l
Real space l a t t i c e v e c to r
See Fig.
1
T-U
0.63^5A
OQ.625^A
See Fi g.
1
L-U
0-1
1. 0337A
See F ig .
1
T-W
°-l
0 . 7220A
See Fi g. 1
X-K
°-l
0 . 3611A
See F ig .
1
r-L
°-l
1. 1500A
See F ig .
1
r-x
0-1
See Fig.
1
r -T
1.4103A
o _i
l.klkkA
See F ig .
1
u-w
s 0-1
0.3611A
See Fig.
1
L-W
1.0951A
See Fi g.
1
L-K
0.8920A-1
See F ig .
1
1.7 0 X 10^ cm/sec
L o n g itu d in a l sound v e lo c it y in (110) d ir e c tio n
2 -35 X 105 cm/sec
L o n g itu d in a l sound v e lo c it y in (110) d ir e c tio n
1 .7 2 X 10^ cm/sec
L o n g itu d in a l sound v e lo c ity in (112) d ir e c tio n
2 .5 5 X 105 cm/sec
L o n g itu d in a l sound v e lo c it y in (111) d ir e c tio n
2. 11 X 10^ cm/sec
L o n g itu d in a l sound v e lo c it y in (100) d ir e c tio n
v 110
v l 10
V 112
Vl l l
O
O
>
a Source:
°-l
C. S. B a r r e tt , Acta C ry s t.
10, 58 (1 957)-
6
Trigonal
(III)
[III]
24
24
(1 1 0 )
42
[ 112]
Bisectrix
(HO)
24
Binary
24
42
DTI]
Fi g.
1
tool]
7
e q u iv a le n ts are the m i r r o r planes
in t h i s system.
is one o f the three tw o-fold r o t a t i o n axes,
j o i n i n g T and U, and [ i l l ]
[ 112]
The [110] d i r e c t i o n
is p a r a l l e l t o the l i n e
is p a r a l l e l t o the l i n e F- T.
Three other
d i r e c t i o n s o f i n t e r e s t are the
direction,
parallel
( 100) d i r e c t i o n , p a r a l l e l to T - L , the ( 110)
«■ . *
•
to P-X, and the [110] d i r e c t i o n , p a r a l l e l t o T-U.
Previous experiments and c a l c u l a t i o n s are in e s s e n t i a l agreement
with a Fermi surface c o n s is ti n g o f a double convex,
zone e l e c t r o n disc centered a t
lens-shaped, second
L on each o f the s i x { l 0 0 ] faces bounding
the f i r s t zone and a m u l t i p l y connected f i r s t zone hole s u r f ac e .
e l e c t r o n lenses are due to an overlapping o f Fermi
zones along the l i n e T- L .
The
spheres from adjacent
The f i r s t zone hole s ur fa ce gives r i s e to a
number of closed o r b i t s along w i t h s e v e r a l s e t s of open o r b i t s . E x p e r i ­
mental data and t h e o r e t i c a l c a l c u l a t i o n s
i n d i c a t e regions of co n ta c t
between the Fermi sphere and the zone boundary at the points X and T
in a d d i t i o n to L.
openings
This contact produces breakthrough regions,
in the hole surface a t
these p o i n t s .
i.e .,
These openings a l l o w fo r
the e x i s t e n c e of s e v e r a l a d d i t i o n a l closed o r b i t s on the hole su rf ace
in a d d i t i o n to r e s t r i c t i n g the angular range of the open o r b i t s .
8
III.
A.
Sample Pr ep ara tio n
The samples used in t h is
purity
i n v e s t i g a t i o n were prepared from high
(99*9999$) mercury obtained from the United M in e ra l and Chemical
Company.
flat,
EXPERIMENT
q
Single c r y s t a l s app roximately 3mm in thickness w it h two smooth,
p a r a l l e l surfaces were grown using a modified Bridgeman technique
in the f o l l o w in g manner.
The sample mold,
shown in Fig.
2, was f i r s t
thoroughly cleaned w i t h methyl al c o h o l making sure t h a t a l l f o r e i g n m at ter
was completely removed.
The mold was then sandwiched between two pre­
v i o u s l y cleaned microscope s l i d e s and clamped a t both ends.
The c a v i t y
of the mold was f i l l e d with reagent grade acetone and the high puri ty
mercury immediately in je c te d d i s p l a c i n g the acetone in the c i r c u l a r
part o f the c a v i t y .
Care was taken to make sure t h a t a l l bubbles were
absent from the surfaces in con tact w i t h the microscope s l i d e s .
The
primary purpose o f the acetone was t o pr ovide a f i l m between the mercury
surface and the glass
in order to prevent the mercury from adhering to
the g l a s s .
f i l l e d w i t h the mercury, was then lowered i n t o a
The mold,
dry ic e- me th y l a l c o h o l bath a t the r a t e o f 3 inches per hour.
mercury was completely immersed in the bath,
Once the
the mold was q u i c k l y
removed, placed i n t o a dish c o n ta i n i n g a s i m i l a r bath o f dry i c e methyl a l c o h o l,
and st o r ed
c a r e f u l l y disassembled, and the mercury c r y s t a l removed
in l i q u i d nitrogen.
Se veral s i n g l e c r y s t a l s were prepared in the previous manner, and
six were chosen f o r
i n v e s t i g a t i o n w it h the normal to the p a r a l l e l surfaces
9
1.75
2 .5 0
.4 0 0
FRONT
- < ------
VIEW
1"
---------►
- * - . 5 ( ) O n- * *
250
ii
-------- ►yv
\
; / /
\
v ' /
*
____ t __l
\ ' 4 I
z
- —
^ .125"
60°
END
Fig.
.37 5 "
2
VIEW
10
d i r e c t e d along the
(110),
tallographic directions,
(110),
(112),
respectively.
[llO ],
( 1 1 1 ) , and (100) c r y s -
The o r i e n t a t i o n of each sample
was determined t o w i t h i n ± 1 ° using standard x - r a y techniques.
A t h in
stream of l i q u i d n itr og en was d i r e c t e d over the c r y s t a l during the x - r a y
time to prevent the m el tin g o f the sample.
P r i o r to being x - r a y e d , each sample was e l e c t r o p o l i s h e d
in a s o l u t i o n
of 1$ by volume o f p e r c h l o r i c acid and 99$ methyl a l c o h o l a t dry ice
tem peratures.^
This served a twofold purpose in t h a t any su rface damage
to the c r y s t a l s was removed rendering b e t t e r x - r a y photographs, and i t
in di ca te d whether or not the sample was a s i n g l e c r y s t a l
since any g r a i n boundaries were rev ealed.
immediately
A l l samples were stored in
l i q u i d n itr og en between runs.
B.
Experimental D e t a i l s
Due to the high a t t e n u a t i o n of ultrasound in mercury, a dual
transducer,
pulse transmission technique s i m i l a r to t h a t described by
Kamm and Bohm^ was employed to observe and record the geometric resonance
signals.
Fig.
3-
A block diagram o f the exp erimental apparatus is shown in
The e l e c t r o n i c equipment used was i d e n t i c a l to t h a t described
by Coon e_t a_l.
12
w i t h the a d d i t i o n of two wideband a m p l i f i e r s cascaded
in f r o n t o f the r e c e i v e r to provide a d d i t i o n a l gai n.
A Z - c u t qu a rtz
13
del ay rod J was used to de la y the received pulse f o r 5 M-sec in order
t o a l lo w the r e c e i v e r to recover s u f f i c i e n t l y from the i n i t i a l
m i t t e r pulse which was c a p a c i t i v e l y coupled to the r e c e i v e r .
gold p l a t e d ,
l/k
inch diameter q u a r t z transducers
1*3
trans­
X-cut,
w it h a fundamental
frequency of 15 MHz were e x c i t e d a t an odd harmonic of the fundamental
to generate and re c e iv e l o n g i t u d i n a l sound waves.
The r e c e i v i n g
SPERRY PRODUCTS
ATTENUATION COMPARATOR
MODEL 5 6 A 0 0 I
100 PPS
TRIGGER
PULSED
OSCILLATOR
GENERAL RADIO
TUNABLE LINE
TRANSDUCER
CRT
5 - 2 0 0 MC
WIDEBAND
RECEIVER
SAMPLE
GENERAL RADIO
TUNABLE LINE
WIDEBAND AMP
H . P . - 4 6 0 BR
DELAY ROD
VARIAN 12"
ELECTROMAGNET
WIDEBAND AMP
H.P. - 4 6 0 BR
-TRANSDUCER
DEWAR
PULSE ECHO
S ELEC TO R DEMODULATOR
H.P. MODEL
410 C
VOLTMETER
ASSEMBLY
MOSELEY MODEL
6 0 B LOGARITHMIC
CONVERTER
VARIAN MAGNET]
SUPPLY
MOSELEY MODEL
X-D-2 RECORDER
VARIAN l / H
SWEEP U N IT
12
transducer was bonded to the delay rod w i t h General E l e c t r i c 7^31 varnish
and allowed to dry fo r approximately two days be fo re each run.
The
technique and a co u st ic bonding agent used to bond the delay rod and
generating transducer to the mercury c r y s t a l was c r i t i c a l and nece ssi ta te s
a d e t a i l e d discussion.
I t was necessary to be ab le to make both of the above bonds w h il e
preventing the mercury c r y s t a l from reaching or approaching i t s m el tin g
point.
The normal bonding m a t e r ia ls such as s i l i c o n e o i I s ,
stopcock
greases, and epoxies e i t h e r s o l i d i f y a t high temperatures or refuse to
adhere to the su rface o f the mercury a t
low temperatures.
Numerous
l i q u i d hydrocarbon compounds were t r i e d w i t h the bes t r e s u l t s being
obtained from a high p u r i t y mixture of 5 parts isopentane ( 2-methylbutane)
and one part 3~methyl pentane (3 MP).
Spectrograph!c grade isopentane
and 3 MP were both obtained from the d i s t i l l a t i o n
liquids.
14
o f te c h n i c a l grade
Both l iq u i d s were stored in separate a i r - t i g h t co n tai ne rs
to prevent evap oration w it h a small q u a n t i t y of magnesium sulphate added
to each c on ta in er to remove any water t h a t might have been absorbed.
This removal o f absorbed w ate r was found t o be a c r i t i c a l step in the
bonding procedure as the bonds tended to crack a t a few degrees above
l i q u i d helium temperatures
performed c a r e f u l l y .
if
the water removal s te p had not been
The l i q u i d s were mixed in the proper p r op o rti on
immediately p r i o r t o use.
The isopentane - 3 MP m ix tu r e remained a
l i q u i d down to l i q u i d ni tr o g e n temperatures although i t became extremely
vi scous.
The above mix tur e was used s u c c e s s fu l ly to si m ultane ously bond
the 3 ’ 0 cm long,
1 . 2 cm diameter quartz de la y rod and a 1/ 4 inch
diameter quartz transducer t o opposite sides of the mercury c r y s t a l .
The technique used was as f o l l o w s .
The mercury sample was placed on a
f l a t pi ec e o f dry ice and both major surfaces cleaned w it h a cot to n
swab, being c a r e f u l to remove a l l
su rf ace s.
t ra ces o f a l c o h o l or acetone from the
The d e l a y rod was placed u p r i g h t
in a dish c o n t a i n i n g l i q u i d
ni tr o g e n whose l e v e l was such t h a t the upper th r e e qu ar te rs o f the
delay rod was above the su rf ace of the l i q u i d .
A small drop of the
isopentane - 3 MP m ix tu re was then placed on the u p r i g h t face of the
mercury c r y s t a l and the transducer a p p l i e d .
