The mean free path of electrons in metals

TL;DR: The mean free path of electrons in metals has been studied in this paper, where the authors show that electrons follow a straight line along the path of the electron in the metal atom.

Abstract: (2001). The mean free path of electrons in metals. Advances in Physics: Vol. 50, No. 6, pp. 499-537.

Summary

Introduction

• Because pedestrian crash rates remain lower than other collision types, surrogate measures such as traffic interactions are now used in road safety research to complement crash history.
• For recorded interactions (n=843), information was collected to characterize the behaviours of involved parties.
• A mixed-effect logit regression model was performed to assess the factors associated with interactions.
• Older adults were those more likely to be involved in an interaction event.
• Given the growing emphasis and adoption of active transportation in many cities, the number of interactions between pedestrians and vehicles during street crossings is likely to increase.

1. PEDESTRIANS CRASH RISK IN CITIES: WHAT TO MEASURE?

• A growing number of North American cities have been actively promoting nonmotorized transportation and developing road infrastructure to support the use of these travel modes.
• Crash statistics show that many unsafe conditions still exist for vulnerable road users such as pedestrians, partly because modern cities were (and are still) mostly built for cars [1-3].
• On the other hand, local pedestrian crashes can be considered “rare events,” at least from a statistical perspective [4, 5].
• As a result, surrogate measures such as traffic conflicts and interactions are increasingly used in road safety research as complementary to crash history [9, 10] in order to have a better portrait of the situation and plan road design accordingly.

1.1 Surrogate measures of crash risk: Traffic conflict techniques and interactions

• The concept of “traffic conflict techniques” was first proposed in the 1960s as a complementary approach to typical collision-based safety analysis.
• Those common pedestrian-vehicle conflicts, referred to as “interactions,” can be seen as part of the road safety continuum shown in the diamond-shaped representation proposed by several authors (see Figure 1) [6, 7, 12].
• Moreover, even when interactions do not lead to injuries, they may be symptomatic of environments that are not adapted to pedestrians.
• Studying interactions can provide insight into the initial circumstances that may lead to crashes (or not).
• It is even more important to have a better understanding of interactions involving the most vulnerable pedestrians, namely children and seniors.

1.2 Objectives

• This paper seeks to provide a better understanding of the individual and environmental determinants associated with the occurrence of interactions between pedestrians and other road users (cars and bikes, other) during pedestrian crossings at intersections.
• As a secondary objective, it seeks to explore differences in interaction characteristics when comparing observed children, adult and senior pedestrians.
• By providing findings related to these objectives, the authors seek to strengthen the research background on pedestrian interactions through an important observational study.

2. INDIVIDUAL AND ENVIRONMENTAL DETERMINANTS OF

• Individual and environmental determinants of pedestrian crashes are well known and have unfortunately changed very little in the past 25 years, especially in the Western hemisphere [17].
• Compared to the general population, ageing pedestrians are overrepresented in crashes compared to their relative proportion of the population [20, 21]; up to 50% of all injured pedestrians in OECD countries are seniors [2].
• The cognitive complexity of dealing with this added infrastructure (needing to look both ways more than once) was thought to cause this effect.
• The presence of vehicles parked at the curb revealed contradictory effects in different studies: while Tom and Granié [43] show that pedestrians display more cautious crossing behaviour when there are no parked vehicles nearby, Yannis et al. [44] found that the presence of illegally parked vehicles at mid-block crossings makes pedestrians more careful because of reduced line of sight.
• In terms of behaviours associated with the number of interactions, the work of Kaparias et al. [3] shed light on the fact that “there are no generic behavioural criteria that can be used to examine lower severity interactions in different traffic situations.”.

3. ANALYTICAL MODEL

• The relatively small number of research on pedestrian-car conflicts and interactions has typically collected data either through video (and post-collection coding grid) or direct observation of behaviours (3; 6; 13; 36; 42).
• In addition, they either focus on one specific age group (e.g. seniors) or do not compare age groups within their sample.
• Finally, their definition of “conflict” is fairly narrow, including only events under a specific threshold such as a short time-to-collision indicator or a small yielding distance.
• In order to assess the occurrence of interactions between pedestrians and other motorized road users as an outcome, the authors posit that information on the individual, behavioural and environmental characteristics as well as location of the crossing must be observed.
• The manner in which the authors organized these concepts around their objectives is presented in Figure 2.

