' THE THERMOELECTRIC POWER OF
COPPER. SILVER AND GOLD AFTER
COLDWORKING
/PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE T E C H N I S C H E HOGESCHOOL TE D E L F T , OP GEZAG VAN DE R E C T O R MAGNIFICUS D R O. BOTTEMA, HOOGLERAAR IN DE AFDELING DER ALGEMENE WETEN-SCHAPPEN, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG
15 MEI 1957 DES NAMIDDAGS T E 4 UUR
DOOR
DIEDERIK JAN VAN OOIJEN
NATUURKUNDIG INGENIEUR GEBOREN TE MIDDELBURG
Dit proefschrift is goedgekeurd door de promotor Prof. Dr. M. J. Druyvesteyn
"Metalen F.O.M.-T.N.0." of the "Stichting voor Fundamenteel On-derzoek der Materie" and was also made possible by financial sup-port from the "Nederlandse Organisatie voor Zuiver Wetenschappe-lijk Onderzoek".
5
C O N T E N T S
Int roduc tion 7
Chapter I. Coldwork and lattice defects 9
Chapter II. The thermoelectric effects 12
§ 1. Summary 12 § 2. a. Definition of the seebeck effect 12
b. Definition of the Thomson coefficient 13 c. Definition of the absolute
thermoelec-tric power 14 § 3. The theoretical expressions for the
abso-lute thermoelectric power 16 § 4i Comparison of theory with the
experimen-tal values for some meexperimen-tals of the Thomson coefficient and the absolute
thermoelec-tric power as taken from literature 19
Chapter III. The change of the absolute thermoelectric
pow-er by lattice defects 23
§ 1. Summary 23 g 2. The theoretical expression for the change
of the absolute thermoelectric power
ac-cording to Kohier and sondheimer 24 § 3. The theoretical expression for the change
of the absolute thermoelectric power
ac-cording to Friedel 29
Chapter IV. Description of the apparatus and the
experi-mental methods 41
§ 1. Methods for coldworklng. The rolling
ap-paratus 41 § 2. The apparatus for the coldworked
thermo-couples 42 § 3. The potentiometer for the thermoelectric
force of the coldworked thermocouples 46 § 4. The method for measuring the
thermoelec-tric force 48 § 5. The method for measuring the electrical
resistivity 50 § 6. The cryostate for recovery treatments 53
Chapter V. Description of the experimental results 56
§ 1. The annealing treatment before
coldwork-lng and the purity of the material used 56 § 2. The thermoelectric power and the
electri-cal resistivity of cu, Ag and Au after
drawing at roomtemperature 57 § 3. The influence of the deformation
tempera-ture on the thermoelectric power and the
electrical resistivity of rolled Cu 62 g 4. The recovery of the change of the
abso-lute thermoelectric power and the recov-ery of the increase of the electrical re-sistivity of Cu, Ag and Au, after rolling
in liquid air 73 § 5. The electrical resistivity of Cu, Ag and
and Au-cu thermocouples in the uDd«formed
s t a t e 81 Chapter VI. Analysis of the experimental results 86
I 1. summary 86 8 2. Analysis of the r e s u l t s of Chapter V, 8 4 86
I 3. A n a l y s i s of the r e s u l t s o f Chapter V, § 2
for the higher temperatures 92 g 4. A n a l y s i s of the r e s u l t s of Chapter V, g 2
for the lower temperatures 102 § 5. A n a l y s i s of the r e s u l t s of Chapter V, § 3 110
§ 6. Some c o n c l u s i o n s from the r e s u l t s of the
preceding paragraphs 112
Summary 115 Samenvatt ing 119 List of symbols 123 References 125
7
I N T R O D U C T I O N
The measurements given by l i t e r a t u r e on the change of t h e absolute thermoelectric power of metals due t o coldwork are i n -cidental and not systematic ' Z ' ^ . Coldworking was mainly applied at roomtemperature and the change of the absolute t h e r m o e l e c t r i c power was in most cases measured over a limited temperature range only. Sometimes the recovery due t o r e c r y s t a l l i z a t i o n of t h e coldworked metal was measured. In most cases the accompanying increase of the e l e c t r i c a l r e s i s t i v i t y was not measured in the same experiment, and the degree of deformation was given only. The t h e o r e t i c a l work on the change of the absolute thermoelectric power due t o coldwork i s only t e n t a t i v e or r e s t r i c t e d t o a spe-c i a l simplified spe-case ' ^ Z " .
In the past few years much work has been done on the influence of coldwork on the e l e c t r i c a l r e s i s t i v i t y 20/22 Measurements on the recovery of the increase of the e l e c t r i c a l r e s i s t i v i t y a f t e r coldwork have shown t h a t t h i s recovery t a k e s p l a c e in s e v e r a l separate s t e p s . I t i s now generally assumed that each s t e p has to be a s s o c i a t e d with one or more types of l a t t i c e d e f e c t s , which. were introduced in the l a t t i c e during the coldworking.
The b e t t e r i n s i g h t in t h e phenomena o c c u r r i n g i f a metal i s coldworked, which i s based on the experiments on the e l e c t r i c a l r e s i s t i v i t y , s t i m u l a t e d t o carry out s y s t e m a t i c experiments on the change of the absolute thermoelectric power due to coldwork, completed by simultaneous measurements on the i n c r e a s e of the e l e c t r i c a l r e s i s t i v i t y .
In t h i s t h e s i s two kinds of experiments will be given. In the f i r s t place experiments on the simultaneous recovery of the change of the absolute t h e r m o e l e c t r i c power and of the i n c r e a s e of the e l e c t r i c a l r e s i s t i v i t y a f t e r coldworking in l i q u i d a i r , will be given. These experiments are c a r r i e d out in order t o in-v e s t i g a t e whether t h i s recoin-very of the change of the a b s o l u t e thermoelectric power can be devided a l s o into d e f i n i t e p a r t s , as was known t o occur for the i n c r e a s e of the e l e c t r i c a l r e s i s t i v -i t y .
In the second place the change of the absolute thermoelectric power a f t e r deformation a t roomtemperature i s measured. The meas-urements are extended over t h e t e m p e r a t u r e range from l i q u i d helium temperature up t o about roomtemperature, as c o n t r a r y to
t h e i n c r e a s e of t h e e l e c t r i c a l r e s i s t i v i t y , which i s measured a l s o , t h e change of the a b s o l u t e t h e r m o e l e c t r i c power depends markedly on temperature.
According t o current theory, the absolute thermoelectric power of a metal i s connected with the e l e c t r i c a l r e s i s t i v i t y . So the analysis of the experiments on the change of the absolute thermo-e l thermo-e c t r i c powthermo-er w i l l bthermo-e basthermo-ed on ththermo-e mthermo-easurthermo-ed i n c r thermo-e a s thermo-e of ththermo-e e l e c t r i c a l r e s i s t i v i t y .
Prom the experimental d a t a , a v a i l a b l e in l i t e r a t u r e , i t i s known t h a t c u r r e n t theory i s i n s u f f i c i e n t t o account for the ab-s o l u t e t h e r m o e l e c t r i c power of pure metalab-s below the Debye tem-perature. In some papers 23/25 n jg suggested t h a t the anomalies
far below the Debye temperature have to be ascribed to a physical e f f e c t , which was h i t h e r t o not considered by c u r r e n t theory. In our a n a l y s i s we w i l l i n v e s t i g a t e whether the measured thermo-e l thermo-e c t r i c thermo-effthermo-ect can bthermo-e dthermo-evidthermo-ed i n t o a p a r t t h a t i s accountthermo-ed for by current theory, and i n t o a p a r t t h a t has to be connected with the anomalies which are known to occur from l i t e r a t u r e .
9
C h a p t e r I
C O L D W O R K A N D L A T T I C E D E F E C T S
By coldworking a pure metal, the e l e c t r i c a l r e s i s t i v i t y i s increased by an amount p j . This i n c r e a s e decreases again p a r t l y or in the whole, as a function of time, over a wide temperature range. The decrease i s c a l l e d recovery. The temperature of de-formation may be included in the recovery range. The increase of the e l e c t r i c a l r e s i s t i v i t y i s experimentally to a high approxima-t i o n independenapproxima-t of approxima-the approxima-temperaapproxima-ture of measuremenapproxima-t (Maapproxima-tapproxima-thiessen's r u l e ) , provided t h a t no recovery occurs. The value of p . depends for a given metal on the degree of deformation and on the temper-ature at which the deformation was applied.
Experiments on the recovery as a function of time and temper-ature have shown t h a t the recovery phenomena can be described by d i f f u s i o n p r o c e s s e s , having d i f f e r e n t a c t i v a t i o n e n e r g i e s of migration. In t h i s way the temperature range over which measure-ment of the recovery can be r e a l i s e d , can be devided into more or l e s s separated r e g i o n s . Over each region the a c t i v a t i o n energy has a constant value and with each region a d e f i n i t e decrease of pj i s connected; t h i s decrease i s called a recovery s t e p .
