Pressure induced tricritical point in the ferroelectric phase transition of potassium dihydrogen
phosphate
by Arthur Boyd Western
A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF
PHILOSOPHY in Physics
Montana State University
© Copyright by Arthur Boyd Western (1976)
Abstract:
Measurement of the net polarization charge of a KH2PO4 crystal as a function of temperature, applied
electric field, and hydrostatic pressure indicates the existence of a tricritical point near 2 kbar of
pressure. This result is based upon static measurements of the polarization response to applied dc field
in a 0.5 K neighborhood of the ferroelectric transition at pressures of 0, 3, and 3 kbar.
In each case the paraelectric region is well described by the Landau equation of state,
E=A0(T-T0)P+BP3+CP5, to within 0.05 K of the transition temperature. Analysis of the data along
lines of constant polarization, which are here called "isopols," indicate that the transition is first-order
at 0 and 1 kbar with the critical field decreasing from 183±60 V/cm at 0 kbar to 43±13 V/cm at 1 kbar.
At 3 kbar the B coefficient is positive which indicates a second-order transition. This observation of a
change in the order of the transition is supported by a change in the behavior of the isothermal
dielectric constant which has a maximum for E>0 at 0.5 kbar and at E=O at 3 kbar.
This thesis contains brief introductions to the thermodynamics of phase transitions, the
phenomenological theory of ferroelectrics, and the structural changes in KH2PO4 produced by the
paraelectric to ferroelectric phase transition. Also included is a survey of recent experiments dealing
with the order of the KH2PO4 transition at ambient pressure. Details are described of experimental
apparatus used to obtain high impedance polarization charge measurement, temperature control to ±2
mK, and hydrostatic pressure generation to 3 kbar with a stability of 10 ppm.
PRESSURE INDUCED TRICRITICAL POINT IN THE
FERROELECTRIC PHASE TRANSITION OF
POTASSIUM DIHYDROGEN PHOSPHATE
by
ARTHUR BOYD WESTERN, JR.
A thesis submitted in p a r tia l fu lfillm e n t
o f the requirements fo r the degree
of
DOCTOR OF PHILOSOPHY
in
Physics
Approved:
Y.
Chairpersdh, Graduate Committee
HWd, Major Dep
tment
Graduate Bean
MONTANA STATE UNIVERSITY
Bozeman, Montana
August, 1976
TO JONNEE
iv
ACKNOWLEDGMENTS
The author is g ra te fu l fo r the support o f the National Science
Foundation under Grant No. DMR74-13220 A01.
I t is a pleasure to
acknowledge the help and guidance o f my thesis advisor, V. Hugo
Schmidt.
I wish to thank him e s p e cia lly fo r giving me the freedom
to develop my own s ty le as a p h y s ic is t, w hile always being a v a ila b le
fo r consultation and providing careful c ritic is m o f my e ffo r ts .
I
am indebted to Alan G. Baker fo r construction and maintenance of the
pressure system and pressure vessel used in th is work and fo r his aid
in taking data a t the cost o f many nights' sleep.
Also thanks to
Richard P o llin a and Charles Bacon fo r assistance taking data.
A
special thank you to Roy Weigand fo r keeping us supplied with the
seven tons o f liq u id nitrogen used over these years.
I have also
benefited g re a tly from the assistance and in stru c tio n o f electro n ic
technicians Fred Blankenburg and Jay Walker, machinist Cecil Badgley,
and Physics Laboratory Supervisor Mark Baldwin, as well as th e ir
many work-study students.
Thanks too, to Mizuho Kaw ajiri fo r trans­
la tin g a paper from Japanese.
In addition I am g ra te fu l to
Dr. Brookeman of Florida State U n iversity fo r supplying the pressure
vessel design used in th is work.
I am g ra te fu l to Sprague, Inc. and
TRW, Inc. fo r supplying samples of polystyrene and experimental
polypropylene capacito rs .
V
I am indebted in less ta n g ib le , but no less im portant, ways
to Kenkichi Okada and Richard P o llin a fo r helpful discussions con­
cerning technical d e ta ils o f the experiment, and to Robert Swenson
and Steven T o rs tv e it fo r th e o re tic a l in put.
In a d d itio n , I am
g re a tfu l to Jack Drumheller fo r informal conversations which did
much to preserve my s a n ity .
Thanks also to Ann Hewitt fo r her careful typing o f the fin a l
manuscript.
F in a lly , h e a r tfe lt thanks to my beloved w ife , Jonnee, whose
confidence, encouragement, and love have sustained me over these
years.
TABLE OF CONTENTS
Page
V I T A ............................................................................... .............................. .. . .
in
ACKNOWLEDGMENT
....................................................................................................
iv
LIST OF TABLES
................................................................................................... v i i i
LIST OF FIGURES................................. . - ..............................................................
ix
ABSTRACT
xi
I.
4.
Thermodynamics o f Phase Transitions .....................................
Description of KHgPO^ ...................................................................
Introduction to Landau-Ginzburg-Devonshire
Theory . ...................................................................................
Review o f Recent Relevant Experiments .................................
EXPERIMENTAL
1.
2.
3.
4.
5.
6.
III.
. . . . .
BACKGROUND...............................................................................................
1.
2.
3.
II.
...........................................................................................
.........................................
Experimental Requirements ..........................................................
Sample P r e p a r a t io n ..........................................' ..........................
Temperature C o n t r o l.........................
Electronics fo r Temperature Control ......................................
E le c tric a l Measurement o f Crystal Properties .................
Pressure S ystem ...............................................................................
EXPERIMENTAL RESULTS
1.
2.
3.
4.
5.
6.
7.
8.
9.
I
I
12
16
25
30
30
31
36
41
45
52
...........................................................................
58
D ie le c tric Measurements of KHgPO^ ..........................................
Is o p o ls ..........................................................................
Isopol Data a t Ambient P re s s u re ..............................................
Isopol Data a t HighP r e s s u r e ...................................
E r r o r s ..............................................................
Isothermal P vs_. E ...............................................................'. .
Time C o n s t a n t s ...............................................................................
Pressure Hysteresis . . . . ......................................................
C r itic a l Exponents .......................................................................
58
61
67
70
78
81
82
85
86
v ii
Page
IV .
DENOUEMENT.................................................................................................
1.
2.
3.
87
Summary . ........................................................................................:
C onclusions.......................................................................................
Significance and Recommendations
fo r Further S tu d y ...........................................................................
87
88
A PPEN D IX................................................................................................................
92
REFERENCES
.
90
100
viii
LIST OF TABLES
Table
I.
II.
III.
Page
Summary o f d e fin itio n s o f c r it ic a l- p o in t
exponents fo r d ie le c tr ic systems ..........................................
9
Summary o f recent published values fo r
the Landau fre e e n e r g y ..............................................................
28
Possible
mixed phase regions in the T-E plane .......................
65
IV.
Data
fo r sample No. 2 a t 0.0016 k b a r .......................................
73
V.
Data
fo r sample No. 2 a t 1.00 k b a r ...........................................
74
V I.
Data
fo r sample No. 2 a t 3.00 k b a r ..........................
75
Landau parameters fo r sample No. 2 as a
function o f pressure .................................................
77
V II .
ix
LIST OF FIGURES
Figure
Page
1.
Phase diagram o f water ..................................................................
5
2.
Phase diagram o f KHgPO^ in the zero pressure plane . . .
7
3.
Topology o f a t r i c r it ic a l p o i n t ..............................................
10
4.
Structure o f K HgPO ^.......................................................................
13
5.
Landau fre e energy as a function of temperature
20
6.
Ideal d ie le c tr ic behavior o f f i r s t - and secondorder t r a n s i t i o n s ..................... .................................................
22
7.
G (T,p,E) as a function o f P in various bias fie ld s . . .
23
.8 .
Visual h isto ry o f sample No. I ...................................................
33
9.
Inverse d ie le c tr ic constant vs_. temperature fo r
sample No. 2 . ....................................................................... .... .
37
10.
Schematic drawing o f c r y o s t a t .........................................
38
11.
Schematic o f coarse temperature c o n tro lle r . .....................
43
12.
V oltag e-con tro lled power supply
..............................................
44
13.
Liquid nitrogen f i l l i n g system ..................................................
46
14.
Liquid nitrogen level control
47
15.
Bridge c ir c u it fo r ac d ie le c tr ic measurements
16.
dc p o la riz a tio n measurement c i r c u i t ............................. ...
17.
Pressure generating s y s te m .................................
18.
Cary-Foster bridge fo r pressure measurement
19.
Isopols as predicted from Landau theory
20.
Isopols o f sample No. I a t 0.001 kbar
. . . .
..................................................
.................
.
49
50
53
.....................
55
.............................
63
..................................
68
X
Figure
Page
21.
Isopols
o f sample No. 2
a t 0.0016 k b a r .................................
69
22.
Isopols
of sample No. 2
a t 1.00 k b a r ........................
71
23.
Isopols
o f sample No. 2
a t 3.00 k b a r .....................................
72
24.
-A 0 (T-T 0 )ZP^ V£.
25.
E ffe c t of th e .v a ria tio n o f Tq
26.
T(E=O) vs. P2
27.
Isothermal P vs. E plots a t | and 3 k b a r ..............................
83
28.
Time constants o f p o la riz a tio n re la xa tio n
84
a t three pressures . . . . . . . . .
76
...................................................
79
....................................................................................
80
Xl
ABSTRACT
Measurement of the net p o la riz a tio n charge of a KH2 PO4 crystal '
as a function o f temperature, applied e le c tr ic f i e l d , and hydrostatic
pressure indicates the existence of a t r i c r it ic a l point near 2 kbar
o f pressure. This re s u lt is based upon s ta tic measurements of the
p o la riz a tio n response to applied dc f ie ld in a 0.5 K neighborhood
o f the fe r r o e le c tr ic tra n s itio n a t pressures of 0 , 1 , and 3 kbar.
In each case the p a ra e le c tric region is w ell described by the Landau
equation o f s ta te , E=A0 (T-T 0 )P+BP3+CP5, to w ith in 0.05 K o f the
tra n s itio n temperature. Analysis of the data along lin e s o f constant
p o la riz a tio n , which are here c a lle d "isopo ls," in d ic a te th a t the
tra n s itio n is fir s t - o r d e r a t 0 and I kbar with the c r it ic a l f ie ld
decreasing from 183+60 V/cm a t 0 kbar to 43+13 V/cm a t I kbar. At
3 kbar the B c o e ffic ie n t is p o s itiv e which indicates a second-order
tr a n s itio n . This observation o f a change in the order o f the
tra n s itio n is supported by a change in the behavior o f the isothermal
d ie le c tr ic constant which has a maximum fo r E>0 a t 0.5 kbar and a t
E=O a t 3 kbar.
This thesis contains b r ie f introductions to the thermodynamics
o f phase tr a n s itio n s , the phenomenological theory of fe rr o e le c tr ic s ,
and the s tru c tu ra l changes in KH2 PO4 produced by the p a ra e le c tric
to fe rr o e le c tr ic phase tr a n s itio n . Also included is a survey of
recent experiments dealing with the order o f the KH2PO4 tra n s itio n
a t ambient pressure. D etails are described o f experimental apparatus
used to obtain high impedance p o la riz a tio n charge measurement,
temperature control to ±2 mK, and hydrostatic pressure generation to
3 kbar with a s t a b ili t y o f 10 ppm.
I.
I.
BACKGROUND
Thermodynamics o f Phase Transitions
This section introduces those concepts which are used la t e r in
the analysis and discussion of experimental re s u lts .
A more complete
review o f the ideas presented below may be found in Introduction to
Phase Transitions and C r itic a l Phenomena by Eugene S tan ley . 1
I 2
Modern treatments o f thermodynamics * adopt the point of view
th a t there exists a thermodynamic p o te n tia l, U, whose magnitude
depends only upon the values o f c e rta in s ta te v a ria b le s .
When the
functional dependence o f U on these s ta te variables is known and they
are sp ecified then the macroscopic s ta te o f the system is specified .
I f these s ta te variab les are held fix e d by external means a ll other
parameters o f the system adjust in such a way as to minimize U.
For a d ie le c tr ic U is a function o f the entropy, S, the s tra in s , x,
and the p o la riz a tio n , P:
U =U (S ,x,p j,
(I)
dU=TdS-Xdx+EdP,
where T is the temperature, X is the stress and E is the external
applied f i e l d .
Some authors re fe r to U as the in te rn a l energy,
others the enthalpy.
This semantic question, which u ltim a te ly depends
upon one's choice of the d iv is io n of energy between the system and
it s environment, is o f no in te re s t here.
Callen
?
and K itte l
O
discuss
the question a t the elementary level fo r magnetic systems.
Leupold
derives s im ila r re su lts s ta rtin g from Maxwell's equations
and trea ts
2
d ie le c tric s e x p lic it ly .
g
I t su ffic e s here to say th a t Li so defined
is a minimum when S, x , and P are held fix e d and is independent of
applied f ie ld fo r p erfe ct p a ra e le c tric s , i . e . when P is a function
o f E/T only.
Experim entally i t is easier to hold the temperature than the
entropy fix e d , h e n c e .it is useful to define a thermodynamic p o ten tial
which is an extremum
fo r constant I , x , and P.
usually re fe rre d to as a fre e energy.
Such a p o ten tial is
I t is re la ted to U via a
Legendre transform:
F=U-TS5
(2)
F=F(T 5X 5P) 5
dF=-SdT-Xdx+EdP.
A second fre e energy, useful when the applied stress is held constant,
is defined
H=U-TS+xX=F+xX, .
(3)
H=H(T5X 5P)
dh=-SdT+xdX+EdP.
F in a lly , in the experiments to be described la te r i t w ill be the
temperature, s tre s s , and e le c tr ic f ie ld which are held constant.
The p o te n tia l which is a minimum under these conditions is
G=U-TSfxX-EP=H-EP5
G=G(T5X 5E) 5
dG=-SdT+xdX-PdE.
(4)
3
A c le a r discussion o f thermodynamic p o te n tia ls and Legendre transforms
appears in C allen.
Leupold has given a complete descriptio n of
Legendre transforms as applied to ferromagnets
4
and d ie le c tr ic s .
5
F ir s t p a r tia l d e riv a tiv e s o f thermodynamic p o te n tia ls with
respect to th e ir proper independent variab les y ie ld values fo r the
conjugate v a ria b le , e .g . 3F /d t jx p=-S and 3F/3P jx ^=E.
Second
d e riv a tiv e s y ie ld the so -c a lle d response functions,
32 F/3T 2 j X ip=TCx ^p and 32F/3P2 | x ^ =
where Cx p is the s p e c ific
heat a t constant s tra in and p o la riz a tio n , and
T 1S the inverse
isothermal clamped d ie le c t r ic s u s c e p tib ility .
A phase is a homogeneous system characterized by c e rta in
macroscopic parameters.
Phase tra n s itio n s are dramatic in th a t fo r
small changes in in ten sive parameters there are la rg e , even divergent,
changes in extensive parameters and/or response functions.
When ice
melts the temperature changes only m inutely from below O C to ju s t
above, y e t there is a discontinuous change in volume.