4. METHODS

• 1 Observation site and data collection Child, adult and senior pedestrians were observed crossing as part of two road safety research projects on the association between street environment and pedestrian road risk (one targeting children and the other, seniors).
• Data collection was undertaken between May and November 2013 in five different cities in the province of Quebec, Canada: Montreal, Laval, Longueuil, Quebec City and Gatineau.
• Those cities were chosen for their representativeness of the North American urban form and for their involvement in walk-to-school programs.
• Observers were given a list of initial intersections within the vicinity of schools to observe children before and after school, but were allowed to move to other nearby locations if few adult and senior pedestrians were crossing there during the day.
• Teams of two observers (one for pedestrians, the other for interactions with vehicles) were positioned, one on each side of the street, slightly set back from the crossing to avoid contact with observed pedestrians (see Figure 3).

4.2 Street crossing environment characteristics

• As previously mentioned, the street crossing environment grid refers to the pedestrian crossing and surroundings at the curb.
• Three individual variables were retained here: age (estimated in 5 categories), gender and observed walking speed (categories from “very slow” to “running” when compared to an average adult speed) of the observed pedestrians (see Table 2).
• Therefore, interactions were recorded when the pedestrian’s path (blue line in Figure 3) and the driver’s path (red line) crossed while the pedestrian was still on the street (on the pavement, not curb).
• This definition recalls the one used by Kaparias et al. (3) in a shared-space context.

4.4 Statistical analyses

• The final database includes pedestrian crossing observations, interaction characteristics (if applicable) and crossing environment characteristics, including a unique ID for observation grouping by location.
• After providing descriptive bivariate analyses, a mixed-effect logit model was used to assess the correlates of interactions with other road users while crossing, accounting for the grouping of observations in selected crossing environments using a random effect [52].
• This model was computed in addition to a basic logistic regression model.
• Finally, Chi-squared tests were used on the interactions’ subset of data (n=843) to highlight significant associations between interaction characteristics and age group.
• All analyses were carried out using Stata 14.

5.1 Descriptive statistics of pedestrian observations and crossing sites

• Data collection resulted in 4,687 pedestrians observed at 278 crossing sites.
• As for children, only 17% of their sample was collected during summer months since most of the observation sites were near schools during school periods.
• Another hypothesis would be that seniors tend to cross more at intersections with traffic lights for a variety of reasons such as better trust in the ability to cross safely, as it was observed before [53].
• Lastly, a smaller number of observations were located near a bike path, located either directly at the crossing or at the intersection (21%) and a third of the observation had parked cars on one or both sides of the streets (Table 2).

5.2 Individual and crosswalk characteristics related to interaction occurrences

• Descriptive statistics and Chi-squared tests are presented in Table 2.
• Almost half of the observed pedestrian in that age group had an interaction with a vehicle or a bike.
• As mentioned earlier, crossing speed of elderly is likely a contributing factor, since very slow and slow speeds are significantly associated with more interactions.
• All crossing site variables are significantly associated with interactions.
• Crossing at intersections with both traffic and pedestrian lights was associated with more interactions.

5.3 Mixed-effect logit model

• To account for the grouping of observations by crossing sites, the authors modeled the binary variable of interactions (yes/no) in a mixed-effect logit model and compared the results to a simple multivariate logistic regression (see Table 3).
• Since nearly 30% of the variance in the logit model is explained by the grouping of observations in studied intersections, the authors can hypothesize that several of the significant results are now amalgamated in this coefficient (i.e.: the crossing site constant is also significant).
• Gender does not influence interaction probability in their models.
• The presence of four characteristics decreased the probability of having an interaction: when the crossing was on a one-way street, when crossing had a different surface material, when a bicycle path directly affected crossing and when a curb extension was present.

5.4 Behaviours during interactions and age difference

• Turning to an analysis of recorded interactions (n=843), Table 4 provides cross tabulations for the behavioural characteristics of interactions across age groups.
• Of the six variables the authors tested, five were significant associated, based on Chi-squared tests.
• Vehicle type was the only variable that was not significant.
• The 65 to 79 years old group were more frequently involved in interactions with vehicles at constant speed and in cases when the pedestrian was at fault, in a proportion similar to adults (17%).

6. DISCUSSION

• This study found that in more than 4,000 observed street crossings in a variety of different environments, interactions with other road users occurred in nearly 18% of cases.
• First, a majority of their observations were taken within school zones (mean distance of 105 meters between observed crossing and nearest elementary school) and during school period (83% of them outside summer time), which means that the road infrastructure, if it follows prescriptions, should be safer.
• Inversely, three variables were associated with fewer interactions: oneway streets, different crossing surface material, and presence of a bike path at the crossing.
• Given the similarities in the factors associated with vehicle-pedestrian interactions and those known to influence collisions occurrence and rate, interactions may be a reasonable proxy for potential collisions.

