Tricritical point in KH2PO4
by Charles Robert Bacon
A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE
in Physics
Montana State University
© Copyright by Charles Robert Bacon (1977)
Abstract:
The presence of a tricritical point is shown to exist near 2.4 kbar by measuring the polarization as a
function of electric field, temperature, and pressure. By applying a dc field across the KH2PO4 crystal
and varying the temperature data points were obtained at preselected polarizations. The behavior of the
electric field, polarization, and the temperature are excellently described by the Landau equation of
state, E = A0(T - T0)P + BP^3 + CP^5. In the T-E plane this equation gives straight lines for constant
polarization. These lines of constant polarization are called isopols. The coefficients in the Landau
equation of state are temperature and/or pressure dependent.
The nature of the B coefficient in particular was an indication of the order of the transition. By
assuming a linear pressure dependence of B, and fitting a line to a graph of B vs. p, the pressure, a
value for the tricritical pressure, pt, was found to be near 2.4 kbar. The isopol method of analysis also
yeilded results that enabled a determination of the δ exponent where δ is defined by the relationship of
the electric field to polarization. The measured values of the δ exponent agree within experimental
accuracy with the predicted mean-field value of 5 at the tricritical point.
TRICRITICAL' POINT IN KH PO
2 4
by
CHARLES ROBERT BACON
'/
A thesis submitted in partial fulfillment
of the requirements for the degree
of
MASTER OF SCIENCE
in
Physics
Approved:
IradrCSte Committee
artment .
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
December, 1977
JI
Jl
STATEMENT OF PERMISSION TO COPY
in presenting this thesis In partial fulfillment of the require­
ments for an advanced degree at Montana State University, I agree that
the Library shall make it freely available for inspection.
I further
agree that permission for extensive copying of this thesis for scholarly
purposes may be granted by my major professor, or, in his absence, by
the Director of Libraries.
It is understood that any copying or publica­
tion of this thesis for financial gain shall not be allowed without my
written permission.
Signature
Date
TO RACHEL
iv
ACKNOWLEDGMENTS
The author wishes to express profound thanks to my thesis advisore
V. Hugo Schmidts for his help and guidance.
Without his criticism and
consultation this work would not have been completed.
I would like to
thank him personally for setting an example as a dedicated scientist
that others could emulate.
Thanks must also go to Alan G. Baker for his
tireless assistance during the bulk of this work, and for his personal
encouragement when it was most needed.
A special thanks must go to
Arthur B, Western for. constructing the apparatus and preparing a foun­
dation for this research to be based upon.
Finally an overall thanks to
the Physics Department at Montana State University is extended for al­
lowing the author the opportunity to learn from its excellent staff.
Special thanks must be given to my wife, Rachel8 for providing
much of the motivation the author has needed.
\
TABLE OF CONTENTS
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Introduction
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Description of KH^PO^* © • © * © © © © * ©
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Samples and Apparatus
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The Landau Theory of Phase
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Critical and Tricritical Exponents
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Conclusions and Recommendations for Further Study
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11
Data Analysis
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vi '
LIST OF TABLES
Table
I.
II.
III.
Tabulated values of the Landau parameters................. ; 24
Predicted values of the tricritical pressure . . . . . . . .
25
Measured values of the 6 exponent and e r r o r s ...............28
vii
LIST OF FIGURES
Figure
1.
Phase diagram of the tricritical point of KHgPO^........
2.
Structure of KHgPO^
3.
Pressure s y s t e m .................................. ■..........8
4.
DC polarization measurement circuit . .....................
5.
Experimental setup............................................ 12
6.
First order transition........................................ 13
7.
Second order t r a n s i t i o n .............................. ..
8.
Isopols predicted by Landau equation of state ..............
9.
Temperature/capacitor sensor linearity....................... 18
10,
T q variation.................................................. 21
11.
Tricritical pressure prediction for linear B vs. p relation­
6
..................... ..
ship......................................
■12.
. ■4
Coordinate transformation ....................
10
. . 13
16
23
. . . . . . .33
viii
ABSTRACT
The presence of a tricritical point is shown to exist near 2.4
k b ar by measuring the polarization•as a function of electric field,
temperature, and pressure.
By applying a dc field across the K ^ P O ,
crystal and varying the temperature data points were obtained at pre­
selected polarizations.
The behavior of the electric field, polariza­
tion, and the temperature are excellently described by the Landau eq­
uation of state, E = A (T - T )P + BR3 + CP^.
in the T-E plane this
equation gives straight lines°for constant polarization.
These lines
of constant polarization are called isopols.
The coefficients in the
Landau equation of state are temperature and/or pressure dependent.
The nature of the B coefficient in particular was an indication of the
order of the transition.
By assuming a linear pressure dependence of
B, and fitting a line to a graph of B vs. p, the pressure, a value
for the tricritical pressure, pt , was found to be near 2.4 kbar. The
isppol'method of analysis also yeilded results that enabled a deter- ■
mination of the 6 exponent where 5 is defined by the relationship of
the electric field to polarization.