The experimental values of the a c t i v a t i o n energies mentioned above suggested to connect the recovery s t e p s with the a n n i h i l a -tion or recombina-tion of the d i f f e r e n t types of l a t t i c e defects, which were introduced by the coldworking. L a t t i c e d e f e c t s are d e f i n i t e i r r e g u l a r i t i e s i n t h e p e r i o d i c a l arrangement of t h e metal atoms. Defects which are l i k e l y to be formed during cold-work are: d i s l o c a t i o n s , vacancies and divacancies **. For t h e i r d e f i n i t i o n s one i s referred to l i t e r a t u r e ^ ' .
The recovery of t h e e l e c t r i c a l r e s i s t i v i t y i n c r e a s e a f t e r coldworking in liquid a i r (about 90°K), i s given schematically in figure 1.1. This figure r e f e r s t o Cu; for Ag and Au q u a l i t a t i v e l y the same behaviour i s found. The curve can be obtained by cold-working a s e r i e s of wires to the same degree. After coldcold-working each of the wires i s kept for some time (for instance 10 minutes) at a constant temperature, in such a way t h a t the whole recovery r a n g e - i s covered. After t h i s recovery t r e a t m e n t , the remaining r e s i s t i v i t y i n c r e a s e of each wire i s measured at a temperature below the recovery region (for instance l i q u i d nitrogen
tempera-400 500 •-T(«>K)
Figure I. 1
The decrease of the e l e c t r i c a l r e s i s t i v i t y , due to coldworking in liquid a i r , versus
the annealing temperature for Cu wires.
t u r e ) . and p l o t t e d as a function of the respective recovery tem-perature.
As i s seen from f i g u r e I . 1 four recovery s t e p s are found, separated by h o r i z o n t a l p a r t s in some c a s e s . Experimentally the h o r i z o n t a l p a r t s are not found e a s i l y , e s p e c i a l l y if a high de-gree of coldworking has been a p p l i e d . The t r a n s i t i o n from one s t e p into the next one i s then found to be nearly continuous, so t h a t they can hardly be separated in t h i s experiment. The tempe-r a t u tempe-r e scale of figutempe-re I . 1 v a tempe-r i e s with the degtempe-ree of defotempe-rmation; the curve s h i f t s to lower temperatures with increasing degree of coldworking.
No general agreement e x i s t s on the question which types of the l a t t i c e d e f e c t s , mentioned above, have to be connected with the four recovery s t e p s of figure 1.1.
As the disappearance of the work hardening takes place simul-taneously with s t e p 4, t h i s s t e p i s ascribed to d i s l o c a t i o n s by most authors.
According t o Manintveld ^° s t e p 2 should be caused by vacan-c i e s . Experiments of Berghout ^^ i n d i vacan-c a t e t h a t s t e p 2 has very l i k e l y t o be a s c r i b e d t o v a c a n c i e s indeed. According t o some authors s t e p 3 should be connected with vacancy diffusion a l s o . As for our experiments, the question of the i d e n t i f i c a t i o n of the recovery s t e p s i s of l i t t l e importance. We did measurements on coldworked Cu, Ag and Au having two d i f f e r e n t i n i t i a l s t a t e s as to the content of l a t t i c e defects.
In one case the metal was deformed a t l i q u i d a i r temperature. No recovery t r e a t m e n t s were given before the measurements were s t a r t e d . In i t s I n i t i a l s t a t e the coldworked metal thus contained a l l recovery steps shown by figure 1.1. These measurements are described in Chapter V, § 4.
11
In the other case the metal was deformed at roomtemperature or at l i q u i d a i r temperature. Before the measurements were s t a r t e d , a recovery treatment was given in order t o have a complete r e -covery of the defects causing step 1 and step 2. This was done by keeping the coldworked metal for some time at a temperature which varied from 50°C to 90°C, thus from about 320°K to 360°K in figure I. 1. After t h i s recovery t r e a t m e n t the l a t t i c e d e f e c t s connected with step 4 are s t i l l present in a l l cases. The l a t t i c e defects connected with s t e p 3 w i l l , at l e a s t p a r t l y , be p r e s e n t too. If t h e recovery t r e a t m e n t was given t o a s e r i e s of wires coldworked to d i f f e r e n t degrees, the same treatment was given t o a l l wires, although t h e i r s t a t e could then no more be considered to be exactly equal, as t h e p o s i t i o n of the temperature regions of the steps in figure I . 1 depends on the degree of coldworking.-The measurements are described in Chapter V, § 2 and g 3.
C h a p t e r I I
T H E T H E R M O E L E C T R I C E F F E C T S
g 1. Suminary
In this chapter the definition of the thermoelectric effects which we need in order to analyze our experiments will be given. These effects are: the Seebeck effect, the Thomson coefficient and the absolute thermoelectric power. A brief discussion of the current theory on these effects will be given; for the extensive treatment one is referred to standard works ^^.
The theory will be compared with the experimental values for some metals of the Thomson coefficient and the absolute thermo-electric power, as given by literature. The discrepancy between theory and experiment, which has been known for long, will be discussed. In view of this discrepancy it is decided to use, for the analysis of our experiments on coldworked Cu, Ag and Au, the experimental values for the absolute thermoelectric power of the metals in the undeformed state, in stead of the values that fol-low from theory.
§ 2. a. Definition of the Seebeck effect
At the ends of a homogeneous wire of a metal A , junctions are
I—^2^-n
M
B B
A
C.J. H.J.
T T+AT
Figure II. 113 made to homogeneous wires of a metal B, «rtiich lead t o a voltmeter
M having a high i n t e r n a l r e s i s t a n c e .
If the temperature of one junction (the hot junction) i s r a i s -ed from r to T + AJ", while the temperature of the o t h e r junction (the cold j u n c t i o n ) i s kept constant at T, the meter M will in-dicate a p o t e n t i a l difference AV^g; AVj^^ only depends on T and AT and i s independent of the t e m p e r a t u r e d i s t r i b u t i o n along t h e wires.
AVAB i s c a l l e d the t h e r m o e l e c t r i c force of the metal A with respect to the metal B and i s expressed in Volts (V).
We define
lin, -J±= S^^iT) ( I I . 1)
AT-o ^ j .
as the t h e r m o e l e c t r i c power of the metal / with r e s p e c t t o the metal B. S^g i s expressed in Volt per degree (V/°K). In general
Sf^B i s a function of the temperature T.
The metal A i s called thermoelectrically p o s i t i v e with respect t o the irretal B, i f a t t h e cold j u n c t i o n the p o s i t i v e e l e c t r i c current flows from the metal A to the metal B.
b. Definition of the Thomson coefficient
Through a wire of metal A passes an e l e c t r i c a l c u r r e n t i . In the meantime, by means of e x t e r n a l sources, a constant tempera-t u r e d i f f e r e n c e A7' i s s e tempera-t up betempera-tween tempera-the p o i n tempera-t s * and x + Aa: of the wire.
T+AT
x+Ax
figure II. 2On the Thomson c o e f f i c i e n t .
I t appears t h a t between the p o i n t s x and x + Ax, in a d d i t i o n to the J o u l e heat, heat i s emitted or absorbed by the e l e c t r i c current, according to
A» = (x^(D i AT ( I I . 2)
Atf = heat per second generated or absorbed (Watt).
The Thomson c o e f f i c i e n t i s a function of the temperature T. I t i s to be taken p o s i t i v e i f h e a t i s emitted when the p o s i t i v e e l e c t r i c current flows from higher to lower temperature.
If the Thomson c o e f f i c i e n t i s considered to be caused by the s p e c i f i c heat of the conduction e l e c t r o n s , i t i s e a s i l y seen that heat will be absorbed when the electron c u r r e n t flows from lower t o higher temperature. Thus in normal cases the Thomson c o e f f i -c i e n t will be negative.
c. Definition of the absolute thermoelectric power
Prom c o n s i d e r a t i o n s , i n v o l v i n g the f i r s t and second law of thermodynamics, Thomson derived a r e l a t i o n between the tempera-t u r e d e r i v a tempera-t i v e of tempera-the tempera-t h e r m o e l e c tempera-t r i c power of tempera-the metempera-tal A witempera-th respect to the metal 6 and the Thomson c o e f f i c i e n t s ^^ and \i.^:
dS^ _ i^A - ^B
^r ' T ^"-^^
A rigorous proof of formula ( I I . 3 ) , which could not been given by Thomson, has been derived in recent years only ^*.
Borelius c. s. ^ ' has integrated formula ( I I . 3) and derived, on account of N e m s t ' s theorem ( t h i r d law of thermodynamics),
( I I . 4 )
( I I . 5 )
i s c a l l e d the absolute t h e r m o e l e c t r i c power and i s expressed in Volt per degree (V/°K).
N e m s t ' s theorem thus involves:
S^B = 0 i f T = QIC , ( I I . 6) for any p a i r of metals.