In addition
the s p e c ific heat diverges as q u a n titie s o f heat are absorbed with
n e g lig ib le change in temperature.
F e rro e le c tric s may show s im ila r
anomalies a t th e ir tra n s itio n temperatures including volume change,
la te n t heat, s tru c tu ra l change, enormous increase in d ie le c tr ic
constant, and sudden appearance o f a large p o la riz a tio n .
One c h a ra c te riza tio n o f phase tra n s itio n s is by the order of
the lowest d e riv a tiv e of a thermodynamic p o te n tia l which suffers a
4
d is c o n tin u ity or a divergence a t the tra n s itio n temperature.
In the
example o f m elting ic e , the f i r s t d e riv a tiv e of F with respect to
\
temperature, the entropy, is discontinuous so the tra n s itio n is
f ir s t - o r d e r .
On the other hand, He^ undergoes a tra n s itio n from a
normal f lu i d , He I , to a s u p e rflu id , He I I , w ith no d isc o n tin u ity
in entropy, but the s p e c ific heat diverges.
second-order.
The tra n s itio n is
The d is tin c tio n between f i r s t - and second-order w ill
be o f key importance la t e r .
The concept o f a c r it ic a l point is introduced by once again
using water as an example.
The p ro jectio n o f the pressure (p ),
volume ( v ) , temperature (T) surface fo r water in to the pT plane.
Fig. I , shows the coexistence lin es between vapor, w ater, and ice
meeting a t the t r ip l e p o in t.
Of p a rtic u la r in te re s t is the coexistence
lin e between vapor and water which terminates in a point known as the
c r it ic a l point (CP).
discontinuously.
I f one crosses th is lin e the density changes
But as one crosses the boundary closer to the
c r it ic a l point the d is c o n tin u ity suffered by the density becomes
sm aller, u n til above the CP no d is c o n tin u ity is observed.
By
tra v e lin g around the CP one can go from gas to liq u id continuously
with no anomalous behavior.
L ettin g
and pg represent the density of the liq u id and gas
phases re s p e c tiv e ly , the d iffe re n c e , p^-Pg, goes to zero as the
c r it ic a l temperature is approached from below.
Such a quantity
5
FUSION
CURVE
SOLID
SUBLIMATION
CURVE ------ >
FIG. I .
LIQUID
VAPOR
PRESSURE
CURVE
TRIPLE
POINT
VAPOR
The projection of the pressure ( p ) , volume ( V ) , and
temperature (T) surface fo r water into the pT plane.
The coexistence
lin e between vapor and liq u id terminates in a c r it ic a l point.
6
which is non-zero below the c r it ic a l temperature and zero above is
a common fe a tu re associated with CP's and is known as the order
parameter fo r the tr a n s itio n .
In a fe rr o e le c tr ic the spontaneous
p o la riza tio n acts as an order parameter.
I f the crys ta l temperature
is lowered through the tra n s itio n temperature,
in Fig. 2 , there
is a jump in the p o la riz a tio n i f the tra n s itio n is f ir s t - o r d e r .
When
an e le c tr ic f ie ld is applied to the crys ta l and the temperature is
lowered a s im ila r but sm aller jump occurs upon crossing the f i r s t
order lin e .
Eventually a t the c r it ic a l f i e l d , Ecr in Fig. 2, the
p o la riz a tio n changes smoothly as the temperature is lowered through
the tr a n s itio n , but the d ie le c tr ic constant diverges.
At even
higher fie ld s the crys ta l is polarized to such an extent th a t there
is no essential d iffe re n c e between the p a ra e le c tric and fe rr o e le c tr ic
states near the tra n s itio n temperature.
Phase tra n s itio n s may be characterized fu rth e r by the way in
which the order parameter tends to zero as the c r it ic a l temperature,
Tc r , is approached.
For example in a liq u id system one w rites
P^(T)-Pq(T)
1ZP(Tq r )
ZP(Trr)
[I+...I
to describe the behavior of the order parameter
Tcr from below.
(5)
T'r
Then B is the c r it ic a l exponent.
exponent, iX, fo r some function f is defined as
P as I approaches
In general a c r it ic a l
PARAE L ECTRI C
FERROELECTRIC
FIG. 2.
e le c tric fie ld
Phase diagram o f KHgPO^ i n the temperature (T) and
(E) plane a t zero pressure.
terminate a t c r i t i c a l
po int s (Tc r , ± Ec r ).
The f i r s t - o r d e r lin e s
8
\ _ Tim
91n f ( e )
9 In e
where e is ( T/T c r )-1 -
( 6)
,
Such exponents appropriately defined may also
ch aracterize divergences a t a c r it ic a l point:
J =
A'
Iim
£->o
91n
9 In £
(7)
,'
where j is the f i r s t divergent d e riv a tiv e o f the thermodynamic
p o te n tia l f .
A l i s t of commonly defined c r it ic a l exponents is
reproduced from S tanley's book
I
in Table I .
The in te re s tin g property of c r it ic a l exponents is th a t they
do not appear to depend upon the d e ta ils of the physical system but
ra th e r upon general features such as the dim ensionality and the
symmetry o f the Hamiltonian.
I f one expands a phase diagram with a c r it ic a l point in to a
th ird dimension the c r it ic a l point may be drawn out in to a lin e of
z-
c r it ic a l points.
in Fig. 3.
One p o s s ib ility , a t r i c r it ic a l p o in t,
is shown
Phase tra n s itio n lin es in Fig. 3 are so lid i f f i r s t -
order and dashed i f second-order.
Note th a t in the plane o f zero
ordering f ie ld the tra n s itio n changes from f i r s t - to second-order
a t the t r i c r it ic a l po int.
T r ic r it ic a l points (TCP's) have a ttra c te d considerable th e o re tic a l
in terest®
in th a t they are expected to e x h ib it d iffe r e n t c r it ic a l
exponents than ordinary c r it ic a l points.
G r iffith s
has proposed
9
TABLE I .
Summary o f d e fin itio n s of c r it ic a l- p o in t exponents fo r
d ie le c tr ic systems, a f t e r S ta n le y .*
D e fin itio n
Conditions
Quantity
E
E
P
<0
0
0
cH ~ e “
>0
0
0
M ~ (-e )3
<0
0
zo
zero f ie ld
p o la riz a tio n
2.
Exponent
Here E=(TlZTc r ) - L
. <0
0
ZO
z e r o -fie ld
isothermal
s u s c e p tib ility
-Y
r
>0
0
0
0
ZO
ZO
c r it ic a l
isotherm
v'
<0
0
ZO
c o rre la tio n
length
V
>0
0
0
0
0
0
■
a
r'
i .
I
•
4=
3
' Ch ~ ( - e ) - * '
Y
X 1- ~
e
6
H~
Tl'
s p e c ific heat
a t constant E
I
Ct1
|M|6 sgn(M)
r(r) ~
I r | - ( d- 2+ri)
p a ir c o rre la ­
tio n length
10
NONORDERING
FI ELD
NONORDERING
FIELD 2
ORDERING
FIELD
FIG. 3.
Topology of a t r i c r i t i c a l point.
are s o lid , second-order dashed.
F irs t-o rd e r lines
A t r i c r i t i c a l point (TCP) occurs
a t the in te rs e c tio n of three lines of c r it ic a l points (C P 's).
11
a notation fo r t r i c r it ic a l exponents in the zero ordering f ie ld plane.
Exponents are subscripted t i f the TCP is considered one.of a series
o f c r it ic a l points, i . e . , part of the dashed lin e in the zero ordering
f ie ld plane of Fig. 3.
The subscript u is used when the TCP is
considered as the terminus of a f i r s t order Tine in analogy to an
"ordinary" c r it ic a l p o in t.
Cl e a rly i f one does not r e s t r ic t one's
s e lf to the zero ordering f ie ld plane i t is possible to define a
large number o f d iffe r e n t kinds o f exponents as the.TCP is approached
along a v a rie ty of paths singled out by the topology o f Fig. 3.
.
The
important point to keep c le a r is th a t every exponent involves a path,
and th a t path must be c le a rly stated before the exponent has meaning.
Experim entally TCP's have been shown to e x is t in a number o f
O
systems:
He -He
I I
A
m ixtures,
i n
magnetic systems (DyAl garnet,
FeClg*3 ) , the s tru c tu ra l tra n s itio n in N H ^ C l and in the fe rro ­
e le c tr ic SbSI.*^
The f i r s t three of these s u ffe r from the drawback
th a t the TCP can be investigated only in the zero ordering f ie ld
plane.
The wing stru ctu re (see Fig. 3) is inaccessible as the fie ld s
which would have to be applied are impossible to produce experimentally.
SbSI suffers from other d i f f i c u lt i e s :
i t grows in long slender
crys ta ls inappropriate fo r d ie le c tr ic studies, and i t tends to
decompose by the evaporation of iodine.
The impetus fo r th is present work was the conjecture by V. H.
Schmidt
Ifi
th a t KHgPO^ (KDP) would have a TCP fo r which the ordering
12
f ie ld
in it s phase diagram
is experim entally a v a ila b le .
The
existence of such a TCP is o f considerable importance as i t would
represent an opportunity to study a TCP in it s e n tire phase space.
Moreover, crys ta ls of high q u a lity are commercially a v a ila b le in
v ir t u a lly any s iz e , and, although hygroscopic, they are otherwise
q u ite .s ta b le .
2.
Description of KH2PO4
Potassium Dihydrogen Phosphate (KDP) is the archetype o f a class
o f isomorphous hydrogen-bonded fe rr o e le c tr ic s .
Since the f i r s t report
of its fe r r o e le c tr ic properties in 1 935^ i t has been the subject
of l i t e r a l l y hundreds of papers.
Only a few of it s most fundamental
properties are ou tlined here.
A more complete introduction to work
IO
p rio r to 1960 may be found in Jona and Shirane.
More recent results
and an extensive review of e x is tin g lit e r a t u r e may be found in
"F e rro e le c tric Hydrogen-Bonded Systems" by V. H. Schmidt.^
KDP is a fe r r o e le c tr ic ; th a t is , i t develops a spontaneous dipole
moment below it s tra n s itio n temperature and th a t p o la riz a tio n can be
reversed by a p p lic a tio n o f an external e le c tr ic f ie ld .
is shown in Fig. 4 and described by Jona and Shirane:
Its structure
18
Each phosphorus atom is surrounded by four oxygen
atoms a t the corners o f a tetrahedron which is
almost regular (being compressed by approximately
2% along the c a x is ). These PO4 groups, together
with the potassium atoms, build up a stru ctu re
13
oK
FIG. 4.
Structure o f KH^PO^, a f t e r W e s t.^
14
in such a way th a t K and P atoms a lte rn a te
with each other a t a distance of c/2 in the
d ire c tio n of the c axis.
Every PO4 is linked
to four other PO4 groups, spaced c /4 apart
along c, by hydrogen bonds. Thus the linkage
is such th a t there is a hydrogen bond between
. one "upper" oxygen of one PO4 group and one
"lower", oxygen o f the neighboring PO4 group,
and each hydrogen bond lie s nearly perpendicular
to the c axis.
Neutron d iffr a c tio n data
20
reveal th a t the hydrogens are located
in one of two o ff-c e n te r positions w ith in the hydrogen bond.
Generally in the four hydrogen bonds associated with a PO^ group,
two o f the hydrogens occupy o ff-c e n te r positions close to the PO^
group and the other two occupy f a r positions (close to the
neighboring PO^ groups).
close or f a r .
I t may happen th a t three hydrogens are
These Takagi
21
configurations are s t a t is t ic a lly
less frequent but are important fo r the crystal dynamics.
S till
r a r e r , and o f no obvious importance, are H^PO^ and PO^ groups.
The remarkable fe a tu re o f the tra n s itio n is th a t, although the two
/
close hydrogens are randomly located amongst the four bonds above
the tr a n s itio n , below the tra n s itio n the close hydrogens are always .
found a t the top of the PO^ tetrahedron (or bottom fo r the opposite
p o la riz a tio n ).
Thus the tra n s itio n is order-disorder with respect
to the position of the hydrogen atoms.^
As pointed out in the description o f the crystal s tru c tu re ,
the hydrogen bonds are perpendicular to the c axis along which the
15
p o la riza tio n is d ire c te d .
Thus i t is not the hydrogen ordering
which is d ir e c tly responsible fo r the production of the dipole
moment o f the c ry s ta l.
Rather i t is the displacement of the P and
K ions along the c axis which accompanies the hydrogen ordering
which is responsible fo r the net charge displacement and the dipole
moment.
Thus the tra n s itio n is a d isp lacive one w ith respect to
the movement of the K, P* and 0 ions.
18
In a d d itio n , the crystal p o la riz a tio n is accompanied by a
shear s tra in perpendicular to the c a x is .
This shear is d ire c tly
proportional to the p o la riz a tio n whether i t is fie ld -in d u c ed above
the tra n s itio n or spontaneous below, and with the same (s lig h tly
temperature dependent) constant of p ro p o rtio n a lity .
IR
The hydrogen ordering has been treated as a tw o-level problem
by considering only the s lig h tly separated ground s ta te energies
of the double-well p o te n tia l hydrogen bond, and then casting the
problem in terms o f two-by-two Pauli m atrices,
any tw o-level problem.
as a pseudo-spin wave.
as can be done fo r
The hydrogen atom dynamics are then described
23
Thus, in the parlance of fundamental
e x c ita tio n s , one has a pseudo-spin wave in te ra c tin g w ith a transverse
o p tic al phonon (associated with the K and P motion) and with an
acoustic phonon (associated with the x-y s h ear).
This ric h v a rie ty
o f phenomena is one of the reasons why KDP has a ttra c te d so much
th e o re tic a l and experimental in te re s t over the past fo r ty years.
16
Moreover, i t appears th a t th is crys ta l is to become of even
fu rth e r in te re s t owing to the existence o f a t r i c r it ic a l point in
it s phase diagram.
In some sense microscopic descriptions o f the
KDP tra n s itio n foreshadowed th is re s u lt.
o f S la te r
24
and Takagi
Schmidt (SUS)
21
In the s t a t is t ic a l theory
as developed by S ilsbee, Uehling, and
and in the theory o f B linc and S v e tin a ^ which takes
proton tunneling in to account, the tra n s itio n is fir s t - o r d e r or
second-order depending upon the values o f a few parameters describing
the microscopic in te ra c tio n s .
Since the tra n s itio n (as w ill be
documented la t e r ) is ju s t barely f ir s t - o r d e r a t atmospheric pressure,
i t is perhaps not surprising th a t small d is to rtio n s o f the crystal
by applied pressure might t ip the scales in favor of a second-order
tra n s itio n thus producing the TCP.
Since much is known regarding
KDP's s tru c tu re , i t should prove f e r t i l e ground fo r th e o ris ts who
would explain the associated c r it ic a l phenomena.
3.
Introduction to Landau-Ginzburg-Devonshire Theory
The experimental re su lts presented in Chapter I I I are analyzed
p rim a rily in terms of the phenomenological theory of Landau.