The measured values of the 6 exp­
onent agree within experimental accuracy with the predicted meanfield value of 5 at the tricritical point.
Introduction
A tr!critical point has been defined as a point where three lines
of critical, points meet.
For ferroelectries which possess tricritical
points, all three parameters of the three-dimensional parameter space of
pressure-temperature-electric field, in which the tricritical point
lies, are experimentally accessible.
This thesis reports on a contin­
uation of studies of critical and tricritical behavior of KH PO (here2 4
after referred to as KDP) by means of static dielectric measurements at
high pressure.
Pyroelectric crystals are defined as crystals whose natural primi­
tive cells have a nonvanishing dipole moment in equilibrium.
The stab­
lest structure of certain crystals is nonpyroelectric above a certain
temperature T
(the Curie temperature) and pyroelectric below it. Not
c
all pyroelectric or ferroelectric crystals exhibit phase transitions.
There can also be a range of temperatures for the pyroelectric phase,
above and below which the crystal is unpolarized.
Ferroelectries are
pyroelectric crystals whose polarization can be reversed by applying a
strong electric field.
In a ferroelectric the distortion of the primi­
tive cell from the unpolarized configuration is small and it is there­
fore possible to reverse the polarization by applying an electric field.
The structure of KDP is presented in the next section.
Potassium dihydrogen phosphate draws interest because the relevant
variables are experimentally accessible, and because it is a simple H-
Ii
I
2
bonded order-disorder ferroelectric.
The large amount of literature
published on KDP also makes it a g ood.research subject.
Many systems
possess experimentally inaccessible ordering fields9 precluding deter­
mination of the behavior in a three-dimensional parameter space in the
vicinity of the tricritical point,
magnetic cubic crystal.
Consider8 for example, an antiferro­
In such a system nearest neighbor cell spins
tend to point in opposite directions, and so spin direction becomes or­
dered and alternates from cell to cell at low temperature.
This can be
viewed as two interpenetrating sublattices8 one with average magnetiza­
tion m and the other with -m.
The average magnetization, usually refer­
red to as the staggered magnetization, vanishes.above the critical temp­
erature Tfi (called the Neel temperature) and so is taken to be the order
parameter for the transitions.
The dependence of m upon the temperature
T defines the critical exponent B (see later section) and is a measur­
able quantity.
However8 for antiferromagnetic systems the dependence
of m on the staggered magnetic field h defines the critical exponent S 8
which is unmeasurable because the direction of h alternates from cell to
cell and therefore h cannot in general be applied externally (in certain
anfciferromagnets 8 however, an applied field H in certain directions also
generates a staggered field h).
This also eliminates the
y
exponent
(determined from the relationship between the zero field susceptibility
and the temperature) from being directly measurable.
This thesis reports on the determination of the Landau parameters
3
(described in a later section) from, the measurements of pressure-temperature-electric field in the vicinity of the phase transition*.
DetI
ermination of the 6 exponent is also reported on.
Previous work
the Y exponent to equal the mean field value of I.
showed
The determination of
these results due to the accessibility of all relevant fields is what
makes KDP experimentally valuable.
For KDP we use the phenomenological theory of Landau (described
in a later section) as a model of the crystal’s behavior.
The equation
of state in the Lazidau theory relates the electric field E to the temp­
erature T and the net polarization P of the crystal.
in this equation of state are pressure-dependent.
The coefficients
In this work we det­
ermined the relations between temperature, electric field, and net pol­
arization near the ferroelectric transition at pressures of 0, 2, and
2,4 kbar.
The data is analyzed by examining the T and E dependence a-
long lines of constant polarization, i»e, isopols,
The origin of the tricritical point is evident from the phase dia­
gram shown in Figure I.
Data points taken along the critical lines (CP)
converge on the tricritical point.
4
nonordering
field
critical point (CP)
tricritical point
>
(TCP)
nonordering
fiel d
i f oordering field
FIGURE I,
Phase diagram of the tricritical point of KI^PO^
Description of KH PO
2 4
Potassium djhydrogen phosphate is a hydrogen-bonded ferroelectric,
that is, it develops a spontaneous electric dipole moment below its tran­
sition temperatureo
As Figure 2 shows, the PO
groups along with pot4
asslum ions make up a structure in which K and P ions alternate in the
direction of the crystallographic c axis.
to four other PO^ groups by hydrogen bonds.
Every PO^ group is connected
The hydrogen bonds are per­
pendicular to the c axis, which is the direction of the polarization.
Above the transition temperature the protons are randomly located
in off-center positions in the four hydrogen bonds associated with a PO^
tetrahedron, subject to the "ice rule" that two protons are near each
PO^6
However, below the transition temperature the two near hydrogens
are always found at the top (or at the bottom) of the PO^ tetrahedron.
The dipole moment is produced by the movement of the K and P ions along
the c axis induced by the ordering of the hydrogen bonds.