The experiments of Borelius c . s . i n d i c a t e t h a t , although the t h e r m o e l e c t r i c power has c o n s i d e r a b l e v a l u e s even a t very low t e m p e r a t u r e s , the c o n d i t i o n ( I I . 6 ) seems to be obeyed indeed. The advantage of introducing the absolute thermoelectric power i s , t h a t i t i s a t h e r m o e l e c t r i c power which r e f e r s to a s i n g l e metal. According t o ( I I . 4 ) t h e absolute t h e r m o e l e c t r i c power of one metal (A) i s s u f f i c i e n t t o determine the absolute
thermoelec-o
The .quantity
" ^ A B
dT dT = 5AB(T) - SAB(O) = 5AB(T) =
o T o 7 * S-.f^.T
15 t r i e power of a l l other metals (B), i f the t h e r m o e l e c t r i c power of A against B (S,^^) i s known.
The absolute t h e r m o e l e c t r i c power can not be measured d i r e c t -ly; t h i s will be shown h e r e a f t e r . I t can be determined according to formula ( I I . 5) if the Thomson c o e f f i c i e n t i s known as a func-t i o n of func-temperafunc-ture down func-to 0°K.
In the preceding s e c t i o n b, we saw t h a t in the normal case \L i s n e g a t i v e . According t o ( I I . 5 ) 5 w i l l then be n e g a t i v e t o o . This can be shown in another way also.
Figure II. 3 "^ '
On the absolute thermoelectric power.
Consider a wire of metal A, the ends of which are kept at con-s t a n t t e m p e r a t u r e con-s 7' and T + AT by meancon-s of e x t e r n a l con-s o u r c e con-s . A p o t e n t i a l d i f f e r e n c e between the ends i s then s e t up, which a r i s e s in the following way.
There i s a continuous flow of heat in the wire A from high to low t e m p e r a t u r e . The p a r t of t h e h e a t which i s c a r r i e d by the l a t t i c e waves w i l l be neglected; only the heat t r a n s p o r t by the e l e c t r o n s i s considered. As the e l e c t r i c a l c u r r e n t i s assumed t o be zero, a r e d i s t r i b u t i o n of the conduction e l e c t r o n s i s necessa-ry to c o u n t e r a c t the d r i f t v e l o c i t i e s of the e l e c t r o n s due t o the heat t r a n s p o r t . This r e d i s t r i b u t i o n causes an e l e c t r i c f i e l d along the wire which g i v e s a p o t e n t i a l d i f f e r e n c e between the ends of the wire. The d i r e c t i o n of t h i s f i e l d must be so, t h a t the e l e c t r o n s are r e p e l l e d from the low temperature region i n t o the high temperature region. So the cold end with temperature T w i l l be n e g a t i v e with r e s p e c t t o the h o t end with temperature
T + Ar.
The potential difference between the cold end and the hot end of wire A is equal to
16
^7+ AT
AKA= ƒ S„dT (II. 7) T
According t o the above considerations we have
AVy^ < 0 ( I I . 8)
for any temperature 7" and any temperature difference AT.
From (II. 7) and (II. 8) it follows then that 5j^ is always nega-tive.
It is not possible toimeasure AVy^ with the voltmeter according to the diagram of figure II.3. For as soon ac the wires B are connected to the ends of wire A the temperature difference AT will be set up in the metal B also, so that we will measure ISV^
-A V B = -AV,B.
The condition that the electrical current is zero, is fulfilled in practice by using a potentiometer in stead of a voltmeter.
§ 3. The theoretical expressions for the absolute thermoelec-tric power
Wilson ^' (p. 204) derives for the absolute thermoelectric power of a metal (5) the following expression:
3e ^ £ " |grad^Ê|'E=5
^ = - " ^ ^ ' " ^ T S S >
k = Boltzmann's constant
- B = charge of an e l e c t r o n
E = Fermi energy of an electron
^ = energy parameter in the Fermi d i s t r i b u t i o n function
V = velocity of an electron
t = time of relaxation
X = wave vector of an electron
dO = element of Fermi surface in Jt-space.
The e x p r e s s i o n s for the e l e c t r i c a l c o n d u c t i v i t y (o) and the thermal conductivity (X) are in the same form given by
,2 ^ TV^dO ( j j jo^
a =
12 71^ E= 5 |grad^£
X = ^ / j ^ ^ (11.11)
36Tt E=c Igrad^fl
supposi-17 t i o n has been made t h a t a u n i v e r s a l time of r e l a x a t i o n e x i s t s , thus T being i d e n t i c a l for e l e c t r i c a l and thermal c o n d u c t i v i t y . This supposition y i e l d s on account of (11.10) and (11.11):
X 1l2t2
with L„, the constant value of the Lorentz number L, being inde-pendent of temperature (Wiedemann - Franz law)v
The formula for S'can be s i m p l i f i e d by s u b s t i t u t i n g (11.10) into ( I I . 9 ) . We then get the « e l l known formula
S--^{1^) or 5 = 4 ! I ( ^ P ) (11.13)
3e oE E=J 3e at &=;;p = 1/a being the electrical resistivity. On account of (II.12) substitution of (11.11) into (II. 9) yielding
5 = i ^ ! ^ ( i ^ - l ^ ) (II. 14)
3e öE E=^
must be identical with (11.13).
Formula (11.13) can be written in a form which makes it apt for comparison with experiment. With the assumption of free elec-trons we have for temperatures above the Debye temperature 9:
P - p (II-15)
So we find • ( ^ 4 ^ ) = - i (II. 16)
at E=C C
S u b s t i t u t i n g t h i s in ( I I . 1 3 ) y i e l d s for the absolute thermo-e l thermo-e c t r i c powthermo-er:
S = -JLU- (11.17)
The Thomson coefficient is, according to (II.5) and (11.17) given by
^ = r ^ = - : ^ = 5 (11.18)
al e^
From (11.18) it follows that S and n are identical. They are always negative, which is in accord with the predictions of § 2b and g 2c of this Chapter. They have a linear temperature depend-ency.
The validity of formula (11.17) ends as soon as the Lorentz number is no longer temperature independent. This will be the case for temperatures below the Debye temperature. The relaxation
time as derived from the electrical resistivity will then differ from the relaxation time as derived from the thermal conductiv-ity.
A formula for the absolute thermoelectric power, which is valid at least as an approximation for all temperatures and' ratios of p^ to p^ is given by
•St,r = - — - — - (11.19)
Pr = residual resistivity
pt = electrical resistivity due to the thermal lattice waves
"T' o ( e ' - l ) ( l - c - ' )
n = number of free electrons per unit volume Hg = number of atoms per unit volume.
The subscript t,r denotes that 5t,, arises from simultaneous thermal and impurity scattering.
This formula is derived by Sondheimer ^*; according to Wilson
'"' (p. 310) we wrote - Jj in stead of + Jj, which is a higher
approximation. The formula in the form as (11.19) is also given by Kohier '^
In the derivation of formula (II.19) both the scattering due to the lattice and the scattering due to impurities have been considered. In formula (II. 13) only the lattice scattering has been accounted for. For the limiting case of p^ = 0 we find from
(11.19), with t,/i) = 2^''^ for monovalent metals: /2 ,ex 2 i_ :il .... ' 2n' T 47t^ Js ,_ . St = — . = a (11.20) ^^ 3^2 ,e,2 1 jy e^ 1 +-
9
27t^ T 2n' J. 5The absolute thermoelectric power of Cu, Ag and Au is calcu-lated according to (11.17) and (11.20) and plotted in figure II.4 as a function of temperature. The theoretical values for C, have been used, being 7.0, 5.5 and 5.5 eV for Cu, Ag and Au
re-19
Figure II.i
The t h e o r e t i c a l absolute thermo-e l thermo-e c t r i c powthermo-er of Cu, Ag and Au versus the absolute temperature.
spectively. The limiting value of a (formula (11.20)) for T = 0°K i s 1/3, so t h a t for low temperatures we get 5t = --K^k^T/SBt^. The l i m i t i n g value of a for high temperatures i s 1, so t h a t we have then 5t = -n^k^T/Bt^, which i s i d e n t i c a l with (11.17). As i s seen from figure I I . 4 , the curves as c a l c u l a t e d according to (11.17) and (11.20) behave s i m i l a r , except for the lower t e m p e r a t u r e s .
§ 4. Comparison of theory with t h e experimental v a l u e s for some metals of the Thomson c o e f f i c i e n t and the a b s o l u t e thermoelectric power as taken from l i t e r a t u r e
Borelius c. s. •"* measured the Thomson c o e f f i c i e n t of a " s i l -veralloy normal" (Ag c o n t a i n i n g 0.37 at % Au) from about room-temperature down to the lowest room-temperature obtainable; t h i s alloy was taken because of the low value of i t s Thomson c o e f f i c i e n t . By using formula ( I I . 5 ) Borelius c a l c u l a t e d the absolute thermo-e l thermo-e c t r i c powthermo-er of ththermo-e "normal" as a function of tthermo-empthermo-eraturthermo-e.