Immediately, another small
drop o f the m ix tu re was placed on the u p r i g h t s ur fa ce of the de l a y rod
and the sample then c a r e f u l l y l i f t e d up and placed on the rod.
The
weight o f the sample was s u f f i c i e n t t o press out the bond q u i t e t h i n .
Liquid ni tr o g e n was added u n t i l the whole assembly was submerged.
The
e n t i r e c r y s t a l assembly could then be handled and placed in the sample
holder submerged in l i q u i d ni tr o g e n .
This bonding arrangement was found to be s a t i s f a c t o r y
in most
cases y i e l d i n g a strong tra nsmission pulse down t o l i q u i d helium
temperatures.
In cases where e i t h e r or both o f the isopentane - 3 MP
bonds f r a c t u r e d upon immersion in the l i q u i d helium the sample holder
c o n t a i n i n g the sample was l i f t e d above the l e v e l o f the l i q u i d helium,
allowed to warm up s l i g h t l y ,
liquid.
and then slow ly lowered back i n t o the
In some instances t h i s procedure had to be repeated s e v er al
times be fo re a s a t i s f a c t o r y bond was obt ained.
A l l data were taken a t
to thermal phonons.
a c ritic a l
Since mercury is a superconductor a t 1-2°K w it h
f i e l d o f 360 G,
magnetic f i e l d s
1.2°K in order to reduce s c a t t e r i n g due
i t was necessary t h a t the data be taken in
in excess o f t h i s va l u e .
The magnetic f i e l d was set
i n i t i a l l y j u s t below the c r i t i c a l va lu e and swept such t h a t H * was a
l i n e a r f u n c t io n of time,
time,
y i e l d i n g geometric resonances p e r i o d i c in
thereby s i m p l i f y i n g the data a n a ly s is .
The v e l o c i t i e s of propagation of l o n g i t u d i n a l sound waves along
the various c r y s t a l l o g r a p h i c axes were determined e x p e r i m e n t a l l y .
were measured a t
1.2°K and are ta b u la t e d in Table I.
They
15
IV.
THEORY AND DATA ANALYSIS
The gen eral theory of magnetoacoustic a t t e n u a t i o n in metals has
been given by Cohen, Harr is on,
and Harrison
the case o f geometric resonances,
c o e fficien t
is p e r i o d i c
and by Pippard.
For
i t was found t h a t the a t t e n u a t i o n
in the r e c i p r o c a l o f the magnetic f i e l d and t h a t
t h i s period can be r e l a t e d t o C, the k-space c a l i p e r o f an e l e c t r o n
o r b i t on the Fermi surface.
For the standard geometry in which the magnetic f i e l d
H is r o t a te d
in a plane p er pe nd ic ul a r to the d i r e c t i o n o f the sound propagation
■4
q,
the r e l a t i o n f o r the k-space c a l i p e r can be expressed as
c = - * V
(1)
hcM~)
where C is twice the " r a d i a l c a l i p e r " o f the Fermi
d irection
q X ft ,
\
surface
in the
M'flj')
' s the
is the sound wavelength, and
period of the o s c i l l a t i o n s
in r e c i p r o c a l f i e l d .
The necessary con­
d i t i o n f o r observing a s e r i e s of geometric resonances is t h a t
and
cd t > 1 , where q is the sound wave v e c t o r , t
mean f r e e path,
is the c y c l o t r o n frequency, and
s c a t t e r i n g time f o r the e l e c t r o n s .
t h a t an e l e c t r o n complete a t
scattered, while
q£ »
1
sev er al sound wavelengths.
The c o n d i t i o n
q£ »
1
is the e l e c t r o n
t
i s tha cha ra ct e r i s ti c
to t > 1
requires
l e a s t one r e a l space o r b i t befor e being
r eq ui r es t h a t the completed o r b i t encompass
In the case o f mercury,
this
l a t t e r condi­
t i o n could be s a t i s f i e d f o r q u i t e reasonable u l t r a s o n i c f re qu en c ie s .
The assumption is g e n e r a l l y made th a t C measures the extremal pro-»
-*
16-II
q X H. However, Pippard
j e c t i o n of the Fermi
surface in the d i r e c t i o n
has in di ca te d th a t a
l i m i t e d ser ie s o f o s c i l l a t i o n s may a r i s e from
regions which are not extremal i f these regions couple s t r o n g l y t o the
sound wave.
As has been pointed ou t,
19
extremal c a l i p e r s can be uniquely con­
v er te d i n t o radius dimensions of the Fermi surface only i f
the given
Fermi surface sheet has s u f f i c i e n t symmetry so t h a t a set of extremal
c a l i p e r s measured on
i t f o r d i f f e r e n t d i r e c t i o n s of if
a l l occur in a
common plane about a
common c en te r .
given sheet of
the Fermi
This occurs i f a
surface has both r e f l e c t i o n symmetry in a plane per pe ndicular
t o q and in version symmetry about some po in t in t h a t plane.
It
is advantageous a t t h i s po in t to b r i e f l y discuss the meanings
of the terms extremal o r b i t and extremal c a l i p e r .
i n t e r s e c t i o n of the Fermi surface w i t h a plane,
—
f
d i c u l a r to H.
An o r b i t
is the
k^ = con sta nt, perpen-
The set of a l l pos sib le c a l i p e r s of the o r b i t may be
obtained by measuring the normal distances between tangents t o the
o r b i t which are p a r a l l e l to q.
An extremal o r b i t as used in geometric
resonance r e f e r s t o an o r b i t which has a c a l i p e r t h a t remains s t a t i o n a r y
w it h respect to small changes in k^.
is extremal
The c a l i p e r f o r which an o r b i t
is r e f e r r e d to as an extremal c a l i p e r of the Fermi
surface.
Throughout t h i s paper a system o f n o t at io n s i m i l a r t o t h a t of
Ref.
19 w i l l be adopted.
The measured c a l i p e r s w i l l be reduced to
r a d i i whenever allowed by symmetry and w i l l be denoted by the symbol k
w it h a s u p e r s c r ip t to i d e n t i f y the d i r e c t i o n and a sub sc rip t to i d e n t i f y
the p a r t i c u l a r sur face.
For example,
L -r
k, _.IC r e f e r s to the r a d i a l
Lh.Nb
c a l i p e r of the lens measured from L toward T.
Ca lip er s obtained from
o r b i t s not having s u f f i c i e n t symmetry t o permit a reduction to r a d ii
w i l l be denoted by the symbol C with an eq u iv a le n t n o t a t io n .
Radial
calipers w il l
h e r e a f t e r be r e f e r r e d to as k - c a l i p e r s w h il e di am et ra l
calipers w i l l
be r e f e r r e d t o as C - c a l i p e r s ,
a k - c a l i p e r where allowed by symmetry.
i.e .,
aC - c a l i p e r
is twice
I t w i l l be assumed t h a t a l l
c a l i p e r s presented here ar e due to ex tr em al o r b i ts unless otherwise
spec i f i ed.
Experimental extremal c a l i p e r s were c a l c u l a t e d using
per iods, A(^), were determined from the experimental data
Eq.
(l).
The
using a
r e l a t i o n of the form
- fiOfjHn + V)
,
(2)
n
where n is the resonance number, and y is a phase f a c t o r which is a
f u n ct io n of n f o r small n but r ap id ly approaches a constant value as n
increases.
Since the a t t e n u a t i o n t h e o r e t i c a l l y reaches a
relative
maximum a t i n t e g r a l values o f n and a r e l a t i v e minimum a t
h a lf integral
val ues , y can be determined from a p l o t o f 1/H^ vs n with
the best
straight
l in e
fit
through these points e x t r a p o l a t e d to X/Hn =
For low magnetic f i e l d s such that the extremal dimension
orbit
is much longer than the wavelength X,
p e r i o d i c in H *.
for
of an
the resonances ar e s t r i c t l y
This corresponds to the high phase region and occurs
large values of n.
fie ld
0.
The low phase regio n occurs when the magnetic
is high enough such t h a t X ?s a s i g n i f i c a n t f r a c t i o n o f the
o r b i t diameter.
The phase o f the o s c i l l a t i o n s then s h i f t s from i t s
low f i e l d asymptotic value and the o s c i l l a t i o n s are no longer s t r i c t l y
p e r i o d i c in H
Deviations from the H * p e r i o d i c i t y occ ur ring in the
18
low phase region are thus e a s i l y det ect ed.
A knowledge of the phase f o r la rg e n is use ful
o s c i l l a t i o n s and can y i e l d
responsible f o r them.
in i d e n t i f y i n g the
information about the nature of the surfaces
T h e o r e t i c a l consid eratio ns
y = O.3 7 5
20
i n d ic a t e th at
Y = 0. 25
f o r a c i r c u l a r c y l i n d e r and
f o r a s p h e r ic a l
sur fa ce .
The data obtained f o r the lens surface of mercury yi el de d a
value of
y = O.2 7 + O.O7 .
19
V.
EXPERIMENTAL RESULTS AND DISCUSSION
A.
1.
E le ct r on Surface
Results f o r q p a r a l l e l t o the (110) d i r e c t i o n .
presents a summary of a l l
( 110) plane; q is in the
Figure k
the C - c a l i p e r s obtained from data
in the
( 110) d i r e c t i o n w h i l e 0 measures the angle
of the c a l i p e r d i r e c t i o n from the t r i g o n a l a x i s .
A t o t a l of ten
d i s t i n c t resonance branches were observed w it h each o f the branches
being designated w i t h a Greek l e t t e r .
two,
namely
Of these ten resonance branches,
and o t have been assigned to the second band e l e c t r o n
surface.
The resonances associated w it h the
branch had the l a r g e s t
amplitude and have been assigned to c a l i p e r s o f a cross sec tio n of the
second zone e l e c t r o n
(110) plane.
po la r p l o t
the
lens surface centered on the po in t L^ in the
The corresponding k - c a l i p e r data f o r a ^ is shown on a
in Fig. 5, along w it h a p r o j e c t i o n of the B r i l l o u i n zone on
( 110) plane w it h the f r e e e l e c t r o n lenses sketched in f o r comparison.
For n o t a t io n
and L^.
purposes,
the ce n te r o f the lenses ar e
The e r r o r s presented r e f e r only
mination of the per iods.
denoted as Lj ,
to u n c e r t a i n t i e s
L^,
in the d e t e r ­
The extremal dimensions of the cross sec tion
centered on L^ have been found to d i f f e r s l i g h t l y from the values
pr e v i o u s ly reported
2
due to a small m i s o r i e n t a t i o n of the c r y s t a l from
which the o r i g i n a l data were taken.
The r e s u l t s of
i n d ic a t e a value f o r the minimum radius,
L-r
k.
LENS
the present data
^
°-1
, of 0 . 1 7 6 ± 0.004A
20
2.4
!......
I
a
2.2
1
13 i
«2 .'
1
Loucks
® B rand t ft
■ Present
Data
*
(in
Calipers
—
1. 6
~ a/ 7 7
--
9
1 ,'
- t
[IT 23
t
(001)
t
[001]
t
[NO]
(3
I
[III]
• • •
—
—
G|
A
'*
—
—
,
—
'•
.8 —
s
.6 —
.
—
.
•
.4
.2
•' t • [II2 ]
. t
(NO)
-
1. 2
»
Experim ental
_
ffl •
1.8
1.0
1
Rayne
2.0
I .4
1
• / i2
f 1 1•a*• . af • ••
«*• •«
—
fi
a
0
I
-90
1
-60
1 _______ L .
-30
Angle
F ig . b
0
From
1
1
1
30
60
90
Trigonal
Axis
.
