Retrospective Theses and Dissertations
1967
Hall effect of gadolinium, lutetium and yttrium
single crystals
Ronald Stanley Lee
Iowa State University
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67-8921
LEE, Ronald Stanley, 1938HALL EFFECT OF GADOLINIUM, LUTETIUM AND
YTTRIUM SINGLE CRYSTALS.
Iowa State University of Science and Technology,
Ph.D., 1967
Physics, solid state
University Microfilms, Inc., Ann Arbor, Michigan
HALL EFFECT OF GADOLINIUM, LUTETIUM AND YTTRIUM SINGLE CRYSTALS
by
Ronald Stanley Lee
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject:
Experimental Physics
Approved:
Signature was redacted for privacy.
in Charge of Major Work
Signature was redacted for privacy.
Hea
^ Mjor Department
Signature was redacted for privacy.
Iowa State University
Of Science and Technology
Ames, Iowa
1967
TABLE OF CONTENTS
Page
I.
II.
III.
IV.
V.
VI.
VII.
VIII.
INTRODUCTION
1
A.
Purpose
1
B.
Previous Experimental Work
2
BASIC CONCEPTS AND EQUATIONS
3
A.
Phenomenologica1 Theory
3
8.
The Hall Effect in Non-magnetic Metals
6
C.
The Hall Effect in Ferromagnets
8
EXPERIMENTAL PROCEDURE
10
A.
Measurement of the Hall Effect - d.c.-Method
10
B.
Measurement of the Hall Effect - a.c. Method
16
C.
Sample Preparation
19
D.
Determination of R and.R. from Isothermal Data
o
1
20
EXPERIMENTAL RESULTS
25
A.
Gadolinium
25
B.
Lutetium and Yttrium
33
DISCUSSION
37
A.
Lutetium and Yttrium
37
B.
Gadolinium
44
BIBLIOGRAPHY
53
ACKNOWLEDGEMENTS
55
APPENDIX
56
A.
Discussion of Errors - 4.c. Method
56
B.
Discussion of Errors - a.c. Method
59
C.
Sample Dimensions and Demagnetizing Factors
60
Page
D.
Sample Impurities
60
E.
Tabulation of Lutetium and Yttrium Data
62
F.
Tabulation of Experimental Data for Gadolinium
64
1
I.
INTRODUCTION
A.
Purpose
H a l l e f f e c t measurements provide information about the number and
types of carriers of electrical current in a metal.
In a ferromagnetic
metal the Hall effect is anomalously large and exhibits a strong temper­
ature dependence.
Although there is general agreement that a spin-orbit
type o f interaction i s responsible for the anomalous Hall effect i n
ferromagnetics, numerical calculations of the effect are very d i f f i c u l t
and have met with only limited success.
Gadolinium is an interesting candidate for Hall effect measurements.
It is ferromagnetic below 293°K (21).
This metal has three electrons with
s-d character occupying the conduction bands.
The seven 4f electrons
which carry the magnetic moment are shielded from direct interaction with
the conduction bands by the filled 5s and 6p shells.
This is in contrast
to the situation with the 3d ferromagnetic transition metals where the
magnetic electrons are not so localized.
Theories have been proposed that explain the qualitative features of
the Hall effect in polycrystal1ine gadolinium (5, 9, 10, |4, l6, 28, 30).
However, recent calculations (4, 11) show the Fermi surface of gadolinium
to be extremely anisotropic.
Measurements on single crystals are necessary
to observe the effect of Fermi surface anisotropy and magnetic anisotropy
on the Hall effect.
The work reported here describes such measurements
for the ferromagnetic metal, gadolinium, and for lutetium and yttrium.
Lutetium and yttrium are two elements which resemble gadolinium very
closely in electronic structure.
Lutetium differs from gadolinium only in
2
the occupation of the 4f shell.
filled.
in lutetium the 4f shell is completely
Yttrium also is very similar to gadolinium.
in yttrium the
conduction electrons occupy overlapping 4d and 5s bands instead of 5d and
6s bands as in gadolinium.
give a magnetic moment.
Yttrium has no partially filled shells to
Thus measurements of the Hall effect in lutetium
and yttrium were undertaken in order to learn to what extent the Hall
effect in gadolinium is influenced by the interaction of the conduction
electrons with the polarized spins of the 4f electrons.
B.
Previous Experimental Work
Hall effect measurements on polycrystal1ine gadolinium were first
made by Kevane et al. (13).
They investigated the Hall effect in poly-
crystalline gadolinium in the temperature range 30° - 350°C.
More recent
measurements by Marotta and Tauer (17), Volkenshtein and Federov (32) and
Babushkina (2) have been made down to 4.2°K.
Volkenshtein je^
(33) have measured the Hall effect of two single
crystals of gadolinium in the temperature range 4.2° - 370°K.
They
reported only values of Rj, the extraordinary Hall coefficient.
Kevane et al. ( I 3 ) measured the Hall effect in polycrystalline
yttrium in the temperature range 20.3° " 300°K.
Anderson _et
(1)
measured the Hall effect in polycrystalline lutetium in the temperature
range 40° - 320°K.
Nigh et al. (21) have measured the magnetization and electrical
resistivity of gadolinium single crystals.
extensively in this investigation.
Their results have been used
The Seebeck effect in gadolinium single
crystals has been measured by Sill and Legvold (27).
3
II.
BASIC CONCEPTS AND EQUATIONS
A.
Phenomenological Theory
If a magnetic field intensity. H, is applied to a conductor which
carries a current density, J, an electric field, Ej^, is set up in a
direction perpendicular to H and J.
This effect is called the Hall effect.
E^ is called the Hall electric field and is related to the current density
and magnetic field, B, by
E^ = -R^ (JXB)
where
(2.1)
is called the ordinary Hall coefficient.
If appreciable magneti­
zation exists in the sample, the Hall effect may be described by
n
= -R
O
(JXH) - R, (JXH)
(2.2)
I
where M is the magnetization and Rj is called the extraordinary Hall
coefficient.
Typically, Rj is one or two orders of magnitude larger than
Ro'
If the Hall effect is measured in anisotropic single crystals, the
observed Hall coefficient will depend on the orientation of the crystal
with respect to J and H.
To understand this orientation dependence one
must consider the dependence of the resistivity tensor on the magnetic
field intensity, ÏÏ, and the magnetization, T(.
One must consider M and TÎ
as independent variables because the part of the Hall effect that depends
on M does not arise from that part of the magnetic induction, B, due to M
but is rather due to a spin-orbit interaction which is dependent on M.
one expands the resistivity tensor as a power series In M and H, one
If
k
obtains to first order in M, H
Pjj(MjH) - p.j(O) +
'^'ijk^k
"
(2.3)
The electric field inside a sample carrying current density J in a magnetic
field intensity H is then
Ej = P;j(0)Jj + Pijk^jHk * P'ijk^jMk
(2.4)
to first order in M, Ïl where in both Equations 2.3 and 2.4 we sum over
repeated indices.
Equation 2.4 may be simplified in several ways.
From Equation 2.1 we
see that the Hall electric field is defined to be that component of the
electric field perpendicular to both current and magnetic field directions.
This eliminates all p.^^ for which i = j, i=korj=k from our consider­
ation, although such terms need not necessarily vanish.
apply Onsager's theorem as stated in (34).
Further we may
This theorem states that
L..(H) = L..(-H) where the L.. are coefficients relating generalized forces
U
J I
IJ
and currents and are defined to make the entropy production a quadratic
form in the thermodynamic forces.
With these simplifications our resis­
tivity tensor may be written
(p|])
[Pij] =
(PI2~P213^3"P'213^3^
(^12*^213^3*^'213^3)
(P22)
13"^^ 132^2"*"^' 123^2^
(P23"P32|H]"P'321^|) (2.5)
(p|3-p]32'^2"P' 132^2)(p23*P32|Hi+P'32]M])
(P33)
where we have assumed that M and TÎ are in the same direction.
From Equations 2.4 and 2.5 we see that the Hall effect is characterized
5
by the direction in which the magnetic field intensity,
is applied,
interchanging the direction of current flow and the direction in which
the Hal] voltage is measured gives the same Hall coefficients.
If the
crystal in which the Hall effect is measured has hexagonal symmetry there
are only two independent Hall coefficients, one for H applied along the
c-axis and one for H along an a-axis or b-axis.
This may be demonstrated
by transforming the resistivity tensor to a coordinate system rotated 60°
about the c-axis and rotating the direction of ïî by the same transforma­
tion.
The sixfold symmetry of the hexagonal lattice requires that the
same fields be observed in the transformed system, which yields the
desired result.
It is also of interest to relate the two Hall coefficients observed
in single crystal samples of hexagonal symmetry to the Hall coefficient
measured in a polycrystal1ine sample of the same material.
Consider a single crystal sample whose crystalline axes have an
arbitrary orientation with respect to a fixed set of axes defined by H, J
and JXH.
If we describe the orientation of the crystalline axes with
respect to the fixed axes by the Buler angles, 0, I, and Y, and if we let
H = (0,0,H), J = (J,0,0), then
E^/J =
is given by
sin^G + R^( 1( )cos^S^H + [Rj(X)sin^0 +
+ G(0,$,Y)
( jj )cos^0]M
(2.6)
Where R^( 1|) and RQ(1) are the ordinary Hall coefficients of a hexagonal
crystal measured with hî applied parallel and perpendicular to the axis of
symmetry and similarly for R|(ll) and Rj(l).
G(0,§,Y) is a function which
vanishes when averaged over the total solid angle.
An average over the
6
total solid angle gives the Hall effect for a polycrystal1ine sample,
Rg/poly.) = 1/3 RqCII) + 2/3 RJL)
R j ( p o l y . ) = 1/3 R ^ ( l l ) + 2/3 R ,a) .
B.
(2.7)
The Hall Effect in Non-magnetic Metals
The origin of the Hall effect can be most simply understood by
considering the forces acting on a charge carrier moving in a conductor
carrying current density, J*, with a magnetic field intensity, H, applied
perpendicular to J,
In the steady state the net transverse force on the
charge carriers must be zero.