6.1 Limitations

• As with any field survey, limitations from the data gathered through observations exist.
• Such field observations are known to be more valid than other automatic traffic-conflict techniques, but they are also vulnerable to intra and inter-observer variability [8].
• Another example lies in the possibility that observers may have misreported more subjective variables such as walking speed and age, even though few categories were used.
• The data is therefore partially representative of all the possible time period a pedestrian can cross streets, such as at night or during the Winter.
• Also, nighttime is a source of insecurity both for children and seniors, leading them to go out less frequently than during daylight hours.

7. CONCLUSION

• The main objective of this paper was first to explore the relationship between individuals and crossing characteristics and interaction occurrence between pedestrians and other road users (mostly vehicles) and second, to see if these particular interactions led to different reactions and behaviors for pedestrians of different age groups.
• The authors unique data set of observations of street crossings in different urban road environments in Quebec, Canada is one of the strengths of this study.
• The authors results provide a better understanding of the interaction between pedestrians and vehicles in different crossing environments and for different age groups: Senior and child pedestrians were found to have very different interaction pattern compared to adults.
• This is where their results are the most valuable: adding to the knowledge of pedestrian-vehicle interactions.
• Such an approach can also help identify which engineering, urban design and enforcement programs are needed to ensure safe pedestrian crossings for all ages.

Figures

Figure 13. Variation with orientation of surface conductivity of single crystals of tin.

Figure 4. Theoretical conductivity of thin wires in a longitudinal magnetic ®eld.

Figure 8. Theoretical variation of resistance of thin metallic ®lms with magnetic ®eld perpendicular to surface of ®lm, assuming di use surface scattering.

Figure 9. Theoretical variation of Hall ®eld Ez across a thin ®lm of thickness a ˆ 0:64 free path, assuming di use scattering at the boundaries

Figure 2. Experimental results on the resistivity of mercury wires at 3.58K and 2.58K, plotted logarithmically against the wire diameter. The full lines are theoretical curves for di use scattering.

Figure 5. Experimental results on a 30 m sodium wire in a longitudinal magnetic ®eld at various temperatures, with theoretical curves.

Figure 6. Experimental results on sodium wires of various diameters in a transverse magnetic ®eld. Diameters: 1, 20 m; 2, 30 m; 3, 66 m. Temperature ˆ 4:2 K.

Figure 11. Experimental results on the surface resistance at low temperatures. (I) Experimental points ®tted to p ˆ 0 curve. (II) Attempt to ®t to p ˆ 1 curve.

Table 3. Estimates of ¼0=l and n=na from observations on the anomalous skin effect.

Table 1. The resistivity of thin metallic ®lms divided by the resistivity of the bulk metal.

Figure 1. Experimental results on the resistivity of tin foils at 3.88K, plotted logarithmically against the foil thickness. The full line is the theoretical curve for di use scattering …p ˆ 0†.

Figure 10. The theoretical high-frequency surface impedance of metals at low temperatures.

Figure 12. Theoretical absorption coe cient of silver at liquid-helium temperatures for p ˆ 1, plotted logarithmically against the wavelength ¶.

Figure 3. Construction for the determination of P…®; Á0† (see text).

Table 2. The resistivity of thin wires divided by the resistivity of the bulk metal.

Figure 7. Theoretical Hall coe cient of thin metallic ®lms (in small magnetic ®elds) divided by the Hall coe cient of the bulk metal.

Full text

The mean free path of electrons in metals*
E. H. Sondheimer{
Royal Society Mond Laboratory, Cambridge
Contents page
1. Introduction 499
2. Conduction in thin ®lms and wires 505
3. Magnetic e ects in thin conductors 515
4. The anomalous skin e ect 524
Acknowledgement 536
References 536
1. Introductio n
1.1. The foundations of the modern electron theory of metals were laid at the
beginning of the present century, when the existence of a gas of free electrons was
postulated by Drude in order to explain the conducting properties of metals; the
behaviour of the electrons was subsequently analysed by Lorentz by means of the
statistical methods of the dynamical theory of gases. The chief success of the Drude±
Lorentz theory was the prediction of the Wiedemann±Frenz law connecting the
electrical and thermal conductivities, but later developments revealed an increasing
number of serious di culties, outstanding among them the inability of the theory to
explain why the conduction electrons do not contribute appreciably to the speci®c
heat of a metal. This paradox was not resolved until the advent of quantum
mechanics, when Pauli and Sommerfeld applied the Fermi±Dirac statistics to the
free electrons in a metal, showing that in this way most of the contradictions could
be reconciled.
The Drude±Lorentz±Sommerfeld theories are essentially formal in character.
They involve as arbitrary parameters the number n of free electrons per unit volume,
which is assumed to be of the same order as the number of atoms per unit volume,
and the mean free path l of the electrons which is to be determined from a
comparison of theory and experiment. So far as the electrical conductivity
¼
0
is
concerned, the results of the Sommerfeld theory are summarized by the formulae
n ˆ
8p
3
m·vv
h
3
;
…1a†
¼
0
ˆ
n
°
2
l
m·vv
;
…1b†
Advances in Physics, 2001, Vol. 50, No. 6, 499±537
Advances in Physics ISSN 0001 ±8732 print/ISSN 1460±6976 online # 2001 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080 /0001873011010218 7
* This article was the ®rst to appear in Advances in Physics, originally published in volume 1,
1952. It had attracted 1030 citations by October 2001, and is ranked 5 in the index of articles attracting
more than 100 citations.
{ Fellow of Trinity College, Cambridge.