The t h e r m o e l e c t r i c power of a number of pure metals (Cu, Ag, Au, Pt, Pd, Fe, Co, Ni and Pb) against the "normal" was measured over about t h e same t e m p e r a t u r e range as mentioned above. By using formula ( I I . 4 ) t h e a b s o l u t e t h e r m o e l e c t r i c power of the metals was calculated. From the absolute thermoelectric power the Thomson c o e f f i c i e n t was derived again according to formula
( I I . 5 ) .
For Cu the measurements of n were extended l a t e r by Nyström ^^ up to about 1100°K.
Lander ^^ measured the Thomson coefficient for a number of metals from roomtemperature up to the melting point. His results in general do not agree with the extrapolated low temperature measurements of other authors. As we are mainly Interested in the
(iOO-'v/'K)
-nw-Figure II.5
The experimental Thomson coefficient versus the absolute temperature for Cu, Ag and Au
according to Borelius 29,3<J.
* 2 « r S(10-'V/«K)
•150
+100
Figure II.6
The experimental absolute thermoelectric power versus the absolute temperature for Cu, Ag and
21 temperature region below roomtemperature, h i s work w i l l not be discussed further.
In figures I I . 5 and 6 the Thomson c o e f f i c i e n t and the absolute thermoelectric power of Cu, Ag and Au according to Borelius c . s . 29.30 ^j,g given. The Debye temperatures of Cu, Ag and Au are respectively 333, 223 and 175°K. So formula (11.17) w i l l , for the temperature range c o n s i d e r e d , not be a p p l i c a b l e t o Cu. For Ag and Au i t covers only a small p a r t of the curves shown. Formula (11.20) however should be v a l i d over the whole temperature range.
I t i s seen t h a t |x and S are not always n e g a t i v e , but mostly p o s i t i v e , and t h a t t h e i r temperature dependence shows anomalies which have not been accounted for by t h e o r y ; see a l s o f i g u r e I I . 4 .
Above about 200°K \i and 5 have a f a i r l y l i n e a r t e m p e r a t u r e dependency. Extrapolation t o low temperatures, of the l i n e a r p a r t of the fi-T curves a c c o r d i n g t o B o r e l i u s c . s . y i e l d s (i = 0 a t 7 = 0 for Au only. This i s t h e case for Cu a l s o , i f Nyström's values for high temperatures are used.
The p o s i t i v e experimental v a l u e s of 5 for Cu, Ag and Au are often ^3 a s c r i b e d to the f a c t t h a t for t h e s e m e t a l s the Fermi surface i s close to the boundaries of the f i r s t B r i l l o u i n zone 3^ The assumption of free e l e c t r o n s , made in d e r i v i n g the formulae
(11.17) and (11.20) i s then no longer v a l i d . Quantitative calcu-l a t i o n s on t h i s effect have however not been c a r r i e d out.
A temperature dependence as shown experimentally by \i and 5 at low temperatures might be explained, at l e a s t p a r t l y , by an ef-fect which has been brought into focus by Gurevich ^^ and o t h e r s ^*'^^. This e f f e c t i s c a l l e d t h e phonon drag e f f e c t ; i t a r i s e s if a non equilibrium d i s t r i b u t i o n of the thermal l a t t i c e waves (phonons) i s considered, thus if the l a t t i c e thermal conductivity i s taken i n t o account a l s o . In the d e r i v a t i o n of (11.17) and (11.20) only the e l e c t r o n i c thermal c o n d u c t i v i t y has been con-sidered (see § 2, section c of t h i s Chapter).
In view of the inadequacy of present theory t o account for the absolute t h e r m o e l e c t r i c power of Cu, Ag and Au, we w i l l use the experimental v a l u e s , as given by B o r e l i u s , in the a n a l y s i s of our experiments, in stead of using the values following from the formulae (11.17) or (11.20).
The anomalous behaviour of |i, having a maximum and a minimum at low temperatures, i s a l s o shown by Pt, Pd, Fe, Co and Ni. At high temperatures, however, (x i s always negative and l i n e a r with temperature, as i s required by theory.
The a b s o l u t e t h e r m o e l e c t r i c power of the a l k a l i metals was measured by MacDonald and Pearson ^^ from about 60°K down t o l i q u i d helium temperature. The absolute t h e r m o e l e c t r i c power of
Na i s n e g a t i v e above about 7°K; below t h i s t e m p e r a t u r e i t i s p o s i t i v e . K has a negative absolute thermoelectric power down to about 12°K. The values of the n e g a t i v e absolute t h e r m o e l e c t r i c powers of Na and K can not be accounted for by formula (11.20). Li and Cs have a p o s i t i v e absolute thermoelectric power over the whole temperature range. The absolute thermoelectric power of Rb i s negative above about 18°K; below 18°K i t i s p o s i t i v e .
Summarizing we may say, t h a t the present theory on the thermo-e l thermo-e c t r i c thermo-effthermo-ects i s in most casthermo-es inadthermo-equatthermo-e.
23 C h a p t e r III T H E C H A N G E O F T H E A B S O L U T E T H E R M O E L E C T R I C P O W E R B Y L A T T I C E D E F E C T S § 1. Summary
In the preceding chapter only the scattering due to the ther-mal lattice waves has been considered. In this chapter the scat-tering by impurity atoms and lattice defects will be taken into account also. This scattering gives rise to an electrical resist-ivity which is, to a high approximation, independent of tempera-ture (Matthiessen's rule) and which can be added to the resist-ivity due to the thermal lattice waves, in order to get the total resistivity of the metal. Starting from formula (11.19) it will be shown, that simply adding of the absolute thermoelectric pow-ers arising from different scattering mechanisms, is not allowed. A substitute of Matthiessen's rule for the absolute thermoelec-tric power will be given (Kohier).
The change of the absolute thermoelectric power of the pure metal due to the residual resistivity is calculated from formula (11.19). A formula for the change of the absolute thermoelectric power due to lattice defects is given also. In this formula the dependency of the electrical resistivity of the lattice defects on the Fermi energy of the electrons is assumed to be the same for all types of lattice defects. There are however both theo-retical *^ and experimental arguments. Indicating that this as-sumption can not be hold.
Therefore other formulae are set up in which the influence of each type of lattice defects is accounted for separately. These formulae are, according to Friedel, based on (11.13), which can be used over the whole temperature range for calculating the ab-solute thermoelectric power due to scattering agents whose elec-trical resistivity is independent of temperature. For these scattering mechanisms a uniquely defined time of relaxation is generally assumed to exist (Wilson *^, p. 286), thus ensuring the validity of (II.13) for this case.
Expressions are derived for the thermoelectric power of three types of coldworked thermocouples on which measurements have
been made. In these expressions the r e s i d u a l r e s i s t i v i t y of the metals in the undeformed s t a t e i s also accounted for.
§ 2. The theoretical expression for the change of the absolute thermoelectric power according to Kohier and Sohdheimer The absolute thermoelectric power of a metal having a residual r e s i s t i v i t y p , i s given by (11.19):
„ . , 1 i; / e N 2 1 ^ 7 - , ^ 2 , 2 y P ' ^ 3 p t { i * ~ . ^ ( 7 ^ -—.j-y
5^ ^ =
- I L U .
L.
(III.
1)
3 e ; 3 ^ ,9x2 1 J 7
The a b s o l u t e t h e r m o e l e c t r i c power due to the s c a t t e r i n g by the l a t t i c e thermal waves only, i s found from ( I I I . 1 ) by p u t t i n g P, = 0:
1 . — '9^2 1 J ,
S . - - ^ ^ ' 2 . - D 7^ 4 . - J 5 ^^^^^^
e^ ^ _ 3 _ 5 / 9 x 2 i_ Jj
* 2n^ D T 2^2 Js
The absolute thermoelectric power due to the residual resist-ivity alone is found from (III. 1) by putting p» = 0:
S, = -ILU- (III.3)
3e^ Note that 5, is independent of p^. We will now show that
5 . , / 5» + S, (III.4)
The thermal r e s i s t i v i t y due to t h e r e s i d u a l r e s i s t i v i t y i s given by
L„ being the Lorentz number, given by (11.12).