This condition may be expressed as
= 0 = q(E^ + JXB/nqc)
(2.8)
giving the simple result
= -R^ jfXB
(2.9)
where
R^ = 1/nqc
.
(2.1O)
in the Equations 2,8, 2.9 and 2.10, n is the number of charge carriers per
unit volume, q is the charge of an individual carrier, c is the velocity
of light and the cgs system of units is used.
R^ is called the ordinary
Hall coefficient and is defined positive for holes and negative for
electrons.
Equation 2.10 is well obeyed for metals whose electrons are
nearly free, but a more careful treatment is required when two or more
bands are involved or when the Fermi surface is non-spherical.
7
it can be shown (22) that for a metal having two Fermi surfaces, each
spherical, one containing n_ electrons; the other containing n^ holes,
the Hall coefficient at low fields is given by
(2.11)
R Q ~ (l/ec)(n_ - n+)/(n_ + n+)^
where e is the charge of the electron.
For a metal with P spherical Fermi surfaces of either electrons or
holes, the arguments used to derive Equation 2.11 yield the result
R
where
ct
~ (e/a^c)
E
j=l
n.p..^
J J
is the total conductivity,
|JL
(2.12)
is the conductivity mobility of the
jth surface and n. is the number of carriers in the jth surface,
n. is
defined negative for holes and positive for electrons.
In the case of the rare earth metals, however, the Fermi surfaces are
decidedly non-spherical (4, 11, 15).
The expressions given above for the
Hall coefficient are valid for the rare earths only in the sense that they
might give an indication of the possible mechanisms by which the observed
results may be explained.
Jones and Zener (7) have proposed a method by which the Hall coeffi­
cient may be calculated for an arbitrary Fermi surface, but in practice
the integrals involved are extremely formidable.
Since Fermi surface
calculations have been made only recently on rare earth metals, no such
calculations exist for them at this writing.
8
C.
The Hall Effect in Ferromagnets
In a ferromagnetic metal the Hall effect may be described by the
equation
e^ = R^H + R,M
(2.13)
where H is the magnetic field intensity inside the sample, M is the
magnetization and e^^ is the Hall electric field per unit current density.
The behavior of
the extraordinary Hall coefficient, cannot be
accounted for on the basis of a simple Lorentz force picture.
There seems
to be general agreement among existing theories that the anomalous
behavior of the Hall effect in ferromagnetics is due to a spin-orbit
interaction (5^ 9, 10, 14, 16, 28, 30).
The first successful treatment of the ferromagnetic Hall effect was
given by Karplus and Luttinger (lO).
They investigated the intrinsic spin-
orbit interaction of polarized electrons.
Their calculations predicted
that R| should be proportional to the square of the resistivity.
Other
investigations have been made by Luttinger (10), Smit (28), Strachan and
Murray (30), Gurevich and Yassievich (5), Kagan and Maksimov (9) and
Leribaux (14).
The only attempts to calculate Rj numerically were made by
Strachan and Murray for Ni (30) and by Leribaux for Fe (14).
It has been pointed out (30) that the p
2
dependence of the ferromag­
netic Hall coefficient, R^, is simply the result of inverting the conduc­
tivity tensor.
If x is the direction of current flow, z the direction of
the magnetization and y the direction of the Hall field, we have the
relation
9
J. = E a.jEj
With
,
(2.14)
= 0 we obtain for the Hall resistivity
°
" "°yx''''xxV
(2-'5)
2
under the condition (a ) « a (j
which is well obeyed experimentally.
yx
XX yy
/
r
/
This
p
2
dependence is quite a general result.
Since
a
yx
depends only on
the spin system and not on the scattering, the temperature dependence will
come mainly from the
term in the denominator of Equation 2.15
unless the magnetization or Fermi surface is strongly temperature
dependent.
10
111,
A.
EXPERIMENTAL PROCEDURE
Measurement of the Hall Effect - d.c. Method
I f a s a m p l e o f t h i c k n e s s , t_, c a r r y i n g c u r r e n t , 1 , i s p l a c e d i n a
magnetic field intensity, H, a Hall voltage, V^, will be developed across
the sample as shown in Figure 1.
V^, is related to 6^,, the Hall electric
M
n
field per unit current density, by
®H " Vyt/I
.
(3.I)
A study of the Hall effect thus involves measuring V^, t, I, H and T, the
sample temperature.
V|^, the Hall voltage, was measured with a Honeywell model 2768 six
dial potentiometer. Guild line Type 5214/9460 photocell galvanometer ampli­
fier and a Guildline Type SR21/9461 secondary galvanometer.
These instru­
ments provided a voltage resolution of 0.01 microvolts.
A drawing of the sample holder is shown in Figure 2.
The sample was
clamped in place between phosphor-bronze strips at the ends of the sample.
These strips also served as current contacts.
The Hall voltage probes
were 0,020 inch diameter tungsten wires on which a knife edge had been
ground.
These were clamped on the underside of the sample holder and
protruded through the sample holder on either side of the sample.
The
wire was springy enough to provide good electrical contact down to liquid
helium temperatures.
All electrical connections in the Hall voltage
measuring circuit were copper-to-copper with the exception of the copperto-tungsten and tungsten-to-sample connections in the sample holder.
Ideally, the Hall probes should be aligned exactly opposite one
I
POTENTIOMETER
/
-O
Figure 1»
CURRENT
SUPPLY
A schematic drawing of the relation between Jj H and E^, and of the
measuring circuit
12
STAINLESS
STEEL
TUBE
GRAY
FIBER
HALL
PROBE
PHOSPHOR
BRONZE
CLAMP
L_/
PHOSPHOR
BRONZE
CLAMP
•(:
I
r
SAMPLE
SAMPLE
HALL
PROBE
FRONT
Figure 2.
BACK
SIDE
Drawing of the sample holder
HALL
PROBE
î
13
another so they touch the sample on the same equipotential line.
practice this is not achieved.
In
Because of the high resistivity of the
rare earth metals, a slight misalignment of the probes produced an
appreciable zero field voltage.
One cannot simply subtract this voltage
from the voltage measured with an applied field because the voltage due
to probe misalignment changes with H due to magnetoresistance.
One must also take into account the thermal emfs produced at the
various junctions in the measuring circuit.
These thermal and resistive voltages were separated from the true
Hall voltage by means of current and magnetic field reversal.
The zero
field voltage was first measured for forward and reverse current.
The
average of these two voltages is the true zero field voltage because the
thermal emfs do not reverse with current.
The magnetic field was then
increased in steps up to the maximum value, the forward and reverse voltage
being measured at each step.
When the maximum field was reached, the
sample was demagnetized by rotating the sample back and forth as the field
was turned back down to zero.
This process was then repeated with the
magnetic field in the opposite direction.
Field reversal was achieved by
rotating the sample 180° from its original position.
By averaging the
results for forward and reverse fields the magnetoresistance effects are
eliminated, since Ap is an even function of B, the magnetic induction.
The true Hall voltage, V^(H), is calculated from the equation
V^(H) = l/4([V+(H)-V_(H)] + [V+(-H)-V_(-H)] - 2[V^(0)-V_(O)]) .
(3.2)
This equation is merely a statement of the averaging procedures described
above.
V^(H) is the Hall voltage with the current forward and the
14
magnetic field in the forward direction.
current
V_(-H) is the Hall voltage with
and field reversed and similarly for V_(H) and
.
V+(0) is
the zero field voltage.
The sample thickness, t, was measured with a Steinmeyer 25-50 mm.
metric micrometer and a 25 mm. gauge block.
This instrument measured to
within O.OOl mm.
The sample current, I, was determined by measuring the voltage across
a 0.01 ohm standard resistor which was connected in series with the
sample.
This voltage was measured with a Rubicon Type K2 potentiometer
and a Leeds and Northrop galvanometer.
The current supply was constructed
in the electronics shop at the Ames Laboratory and was regulated to 0.01%
Sample currents were about 0.125 amperes.
Larger currents than this
heated the sample and resulted in excessive thermal emfs.
The magnetic field was produced by a Harvey-Wells L-158 electromagnet
and HS-200 kw. power supply.
30.7 kOe.
This magnet produced a maximum field of
Field calibration was made with a Rawson Type 501, 0.1%,
rotating coil gaussmeter.
Alignment of the sample perpendicular to the magnetic field was made
by means of a rotating sample holder assembly as described by Rhyne (25).
This device allowed the sample to be rotated in the magnetic field by
remote control.
The sample was put into the cryostat with the approximate
orientation for the forward magnetic field direction.
The rotating sample
holder assembly was set to 0° and the magnet was rotated about the verti­
cal axis to give a maximum Hall voltage.
The 0° setting then gave the
forward field and the 180° setting gave the reverse field.
The variation
of the Hall voltage with angle was symmetrical about 0° and 180° when the
15
sample was aligned in this way.
The sample temperature, T, was measured by thermocouples mounted
directly beneath the sample.
A Cu versus constantan thermocouple was used
about 20°K and a Au-Fe versus Cu thermocouple was used below 20°K.
The
calibration used for the constantan wire was that reported by Powell et al.
(23).
Two different Au-Fe versus Cu thermocouples were used because one
of the thermocouples broke during the course of the experiment.
The
composition of the first was Au O.O7 at.% Fe and the calibration on this
wire was performed by Finnemore et al. (3).
The thermocouple which
replaced it had composition Au O.O3 at .% Fe.
This roll of wire was
calibrated by W. Gray of this laboratory by comparison with a calibrated
germanium resistor.
Individual thermocouples prepared from the same roll of thermocouple
wire differ by several microvolts at low temperatures, presumably due to
inhomogeneous distribution of impurities in the wire.
The thermocouples
used in this experiment were normalized to the calibration curves by the
method described by Rhyne (25).
soft solder.