21
Y
-6 0 °
T
T
1
T
7
60°
90
\
A
K
A
0 .5 A
30
(
( 112)
112 )
Fig. 5
22
L-u
and a value f o r the maximum radius,
o_ i
0-538 ± 0.010A , whereas
L_r
o_ i
the values reported i n i t i a l l y were 1<leNS = 0. 180A
respectively.
Loucks
21
l-U
o_ j
and l<LENS = O.565A
,
has reported dimensions obtained from his RAPW
L_ r
o_ 1
c a l c u l a t i o n of I<lenS = 0 . 215A
and
l-U
e x c e l l e n t agreement on the value of ^ ^ ^ 5 ’
0-1
= O.5 4 OA ,
thus p r ov id in g
The cross s e c t i o n a l area
of t h i s sec tio n of the lens was determined g r a p h i c a l l y and found to be
O -p
0.299A
O- 2
which is in good agreement w i t h the value of O.3 O5A
by Brandt and Rayne from dHvA data.
determined
A comparison of the extremal areas
of t h i s cross sec tion, along w it h the major and minor c a l i p e r s as d e t e r ­
mined by d i f f e r e n t methods,
is shown in Table I I .
the r e s u l t s of a k pseudowave
PW) c a l c u l a t i o n to be discussed l a t e r .
The C - c a l i p e r s designated o>
in Fig. k have been assigned t o a
p r o j e c t i o n o f one of the e l e c t r o n lenses onto the
projection
jection
is shown centered on the po in t
in Fig. 5 .
Resonances
This t a b l e includes
or
(110) plane.
This
on the zone pro­
corresponding to these c a l i p e r s were
observed over an i n t e r v a l of about 22° in the v i c i n i t y of the (0 0 1 )
direction.
o_l
The C - c a l i p e r s ranged from a value o f 1.08 ± O.OA-A
to
0-1
1.20 ± 0.02A
w ith con siderable s c a t t e r in the data.
k - c a l i p e r values are a l s o shown in p o l a r p l o t
2.
The corresponding
In Fig. 5*
Results f o r q p a r a l l e l t o the (110) d i r e c t i o n .
a p o la r p l o t of the data f o r q in the
the (110) plane.
A projection
(110) d i r e c t i o n
Figure 6 shows
and H ro t a te d
of the B r i l l o u i n zone onto
in
the (110)
plane w it h the f r e e e l e c t r o n lenses sketched in f o r comparison is shown
in the same f i g u r e .
Only two d i s t i n c t resonance branches were observed
f o r t h i s o r i e n t a t i o n as a r e s u l t o f the complete domination of the
s i g n a l by resonances due t o the lens.
p r o j e c t i o n o f the lens centered on
One branch was assigned t o the
onto the
( 110) plane w it h the
23
TABLE I I .
Comparison o f c a l i p e r s and areas o f lens
obtained by d i f f e r e n t methods in ( 110) plane.
1_ T
Source
o_i
k LENS <A
>
k L“ U ( A ' 1)
LENS
’
Area
°-p.
(A )
Free e l e c t r o n
0.220
0 .7 3 0
0. 441
dHvA 3 PWa
0.204
0 .600
0.344
8 PWb
0 .2 2
0.64
0.423
RAPWC
0.215
0.540
0.354
O . 3O5
dHvA
Present c a l c u l a t i o n
o. 176
0.540
0 .3 0 0
Present experiment
0 .1 7 6
0 .5 3 8
0.29 9
a See Ref.
1.
^See Ref. 5*
c
See Ref. 21*
2k
1
T
T
T
60°
90'
A
to o i]
0.5 A
/ I
30*
( 110)
ml
•••
.*
Fi g. 6
*+-4
t-. ***
i-T*1
25
remaining branch being assigned to a s i m i l a r p r o j e c t i o n of the lenses
centered on
and L^*
of the sample,
the p r o j e c t i o n s o f the surfaces centered on
should be i d e n t i c a l
I t should be noted t h a t f o r p e r f e c t alignment
and
i f the lenses ar e surfaces of r e v o l u t i o n about the
T” L l i n e .
With the magnetic f i e l d d i r e c t e d along the [ 0 0 l ] d i r e c t i o n ,
maximum c a l i p e r o f the p r o j e c t i o n o f the su rf ace centered on
0°
°-1
C.
= 1.28 ± O.O3A .
found to have a value of
the
was
This c a l i p e r corresponds
2
to the remaining symmetry a x i s c a l i p e r not obtained from the
With the magnetic f i e l d
in the
(110) d i r e c t i o n ,
of the p r o j e c t i o n of the lens centered on
°-l
O. 5 2 ± 0.01A .
( 110) data.
the minimum c a l i p e r
was found to be
0C
90°
2
A consistency check of t h i s dimension can be made from
a p r o j e c t i o n o f the cross se c ti o n of the lens obtained from the bi na ry
data onto a plane pe rpe ndi cul ar to the (110) d i r e c t i o n .
Such a pro-
°-1
j e c t i o n y i e l d e d a value O.5 I ± Q.01A , which compares q u i t e w e l l w ith
the above c a l i p e r .
The maximum dimension of the p r o j e c t i o n o f the lenses centered on
Lj and
was obtained w i t h the magnetic f i e l d
The data i n d i c a t e
qno
qno
C.
= C.
-
L1
d i f f e r s s l i g h t l y from
90°
C.
Li
l3
and
in the
o_i
1.20 ± O.O3A .
90°
C,
L3
( 110) d i r e c t i o n .
no
The f a c t t h a t C.
l2
is evidence t h a t the lenses
are not e x a c t l y c i r c u l a r since a c i r c u l a r shape would imply t h a t a l l
three of these dimensions ar e equ al .
The minimum dimension
wa s obtained w i t h the magnetic f i e l d
in the [OOl] d i r e c t i o n .
t h i s region the data s p l i t
0°
0°
C.
.
Ll> l 3
In
i n t o two branches y i e l d i n g two separate
values f o r C.
. .
One s et of data in d ic a t e d a value of 0. 81 ±
l 1>l 3
o_ i
A
o_i
0.02A , w hi le the other s et gave a value of 0 .8 3 ± 0.02A .
This
sp lit
in the data can be a t t r i b u t e d t o a misalignment o f the sample
w i t h the sound propagating about 1° o f f the (110) d i r e c t i o n toward
the
(110) d i r e c t i o n .
3.
Results f o r q p a r a l l e l to the (112) d i r e c t i o n .
A t o t a l of
f i v e d i s t i n c t resonance branches were obtained f o r q p a r a l l e l
(112) d i r e c t i o n and H r o t a te d in the (112) plane.
branches are shown in a p o la r p l o t
in Fig.
to the
The two primary
7 along w i t h a p r o j e c t i o n of
the B r i l l o u i n zone onto the (112) plane w it h the f r e e e l e c t r o n lenses
sketched in f o r comparison.
One of these primary branches has been
assigned to a p r o j e c t i o n of the lens centered on Lj onto the
(112)
plane w it h the other primary branch being assigned to a s i m i l a r pro­
j e c t i o n of the lens centered on
Both of these p r o j e c t i o n s should
be symmetrical and should i n t e r s e c t a t 0 ° and 90°
s e n ta t i o n of Fig.
7 i f the c ry s ta l
in the p o l a r r e p r e ­
is pr o p e rl y o r i e n t e d .
There is no
d e f i n i t e evidence of resonances corresponding to c a l i p e r s of the
p r o j e c t i o n of the t h i r d
lens centered on L^-
However,
the o r b i t s
which should give r i s e t o these resonances have a small radius of
cu r va tu r e over the region to be c a l i p e r e d .
I t has been pointed out
22
t h a t the amplitude of the geometric resonances should depend q u i t e
s t r o n g l y on the radius of cur va tu re of the o r b i t .
radii
Orb its w i t h lar ge
of cur va tu re should produce l a r g e r amplitude resonances than
those w it h small r a d i i
the sig nal s from the
of c u r v a tu r e .
In l i g h t o f t h i s f a c t one expects
resonance to be weak.
The c r y s t a l from which the data in Fig.
7 was obtained was mis-
or i e n t e d w it h the b i s e c t r i x plane ro ta te d about 5 degrees about the
trigonal axis.
This m i s o r i e n t a t i o n is r e a d i l y obvious from the data
since the shapes and the areas of the p r o j e c t io n s of the two lenses.
2?
60
90
-60
0.5 A
30
-30
•*
%
&
( 110)
Fig .
T
28
Lj and Lg, d i f f e r co n s id e r a b l y .
the two p r o j e c t i o n s
of 0 ° and 90°.
The e x p er im en ta l data
indicate that
i n t e r s e c t a t p o l a r angles o f 20° and 88 ° instead
The minimum C - c a l i p e r dimension o f the
projection
o_ i
was found to be 0 . 6 9 ± 0.01A , w h i l e the minimum dimension of the
o—l
p r o j e c t i o n was found to be O.5O ± 0. 01 A .
The c r y s t a l misalignment
consequently accounts f o r a d i f f e r e n c e of roughly 39$ in the minimum
dimensions.
In c o n t r a s t ,
the maximum d i a m e t r a l dimensions of the two
p r o j e c t i o n s agree q u i t e w e l l w it h th e
° - l
of 1 . 1 2 ± 0.02A
0-1
0.02A
h.
p r o j e c t i o n having a dimension
and the Lj p r o j e c t i o n having a dimension o f 1.09 ±
.
Results f o r q p a r a l l e l to t h e
(10 0) ,
(111),
and [ 1 1 0 ] d i r e c t i o n s .
In a d d i t i o n t o the t hr ee o r i e n t a t i o n s j u s t discussed, an at te m p t was
made to ob ta in data on th e lens s ur fa ce s fo r q d i r e c t e d in the
(ill),
( 100) ,
and [ 110] d i r e c t i o n s .
The (100) d i r e c t i o n
is most i n t e r e s t i n g si nc e w it h H r o t a t e d
in
the ( 100) plane one should be able t o obt ai n the main cross s e c ti o n of
the lens cut by the
(100) plane.
However, due t o the st ro ng open o r b i t
abs or pt io n t h a t e x i s t s
in t h i s d i r e c t i o n ,
the maximum frequency
o b t a i n a b l e was 45 MHz,
in which case onl y a maximum of about two
o s c i l l a t i o n s were observed before th e s i gn al s a t u r a t e d .
At higher
frequencies the s i g n a l s a t u r a t e d w i t h no resonances being observed.
The lack of amplitude o f the lens resonance in t h i s plane could a l s o
be a t t r i b u t e d
t o the l a r g e cu r v a tu r e of the o r b i t s over the region to
be c a l i p e r e d as was the case w ith the lens ce n te re d on
plane.
tion.
Thus no a d d i t i o n a l
in the
( 112)
info rm at io n was ob ta in ed from t h i s o r i e n t a ­
29
An attempt was made to propagate the sound along
and obta in data f o r H in the [ l l O ] plane.
the [ 1 1 0 ] d i r e c t i o n
This d i r e c t i o n
is of
i n t e r e s t because i t would y i e l d the symmetric cross s e c ti o n of the lens
cut by the. II.W plane.
This cross s e c ti o n forms, along w i t h the two
cross sections from the ( 110) and ( 100) planes,
orthogonal cuts of the lens.
the set o f three
0°
A consistency check on the val ue o f
'
C.
-*
could have been made since w it h q p a r a l l e l to [ 1 1 0 ] and H p a r a l l e l t o
(1 12 ),
2
0°
C.
2
the r e s u l t of not
the c a l i p e r obtained from the lens should be i d e n t i c a l t o
obtained from the (110) dat a.
This at te m p t f a i l e d as
being ab l e to make a successful a co u st ic bond between
the delay rod
and the [ 110] sample.