6
H
PO 4
/O
K
/\
FIGURE 2.
Structure of KH2PO 4°
2
C
Samples and Apparatus
The KDP sample used was obtained from Cleveland Crystals, Inc.
The crystal dimensions were I x I x 0.2 cm with the large faces covered
with chrome-gold-plated electrodes by the supplier.
lowed to hang freely in the pressure vessel.
The crystal was al­
The crystal was connected
to a coaxial three-conductor cable by two tiny wires connected to the
chrome-gold-plated faces of the crystal.
The experimental apparatus is excellently described in the thesis
I
of A. Western and will only briefly be described here.
TIie cryostat
assembly consists of three separate cans, the innermost being the pres­
sure vessel, surrounded by two other cans.
The outermost can is vacuum
tight and connected directly to the vacuum system.
The electrical leads
are fed to the pressure vessel through high pressure tubing.
This en­
tire assembly is immersed in a liquid nitrogen bath that is filled auto­
matically from an external source.
The can surrounding the pressure
vessel is temperature regulated to approximately 0.1 °K, and kept about
one degree below the pressure vessel temperature.
The pressure vessel
temperature is regulated by a temperature-sensitive capacitor which con­
trols a heater wound around the top of the pressure vessel.
acitor’s temperature was kept constant, to within 4 2 mK.
This cap­
A second cap­
acitor is placed at the bottom of the pressure vessel where the tempera­
ture is presumed to approximately that of the crystal.
A diagram of the pressure generating system is shown in Figure 3»
manganin cell
filter
A C Z
rupture
disk
V-I
H
H ^
m
h
2N2K^)
He
source
.V
trap
V-2
9
gas
intensifier
-O
V-7
I iquid
inten­
sifier
V-9
FIGURE 3.
Pressure system
V-8
pump
9
A liquid-gas system was used to produce pressures up to 45,000 psi,
Pressurising by means of the remote head produced pressures of nearly
20,000 psi, and then switching to the gas lntensifler enabled us to re­
ach pressures necessary to determine the tricritical point of KDP.
The setup used in collecting data at constant polarization is shown
in Figure
4.
An electric field was supplied to the crystal by means
of a battery and voltage divider, and the polarization charge was then
stored on ah 8 yf polystyrene Capacitor6
The voltage on this capacitor
was measured by a vibrating reed electrometer, and the recorder output
was kept constant during measurements at constant polarization while
changing the electric field and temperature6
10
ON-OFF
200 V
100 K
A A A / V V V
SAMPLE
ELEC
POLYSTYRENE
FIGURE 4.
DC polarization measurement circuit.
The Landau Theory of Phase Transitions
In this Section the phenomenological theory of Landau will be dis­
cussed.
Referring to Figure 5 as a crude model of the experimental set­
up being dealt with, we can write the first law of Thermodynamics in
differential form,
(1)
dU ■= dQ - dW
or, in terms of quantities which are represented in the Figure,
(2)
dU - TdS + EdP .
The Legendre transform of this expression yields the Helmholtz free energy, F,
(3)
F = U - TS .
The differential form of the Helmholtz free energy can be written as
(4)
dF = EdP - SdT
so we see that the free energy is a function of temperature and polari­
zation, F(T,P).
By definition the ordering field is the first partial
derivative of the free energy, E =» 8F/3P, and the response function is
- 1 2
2
the second partial derivative of the free energy, x
= 9 F/3P . The
—I
behavior of the inverse susceptibility, % $ determines whether the tran­
sition is first or second order, as shown in Figures 6 and 7, respec­
tively.
For dielectrics the susceptibility must be greater than or e—1
qual to zero, and so when x
m 0 the following conditions must be ful­
filled:
(5)
(6)
3
3
3 F/3P - 0,
4
4
3 F/3P > 0 .
system
FIGURE 5. Experimental setup.
13
FIGURE 6,, First order transition
FIGURE 7.
Second order transition.
14
•
The foundation of the Landau theory of phase transitions Is the as­
sumption that the free energy* F(T, P)8 can be expanded-In an infinite
series, such that
(7)
F(T, P) ■ LA (T)Pn .
n n
Due to the symmetry of KDP the free energy is an even function of pol­
arization, F(T, P) «>» F(T, -=P), so
^ 0 for n odd.
The Landau expan­
sion for the free energy thus takes the form
(8)
F(T, P) - A(T)P2/2 4-B(T)P4/4 + C(T)P6/6 + . . .
and the ordering field is given by,
(9)
.
E = AP -}• BP3 -$• CP5 4- . . .
In order to predict Curie-Weiss behavior in the paraeleetric region it
is necessary that A = A
(T-T
O
ture.
), where T
0
is the Curie-Weiss temperao
The Landau equation of state becomes
(10)
E - Aq (T - Tq )P 4- BP
3
4- CF
5
,
where B and C may be functions of temperature and/or pressure.
of B is an indication of the order, of the transition.
The sign
A negative B in­
dicates a first order transition, whereas a positive B indicates a sec­
ond order transition.