in which h is Planck’s constant, ¡
°
is the charge and m is the mass of an electron, and
·vv is the velocity of an electron at the surface of the Fermi distribution.
¼
0
therefore
depends on n, l and fundamental constants only. In order to obtain the correct order
of magnitude for the conductivity and the correct temperature variation, it is
necessary to assume that l is of the order of several hundred interatomic distances
at ordinary temperatures, and increases rapidly in a pure metal towards very low
temperatures.
Such long free paths whic h vary with temperature are very di cult to explain on
classical theory, bu t they can be understood on the basis of the recent developments
in the theory of metals. These were initiated by Bloch and elaborated by a great
many authors; they have been concerned with a detailed quantum-mechanica l
analysis of the motion of electrons in a crystal latice , and they have made it possible
to give precise meanings to the two fundamental concepts of the `number of free
electrons’ and the ` mean free path’ and to obtain numerical estimates of these
quantities in certain cases. The electrons in a metal are regarded as distributed over a
number of energy bands, ®lling most of them completely. All the electrons are free to
move through the lattice, but only those which are contained in incompletely ®lled
energy bands can contribute to the resultant current and are to be regarded as free
electrons for the purposes of conduction theory. The number of free electrons in a
metal is of the same order as the number of atoms: the precise number, however,
depends on the detailed con®guration of the energy bands and need not be a simple
multiple or submultiple of the number of atoms. The variation of n w ith temperature
is negligible, since at ordinary temperatures the fre e electrons form a highly
degenerate Fermi±Dirac gas.
An electron can move freely through a perfect and rigid crystal lattice and ther e is
no resistance. In a pure metal a ®nite free path is caused by the thermal vibrations of
the lattice and is of the same order as the average wavelength of the sound waves in
the metal, which is large compared with the interatomic distance and is increased by
lowering the temperature. The free path does not increase inde®nitely, however, as
the temperature is lowered, and at very low temperatres it tends to a constant
`residual’ value l
r
which is determined by static lattice imperfections such as the
presence of impurity atoms, and which is of the order of the distance between the
impurities.
The following theoretical formula may be deduced for the electrical conductivity
of a metal, subject to many simplifying assumptions concerning the interaction
between the electrons and the lattice vibrations (see, for example, Wilson 1936,
Chapter VI):
1
¼
0
ˆ
m·vv
n
°
2
l
r
‡
m
2
± ²
1=2
9ph
2
C
2
8n
D°
2
Mk
Y±
3=2
T
Y
5
…
³=T
0
z
5
dz
…e
z
¡ 1†…1 ¡ e
¡z
†
:
…2†
Here k is Boltzmann’s constant,
±
is the Fermi energy level (
±
ˆ
1
2
m·vv
2
†,
Y
is the
Debye temperature, M is the mass of an atom,
D
is the volume of the unit cell, and C
is a constant which determines the interaction between th e electrons and the lattice.
According to this formula the `ideal’ and `residual’ resistances are additive, and the
ideal resistance is proportional to T at high and to T
5
at very low temperatures. (For
commercially pure metal s the residual resistance can be neglected at ordinary
temperatures.) The temperature variation of the resistance predicted by (2) is,
generally speaking, in good agreement with observation, particularly for the
E. H. Sondheimer500