With q p a r a l l e l to the (111) d i r e c t i o n and H r o t a te d
plane,
in the
(111)
resonances were observed which corresponded to c a l i p e r s o f the
th re e i d e n t i c a l e l l i p t i c a l cross sections of the e l e c t r o n
lens.
Since
these three sections give r i s e to resonances a l l o f about the same
amplitude w it h two o f the sec tio ns, and in some instances a l l
having ap pro ximately the same c a l i p e r va l u e ,
the a d d i t i o n a l
obtained from the present data was not of s u f f i c i e n t
three,
info rm at io n
i n t e r e s t t o be
repo rted .
5.
Summary of e l e c t r o n d a t a .
In summary,
t hr ee major symmetry
dimensions of the second zone e l e c t r o n lens have been determined
e x p e r im e n t a l ly .
The values
obtained a r e
L- r
°-1
= 0. 176 ± O.GQUA ,
L-U
°-l
0°
°-1
i-u
kLENS = 0 ‘ 538 ± ° - 010A > and c l
= 1-28 * O.O3A
or k^EjJs = 0 . 6 4 ±
0-1
2
0.015A , r e s p e c t i v e l y .
The f i r s t two dimensions, along w it h i n t e r ­
mediate c a l i p e r s ,
give a cross s e c t i o n a l area
in the ( 110) plane' t h a t
agrees w i t h i n 2$ of the experimental dHvA data.
dimension,
0°
The remaining
, gives the maximum diameter of the lens along the L-W
30
line.
2
The f a c t th at the diameters of the lens along L-U and L-W,
LhN5
i.e .,
and C? , d i f f e r by about 8$ in di c at es t h a t the lens is not a
Lg
surfa ce of r e v o l u t io n about the l i n e T- L.
6.
Pseudopotential c o e f f i c i e n t s .
The pse udopotential c o e f f i c i e n t s
corresponding to the planes bounding the f i r s t B r i l l o u i n zone in mercury
have been estimated by comparing the geometric resonance data obtained
from the lens wit h the r e s u l t s of a fou r pseudowave c a l c u l a t i o n neg le c tin g
spin o r b i t coupling.
sation.
Furt her ,
th ere was no attempt to ma in ta in compen­
D e t a i l s of the c a l c u l a t i o n are presented in Appendix A.
The r e s u l t s of the c a l c u l a t i o n f o r the
shown in Fig.
8 compared to the data.
(110) cross sec tio n ar e
I t should be noted t h a t the 4 PW
method y i e l d s reasonable agreement w it h the data
i f the Fermi
level
depressed to O.5O5 Ry, about 4 . 0 $ below the f r e e e l e c t r o n va lu e.
is
The
dHvA area f o r the (110) cross sec tio n of the e l e c t r o n su rface and t h a t
in di ca te d by the present measurements ar e in e x c e l l e n t agreement, both
w i t h each ot her and the present 4 PW r e s u l t s .
Experimental and present t h e o r e t i c a l r e s u l t s fo r both the e l e c t r o n
and hole bands were compared a t sev er al points of i n t e r e s t
and ar e summarized in Tables I I ,
III,
and IV.
in the zone
A survey o f the r e s u l t s
o f the c a l c u l a t i o n shows general agreement w it h the t o p o l o g i c a l f e a t u r e s
of the Fermi surface of mercury as suggested by ot her workers as w e l l
as by the present experiment;
ment.
Indeed,
however,
the present c a l c u l a t i o n
f o r a rigorous band c a l c u l a t i o n ,
t her e remains need f o r
improve­
is not presented as a s u b s t i t u t e
but only as a method by which some
of the a v a i l a b l e experimental data may be compared.
The 4 PW r e s u l t s , along w i t h other e x i s t i n g c a l c u l a t i o n s , w i l l be
compared to the data from the hole su rface in the f o l l o w i n g s e c ti on .
31
( II I)
0.5 A'1
(001)
[no]
Fig. 8
32
TABLE 111.
Comparison of r a d i i of openings in
f i r s t zone hole sur fa ce .
*
Values a r e
O-1
in A .
X-K
. L-U
1N
. X-U
IN
. T-U
1N
RAPWa
0.948
0 .1 1 7
0 .Il7 f
0 .117
HAAa
0.831
0,354
0 . 382 ^
0 . 382^
0.347
0 .2 6 0
0 . 198t
0 . 198f
0.20 9
Model
MAG Va ...........
--------0 .8 6 0 -----
T-W
IN
kin
t
0 .1 1 5
8 PWa
0.874
0 .1 7 7
0. 0 7l t
0 . 0711,
0.2 09
dHvA 3 PWa
0. 897
0.299
0-324t
0 .324^
0. 182
Present
c a l c u l a t i on
O.87O
0.300
0.2 7 0
0 .3 4 0
0 .1 8 5
Present
experiment
O.9 0
0 .3 0
For an ex p la n a ti o n of n o t at io n see Figs.
a See Ref. 5 .
t
C i r c u l a r approximation.
10 and 11.
33
Comparison of outside dimens i ons o f the T - s e c t i on
TABLE IV.
*
as measured from L, T, and X.
°-1
Values are given 1n A .
. X-U
OUT
T-W
OUT
O.78
O.765
0-935
1.18
0.79
0-79
1.00
1. Ik
O.7 7
0 .7 6 5
0 .9 0
k L_U
*0UT
T-U
OUT
dHvA 3 PW3
1.16
Present c a l c u l a t i o n
Present experiment
Method
"For an ex p la n a ti o n of n o t a t i o n ,
aSee Ref.
1.
see Figs.
10 and 11.
Furt her ,
the k PW c a l c u l a t i o n w i l l a l lo w some e x t r a p o l a t i o n of the
exp erim ent al r e s u l t s to c a l i p e r s not d i r e c t l y observed.
theory is f i t
The present
to the e l e c t r o n su rf ac e , as opposed to the method of
e a r l i e r workers.
Thus i t
is to be expected t h a t the l a r g e s t discrepancies
between theory and experiment w i l l be found f o r the sm al le r o r b i t s on
the hole surface such as the (3 o r b i t s which w i l l be discussed l a t e r .
As w i l l be evidenced in the next s e c t i o n ,
the l a r g e r hole surface c a l i p e r s
in general show b e t t e r agreement between the present c a l c u l a t i o n and
experiment than do those due to the sm alle r sections of the hole
surface.
B.
1.
Hole Surface
Results f o r q p a r a l l e l to the (110) d i r e c t i o n .
branches designated by p,
e^,
e^, 6, y, p^, p^, anc*
The resonance
shown in Fig. k
have a l l been assigned to C - c a l i p e r s ass oci ate d w i t h various o r b i t s on
the f i r s t zone hole sur fa ce .
w i t h the labeled o r b i t s
A r e p r e s e n t a t i o n of t h i s surface along
is shown in Fig. 9-
Among the more i n t e r e s t i n g
o r b i t s from t h i s set are those designated by e^,
denoted by
g^
e^, &, and Tj.
The branch
has been assigned as the c a l i p e r of an o r b i t centered on
L around the insid e of the hole su rface w it h H in the v i c i n i t y of the
(001) d i r e c t i o n .
This o r b i t was observed over a range of about 6°
and y i e l d e d a r a d i a l c a l i p e r
°-l
0.02A .
° -
t h a t v a r i e d from 0.91 ± 0.02A
The branch denoted by
in the same plane as
g ^,
1
to 1.03 ±
has been assigned t o an o r b i t l y i n g
but moving on the outside o f the hole surface
and threading through the openings in the T faces.
observed over the same range as
,
o_I
v a r i e d from 1.16 ± 0 . 02A
g^
This o r b i t was
and y i e l d e d a r a d i a l c a l i p e r t h a t
o_ 1
to 1.18 ± 0 . 02A
35
m m S m
Fi g. 9
The 7] branch has been assigned t o a hexagonal shaped o r b i t centered
a t T completely en cl os ing the T face which occurs when the magnetic
fie ld
is d i r e c t e d
in the v i c i n i t y of the t r i g o n a l ax i s .
This o r b i t
was observed over an angular range o f about 12° on e i t h e r side o f the
( 111) d i r e c t i o n and gave r a d i a l c a l i p e r s th a t v a r i e d from a maximum
o-l
,,
°-1
o f O.8 3 ± 0.02A
to a minimum value o f 0 .6 6 ± 0.02A
w i t h the f i e l d
tilte d
12° toward [ 0 0 l ] .
The r a d i a l c a l i p e r obtained w i t h the f i e l d
o_ 1
along the (111) d i r e c t i o n was O. 7 7 ± 0.02A .
The 6 branch has been assigned t o an o r b i t on the i n si d e o f the
opening on the X face centered on the point X.
w it h the magnetic f i e l d
This o r b i t was observed
in the v i c i n i t y of the
( 110) d i r e c t i o n and was
d e t e c t a b l e over .a range of approximately. 80 on ei ther s i de of the
( 110)
o_i
d i r e c t i o n gi v i n g a r a d i a l c a l i p e r of O . 3O ± 0.01A
fo r t h i s p a r t i c u l a r
direction.
The data obtained from these o r b i t s are p l o t t e d
plane in Fig.
8 along w i t h the ff
c a l i p e r s obtained from
data f o r comparison.
Note t h a t
e^, T|, and 6 almost com plet ely determine
the cross sec tion of the hole surface centered on U.
sec tio n corresponds to the area enclosed by the
Rayne.
in the (110)
The cross sec tio n generated by the
t
t
This cross
o r b i t o f Brandt and
o r b i t w i l l be r e f e r r e d
to as the T - s e c t i o n .
The r e s u l t s of the 4 PW c a l c u l a t i o n
the present data are shown in Fig.
8.
in the
(110) plane based on
As is e v i d e n t ,
the f i t
to the
lens section is ext remely good w it h the extremal dimensions and
enclosed area agreeing w i t h i n exp erim ent al e r r o r .
T - s e c t i o n is f a i r l y good w it h the c a l c u l a t i o n
s l i g h t l y outsi de the exp erim ent al data.
The f i t
to the
l y i n g f o r the most parjt
I t should be noted t h a t most
o f the experimental data assigned to the T - s e c t i o n was taken from
37
o rb its
w ith
d im en sio n s
the
rath er
of
larg e
these
T -s e c tio n .
The
d im en sio n s .
o rb its
area
im ply
of
the
Thus
la rg e
errors
errors
T -s e c tio n
has
o -c
present c a l c u l a t i o n and found to be 0 . 132A
.
on
in
the
been
d eterm in in g
the
sm all
of
s c ale
d eterm in ed
This
from
the
is co ns id er abl y
°-2
sm alle r than the exp er imen ta l dHvA area of 0 . 151A , but agrees w it h
0-2
the RAPW c a l c u l a t i o n of 0 . I 32A .
The dHvA 3 PW c a l c u l a t i o n gives a
o_ p
value of 0 . IO7A
o_2
w h i l e the 8 PW c a l c u l a t i o n y i e l d s a value o f 0 . I 5 IA
in agreement with the ex pe rim ent al dHvA data.
w h il e the 8 PW c a l c u l a t i o n f i t s
T-section,
it
area f o r the
It
is to be noted t h a t
the dHvA ex pe rim ent al data f o r the
is in disagreement w i t h the corresponding exp erimental
( 110) lens sec tio n by about 27$, g i v i n g a value of
o_2
o_2
0.423A
as compared to the exp erimental area of O . 3 O5A .
The present
c a l c u l a t i o n gives good agreement f o r the area of the lens but is in
disagreement wit h the measured dHvA area o f the T - s e c t i o n by about 15$.
There ar e sev er al
a
know ledge
of
the
important dimensions th at can be obtained from
T -s e c tio n .
in the X, T, and L faces.