A b r ie f introduction is given here.
is re fe rred to Landau,
26
Ginzburg,
27
For fu rth e r d e ta ils the reader
and Devonshire.
28
A complete
account of the phenomenological theory o f fe rro e le c tric s may be found
in G rindlay's book^ as well as th a t of Fatuzzo and M e r z .^
17
The central point o f the Landau-Ginzburg-Devonshire theory is
th a t the fre e energy H(T 5XsP) may be expanded in a power series in
the stress and the p o la riz a tio n .
H(T!X ,P )= H (T ,0 ,0 )+| I l.j Yj j .P1 Pj + J I 1jk t£ ljk J ,P.PJ.PkPJ
(8)
+5i i j k r'ijk!i,mnPi Pj PkP)lPmPn+ 2^ 'jk ;iSi j k ; Xi j Zk?,'l7'i'jk ai j k Xi‘/ k
&mn
+2^i j Jlmc*i j Jlm^i j ^Jl^m.
The
c o e ffic ie n ts are tensors o f e la s tic compliances, a^.^ are
p ie zo e la s tic tensors, and Q1-J ^ are e le c tr o s tr ic tiv e tensors.
Higher-
order terms may o f course be included, but the ones shown are
s u ffic ie n t to describe a ll known fe r r o e le c tr ic phenomena.'
Odd
powers of the p o la riz a tio n have been omitted as the crys ta l symmetry
demands.
31
I f one r e s tr ic ts consideration to KDP, a un iaxial
fe r r o e le c t r ic , under hydrostatic pressure the equation g re a tly
s im p lifie s :
H (T ,X ,P )= H (T ,0 ,0 )+ |A 'P 2+£BP4+£CP6+±SX2+^QXP2 ,
(9)
where the p ie z o e le c tric term is also omitted as KDP has no lin e a r
p ie zo e la s tic coupling in the high temperature phase due to symmetry.
A fu rth e r s im p lific a tio n can be made i f one suppresses the display
o f the purely e la s tic energy, |SX , and the e le c tr o s tr ic tiv e term,
2
iQXP .
The fre e energy is then
H=lAP
2+fBP
4+lCP
6
Z
If
D
(10)
18
18
For a fix e d pressure th is fre e energy function describes the thermo­
dynamic s ta te of the c ry s ta l.
The c o e ffic ie n ts A, B, and C may be
functions o f temperature and/or pressure.
In p a r tic u la r , the inverse
isothermal d ie le c tr ic constant decreases lin e a rly with temperature
in the p a ra e le c tric phase (C u rie-Weiss law) extrap o latin g to zero
a t Tq, the Curie-Weiss temperature.
This behavior can be described
in the fre e energy by w ritin g A=Aq (T-Tq) , so th a t
H=|Aq (T-T q ) P ^ B P 4+JCP6.
( 11)
The point of view taken in th is thesis is to describe the
temperature and pressure dependences of the parameters Aq ,T q ,B, and
C, suppressing e x p lic it terms depending upon the pressure in the free
energy.
This seems d esirable in an experimental paper as there
remains lack o f agreement amongst th e o re tic a l treatments o f the KDP
tr a n s itio n .
For example, Hegenbarth and Ullwer
e l e c tr o s tr ic tiv e term w ith Aq (T-Tq)P
32
lump the
and show th a t the e le c tro s tric tio n
accounts fo r the lowering o f the tra n s itio n temperature w ith pressure.
On the other hand Vaks and Sidnenko
33
take th is same e le c tr o s tr ic tiv e 2
2
term, argue th a t s tra in its proportional to P , and hence lump i t with
the BP4 term and report th a t i t is responsible fo r the f ir s t-o r d e r
nature o f the tr a n s itio n .
While both treatments may be c o rre c t, the
experiments described here are not s u ffic ie n t to separate e le c tro ­
s t r i c tiv e energy from other pressure e ffe c ts , le t alone separate i t
19
in to a quadratic and a q u artic p a rt.
Hence e x p lic it pressure
dependence due to separate mechanisms is suppressed.
Experim entally i t is the external f i e l d , E, which is held
constant and thus
G(T,X,E)=H(T,X,P)-EP
(12)
is the appropriate thermodynamic p o te n tia l to minimize.
however, there is no d iffe re n c e between H and G.
For E=O,
Thus fo r a crystal
whose faces are shorted together one may minimize H a t fix e d T, p,
and E=O with respect to P to fin d the equilibrium s ta te o f the
system.
Fig. 5 shows H ys_. the parameter P fo r various temperatures
and a p o s itiv e and negative B.
I f the crystal passes only through
absolutely stable thermodynamic s ta te s , the value o f P corresponding
to the minimum o f H is a c tu a lly obtained.
At the tra n s itio n temperature, Tc , Fig. 5a shows th a t three
d is tin c t values of the p o la riz a tio n minimize the fre e energy, so
these p o la riza tio n s c o exist.
S lig h tly above Tc the p o la riz a tio n must
be zero, w hile s lig h tly below i t must jump to a non-zero value.
This
jump in p o la riz a tio n a t the tra n s itio n temperature is a c h a ra c te ris tic
o f a fir s t- o r d e r tr a n s itio n .
The two p o sitiv e h i lls in the free
energy o f Fig. 5a are due to the n e g a tiv ity of B.
In Fig. 5b where
B is assumed p o s itiv e no such humps appear; P changes ra p id ly with
temperature ju s t below T 1 but i t does not jump discontinuously.
FIG. 5.
H-^Ao (T-To )P2+iBP4+|CP6 vs. P f o r (a) B<0 and (b) B>0.
Tq i s the Curie-Weiss temperature, T the Curie temperature, T1 the
L
I
m e t a s t a b i l i t y l i m i t , and Tc r the c r i t i c a l temperature.
21
I t can be shown e a s ily
th a t the inverse d ie le c tr ic constant
(second d e riv a tiv e o f H with respect to P) jumps discontinuously a t
Tc i f B is negative ( f ir s t - o r d e r t r a n s it io n ) , but is continuous with
a discontinuous slope a t Tc i f B is p o s itiv e (second-order t r a n s it io n ) .
Graphs of these two ideal types of behavior are shown in Fig. 6 .
I f metastable thermodynamic states are allowed, a fir s t - o r d e r
tra n s itio n may e x h ib it thermal hysteresis.
The p ictu re here is th a t
the crys ta l may remain in a local minimum well of the fre e energy
even when the bottom o f th a t well has higher energy than the absolute
minimum.
That is , i f the c rys ta l is polarized to P^O a t T<TC and
the temperature ra is e d , the p o s s ib ility exists th a t the crys ta l w ill
remain polarized fo r T>TC i f the thermal flu c tu a tio n s are not
s u ffic ie n t to allow the crystal to "jump the h i l l " separating i t
from the absolute minimum a t P=0.
The reverse s itu a tio n may a rise
when the temperature is lowered.
No such p o s s ib ility exists when
the tra n s itio n is second-order.
The maximum possible extent of the
hysteresis is o f course when the metastable minimum disappears
(becomes lo c a lly un stable).
The high and low temperature lim its
fo r hysteresis are c a lle d T^ and Tq follow ing Fatuzzo and Merz.
ID
I f . a non-zero e le c tr ic f ie ld is applied to the c rys ta l the f u ll
p o te n tia l G (T,p,E) must be used.
Although G(T,p,E) is not a proper
thermodynamic function o f the p o la riz a tio n , i t is s t i l l in s tru c tiv e
to graph it s parametric dependence on P.
This is shown in Fig. 7
22
(a) B<0
FIG. 6.
Id e a liz e d behavior o f the p o l a r i z a t i o n and inverse
isothermal s u s c e p t i b i l i t y f o r (a) f i r s t - and (b) second-order
t r a n s i t i o n s as pre dict ed from the f r e e energy H=^Aq ( T-T q )P2+£BP4+^CP6 .
The order o f the t r a n s i t i o n depends upon the sign o f B.
t = (TZTq ) - I .
(After G rin d la y^)
Here
23
E =O
(a)
0< E < E
(b)
E=E
E> E
(d)
FIG. 7.
Tc<T<V
The fr e e energy G=H-EP f o r fo u r bias f i e l d s a t constant
24
fo r various values o f E a t a fix e d temperature I , I <T<T
.
A
fir s t-o r d e r tra n s itio n may now be induced by the a p p licatio n o f a
s u ffic ie n tly large e le c tr ic f ie ld .
For fie ld s less than the tra n s itio n
f i e l d , Et r , the minimum of G is a t P=O.; fo r E>Et r the minimum is a t
P>0 and the crys ta l p o la riz a tio n jumps discontinuously to th is value
(perhaps with h y s te re s is ).
This construction is completely equivalent
to the bi-tan g en t construction to the fre e energy H which may be more
fa m ilia r to some readers.
I f the temperature is raised above Tcr then the two minima
in Fig. 7c coalesce in to one and there can be no fu rth e r f i r s t order tra n s itio n .
The temperature, T
, and f i e l d , Ec r , where th is
occurs are the coordinates of the c r it ic a l point a t the terminus
of the fir s t - o r d e r lin e in the te m p e ra tu re -ele ctric f ie ld plane.
This point has the properties o f a second-order tra n s itio n .
One can fin d exact expressions fo r the various special temperatures
and other q u a n titie s o f in te re s t d ir e c tly from the expression fo r the
fre e energy.
One finds Tc by requirin g H=0, P^O have a sing le
p o s itiv e root.
The temperature T
is where the two points of
cr
2
2
in fle c tio n o f H merge in to one, hence d H/dP =0 has only one po sitive
ro o t.
I t is then a simple m atter of algebra to solve fo r the other
q u a n titie s o f in te re s t.
The re su lts are:
W
9B V O A oc
( 13)
Ec r= ( 2 ( - B /5 ) 5/ ( C / 3 ) 3) 1/2
Apspon(Tc>= < -3B/ 4c) 1/2
where AP
(T ) is the jump in p o la riz a tio n occurring a t T .
ojJU11 U
V
Standard mean f ie ld exponents may be derived by considering
the dependence o f such q u a n titie s on T-Tc .
The re su lts are:"*'
a^=a=0 ,B=iSsY^=Y=I ,6=3.
In the context o f the m ean-field Landau-Ginzburg-Devonshire free
energy expression, the condition fo r a t r i c r it ic a l point is th a t B
change continuously from p o s itiv e to negative as a function o f some
parameter.
The TCP occurs when B=O.
In Chapter I I I re su lts are
presented which in d ic a te th a t a TCP does e x is t in KDP a t high pressure
where the c o e ffic ie n t B is driven to zero.
4.
Review o f Recent Relevant Experiments
P rio r to 1969 the tra n s itio n in KDP was generally thought to
be second-order.
18
Recent re su lts in d ic a te th a t i t is in fa c t f i r s t -
order but qu ite close to being second-order.
This opinion is now
supported by a number o f experiments done in various countries.
In
Russia, Strukov3Z* exploited KDP1s large e le c tro c a lo ric e ffe c t and
measured the temperature change produced by the sudden application
of an e le c tr ic f ie ld .
G ladkii and Sidnenko
35
measured the p o la riza tio n
26
vs. temperature o f the crys ta l in various e le c tric f ie ld s .
Garber
and Smolenko 0 made painstaking d ila to m e tric measurements of the
07
crystal dimensions near the tra n s itio n .
Vallade and coworkers
in France have measured the p o la riz a tio n vs_. temperature dependence
o f KDP by o p tic a l birefrin g en ce.
Okada and Sugie’' and others in
Japan have studied the KDP tra n s itio n exten sively.
They have
reported on the temperature sweep ra te dependence of the thermal
hysteresis,
38
the d iffe re n c e between the adiabatic and isothermal
d ie le c tr ic constant,
39
and hysteresis loop measurements o f the
p o la riz a tio n ys^. applied f ie ld a t constant temperature.
In th is
country, Reese has studied the tra n s itio n in a number o f c a re fu lly
done c a lo rim e tric e x p e rim e n ts .^ ’ ^
The most recent re su lts of a ll
of these groups are in f a i r agreement as to the coordinates of the
c r it ic a l point a t the end o f the f ir s t - o r d e r lin e (200-300 V/cm)
and the fa c t th a t KDP obeys the phenomenological theory o f Landau
to w ith in a t le a s t 0.1 K o f the tra n s itio n temperature.
Th ere.are, however, three experiments described in the lit e r a t u r e
which yield ed markedly d iffe r e n t re s u lts .
The f i r s t o f these is an
x -ra y d i latom etric study by Kobayashi e t a l .
43
who found a c r it ic a l
f ie ld o f 8500.V/cm, much higher than the c r it ic a l fie ld s of 200-300
V/cm found in the experiments described above.
Matsuda and Abe44
c a lc u la te the B c o e ffic ie n t in the Landau fre e energy from measure­
ments of the th ird harmonic o f a I kHz ac e le c tr ic f ie ld applied
27
to the c ry s ta l.
Their value is two orders of magnitude la rg e r (in
absolute value) than reported in the papers c ite d e a r lie r .
F in a lly ,
Eberhard and H orn^ (EH) studied the thermal hysteresis o f the
tra n s itio n a t various applied fie ld s and concluded Ec r=6500 V/cm.
There is reason to believe th is value should be revised downward
closer to 300 V /c m .^
EH have rece n tly revised th a t re s u lt to
1200 V/cm. 47
The la te s t published re su lts of a ll o f these workers is displayed
AQ
in Table I I along w ith published
and unpublished re su lts o f the
Montana State U n iversity group.
Far less work has been done on the KDP tra n s itio n a t elevated
pressures.
Samara
49
evaluated the change of T
and Aq with pressure
using a sm all-signal ac f ie ld to measure the d ie le c tr ic response of
the c ry s ta l.
He found dT^/dp=-4.6 K/kbar and (dAo/d p )/A o= -7 x l0 ~3 kbar
a t pressures to about 7 kbar.
The tra n s itio n temperature then
decreased ra p id ly to 0 K a t 17 kbar.
The tra n s itio n temperature
appeared to approach 0 K w ith an i n f in i t e slope.
OO
Hegenbarth and Ullwer
(HU) performed e s s e n tia lly the same
experiment up to 1.6 kbar and found the i n i t i a l decrease o f Tc
with pressure to be -5 .6 K/kbar.
HU a ttrib u te d the change in Tc
to the e l e c tr o s tr ic tiv e term in the fre e energy and found reasonable
agreement with the required magnitude o f the appropriate e la s tic
constants.
TABLE I I .
Summary o f recent published values of the parameters in the free
energy H=|A0 (T-T 0 )P^+^BP^+-|CP®+^DP^ and derived coordinates o f the c r it ic a l point
10~3 . esu
IO " 11 esu
IO " 19 esu
S4
Strukov 04
3.9
-1 .9
6.3
0
120
0.07
Sidnenko^
3 .8 ± 0 .I
-3 .0 ± 0 .8
6 . 5±1. I
0
370
0.16
3 .8 ± 0 .I
- 0 . 5 ±0 .3
0
3 .8 ± 0 .4
87
0.036
3.9
- 0 . 54±0.05
0
2 .8 5 ± 0 .10
124
0.046
3.9
- I .8 5 ± 0 .25
3.3+0.5
0.87+0.5
280
0.11
. 4 . 2 ± 0 .1
- I . 9 ± 0 .I
5.4+0.4
160
0.07
(3 .8 1 )
-0 .4 4
84
0.055
Vallade 37
Okada^
Benepe
42
Kobayashi
Matsuda
43
44
Eberhard^3
MSU
an
Sample V 0
Sample 2
(3 .8 6 )
—
(7 .3 )4 6
4 . 3 ±0 .2
4 . 0 ± 0 .2
-1 1 .9
- 110.