Thermocouple junctions were made with
Thermocouple voltages were measured with a Rubicon Type.K2
potentiometer and a Honeywell galvanometer.
All thermocouple leads and all
leads coming to the sample were thermally anchored by wrapping the leads
several times around the bottom of the stainless steel sample tube and
cementing them to the tube with General Electric 7031 varnish.
The
absolute accuracy of the temperature measurements is estimated to be
+ 0.5°K and the relative accuracy is about 0,I°K over the range 4.2°K to
350°K.
The temperature of the sample was controlled to + 0.1°K by means of a
16
temperature control amplifier and bridge as described in (20, 25, 27).
nitrogen shielded dewar was used for these measurements.
dewar used by Rhyne and is described in (25).
A
This was the same
Measurements were made over
the temperature range 4.2°K to 340°K.
B.
Measurement of the Hall Effect - a.c. Method
Hall effect measurements for the lutetium and yttrium samples were
made on an a.c. Hall effect apparatus.
This apparatus was designed and
constructed by L. D. Muhlestein and is described in detail
dissertation (18).
in his Ph.D.
It was necessary to resort to using an a.c. technique
on the lutetium and yttrium samples because the Hall effect in these
metals was too small for the d.c. apparatus.
Figure 3 shows a block diagram of the a.c. Hall apparatus.
A 386
cycle per second signal was produced by a Hewlett-Packard Model 203A signal
generator.
This signal was amplified by a Hewlett-Packard Model 467A
power amplifier to supply the a.c. current to the sample.
The current was
determined from a measurement of the voltage drop across a 0,1 ohm
standard resistor with a Fluke Model 883AB differential voltmeter.
The a.c. signal from the signal generator was also used as the
reference signal for a Princeton Applied Research Model JB-5 phase sensi­
tive lock-in amplifier as well as for two bucking circuits.
The first
bucking circuit was used to buckout the zero field voltage due to misalign­
ment of the Hall probes.
The second bucking circuit contained a phase
shifter followed by a Fluke Model 883 AS differential voltmeter and a
Dekatran Model DT72A decade transformer.
The second bucking circuit
measured the unknown signal from the Hall probes or the resistivity probes
hp 203A
FUNCTION
GENERATOR
AMPLIFIER
"A) 0-1 AMP A C.
'ST
SCOPE .1
OUT -trrrr
PHASE
SHIFT-R a I
ATTEI UATOf 1
I
SCOPE OUT
PHASE
SHIFTER
DECADE
I TRANSFORMER
L_
1.25 Û
AM-I
24-1
PAR CR-4
PRE
AMPLIFIER
LOCK-IK
AMPLIFIER
A C. HALL APPARATUS
GROUND
HIGH
Figure 3.
Block diagram of the a.c. Hall effect apparatus
18
by providing a signal IBO degrees out of phase with the unknown signal and
of equal magnitude.
The null was detected by a Princeton Applied Research
Model CR-4 pre-amplifier and a Model JB-5 lock-in amplifier.
-9
provided a voltage resolution of 10
This apparatus
volt.
The Hall voltage was measured at the maximum field and the Hall
coefficient was computed from the equation
Ro =
t/IH
.
(3.3)
It was not practical to measure the Hall voltage as a function of H because
of the extremely small signals.
The linearity of
with H was checked on
the yttrium c-axis sample and
was found to be linear with H within the
limits of error.
The sample holder of the a.c. apparatus is described in detail in
(18).
This sample holder differs somewhat in design but is equivalent in
operation to the sample holder used for the d.c. measurements.
Temperatures were measured with a copper versus constantan and a
copper versus Au 0.07% Fe thermocouple.
The magnet used for the a.c. measurements was a six inch Varian with
a Varian V2100B power supply.
7.8 kOe.
This system provided magnetic fields up to
The magnetic field was measured with a Radio Frequency Labora­
tories Model 1890 gaussmeter.
This gaussmeter was calibrated against a
0,1% Rawson Type 501 rotating coll gaussmeter.
All of the determinations of R_ were averaged over forward and
reverse field directions.
the samples.
The resistivity ratio was measured for all of
19
C.
Sample Preparation
As was mentioned earlier, it is necessary to study single crystal
specimens to observe the effects of Fermi surface anisotropy and magnetic
anisotropy on the Hall effect.
The rare earth metals used in this investigation were produced at the
Ames Laboratory by use of an ion exchange process for the separation of the
rare earth compounds (29) and a reduction process for the preparation of
the pure metal from the fluoride.
The single crystals from which the samples were cut were grown from
arc-melted buttons of pure metal by a thermal-strain-anneal technique
described by Nigh (19).
Thermal strains which were frozen Into the button
by the arc-melting process were removed by annealing the metal button at
a temperature slightly below the melting point.
The gadolinium crystals
used In this Investigation were those from which Sill (27) had obtained
his samples.
The lutetlum and yttrium crystals were grown by the author.
The crystals grown by this technique were aligned by Laue backreflection of x-rays.
The samples were cut to approximate dimensions by a
Servomet spark-erosion apparatus.
The samples were then hand lapped to
final dimensions with carborundum paper and were finally etched or electropolished to ensure a surface to which good electrical contact could be
made.
The samples were cut In the form of a rectangular parallelepiped.
The
gadolinium samples had dimensions of approximately 1 cm. x 2.5 mm. x 0.5 mm.
The lutetlum and yttrium samples were somewhat smaller so they would fit
Into the a.c. Hall effect apparatus sample holder.
The ratio of length to
width of all of the samples was at least 4:1 to eliminate shorting effects
20
(6).
The thinness of the samples was necessary to produce a larger
but
the thin samples also require a relatively large correction for demagne­
tizing fields.
The demagnetization factor for each sample was calculated
by the method of Joseph and Schlomann (8).
Sample dimensions and demagne­
tization factors are listed in the Appendix.
The rare earth metals used in this investigation were of the best
purity available at the Ames Laboratory.
Scraps of metal left over after
the samples were cut were heavily etched to remove surface impurities and
were analyzed for impurities by a semi-quantitative spectrograph Ic analysis
and a quantitative vacuum fusion analysis for dissolved gas impurities.
These analyses were made at the Laboratory.
20K^ were measured for all the samples.
Resistivity ratios, PgQQOR/
Information regarding sample
purity is tabulated In the Appendix.
D.
Determination of R and R, from Isothermal Data
o
I
In this investigation the Hall effect was measured isothermally,
i.e., the temperature was held constant and the variation of the Hall
voltage with magnetic field was measured.
If the sample is paramagnetic
the ordinary Hall coefficient is given by Equation 3.3 and Is independent
of H.
If the sample is ferromagnetic, a more complex procedure Is necessary
to extract the coefficients R^ and Rj from the Isothermal data.
For a
ferromagnetic material we must use the equation
e. = R H + R,M
n
o
1
.
(3.4)
21
it is convenient to write this in terms of the applied field, H
% - R."app.
app.
, as
+
where we assume we can write
H = H
app.
- NM
.
(3.6)
N is called the demagnetizing factor and is, in reality, a complicated
tensor quantity.
soidal samples.
Equation 3.6 holds only to first order for non-ellip­
Joseph and Schlomann (8) have calculated N as a power
series In ascending powers of M/H.
Their result has been assumed to be
good to a first approximation above magnetic saturation where M/H will be
small.
Since N also varies with position inside the sample, the value of
N used for each sample represents an average over the sample.
shows the variation of
Figure 4
In various directions from the center of a
sample with dimensions 1 cm. x 2.5 mm. x 0.4 mm.
Above saturation M/H = 0 so
is given by
"o =
(3.7)
where H^ Is the applied field necessary to saturate the sample and H is
now taken to be the applied field.
We can determine Rj also, for
:H|H > H; = *o" + (*, - "Ko'Ms
where M^ is M(H = <=°), the saturation magnetization.
gadolinium were obtained from the data of Nigh (20).
'3.8)
Values of M^ for
Equation 3.8 is
22
Nzz (0,y,0)
12.0
8.0
4.0
Y(mm)
-1.25
-0.75
-0.25
0.75
.25
12.0
0.0
4.0
z(mm)
-020
-0.12
-004
12.0
0.04
0.12
0.20
Nzz (X.O.O)
8.0
4.0
x(mm)
-5.0
Figure 4.
-3.0
The variation of
-1.0
1.0
3.0
5.0
in various directions from the center of a
sample with dimensions 1 cm. x 2,5 mm. x 0.4 mm.
23
linear and if it is extrapolated to H = 0 one obtains the result
41= (R, - NR )M
1
O
S
(3.9)
where «Û. is the intercept of the linear portion of the e^ versus H curve
ri
on the 6^ axis.
From this we have
R, = (^+ NR M )/M
I
O S S
(3.10)
A correction to Equation 3.7 must be made to take into account the
high field susceptibility, Xg^ of the sample above saturation.
Because
of the small linear dependence of M on H above saturation, we must include
a term depending on Rj in the expression for R^.
We then have
(3.11)
Since the term depending on R^ in Equation 3.10 is small compared to Rj in
ferromagnetic materials, we may substitute de^/ôHj^ > ^
for R^ in this
S
equation with little error.
The value of Rj thus computed may then be
used in Equation 3.11 to determine R^.
In gadolinium R^Xg is as large as
30% of R^ near the ferromagnetic Curie temperature, 8^, so this is an
important correction.
Xg for gadolinium was determined from the magne­
tization data of Nigh by computing the slope of the M versus H curves
above saturation.
in Figure 5.
Xg for gadolinium Is shown as a function of temperature
24
HIGH FIELD SUSCEPTIBILITY OF 6d
o C AXIS
A B AXIS
• A AXIS
A A AAA •••O
40
60
120
160
200
240 280 320
T*K
Figure 5.
Xg for gadolinium single crystals as a function of temperature
25
IV.
EXPERIMENTAL RESULTS
Ao
The Hall voltage,
Gadolinium
was measured for four single crystals of gado­
linium over the temperature range 4.2° to 340°K in magnetic fields up to
30.7 kOe.