These
in c lu d e
the
w id th
of
the
openings
Figure 10 indicates these dimensions along
w it h the corresponding n o t a t io n w h i l e the values a r e given In Table I I I .
Data obtained from the 6- o r b i t w i t h the magnetic f i e l d
in the
(110)
d i r e c t i o n y i e l d a value f o r the major dimension of the opening in the
X-u
o_i
X face of k|^ = O.3O ± 0.01A
which is p r e c i s e l y the same val ue
obtained from the k PW c a l c u l a t i o n .
The dimensions o f the opening in
the T face were determined e x c l u s i v e l y from the k PW c a l c u l a t i o n
since u l t r a s o n i c data could not be obtained fo r t h i s d i r e c t i o n .
The
lack of data from o r b i t s from which the s i z e of the opening could be
determined was due to the f a c t t h a t resonances from the lens and T|
o r b i t s completely dominated the a t t e n u a t i o n and consequently obscured
38
\ X
T-U
OUT
T- U
Fig.
10
39
weaker resonances.
The c a l c u l a t i o n
in d ic a t e s a minimum va l u e f o r the
j_U
o_j
j_ y
wid th of the opening o f k |^ = O.27A
and a maximum w idth o f k| ^
=
o_ 1
0.340A •
A c i r c u l a r approximation t o the opening has been made in
T-U
T-W
°-1
the 8 PW c a l c u l a t i o n and gives a value o f k... = k...
= O.O7 IA , w h i l e
I N I N
o_l
the dHvA 3 PW c a l c u l a t i o n gives a r e s u l t o f 0.324A
f o r the same
dimension a l s o in a c i r c u l a r app ro xim at io n.
The w id t h of the opening
in the L face should be measured d i r e c t l y from the
orbits.
However,
t h i s opening can be b e t t e r de f i n e d from the T] o r b i t centered on T as
is e v i d e n t from Fig.
3
0-1
0.02A
8.
The data
i n d i c a t e a value o f k^.,^ = 0 . 9 0 ±
IN
l-u
as compared t o a c a l c u l a t e d h PW va lu e o f k ^
° - 1
= O.87OA
A comparison o f the va ri ous breakthrough dimensions on the hole s ur fa ce
as determined by d i f f e r e n t methods is given in Table 111.
Three o th er dimensions t h a t can be determined from the e x p e r i ­
mental data a r e the ou t s i d e c a l i p e r s o f the T - s e c t i o n measured from
the poi nts X, T, and L.
kQuj as i n d i c a t e d
These c a l i p e r s a r e denoted as kgg!j!, ^OUT* anc*
in Fig.
10.
The e x p er im en ta l r e s u l t s a r e t a b u l a t e d
in Table IV along w i t h the corresponding k PW and dHvA 3 PW dimensions
T-U
The dimension k g ^ was determined d i r e c t l y from the
f o r comparison.
X-u
T] o r b i t s w h i l e k g ^ was determined i n d i r e c t l y from the
e x p e r im e n t a l d e t e r m i n a t i o n o f kjr.,1^ u t i l i z e s
OUT
exp lai ne d l a t e r
in t h i s
orbits.
The
the dimension C*- *"* as
t
text.
The c a l i p e r s designated f3 have been assigned to o r b i t s around
the arms on the hole su r f a c e ext en d in g along the (lOO) d i r e c t i o n s as
I n d ic a t e d in Fig.
9-
Resonances corresponding t o these c a l i p e r s were
observed over an an g ul a r range of about 18° and gave a value o f . 0 6 5 ±
o-l
0.010A .
The minimum c a l i p e r should occur w i t h the magnetic f i e l d
d i r e c t e d along the [lOO] d i r e c t i o n ,
but u n f o r t u n a t e l y the
resonances
40
were obscured about 8° from t h i s p o i n t .
in d ic a t e s a minimum c a l i p e r
The dHvA 3 PW c a l c u l a t i o n
in a c i r c u l a r approximation of ap pr ox im at el y
°-1
o_ i
0.04A
w h i l e the present c a l c u l a t i o n gives a value of 0.06A .
It
might be pointed out t h a t dHvA type o f o s c i l l a t i o n s have been observed
in the a co u st ic a t t e n u a t i o n w i t h the magnetic f i e l d
in the range of
7 t o 12 kG in a separate obliq ue f i e l d experiment.
These o s c i l l a t i o n s
give areas assignable to the p o r b i t s which agree w i t h the data of
Brandt and Rayne.
The c a l i p e r s denoted by y were assigned to an o r b i t t h a t passes
through the opening in the T face as i l l u s t r a t e d
in Fig.
9-
Ex peri­
m e n t a l l y t h i s o r b i t was observed over a range o f app ro xi m at el y 4 0 °
o_ 1
o_ 1
g i v i n g C - c a l i p e r s t h a t v a r i e d from O.3 8 ± 0.04A
to O.58 ± 0.04A
When the magnetic f i e l d
is in the (112) d i r e c t i o n ,
be r e f e r r e d to as a t 1 o r b i t
in which case the height of the o r b i t
o_ 1
11 was found to be 0 . 4 7 5 ^ 0.04A
as compared
o_ 1
o„ 1
t o the 4 PW value of O.52OA
and a dHvA 3 PW value of 0.420A
denoted
(H I)
C^,
the y o r b i t w i l l
in Fig.
The two remaining resonance branches In the (110) plane have been
assigned t o c a l i p e r s associated w it h o r b i t s p^ and p^, w it h p^ being
g
f i r s t proposed by Keeton and Loucks.
p^ was observed over a range of
approximately 3 0 ° w i t h C - c a l i p e r values ranging from a minimum of
°_ 1
o_ 1
0. 20 0 ± 0.020A
t o a maximum of 0 .4 40 ± 0.040A .
fie ld
With the magnetic
in the (001) d i r e c t i o n the C - c a l i p e r obtained f o r the p^ o r b i t
w i l l c l o s e l y match the d i a m e t r a l c a l i p e r of the T - s e c t i o n along the
L-U l i n e .
Fig.
This dimension is denoted as
10 t h a t kjr..!^ = k^..^ + C*"
OUT
IN
T
C^
T
and i t can be seen from
*
The ex pe rim ent al data i n d i c a t e a
value of C^ ^ of 0. 240 ± 0.020A * as compared to the dHvA 3 PW value
o_l
o_ I
o f 0.2o0A
and a value of 0 . 3 IA
obtained from the present 4 PW
kl
T-W
T-W
OUT
Fig.
11
42
calculation.
l__U
then
Using the values of
o_ i
= 1. 14A
and
determined experimenta 1 l y ,
is obtai ned which is in reasonable agreement w i t h
o_l
the present c a l c u l a t e d va lu e of 1 . 18A .
The resonance branch
been assigned t o a "bowtie o r b i t " as i n d i c a t e d in Fig.
9-
^as
This o r b i t
was observed over an angular range of 10° w i t h C - c a l i p e r s
ranging from
0 .2 58 ± 0.020A" 1 to 0. 16 6 ± 0. 02 0A " 1.
The present c a l c u l a t i o n f o r the w i d t h of the openings
X faces
i n d i c a t e th a t the H || (110) open o r b i t ,
Dishman and Rayne,
5
can e x i s t .
in the T and
f i r s t observed by
I t was impossible to d e t e c t any
resonance due to t h i s open o r b i t since the a t t e n u a t i o n a t f i e l d s
corresponding to low i n te g e r values o f the open o r b i t resonance was
very high.
2-
Results f o r q p a r a l l e l
to the
(112) d i r e c t i o n .
Three resonance
branches were observed in t h i s d i r e c t i o n which could be assigned to
o r b i t s on the hole su rf ac e .
There was c on si d er ab le evidence f o r the
e x i s t e n c e o f the (3 o r b i t s w i t h the data
indicating ra d ia l calipers
°-l
t h a t ranged from a minimum o f 0 . 0 8 ± 0. 01A
°-l
0.020A .
With the magnetic f i e l d
in the
to a maximum o f 0.181 ±
(112) plane,
a direction
was never assumed f o r which the'minimum r a d i a l c a l i p e r o f
could be obt ain ed.
As was poi nt ed out e a r l i e r ,
the (3 o r b i t
the minimum r a d i a l
o_ 1
c a l i p e r f o r t h i s o r b i t should be a p p ro xi m at el y 0.06A
With the magnetic f i e l d
In the v i c i n i t y of the (111)
direction
th er e was strong evidence of the *1] o r b i t discussed p r e v i o u s l y .
This
o r b i t was observed over an i n t e r v a l o f 10° and gave a r a d i a l c a l i p e r
of
= 0 . 9 0 ± 0.02A 1 wi th H in the
° -
3 PW c a l c u l a t i o n gives 0.93A
°-l
gives a value of 1.00A
1
(111)
direction.
The dHvA
w h il e the prese nt 4 PW c a l c u l a t i o n
f o r the same dimension.
The t h i r d resonance branch was observed w i t h the magnetic f i e l d
the v i c i n i t y of the
( 110) d i r e c t i o n y i e l d i n g c a l i p e r s
the i n t e r s e c t i o n o f the
and
t h i s area were very d i f f i c u l t
of three or more per io ds.
p r o j e c t i o n s of Fig.
in
in the regi on of
7*
The data in
to analyze due to the complicated mixing
I t was almost impossible to reso lve more
than one period w i t h any accuracy.
With the magnetic f i e l d
in the
region from 4 ° to appro xima tely 16° from the ( 110) d i r e c t i o n ,
the
dominant resonance y i e l d e d C - c a l i p e r values t h a t ranged from 1. 02 ±
o_i
o_l
0.02A
to 1.20 ± O.O3A
1*2 p r o j e c t i o n s .
It
,
too large to be associ ated w ith the Lj or
is pos sib le t h a t these c a l i p e r s might be assigned
t o the open o r b i t t h a t e x i s t s w it h the f i e l d
Ideally,
in the ( 110) d i r e c t i o n .
the c a l i p e r o f the open o r b i t should gi ve a value e q u i v a l e n t
° -
to 1/3 the zone he ig ht or 0.9^3A
1
.
However,
i t could be p o s si b l e
to obtain a l a r g e r valu e as a r e s u l t of the m i s o r i e n t a t i o n of the
crystal.
3.
from
in
The primary info rm at io n obtained
the various hole o r b i t c a l i p e r s was a d e s c r i p t i o n
the
h ole
Summary of hole r e s u l t s .
(110)
p la n e
surface.
The
alo n g
data
w ith
fo r
several
the
breakthrough
T -s e c tio n
is
in
of
the T - s e c t i o n
dim en sion s
general
on
agreem ent
the
w ith
the r e s u l t s of a k PW c a l c u l a t i o n although the c a l c u l a t e d area
is s t i l l
about 15$ too small to agree w i t h the exp er im en ta l dHvA d at a.
In
order to f i t
the dHvA area of Brandt and Rayne, a breakthrough dimension
in the T face much sm al le r than t h a t p r e d i c t e d from the present k PW
calculation
is req ui red .
pre dicts such a dimension,
The 8 PW approach o f Dishman and Rayne
i.e .,
°-1
O.O7 IA .
However, as discussed by
Dishman and Rayne, a breakthrough dimension of t h i s s i z e in the T
face is not compatible w ith t h e i r magnetoresistance open o r b i t data.
kk
In view of t h i s t her e is s t i l l need f o r a f u r t h e r d e t e r m in a ti o n ,
e x p e r i m e n t a l l y and t h e o r e t i c a l l y ,
the dimension
both
of the T cross s e c tio n and in p a r t i c u l a r
The agreement between the remaining breakthrough
dimensions determined e x p e r i m e n t a l l y from the hole o r b i t s and those
c a l c u l a t e d using the 4 PW method is g e n e r a l l y good wherever such
comparisons can be made.