A formula for the thermal r e s i s t i v i t y due to the l a t t i c e waves which i s , a t l e a s t as an approximation, v a l i d over a l l
The electrical resistivity due to the thermal lattice waves is given, at least as an approximation, for all temperature ranges by Wilson ^^ (p. 285):
Pt = 4>1 (|)' J5 (III. 7) with ^ = c o j ^ ^jjj g^ Substituting (III.7) into (III.6) yields
Note that from (III. 9) follows:
L - ^ - ^-^ (III. 11)
2n^ D T 271^ Js
L reduces to the constant value L„ only for temperatures above
the Debye temperature (T > 9 ) . For T = 0°K i s L = 0. Formula ( I I I . 1) can be written as
n ^ , ^ „ . LJ 2n' D T 4 n ' J s •^t.r - oc-r • " 5 : F: \ s r a 2 Ï J : ~ ( I I I - 1 2 ) 3 e ; • Pr , Pt . 3 ; ,9x2 1 J -L„T -L„T 271^ i) T 27c2 J s ^ I n s e r t i n g ( I I I . 5) and ( H I . 9) into ( I I I . 12) y i e l d s
1 + J _ ^
(6)'
_ _L
:£i
nhh 1 nk^T 2n^ D T 47i2 J^ 1 3e^ X, 3e^ 3 ^ 0.2 l J , Xt ^ 2n^ D y 2^2 Js ' S t , r = - 1 J ( I I I . 13) X, XtOn account of (III.2) and (III.3), (III.13) can be written as S,/X, + St/Xt
St.r
or _ L I J : = _ L + _ L (III. 14) • S t , r 1 •St ^ s, Xt X , 1 1 Xt X , with = 7 — + T — ( I I I . 15) Xt,r Xt Xr
and 1/Xt.r being the t o t a l thermal r e s i s t i v i t y .
By comparing ( I I I . 4 ) t o ( I I I . 1 4 ) we see t h a t simply adding of the absolute thermoelectric powers i s not allowed. Adding i s only allowed a f t e r dividing t h e absolute thermoelectric powers by the respective thermal c o n d u c t i v i t i e s .
Formula ( I I I . 1 4 ) has been derived also, in a more complicated way by Kohier '^ as being the s u b s t i t u t e for M a t t h i e s s e n ' s r u l e for the thermoelectric power.
We w i l l now c a l c u l a t e t h e change of t h e t h e r m o e l e c t r i c power of the pure metal (A 5 , ) due t o t h e r e s i d u a l r e s i s t i v i t y . This change i s measured on a thermocouple c o n s i s t i n g of a pure wire and a wire c o n t a i n i n g i m p u r i t i e s . The a b s o l u t e t h e r m o e l e c t r i c powers of ^hese w i r e s a r e then r e s p e c t i v e l y 5 t and 5 t , r - The thermopower measured on t h e couple, i s then according t o ( I I . 4 ) given by
A 5 , = St,r -St (III. 16) S,,, and St follow from (III. 1) and (III. 2).
The expression for A 5 , can be simplified in the following way. Eliminating Xj from (III. 14) with (III. 15) yields:
St,r -
Xt.. {St ( ^ - f ) -T^} =St - ^ ( - S , -St)
S - S
Thus: A S, = St.r --St = - 4
'
(III. 17)
Xr J^t.r Prom (III.15) follows:
Xr , X,
1 + —
Xt.r Xt
which on account of (III. 5) and (III.9) can be written as:
Xt Pr
Inserting this in (III.17) yields:
S, - St
27 Using ( I I I . 3 ) we get
-jfÉl.St
A S , = ^^^ , ( I I I . 19)1 . ±lx
PrNote t h a t A S , i s , contrary to S , , a function of p . . A S , can be calculated i f St and p , are known.
Next the change of the absolute t h e r m o e l e c t r i c power due t o coldworking will be c a l c u l a t e d . The r e s i s t i v i t y due to the l a t -t i c e d e f e c -t s (pj) i s independen-t of -tempera-ture and can be added to p , , the r e s i d u a l r e s i s t i v i t y of the metal in the undeformed s t a t e . For the absolute t h e r m o e l e c t r i c power of the coldworked metal we then get:
•St.r.d = -St + A S , . d = St t . ^ ^ ^ , ( I I I . 2 0 ) 1 + *^^ Pr+ Pd 7c2fe2j. S. with A S, d = ^ ( I I I . 21)
l.J^
Pr+ Pd which i s analogous to ( I I I . 19).The absolute t h e r m o e l e c t r i c power due to the l a t t i c e d e f e c t s alone, i s according to ( I I I . 3) given by
- n^k^T
Sd = , , ( I I I - 22)
The change of tne a b s o l u t e t h e r m o e l e c t r i c power of the pure metal, due to the l a t t i c e defects alone i s , analogous to ( I I I . 19) given by
37c *
ASd= ^ ^ S - j (III. 23)
1 .i^
Pd
Actually t h i s change can not be measured, as the metal in the undeformed s t a t e always had a residual r e s i s t i v i t y which was comparable with or even l a r g e r than the r e s i s t i v i t y due to the l a t -t i c e defec-ts.
In f a c t , the change of the a b s o l u t e t h e r m o e l e c t r i c power of the undeformed wire due to coldwork, having a r e s i d u a l r e s i s t i v i
-ty p^, was measured on a thermocouple c o n s i s t i n g of a coldworked wire and a wire in t h e undeformed s t a t e . The a b s o l u t e thermo-e l thermo-e c t r i c powthermo-ers of ththermo-esthermo-e wirthermo-es arthermo-e givthermo-en by S^ ^ ^ and S j ,- Ththermo-e thermopower measured i s then given by:
ASS = S t . r . d - 5 t . , ( I I I . 2 4 ) With ( I I I . 20) and ( I I I . 16) we get:
A S j = A S , , d - A S, ( I I I . 2 5 ) This expression can be simplified i n t o :
lek'T - • S t . r Ott-, , ^'^X * Pr 1 + A s ; = ^^. — ( I I I . 26) Pd
which resembles (III.23) and Is equal to i t if p, = 0.
We will now give the proof of (III. 26). From (III. 25), (III. 21) and ( I I I . 19) follows: A S * - (.ÉÈ^-s) (—^ i—) = ^^^ ^ 3e; ^^> \ , 4>tx i , W Pr "• Pd P r / Tt^fe^T ^ X / P r -^ Pd P r ^ ^ 3er, * Pd + 4/lx + p , 4,4x + p , ( ^ ' f e ' ^ s ) ( Pd'^^X ) -" ^ 3 e ; •^'' \ p , + iAx* p , ) ( 4 A x * p , ) ' = ( _ Z ^ - S t ) , , ^ X _ i
3eC, 4Ax + p , 4Ax + p , 1 +
Pd
= (S, - S t ) — 3 - / - - ( I I I . 2 7 ) Pr , 44X + P r
^^IX Pd From ( I I I . 1 4 ) and ( I I I . 15) we deduce
(S, - S t ) = ( S , - S , , , ) (1 + ^ ) . r
On account of ( I I I . 5) and ( I I I . 9 ) t h i s can be written as (S, - S t ) = ( S , - St , ) (1 + - ^ ) ( I I I . 28)
29 I n s e r t i n g ( I I I . 2 8 ) in ( I I I . 2 7 ) y i e l d s , with ( I I I . 3 ) : 7i2fe2r A S !
= ^ ^
- • S t , , •"A1 +ifLl£i
Pd which i s equal to ( I I I . 2 6 ) .In Chapter VI formula ( I I I . 2 6 ) w i l l be used t o analyse our experimental r e s u l t s . I t w i l l be shown t h e r e , t h a t t h i s formula i s not able to account for the observed phenomena. One reason why t h i s formula i s inadequate, i s the fact t h a t in deriving i t , the r e s i s t i v i t i e s of a l l types of l a t t i c e defects are handled in the same way as the r e s i d u a l r e s i s t i v i t y was. However t h i s seems not to be allowed. Therefore in t h e n e x t paragraph the refinement will be made, t h a t the influence of each type of l a t t i c e d e f e c t s on the absolute t h e r m o e l e c t r i c power w i l l be accounted for sepa-r a t e l y .
§ 3 . The t h e o r e t i c a l e x p r e s s i o n for the change o f the a b s o -l u t e t h e r m o e -l e c t r i c power according to F r i e d e -l
Section a
If in a metal having a residual resistivity p, only the scat-tering due to the impurities is considered, a uniquely defined time of relaxation always exists. So formula (II.13) can be ap-plied at once. The absolute thermoelectric power, if impurity scattering alone is effective, is then given by
S, = 4 ! I ( 1 4 P I ) (III. 29)
3e B£ z=i
The residual resistivity is, as for its dependence on the Fermi energy of the electrons given by (Wilson ^^, p. 269. 285):
p = const i (III. 30) Hence we deduce
( i ^ ) = J _ ( ^ ) = - 1 (111.31)
I n s e r t i n g ( I I I . 3 1 ) i n t o ( I I I . 2 9 ) y i e l d s : n2fe2. 3et, S, = - I L l - L ( I I I . 32)This expression i s I d e n t i c a l with formula ( I I I . 3 ) .
We note t h a t formula ( I I I . 3 0 ) i s only a crude approximation, as the dependence of the s c a t t e r i n g cross s e c t i o n of the imper-fections causing p , , on the electron energy E i s neglected * ' .