Table 1.
The orientations of the four crystals are listed in Table 1.
Orientations of gadolinium single crystal samples.
Sample
H
J
e^
Designation
Gd I
b-axis
c-axis
a-axis
b-axis
Gd I I
a-axis
c-axi s
b-axi s
a-axi s
Gd III
a-axis
b-axi s
c-axi s
a-axi s
Gd IV
c-axis
b-axi s
a-axi s
c-axi s
The data were taken isothermally, that is, the Hall voltage was
measured as a function of applied magnetic field while the sample temper­
ature was held constant.
For each temperature the magnetic field was
varied from 0 to 30,7 kOe.
c-axis sample.
sample.
Figure 6 shows representative isotherms for the
Figure 7 shows representative isotherms for an a-axis
One can see from the linear portions of these curves at high
fields that the samples are magnetically saturated in spite of the large
demagnetizing fields.
Appendix.
All of the isothermal data are tabulated in the
Figure 8 is a plot of e|^ versus T for H = 30.7 kOe.
all of the samples are plotted on the same graph.
Data for
A large anisotropy is
observed between the c-axis sample and the a and b-axis samples which will
from now on be called the basal plane samples.
is observed.
No basal plane anisotropy
This presentation of the data is useful to show the size of
the Hall effect as a function of temperature.
1.00
GD ISOTHERMS
0.80 —
TT ALONG C-AXIS
T
ALONG A-AX IS
J.5°K
166.2 ®K
292.8 °K
O
0.40
to
o\
0.20
341.5 «K94.0 ®K
77.6 °K
47.0 ®K
K
-0.20
4.00
Figure 6„
6.00
12.00
18.00 20.00 24.00
H(kOe)
28.00
Isotherms for the gadolinium c-axis sample
32.00 36.00
(OHM CM) X 10®
o
o
!\)
o
o
O
CD
o
ro
o
m
o
g
o
è
poo
R
o
00
b OO o
(DO O
ro
b
O)
o
a.
b
>
5
g
ro
§
§
X
n
70
>
X
CO
Ê
g
o o
N
o< ro
fo
^ ?c
œ 0)
OJ
m
o
Zz
ro
z
w
1.20
1.00
0.80
0.60
I
O I H ALONG A-AXIS
\T ALONG C-AXIS
p f FT ALONG B-AXIS
J ALONG C-AXIS
ÎT ALONG C-AXIS
T ALONG A-AXIS
^FT ALONG A-AXIS
T ALONG B-AXIS
GD ISOFIELDS
H = 30.7 kOe
s
o
S
X
o
tn
g
X
X
0
1
ro
0.40
00
020
0.20
1
40.00
1
120.00
1
200.00
280.00
360.00
T («K)
Figure 8.
e^, versus T for H = 30*7 kOe.
Data from all of the samples are plotted on the same graph
29
the ordinary Hall coefficient, is plotted as a function of temp­
erature in Figure 9.
A large anisotropy between the c-axis and basal
plane samples is observed but no basal plane anisotropy can be observed.
Since the data for Gd II and Gd III are the same, within the limits of
error, it can be seen that the direction of J in the crystal does not
affect the value of R^.
This is in agreement with the prediction of the
phenomenological theory that the direction of H and M determined which
component of the resistivity tensor would be measured.
Interesting features of these curves are the sharp rise in R^ near
8g and the changes in sign of R^ at 195°K and 277°K for the basal plane
samples and at 130°K for the c-axis sample.
Figure 10 shows Rj, the extraordinary Hall coefficient, plotted as a
function of temperature.
Again one observes a large anisotropy between
the c-axis and basal plane samples and no basal plane anisotropy.
Rj does
not have its maximum value at 0^, but rather at about 280°K for the basal
plane samples and about 250°K for the c-axis sample.
Figure 11 shows the values of R^ for single crystals of gadolinium as
measured by Volkenshtein et al. (33) compared to the findings of the
present investigation.
The agreement is relatively good for the c-axis
sample but is not so good for the basal plane samples.
Volkenshtein et a1.
(33) report no values for R^.
R^ and Rj have been determined only up to 293°K because above the
Curie temperature it is not possible to separate the effects due to R^ and
5.0
GADOLINIUM ORDINARY HALL COEFFICIENT
ALONG A AXIS
ALONG C AXIS
FT ALONG B AXIS
4.0
3.0
2
R ALONG C AXIS
IT ALONG
JT ALONG
f FT ALONG
^ W ALONG
C AXIS
A AXIS
A AXIS
B AXIS
2.0
-1.0
40
80
120
160
200
240
280
320
T (®K)
Figure 9»
The ordinary Hal 1 coefficient of gadolinium
360
1
.0 —
1
1
T"
GADOLINIUM EXTRAORDINARY
HALL COEFFICIENT
H ALONG
J ALONG
PR ALONG
J" ALONG
FT ALONG
TALONG
VCALONG
(TALONG
40
80
120
160
200
240
280
320
360
T (®K)
Figure 10,
The extraordinary Hall coefficient of gadolinium
A AXIS
C AXIS
B AXIS
C AXIS
C AXIS
A AXIS
A AXIS
B AXIS
6.0
1
X SINGLE CRYSTAL DATA - VOLKEiMSifTElN
5.0
r
X
'O—f
"O
%
\
t >TH1S v;
8 4.0
ZD
S
\
O 3.0
X
H ALONG C-AXIS
2
X
s
OJ
N5
% 2.0
X
X
cr
qT
' 1.0
X
\
—x___Xd^—Q
0
H IN BASAL PLANE
-1.0
0
40
80
120
160
200
240
280
320
T MO
Figure 11
Comparison of the single crystal data of Volkenshtein et al, (33) with the present
investigation
33
Bo
The Hall voltage,
Lutetium and Yttrium
was measured for two single crysta Is of lutetium
and two single crystals of yttrium over the temperature range from 8° to
c
MC
o
o
magnetic fields up to 7.8 kOe.
The orientations of the four
crystal s are listed in Table 2.
Table 2.
Orientations of lutetium and yttrium single crystal samples
Sample
H
J
Des ignat ion
®H
Lu 1
a-axis
b-axis
c-axi s
Lu a-axis
Lu il
c-axis
b-axis
a-axis
Lu c-axis
Y 1
b-axis
a-axis
c-axi s
Y b-axis
Y II
c-axis
a-axis
b-axis
Y c-axis
The data were taken isothermally.
These measurements differ from the
measurements made on gadolinium in that the Hall voltages were so small it
was not practical to measure
as a function of H.
yttrium have extremely small susceptibilities, so
was determined from the value of
Both lutetium and
is independent of H.
at the maximum field.
Below 8°K a spurious signal was produced in the a.c. Hall effect
apparatus so it was not possible to make measurements below this tempera­
ture.
The explanation of this spurious signal is still a mystery, both to
the author and to L. Muhlestein, who constructed the apparatus.
Above 8°K
the apparatus worked properly and reproduced known results for copper and
silver (18).
Above 320°K the adhesive used to fasten down the wires in the
sample holder became soft and permitted the wires to vibrate.
This intro­
duced a spurious signal, so measurements were restricted to the range 8° to
34
320°K.
Figure 12 shows a plot of
versus T for the yttrium crystals.
The
interesting thing about these curves is the large anisotropy between the
c-axis and basal plane, and the strong temperature dependence.
Figure 13 shows a plot of
versus T for the lutetium crystals.
Again there is a large anisotropy and even more temperature dependence
than for the yttrium crystals.
Even more striking is the fact that R^ is
positive for the basal plane sample.
The c-axis curve for lutetium is
qualitatively similar to that of yttrium.
Measurements were made on only one basal plane sample for yttrium and
one for lutetium because both of these metals have hexagonal close-packed
crystal structure and according to the phenomenological theory, there
should be no basal plane anisotropy.
This prediction was verified experi­
mentally for gadolinium so it was not deemed necessary to measure the Hall
effect in both a and b-axis samples, since they would give equivalent
results.
It was not possible to measure the sign of
with the a.c. apparatus,
so the sign of the effect was checked with the d.c. equipment.
R^ was
negative for both of the yttrium crystals and the c-axis lutetium crystal,
but was positive for the lutetium basal plane sample.
28.0
YTTRIUM HALL COEFFICIENT
24.0
^ 20.0
16.0
ro
H ALONG C-AXIS
J" ALONG A-AXIS
12.0
H ALONG B-AXIS
.T ALONG A-AXIS
8.0
4.0
40
80
Figure 12.
120
160
200
240
280
320
360
versus T for yttrium b and c-axis crystals
1
28.0
.A-MA
-ô-
24.0
-
Sa (f)
o CO
3
:c
oO
ro
O
20.0
O - H ALONG A-AXIS
A- H ALONG C-AXIS
16.0
12.0
VjJ
cr\
X
0
(T
8.0
1
4.0
0.0
-Q-O-
-40
"O.
O
0
J
I
I
40
60
120
J.
160
200
240
280 320
T{°K)
Figure 1 3 .
versus T for lutetîum a and c-axis crystals
360
37
Mo
A.
DISCUSSION
Lutetium and Yttrium
The Hall effect in lutetium and yttrium will be discussed first, since
these results will have a bearing on the discussion of the Hall effect in
gadolinium.
The features of the Hall effect in lutetium and yttrium that
require discussion are the large anisotropy, the strong temperature
dependence and the positive
observed in the lutetium basal plane
sample.
The anisotropy of the Hall effect in lutetium and yttrium is almost
certainly due to the extremely anisotropic Fermi surfaces of these metals.
Figure 14 shows a model of the Fermi surface of gadolinium as calculated
by Freeman et_ a_l_. (4).
Calculations by Loucks (15) and Keeton and Loucks
(11) show that the Fermi surfaces of lutetium and yttrium are very similar
to the Fermi surface of gadolinium in the paramagnetic state.
The Fermi
surface of gadolinium is thus a good approximation to the Fermi surfaces
of lutetium and yttrium.