0
11.0
—
0
2.96
0
—
- 2.2
0.6
0
2 . 35±0.4
I .4 8 ± 0 .2
5. 91±1.5
3 .U 0 .4
0
0
8500
—
6500
232±70
186±60
*
O
Ec r ’ .
V/cm
iO"27 esu
O
D,
. C,'
B,
.. A0 ,
H
a t ambient pressure.
1.50
—
0.51
0 . I 0±0.03
0 .08±0.03
I
29
In n e ith er of the above two experiments was the temperature
resolution s u ffic ie n t to monitor the order o f the tra n s itio n .
The purpose o f the experiments described in th is thesis was
to ( I ) resolve the controversy regarding the coordinates of the
c r it ic a l point a t zero pressure, ( 2 ) to monitor those coordinates
w ith increasing pressure, and (3) determine i f the tra n s itio n becomes
second-order a t high pressure, thus in d ic a tin g the existence of a
t r i c r it ic a l point in the phase diagram o f KDP.
II.
EXPERIMENTAL
The accomplishment o f the goals set fo rth a t the end of
Chapter I required considerable care in the measurement of. p o la ri­
z a tio n , e le c tr ic f i e l d , temperature, and pressure.
Results of other
workers which in d ic a te the precision necessary are reviewed below.
Then fo llo w descriptions o f the actual hardware used to obtain that
precision.
I.
Experimental Requirements
One measure o f the f ir s t - o r d e r nature of a tra n s itio n is the
d iffe re n c e between Tq and T^.
Here T^ is th a t temperature where the
inverse d ie le c tr ic constant in the p a ra e le c tric region extrapolates
to zero, and Tc is the actual tra n s itio n temperature.
For KDP,
T0-T c values from 0.01 K to 0.63 K appear in the l i t e r a t u r e . ^ - ^
C learly i f the T gap was as small as a few hundredths o f a degree,
temperature reso lu tio n on the order o f millid e g re e s is ca lle d fo r.
Moreover, Okada
38
, and Garber and Smolenko
36
have shown th a t temperature
d r i f t rates as small as 0.1 K/hr would be too fa s t to e x h ib it
equilib rium properties of the c ry s ta l.
Thus the temperature would
have to be stable to mi 11i degree accuracy.
In e a r lie r pressure studies Samara,
49
and Hegenbarth and Ullwer
32
found th a t the tra n s itio n temperature o f KDP changed on the order of
-5 K/kbar.
Thus pressure s t a b ilit y o f ±0.25 bar would be necessary to
match temperature s t a b ili t y of ±2.5 mK.
This is 10 ppm a t 2.5 kbar,
31
a s trin g e n t requirement on a small volume system.
This d i f f i c u lt y is
compounded by the use of helium gas as a pressure medium which was
necessary in order to m aintain hydrostatic pressure up to 7 kbar a t
temperatures as low as 70 K.
For reasons discussed in the beginning o f Chapter I I I , s ta tic
.
d ie le c tr ic measurements were necessary to determine the order of the
tr a n s itio n .
To cover any reasonable temperature range (e .g . 0.5 K) in
steps o f, say, a few hundredths o f a degree, a high impedance
electrom eter was necessary to measure the p o la riz a tio n charge without
draining th a t charge to ground excessively over a period of days.
In addition th is charge would have to be stored on a large low-leakage
capacitor.
An e ffe c tiv e RC time constant of 8x10® sec is the best so
fa r obtained.
This re su lts in a leakage o f 1.1% in 24 hr.
Guarded
c ir c u itr y and Teflon in s u la tio n were used to insure th a t leakage
currents to ground did not d e te rio ra te th is value.
2.
Sample Preparation
Sample No. I was obtained from In te ra c tiv e R adiation, Inc.
50
in May o f 1974 and stored in a small v ia l with calcium s u lfa te
desiccant u n til June o f 1975.
Crystal dimensions were 1x1x0.2 cm,
the large faces being perpendicular to the fe rr o e le c tr ic c axis.
Gold electrodes were evaporated onto the large faces in a vacuum of
10
-4
to r r .
The evaporator consisted of a tungsten filam en t wound.
32
with gold w ire.
The filam en t was heated white hot with a large
current fed in to the vacuum chamber via spark plug feedthroughs.
On one large face o f the crys ta l a c ir c le o f 32 gauge w ire was used
as a mask during evaporation to create a guard ring configuration.
The average diameter o f the nearly c ir c u la r center electrode was
0 . 671±0.004 cm and the unplated annular guard’ ring width was
0 . 0377±0.007 cm.
This yield ed an e ffe c tiv e center electrode area
o f 0.40+0.02 cm2 .
The temperature of sample No. I was lowered to the tra n s itio n
region fiv e times between July and December o f 1975.
three hours on each occasion.
This took about
No measurements were taken on the f i r s t
occasion due to cryo stat problems.
On the second t r i a l the small
signal (0.05 V/cm) ac d ie le c tr ic constant was measured a t I kHz.
A rounded tra n s itio n was observed as shown in Rig. 8 .
d ie le c tr ic constant was 10 000.
The maximum
This run was then term inated.
Upon
inspecting the crys ta l i t was found th a t each of the four corners had
broken o f f (see Fig. 8 ).
I t was then discovered th a t the compression
spring of the spring-loaded sample holder was f u lly compressed causing
considerable pressure a t the center o f the crystal from the approxi2
mately I mm central contact of the crys ta l holder. This condition
was re lie v e d and the crys ta l held as lig h t ly as the holder and the
a v a ila b le manual d e x te rity would allow .
Run No. 3 was aborted when
the ac measurements indicated a broken lead w ire.
33
□
m
RUN
NO. 4
H-0 .0 4 K-H
RUN
,°o NO. 2
/
O
O
O\
|oq\
= I
°§
d> D
D
□
° o D
°
■5
/
a
□
O
\ D
D
□.
O r ig in a l c r y s ta l w ith
guard r in g e le ctro d e
A f t e r run No. 2
A f t e r run No. 4
FIG. 8.
A v is u a l h is t o r y o f sample No. I .
34
On run No. 4 the ac d ie le c tr ic constant was higher ( £max=12 500)
and the tra n s itio n was hot rounded w ith in the a v a ila b le resolution
(see Fig. 8 ) .
A graph o f e * is e s p e c ia lly in te re s tin g fo r th is
crystal as i t indicates a downward jump d isco n tin u ity in e a t the
tra n s itio n followed by a continuous decrease in e below Tc -
This
type o f behavior is predicted by Landau theory but is not the usual
behavior o f KDP; in fa c t we have seen i t in no other c ry s ta ls .
Run No. 4 lasted from 10 September to 8 November during which time
the crys ta l remained near the tra n s itio n temperature, although i t
was a c tu a lly brought through the tra n s itio n fewer than a dozen times.
Run No. 4 was terminated when a buildup o f fr o s t around the dewar
top put s u ffic ie n t pressure on the b a c k -to -a ir valve to break the
solder jo in t and destroy the vacuum.
Routine inspection showed th a t sample No. I was badly cracked
around the outside o f the guard rin g ; nearly a th ird of the outer
area had broken away.
This is lik e ly due to the spring pressure
o f the guard ring contact of the crystal holder.
area appeared to be fre e o f cracks.
The center electrode
A subsequent attempt to take
fu rth e r data on sample Noll I led to inconsistencies w ith previous
data which could not be reconciled by minor area corrections.
Cl
Sample No. 2 was obtained from Cleveland C rystals, Inc.
in
May of 1974 and stored in a small v ia l with calcium s u lfa te desiccant
u n til 13 January 1976.
Crystal dimensions were 1x1x0.2 cm, the large
35
faces being perpendicular to the fe r r o e le c tr ic c a x is .
Chrome-gold
electrodes had been evaporated on the surface by the s u p p lier.
guard ring was used.
No
The 0.0025 in . diameter s o lid copperweld center
wires o f Type A U ltram in iatu re Coaxial C ab le^ were attached to the
CO
evaporated electrode faces by means o f s ilv e r paint
2
small (2 mm ) dots on e ith e r side.
applied as
Five coats were used in an e f f o r t
to increase the strength of the bond as these lead wires were the
sole support fo r the c ry s ta l.
Thus the only stress on the fre e ly
hanging c rys ta l was it s own weight.
The crystal face area was
0.995+0.02 cm^; it s thickness was 0.1962+0.0002 cm. .
Sample No. 2 was pressurized w ith He gas, then vented to
ambient pressure a number of times a t room temperature in order to
flu sh the pressure vessel of moisture and a ir :
four cycles to 500
p s i, three cycles to 1000 p s i, two cycles to 1500 p s i, and once to
15 000 p s i.
The sample temperature was then lowered to near Tq in
a period o f about fiv e hours.
The small signal (0 .0 5 V/cm) ac d ie le c tr ic constant ( e ) was
measured a t I kHz. For 0 .3 K above Tc a s tra ig h t lin e was obtained
I
fo r £
VSy T in accordance with the Curie-Weiss law. A s lig h t
decrease in e occurred 40 mK above T , then the d ie le c tr ic constant
o
rose to the in c re d ib ly high value of 360 000 a fte r which i t stayed
f l a t fo r a t le a s t 0.2 K above Tq.
This high d ie le c tr ic constant was
checked c a re fu lly below Tq with even sm aller ac f ie ld (0.005 V/cm)
36
and found to be a t le a s t 300 000.
No. 2.
Fig. 9 shows e~* vs_. I fo r sample
The crys ta l remained near Tc a t ambient pressure u n til
4 March when i t was pressurized to I kbar.
The crys ta l then remained
near the I kbar tra n s itio n temperature u n til 4 May when i t was
returned to ambient pressure and raised to room temperature.
The c r y s t a l, upon in spection, showed no v is ib le d e te rio ra tio n .
The crys ta l was again lowered to the tra n s itio n temperature, th is
time over a period of four days.
E le c tric a l measurements a t I bar
were made, and the crys ta l was then pressurized to three kbar.
temperature was then lowered 15 K in about twelve hours.
remained near the 3 kbar tra n s itio n u n til 19 June.
The
The crystal
The crys ta l was
then heated fiv e degrees, and the pressure lowered to 2 kbar.
On
30 June the c rys ta l was raised to room temperature, and the system'
was vented to ambient pressure over a period o f days.
3.
Temperature Control
The cryo stat used to control the pressure vessel temperature
consists o f three concentric cylinders with the 0.25 in . highpressure tubing acting as the central connecting axis.
drawing of the e n tire assembly is shown in Fig. 10.
A schematic
The innermost
piece is the pressure vessel (PV) i t s e l f a t the term ination o f the
pressure tubing.
and 12 in .
long.
Surrounding the PV is a copper can 3 in . in diameter
The top o f the can is attached to the pressure
37
KDP SAMPLE NO. 2
€ max = 3 6 0 OOO
I KHZ
• 0. 0 5 V/CM
o 0. 0 0 5 V / C M
Tn - T n = 2 4 m K
T-To , K
FIG. 9.
No. 2.
Inverse d i e l e c t r i c co nstant vs^ temperature f o r sample
H ysteresis is in s tru m e n ta l.
38
FEED­
THROUGH
VACUUM
FIG. 10.
Schematic drawing o f c r y o s ta t assembly.
areas show lo c a t io n o f temperature sensors.
Dark re c ta n g u la r
39
tubing with s o ft solder and the body of the can slipped over the PV
from below and attached to the can top with six screws.
The top
of the can has three | in . holes allowing the space between the can
and the PV to be evacuated and the e le c tr ic a l leads to reach the
outside o f the PV.
Surrounding th is inner can is a 5 in . diameter brass can whose
top is s o ft soldered to the pressure tubing.
attached to it s top by a vacuum flang e.
The large can is
The top h a lf o f the flange
is f l a t save a small ridge I mm wide and | mm high.
flange is completely f l a t .
The lower h a lf
An aluminum f o i l gasket is pinched between
the two pieces with pressure from a b o lt rin g .
The extruded aluminum
f o i l has acted as a vacuum tig h t gasket under liq u id nitrogen fo r
months without tro u b le.
O ff center on the top o f the outer can is
s ilv e r soldered a 1.5 in . diameter s ta in less steel tube through which
the assembly is evacuated and leads fo r temperature sensors and heaters
leave the cryo sta t.
The e n tire assembly is supported by th is tube
which terminates in a tee.
One side o f th is tee leads to the vacuum
pump, and the other to a plexiglas flange through which shielded
e le c tr ic a l leads are brought to ambient pressure.
F in a lly the large
brass can is immersed in a glass dewar of liq u id nitrogen (LN ), the
top of the can being kept a t le a s t 4 in . below the LN surface.
The actual temperature regulatio n occurs in two steps.
F irs t
the inner can is "roughly" regulated to ±0.1 K by means of a heater
40
wound around a 0.75 in . copper rod concentric with the high-pressure
tubing and attached to the li d o f the inner can with s ilv e r solder.
The heater current is regulated by the voltage of a grounded copperconstantan thermocouple attached to the copper rod.
This point is
operated about one degree below the PV temperature.
This gives a
point o f approximately constant temperature along the pressure tubing
and minimizes temperature gradients on the PV i t s e l f by minimizing
heat tra n s fe r w ith it s surroundings.
The temperature gradient between
the top and bottom of the inner can was measured w ith a d iffe r e n tia l
thermocouple and found to be less than 0.1 K.
The second stage of regulatio n occurs a t the top of the PV
its e lf.
A heater is wound around the outside of the PV and a
tem perature-sensitive capacitor controls the heater curren t.
The
capacitance sensor has a reso lutio n of ± 2 mK and the top o f the PV
is stable to w ith in th is re s o lu tio n .
The temperature o f the sample
is assumed to be th a t o f a second capacitance sensor located a t the
bottom o f the PV in a hole d r ille d in to the center o f the closure
plug.
Extensive time h is to rie s were taken comparing the PV tempera­
ture w ith the sample p rop erties.
At the fa s te s t scan rates used
(~ 25 mK/hr) no more than a 5 min lag could be detected between
heating and cooling.
The d iffe re n c e between the top and the bottom
sensor capacitance was s e n s itiv e to the amount o f heater current
supplied to the PV.
When taking data, care was taken th a t th is
41
d iffe re n c e was nearly constant, in order to maintain a constant
temperature d iffe re n c e between the sample and the lower sensor.
In
immediate ju x ta p o s itio n to the capacitance sensor in the closure plug
hole was a copper-constantan thermocouple used to c a lib ra te the.
capacitance sensor.
A ll heater wires and sensor leads were separately shielded.