The t o t a l e l e c t r i c a l r e s i s t i v i t y of the metal i s , according to Matthiessen's rule, given by
P t , r = P t + P r ( I I I - 3 3 ) The absolute t h e r m o « l e c t r i c power of the metal i f s c a t t e r i n g
on the thermal l a t t i c e waves and on the impurities occurs simul-taneously i s , according to Friedel *' on account of ( I I . 1 3 ) given
' • ' 3et, B In £ E=C
This formula can be written in another form: 5 TC^fe'r r ^ In (Pt ^ P r ) . *•' ~ 3eC, B In £ E=C " n^k^T r - E ^(Pt ^ P r ) | 3et, p^ + p^ B£ E=C n^k^T r - E /Bpt , ^Prxx 3et, p^ + p^ ^B£ * B£ E=S ' S . , = - I ^ P l ^ ^ l ± ^ ( I I I . 3 5 ) 3s!; P J + p , t E ^Pt) ( ^ I n p t x . „ T •^c^ with ( ^ j = ( - - - ; = X ( I I I . 3 6 ) Pt o£ E=C d In £ E=5
, £ Bprx / ^ In Pr,
and - - ^ = - ^ ^) = *r ( I I I . 37) p, B£ E=C B In £ E=^
The quantity x i s introduced by Mott and Jones ^*. Using ( I I I . 3 6 ) we write for (11.13):
_g _ Tt2fe2y 3 J n _ p t x _ _ n^k^T / 3 In ptx * ' 3e B£ E=4 ~ 3et, 3 In £ E=C '
( I I I . 3 9 ) 31 With ( I I I . 3 7 ) formula ( I I I . 2 9 ) can be converted i n t o :
. •^k^T /B In p,x _ n^k^T /_ ^ In p
' ' 3e ^ B£ E=C ' 3ei; ^ B In £ E=S '
= X ,
3e^
Prom ( I I I . 3 5 ) we deduce:
S t , r (Pt - Pr) = - - ^ ^ ' ^ P t - ^ ^ ^ ' P ' Which with ( I I I . 3 3 ) , ( I I I . 3 8 ) and ( I I I . 3 9 ) y i e l d s :
•St,, P t . r = St Pt + S , p , ( I I I . 4 0 ) This formula i s analogous to formula ( I I I . 14) and again a
sub-s t i t u t e for M a t t h i e sub-s sub-s e n ' sub-s r u l e for t h e a b sub-s o l u t e t h e r m o e l e c t r i c power. We note t h a t ( I I I . 4 0 ) follows from ( I I I . 14) only i f X^p^T = L„ which i s according t o ( I I I . 1 1 ) t r u e i f ?• > 0. i t i s i n t e r -e s t i n g t h a t ( I I I . 1 4 ) can b-e d-eriv-ed at onc-e, i f w-e put
Tt^k^T /3 In 1/Xt rx
according to (11.14) and analogous t o ( I I I . 34). However the s u b s t i t u t i o n
Tt^fe2T ,B In 1/Xt,
•5t=^^( ^ ^ ) , (III. 42) 3e B£ E=C
using (III.9) for 1/Xt does not yield the expression (III.2).
The change of the absolute thermoelectric power of the pure
metal due to the residual resistivity is again given by:
A S, = St,, - St (III.43)
From ( I I I . 35) and ( I I I . 38) i t follows:
A Q «: Q ^''fe^r , P t « t + Pr *r X A S , = S t , r - S t = — — ( Xt) = 3eC, p^ + p^ = -JLÉL (Pt ^t ^- Pr ^r - Pt *t - Pr ^tx 3 e ^ Pt + P , •^k^T /Pr ^r - Pr ^tx ^ _ Tl^fe^T ,X^ - X^, 3 e t , p^ H. p^ 3 e t , p^ 1 + — Pr
With ( I I I . 3 0 ) , ( I I I . 3 8 ) and ( I J I . 3 9 ) we get
A S , = ^ '- ( I I I . 44)
Pr
With * , = + 1 ( I I I . 4 5 ) In comparing ( I I I . 4 4 ) to ( I I I . 1 9 ) we see t h a t t h e s e formulae
are i d e n t i c a l , except for the s u b s t i t u t i o n of 4i4x by p^ in ( I I I . 4 4 ) . The consequence of t h i s s u b s t i t u t i o n can be i n v e s t i g a t -ed in the following way.
According to ( I I I . 7 ) p^ i s given by:
Pt =4A {y Js Putting T = e y i e l d s : Pg = iA X 1 X 0.23662 ( I I I . 45a) Dividing ( I I I . 7) by ( I I I . 45a) y i e l d s : p, 44 (T/Q)^ Js 1 TxS , ^ = ^ = (-) Jc ( I I I . 45b) p^ 4/10.23662 0.23662 9
The expression ( I I I . 4 5 b ) has been tabled as a function of (7/9) by Grüneisen ^^.
Dividing 4/lx ^y ^^^ r i g h t hand side of ( I I I . 4 5 a ) y i e l d s :
'^X X
4/1 0.23662 0.23662 (111.45c) The expression for x ( I I I - 1 0 ) has been t a b l e d as a function of (7/9) by Makinson ^ ' , t a k i n g t,/D=2'^^ for monovalent metals.
In figure I I I . 1 Pj/pg and x/0- 23662 are p l o t t e d as a function of T/9. We see t h a t the two f u n c t i o n s d i f f e r markedly a t low temperatures only. Q u a l i t a t i v e d i f f e r e n c e s between the formulae
( I I I . 44) and ( I I I . 19), however, are not to be expected.
Section b
We will now give the expressions for the thermoelectric power of three types of coldworked thermocouples. If more types of lattice defects are present simultaneously in the coldworked wire, the influence of each type will be accounted for separate-ly. The influence due to elastic deformations eventually set up by a lattice defect, will not be taken into account.
figure III. 1
See formula (III.45b) and (III.45c).
figure /ƒƒ.2
Coldworked thermocouple, case A; the thermoelectric power is
given by formula (III.54). Case A
Consider a thermocouple c o n s i s t i n g of one leg in the undeform-ed s t a t e (A) and the o t h e r one in a deformundeform-ed s t a t e so as to con-t a i n only one con-type of l a con-t con-t i c e defeccon-ts.
The e l e c t r i c a l r e s i s t i v i t y of l e g A i s then Pt r ^ Pt •*" Pr ^nd the e l e c t r i c a l r e s i s t i v i t y of leg B i s p^ r d " Pt * Pr * Pd- Pd being the e l e c t r i c a l r e s i s t i v i t y due t o the l a t t i c e defect con-sidered.
The absolute thermoelectric power of leg A i s given by formula ( I I I . 3 5 ) ,; n^k^T Pt ^t * Pr ^r Tc2fe'r O t , = ^ — — X 3eC, 3et, t , r ( I I I . 46) with P t ^ t + Pr ^ r P t * P r ( I I I . 47)
The absolute t h e r m o e l e c t r i c power of leg B i s , analogous to ( I I I . 34) given by
n^k^T r 3 I n ( p , + p , + p j ) ,
•St r d = - ^^^ { T TT ^ > ( I I I . 4 8 ) *•'•'* 3eJ; B In £ E=?
scattering by the lattice waves and by the imperfections causing Pr and pj.
Formula (III.48) can be rewritten according:
.?t,r,d - - ^ { , - \ | r ( P t - P r ^ Pd)Vr =
3e;, Pt + P , + Pd ^£ ^'^ n^k'T f -E /3Pt ^ 'Spr , ^ P d u 3e; Pt + P, + Prf B£ B£ B£ EH 3 e ; Pt * P, * Pd ^ ' Pt B £ ' ^' P 7 B £ ' Pd o£ E-Cor on account of (III.36) and (III.37):
n ^ P t ^ t ^ P r ^ r - P d ^ d (JJJ49) '•'•" 3e; Pt * P, ^ Pd
,ith (Zl^) - {-\^) -X, (III. 50)
Pj B£ E=C B m £ E=C ''
Prom (III.49) follows, on account of ( I I I . 3 8 ) and ( I I I . 3 9 ) :
' S t . r . d P t , r , d =-St Pt ^ ^ r P, ^ ^ d Pd d l l - S D
with Sd = - - ^ ^ r V - ' d ( I I I . 52) Sd denotes the absolute t h e r m o e l e c t r i c power i f only s c a t t e r -ing on t h e l a t t i c e d e f e c t s o c c u r s . Formula ( I I I . 5 1 ) i s again M a t t h i e s s e n ' s r u l e for the a b s o l u t e t h e r m o e l e c t r i c power (see a l s o ( I I I . 40)).
Ttie thermoelectric power measured on the thermocouple of fig-ure III. 2 i s according to formula ( I I . 4) given by
A-Sd = ' S t . , . d - 5 t . , (III.53) S,.,.dand St., are given by (III.49) and (III.46).