One can see from Figure l4 that the electron and hole orbits with the
magnetic field applied along the c-axis will be very much different from
the orbits with the field applied in the basal plane.
With such different
orbits for the different field directions, it is not surprising that a
large anisotropy in the Hall effect is observed.
The temperature dependence of the Hall effect in lutetium and yttrium
can be qualitatively understood in terms of a two-band free-electron model.
The reason that a free-electron model gives qualitative results when the
Fermi surface is so obviously non-spherical is that the temperature
•'U-V
Figure 14,
The complete Fermi surface for holes in gadolinium metal in the
double zone representation as calculated by Freeman _et a^. (4)
39
dependence does not arise from the Fermi surface geometry.
The mechanism
for the temperature dependence seems to be the transition from impurity
scattering at low temperatures to phonon scattering at higher temperatures.
Equation 2.12 states that the Hall coefficient for a metal having two
spherical Fermi surfaces, one containing n^ holes, the other containing n_
electrons per unit volume, may be written
RQ = (e/a^c)
where
- n_(i^)
(5.1)
is the hole conductivity mobility and pL_ is the electron conduc­
tivity mobility.
It is reasonable to assume that the electron and hole
mobilities will have different temperature dependence.
We write this as
M.+ = (a + bT) '
H_ = (c + dT)"^
(5.2)
.
(5.3)
If we substitute 5.2 and 5.3 into Equation 5.1 and assume we have a
compensated metal, i.e., n_ = n^ = n, we obtain
necR .
°
(c
'ab)? +
.
+ a ) + 2(cd + ab)T + (d
( 5 .4)
+ b )T
The temperature dependence of this expression is determined by the values
of the coefficients a,b,c and d.
One may choose initially the ratio d/b.
One may then determine the ratio c/a by the relation
R^(T = 0°K)
(c^ - a^)(d^ + b^)
«[(T
.300 K)= (c? + a2)(d2.b2)
'
R^(T = 0)/R^(T = 300°) may be determined from the experimental data.
One
ho
may determine the ratio a/b by writing the total conductivity in terms of
the electron and hole mobilities and calculating the resistivity ratio,
PZf 2OK^^300OK' which has been determined experimentally for each sample.
Figure 15 shows the results of some calculations using Equation 5.4 and
values a,b,c and d estimated as described above.
It would be pointless to try to fit these curves to the experimental
data since the actual Fermi surfaces of the rare earth metals are so far
from spherical, but this argument does demonstrate in a qualitative manner,
a possible mechanism for the temperature dependence of
in yttrium and
lutetium.
The difference in sign of
samples is an interesting result.
between the c-axis and a-axis lutetium
Scovil (26) reports a similar result for
single crystals of titanium, where R^ is positive with the field applied
along the c-axis and negative with the field applied in the basal plane.
This sign change of R^ must be due to the Fermi surface geometry.
One
might wonder why this sign change is not also observed in yttrium, but the
difference between the Hall effects in these two metals is not as drastic
as one might think.
From Equation 5.1 we see that for a compensated metal
the sign of R^ is determined by the relative magnitudes of the electron
and hole mobilities.
For both lutetium and yttrium R^ is nearly zero with
the field applied in the basal plane, so the electron and hole mobilities
in this direction must be nearly equal.
The Fermi surfaces of lutetium
and yttrium are quite similar, but they are not identical so a relatively
small difference in Fermi surface geometry might change the mobilities
sufficiently to change the sign of R^.
Figures l6 and 17 show R^ for polycrystal1ine lutetium and yttrium.
41
40
°4.2°K
2.0 -V
25
c = 1.04a
"300°K
RgOOCK)
10
Ro(0»K)
80
160
240
40
25
°500''K
Ro(300''K)
£r° 2.0
o
o
c
RQ^O'K)
)0-
d = l.5b
0 = 12.2 b
320
d = l.Oi b
c =I.OIa
a =8.12b
=1
-7-0-
80
160
240
320
40
°AZ''K
(T'socx
2.0
25
a "8.12b
Ro(300»K)
Ro(O'K)
80
160
d = 1.5b
c = 1.5a
•I
240
320
T (°K)
Figure 15.
necR^ calculated from Equation 5.4 for various values of the
parameters a^b^c and d
to
o_
O
CO
4.0
A
-THIS VYORK
O
- KEVANE
-o—
eî al
fM
0
40
80
120
160
200
240
1
280
1
320 360
T(°K)
Figure 16.
Comparison of
for 1utetiurn, calculated from single crystal data, wi tli the polycrystal
line data of Kevane^
(13)
1
14.0
12.0
;â^-g
s^
o
s
10.0
3
X O
o
8.0
ro
O
<0
o
Q
.O
0
OC
6.0
Rq FOR POLYCRYSTALLINE YTTRIUM
,o
1
o—o—o—^
O CALCULATED FROM SINGLE
CRYSTAL DATA
4.0
ANDERSON ot. ai.
2.0
0
0
40
80
120
160
200
240
280
320
T (°1<)
Figure 17c
Comparison of
for yttrium, calculated from single crystal data, with
data of Anderson et a 1. (])
the polycrysta]1ine
44
calculated from the single crystal data, compared with the polycrystal1ine
data of Kevane ^
(13) and Anderson et al. (l).
The agreement is
reasonably good considering the extremely small voltages being measured
and the difficulty of making a polycrystalline sample with truly random
orientation.
Because of the large anisotropy of the Hall effect in these
metals, preferred orientation in the polycrystalline specimens can change
the results considerably.
B.
Gadolinium
The ordinary Hall coefficient, R Q, of gadolinium also exhibits a large
anisotropy and temperature dependence.
In addition,
for gadolinium is
one to two orders of magnitude larger than R^ for yttrium and lutetium
over the temperature range 4.2°K to 300°K.
The anisotropy of the Hall effect observed in gadolinium is probably
due to the anisotropic Fermi surface as was the case for yttrium and
1utet i um.
A two band model
is not adequate to explain either the large values of
or the changes in sign of R^ observed in gadolinium.
Kevane et al. (13)
were able to interpret their polycrystalline Hall effect data above 100°C
by assuming a constant value of R^, taking R^ = -0.4 x 10^^, and using the
values of magnetic susceptibility of Trombe (31).
These results indicate
that at temperatures sufficiently high above the Curie temperature, the
Hall coefficient of gadolinium is of the same order of magnitude as the
Hall coefficients of yttrium and lutetium.
This is what one would expect,
since the electronic structures of these metals are so similar in the
absence of magnetic ordering.
One must conclude that the anomalously
45
large values of the Hall coefficient observed in gadolinium below the Curie
temperature are the result of magnetic ordering.
The resistivity measurements of Nigh et al. (21) and the Seebeck
effect measurements of Sill and Legvold (27) indicate that the scattering
varies rather smoothly through the Curie temperature.
Since the resistivity
and Seebeck effect are much more sensitive to scattering than the Hall
effect, one must look elsewhere to account for the behavior of
A mechanism which might provide an explanation for the anomalous
behavior of
near and below the Curie temperature is the splitting of the
conduction band of gadolinium at low temperatures through an exhange
interaction with the 4f electrons.
Freeman et al. (4) estimate the exchange
splitting for gadolinium from their calculated density of states at the
Fermi surface and the observed saturation magnetization of 7.55Mg/atom.
Their estimate is 0,61 ev.
One would expect this exchange splitting to
have the same temperature variation as the spontaneous magnetization.
Pugh (24) proposed a similar explanation for the Hall effect in the ferro­
magnetic 3d transition metals.
Figure 18 shows E(k) curves for the conduction bands of gadolinium
metal along the symmetry directions T-K-H-A.
Consider the effect of the
exchange splitting on the Fermi surfaces for spin-up and spin-down elec­
trons.
The dashed lines in Figure 18 are the Fermi energies for spin-up
and spin-down electrons at T = 0°K.
inspection of the E(k) curves in
Figure 18 shows that the Fermi surface undergoes considerable distortion
as the temperature is lowered, if one assumes that the E(k) curves are not
themselves altered by the exchange interaction.
46
H, GADOLINIUM
Figure 18.
E(k) curves for the conduction bands of gadolinium metal along
the symmetry directions T-K-H-A as calculated by Freeman et al.
(4)
47
For example, the spin-up 3rd and 4th zone hole surfaces along H-A will
move rapidly toward A as the temperature is lowered.
Similarly, the spin-
down surface will suffer considerable distortion in the region F-K-H.
The observed saturation magnetization of 7.55 M,,/atom also indicates
D
that the occupation of the spin-up and spin-down Fermi surfaces changes as
the temperature is lowered.
A model based on a Fermi surface of at least four sheets would be
required to account for the Hall effect in gadolinium.
Even with such a
model it is not feasible to calculate the temperature dependence of
was done with the two band model for lutetium and yttrium.
as
The rapidly
changing Fermi surface makes it impossible to predict a reasonable temper­
ature dependence for the mobilities of the various sheets of the Fermi
surface.
The picture is further complicated by the changing occupations
of the spin-up and spin-down bands.
It does appear, however, that the changes in the Fermi surface caused
by the exchange splitting of the conduction bands provide a possible
explanation for the large magnitude and changes in sign of
gadolinium below the Curie temperature.
observed in
Relativistic calculations of the
Fermi surface of gadolinium have been made recently by Keeton and Loucks
(11).
Their results do not differ appreciably from those of Freeman et al.
(4) except that the degeneracy in the plane A-L-H is removed.
The sharp rise in
near the Curie temperature is somewhat peculiar,
since all the other transport properties vary smoothly through the transi­
tion.
The method of determining R^ and R^ seems to still be valid near
the Curie temperature, since the high-field portions of the e^ versus H
48
curves are still linear and a correction for the high field susceptibility
has been made.
It may be that
the Curie temperature.
peaks smoothly at some temperature above
Distortion of the Fermi surface from exchange
splitting may still persist above the Curie temperature due to short-range
ordering effects.
