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Determination of the Mean Free Path of Conduction Electrons.

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Determination of the Mean Free Path of
Conduction Electrons')
B y R. K l e i n b e r g and F . J . B l a t t
Abstract
Aii earlier paper by S c h e f f e r s [Ann. Physik 33, 29 (1956)l is criticized.
It is shown that S c h e f f e r s result for the magnetoresistance ratio of a metal
is based on incorrect boundary conditions, and that the agreement with experiment iii the case of the noble metals is fortuitous.
Scheffers2) suggested a method for calculating the number of free electrons, N , and the electronic mean free path, I, in a metal which has a n assumed
spherical energy surface. He started wit>htJhs classical equation of motions)
mr f my i= e
[E + :
x H]
(1)
with
E
= (Ez,
0, 0),
H
= ( 0 , 0, H z ) , J , = Ne C, = (T E ,
and showed that
(T"=--
y
Ne2
m y2
+ (2 w # '
From t,his he obtained
Combining this result wit'h the well known expression
v
0 = -m-
Ne2 1
one obtains
and
Using the K o h l e r diagrams for copper, silver, and gold, S c h e f f e r s
deduced numerical values for N and eoIo which are in very good agreement
with those obtained from anomalous skin effect measurements.
~-
~
Supported in part by the Office of Scientific Research, US Air Force, and the
Office of Ordnance Research, US Army.
2) Helmut Scheffers, Ann. Physik 18, 29 (1956).
3, The notation is that adopted by Scheffers.
1)
R. Kleinberg and 3’. J. Blatt: Determination of tbe iMeun Free Path of Electrons 63
It is now firmly established that the F e r m i surface in copper touches the
B r i l l o u i n Zone a t thc hexagonal zone faces4). Thus, it deviates considerably
from the spherical shape assumed by S c h e f f e r s . It is most probable that the
F e r m i surfaces in silver andgold resemble closely thatin copper 6). Moreover, it is
also well known that standard free electron theory predicts a vanishing magnetoresistance under the assumption of a spherical energy surfaces). Indeed, a
theoretical account of the observed magnetoresistance of copper demands a
pronounced deviation from sphericity’). It is, therefore, curious that Sc hef f e r s
should have arrived a t a non-vanishing result for A@/?== 0, and most surprising
that his result is in good agreement with experimental observation.
We wish t o point out that the non-vanishing result obtained by Schef f e r s
is a direct consequence of his choice of incorrect boundary conditions. The
numerical agreement is thus entirely fortuitous.
I n the normal experimental arrangement for measurement of magnetoresistance the boundary conditions imposed are
J
= (Jx,
0,
H = (0, 0, H,).
0)
This situation leads t o a H a l l field E , in addition to the externallyimposed
field E,, i. e. E = (Es, E,, 0). The resistivity is defined by
E , E Cose
e=K
(7)
J-
where 8 is the H a l l angle. I n the range of applicability of t,he B o l t z m a n n
transport treatment 8 is small and proportional t o H .
I n contrast, S c h e f f e r s imposes the boundary condition E = (Ex,
0, 0).
It follows that, for H,
0, J must have a transverse component, J , as well as
longitudinal component J1:8).
S c h e f f e r s now defines the resistivity by
+
Thus, we have
<
+
@#) = C, sec2 8 = Q [I
a ~ 2 1 8,
I.
(9)
I n this way, even though A g / e H = E 0, S c h e f f e r s obtains a nonvanishing
magnetoresistance ratio
where R = 1/Ne c is the H a l l constant.
4, A. B. PiDDard. %D.
Proaess
in Phvsics. Vol. XXIII. D. 176 (The Phvaical
”
Society, London-1960).
s, P. G. Klemens, Handbuch der Physik, Vol. XIV, p. 242. (Springer Verlag, 1956).
6) A. H. Wilson, The Theory of Metale (Cambridge University Press, 1963).
7 ) M. G. Priestlev, Philos. Mag. 6, 111 (1960).
*) ,,Dabei ist nadich angenommen, dad Strom und konstante Feldstcirke E die
x-Richtung haben und das Magnetfeld H = H , senkrecht dazu wirkt.“ This “assumption"
presupposes a physical impossibility. Near the end of his paper Scheffers reproduces the
standard treatment of the Hall effect and here this assumption is disregarded.
“
A
I
,
I
64
Annalen. der Phyeik. 7. Fdge. Band 9. 1961
Of all the monovalent metals sodium probably has a F e r m i surface most
nearly spherical. Using S cheff e r s expression, Eq. ( 5 ) , and the experimental
datas) we obtain values for N ranging between 10 and 20 lom ~ m - ~
This
. is
to be compared with N = 2.3 . 1022 om4 found by B a b i s k i n and S i e b e n m a n n l O )and N = 2.5. loz20111-3 from the H a l l coefficients). Evidently the
agreement becomes worse just where S c hef f e r s should expect it t o be best.
The rather good agreement in the cases of copper, silver and gold is clearly
fortuitous.
-
~~
D. K. C. MacDonald, Philos. Mag. 5,W (1957);Proc. physic. SOC.@ A , 290 (1950).
lo) Babiskin and Siebenmann, Ph.ysic. Rev. 107, 1249 (1957).
O)
E a s t L a n s i n g , Michigan (USA). Physics Department, Michigan State
University.
Bei der Redaktion eingegangen am 10. April 1961.
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electrons, free, mean, determination, conducting, path
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