They were also thermalIy anchored to the cryostat a t each stage of
the temperature regulation by wrapping them several times around the
appropriate can and the pressure tubing.
No heat-conducting medium
was smeared over them, however, as the heat tra n s fe r via the small
diameter wires was calculated to be small compared to heat flow
along the pressure tubing which acted as the main heat path from
the PV.
4.
Electronics fo r Temperature Control
The current to the shield can heater was controlled by the
voltage of a copper-constantan thermocouple anchored close to the
heater c o i l .
The thermocouple voltage was compared to the voltage
on a potentiom etric voltage d iv id e r.
The d iffe re n c e voltage was
then measured by a Leeds and Northrup Model 9834 dc null detector.
The recorder output of the null detector was in turn fed to the input
of a vo lta g e -c o n tro lle d power supply which supplied the actual
heater curren t.
The vo lta g e -c o n tro lle d power supply was b u ilt
42
in th is laboratory a f t e r a design by Kepco.
Schematics o f th is
arrangement appear in Figs. 11 and 12 which are taken from the Ph.D.
thesis of R. S. Parker.
S4
The fin e temperature control and measurement of the PV
temperature employs a Model CSC 400 capacitance temperature c o n tro lle r
marketed by Lake Shore Cryotronics.
52
This c o n tro lle r employs
capacitance sensors made up o f glass encapsulated strontium
tita n a te .^ ’ ^
The sensor capacitance varies with the temperature
dependent strontium tita n a te d ie le c tr ic constant.
The useful range
o f th is c o n tro lle r with the present reference capacitor is from 200 K
to I K w ith the exception o f a region around 65 K where the strontium
tita n a te i t s e l f undergoes a phase tra n s itio n and the slope o f the
heater current yjs. temperature curve must be reversed.
(A switch
fo r th is purpose is provided on the re ar o f the chassis.)
An
advantage o f th is c o n tro lle r is th a t the temperature c a lib ra tio n is
not a ffected by magnetic f ie ld s , making i t s u ita b le fo r use in
nuclear magnetic resonance experiments.
In the temperature range
near the KDP tra n s itio n th is instrument has a resolution o f ± 2 mK.
The capacitance sensors were c a lib ra te d in s itu against copperconstantan thermocouples using a Leeds and Northrup K-5 potentiometer
and a d is t ille d water ice bath reference.
An automatic f i l l e r was employed to maintain the liq u id
nitrogen (LN) level in the glass dewar in which the cryo stat was
I kft
Ten
Turn
r
~
U
D C ell
Ice Bath
L
_ T
DC Power
L & N Null
Supply
Detector
0-20 V
0-5 A
^
FIG. T l.
Parker.
54
Heater
Schematic o f coarse temperature c o n tro l system f o r s h ie ld can, a f t e r
9 0 MA OIODES
SOMA OIOOES
« —AW-^-AW-----1, '<V
15 2NI78,
IO A
'2NI7X
DIODES
CONTROLLED
INlOBS
*)— r
o~"T
12.
Voltage controlled power supply, a fte r Parker
54
OUTPUT
\
45
immersed.
The c o n tro lle r was assembled in the Rhysics Department
electro n ics shop a f t e r a design by V. H. Schmidt.
I t employs a
latching re la y configuration c o n tro lled by two zener diodes used as
level sensors.
The zener voltage changes by a few tenths o f a v o lt
depending upon whether the diode is above or below the LN surface;
th is change is am plified by two cascaded tran s is to rs which control
the re la y supplying power to the solenoids c o n tro llin g the a ir
pressure above a re s e rv o ir dewar.
Schematics o f the f i l l i n g system
and the diode f i l l e r control are shown in Figs. 13 and 14.
5.
E le c tric a l Measurement o f Crystal Properties
D ie le c tric properties o f the crys ta l were measured in one of
two configurations:
config u ratio n .
a small signal ac bridge and a dc Sawyer-Tower
57
.
ac Bridge
The small signal ac d ie le c tr ic constant was measured using an
ac Wheatstone bridge which employed a r a tio transformer as two arms
o f the bridge.
A Princeton Applied Research Model HR-8 lo c k -in
a m p lifie r was used as a phase s e n s itiv e null detector.
No phase-
compensating re s is to r was used in the bridge c ir c u it as no phase
d iffe re n c e was measurable between signal points A and B (see Fig. 1 3 ),
and the small voltage perpendicular to the reactance remained
constant throughout the tra n s itio n region.
No phase s h if t was
SAND
TO I S p s i A I R
ZENER
"d i o d e s
VENT
RESERV OI R
FIG. 13.
C ON T R OL L E R
L iq u id n itro g e n f i l l i n g
system.
SAFETY
HOUSE
SOLENOID
2 50 V
250 V
DPST
FULLWAVE
BRIDGE
120 V
2000
2 5 WVDC
Zi= 11.3 V, I W
2200
2200
500 K
500K.
IO ;
TURN
10 TURN , grn
9 .4 V
FIG. 14.
POWER
RELAY
L iq u id n itro g e n le v e l c o n t r o l.
48
expected because of the low conductivity o f KDP and the polystyrene
reference c a p a cito r, and the high Q o f the ra tio tranform er.
As can
be seen from Fig. 15, the value and po sition of the reference capacitor
were chosen so th a t the input impedance o f the HR-8 preamp and stray
capacitance to ground had a n e g lig ib le e ffe c t upon the measured
values.
dc Measurements
The configuration used fo r q u a s i-s ta tic hysteresis loop tra c in g ,
is o p o l, and time constant measurements is shown in Fig. 16.
An
e le c tr ic f ie ld is supplied to the sample using a b attery and voltage
d iv id e r.
capacitor.
The p o la riz a tio n charge is stored on an 8 p f polystyrene
The voltage thus produced is measured by a Cary 401
v ib ra tin g reed electrom eter with an input impedance o f 10
12
ohm.
The recorder output o f the electrom eter was monitored w ith a Fluke
Model 881A d if fe r e n t ia l voltm eter or a Moseley Model 7000AR x-y
recorder.
The recorder output of the electrom eter is also fed to
a voltage fo llo w er and d iv id e r c ir c u it which supplies a guard
voltage equal to the input voltage seen by the electrom eter.
Leads
to the electrom eter input are guarded; sample No. I also included a
guard ring configuration .
In a d d itio n , a ll surfaces which might
provide a charge path between the electrom eter input and the guard
or between guard and ground were scrupulously cleaned with acetone.
SAMPLE - I
UJUUUUUUUUUUUUUUUUUUUU
UUUU
0
jr
ABB
FIG. 15.
RATIO TRANSFORMER
H R - 8 RE FERENCE OUT
POLYSTYRENE C A P A C I T O R BOX
H R - B T Y P E A P R E A MP , A-B MODE
Bridge c i r c u i t fo r ac d i e l e c t r ic constant measurements.
50
ON-OFF
200V
DPDT
” “ REVERSING
TEN TURN
DVM
ELEC
_____
DVM
8
POLYSTYRENE
FIG
16.
Schematic drawing of dc po la riza tion measurement c ir c u it .
51
This included disassembly of BNC connectors and removal o f a ll
remnants of solder flu x and fin g e rp rin ts .
Only Teflon in su la tio n
was used in th is portion of the c ir c u it with the exception of short
(~ 4 i n . ) sections of w ire a t the h ig h -p re s s u re -to -a ir feedthroughs.
Several e p o x y -fille d feedthroughs were constructed (d e ta ils are given
in the Appendix) and only one had s u ffic ie n tly high in su la tio n
resistance (10
ohm) to be s u ita b le fo r use.
Getting such high
resistance appears to be a m atter of good luck, as fa b ric a tio n
techniques did not change from one feedthrough to the next.
The
above precautions were s u ffic ie n t to m aintain leakage currents to
a minimum and on the same order as the "leakage" to the electrom eter
input.
A block diagram of th is arrangement and a schematic fo r the
voltage fo llo w e r appear in Fig. 16.
During hysteresis loop measurements the potentiometer supplying
the bias f ie ld to the crys ta l was varied with a clock d riv e .
A
fra c tio n of the actual bias voltage was fed to the x input of an
x-y recorder w hile the recorder output o f the electrom eter was fed
to the y axis inputs.
During isopol measurements the E f ie ld was changed manually
keeping the electrom eter output, as measured by a Fluke Model 881A
d if fe r e n t ia l vo ltm eter, constant.
During measurements o f the p o la riz a tio n re la x a tio n time the
electrom eter output was once again connected to the y axis of the
52
x-y reco rd er, w hile the x axis was driven lin e a r ly in tim e.
A step
change in the E f ie ld (~ 5 V/cm) was effected manually, and the
re s u ltin g change in p o la riz a tio n measured as a function o f time.
6.
Pressure System
The o v e rall layout o f the pressure generating system is shown
in Fig. 17.
There are two high pressure (HP) o u tle ts :
liq u id pressure medium, the other gas.
one using a
The liq u id system, shown in
the heavier lin e s in Fig. 17, was used to te s t the in te g r ity of
PVs before use in the gas system.
d ir e c tly w ith the hand pump.
The PV may be charged to 20 000 psi
Higher pressures are obtained by closing
the d ire c t lin e from the pump to the PV (V - 8 ) and opening value V -4,
then pumping on the large diameter piston o f the pressure in t e n s if ie r .
The pressure a t the small diameter end is roughly m u ltip lie d by the
r a tio o f large piston area to small piston area.
Two precautions
should be mentioned here: ( I ) Since the area r a tio o f the liq u id
in te n s ifie r is 23:1 and the hand pump could produce 20 000 p s i, the
re s u ltin g pressure on the HP end could be as high as 400 000 psi.
This exceeds the pressure ra tin g o f a ll components including the
in te n s ifie r . (2) Many commonly used hydraulic media s o lid ify a t high
pressure (50 000 psi fo r IOW motor o i l ) .
f ie r s , e tc . occurs in such cases.
Damage to gauges, in te n s i-
One must be sure o f the pressure
medium being used and it s lim ita tio n s .
I t is Strongly advised th a t
M ANGANI N C EL L
VH-I
VP- I
FILTER
[REMOTE1
I HEAD>
CHECK
VALVE
VH-2
VP-2
WORK
POI NT
TRAP
GAS
INTENSI FIER
LI QUI D
INTENSIFIER
LIQUID
WORK
POI NT
FIG. 17.
Pressure generating system.
PUMP
54
no attempt to use any p a rt of the system be made without reading the
Appendix which serves as an operating manual fo r th is system.
In the gas system (lig h t e r lin es in Fig. 17) the gas flows
from rig h t to l e f t s ta rtin g a t a b o ttle pressure of 3500 p s i.
is immediately reduced by a standard regulator to 1450 p s i.
This
The
gas is then passed through an L N -fille d cold trap and a f i l t e r to
the remote head.
The remote head is a diaphragm pump much lik e the
fuel pump in an automobile.
Two check valves on the in le t and
o u tle t re sp ec tiv e ly allow gas to pass only from r ig h t to l e f t .
Gas
is drawn in to and forced out of the head by means o f a diaphragm
driven by an o il lin e from the.hand pump.
The gas pressure downstream
o f the remote head may b& raised to 14 000 psi by th is means.
Further increase in pressure is effected by means of the gas
in te n s ifie r located downstream.
D e ta ils , operating in s tru c tio n s ,
and precautions are found in the Appendix.
The pressure is monitored by measuring the resistance change
o f a c o il o f manganin w ire which has a nominal pressure c o e ffic ie n t
o f resistance of (AR/R)/AP=a=1.67x 10
-7
bar
-I
.
These small resistance
changes are measured by means o f a Cary-Foster type
bridge arrangement, as shown in Fig. 18.
58
Wheatstone
In an actual Cary-Foster
bridge the reference and sample re s is to rs are interchanged during
measurement, allowing the e ffe c t of contact EMF's to be subtracted
out.
In p ra c tic e i t was found th a t the necessary reversing switches
55
REFERENCE
ACTIVE
R=120.51 ft
A+B+r
X =120.36 ft
o
_ -j
a = l . 651x10 ft/ft/psi
S=A- 1 R
r = 5 .526 ft
C=CtXo u P"
C
l+c
R+Xoj
A=B=IOO.00 ft
=xo<% pV >
A=BxlO"^ ft/d i v
6=No. of d iv . on s lid e w ire
£=resistance change
of a c tiv e c o il
P=pressure in psi
NANOVOLT­
METER
FIG. 18.
Cary-Foster bridge fo r pressure measurement.
56
introduced a great deal o f noise in to the null
d e te c to r, and
produced large u n certain ties in contact resistance and thermal EMF1s
due to the use o f d is s im ila r metals.
Thus, i t was decided to dispense
with th a t fa c e t o f the bridge op eration, p re fe rrin g an increased
a b il it y to monitor small changes in pressure over a somewhat
increased absolute accuracy.
The s lid e w ire employed was from a Leeds and Northrup Type K
3
potentiom eter. This 5 ohm, te n -tu rn s lid e w ire had 2x10 divisions
and could be in terp o la ted to give e ffe c tiv e ly IO^ p a rts. Thus the
_O
e ffe c tiv e resistance reso lu tio n was 10" ohm. (A change o f one
d iv is io n added
o th e r.)
5x10
-4
ohm to one side and subtracted i t from the
The manganin c o il used5^ had an ambient pressure resistance
o f 120 ohm, hence the pressure reso lu tio n was AP=(AR/R)/a- 50 psi.
On the other hand, the n u ll detector used was a K eithley Model 148
nanovoltmeter.
Operating on the 0.01 m ill iv o l t f u ll- s c a le range the
peak-to-peak noise was equivalent to approximately I p s i.
Thus the
s t a b ili t y o f the pressure, once s e t, could be monitored to a. much
higher precision than the actual pressure was known.
The bridge c ir c u it was nominally compensated fo r ambient
temperature changes by enclosing a matched reference c o il o f manganin
wire in a cap atop the vessel containing the pressure sensing c o i l .
Nonetheless, a substantial improvement in s t a b ilit y was obtained by
enclosing the e n tire assembly in a styrofoam box f i l l e d with fib erg las
57
in s u la tio n , and thermosta tin g the manganin c e ll housing.
The
temperature c o n tro lle r used was a commercial unit®^ w ith an estimated
s e n s itiv ity o f 0.05 K.
The long term e le c tr ic a l s t a b ili t y o f the
e n tire pressure sensing u n it was monitored a t a known pressure of
one atmosphere and found to be equivalent to ± 0.7 bar over a period
o f three days.
III.
EXPERIMENTAL RESULTS
Section I is a discussion o f the facto rs which led to the choice
of s ta tic d ie le c tr ic measurements as a means to determine the order
o f the KHgPO^ (KDP) phase tr a n s itio n .
Section 2 is an introduction to
the in te rp re ta tio n o f constant p o la riz a tio n , isop ol, data.
In
Sections 3 and 4 actual isopol data is presented fo r ambient and
high pressure re sp e c tiv e ly .
Section 5 contains a discussion of errors
Sections 6 , 7, 8 , and 9 are b r ie f accounts o f isothermal P vs^. E,
time constant, pressure h y s te re s is , and c r it ic a l exponent measurements
I.