A zero value of B occurs at the trlcritieal pres­
sure.
A graph of the equation of state in the T-E plane for constant pol­
arization yields straight-line plots called isopols, as shown in Figure
8
I
Isopols
As was shown earlier the Landau equation of state enables us to
graph straight-line plots in the T-E plane.
These straight lines at
constant polarization are referred to as isopols.
An isopol plot is
shown in Figure 8 as predicted by the Landau equation of state.
small P the terms in P
3
and P
5
For
become negligibly small so that for E raO
the T intercept tends to Tq .
The sign of B affects the nature of the isopol plots.
For negative
B and increasing P the isopol intercepts rise above Tq , while increasing
P further drives the'E m O intercepts below T q due to the onset of dom­
ination by the CP"* term in the equation of state.
E = O
For positive B the
intercepts simply fall with increasing P.
The order of the transition is determined by the sign of B.
A neg­
ative B indicates a first order transition, and a positive B indicates a
second order transition.
The tricritical point occurs at B = 0.
Li
I
/
V
Data Analysis
The experimental results subjected to analysis were the measured
values of the temperature T, the electric field E 9 the polarization P 9
and the pressure p.
.
A standard least squares analysis was done on the lsopols to deter­
mine the slope* intercept, and mean values of T and E.
The standard
deviations of these quantities were also found.
Two methods were used to determine the Landau parameters and the
Curie-Weiss temperature.
In both methods A q is found as the weighted
average of the Aq^ for the various lsopols.
associated
(11)
To each isopol there is an
From the Landau equation of state
E = A (T - T )P + BP3 + CP5
o
o
we get
(12)
3T/3E - !/Aq 1P1 .
Using the chain rule we can write
(13)
3T/3E = (3T/3B')(3B'/3E')(3E'/3E)
where B f is the capacitor sensor reading of the temperature.
This read­
ing was defined to be in ltB units” and could be converted into degrees
Kelvin from an experimentally determined linear relationship between the
two.
.Figure 9
shows this linearity between the capacitor sensor read­
ing and the actual temperature in degrees Kelvin.
The slope of this
line is 3T/3B’ = -4,25 °K/"B unit” in this temperature range.
The voltage applied across the 0.2 cm crystal is E 1 in Equation
(13).
To put this quantity in electrostatic units (esu), as the polar-
18
B 11 u n i t s
FIGURE 9
Temperature/capacitor sensor linearity
19
Izatlon Is, we use I esu volt « 300 volts, and so
(14)
8 E V 9 E - (I volt)/((I esu/300)/C0o2 cm))
The final quantity in Equation (13) is 8B*/8E%
.
The parameters B e and
E* are precisely the measured values that are plotted to graph the isopels.
For each constant polarization P- , the slope of the ith isopol is
SB'/SE’o
Combining these results we rewrite Equation (13) as
(15)
Aq1 ■ (P1)-1 OTZ a E ) * 1
» (P4)-1C1B Unit11M
. 25 °K) x
(esu/(60 V/cm)) O B ' / 3 E ,)“1
.
The error is
(16)
AA . ■ - ( A B V E t)A /(SBtZaEt)
oi
o
“ (AS5/ E t)/255(SB’/ SEt)2P1
where ABtZEt is the error in the slope of the ith isopol.
These results
are averaged to give A q and AAq .
The first method used to obtain the Landau parameters is one described in detail in a doctoral thesis by A. Western.
X
Briefly, his
method involves finding the Curie-Weiss temperature by plotting T(E « 0)
vs. P
2
for low polarization.
A straight-line fit to this graph yields
an intercept which is identified as Tq .
The remaining Landau parameters
B and C are obtained by doing a straight-line analysis of a graph of
-A-(T(E » 0 ) - T )/P
o
o
slope is equal to C«
2
9
VSe P e
The intercept at P
2
® 0 is B 9 and the
Results from this method were used as a check on
the second method, yet to be described.
20
The first method has certain drawbacks«
finding A
o
While the procedure for
is very accurate, it is not known beforehand whether the T
o
determination is also as accurate.
The size of the contribution of the
CF"* term is uncertain so the discarding of the higher P isopols could be
crucial.
parameter.
This leads to another drawback in the determination of the B
The value of B is strongly sensitive to the value of Tq
used, as Figure 10 exhibits.
duces a large change in B.
In this Figure a small variation of Tq in­
The C parameter, given by the slope of the
line, is less sensitive to the value of Tq .
The second method, describ­
ed below, was developed to eliminate these problems.
In this approach we solve for A q in the same manner, and then solve
for T , B, and C simultaneously.
o
We use as input the average tempera-
tures of the isopols, T^, instead of the T(E = 0) values which have lar­
ger standard deviations.
Writing the Landau equation of state in the
following form,
(17)
T1 = To +■ (E1ZA0P1) -. (PjpZA0 ) - (P1CZAo )
the best fit values of the parameters T0 , B, and C maxmize the probab­
ility
(18)
W(To , B, C) = c exp(-I(T. - T q - E1 ZAq P 1 + BP1 ZAq +
GP1 Za q )2 Z2 (AT1)2
that the parameters have their true values.