V I.
The r e s u l t s of t h i s
CONCLUSION
investigation
i n d ic a t e t h a t the Fermi su rface
of mercury consists of t hr ee e l e c t r o n lenses belonging t o the second
zone and a m u l t i p l y connected f i r s t zone hole sur fa ce .
w ith the model t h a t has been p r e v i o u s ly proposed.
This agrees
The hole surface
has been found t o con tac t a l l faces o f the B r i l l o u i n zone producing
breakthrough regions or openings in the sur fa ce .
The 4 PW c a l c u l a t i o n
performed is in general agreement w i t h the data obtained from both
the e l e c t r o n and hole sur face.
I t should be noted,
however,
that
t h i s c a l c u l a t i o n is not found to be su pe rio r to previous c a l c u l a t i o n s .
Rather,
the primary consequence of performing the c a l c u l a t i o n
i t suggests no b e t t e r f i t
is t h a t
to the a v a i l a b l e data f o r the Fermi surface
can be obtained by choosing pse udopotential c o e f f i c i e n t s
to f i t
only
the e l e c t r o n data than was found by previous workers when they f i t
on ly the hole surface data.
fit
For example,
the h PW c a l c u l a t i o n was
to the experimental data on the second band e l e c t r o n surface and
the r e s u l t i n g hole surfac e was a l s o generated.
Consequently,
the f i t
to the e l e c t r o n surface was extremely good, but the agreement w it h the
{3
and
t
o r b i t data was not as s a t i s f a c t o r y .
The dHvA 3 PW and RAPW
c a l c u l a t i o n s which placed emphasis on f i t t i n g
hole su rf ac e ,
in p a r t i c u l a r to the p o r b i t ,
the dHvA data t o the
both give e s s e n t i a l l y
the same area f o r the T - s e c t i o n which g e n e r a l l y compares w i t h t h a t of
the present data.
These r e s u l t s ,
however, d i f f e r markedly from the
dHvA data both f o r the T - s e c t i o n and the e l e c t r o n su rf ac e .
The 8 PW
46
c a l c u l a t i o n on the other hand f i t s
p - se ct io n to the dHvA data e x a c t l y ,
the T - s e c t i o n as w e l l as the
but in the process worsens the
disagreement between the area of the e l e c t r o n sec tion and the c o r r e ­
sponding data as w e l l as c o n f l i c t i n g w it h the open o r b i t data.
I t would appear t h e r e f o r e t h a t there is need f o r f u r t h e r d e t a i l e d
band c a l c u l a t i o n s to resolve the apparent discrepancies between the
e x i s t i n g exp erimental and t h e o r e t i c a l r e s u l t s .
47
APPENDIX A
A Discussion o f the 4 PW C a lc u la t i o n
po
Recently Harrison
has pointed out t h a t the Fermi surface of a
number of metals can be c a l c u l a t e d q u i t e w e l l
in the n e a r l y fr e e
e l e c t r o n approximation where the de v ia t i o n s from the f r e e e l e c t r o n
behavior can be expressed in terms of m a t ri x elements between plane
wave s t a t e s of an e f f e c t i v e pseudopotential
crystal potential.
in place o f the r e a l
This method has been discussed e x t e n s i v e l y by
Harrison and has been a p p lie d to a number of metals to give good
agreement w it h exp erimental data.
No attempt w i l l be made here to
j u s t i f y why t h i s method works or to present a d e t a i l e d development
o f the theory, but r a t h e r emphasis w i l l be placed on the mechanics of
the c a l c u l a t i o n along w ith a b r i e f i n t r o d u c t i o n t o the pseudopotential
method.
The reader is r e f e r r e d to the above re fe re nc e f o r a more
d e t a i l e d discussion.
The conduction e l e c t r o n wave f u n c t i o n s , s a t i s f y
Schroedinger equ ation ,
the
i.e.,
2
V2* + V(7))|r = Ei|f
where
<—
I
V( r)
,
(1)
is the p e r i o d i c p o t e n t i a l due to the ion cores and the
s e l f - c o n s i s t e n t f i e l d of a l l the e l e c t r o n s .
V( r )
is a r a p i d l y
vary ing fun ct io n wi t h a strong a t t r a c t i v e p a r t clo se t o the ion cores.
It
is assumed t h a t the core wavefunctions ar e the same as the is o l a te d
48
ion although t h e i r energies ar e d i f f e r e n t .
The conduction e l e c t r o n
s t a t e s i|r are orthogonal to the core s t a t e s .
I t has been shown t h a t
t h i s c o n s t r a i n t fo rc es the conduction e l e c t r o n s to sample
V(r)
in a
very sp e c ia l way which can be represented by a new wave equation which
2lf
“4
is i d e n t i c a l to ( 1) but with an e f f e c t i v e or ps e udo pot ent ia l, Ve ^ ^ ( r ) ,
such th a t
2
V2$ + Ve f f f f , k ) . 4 =
This pseudopotential
v ar y in g.
is much weaker than
E$
V(r)
.
(2 )
and more smoothly
The pseudowave f u n ct io n $ is a smooth fun ct io n which does
not include the t y p i c a l at omic -core o s c i l l a t i o n s which insure t h a t
is orthogonal to the core s t a t e s .
As a r e s u l t ,
plane waves converges rath er r a p i d l y .
represented by a F o u ri e r expansion,
In ge n e r a l,
where the c o e f f i c i e n t s
-
■
G.
is a r e c i p r o c a l
are
V?
u.
Ve^ ^ ( r , k )
can be
i.e .,
—►
Ve f f (7 ,k ) =
an expansion of $ in
S V J ( k ) e ' G|
G.
I
I
—♦
"
(3 )
ar e the pseudopotential c o e f f i c i e n t s and
1
l a t t i c e vector.
in general momentum dependent;
The pse udopotential c o e f f i c i e n t s
however,
in a l o ca l approximation,
they may bet r e a t e d as being independent of momentum.
This
expansion
u s u a l l y converges r a p i d l y since the c o e f f i c i e n t s decrease in magnitude
w ith
increasing magnitude of the r e c i p r o c a l
l a t t i c e ve c to r
consequently can be truncated a f t e r a few terms.
G.
and
49
The Hamiltonian f o r the system may be expressed as
P2
-
I V
H = a : + ' ' e f f (r) - " o
In
order t o determine the energy as
it
is necessary
+
P
Gi
r
V
1
(lt)
a fu n c t io n o f the
to d i a g o n a li z e the m a t r i x o f H.
wave
If
it
ve c to r k,
is assumed
that
the pseudowave f u n c t io n $ can be w r i t t e n as a l i n e a r combination of
plane waves o f the form
' 7
gj
then the secular equation can bew r i t t e n
E t C ( k +G . ) 2 - X ] 6
Gj
-
<5)
■>
i j
+ U■
i
as
}a
=0
j
.
(6 )
i
The diagonal terms involve only the energy q u a d r a ti c
in the wave
vec tor k, w h i l e the o f f - d i a g o n a l terms involve only the pseudopotential
coefficients.
The diagonal terms of the pse ud op ot en ti al , Uq, have been
absorbed i nto
The pseudopotential c o e f f i c i e n t s corresponding to the planes
bounding the f i r s t B r i l l o u i n zone in mercury have been estimated by
comparing the geometric resonance data obtained from the e l e c t r o n lens
w it h the r e s u l t s of a c a l c u l a t i o n using f i r s t
two pseudowaves to
determine Ujqq and then using four pseudowaves t o determine U^j^ and
I t should be pointed out t h a t spin o r b i t coupling was neglected,
and f u r t h e r ,
there was no attempt to mai nta in compensation.
i t appears t h a t to f i r s t orde r, compensation was maintained.
However,
The
50
Fermi energy, X, as w e l l as the pseu dop ot ent ia l c o e f f i c i e n t s , were
t r e a t e d as a d j u s t a b l e parameters w it h the r e s t r i c t i o n
t h a t 0 ^ ^ and
U i j o would be equal as the magnitudes o f the ( 111) and ( 110) r e c i p r o c a l
l a t t i c e vectors d i f f e r by less than 0 . 5$.
t h a t since the pse udopotential and i t s
then
= Ug
if
I t has been pointed o u t ^
Fo u rie r transform are continuous,
|G. | = | G j | .
Since the data f o r the second band e l e c t r o n surface is the most
reliab le,
the approach was to f i r s t seek agreement between the c a l c u ­
l a t i o n and t h i s data.
The e l e c t r o n su rface resembles a deformed lens
centered on the p o i n t L; hence i t was found convenient to s t a r t w it h
on ly two pseudowaves in the (110) plane.
Thus an i n i t i a l value of
Uqqj was obtained from a r e l a t i o n of the form
=
100"
where
1
LENS
(k l ‘ u ) 2 ^ LENS
k. = t ; ( 100)
1
2
and
_
LENS
( k L- r >2
^ LENS
kk ^_ and
LENS
f o r the e l e c t r o n lens c a l i p e r s .
^ LENS
2
kk-Fr
LENS
a r e the ex pe rim ent al values
The Fermi energy, X, was then c a l c u ­
l at e d from the expression
*
The value of
variation.
2
„ L - r \2
r,
k l + ^LENS^ + ^
, L - r \2
2
1 /2
k l k LENS^ + U100^
was then v a r i e d and X r e c a l c u l a t e d f o r each
I t was thus determined t h a t X v a r i e d very sl ow ly,
the order of 0. 1 $ , as U jqq was v a r i e d by appro xim at el y 10$.
each p a i r of values f o r
and X,
L -r
on
For
was held f i x e d and the
value o f kf-- ^ . was r e c a l c u l a t e d , and r e s t r i c t i o n s placed on
LENS
the maximum and minimum values of Uggj such t h a t the c a l c u l a t e d value
of
remained w i t h i n ± 1$ o f the e x p e r i m e n t a l l y determined value.
The c a l c u l a t i o n was then extended to f ou r pseudowaves about the
p o in t L in the 3TLW plane which allowed the best values f o r U j j q and
U jjj
to be determined as w e l l as the f i n a l value f o r the Fermi energy
which was c o n s is te n t w i t h the second band data.
The c a l c u l a t i o n was
performed by s e l e c t i n g the four plane wave s t at es which had the lowest
f r e e e l e c t r o n energies when r e s t r i c t e d to regions near the p o i n t of
interest.
In t h i s case,
the c a l c u l a t i o n was done along the l i n e L-W
w ith the plane wave s t a t e s corresponding to k values o f
= -^(001) ,
k^ = k^ + (001) , k^ = k j + (010) , and
k^ = k j + (101). This
nec ess itate d sol vi ng a 4 X 4 determinant
which was done on
7040 d i g i t a l computer.
The f i n a l values obtained f o r the pseudo­
p o t e n t i a l c o e f f i c i e n t s are
-O.O546 Ry.
anIBM
V
= V j j j = 0 . 0 6 l 6 Ry and
V jqq =
These values are in reasonable agreement w it h the work
of Animalu and H e i n e ^ and are compared w i t h t h e i r
The r e s u l t s o f Brandt and Rayne are a l s o
results
in Fig.
shown f o r comparison.
Once the pseudopotential c o e f f i c i e n t s were known,
the c a l c u l a t i o n
was extended t o obta in p r o f i l e s of sections of the Fermi su rface in
planes of i n t e r e s t .
Again the four plane wave s t at es which had the
lowest f r e e e l e c t r o n energies when r e s t r i c t e d to regions near the
po in t o f i n t e r e s t were s e l e c t e d .