Figure 19 shows R^ for the basal plane and c-axis samples plotted as
2
a function of p{basal plane)» p(c-axis) and p (basal plane) respectively,
where p is the zero field resistivity as reported by Nigh e_t a_l_. (21),
Rj(poly.) as calculated from the single crystal data is plotted as a
function of p (poly.) in the same figure.
According to Equation 2.1$, R, oco /a a
= a p p .
^
1
yx XX yy
yx xx^yy
This rela-
tion is obeyed below 160°K for the c-axis sample and below 180°K for the
basal plane samples.
An analysis of the
dependence of
below 50°K
has not been made because impurity scattering becomes important at these
temperatures and resistivity measurements were not made on the same
samples used for the Hall effect measurements,
Rj(poly.) versus p^(poly.) shows a linear relationship from 250° to
50°K.
The fact that Rj(poly„) shows a p^ dependence over a much wider
temperature range is probably due to an averaging of the Fermi surface
effects in the polycrystal1ine material.
p
2
Conversely, the deviation from
dependence in the single crystal samples is probably the result of the
Fermi surface changing with temperature.
The decrease in magnitude of Rj near the Curie temperature-may be due
to the rapid decrease of the spontaneous magnetization in this temperature
range.
Some comments regarding the single crystal data of Volkenshtein et al.
o X=/)^,R,= R,(C-AXIS)
6.0
à X = /)jj/)ç R,= R,(BASAL PLANE)
O X = /)^(POLY),R,= R,(POLY)
5.0
S
X
oi j r
4.0
290°K
en
O
fVJD
3.0
a:
I
2.0
A ^A
1.0
0
O'
O^n-O
. A
A
aS&-A-A-A-A-A-AT-A-/\M80°K
O
0.5
1.0
1.5
(ft OHM CM) X lO"'^'
Figure 19.
R, for the basai plane and c-axis samples versus p(basai plane)- p(c-axis) and p (basai
2
plane) respectively; R|(poly.) versus p (poly.)
50
(33) are in order, since tine agreement with their results is not as good
as one might hope.
Their values for
temperature instead of dropping.
increase sharply near the Curie
This seems to be the result of an
erroneous equation used to calculate
.
In an earlier paper Volkenshtein
and Federov (32) calculated R^ from the equation
R, = e^(0) /
(0,T)]
where M^(0,T) is the spontaneous magnetization.
obtain this expression is incorrect.
(5.6)
The argument used to
The correct result is equation 3,10
where the quantity in the denominator is M(oo,T) ^ the saturation magnetiza­
tion.
If one uses the spontaneous magnetization, as they did,
diverge at 0^ because M^(0,6^) = 0.
T and
^ versus T.
cr^ j and
Figure 20 shows a plot of
will
^ versus
^ are the spontaneous magnetization per
gram and the saturation magnetization per gram calculated from the data of
Nigh (20).
This graph illustrates the difference between using the
spontaneous magnetization and using the saturation magnetization in the
denominator of Equation 3.10,
At lower temperatures the difference
becomes less serious.
Volkenshtein et al. (33) made no correction for demagnetizing fields.
This is not a large correction as regards the calculation of Rj, however
their plots of e^^ versus B are misleading since demagnetizing fields are
very large for samples of the shape used for Hall effect measurements.
Figure 11 shows that the c-axis data of Volkenshtein e_t aj_, and the
present investigation agree reasonably well except near the Curie temper­
ature.
The basal plane samples do not agree so well.
It is not likely
51
250
FOR 6d b-AXIS CRYSTAL
200
150
(CGS UNITS)
100
50
100
150
200
250
300
T(°K)
gure 20.
a
versus T and a
o,T
^ versus T for gadolinium single crystals
52
that this is due to a systematic error, since the c-axis samples agree.
The results of the present investigation seem more trustworthy, however,
since they were obtained from measurements on three different crystals
which all yielded the same results.
It does not seem likely that magnetic anisotropy plays a large role
in the anisotropy and temperature dependence of
. All of the informa­
tion about R, is obtained from the e^, versus H curves for values of H
I
above magnetic saturation.
in the magnetization.
n
In this region there is not much anisotropy
53
VI.
BIBLIOGRAPHY
1.
Anderson, G. S.S. Legvold and F. Ho Spedding, Phys, Rev., Ill, 1257
(1958).
2.
Babushkina, N. A., Soviet Phys. - Solid State,
3.
Finnemore, D. K., J. E. Ostenson and T. F. Stromberg. U.S. Atomic
Energy Commission Report IS-1046|lowa State University of Science and
Technology, Ames, lowa. Inst. for Atomic Research] (1964).
4.
Freeman, A. J,, J. 0. Dimmock and R« E. Watson. [The augmented plane
wave method and the electronic properties of rare earth metals. To be
published in Lowdin, P. 0., ed. ça. 1967].
5.
Gurevich, L. E. and 1. N. Yassievich, Soviet Phys. - Solid State, 4,
2091 (1963).
6.
Jan, J. P., Solid State Physics
7.
Jones, H. and C. Zener, Proc. Roy. Soc. (London), Al45, 269 (1934).
8.
Joseph, R. I. and E. Schlomann, J. Appl. Phys.,
9.
Kagan, Yu. and L. A, Maksimov, Soviet Phys. - Solid State, 2.) ^22 ( I 9 6 5 ) .
2450 (1966),
I (1957).
1579 (1965),
10.
Karplus, R. and J. M. Luttinger, Phys. Rev.,
1154 (1954).
11.
Keeton, S. and T. L. Loucks, [The relation between antiferromagnetism
and the Fermi surface in some heavy rare earths.] Phys. Rev. [To be
published ça. I967].
12.
Kevane, C. J., Hall effect at low temperatures for rare earth metals,
unpublished Ph.D. thesis, Ames, lowa. Library, lowa State University
of Science and Technology, 1953»
13.
Kevane, C. J., S. Legvold and F. H. Spedding, Phys. Rev., 91, 1372
(1953).
14.
Leribaux, H., Phys. Rev., 150, 384 (1966).
15.
Loucks, T. L., Phys, Rev., 144, 504 (1966).
16.
Luttinger, J. M., Phys. Rev., 112, 7 3 9 (1958).
17'.'" Marotta, A. S, and K. J. Tauer, J. Phys. Soc. Japan, 18, 310 (1963).
18.
Muhlestein, L. D., Electrical properties of metallic sodium tungsten
bronze, unpublished Ph.D. thesis, Ames, lowa. Library, lowa State
University of Science and Technology, I966.
54
19.
Nigh, H. E., J. Appl, Phys., j4. 3323 (1963).
20.
Nigh, H. E.j Magnetization and electrical resistivity of gadolinium
single crystals, unpublished Ph.D. thesis, Ames, Iowa, Library, Iowa
State University of Science and Technology, I963.
21.
Nigh, H. E., S. Legvold and F. H. Spedding, Phys. Rev., 132. 1092
(1963).
22.
Pippard, A. B., The dynamics of conduction electrons. New York, New
York, Gordon and Breach, 1965.
23.
Powell, R. L., M. Do Bunch and R. J. Corruccini, Cryogenics, J_, 139
(I96I).
24.
Pugh, E. M., Phys. Rev.,
64? (1955).
25.
Rhyne, J. J., Magnetostriction of dysprosium, erbium, and terbium
single crystals, unpublished Ph.D. thesis, Ames, Iowa, Library, Iowa
State University of Science and Technology, I965.
26.
Scovil, G. W., Appl. Phys. Letters,
27.
Sill, L. R. and S. Legvold, Phys, Rev., 137, A1139 (1965).
28.
Smit, J., Physica, 2J_, 8 7 7 (1955).
29.
Spedding, F. J. and J. E. Powell, J. Metals,
30.
Strachan, C. and A. M. Murray, Proc. Phys. Soc. (London), 73, 433
(1959).
31.
Trombe, F., Annales Physique, 2, 383 (1937).
32.
Volkenshtein, N. V. and G. V. Federov, Phys. Metals Metal log., 18, 26
(1964).
33.
Volkenshtein, N. V., G. V. Grigorova and G. V. Fyodorova, Zhur. Eksp.
jjeoret. Fiz.,
1505 ( I 9 6 6 ) .
34.
ZIman, J. M., Electrons and phonons, London, England, Oxford University
Press, i960.
24? (1966).
I I 3 I (1954).
55
VII.
ACKNOWLEDGEMENTS
The author wishes to express his sincere thanks to Dr. Sam Legvold for
suggesting this problem and for his interest and assistance throughout the
course of this investigation.
Thanks are also extended to Mr. P. Palmer and Mr. B. J. Beaudry for
the preparation of the rare earth metals; to the Ames Laboratory graphics
department for the preparation of the figures; to Mr. W, Sylvester and
Mr. R. Brown for aid in constructing experimental apparatus and to
Mr. G. Erskine for aid in maintaining the equipment and processing data.
Special thanks are extended to Dr. J. Rhyne who constructed some of
the equipment used for this investigation and who was of great assistance in
writing the computer programs used to process the data.
The author gratefully acknowledges the assistance of Dr. L. Muhlestein
who permitted the use of his a.c. Hall effect apparatus for part of the
experiment.
It is a pleasure to acknowledge the help of many of the author's Ames
Laboratory colleagues:
Dr. L. R. Sill for growing some of the single
crystals; Mr. D. Richards, Mr. L. R. Edwards, Mr. D. Boys, Mr. W. Nell is,
Mr. C. M. Cornforth, Dr. S. Keeton and Dr. R. Williams for numerous helpful
discussions.
Thanks also to Dr. A. V. Gold, Dr. S. Liu and Dr. R. Fivaz for many
helpful discussions.
Certainly not least has been the assistance of the author's wife, Jean,
for her understanding and encouragement during the course of this investi­
gation.
56
VI 11.
A.