Then Tq8 B, and G are found
by setting the partial derivatives of W(T0 , B, C) with respect to TQ , B,
and C equal to zero.
This gives three equations in three unknowns,
M T
FIGURE 1 0 . variation.
- TJ / P
2.0
01
K
1.5
ND
1.0
H
T + .005
+ .01 u K
0.5
0.0
P2 9, IO7 esu
.4
.9
1.6
2.5
3.6'
I
.
.
22
.
UCfi
(19)
- T o - E1ZAoP1 + P^BZA0 + P^CZAo )Z(AT1)2) - O
,
(20)
U Cf,
(21)
K((T1 - T o - E1ZAoP1 + P 2BZA0 + P^CZAq ) P ^ A q (AT1)2) - O
- T 0
E /A P
X o i
+
p ^ b Za
i
°
+
p ^ c Za
x
o
)p ^Za
x o
(a t )2) i
o
,
,
which are solves simultaneously for the best fit values of Tq9 B 9 and C.
The analysis of the experimental data yields the results listed in
Table I.
These results when compared with the results obtained by West5
ern et al„ show the characteristic change of sign of the B parameter
indicating a second order phase transition at higher pressure.
A graph of the B parameter vs. the pressure* p„ is shown in Figure
11' A least squares fit to the data indicates values of the tricritical
pressure.
These values are listed in.Table II9 (a)9 (b)9 and (c).
The
variation of the tricritical pressure is from 2.16 kbar to 2.44 lcbar
with a weighted average of 2.35 kbar.
The results are based on the as­
sumption that the pressure dependence of B is linear.
2
I
11
3
i
, esu
P , kbar
-I
4-0+
B x 10
k
FIGURE 11.
Tricrltical pressure prediction for linear B vs p relationship
N)
U>
TABLE I.
p* kbar
Tabulated Values of the Landau Parameters
T f °K
A , 10 3 esu
B, IO"11 esu C, 10 19 esu T
- T , °K E
, V/cm
q
________o___________________ ._________________________cr____ o______ cr_____
0.00
122.12 + 0.01
4.273 + 0.029
-1.83 + 0.03
5.66 + 0.13
0.062
„ 132.8
2.00
113.12 + 0.01
3.751 + 0.041
-0.22 + 0.17
4.24 + 0.06
o.ooi-
2.8
2.40
111.22 + 0.01
3.975 + 0.023
-0.00 + 0.08
4.08 + 0.33
NA
NA
111.23 + 0.01
3.804 + 0.039
0.22 + 0.35
4.83 + 1,90
NA
NA ■
111.21 + 0.01
3.255 + 0.077
0.12 + 0.23
3.65 + 0.62
NA
NA
A
Stable to within + 0.0005 kbar^Absolute calibration not tied to National Bureau of Standards
ND
4N
NA-Ndt applicable,'transition is second-order
25
TABLE TI.
Predicted Values of the
Tricritical Pressure
Run No.
Tricritical Pressure
1
2.24
2
2.16
3
2.44
Critical and Tricritical.Exponents
The Landau expansion of the free energy is a mean field treatment
of the crystal behavior in the paraelectric region.
Mean field theory
is accurate for ferroelectrics because of the long-range nature of the
electric dipole interaction.
To find the critical exponents one makes a Taylor series expansion
of the free energy, F, about the critical polarization P ^ ,
(22)
F(T, P) = F
-J*(A-A
cr
cr
)P (P - P ) -S- Bp 3 (P - P ) + CP^ x
cr
cr
cr
cr
cr
(P - Pgr) + (A - Acr).(P - Pcr)2/2 + S B P ^ C P - P ^ ) 2/2
-J- 5 C P ^ ( P - Pcr)2/2 + . . .
(23)
F(T, P) - Fcr 4- ((A - A cr)Pcr + Ecr) (P -
+ (A - A ^ ) (P - P ^ )
x 1/2 + 1/4(P + S C P 2 /2)(P - P )4 + CP
(P - P )5 +
cr
cr
cr
cr
C(P - P c r )6 /6
.
The Y exponent is defined by
(24)
((T - Tcr)/Tcrf Y .
The mean field value of y
I is obtained from Equation (23).
The exponent of prime interest in this work was the tricritical ex­
ponent 5.
The 6 exponent is defined by
6
(25)
(E - Ecr) = (P - Pcr)
.
AT the tricritical point and for pressures greater than the tricritical
pressure p , Ecr « Pcr = 0 so Equation (25) becomes
(26)
E = P6
.
Since E = 3F/3P, Equation (23) gives
3
(27)
E “ (B + 5CP
/2)(P - P )
CIT
ClT
4
+ 5CP (P - P )
•*
cr
cr
+ C(P - P
Ct
5
)
,
I
27
so that
(28)
'E - Ecr - .(B + 5CPcr/2) (P - P c r ) 3 + . . .