The se c u la r determinant was
constructed from (6 ) and a search was performed a t a f i x e d Fermi
energy f o r those values o f k which caused the determinant to vanish.
The r e s u l t s were subsequently p l o t t e d to give p r o f i l e s of the
p o r t io n of the Fermi su rface under i n v e s t i g a t i o n .
This procedure
was found to be convenient in t h a t c a l i p e r s not along symmetry lin es
12.
52
• PRESENT
V ( k ) (rydbergs)
▼ BRANDT 8
WORK
RAYNE
0
*
.2
3
2
3
k/2 k F
F ig .
12
4
5
6
53
could be as e a s i l y determined as those along l in e s of high symmetry,
and a comparison of dHvA areas could a l s o be accomplished.
A p l o t o f nine d i f f e r e n t cross sections o f the T - s e c t i o n centered
on U in the (110) plane is shown in Fig.
values of X,
anc*
exp erimental data points a r e
the be s t f i t
' nc* ' catec*'
ind ic at ed .
13 w i t h the corresponding
F°r comparison purposes, the
As has a l r e a d y been i n d ic a t e d ,
t o the exp erim ent al data was f o r a value of V j j q = 0 . 0 6 l 6 Ry
and X = 0.505 Ry-
= .5033
V,oo = --05Z0
V110 - .0616
X =.5050
V|QO=--0546
V|10 =.0616
X =.5038
X
W
- 0532
V „ 0 = .06I6
X= . 5 0 5 0
X= . 5 0 3 8
w - 0532
X= . 5 0 3 3
V|00 = - 0 5 4 6
Vu o = . 0 6 4 4
Vno = . 0 6 4 4
Vl|0 - . 0 6 4 4
( d)
X» .5033
V,00=-.O52O
V)10=.0672
X * .5050
X= . 5 0 3 8
V100-.O B48
V|00a- ° 532
Vllo =.0672
v „ o c-0672
F ig .
13
55
APPENDIX B
C a l c u l a t i o n o f the A t t e n u a t i o n C o e f f i c i e n t
in Transverse Magnetic Field s
In the f o l l o w i n g discussion a se m i c l a s s i c a l approach to the
c a l c u l a t i o n of the a t t e n u a t i o n c o e f f i c i e n t of a sound wave in the
presence of a transverse magnetic f i e l d
a method due to S i e v e r t .
co e ffic ien t
27
is c a r r i e d out f o l l o w i n g
This method y i e l d s the a t t e n u a t i o n
in a d i r e c t manner as the r e s u l t o f the s o l u t i o n o f
the wave equation f o r a sound wave propagating in a metal.
Extensive
use is a l s o made o f s e v e r a l r e s u l t s of the c a l c u l a t i o n of the
a t t e n u a t i o n c o e f f i c i e n t by Cohen, Harrison and Harrison
(CHH).
15
Consider as a model f o r a metal a f r e e e l e c t r o n gas c o n s is ti n g
o f Nq e l e c t r o n s per u n i t volume moving through a uniform background
of p o s i t i v e
ions o f the same p a r t i c l e d e n s it y .
A sound wave is
introduced i n t o the system which causes the ions t o be moved in a
p e r i o d i c fashion over a macroscopica1ly small region of the metal.
This motion o f the ions causes an e l e c t r i c
f i e l d t o be produced
which w i l l then a c t as a p e r t u r b a t i o n on the f r e e e l e c t r o n gas.
It
is t h i s
i n t e r a c t i o n w i t h the f r e e e l e c t r o n s t h a t gives r i s e
t o the a t t e n u a t i o n of the sound wave by the e l e c t r o n gas.
be shown,
As w i l l
the a t t e n u a t i o n is governed p r i n c i p a l l y by the c o n d u c t i v i t y
of the e l e c t r o n gas and accounts f o r the dependence of the a t t e n u a t i o n
c o e f f i c i e n t upon the magnetic f i e l d .
An expression is now sought f o r the a t t e n u a t i o n c o e f f i c i e n t of
a sound wave propagating in a metal in a transverse magnetic f i e l d .
A co o rdin ate system is chosen such t h a t
H = Hz
the sound wave is propagated in the x d i r e c t i o n .
f o r sound propagation in the metal
afs
and
q = qx,
i.e .,
The wave equation
is given by
2 sfs
( 1)
Bt2 " " S dx2 = N0M
where M is the io n ic mass,
F is the f o r ce per u n i t volume a c t i n g on
the
ions, Nq is the io nic
d e n s it y , v g is the v e l o c i t y of soundbefor e
the
i n t e r a c t i o n s w it h the
conduction e l e c t r o n s ar e included, and
S is the i o n ic displacement f i e l d .
wave, u ( r , t )
The v e l o c i t y f i e l d of the sound
is assumed to have the dependence
u ( r , t ) « e x p { i [ ( q + iA)x - cot]}
(2)
where iA is the d e v i a t i o n o f the wave number from
q = —
induced
vs
A can be regarded as the
by the i n t e r a c t i o n w i t h the e l e c t r o n s .
attenuation c o e f f ic ie n t .
From (2)
S - ^
co
it
is
evi de n t t h a t
,
and ( 1) can be wri t t e n as
2
The force F on the system,
neg le c tin g the deformation p o t e n t i a l ,
(3)
57
i s given by
F = N J e l (E + - u X H) + F
O' 1
c
c
where
E
is the s e l f - c o n s i s t e n t e l e c t r i c f i e l d ,
(5 )
H is the e x t e r n a l
magnetic f i e l d , and F 'is the force per u n i t volume which feeds
energy c o h e r e n t ly from the e l e c t r o n s - back i n t o the ion system.
This
force a r i s e s from the f a c t t h a t the average e l e c t r o n v e l o c i t y (v )
d i f f e r s from t h a t of the ions, u .
The e l e c t r o n s c o l l i d e w it h the
ions, and momentum is t r a n s f e r r e d from the e l e c t r o n s to the ion system.
The net f or ce exerted by the e l e c t r o n s on a u n i t volume of the p o s i t i v e
charge is given by
(6 )
Here
t
is the r e l a x a t i o n time or the c h a r a c t e r i s t i c s c a t t e r i n g time
f o r the e l e c t r o n s and m is the mass of the e l e c t r o n .
v e l o c i t y o f the e l e c t r o n s
where j
j
The average
is given by
is the e l e c t r o n c u r r e n t den sity .
The t o t a l c u r r e n t d e n s it y ,
, of the system is the sum of the e l e c t r o n i c cu rr en t d e n si ty and a
cu rr en t
Ng|e|u
due t o the background of p o s i t i v e ions,
j
= j e + N0 l e l “
The t o t a l c u r r e n t is r e l a t e d to the s e l f - c o n s i s t e n t e l e c t r i c f i e l d ,
(8 )
E , by Maxwell's equation and may be expressed as
Ei -
<9 >
<$*><V
E2
2 = ! ^.
The subscripts
1 and 2 r e f e r
(^Vg) 2"g J 2
2
to components p a r a l l e l and perpe ndi cul ar
to the d i r e c t i o n of propagation,
respectively.
i.e .,
the x and y d i r e c t i o n s ,
Thus the t o t a l cu rr en t may be w r i t t e n as
j
where f o r
<10>
H = Hz
and
= -<70B • E
(11)
q = qx ,
( 12)
wi th
Y = 0 (— )
c
and
Og
and
£ =
— p
Ifita.v2
0 s
(13)?
( 1*0
is the dc c o n d u c t i v i t y given by
M
N e
an =
0
2
t
.
m
I t should be noted t h a t the q u a n t i t y
Vs 2
(— )
in the denominator of
(10) was neglected as being small in comparison to u n i t y .
c u r r e n t de n s it y ,
j
0
( 15)
The e l e c t r o n i c
, has been determined by CHH as the r e s u l t of
s o l v i n g the Boltzmann t r a n s p o r t equation in the r e l a x a t i o n time
approximation.
T h e ir r e s u l t
is
■?
_
I = g jj' * ( E
Je
0
'
mu*
;
e-r
where
ys
^
o' = (1 - R)
|
'
• a /a Q
>s
and
R
is a tensor whose components ar e given by
R. . = R. 6 , .
iJ
i
lj
The vector R and the c o n d u c t i v i t y tensor a ar e given as follows:
R =
v K (v )(^ )d v
3 N0 - s
and
ct
= J ev • J( v) (-=“
)dv
where
t
(J(v),K (v))
= J* {-ev 1j l ) e x p { i [q • ( r ' CO
a )(t'-t)
in Eq.
(19)
-
r) -
3d t *
is the Fermi energy, and
fg
in both expressions
the Fer mi-Dirac d i s t r i b u t i o n f u n c t i o n .
-4
E
fie ld
The s e l f - c o n s i s t e n t e l e c t r i c
-4
u,
should be r e l a t e d l i n e a r l y to
E = w • - 2-
u
i.e .,
.
(22)
ao
A
An e x p l i c i t expression f o r
( 16) ,
W may be obtained by combining (8 ) ,
(11),
and ( 17) t o y i e l d
A
A
A
W = -[o '
Now using (7 ) ,
(8 ) ,
(11),
1
+ B]
A
A
[l
and (22)
- a ']
(2 3 )
in (5 ) and (6 ) , the f o r c e
F
becomes
2 2
Nfe
A
N.e
F = - 2 — [1 + B] • W• u + - 2 - u X
A.
Using
(2 3 ) ,
A
the term
A
[l +
A
H
.
(2*0
A
B]
A
• W may be w r i t t e n as
A
A
[1 + B] •. W = - [ ( 1 + B) • (a' + B)
1
A
A
A
(1 + B) -
A
(1 + B)]
.
(2 5 )
Now defi ne
A
S =
Consequently,
A
A
A
- [ 1 + B] ■ W=
A
A
A
A
i
(1 + B ) [ ( a ' + B)
A
•
(1 +
A
B) -
l]
.
(24) becomes
N2e 2
Nne
—
♦ - O x
—
*
0
*
F = - ( -2 — ) S • u + - 2 - u X H
0
C
S u b s t i t u t i n g t h i s expression f o r
q u a d r a t i c in A, one gets
F
.
/ \
(27)
i n t o (4) and n e g le c ti n g terms
61
where
eu
oi = —
c
me
and the mean f r e e path
r
<6
is given by
-
v ct
F
.
If
one d efin es
' 0
1
G = I
|
■1
(29)
0
then (2 8) can be w r i t t e n as
1
r
A
2 ( m7 T )C s - v
G] ’ u = Au
■
(30)
L e t t i ng
a
1
mv _
a
° = 5 ( m7 T ) [ s - V
S
Eq*
G]
(31)
(3 0 ) can be expressed as
D • u = Au
(32)
which is an ei genv alue equation f o r the complex a t t e n u a t i o n c o e f f i c i e n t
A.
The sec ula r determinant
is
° i r A
“ is
= 0
D21
“a
.
( 33)
A
Solving f o r the eigenvalues y i e l d s
» , i - y
y
62
4 D12D21
The term ----------------- 0 a r i s e s from a mixing of the l o n g i t u d i n a l and
(Dn - D22 >
transverse modes o f v i b r a t i o n .