APPENDIX
Discussion of Errors - d.c. Method
Above magnetic saturation
may be determined from the equation
eu(H ) - eu(H,)
where H2 and
sample.
are both larger than
, the field required to saturate the
In terms of experimentally measured quantities. Equation 7.1 may
be wr i tten
R q = Vt/Hl
,
where V = VfHg) - V(Hj) and H = H2 -
(7.2)
.
The error in the measurement of
R is thus
o
ARo/Ro = AV/V + At/t - AH/H - A I/I
.
The estimated error in the thickness, t, is about 1%.
(7.3)
The estimated
error in the measurement of the magnetic field intensity, H, is less than
1% at fields above H^.
The estimated error in I is less than 0.1%.
The estimated error in V depends on the magnitude of V.
For values of
V larger than 0.2 microvolts the error is less than 5% since the potentio­
meter will measure to 0.01 microvolt.
The error is probably considerably
less than 5% because each measurement of e^ represents an average over
forward and reverse current and field directions.
R^ was determined by
drawing a straight line through 5 to 10 points on the high field portion of
the ey versus H curve, so random errors will tend to be averaged out.
values of V less than 0.02 microvolts, corresponding to R^ of 10
For
or less.
57
one has obviously reached the limits of measurement and the relative error
will be quite large.
Examination of the data shows that even the very small
values of R fit fairly smoothly onto the R versus T curve.
o
'
o
The scatter
of the points from a smooth curve is certainly less than 10% over almost all
of the temperature range and is better than this for most of the temperatures
measured.
One source of error that is very difficult to estimate for a d.c.
measurement is the Ettingshausen voltage.
This voltage is typically very
small compared to the Hall voltage in metals.
of the polycrystalline data of Kevane e^
gation.
Figure 21 shows a comparison
(I3) with the present investi­
The agreement is quite good over the range of temperatures where
comparison is possible.
Since the Ettingshausen voltage does not appear in
an a.c. measurement, as was made by Kevane, this indicates that the
Ettingshausen voltage does not represent a serious error.
With these considerations, the error in
is principally due to the
error in the measurement of V.
R^ is determined from the equation
R
where
is the mass density and
(7.4)
is the saturation magnetization per gram.
the intercept of the linear portion of the
versus H curve on the H axis,
may be determined from the equation
(7.5)
where
is the maximum applied field.
Rj is then given by the equation
120
_
(O
1
o
100 -
CO
0
1
0
O KEVANE
A PRESENT INVESTIGATION
3
g
o
80
1
60
00
X
o
ÇJ
0
OA
40
X
£
>
20
1
SL
-O
0
0
40
80
120
160
200
240
280
320
360
T(°C)
Figure 21.
Comparison of the d.c. measurements on gadolinium with the a.c. measurements of Kevane
et aj_. (13)
59
(7.6)
In most temperature ranges, e^(H^) is much larger than the second term in
7.6 so the largest part of the error in
of e^j.
will come from the measurement
Since the maximum voltage measured over almost all of the tempera­
ture range exceeded 0.5 microvolts, one can conservatively state that the
error in e^(H^) was less than 2%.
In temperature ranges where the second
term in 7.6 becomes comparable to e^, the error in R, is of the same order
n
I
as the error in R^, but this occurs only at temperatures below IOO°K.
error,in
is less than 0.1%.
Nigh (20) estimates the error in
The
to be
less than 1%,
The temperature was controlled to within 0.1°K.
Absolute accuracy of
the temperature data Is estimated to be +0.5°K, .
B.
Discussion of Errors - a.c. Method
The error in the measurement of
Equation 7.3.
by the a.c. method is also given by
The estimated error in the thickness, t, is about 1%.
estimated error in 1 is less than 0.1%.
The
The estimated error in H is 1%.
—Û
The limit of resolution of the system was 10
—8
voltages measured were of the order of 10
volts and the smallest
volts, so the error in V is
estimated to be less than 10% and is better than this for the c-axis
lutetium and yttrium samples, where larger voltages were measured.
L. Muhlestein reports (18) that his apparatus reproduced published values
of R^ for copper to about 6% and reproduced published values of R^ for
silver to about 7%.
In both of these metals
is of the same order of
60
magn i tude as for lutetium and yttrium.
C.
Sample Dimensions and Demagnetizing Factors
Sampl e dimensions for all of the samples used in this study are 1i sted
in Table 3.
Demagnetizing factors were calculated only for the gadolinium
samples.
Table 3»
Sample dimensions and demagnetizing factors
Sample
Length
(cm)
Width
(mm)
Gd 1
1.4
GD 11
Th i ckness
(mm)
Dernag,
Factor
2.2
0.397
10.9
1.2
2.5
0.554
10.6
GD 1 11
1.2
2.8
0.387
11.4
Gd IV
1.3
2.4
0.550
10.5
Lu 1
0.77
1.6
0,634
Lu 11
0.75
1.3
Oo 666
Y 1
0.75
1.7
0,572
Y 11
0.75
1.8
0.504
Do
Table 4.
Sample Impurities
Gadolinium impurities
Gd il and Gd III
Impuri ty
Gd 1
Sm
<200
<200
<200
Y
< 20
< 20
<100
Dy
<500
<500
<500
Fe
< 30
< 30
< 30
Ni
500
500
500
Si
< 50
< 50
< 50
Ta
2000
2000
2000
Mg
< 20
< 20
< 20
Cu
< 20
<
5
< 20
W
<500
<500
<500
Gd IV
61
Table 4.
(Continued)
Impurity
Gd 1
Ca
< 30
< 30
< 30
A1
< 30
< 30
< 30
Cr
< 10
< 10
< 10
02
l6l 1
1489
1588
H2
1
2
3
183
181
171
"2
Gd II and Gd 111
Gd IV
Analysis was made on scraps left over after cutting out the samples.
Gd II and Gd III were cut from adjacent areas of the same crystal, so
analysis was made on Gd III scraps.
mined by vacuum fusion analysis.
0^ and
impurities were deter­
Other impurities were determined by
semiquantitative spectrographic analysis and are accurate to 20%.
quoted are In parts per million.
Table 5.
Lutetium and yttrium impurities
Impuri ty
Lutet i um
Er
<100
<100
Yb
<
<100
Tm
<100
Sc
<
Y
<300
5
Yttr i um
5
Tb
<200
Ho
<100
Dy
<100
Gd
<400
Sm
<200
Nd
<200
Ta
<200
<400
Mg
< 10
< 20
Values
62
Table 5-
(Continued)
Lutetium
1mpur i ty
Yttrium
<100
Fe
< 30
Si
< 30
Cu
< 10
Ni
< 10
Al
< 30
Ca
< 10
< 30
Cr
< 10
< 40
< 70
< 30
Ti
O2
1024
991
N2
45
42
46
25
"2
Analysis was made on scraps left over after cutting out the samples.
Both lutetium samples were cut from the same crystal, so only one analysis'
was made.
This was also the case for the yttrium samples.
Analysis was
made by the same methods as for the gadol inium samples.
E.
Table 6 ,
T°K
Tabulation of Lutetium and Yttrium Data
Experimental data for the lutetium a-axis crystal (Lu 0
R
X 10^3
T°K
0
R X 10^3
0
T°K
R
X 10^3
0
311.4
5.11
278.6
3.68
109.9
1,06
302.4
5.06
259.0
2.90
80.8
1.00
303.3
4,82
239.5
2.52
79.8
0.70
303.6
4.97
232.3
2.28
50.0
1.21
303.6
4.74
205.4
1.94
32.8
2.32
303.6
5.00
188.8
2.02
14.4
3.32
302.9
4.72
163.2
1.63
8.8
4.04
293.6
4.28
136.0
1.14
63
Tab 1 e 7.
T°K
Exper i menta 1 data for the lutetium c-axis crystal (Lu 1 1 )
R
X lo'3
T°K
0
R
0
X 10'^
T°K
X 10^3
R
0
314,8
-25.72
244.4
-24.46
80.1
-16.64
308.1
-26.04
206, 2
-23.54
59.5
-14.24
301.2
-26.09
187.7
-21.87
43.0
-10.46
286.0
-25.80
151.9
-20.37
28.3
- 6.57
269.0
-25.28
144.7
-18.46
15.6
- 1.89
268.2
-24. 9 8
83.0
-16.27
8.4
- 1.56
Table 8 .
T°K
Experimental data for the yttrium b-axis crystal (Y l)
R
X 10^3
T°K
0
R
X 10^3
T°K
X lo'3
0
319.0
-4.20
245.4
-1. 0 0 .
36.4
-4.32
312.2
-3.66
203.4
-1. 0 6
21.7
-4.59
301.1
-2.76
171.4
-1.44
12.5
-4.50
301.1
-2.86
142.2
-1.88
7.9
- 4 .49
283.2
-1.93
80.6
-3.12
270.0
-1.46
49.5
-4.05
Table 9.
T°K
Exper imental data for the yttrium c-axis crystal (Y il)
R
X 10/3
T°K
R
X 10^3
T°K
0
0
«0
X 10^3
323.6
-16.90
155.8
-17.84
51.6
-14.90
296.2
-17.08
123.0
-17.00
44.8
-13.68
268.2
-17.78
80.7
-17.29
24.9
- 9.17
244.5
-17.92
80.5
-17.12
16.0
- 7.22
200.0
-17.63
80.3
-16.54
8.2
- 6.84
188.1
-17.74
64.4
-16.14
Resistivity ratios were as follows:
15.7; Y II, 14.4.
Lu i , 25.7; Lu I 1, 22.4; Y I,
64
Fo
Tabulation of Experimental Data for Gadolinium
The experimental data for the gadolinium single crystals are tabulated
in Figures 22 - 31.
These tables were printed out by the computer, so the
notation requires some explanation.
called e^j the Hall resistivity.
The quantity, E, is what was formerly
It was not possible to print a lower
case e or a subscript with the printer for the computer.
^O
The notation -0.4689E-08 means -0,4689 x 10 , etc.
This is again a
case of having to accept what the computer was capable of printing.
Resistivity ratios were as follows:
Gd I, 46; Gd 11, 42; Gd IV, 34.