We can simplify (III.53) into:
*d -^t.r 3s^ 1 + A S ; = ^^ (III.54) Pt,r Pd
35 The advantage of w r i t i n g ( I I I . 5 4 ) i s , t h a t i t resembles the formula for A Sd, which denotes the change, due to the l a t t i c e d e f e c t s , of the absolute t h e r m o e l e c t r i c power of the pure metal (having p^ = 0 ) . The formula for A Sd i s , analogous t o ( I I I . 4 4 ) given by •ri'k^T ^ 3 7 r ~ * d ~ t A Sd = ^^ ( I I I . 55) 1 . l l . Pd
We w i l l now prove formula ( I I I . 5 4 ) . From ( I I I . 5 3 ) , ( I I I . 4 9 ) and ( I I I . 46) follows: . ^ „ „ _ Tl2fe2r ^PtXt + Pr^r^-Pd^^d P t f t + P r f r x A 5 - d = S t . , . d - S t . , = - ^ ^ ( — ^ - ; ^ - ; ^ - — - p^^p^ ; 3 e ; ( P t ^ P r ) ( P t * t ^ P r ^ r ) ^ ( P t + P r ) P d ^ d - ( P t ^ P r ) ( P t * t + P r * r ) - p d ( P t * t ^ P r * r ) ( P t * P r ^ P d ) ( P t * P r ) = Tt^fe^T ( P t ^ P r ) P d ^ d - ( P t ^ t - ^ P r ^ r ) P d 3 s ; ( p t + p , + P d ) ( p t * P r ) P t * t + P r * r ^ P t * t + P r * r Pd*d rrZ Pd •y.o^^'d-n'k'T Pt^Pr It'k'T Pt^Pr 3ei; p t + p , + P d 3 e i ; ^ ^ pt+Pr
On account of ( I I I . 4 6 ) and ( I I I . 3 3 ) we get:
A S : = - ^ i ^
•*d ~ "St . r 1 +
which i s equal t o ( I I I . 5 4 ) . Case B
The thermocouple now c o n s i s t s of one l e g i n the undeformed s t a t e (A) and the o t h e r one (C) having more than one t y p e of l a t t i c e defects, each characterized by i t s pj and x^.
Figure III.3
Coldworked thermocouple, case B; the thermoelectric power is given
by formula ( I I I . 6 0 ) . Pt,r,Sd = Pt * Pr * S pj, with S Pd = P^^ + Pd;3 * Pdr -^ ••• The a b s o l u t e t h e r m o e l e c t r i c p o w e r o f l e g C is, a n a l o g o u s to (III. 3 4 ) given by T^^k^T B In (Pt + p , + S p d ) «t.r, 2d 3e!;
{-B In £ } (III.56) E = C S t , , _ 2 d d e n o t e s t h e a b s o l u t e t h e r m o e l e c t r i c power c a u s e d by s i -m u l t a n e o u s s c a t t e r i n g by t h e l a t t i c e waves and t h e i -m p e r f e c t i o n s c a u s i n g p^ and Sp^. Formula ( I I I . 56) can be w r i t t e n a s : n^k^T - £ (!PI
. ^ . E
BPd)} 3eJ; Pt + P , + Spj B£ B£ B£ E=C=rr
ff '
[p. ( - ^ )
^ p^
( - ^ )
^
3eC, Pt + P , + SPd ' Pt ^ £ Pr S * Pd B£ E=S*2{p, P ^ ) } ] ] ]
o r , on a c c o u n t of ( I I I . 3 6 ) , ( I I I . 37) and ( I I I . 50) n^k'T Pt ^ t ^ Pr ^r ^ ^ (Pd ^ d ) 3 e ; Pt + Pr * Spd S t , r , 2d ( I I I . 57)37 From (III.57) follows, according to (III.38), (III.39) and (III. 52):
•St,r,ad Pt,r,2d = 5, p, + S, p, + 2 (S, p,) (III.58) Formula ( I I I . 5 8 ) i s an extended form of formula ( I I I . 5 1 ) .
The thermoelectric power measured on the thermocouple of f i g -ure III. 3 i s , according to formula ( I I . 4) given by:
^ • S ^ d = 5 t . , . s d - > S t . , (III-59) St.,,2d and St., are given by (III.57) and (III.46).
We can simplify (III. 59) again into:
X-r
s.
"ieT S d t,r A S* = ^ (III. 60) ^ p» 2 pj S (P^ X.) with x„^ = LÉ—L- (III. 61) ^ PdA Svj denotes the change of the absolute thermoelectric power, due to the lattice defects, of the metal in the undeformed state. The change of the absolute thermoelectric power of the pure met-al, having no residual resistivity, is given by
n'k'T _ ^ •ipr *2d "^t A Sy. = ^^ (III. 62) P. 1 + 1±-SPd
Next the proof of ( I I I . 6 0 ) i s given. From ( I I I . 5 9 ) . ( I I I . 5 7 ) and (III.46) follows:
. « J . ^ «J Tt^fe'r / P t * t + P r * r ^ ^ ( P d ^ _ P t ^ t ^ P r ^ r x ^ ^ 2 d = - S t , r . S d - 5 t , r = - 3 , ^ i ^^;^^^ p ^ , p ^ > 3 e ; ( P t + P r ) ( P t * t + P r * r ) + ( P t + P r ) S ( p d * d ) - ( P t + P r ) ( P t * t + P r * r ) - ( P t * t + P r * r ) 2 p d ( p t * P r + S P d ) ( p t + P r > n^k^T ( P t + P , ) S ( P d ^ d ) - ( P t J C t + P r ^ r ) S P d 3 e ; ( P t + P r + 2 p d ) ( P t + P r )
, , , , , S(p,x,) - P* 'f' ' 2pd
Tl2fe2r Pt-^Pr 3 e ; Pt+p,+2pd S ( P d « d ) P t * t + P r * r 3 e ; Pt + Pr 1 + Pt-^Pr SPdOn account of (III. 61), (III.46) and (III.33) this can be written
as: A-S^d which i s equal to ( I I I . 60). ^^^x - S 3 e ; *2d -^t.r 1 + P t , r Spd Case c
The thermocouple now has one leg (D) c o n t a i n i n g one type of l a t t i c e defects, with r e s i s t i v i t y p . and the other one (E) hav-ing, b e s i d e s the same type as occurs in (D), s t i l l more t y p e s .
Pf^Pp-'PcJo^^Pd
c:
Pt + Pf-^Pdo
Figure Ill.i Coldworked thermocouple, case C; the thermoelectric power is given
by formula (III.67).
The e l e c t r i c a l r e s i s t i v i t y of leg D i s given by p^ j. ^ = Pt •••
The absolute t h e r m o e l e c t r i c power of leg D i s , analogous to ( I I I . 4 9 ) given by: . ^ 2 L 2 •St , r , d o = Tt'fe'T Pt ^ t + Pr * r + Pdo * d o 3 e ^ Pt + Pr * Pd •nH^T 3 e ; t . r . d o ( I I I . 6 3 ) ^ Pt * t ^ Pr ^r ^ Pdc *--Pt + Pr * Pdo With ^ t . r , d o ='''"' ''' ~' ^''° ''° ( I I I . 6 4 )
The absolute t h e r m o e l e c t r i c power of leg E can e a s i l y be de-rived from ( I I I . 57) and i s given by:
Tt^fe'r Pt ^t ^ Pr ^r ^ Pdo ^do + ^ (Pd * d ) 3 ^ ; Pt * Pr + P d o * 2 p ,
<j U-R-i Kt -^t " b-r •'•r " Kdo -^do ^ ^ VPd -^d' ,.,^, -^,
•^t.r.do,2d " ~ - : 7 - 7 - \ : -z: (III.65)
The thermoelectric power, measured on the thermocouple of fig-ure I I I . 4, i s according to ( I I . 4) given by:
^ ^ d = - S t , r , d o , 2 d - ' S t . , . d o ( I I I - 6 6 ) •^t.r.do,2d ^ d S t _ , , d o are given by ( I I I . 65) and ( I I I . 6 3 ) .
Formula ( I I I . 6 6 ) can be simplified again to
-^^^^x -S . ^ 3 e ; 2d -^t.r.do
A ^ d = ^2 ( I I I . 67)
1 4. Pt , r , do
Spd
With xgd given by ( I I I . 6 1 ) . A proof of formula ( I I I . 6 7 ) i s not given here.
We now have derived the basic formulae ( I I I . 5 4 ) , ( I I I . 6 0 ) and ( I I I . 6 7 ) , which w i l l be used in analyzing our experiments. This analysis will be carried out in Chapter VI.
If we assume t h a t for each type of l a t t i c e d e f e c t s the e l e c -t r i c a l r e s i s -t i v i -t y , as for i -t s dependence on -the Fermi energy of the e l e c t r o n s , i s given by ( I I I . 3 0 ) , so that
Pj = const I ( I I I . 68) we get, on account of ( I I I . 50) and ( I I I . 4 5 )
Xd = a:, = + 1 ( I I I . 69) for each type of l a t t i c e defects.