27
This term has been c a l c u l a t e d
f o r the case o f simple c i r c u l a r o r b i t s and open o r b i t s .
both
In both
—Q
cases the r e a l and imaginary part s were no g r e a t e r than 10 J f o r the
magnetic f i e l d s of i n t e r e s t .
neglected in (3*0 •
Consequently,
t h i s term may be
Thus
A± - H i p e
± (! u ^ g g l
os.
f o r the case o f a pure l o n g i t u d i n a l mode in the 1 d i r e c t i o n ,
D21 = 0, and
A+ = Dn
The o r d i n a r y a t t e n u a t i o n c o e f f i c i e n t
.
ct
(36)
may be defined as
a = Re A+ = Re 0 U
(37)
From (31)
mv
°11 = Mv" Z Csu
s
But from ( 2 9 ) ,
- OTG11]
(38)
= 0
i
mvF
“irliTi'n
(3
9
>
from which (3 7 ) becomes
1
=
mvF
i s r ! Re<s n>
(<t0 )
63
From (26)
A
A
S11 = (1 +
+
+ Bn ) -
(1 + Bn )
(41)
or
Re(Sn ) = Re (l + Bn ) 2[ ( a l + B ) j j ] - 1
s i nee
Re Bj ^ = 0
.
I
" =
(42)
Thus
FDVp
n
s m TT
A
A • 1
tRe(1 + B11> C(a‘ + B )11] ■ 1]
f o r l o n g i t u d i n a l waves
|Bn | - V ■=
■
“5
Since p is t y p i c a l l y of the order 10 y f o r metals and
order 10 ^ ,
then
W
vs
—
of the
may be neglected f o r a l l p r a c t i c a l purposes.
The refore,
.
mv
~
A -1
“ 4 ' l « ^ E Re[(CT' + B ) l l J '
This may be w r i t t e n
a,
4
in terms of
mvr
.
o'
1
(45)
1
(46)
as
a4o + 'P
= | ; r —r Ret----------- — ---------- :-------- ;— r ) 2 " V
•
° 'n (V 2 2 + i p )
+
(a 12) 2
where use has been made of the f a c t that
12
21
(47)
The magnetic f i e l d dependence o f the a t t e n u a t i o n
expressions f o r the cr's in ( 4 6 ) .
is i m p l i c i t
in the
They have been c a l c u l a t e d by CHH
and ar e as f o ll o w s :
-3iayr(l-iGorr)[l-g0 (X)]
(48)
11
CT.
q 2-t2[ 1- i c o t- gQ (X ) ]
= -a.
22
=
1-iarr
- 3 iayrg (X)
2----------2 q ' t [ l - i o T - g Q( X ) ]
( S + --------2----------- j
v 0
, .
,v v
l - i a y r - g 0 (X)
(4 9 )
(5 0 )
where
rt/2
gQ(X) =
J
Jq(X s i n 0 ) s i n 0 d0
(51)
0
9q(X) = ( j f ) 9 0 (X)
(52).
rt/2
SqOO =
J*
CJg(X s i n e ) ] 2 Sin3 0 d0
(53)
0
and
q vr
v_
X = — - = — ~
CJD
V O
c
s c
9
is the p o l a r angle
in
v
space.
= qR
n
The exp ression f o r the a ' s
were
c a l c u l a t e d under the assumption t h a t the sound wavelength is the order
65
of the d a s s i c a l o r b i t r a d i u s ,
i.e .,
X ~ 1.
Also,
the assumption
was made t h a t
|m T/ (1 “ i cut) |
c
When the expressions f o r the
ct' s
attenuation c o e ffic ie n t
»
1
' (54)
a re s u b s t i t u t e d
into
(46),
the
may be w r i t t e n as
at, o= ■ -q- \
*
2
3 ( 1+ci
-
(
l-
)
t
-
-
1)
.
(5 5 )
(g ')2
l-9n+
S0
The f i e l d dependent p a r t which is included w i t h i n the brack et s above
has been c a l c u l a t e d by CHH and the r e s u l t s ar e shown in Fig.
t h a t the abscissa
14.
Note
is p r o p o r t i o n a l t o the r e c i p r o c a l o f the magnetic
f i e l d w h i l e the o r d i n a t e e x h i b i t s strong o s c i l l a t i o n w i t h maxima
and minima oc c u r ri n g whenever
9q(X)
vanishes.
For pure tr a n s v e rs e propagation we o b t a i n from (34)
A_ = D22
(56 )
or
<*t
= Re ° 2 2 “
^ 7)
Re S2 2
and in a manner s i m i l a r to t h a t used f o r det er min ing
a
cc [ ----------1--------- _
-
i]
Qf^
.
we f i n d t h a t
(5 8 )
RELATIVE
O
ATTENUATION
ro
ro
Fig.
l*f
<D
a>
ro
cr\
ON
67
As f o r the l o n g i t u d i n a l case,
CHH and a p l o t
t h i s expression has been e v a lu a te d by
is given in Fig.
I t should be noted t h a t
15*
in both Fig.
Ik and Fig.
f i e l d dependent f a c t o r contained in the brackets of
has been p l o t t e d .
6 «
I-
(5 5 ) and (5 8 )
Both these f a c t o r s a r e independent of parameters
ass oci ate d w i t h the m a t e r i a l provided t h a t
and
15 o nl y the
*di Y »
1 ,
Thus both the a t t e n u a t i o n c o e f f i c i e n t s
q£ »
1 ,
for longitudinal
and t ra nsv er se p o l a r i z e d sound waves e x h i b i t o s c i l l a t o r y behavior as
a f u n c t i o n of the r e c i p r o c a l of the magnetic f i e l d .
These geometric
resonances in the a t t e n u a t i o n a r e ass oci ate d w i t h the Bessel
fun ct ion s
in the c o n d u c t i v i t y tensor.
The geometric resonances can be understood in simple p h y s ic a l
terms f o r tra ns ve rs e waves p o l a r i z e d p e r p e n d i c u la r to the magnetic
fie ld at
low,
low f re qu en c ie s .
I f the frequency o f the sound wave is
the e l e c t r o n s are a b le t o f o l l o w the i o n ic motion in a manner
such as t o n e u t r a l i z e the e l e c t r i c f i e l d almost co m pl et el y and cause
the t o t a l c u r r e n t o f the system t o van ish ,
that
is,
j
0.
From
(8 ) then
(59)
Energy is t r a n s f e r r e d from the sound wave t o the e l e c t r o n system a t
a rate
—>
—>
j
• E .
The a t t e n u a t i o n c o e f f i c i e n t is given by
28
.
(6 0 )
a
M u u c
s
RELATIVE
r
o
^
&
o
o
O
ATTENUATION
r
o
-
^
C
D
O
o
o
Since the e l e c t r o n c u r r e n t response is f i x e d from (59).> the system
is one of cons tan t c u r r e n t and the e l e c t r i c f i e l d may be w r i t t e n as
E = p(q,H)
There for e
e
•
(61)
(60) can be w r i t t e n as
« = —
Using ( 59 ) ,
• 7
p(q,H) • j j
H s —
:— —
M u u c
s
•
(6 2 )
the previous expression can be expressed as
< *«p (q ,H )
(6 3 )
or s c h e m a t ic a ll y
a « ——*-----cr(q,H)
Thus the a t t e n u a t i o n c o e f f i c i e n t
conductivity.
is
.
( 6^4)
i n v e r s e l y p r o p o r t i o n a l t o the
Assuming t h a t the sample is pure enough and t h a t the
temperature o f the sample kept s u f f i c i e n t l y
low to reduce the number
o f thermal phonons which s c a t t e r the e l e c t r o n s ,
then the e l e c t r o n s
the m e t a l l i c sample ar e ab l e to execute simple closed o r b i t s
plane p er p e n d ic u la r t o the magnetic f i e l d .
Fig.
in
in a
These are i l l u s t r a t e d
in
16 where the v e r t i c a l arrows correspond to the s e l f - c o n s i s t e n t
e l e c t r i c f i e l d as so ci ate d w it h the l a t t i c e wave.
The magnetic f i e l d
is i n t o the page and the sound is propagated in the
Hj the e l e c t r o n
x direction.
is a l t e r n a t e l y a c c e l e r a t e d and d e c e le r a te d by the
For
70
CM
j
,
F ig .
16
e l e c t r i c f i e l d as i t
t ra v e r s e s
its o rb it.
Thus th ere
in the v e l o c i t y of the e l e c t r o n per c y c l e .
is no net increase
This corresponds t o a
small c u r r e n t response, a low c o n d u c t i v i t y and hence a high a t t e n ­
uation.
For H^,
the component of the f i e l d
in the d i r e c t i o n o f the
e l e c t r o n motion is neg ati ve and the e l e c t r o n ' s v e l o c i t y
w i t h each passage.
This corresponds t o a s i g n i f i c a n t
increases
increase in the
c u r r e n t response, a la rg e c o n d u c t i v i t y and hence a low a t t e n u a t i o n .
The a t t e n u a t i o n thus passes through an extrema whenever the o r b i t
diameter encompasses an i n t e g r a l or h a l f
i n t e g r a l number o f sound
wavelengths.
The info rm at io n de ri ved from the peri od of these o s c i l l a t i o n s
is obtained in the f o l l o w i n g manner.
k =
The o r b i t of the e l e c t r o n
space except t h a t
9 0 ° about H.
This
it
In
k
space we have
7 X H
..
.
(6 5 )
in k-space is the same as the path in r e a l
is m u l t i p l i e d by the f a c t o r
is i n d i c a t e d in Fig.
15*
6H
-j—■ and r o t a t e d
The radius
in r e a l
space is then given by
r
x
= “U k
eH y
.
Now consider what happens i f we examine t h i s expression a t two
a d ja ce nt maxima in the a t t e n u a t i o n .
have enclosed
n
At the f i e l d val ue Hn, we
wavelengths of sound.
(6 6 )
w h il e at a lower value of the f i e l d
a d d i t i o n a l wavelength.
^n+i
we have spanned one
Thus
- 2rx -
\
Subtr ac ting these two equations we get
2k
■> —
------
» U (|)
where
* / 1\
Ti
1
1
T " TT
n+1
n
h~
Thus, from a knowledge of the frequency of the sound wave, the
v e l o c i t y of sound, and the period of the o s c i l l a t i o n s of the
a t t e n u a t i o n c o e f f i c i e n t ,> the diameter,>
k-space can be determined.
2k y
o f the o r b i t
in
73
REFERENCES
1.
G. B* Brandt and J. A.
2.
T* E. Bogle, C. G.
Soc.
12,
148, 644 (1966 ).
G r e n ie r , and J. M. Reynolds,
Bull.
Am.
Phys.
183 (19 67 ).
3.
A. E. Dixon and W.
4.
W. R. Datars and A. E.
5.
J. M. Dishman and J.
6.
S. C. Keeton and T.
7.
N.
F- Mott and H.
R.
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A. Rayne,
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done by Dr.
J.
Holliston,
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R. McDonald of the Department o f Chemistry,
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74
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VITA
Tommy E. Bogle was born September 4,
Louisiana.
1940 in Logansport,
His secondary edu cation was completed in Logansport,
Louisiana,
and in September,
In stitu te,
Ruston,
1958 he e n r o l l e d
Louisiana.
He graduated w i t h a Bachelor o f Science
degree in physics from Louisiana Po lyt ech nic
and was married t o the former Judy Dian
year.
in Louisiana Po lyt ech nic
H ill
Institute
in May,
1962
in June of the same
A f t e r having been employed f o r one year a t Texas Instruments
In corporated,
Dallas,
Texas,
Louisiana S t at e U n i v e r s i t y ,
he ent ere d the Graduate School o f
Baton Rouge,
Louisiana in September,
He is a member of the American Physical So c ie ty ,
19&3*
the American I n s t i t u t e
of Physics, and is p r e s e n t l y a candidate f o r the degree of Doctor of
Philosophy.
75
EXAMINATION AND THESIS REPORT
Candidate:
Major Field:
Tommy Earl Bogle
Phys i cs
Title of Thesis:
The Magnetoacoustic E f f e c t
In Mercury
Approved:
Dean of the Graduate School
E X A M IN IN G C O M M IT T E E :
D ate o f E xa m in a tio n :
July 12,
1968