At the tricritical point, B = O ,
(29)
.
giving
E = CP 5
so the mean field tricritical exponent 6 = 5 «
From the data analysis
(see Appendix A ) * values for the 6 exponent are determined from the high
polarization data.
Table III shows the values predicted from the data
for the three separate runs done with the sample.
Within experimental
accuracy a value of 6 = 5 is indicated by each data run.
The exponent $ is defined by
(30)
(P - Pcr) = (T - T c r ) 6
.
The Landau equation of state can be written so that
(31)
2
4
(T - Tcr) « B(P - Per)Z + C(P - Pcrr
.
This gives B = 1/2 for the critical lines and B = 1/4 at the tricritical
point.
Recent results by a group in France^ working with y-ray diffrac­
tion to study the lattice parameter (which is related to the spontaneous
polarization) below Tfi showed the tricritical exponent B
and the critical exponent B G 1/2 at 3.5 kbar.
a
1/4 at 2 kbar
They concluded that the
tricritical pressure p t = 2 kbar, in general agreement with our results.
28
TABLE III.
Measured Values of the 6
Exponent and Errors
Pressure, kbar
. 6
AS
2.00
3.13
0.04
2.40
5.74'
1.06
2.40
4.91
0.40
Summary
Using the method of "isopols" to analyze the static dielectric
data obtained for KH^PO^ at 0, 2, and 2.4 kbar we find values for the
tricritical pressure, p^, and the 6 exponent.. The method of "isopols"
was developed from consideration,
of the Landau free energy expansion.
Analysis of the data along these isopols yielded values of the Landau
parameters.
Assuming a dominant linear relationship between the pres­
sure, p, and the Landau B coefficient we calculate a tricritical pres­
sure near 2.4 kbar.
By expanding the free energy in a Taylor series expansion about
the critical polarization, P ^ , we arrive at an expression which de­
fines the 6 exponent.
The Landau theory is a mean-field treatment and
predicts a value of 5 for the 6 exponent.
It has been shown that at
2.4 kbar we have calculated a 6 exponent which is consistent with the
predicted mean-field value.
Conclusions and Recommendations for Further Study
Our results have shown us that the data analysis based on the
Landau theory of phase transitions does indeed give the anticipated
values for the Landau coefficients and the critical exponents.
■
This
I
study and previous work
confirms that the transition at 2 kbar is
first order and that it is second order at 3 kbar.
sults data was obtained at 2.4 kbar.
Based on these re­
At this pressure we found that
we were very near t h e 'tricritical pressure which was calculated to be
2.4 + 0.2 kbar.
The calculation of the 6 exponent served as another
confirmation that we were near the tricritical pressured
In going from
first to second order the exponent is expected to change from a value
of 3 to 5 respectively, which is experimentally seen in the data.
We
conclude that the tricritical point does exist for KH^FO^, and that the
tricritical pressure, p fc, is 2.4 + 0.2 kbar.
A further significance of this study was the measurement of the 5
exponent.
This exponent had never been determined experimentally in
any ferroelectric previously.
Our predictions of the tricritical pressure were based on assump­
tions about the pressure dependence of the B coefficient.
Further study
at higher pressures would be useful in order to verify these assump­
tions.
••
Since our measurement of the 6 exponent was the first of its kind
31
determination of the exponent in other substances would be very inter­
esting.
One such, substance is SbSI1 but because of the way the cry­
stal grows, i.e. needle-like, it has so far been too hard to experi­
mentally handle.
When these problems are overcome an experiment sim­
ilar to the one reported on in this thesis would yeild valuable re­
sults that could answer questions as to the validity of the use of the
Landau theory in a region very close to the transition.
I
APPENDIX
One method used to analyze the KDP isopol data is presented here in
detail.
While the presentation will be in terms of the Landau theory
parameters, the results should have general applicability.
We will consider the following three problems:
A.
Find
in the Landau equation of state".
B.
Solve for T „ B 8 and C simultaneously.
C.
Determine the exponent 6 in E - P .
6
In order to deal with A. we will, review the problem of obtaining
the best fit straight line to a collection of data points.
Figure A-I we assume that
viation Ay^.
Referring to
is known exactly, and y^ has a standard de­
The probability that a given y is the true value y^ is
given by
i
W(y) « (Ay
(A-I)
/2tt)
which is properly normalized.
2
2
exp(-(y - y^) /2(Ay^) )
,
For an arbitrary straight line y «= ax + b
the probability that it represents the true data points is the joint
probability,
2
(A-2)
2
W(a„ b.) ® c Iexp (- (ax^ -S- b - y^) /2(Ay^) )
. i
O1 - V 1
The two conditions that maximize W(a, b) are
(A-3)
I ((ax
i
(A-4)
Jl
-f- b - y.)x /a ) ® 0, and
._
i i i
I((ax^ + b - F1)Z01) 13 0
where a and b represent the best fit values.