D ie le c tric Measurements of KHgPO^
The KDP fe r r o e le c tr ic phase tra n s itio n appears unsuited fo r
analysis by ac d ie le c tr ic measurements.
This is indeed unfortunate
in th a t ac measurements near I kHz are r e la t iv e ly simple and may be
c a rrie d out to high precision .
Moreover, the d is tin c tio n between
f i r s t and second-order behavior o f the d ie le c tr ic constant in zero
bias f ie ld is qu ite pronounced, as can be seen in Fig. 6 .
This
d iffe re n c e is , however, obscured in the case of KDP by the fa c t
th a t the d ie le c tr ic constant in the fe rr o e le c tr ic phase immediately
below the tra n s itio n temperature, Tq , is q u ite high and e s s e n tia lly
constant.
The d i f f i c u lt y can be seen by comparing Fig. 9 with Fig. 6 .
The conventional wisdom is th a t th is behavior is due to domain wall
motion.
B ornarel, Fouskova, Gagon, and Lajzerowicz
61
showed th at
the d ie le c tr ic constant 12 K below Tc could be reduced from
59
4.5x10
4
to 1.4x10
4 ■
by reducing the s ize o f the applied ac f ie ld from
2 V/cm to 0 .5 V/cm.
explanation.
This would seem to support the domain wall
In the course o f th is in ve s tig atio n ac fie ld s as small
as 0.005 V/cm were used 0.05 K below Tc with no important reduction
in d ie le c tr ic constant.
I f domain w all motion is responsible fo r
the high d ie le c tr ic constant in the fe r r o e le c tr ic region, the walls
are very mobile indeed immediately below T .'
In addition the
d ie le c tr ic constant could not be lowered by increasing the frequency
to 10 kHz, above which the measurements become suspect owing to the
d is trib u te d reactance o f the great lengths of cable used to reach
the c rys ta l in the pressure vessel.
One might, nonetheless, hope to determine the order o f the
tra n s itio n by ac measurements in the presence o f bias fie ld s and
confined to the p a ra e le c tric region.
another o f KDP1s p ro p erties:
This hope is thwarted by yet
a large e le c tro c a lo ric e ffe c t.
This
e ffe c t manifests i t s e l f in the case of small signal ac measurements
as an a d ia b a tic c o rrec tio n .
18
The equation o f s ta te
E=Ao(T~To ) P+Bp3+Cp5
(14)
is obtained by d iffe r e n tia tin g H as in Eq. 13 with respect to P at
constant T and X.
One can then c a lc u la te the expressions fo r the
a d iab atic and isothermal d ie le c tr ic constants,
(£T) ' 1= ( 47r)‘ 1 (Ao(T-To)+ 3BP2+ 5CP4 )
(£S) ' 1= ( 4 ^ ) ' 1 (A0 (T-T 0 )+(3B+TA02/C P)P 2+5CP4 ) ,
(15)
60
by taking the d e riv a tiv e of E with respect to P while holding the
temperature and the entropy re sp ectively constant.
The d ifferen ce
stems from the d iffe re n c e in d e riv a tiv e s :
(3E/3P) s=(9E /8P ) t + ( 8P/8T)e .
Jona and Shirane
as (TA
18
(16)
show th a t the extra term on the r ig h t may be w ritte n
This ad iab atic correction is la rg e r than 3 |B| thus
making i t impossible to determine the sign o f B, and hence the order
o f the tr a n s itio n , in a straightforw ard adiabatic d ie le c tr ic
experiment.
Eberhard and H orn^ have attempted to measure the order of the
tra n s itio n in an ac experiment using the thermal hysteresis of the
tr a n s itio n .
Such experiments re s t On the assumption th a t the ra tio
o f thermal hysteresis a c tu a lly observed to th a t possible thermo­
dynamically remains constant in increased bias f ie ld s .
This is a
point not in evidence and, indeed, u n lik e ly , as the height of the
fre e energy b a r r ie r separating stab le and metastable states changes
markedly w ith applied f ie ld and temperature.
Thus isothermal d ie le c tr ic measurements w ith in the p a ra e le c tric
region appeared to the most promising fo r analyzing KDP.
Okada
40
has
determined the order of the KDP tra n s itio n a t ambient pressure by
analyzing hysteresis loops taken a t constant temperature and with
very slow (200V/cm-hr) e le c tr ic f ie ld sweeps.
S im ilar measurements
have been repeated in th is laboratory and are reported in Section 5.
61
They are in general agreement with the constant p o la riz a tio n measure­
ments described next.
However, i t has been found th a t even a t such
slow sweep speeds the re su lts appear ra te dependent.
Attempts to
repeat such measurements a t high pressure were hampered by the need
to p e rio d ic a lly pump up the pressure to compensate fo r f i n i t e leak
ra te s .
Such pumping d is to rts the hysteresis loops.
The dilemma is
one o f e ith e r reducing the value o f the loops by tracing fa s te r ,
or allowing the pressure to change by an amount great enough to
a lt e r T
by more than 2 mK, the experimental temperature reso lutio n.
Owing to the above d i f f i c u lt i e s w ith previously used methods fo r
determining the order o f the KDP tra n s itio n from d ie le c tr ic measure­
ments, i t was decided to take equilib rium measurements of P, E, p,
and T.
The methods o f analysis used are outlined in Section 2, and
data is presented in. Sections 3 and 4.2
2.
Isopols
There is a tr a d itio n in the lit e r a t u r e of measuring p o la riza tio n
as a function of f ie ld
(E) along isotherms, or as a function of
temperature (T) along isochamps.
Most o f the data to be presented
here is displayed as a function o f T and E along lin e s o f constant
p o la riz a tio n , i . e . along isopols.
This approach appears to be new
and thus the follow ing discussion is devoted to the in te rp re ta tio n
o f such p lo ts .
This discussion is based upon the Landau equation
62
o f s ta te (Eq. 14); however, many o f the conclusions based on isopol
plots are independent o f this equation o f s ta te .
The reader is
cautioned not to equate the v a lid it y o f a l l conclusions w ith the
v a lid it y o f the Landau expansion which is used here merely as a
vehicle fo r introducing the isopol p ic tu re .
The equation o f s ta te which follow s from the Landau fre e energy
(Eq. 1 4 ), when re w ritte n , indicates th a t the isopols are s tra ig h t
I
lin es in the T-E plane w ith slopes (AqP)- - and E=O intercepts
T0-(BP 2+CP4 )/A 0 :
T=(A0 P- 1 )E+To-(BP 2+CP4 )/A o.
In the lim it of small P the E=O intercepts tend to Tq.
(17)
If B,
is negative, as P increases the in tercepts ris e above Tq , and then
C
f a l l as P increases fu rth e r and the CPd term begins to dominate.
For
B p o sitiv e the in tercepts simply f a l l fa rth e r and fa rth e r below Tq
a s .P increases.
The case fo r negative B is shown in Fig. 19.
The
i n i t i a l increase and subsequent decrease in the T in tercep ts creates
a region where isopols in te rs e c t.
The f i r s t order lin e , FD in Fig. 19,
lie s w ith in the overlap region which is bounded by caustics ACDB.
BCE is the extension of a lin e s im ila r to BD in the negative E
h a lf-p la n e not shown in Fig. 19.
The c r it ic a l po in t, D, is a t a
vertex of the c u rv ilin e a r tria n g le formed by the caustics o f in te r ­
secting isopols .
Isopols are shown as s o lid lines when they correspond
to an absolute minimum of the fre e energy.
A fte r crossing the f i r s t
63
FIG. 19.
Isopols as predicted from Landau equation of state:
E=/V T- To)P+Bp3+Cp5-
64
order lin e they are shown as dashed lin e s and correspond to metastable
thermodynamic states o f the Landau fre e energy.
I f metastable states are a c tu a lly manifested by.the crystal the
regions o f isopol overlap represent mixed phases.
Denoting regions
o f p o la riz a tio n p a ra lle l to +E as "up", the mixed phase regions may
be characterized as in Table I I I .
I t w ill be seen in data presented
in Section 3, th at when an isopol crosses a lin e such as CE in Fig. 19,
i t changes d ire c tio n , heading almost v e r t ic a lly downward.
in terp re te d as the formation o f domains w ith in the c ry s ta l.
This is
These
are allowed by the Landau equation o f s ta te even though, on the basis
o f the fre e energy alone, such states are not e n e rg e tic a lly favorable.
In the Landau equation o f s ta te the order o f the tra n s itio n is
indicated by the sign o f the c o e ffic ie n t B, being negative fo r f i r s t order and p o s itiv e fo r second-order.
In an isopol p lo t th is d i f ­
ference manifests i t s e l f as a convergence or non-convergence of
isopols , re sp ec tiv e ly .
The deduction o f the order o f the tra n s itio n
from the behavior o f isopols is , however, independent of the Landau
equation, o f s ta te .
This may be seen by considering an isotherm
drawn through an isopol p lo t ju s t above Tcr (see Fig. 19).
I f the
isopols are converging toward a E^O p o in t, then the isotherm w ill
encounter a large change in p o la riz a tio n fo r a small change in f ie ld
near th a t point o f convergence, i . e . the d ie le c tr ic constant w ill
be high.
A d ie le c tr ic constant higher a t E>0 than a t E=O indicates
65
TABLE I I I .
Possible mixed phase regions in Fig. 19.
Phases
corresponding to absolutely stable minima o f the fre e energy are
denoted S, states corresponding to r e la t iv e minima are denoted M.
The p o la riz a tio n is "up" when p a ra lle l to the applied E f ie ld .
Region
Parae le c tr ic
BDG
S
CDG
■ M
BGF
AFCG
ACE
Ferro­
e le c tric
"up"
Ferro­
e le c tric
"down"
M .
S
S.
M
M
■M.
S
M
S
M
66
a c r it ic a l p o in t, and thus a f i r s t order tra n s itio n is im plied.
On
the other hand, i f the isopols do not converge except fo r E=O,
a second-order tra n s itio n is indicated.
The advantage o f displaying data in isopol plots ra th e r than
as maxima in the d ie le c tr ic constant is the numerical convenience
and graphical c la r it y afforded by f i t t i n g s tra ig h t lin e s .
In a d d itio n ,
deviation from the simple Landau p ic tu re becomes obvious when the
actual crys ta l isopols deviate from the high temperature extrapola­
tio n s.
The method fo r e x tra c tin g Landau parameters from isopol data
is as follow s.
A le a s t square s tra ig h t lin e f i t to p a ra e le c tric isopols
is c alcu lated .
As follows from Eq. 14, Aq is obtained from the slopes
o f the isopols using
Ao= O E /9 T )p/P .
(18)
An approximate value o f Tq is found from a p lo t of the extrapolated
T(E=O) in tercepts ys^. P
fo r the three sm allest isopols.
This p lo t
is a s tra ig h t lin e and should e x trap o late to Tq (see Eq. 17 and Fig.
2 6 ).
The parameters B and C are then deduced from the in te rc e p t and
slope re sp ec tiv e ly o f a graph of -A q (T-T0 )ZP
T
v£.
P •
In practice
is then varied a small amount (w ith in the experimental
produce the best s tra ig h t lin e on th is graph.
e rro r) to
In th is la t t e r pro­
cedure, points from the higher p o la riz a tio n isopols are heavily
/
67
weighted owing to the large s c a tte r produced on th is type of graph
by the low p o la riz a tio n isopols fo r which T-T q is q u ite small.
3.
Isopol Data a t Ambient Pressure
Actual data fo r two d iffe r e n t samples o f KDP a t ambient pressure
are shown in Figs. 20 and 21.
Convergence o f isopols from the high
temperature region toward a E^O point is obvious in both cases.
This suggests th a t the tra n s itio n is indeed fir s t - o r d e r a t ambient
pressure.
The s o lid lin e s drawn in Figs. 20 and 21 are the isopols
as predicted by the Landau equation o f s ta te fo r the best f i t
parameters Aq 1Tq , B, and C as given in Table I I .
As can be seen, the
Landau expansion gives a good representation o f the data throughout
the p a ra e le c tric region.
In the fe r r o e le c tr ic region major deviations
occur when the isopols turn nearly v e r t ic a lly downward.
This behavior
is a ttrib u te d to the formation o f domains w ith in the c ry s ta l.
This
b e lie f is supported by the proxim ity of the "bends" to the mixed .
phase boundaries described in Section 2 o f th is Chapter.
The properties o f the crystal as deduced from isopol analysis
are in good agreement w ith properties measured by other techniques
in a number o f la b o rato ries (see Table I I ) .
This gives increased
confidence th a t deductions based upon isopol plots are v a lid , and
strengthens
arguments made a t higher pressures where other data are
as y e t not a v a ila b le fo r comparison.
68
1000
E , V/ c m
FIG. 20.
Isopols of sample No. I a t 0.001 kbar.
1500 2000
69
x I O esu
)
500
750
E , V/ c m
FIG. 21.
Isopols of sample No. 2 a t 0.0016 kbar.
1 0 00
70
4.
Isopol Data a t High Pressure
Data fo r sample No. 2 a t pressures o f 0.0016, 1 .0 0 , and 3.00
kbar are presented in Tables IV -V I and p lo tted in Figs. 21-23.
The
ambient pressure c r it ic a l f ie ld o f 186 ±60 V/cm is reduced to 43+
13 V/cm by I kbar of hydrostatic pressure based on the ca lc u la tio n
o f Ecr from the best f i t Landau parameters.
The Landau equation of
s ta te once again provides a good descriptio n of the data in the
p a ra e le c tric region.
At three kbar the tra n s itio n appears to be second order, th a t
is , the extrapolated isopol in tercepts a ll f a l l below Tq.
The ,
d ie le c tr ic constant has it s maximum value a t E=O.
Fig. 24 shows a graph o f ~A0 (T ~T0 ) / p2 vs... P2 fo r sample No. 2
a t three d iffe r e n t pressures.
The y - in te rc e p t o f these graphs
corresponds to B; the slope is C.
Table V II summarizes the Landau
parameters a t the three pressures as deduced from the data using the
procedure ou tlined a t the end of Section 2 o f th is Chapter.
Table V II and Fig. 24 show th a t B values show a system atic,
although apparently n o n -lin e a r, increase with pressure.
Comparison
of the .1 and 3 kbar B values indicates B=O a t about 2 kbar, while
a lin e a r e x trap o latio n o f the decrease in B from 0 to I kbar indicates
B=O a t about 2.5 kbar.
between.
The actual value probably lie s somewhere in
x IO esu
xIO esu
500
E , V/cm
FIG. 22.
Isopols of sample No. 2 a t 1.00 kbar.
750
IOOO
72
IOOO
E l VZcm
FIG. 23.
IsopoIs of sample No. 2 a t 3.00 kbar.
150 0 2 0 0 0
TABLE IV.
Data fo r Sample No. 2 a t 0.0016 kbar.