Formula ( I I I ; 5 4 ) of case A, then reduces to 712^27-AS: = or, on account of ( I I I . 3 3 ) 3 e ; - 5 t . ,
l . £ i
-Pd 3 e ; " ' ^ ' • ' Pt * Pr x^^ = ^-^—2- = + 1 ( I I I . 71) A S* = —^ ( I I I . 70) 1 + PdThis formula i s i d e n t i c a l to ( I I I . 2 6 ) , except for the s u b s t i t u -t i o n of 4/lx by P f
I n s e r t i n g of ( I I I . 6 9 ) i n t o formula ( I I I . 61) y i e l d s S (pd ' d )
2 p d and ( I I I . 6 0 ) of case B reduces to:
A S* = ^^ ( I I I . 72) ^ P» + P
2 Pd
This formula can be derived at once from ( I I I . 7 0 ) by s u b s t i t u t i n g Pd by
SPd-The refinement introduced in t h i s paragraph with r e s p e c t t o the preceding paragraph, i s thus based on the assumption t h a t x^ has a d i f f e r e n t value for each type of l a t t i c e d e f e c t s , and i s not given by ( I I I . 6 9 ) as was assumed in the preceding paragraph, where the formula ( I I I . 1) of Kohier and Sondheimer was used,
In Chapter VI the value of x^ will be calculated from the ex-periments. On the o t h e r hand, i t i s now p o s s i b l e to d i v i d e the t o t a l e l e c t r i c a l r e s i s t i v i t y introduced by coldwork, i n t o d e f i -n i t e p a r t s , each of which belo-ngs to o-ne type of l a t t i c e defects, characterized by i t s own value of x^.
41
C h a p t e r IV
D E S C R I P T I O N OF T H E A P P A R A T U S A N D T H E E X P E R I M E N T A L M E T H O D S
§ 1. Methods for coldworking. The r o l l i n g apparatus
Metal wires can be coldworked by s t r e t c h i n g (maximum elonga-t i o n abouelonga-t 30% aelonga-t l i q u i d a i r elonga-temperaelonga-ture for elonga-the d u c elonga-t i l e meelonga-tals), torsion (deformation inhomogeneous over the cross area), drawing through d i e s (not e a s i l y c a r r i e d out in l i q u i d a i r ) and by r o l l -ing.
For our purposes a high degree of coldworking i s d e s i r a b l e in view of t h e s m a l l n e s s of the t h e r m o e l e c t r i c e f f e c t s . In some cases the method must be applicable in l i q u i d a i r without l i m i t a -t i o n as -to -the degree of coldworking. In a l l cases a deforma-tion homogeneous over the cross area has to be preferred.
On account of these considerations, coldworking in l i q u i d a i r was done by r o l l i n g . Coldworking at roomtemperature was e s t a b -l i s h e d by drawing through Widia d i e s (tungsten carbide) and in some cases by r o l l i n g .
The r o l l e r actually used had r o l l s of s t a i n l e s s s t e e l , with a diameter of 10 mn. The distance between the r o l l s could be varied with the r o l l e r being in l i q u i d a i r . The bearings were made of Akulon, as any o i l or grease f a i l s at liquid a i r temperature. The disadvantage of a s l i g h t v a r i a t i o n in the thickness of the r o l l e d wire, due to e l a s t i c e f f e c t s in the bearing m a t e r i a l , could not be avoided. The r o l l e r was driven e l e c t r i c a l l y , with a speed of about 6 mm/sec.
The whole arrangement i s given in f i g u r e I V . 1 . In o r d e r to get a s t r a i g h t r o l l i n g , the wire to be r o l l e d was guided by the nylon wires N which were, by means of p u l l e y s , s t r a i g h t e n e d by the weights W. Before f i l l i n g the container with l i q u i d a i r , a l l spurs of moisture or o i l on the moving p a r t s of the r o l l e r had to be removed carefully, so as t o ensure good operation. In 7 to 10 runs the wire was rolled from i t s o r i g i n a l diameter of 0,58 mm t o i t s final thickness of about 0.1 mm. The elongation was indicated on the s c a l e S. After r o l l i n g , the r o l l e r was removed from the container together with i t s mounting desk MD and the pulleys. The rolled wire s t i l l remained in liquid a i r . Pieces of the
appropri-^7« 1 2 1 U 5 '\ S
w
-Figure ÏV. 1 The r o l l i n g apparatus. N = nylon wires W = weights S = scale R = r o l l e r (schematically) MD = mounting desk for r o l l e r LA = liquid a i rTI = thermal i n s u l a t i o n .
ate s i z e were cut off and mounted in the thermocouple ^ p a r a t u s (§ 2) and in the r e s i s t i v i t y apparatus ( § 5 ) , which were placed on the bottom of the container after r o l l i n g was finished. During a l l these t r e a t m e n t s care had t o be taken t h a t the r o l l e d wire always remained below the l i q u i d a i r surface so as t o avoid any uncontrolled recovery.
§ 2. The apparatus for the coldworked thermocouples
The q u a n t i t i e s t o be measured a r e : the cold j u n c t i o n tempe-r a t u tempe-r e , the hot junction tempetempe-ratutempe-re and the thetempe-rmoelecttempe-ric fotempe-rce of the coldworked thermocouple. In a l l measurements the cold j u n c t i o n t e m p e r a t u r e w a s ' k e p t c o n s t a n t in l i q u i d n i t r o g e n o r l i q u i d helium under atmosferic pressure (77 and 4.2°K
respective-43
figure IV. 2
The helium thermocouple apparatus. B = insulating bar RT P = Insulating plate N T = german silver tube ID HC = constantan heating coil CD
RS = radiation shield SC C = german silver cap
rubber toroid nut
inner Dewar vessel outer Dewar vessel sealed cork
l y ) . The hot j u n c t i o n temperature was determined by means of a c a l i b r a t e d copper constantan thermocouple, with i t s cold j u n c t i o n a l s o in l i q u i d n i t r o g e n o r helium. The t h e r m o e l e c t r i c force of the coldworked couple was measured with a p o t e n t i o m e t e r ( § 3 ) . We will now give a d e s c r i p t i o n of the two apparatus used for mounting of the coldworked thermocouples, one apparatus was de-signed for measurements down t o l i q u i d helium temperature. The o t h e r apparatus was designed for mounting of the coldworked wire in liquid a i r .
The thermocouple apparatus for measurements down to liquid helium temperature
This apparatus was used for measuring the thermoelectric force of wires drawn at roomtemperature. The cold j u n c t i o n s were kept
in l i q u i d helium o r n i t r o g e n . The apparatus w i l l be r e f e r r e d t o sometimes as the helium thermocouple apparatus.
Four coldworked wires and two wires in the undeformed s t a t e were soldered c a r e f u l l y with Wood's metal t o a s i l v e r soldered j u n c t i o n of a copper c o n s t a n t a n thermocouple on top of an i n -s u l a t i n g bar B. The end-s of the-se wire-s were -soldered, again with Wood's metal, to small copper d i s c s on p l a t e P. So were the ends of the copper constantan couple. To each d i s c a copper wire was s o l d e r e d which, through a german s i l v e r tube T, leaded to t h e potentiometer. The j u n c t i o n s on top of bar B are the hot j u n c -t i o n s . The cold j u n c -t i o n s are formed on -the copper d i s c s . Wood's metal was used for s o l d e r i n g the coldworked wires in view of i t s low melting temperature (about 70°C, 340°K). So uncontrolled r e -covery of the coldworked wires was prevented.
The hot j u n c t i o n s were warmed up by a constantan heating c o i l HC. A r a d i a t i o n s h i e l d RS was placed between the h e a t i n g c o i l and the wires leading to the potentiometer. The tube T could be moved up and down through t h e german s i l v e r cap C; i t could be locked vacuumtight with a rubber toroid RT and a nut N. The wires to the d i s c s and the h e a t i n g c o i l entered the tube T through a conical glass tube, which was closed vacuumtight by a sealed cork SC.
The cap C was mounted vacuumtight on the inner Dewar v e s s e l , which contained the l i q u i d helium. The o u t e r Dewar v e s s e l was f i l l e d with l i q u i d a i r . The height of the helium l e v e l could be checked visually through an opening over the whole length in the coating of the Dewar v e s s e l s .
45 The nitrogen thermocouple apparatus (allowing mounting of the coldworked wire in liquid air)
This ^paratus was only used with the cold junctions in liquid nitrogen. Therefore i t will be referred to sometimes as the n i -trogen thermocouple apparatus.
CCW uw RW MS CP ML SSA Figure IV. 3
Sideview and frontview of the nitrogen thermocouple apparatus. undeformed wire B rolled wire CCW mounting screws HE copper plates P mica layers L soft solder area
bolts
copper constantan wires heating element
pin for HE
leads to potentiometer and spotlight galvanometer. As soldering can not be carried out in liquid air, the junc-tions of the coldworked thermocouple were made by pressing. This may cause an extra deformation, especially of the soft undeformed wire. Only if a temperature gradient exists over this deformed