(A-5)
*
Solving for these we find,
2 ----- 2 V
, 2 _ z , 2 ii,,_,
a - (I(x,y,/o;)I(l/cQ - Z(x,/o,)I(y,/o2))/(I(x,/o,)I(l/o,)
I 1* 1 1 !
^
^ I I 4i 1x 1
4
I
I X
x xI 4
I
33
weighted mean
data
F I G U R E 12.
Coordinate transformation
(A-5) cent.
(A-6)
b -
(Z(x,/o^))2)
i 1 i
(
.
i
H
y
oZ.)
- ZCx 1Z p 2)E(x,y,/a^))/(Z(x^/a^)Z(l/a„2)
£->
i
" (ZCx1 Za2))2)
,
1 I
1
I
1 1 I
1
.
i
If we make a linear transformation to new variables (see Figure A~l)„
(A-7)
U = X - X
»
v =■ y - b
where
(A-8)
x«
u^hhfHHoh
i
1
i
so that
(A-9)
Z(u.Zo2) = 0
I
1
In terms of these new variables we have,
(A-IO)
.
b(u1 , V 1) = 0
,
and
(A-Il)
a(u1# V 1) = Z(U1V 1Za1)ZCZCu^Za2))
.
To find the standard deviations in a and b, define a = a - a and
3 *« b - b, and use the u, v variables, so that b = 0.
The joint prob­
ability becomes,
(A-12)
_
n
9
W(a, 6) = C exp-Z((au. - v. + au + $) Z2a.
JL
X
X
= C BxpC-ZCau1 - V1)^ZSa1)exp(-Zg2ZZa2) x
exp(-Z(a2u2Z2a2)) .
i
1
where terms in a, g, and ag vanish as a result of Equation (A-S)e
standard deviations are
(A-13)
• Aa = IZZ(u Za )2
I 1 1
The
35
and, .
(A-IA)
Ab - l/Z(l/of)
I
*As an example we apply these results to the individual isopols.
n
EXj/n
j=l
(A-15)
,
_
_
y «= b =
(A-17)
(A-18)
E -
n
Z E./n
n
E y./n
J-I
or
(A-16)
n
,
T = E
T 1Zn
_
J-I
_
- EJCT 1 - T))/E(E.
E)2
5 3 »
■
_ 2
I 3
,
Aa = a/v E (E, - E)
, Ab == o//n
J-I
a - E((E
j-1
Assume
,
J
where,
a *= ( E (T4
a(E,
E))2/ (n
j°l
In order to find A q in step A e we find from the equation,
(A-19)
(A-20)
T=
E/A ,P, + (T - BP
V kO1
oi i
o
-
CPi/Aoi>
that
(A-21)
3T/3E = !/A q 1 P 1
.
Here A 1 is obtained from Equation (A-17) with
dard deviation, is,
(A-22)
= l/A^^P^.
Ti :
AA0 1 = 1/255(Aa11Za1 1 (!/PiS11))
where Aa1 1 is obtained from Equation (A-18).
From this we find the
weighted average for A q and its standard deviation as
(A-23)
Ao - E(Aol)2 )/E(l/(AAoi)2)
and
2 i /2
(A-24)
The stan­
AAo «= (I/ElZ(AAoi) )
,
36
respectively.
Having determined A q1 we now want to find Tq8 B 8 and C simultan­
eously,
We rearrange the .Landau equation of state such that
(A-25)
T 1 = T q + I 1 ZAq P 1 - P 1 BZA0 - P^CZA^
.
The best fit values of Tq8 B 8 and C maximize the probability
(A-26)
W(T , B 8 C) -■ c exp(-E(1Tt - T O
^
J
O
Eji k
P, + P^BZA. + P ^ C Z A ri) 2 x
O
X
X
O
i
O
IZa(AT1)2)
that the parameters have their true values.
,
Having previously determin­
ed A 8 we define
o
(A-27)
U1 - E1ZAq P1 8 B1 - P1ZAo
P j Za „ 8 Y1 - PfZA0 , a± - AT1
in order to simplify the notation. ■ Setting the partial derivatives of
W with respect to T q8 B 8 and C equal to zero, we obtain three equations
(A— 28)
Cx1 + B1B + Y1O Z a i ) » O
E((T.
(A-29)
! ( ( T 1 - Tq - O1 + B1B + Y1O e i Za1) = O
(A-30)
E( ( T 1 - T q - O1 + B1B -5- Y1C) Y1Za2)
- O
,
which enables us to solve for the best fit values of T q8 B 8 and C.
In order to determine the form of the standard deviations we return
to the general problem of the three parameter probability W(x 8 y 8 z) „
such that.
(A-31)
W(x 8 y,
z)
= c exp(-E(-S1 + b ix +
c^y
2
+ d^)
As an example we will find the standard deviation of x.
)
.
In order to
find O^c we must integrate out the contributions that y and z make, in the
probability.
This will leave us with the single function probability
37
W(x) from which we can find the standard deviation by inspection.