Aq ,
p,
Standard
Error
10 ^esu
95%
Confidence
In te rv a l
T- To ’
mK
, Standard
E rror,
1mK
95%
Confidence
■ In terv a l
IO^ esu
N'
■
10""^esu
0.621
7
3 .9 1 -
0.05
0.13
9.8
3.0
7.7
1.194
8
3.90
0.047
0.12
4.9
. 2.8
6.9
2.388
6
4.07
0.06
0.17
27.2
2,8
7.8
3.582
5
4.21
0.029
0.09
47.3
4.776
4
3.95
0.071
0.31
41.9
7.7
22.8
5.97
16
3.52
0.144
0.31
21.0
4.3
9.1
7.167
9
4.09
0.375
0.89
-1 2 .5
6.7
15.8
A
4.01
Zvar=O.13
.19
TABLE V.
Data fo r Sample No. 2 a t 1.00 kbar.
p,
IO^esu
A0 >
IO- ^esu
N
Standard
Error
IO- ^esu
95%
Confidence
In te rv a l
T-T0 ,
. mK
Standard
E rror,
mK
95%
Confidence
In terv a l
Decreasing Temperature
1.00
12
3.62
0.062
0.14
2.7
4.2
9.4
2.00
10
3.61
0.081
0.19
7.6
5.5
12.7
3.00
6
3.81
0.15
0.42
21.4
6.7
18.6
4.00
4
3.80
0.14
0.60
23.2
4.5
19.4
Increasing Temperature
1.00
8
3.62
0.084
0.21
3.1
3.6
8.8
2.00
6
3.56
0.114
0.32
7.6
4.5
12.5
3.00
6
3.74
0.12
0.33
. 20.5
4.5
12.5
4.00
6
3.60
0.05
0.14
15.6
2.0
5.6
5.00
9
3.52
0.064
0.15
-
2.7
2.2
5.2
6.00
8
3.56
0.165
0.16
-4 3 .7
1.8
4.3
7.167
not calculated
-147.5
NA
NA
Aq
=
3.64
assuming
Zvar=O.11
Aq
0.08
TABLE V I.
Data fo r Sample No. 2 a t 3.00 kbar.
p,
A0 -
IO^esu
N
. IO- ^esu
0.50
14
4.23
1.00
15
2.00
Standard
Error
10" 3esu.
95%
Confidence
In te rv a l
T-T0 ,
mK
Standard
Error,
mK ■
95%
Confidence
In terv a l
0.085
0.19
+ 10.6
5.8
12.6
4.13
0.071
0.15
- 2.4
4.8
10.4
16
4.09
0.054
0.12
-1 5 .8
3.8
8.1
3.00
16
4.10
0.054
0.12
-3 0 .0
4.0
8.6
4.00
9
3.90
0.085
0.20
-8 2 .4
5.2
12.3
5.00
5
3.74
0.039
0.12
-169
2.4
7.
6.00
3
3.53
0.106
1.35
-309
8.0
101.
<
I
.
4.04
V var=O .18
0.20
4
v O kbar
A I kbar
o 3 kbar
FIG. 24. - A0 (T-T q )ZP^ v£.
a t 3 pressures.
TABLE V I I .
Pressure*
kbar
0.0016
Landau parameters fo r Sample No. 2 as a function of pressure.
.
Tof
K
d.T0/dp
K/kbar
122. 12+0.1
B
A0
IO- ^esu
IO^l l BSU
C
10' 19esu
Tqr - T0
Ecr
K
V/cm
3 . 93±0.2
t 1 .4 8 ± 0 .I
3 . 1±0.4
0.081+0.03
186±60
3 . 64±0.I
-Q .8 9 ± 0 .I
3 .6±0.4
0 . 027±0.01
43±13
4 . 04±0.2
+ 0.9 0 ± 0 .I
6 . 1±0.4
- 4 . 54±0.05
1. 00+0.002
117.7 2±0.I
3 .00±0 . 002
108.24±0.I
t
*Stable to ±0.0005 kbar
#Absolute c a lib ra tio n not tie d to National Bureau of Standards
+Not measured d i f f e r e n t ia lIy due to scale change
NA Not a p p lic a b le , tra n s itio n is second-order
NA
NA
78
5.
Errors
Standard errors and confidence in te rv a ls fo r the slopes and
intercepts o f isopols were calculated in the usual way.
The Montana
State U n iversity M ath-Stat lib r a r y program MREG®^ was used.
Standard
errors are in general agreement with the claimed temperature resolution
o f ±2 mK combined with pressure induced Tq v a ria tio n o f ±2 mK.
Error bars are d e lib e ra te ly omitted from Fig. 24 in favor of
Fig. 25 which shows the e ffe c t o f varying Tq by ±0.01 K about the
best value.
S im ila r re su lts obtain i f Tq is held fix e d and the isopol
in tercepts varied .
When T-T q is on the order of the standard error
in T the s c a tte r in such a p lo t is large indeed.
This is not to
suggest, however, th a t in te rc e p t data from the lowest p o la riz a tio n
isopols is not s ig n ific a n t in in d ic a tin g the tra n s itio n order.
instead, one chooses to graph T(E=O) vs. P
If,
fo r the lowest p o la ri­
zation isopols, as in Fig. 26, one finds a d iffe re n c e in the sign
o f the slopes between the f ir s t -o r d e r (0 and I kbar) and the secondorder (3 kbar) data which is unequivocal even i f 95% confidence
in te rv a ls are used as e rro r bars.
Because o f the large uncertainty in the T-T q values fo r the low
p o la riz a tio n isopols only the high p o la riz a tio n isopols were used
in the le a s t squares f i t s from which best values of the B and C
parameters were deduced.
Then Tq was varied s lig h tly to bring the
low p o la riz a tio n in tercepts in to agreement.
The fin a l ju s t if ic a tio n
■>sj
kO
FIG. 25.
E ffe c t of the v a ria tio n of Tq by ± 0 . 0 1
k on Fig. 24.
80
122.1
O k bar
Ikbar
I 17. 6
0.1 K
3 kbar
108.3
6 10
O e s u
FIG. 26. T(E=O) intercepts of low p o la riz a tio n isopols, described
2
by T(E=0)=To+(B/Ao )P . R elative displacements on v e rtic a l scale
a r b itr a ry .
Slope change indicates change in the sign of B.
8.1
fo r th is process is the good agreement with the o rig in a l p o la riza tio n
data which th is procedure produced.
6.
Isothermal P vs E
In addition to the above isopol p lo ts , isothermal measurements
of the p o la riz a tio n as a function o f slowly varying E f ie ld were made
a t 0 .5 and 3 kbar.
Double hysteresis loops s im ila r to those reported
fo r KDP a t ambient pressure were produced a t 0.5 kbar.
These have
CO
been in terp re te d as in d ic a tin g a f ir s t - o r d e r tra n s itio n .
In
a d d itio n , the P-E curve s ta rtin g from P=0, E=O is a graph o f the
equation o f s ta te .
as Okada
G la d k ii.
40
35
From th is curve values o f B and C may be deduced
d id , follow ing a method due o r ig in a lly to Sidnenko and
This technique was used and the re s u ltin g Landau parameters
found to be in reasonable agreement w ith those deduced from isopol
p lo ts .
However, these values appeared to be sweep ra te dependent even
a t E f ie ld sweep rates as slow as 240 V/cm-hr.
For th is reason a
great deal o f confidence is not placed in the exact value o f the
parameters so deduced.
Nonetheless, the general shape o f these
curves is highly s ig n ific a n t.
I f on a P y£. E graph the actual
curve rises above the e xtrap o latio n o f the s tra ig h t lin e portion
o f the curve near the o r ig in , the d ie le c t r ic constant is necessarily
greater during th a t r is e in d ic a tin g a E^O c r it ic a l p o in t.
I f the
curve f a l l s only below the extrapolated s tra ig h t lin e no such point
82
is in dicated.
Fig. 27 shows ju s t th is d iffe re n c e between the 0.5
and 3 kbar P-E traces.
7.
Time Constants
Time constants of the p o la riz a tio n response were measured a t
ambient pressure.
The hope was th a t these could be used to id e n tify
the c r it ic a l po int and then compared to s im ila r measurements a t
high pressure.
Measurements were made by perturbing an equilibrium
T, E, P configuration by manually increasing the f ie ld in a small
step (AE~5 V/cm) and observing the retu rn to equilibrium as a function
o f time ( t ) .
The 1/e time constants were then deduced from a p lo t
o f log P y s / t .
generally
The values found are shown in Fig. 28.
4x longer than those found by Okada.
39
They are
I t is noted th at
lin e s of constant time constant are very nearly isopols in the
p a ra e le c tric region.
Measurements o f time constants in the immediate
neighborhood o f the c r it ic a l point were d i f f i c u l t to obtain owing to
th e ir increased length.
On one occasion (marked °° in Fig. 28) the
change in p o la riz a tio n produced by AE was la rg e r than the i n i t i a l
p o la riz a tio n and showed no sign of h a ltin g it s upward creep in over
a h a lf hour.
At elevated pressures time constant data was not taken owing to
the lack o f complete equilibrium caused by f i n i t e leak rates and
perturbations caused by in te rm itte n t pumping.
83
IO esu
T - T 0 = 0 .07 K
d E / d t = 6 0 0 V/cm-hr
E , V/cm
10
T - T 0 = 0 . 11 K
d E / d t = 2 3 4 V/cm-hr
esu
E t VZcm
T-T0 = 0.12 K
d E / d t = 2 2 3 V/ cm-hr
r> 0
ONLY
E t VZcm
FIG. 27.
Isothermal P VSl . E plots a t 0.5 and 3 kbar.
r=E-Ao (T-To)P=BP3+CP5.
Here
Spikes a t 3 kbar are due to pressure pumps.
84
59
/
58
'o 2
/
105 sec
> 39 / 1 0 8 /
/ 4 9 / 8> 162
87 sec
500
750
1000
E,V/cm
FIG. 28.
Time constants fo r p o la riz a tio n re la xa tio n a t O kbar.
The + marks the c r it ic a l point as calculated from isopol data.
85
8.
Pressure Hysteresis
In addition i t should be noted th a t a pressure hysteresis in
the value of Tq , and in some cases Aq , has been noticed.
In order
to m aintain the pressure o f the system constant to w ith in ±0.25 bar,
i t was necessary to pump the system a t in te rv a ls ranging from 30 min
a t 3 kbar to 2 hrs a t I kbar.
I f a set of isopols was begun one
day, and the system l e f t overnight without pumping, upon returning to
the standard pressure the crystal would almost in v a ria b ly show a s h if t
in Tq as deduced by a systematic s h if t in the isopols.
Unless
otherwise noted a ll o f the data presented re fe rs to sin g le runs
la s tin g around the clock, often fo r several days.
In the case o f the I kbar data two sets of isopols taken a fu ll,
week apart show id e n tic a l (to w ith in experimental e rro r) isopol
stru ctu re except fo r Tq even though one was taken as temperature was
increased and the other as temperature decreased.
On the other hand,
the three kbar data showed a s h if t in both Tq and Aq when l e f t
unpumped only 8 hours.
This type o f hysteresis makes i t d i f f i c u l t
to determine the parameters of the crys ta l unambiguously as a function
o f pressure.
86
9.
.
C r itic a l Exponents
The only c r it ic a l exponent d ir e c tly accessible to measurements
made here is Y, the exponent fo r the d ie le c tr ic s u s c e p tib ility :
X ~ ((T -T c) /T cr Y=e-y .
(19)
Since x=dP/dE one may w rite fo r small AR and AE th a t ae=APe ~y .
Thus
i t can be seen th a t the mean f ie ld value o f I fo r y follows from
s tra ig h t lin e isopols.
At 3 kbar the P=O.SxlO3 esu isopol indicates
Y=-d(log E )/d (lo g e )= 1 .0 3 ±0 .0 4 to w ith in 0.02 K o f Tf,.
in te rv a l is the 95% confidence in te r v a l.
The e rro r
In d ir e c tly , mean f ie ld
values fo r other exponents in the p a ra e le c tric region are implied by
the fa c t th a t the data can successfully be f i t by the mean f ie ld
Landau expansion.
Data in the fe r r o e le c tr ic region does not obey a simple equation
of s ta te , n e ith e r does i t scale w ith mean f ie ld exponents.
This
behavior is a ttrib u te d to the formation o f domains w ith in the crystal
and one's subsequent in a b ilit y to measure the true intradomain
p o la riz a tio n , ra th e r than.any breakdown o f mean f ie ld theory or
scaling laws.
IV .
I.
DENOUEMENT
Summary
This in v e s tig a tio n began with the conjecture by V. H. Schmidt16
th a t a t r i c r it ic a l point might be produced in the phase diagram of
KHgPO^ (KDP) by the ap p lic a tio n o f hydrostatic pressure.
I f this
were the case, KDP's p a ra e le c tric to fe rr o e le c tr ic tra n s itio n would
a lt e r from fir s t - o r d e r a t ambient pressure to second-order a t high
pressure, the point of cross-over being the t r i c r it ic a l point (TCP).
I f there were a TCP the two c r it ic a l points a t the term inations of
the f ir s t - o r d e r lin e in the temperature (T) and e le c tr ic f ie ld (E)
plane would merge a t E=0.
At the time of th is conjecture, however,
there was some doubt (in the mind of a t le a s t one review er) th a t the
tra n s itio n was in fa c t f ir s t - o r d e r a t ambient pressure.
Even amongst
those experiments which supported a f ir s t - o r d e r tra n s itio n a t ambient
pressure, there was disagreement as to the coordinates o f the c r it ic a l
points.
Hence the purpose o f the experiments described in th is thesis
was to ( I ) resolve the controversy regarding the coordinates of the
c r it ic a l point a t zero pressure, (2 ) to monitor those coordinates
with increasing pressure, and (3) determine i f the tra n s itio n becomes
second-order a t high pressure, thus in d ic a tin g the existence of a
t r i c r it ic a l point in the phase diagram o f KDP.
A review o f recent high precision experiments indicated th a t
even a "simple" d ie le c tr ic experiment which could meet the above goals
88
would require temperature resolution on the order o f ±2 mK, pressure
s t a b ilit y o f ±10 ppm a t pressures o f several kbar, and high impedance
s ta tic p o la riz a tio n measurements.
In a d d itio n , measurements would
have to be made as near to equilibrium as possible owing to the
d is to rtio n of resu lts when variables are allowed to d r i f t "quickly"
hear the tr a n s itio n ; th is re s u lt has long been known to workers in
c r tic ia l. phenomena but has not generally been appreciated by workers
in f e r r o e le c t r ic it y .
Apparatus was assembled to meet the above requirements.
The
major components were a two stage cryo stat fo r temperature c o n tro l,
guarded c ir c u it r y fo r p o la riz a tio n charge measurements, a
beryllium -copper pressure vessel, and a pressure generating system
fo r a p p lic a tio n o f hydrostatic pressure using He gas.
The temperature and e le c tr ic f ie ld dependence o f the net
p o la riz a tio n o f a KH2PO4 crystal was mapped out in the E>0 h a lf
plane in a 0 .5 K neighborhood of its fe rr o e le c tr ic tra n s itio n at
pressures of 0.0016, I , and 3 kbar.