The resistivity ratio was not measured for Gd III.
3
4
5
6
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T=329. 50EG.K
H ( H O E ) E lOHM C M . )
0.0
0.0
1.0 -0.3173E-07
2.0 -0.6427E-07
3.0 -0.9681E-07
4.0 -0.1277E-06
5.0 -Q.1595E-06
6.0 -0.1904E-06
8.0 -0.2530E-06
10.0 -0.3148E-06
13.0 -0.40UE-06
16.0 -0.4897E-06
20. 0 -0.5996E-06
25.0 -0.7273E-06
30.7 -0.8729E-06
T=342. IDEG.K
H ( K O E I E lOHM C M . )
0.0
0.0
1.0 -0.2440E-07
Z.O —0.5126E—07
3.0 -0.7159E-07
4.0 -0.9600E-07
5.0 -0.1204E-06
6.0 -0.144CE-06
8 . 0 - 0 .1879E-06.
10.0 -0.2367E-Ô6
13.0 -0.301BE-06
16.0 -0.3750E-06
20.0 —0.4588E—06
25.0 -0.5687E—06
30.7 —0.6842E—06
Figure 24.
T=341. 5DEG.K
H ( K O E ) E (OHM C H .
0.0
0.0
1.0 -0.5047F-07
2.0 -0.8811E-07
3.0 -0.1250E-06
4.0 -0.160?E-06
5.C -0.1987E-06
6. 0 -0.2347E-06
8.0 -0.3108E-06
10.0 -0.3845E-06
13.0 -0.4942E-06
1 6 . 0 —0.6024E—06
20.C -0.7474E-06
21.0 -0.7850E-06
22.0 -0.8203E-06
23.0 -0.8539E-06
24.0 -0.8892E-06
25.0 -0.9244E-06
26.0 -0.9589E-06
27.0 -0.9941E-06
28.0 -0.1029E-05
29.0 -0.1C63E-05
30.0 -0.1Q97E-05
30.7 -0.1122E-05
Experimental data for the gadolinium b-axis sample (Gd l)
T- 4.TOEG.K
H C K O E ) E I OHM C M . )
0 .0
0 .0
I .0 - 0 . 1 2 0 3 F - 0 7
2 . 0 - 0 .1 0 9 4 E - 0 7
3 . 0 - 0 .1 3 6 ? F - 0 7
4 . 0 - 0 .2 4 C 7 C - 0 7
5 . 0 - 0 .2844E- 07
6 . 0 - 0 . 3 2 a 2 F -• 0 7
8 . 0 - 0 . 4 1 5 7 E -• 0 7
10.0 - 0 .4923E- 07
1 3 . 0 - 0 .6 3 4 5 E - 0 7
1 6 . 0 - 0 .7 8 7 7 F - 0 7
2 0 . 0 - 0 .1 0 1 7 E - 0 6
21. 0 - 0 .1C83E- 06
2 2 . 0 - 0 .1 1 6 P E - 0 6
2 3 . 0 - 0 . 1 2 C 3 E -• 0 6
2 4 . 0 - 0 .1 2 4 7 E - 0 6
2 5 .0 - 0 .1291E- 0 6
2 6 . 0 - 0 .1 3 4 6 E - 0 6
2 7 . 0 - 0 .1 3 7 8 E - 0 6
28.0 - 0 . 1 4 2 2 E - 0 6
29.0 - 0 .1466E- 06
3 0 . 0 - 0 .1 4 8 8 E - 0 6
3 0 . 7 - 0 . 1 5 2 1 E -• 0 6
T= 53.4DEG.K
H ( K O E I E (OHM C M . I
0 .0
0 .0
I .0 - 0 . 1 2 0 1 E - 0 7
2 . 0 - 0 .1 6 4 1 F - 0 7
3 . 0 - 0 .2516F-07
4 .0 - 0 .3173E-07
5 .0 - 0 .3829E-07
6 .0 - 0 .4485E- 07
8 .0 - 0 . 5689E- 07
1 0 . 0 - 0 .6 8 9 2 E - 0 7
1 3 . 0 - 0 .8 S 6 1 E - 0 7
16. 0 - 0 .9955E- 07
70. 0 - 0 .1247E- 06
2 1 . 0 - 0 .1 3 C 2 E - 0 6
22.0 - 0 .1357E- 06
2 3 . 0 - 0 . 1 3 8 9 E -• 0 6
2 4 . 0 - 0 .1 4 5 5 E - 0 6
2 5 . 0 - 0 .1 4 7 7 E - 0 6
26. 0 - 0 .1521E- 06
2 7 . 0 - 0 . 1 5 5 3 E -• 0 6
2 8 . 0 - 0 . 1 5 9 7 E -• 0 6
2 9 . 0 - 0 . 1 6 3 0 E -• 0 6
3 0 . 0 - 0 .1663E- 0 6
3 0 . 7 - 0 .1 6 9 6 E - 0 6
T=122.70EG.K
H ( K O E ) E (OHM C M . )
0.0
0.0
1.0 -0.2188E -07
2.0 -C.4704F -07
3.0 -0.6673E -07
4.0 -0.7877E -07
5.0 -0.91B9F -07
6.0 -0.1050E -06
8.0 -0.1346F-06
10.0 -0.1707F-06
13.0 -0.2243E-06
16.0 -0.26806-06
20.C -0.3085E -06
21 . 0 -0.31676 -06
22.0 -0.3249F-06
23.0 -0.3304E -06
24.0 -0.33696-06
25.0 -0.34026-06
26.0 -0.34466-06
27.0 -0.35126-06
28.0 -0.35556-06
29.0 -0.35996-06
30.0 -0.36436-06
30.7 -0.36766-06
T=16 6.3DEG.K
H ( K O E ) E (OHM C M . I
0.0
0.0
1. 0 -0.404BE-07
2.0 — 0.66736— 07
3 .0 -0.9955E- 07
4 .0 - 0 . 1 1 6 0 E - 0 6
5 .0 -0.1324E-06
6. 0 -0.1543E-06
8. 0 -0.2046F-06
10. 0 -0.2505E- 06
13.0 -0.311BE- 06
16.0 -0.36B7E- 06
20.0 -0.4157E- 06
21.0 -0.42456- 06
22.0 -0.42886-06
2 3 . 0 - 0 . 4 3 3 2 E -• 0 6
24.0 -0.4387E- 06
2 5 . 0 - 0 . 4 4 3 1 E -• 0 6
2 6 . 0 - 0 . 4 4 7 4 E -• 0 6
2 7 . 0 - 0 . 4 4 9 6 E -• 0 6
2 B . 0 - 0 . 4 5 2 9 E -• 0 6
29. 0 -0.4573F- 06
3 0 . 0 - 0 . 4 5 9 5 E -• 0 6
3 0 . 7 - 0 . 4 6 1 7 E -• 0 6
= 202.4nEG .K
1 ( K O E l E (OHM C M . 1
0.0
0.0
1 . 0 -C. B862E-07
2 . 0 - 0 .1 3 8 9 F - 06
3 . 0 - 0 . 1360E-06
4 . 0 - 0 .2243E-06
5 . 0 - 0 .2735E-06
6.C - 0 . 3205E- 06
8.0 - 0 .4113E- 06
10.0 - 0 .4973E- 06
13.C - 0 . 5929E- 06
1 6 . 0 - 0 . 6 5 7 5 E -• 0 6
2 0 . 0 - 0 .7 0 4 5 6 - 0 6
21. 0 - 0 .70896- 06
22.0 - 0 . 7144E- 06
2 3 . 0 - 0 . 7 1 7 7 6 -• 0 6
24.0 - 0 . 71986- 06
2 5 . 0 - 0 . 7 2 2 0 E -• 0 6
2 6 . 0 - 0 . 7 2 3 1 E -• 0 6
2 7 . 0 - 0 . 7 2 4 2 E -• 0 6
2 8 . 0 - 0 . 7 2 6 4 E -• 0 6
2 9 . 0 - 0 . 7 2 6 4 E -• 0 6
3 0 . 0 - 0 . 7 2 7 5 E -• 0 6
30.7 - 0 . 7275E- 06
T= 19.90EG.K
H ( K O E ) E (OHM C M . )
T= 78.1DEG.K
H ( K O E I E (OHM C M . )
0 .0
0 .0
1 . 0 - 0 .1 3 1 3 E - 0 7
2 . 0 - 0 .2 5 1 6 E - 0 7
3 . 0 - c .35C1E-07
4 .0 - 0 .4485E- 07
5 .0 - 0 .5361E- 07
6 . 0 - 0 .6 3 4 5 E - 0 7
B . 0 - 0 .8 2 0 5 E - 0 7
1 0 . 0 - 0 .1 0 0 6 E - 0 6
1 3 . C - 0 .1 2 6 9 E - 0 6
16. 0 - 0 .1521E- 06
2 0 . 0 - 0 .1 8 8 2 E - 0 6
2 1 . 0 - 0 . 1 9 4 7 E - OA
2 2 .0 - 0 .20C2E- 0 6
23. 0 - 0 .2068E- 06
2 4 . 0 - 0 .2122F- 06
2 5 . 0 - 0 .2166F- 0 6
26.C - 0 .2221E- 06
2 7 .0 - 0 . 2 2 6 5 E - 0 6
2 8 . 0 - 0 . 2 3 1 9 F -• 0 6
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T=124. 8DEG.K
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4.0 -0.57986-07
5.0 -0.7549F-07
6.0 -0.89716-07
8.0 -0.12256-06
10.0 -0.14996-06
13.0 -0.19256-06
16.0 -0.22976-06
20,0 -0.27246-06
21.0 -0.28126-06
22.0 -0.28886-06
23.0 -0.29546-06
24.0 -0.30196-06
25.0 -0.3096E-06
26.0 -0.31626-06
27.0 -0.32166-06
28.0 -0.3271E-06
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Figure 25.
Experimental data for a gado1i n i
0.0
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