The
mathematical derivation begins with
(A-32)
W(x) » /W(x, y)dy =« //W(x 9 y e z)dzdy
.
Putting the form of W(xp y p z)'gives.
y(x) s> //c exp(-E(-a
(A-33)
+ b^x + e^y. + d^z) )dzdy
Carrying out the integration over z Ieavess
(A-34)
W(x) a C 8 Z exp (-Z (-a
,
4- b x + c.y)
I,
+
.
1
(I (a,
L 1
b^x -
°
Since we are interested only in terms in x and x
the integration over
y is carried out and all irrelevant factors are place in the constant
coefficient.
(A-35)
The probability W(x) becomes,
W(x) = c 88 exp((-Zb^ + ( Z b ^ i)2ZzdI + ((Zb^i)2 - Z Z b ^ ^ b j x
7 Zd?))x2 + 2(Za,b. - Za.d.Zb.d,/Zd2 H- (Za,c. ,i I
I
I
i I
Za^d Zc.d/ZdZ)(-Zb.c
i 1 1
J J k ^
,1
I,
+■Zbid1r c d / Z d ‘)/(ZcJi 1 1
j j k K
i 1
(Ze,d,)z/Zdf))x
I
^
i
which gives for the standard deviation in the x direction
(A-36)
a
x
- l/2(Zc?ld 2 - (Zc.d,)2 )/(Zb2 Zc2 Zd2 - Zb2 (ZCid .)2 i i %
i i i
i
i
*■
Zc|(Zbidi)^ “ ZdI(ZbiCi)Z + 2Zbici Zbidi Zcidi) .
In terms of the parameters of the Landau equation of state the probabil­
ity can be written as,
(A-37)
_
'
W(T0 , B, C) * c exp (-Z (-T1 - E ^ A ^ )
1/2(AT1)2)
,
2
H-Tq - P ^ / A ^ - PiC/AQ )
x
38
therefore we can make the identifications.
2x1/2
(A-3.8)
a± = (T, - E 4 /AflP,)/(2(AT4)‘)
"I'
i'
(A-39)
bi - I/(2(AT1)2)1/2
(A-40)
cI " " p Z a 0 <2 Cat1)2)1 / 2
di = - P ^ M o (2 (ATi)2)1/2
(A-41)
with x ^ T^„ y = B s, and z =• C c
e
From (A-36) we can find oT , Og , and O^ 0
To get the standard deviation of B from (A-36) replace, b with c, and c
with b leaving
d unchanged.
(A-36) replace
b with d* d with C8 and c with b,
The final
onent,
(A-42)
Let X 1
To obtain the standard deviation ofC from
application of these" results is in
determining the 6 exp­
= log P 1 J1 y 1 «= log E 18 and define ,
i - Iog ^
1 + E, 1 )/(E , 1 _ E^1)
.
We find Eol from the equation
(A-43)
T1 - T1 =
I1(E1
- E1)
for the ith isopol in terms of (u8 v)-variables, and evaluate it at
T. = T , the best fit T from all the isopol data.
I O
o
Equation (A-43) be-
comes
(A-44)
Eoi “ E 1 + (To - T1)Za 1
.
Since E 1 and B 1 are independent parameters
(A-45)
AE0 1 - ((AE1 ) 2 + ((Tq - Tj,) I 1 /a 1 ) 2 ) 1 / 2
„
where AE1 is Ab in Equation (A-14) and Aa1 is Aa in Equation (A-13).
Using these results we find the weighted mean values
(A-46)
log Eo « E(log EolZa2 )/E(l/o2)
39
(A-46) cont.
log P =» Z(log P /o^)/E(l/a>)
I
l i I
From this we find the best fit exponent,
(A-47)
T o = (I(log (P1 ZP)Iog(EolZEo)Zo^)/(Z(Iog(PiZP)Oi)2
.
The standard deviation is
(A-48)
AS = (E(Iog(P1 ZP)AolPi)AToZE(Iog(P1 ZP)Zai ) 2 ) 2 +
IZE(Iog(P1ZP)Zo1)2 .
REFERENCES
I
A. Western, unpublished doctoral thesis (1976)
2J. West, Z. K r i s t '74, 306 (1930).
3
Cleveland Crystals, Inc., Box 3157,.Cleveland, Ohio
4
. Collected Papers of L. D. Landau, D. ter Haar, Ed.
44117.
(Gordon and
Breach, •New York, 1965), pp. 193-216..
^V. H. Schmidt, A. B . Western, A. G. Baker, and C. R. Bacon, to be
published in Ferroelectrics.
^P. Bastie, M. Vallade, C. Vettier, and C. M. E
ed in Ferroelectrics.
Zeyen, to be publish­
MONTANA STATE UNIVERSITY LIBRARIES
1762 100
BI32
cop.2
Bacon, Charles R
Trlcritical point in
KHgPOi;
ISSUED TO
I
DATE
G A Y _O RP
984 9