The data were analyzed by the
apparently new technique o f considering the T and E dependence along
lines o f constant p o la riz a tio n , isopols.
2.
Conclusions
On the basis o f the s ta tic measurements of the net p o la riz a tio n
o f KDP as a function o f temperature and e le c tric f ie ld in a 0.5 K
89
neighborhood o f it s fe r r o e le c tr ic tra n s itio n a t pressures o f 0.0016,
1 .0 0 , and 3.00 kbar, the follow ing conclusions appear v a lid .
1.
In the p a ra e le c tric region the p o la riz a tio n is w ell described
by the Landau equation of s ta te
E=V T_To} p+Bp3+Cp5
to w ith in 0.05 K o f the tra n s itio n temperature a t hydrostatic
pressures of 0.0016, 1 .0 0 , and 3.00 kbar.
2.
Based on the best f i t Landau parameters to the p o la riza tio n
data on two c ry s ta ls , the fe r r o e le c tr ic tra n s itio n is f ir s t-o r d e r
a t ambient pressure with the c r it ic a l point a t the term ination of
the f i r s t order lin e located a t T -T =0.08+0.03 K and E =200+60 V/cm.
cr o
cr
This conclusion is supported by in creases.in the isothermal d ie le c tr ic
constant and the p o la riz a tio n re la x a tio n time constant in the v ic in ity
o f the c r it ic a l po int.
3.
At 1.00 kbar Tq o f sample No. 2 changed by -4.5,4+0.05 K,
and the c r it ic a l point moved s u b s ta n tia lly closer to the temperature
a xis:
Tcye-T 0 = 0.027 ± 0,01 K, Ec r=43±13 V/cm.
4.
At 3,00 kbar the tra n s itio n is second-order as based upon
the analysis o f isopols which indicates a p o sitiv e B in the Landau
expansion, and supported by the isothermal d ie le c tr ic constant
maximum.occurring a t E=0.
*
90
5.
A t r i c r it ic a l point is expected to e x is t a t 2 .0±0.5 kbar
based upon the in terp o lated behavior o f the best f i t Landau parameters
as a function o f pressure.
3.
S ig nificance and Recommendations
fo r Further Study
The re su lts presented in th is thesis have s ig n ifican ce beyond
merely mapping out the phase space o f KDP.
I t is believed th a t th is
is the f i r s t m aterial w ith an indicated t r i c r it ic a l point (TCP) whose
e n tire three-dimensional phase space is experim entally accessible.
This should allow c r it ic a l exponents fo r wing c r it ic a l points to
soon be measured fo r the f i r s t time.
Other members o f the KDP fam ily whose tra n s itio n s are barely
f i r s t order a t ambient pressure (e .g . Cesium D ihydrogen Arsenate)
may have TCP's thus allow ing the e f f e c t , i f any, o f chemical substi­
tu tio n to be studied.
The present re s u lt combined with Peercy1s^
recent discovery of a TCP in SbSI, may encourage a search fo r
other fe rro e le c tric s with TCP's.
Results from a number o f such
fe rro e le c tric s would allow fu rth e r tests of u n iv e rs a lity .
The fa c t th a t the isopol analysis outlined in Chapter I I I
has produced re su lts in good agreement with other methods is
s ig n ific a n t.
D ie le c tric measurements are in general experim entally
simpler a t high pressure than neutron or x-ra y d if fr a c t io n , or lig h t
s c atte rin g techniques.
Thus a convenient means is afforded to check
91
the order o f phase tra n s itio n s which are b o rd e r-lin e f i r s t or secondorder.
As fo r KDP i t s e l f the findings reported here in d ic a te th a t the
tra n s itio n region around 2 kbar should be thoroughly explored.
Taylor
expansions of microscopic descriptions of KDP produce a "B" c o e ffic ie n t
which is the sum o f a number o f competing plus and minus term s.^
Careful measurement of the pressure dependence of microscopic
parameters near the tra n s itio n temperature would aid in the develop­
ment of a microscopic p ictu re o f the TCP.
I t is f e l t th a t measurements of the noise voltage and current of
KDP near the tra n s itio n might provide an e x ce lle n t means o f id e n tify in g
both CP's and TCP's.
In a d d itio n , a u to -c o rre la tio n o f the noise
voltage should provide a means fo r d ire c t examination o f the c o rre la tio n
time in various regions o f the phase space.
I t is c le a r th a t the fin a l chapter of th is in v e s tig a tio n has
not y e t been w ritte n .
APPENDIX .
Pressure System Manual
This Appendix is not intended as a complete service manual fo r
the e n tire pressure system.
D etailed descriptions o f each component,
it s operation, and re p a ir are given in separate manuals published by
the manufacturer.
I t is suggested th a t these be consulted before
operating any component for. the f i r s t time.
What th is Appendix does contain is an overview of the e n tire
system operation, valve open-closed conditions fo r various modes of
operation, general precautions, and other items p e c u lia r to th is
system which cannot be found in the manufacturers' lit e r a t u r e . .
The system consists o f two major subsystems:
one employing a
liq u id hydraulic medium a t the work p o in t, the other supplying a
gas medium.
The two subsystems are described separately.
Liquid System
The liq u id system is shown in the darker lin e s of Fig. 17.
The
hydraulic f lu id used to 30 000 psi may be as simple as SAE IOW HD
motor o i l .
Higher pressure, up to 100 000 p s i, requires a mixture of.
I part kerosene and 2 parts SAE 10W o i l .
s e ll diesel oil. as kerosene.
(Caution:
Kerosene is c o lo rle s s .)
many o u tlets now
Pure 10W o il
s o lid ifie s a t ~50 000 psi causing considerable system damage.
o f the pressure medium.
Be sure
94
The system should be charged with hydraulic f lu i d , taking care
to remove as much a i r as possible from the lin e s .
This may be done
by cycling the in t e n s if ie r several times u n til no a ir appears a t the
discharge lin e or the work po int.
Procedure A:
1)
The procedure is as follow s.
Removal o f a ir from liq u id system.
Close valves V -3, V-4, V -5, and V -7, open V-8 and V -9, and pump
with the work point open u n til no a ir bubbles appear in the discharge
lin e .
2)
Close V -9, open V-5 and continue pumping.
in te n s ifie r .
This re tra c ts the
A very rapid permanent pressure increase a t gauge G -I
w ill occur when the in te n s ifie r is f u l ly re tra c te d .
3)
When the in t e n s if ie r is f u l ly re tra c te d , close V-5 and V -8, open
V-9 and V -4, and pump to advance the in t e n s if ie r .
Again pressure
w ill ris e dram atically a t the end o f the stroke.
4)
Repeat the procedure from step I u n til no a ir appears a t the
work point or the discharge lin e a t any point of the operation.
Procedure B:
Pressure production w ith the liq u id system.
1)
Close valves V-3 and V-6.
2)
Remove a ir from system follow ing procedure A.
3)
R etract the in t e n s if ie r :
close V -4, V -9, open V-5 and V -8, and
pump u n til a sharp permanent pressure r is e occurs a t G - I.
4)
Close V-4, open V-8 and V -9, and pump to a maximum o f 15 000 psi
a t G-I .
95
5)
I f higher, pressure is desired, close V -8, and, with V -5 open,
open V-4.
a t G-2.
Now close V -5 and pump to a maximum pressure o f 100 000 psi
N.B.
The pressures on G -I and G-2 should be in the approxi­
mate r a tio o f the in t e n s if ie r piston areas, 1:23.
deviates appreciably (say 15%), stop:
I f th is r a tio
something is wrong.
6)
To re lie v e the pressure crack V-5 open.
7)
I f necessary r e tr a c t the in t e n s if ie r as per step 3.
Gas System
The gas system is comprised o f the lig h te r lin es in Fig. 17.
Liquid lin e s through valve V-3, V -6, and V-7 are also used.
Before
pressurizing the gas lin e , one should check th a t the liq u id pressure
system has been bled (see procedure A o f th is Appendix) and th a t the
remote head has been primed (see the manufacturer's manual).
The
follow ing procedure may then be used to obtain gas pressures to
100 000 psi.
Procedure C:
Gas p re s su riza tio n .
1)
F i l l the liq u id nitrogen trap with liq u id nitrogen.
2)
Close V -2, V -4, V -5, V -6, V -7, V -8, V -9, VH-2, VP-2, and open
VP-1, VH-I , V -I and V-3.
3)
Close the small unlabeled through-valve immediately downstream
o f the reg u lato r on the He b o ttle , open the main b o ttle va lv e , and
set the reg u lato r to 1450 p si.
96
4)
Crack the small through-valve on the reg u lato r open, thereby
adm itting He to the system.
b o il.
5)
The liq u id nitrogen in the trap w ill
When the b o ilin g stops, open the small valve several turns.
The system is now pressurized to approximately 1450 p s i.
At
th is point the gas in t e n s if ie r can be re tra c ted i f i t is not already
in it s re tra c ted p o sitio n .
A meter showing the position o f the
in te n s ifie r piston in it s stroke is located on the fro n t o f the
pressure panel.
R etraction is accomplished by cracking V -7 , then
w aiting fo r the piston motion to stop.
Valves V-7 and V-5 are then
opened f u l ly and V-4 cracked, and the in te n s ifie r allowed to r e tra c t
f u lly .
The in t e n s if ie r is f u lly re tra c te d when G -I reads zero and
no o il flows in the return lin e .
The object here is to prevent the
in te n s ifie r from moving too ra p id ly .
The speed is about rig h t when
a slow but continuous stream o f o il flows in the return lin e to the
re s e rv o ir.
6)
Valves V -4, V-5 and V-7 should then be closed.
The system downstream o f the remote head may now be pressurized
to 14 000 psi by pumping.
Be sure V-6 and V-7 are closed, and V-3
open.
7)
I f higher pressure is desired, advance the gas in t e n s if ie r by
closing V -3, opening V-7 and pumping.
8)
I f pressure higher than th a t a ttain e d by a sing le stroke o f the
in te n s ifie r is desired, the in te n s ifie r must be recycled.
note the pressure on G-I . .
C arefully
97
9)
Close VP-I and re tra c t, the in t e n s if ie r as in step 5.
10) Repressurize the in te n s ifie r volume with the remote head as in
step 6.
11) Close V -3, open and close V -6, open V -7, and pump to advance the
in te n s ifie r u n til the pressure reading on G -I matches the value noted
in step 8.
Open VP-I and continue pumping to increase the. pressure
a t the work po int.
12) Repeat steps 8-11 as needed to reach the desired pressure.
The equipment upstream o f VH-I should not be l e f t pressurized
fo r great lengths of tim e.
High pressure gas leaks back through
the check valves and the low pressure end o f the system can reach
pressures great enough to blow the p ro te c tiv e rupture disks.
The
manufacturer also warns against leaving the remote head under pressure.
The procedure fo r venting the low pressure lin e follow s.
Procedure D:
Venting the low pressure lin e .
1)
Close the main valve on the supply tank.
2)
Crack VH-2, venting the system up to check valve I .
3)
Close VH-I and complete venting by opening V-2 u n til the gas
b o ttle reg u lato r reads less than 50 p s i.
4)
Pressure on the high pressure side of VH-I may be maintained by
periodic pumping on the in te n s ifie r through V-7.
I t has been found
th a t hanging 5 to 20 kg on the end o f the pump handle is an aid to
98
making small pressure corrections when some delicacy but considerable
force is needed.
High Pressure E le c tric a l Feedthroughs
E le c tric a l feedthroughs were fa b ric a ted from Harwood 3-M stainless
steel high-pressure tubing.
The tubing was coned and threaded on one
end, cut square on the o th e r, and etched inside by heating fo r 20-30
minutes in a Kel-F beaker containing 25 ml tap w ater, 10 ml concen­
tra te d HNOg, and 15 ml concentrated HF.
The acid solution was
p e rio d ic a lly c irc u la te d through the tubing with an eye dropper.
The center conductor was 27 gauge enameled magnet w ire .
Eccobond 104
epoxy to which 5 percent by weight alumina powder had been added
was thoroughly mixed and then out-gassed by pumping on the mixture
w ith a mechanical fore-pump fo r 15 to 20 min.
A Tygon tube was then
f i l l e d w ith heated epoxy and attached to the pressure tube by a hose
clamp.
The epoxy was then squeezed in to the s ta in le s s -s te e l tube
containing the center w ire .
A fte r the epoxy s e t, standard m ale-to-
cable BNC f it t in g s were soldered to the pressure tubing with copper
spacers employed to match the pressure tubing OD to the BNC ID.
Later
experience showed i t to be considerably easier to solder the spacers
to the tubing before f i l l i n g the tubing with epoxy.
Connection to
other male-ended cables was made using a female-female adapter which
was never removed from the feedthrough side in order to prevent
99
s tra in in g the somewhat d e lic a te center pin connections on the
feedthrough BNCs.
REFERENCES
REFERENCES
1
H. Eugene Stanley, Introduction to Phase Transitions and
C r itic a l Phenomena (Oxford U n iversity Press, New York, 1971), Parts
I and I I ,
2
H. B. C allen, Thermodynamics (W iley, New York, 1960), Part I .
3C. K it t e l, Thermal Physics (W iley, New York, 1969), Chap. 23.
4
H. A. Leopold, Am. J. Phys. 37, 1047 (1 9 69 ), Erratum:
J. Phys. 39, 1094 (1971).
Am.
.
5H.' A. Leopold, Am. J. Phys. 39, 1099 (1971). .
5R. B. G r if f it h s , Phys. Rev. Letters 24, 715 (1970).
7R. B. G r if f it h s , Phys. Rev. B 1_, 545 (1973).
8F. J. Wegner and E. K. R iedel, Phys. Rev. B _7, 248 (1973).
9L. S. Schulman, Phys. Rev. B _7, 1960 (1973).
10E. K. Riedel and F. J. Wegner, Phys. Rev. Letters 29^, 349 (1972)
^ G . Ahlers in The Physics o f Liquid and Solid Helium, K. H.
Benneman and J. B. K etterson, Eds. (W iley, New York, 1976), V ol. I .
^ N . Giordano and W. P. W olf, Phys. Rev. Letters 35, 799 (1975).
13
R. J. Birgeneau, G. Shirane, M. Blume, and W. C. Koehler,
Phys. Rev. Letters 33^, 1098 (1974).
■ 14
15
'
B. B. Weiner and C. W. Garland, J. Chem. Phys. .56, 155 (1972).
P. S. Peercy, Phys. Rev. Letters _35, 1581 (1975).
16V. H. Schmidt, B u ll. Am. Phys. Soc. 19, 649 (1974).
17G. Busch and P. S cherrer, Naturwiss. 23^, 737 (1935).
18
F. Jona and G. Shirane, F e rro e le c tric Crystals ( Pergamon,
Oxford, 1962), Chap. I I I .
102
19
V. H. Schmidt, " F e rro e le c tric Hydrogen Bonded Systems",in The
Hydrogen Bond: Recent Developments in Theory and Experiments, P.
Schuster, G. Zundel, and C. Sandorfy, Eds. (N orth-H oiland, Amsterdam,
1976) VoL I I I , Chap. 23.
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