monovalent metals. The absolute value of the free path at any temperature can be
estimated by combining (1) and (2) and substituting reasonable value s for the
parameters. The chief uncertainty concerns the magnitude of the interaction
constant C, estimates of which can be obtained by numerical integration if the wave
functions of the conduction electrons are known; it is of the same order as the Fermi
energy
±
. Th e estimates of the free path are of the right order of magnitude, but
precise numerical values cannot be obtained in this way.
1.2. An important conclusion to be drawn from the detailed quantum-mechan-
ical theory is that the simple Sommerfeld treatment of the conduction phenomena
remains co rrect within certain limits. In the Sommerfeld theory the electrons are
regarded as perfectly free, their energy being proportional to the square of the
velocity; this remains approximately true for electrons moving in a lattice, but th e
mass m which appears in equations (1) and (2) must be regarded as an e ective mass
which is of the same order of magnitude as, but not necessarily equal to, the mass of
a free electron. This model of `quasi-free’ electrons, which is implicit in the derivation
of (2), applies most closely to the monovalent metals in which the conduction
electrons are all contained in a single energy band; for these metals, moreover, the
number of conduction electrons should be precisely one per atom. In multivalent
metals, in which the electrons occupy more than one band, the model may still be
used to give a semi-quantitative description of the simpler conduction phenomena,
but quantities such as n and m must then be regarded as representing certain
averages of the numbers of electrons and the e ective masse s of the electrons in the
various bands, and the precise numerical values have no immediate physical
signi®cance. It would, of course, be possible to consider more complicated models,
for example one in which the conduction electrons are contained in two overlapping
energy bands. In the present state of the theory of metals, however, it is impossible to
work out a theory which fully takes into account the electronic structure peculiar to
any particular metal, and instead of introducing a large number of parameters of
doubtful physical signi®cance it is best to work with the simplest model which gives
reasonable results. For more complicated conduction phenomena, however, par-
ticularly those which are associated with the presence of a magnetic ®eld or with
anisotropy e ects, the free-electron model is entirely inadequate (it leads, for
example, to a zero magneto-resistanc e e ect), and eve n a qualitative theory can
only be obtained by using a model in which the energy surfaces do not form a singly-
connected set of spheres.
In the Sommerfeld theory the free path is most conveniently introduced through
the time of relaxation
½
, which is de®ned as follows. Let
v
ˆ …v
x
;
v
y
;
v
z
† be the
velocity of an electron, and let 2…m
=
h†
3
f …
v; r
†d
r
d
v
be the number of electrons in
the volume element d
r
ˆ dx dy dz which have their velocities in the range
d
v
ˆ dv
x
dv
y
dv
z
; f is the distribution function as usually de®ned in Fermi±Dirac
statistics. Suppose that some non-equilibrium distribution function is set up by a
system of external forces which are suddenly removed; the rate of approach
to equilibrium under the in¯uence of collisions alone is then supposed to be
given by
@
f
@
t
µ ¶
coll
ˆ ¡
f ¡ f
0
½
;
…3†
Mean free path of electrons in metals 501

where f
0
is the equilibrium distribution function. The time of relaxation need not be a
constant and may depend, for example, on the velocity; if v is the mea n velocity of
those electrons to which
½
refers, the corresponding free path l is de®ned by l ˆ v
½
.
In general ‰
@
f
=@
tŠ
coll
takes the form of an integral operator which does not reduce to
the simple form (3); it is then impossible to de®ne a free path in any natural way. The
detailed theory of the conduction mechanism in metals shows, however, that a free
path does exis t in the above sense under certain conditions for quasi-free conduction
electrons; thus it can always be de®ned for scattering by randomly distributed
impurity atoms, and it also exists for scattering by lattice vibrations if the
temperature is above the Debye temperature. So far as the electrical conductivity
is concerned, it is a reasonble approximation to assume that a free path can be
de®ned for all temperatures, so that equation (1) holds and the temperature variation
of the free path is the same as that of the electrical conductivity. An approximate
theoretical expression for l in terms of the atomic constants of the metal can be
obtained by combining equations (1) and (2). It must be borne in mind, however,
that the free path associated in this way with the electrical conductivity is not
necessarily the same as the free path associated, for example, with the thermal
conductivity.
1.3. With all the provisos mentioned, then, the simple Sommerfeld picture of the
free electrons in a metal and their mean free path retains its validit y and may be used
to discuss the conduction phenomena. It is therefore an important problem to
determine the various parameters of the theory by as many independent methods as
possible; such estimates are of interest in themselves in providing information about
the electronic structure of metals, and they serve as a valuable check on th e
consistency of the free-electron theory. Furthermore, since the calculation of the
free path from fundamental principles is highly complicated and involves many
drastic approximations , it is desirable to have methods by which l may be estimated
directly from observational data.
The simplest procedure (Mott and Jones 1936) is to compare equation (1) with
the observed electrical conductivity, but this gives only n
2=3
l and l cannot be
obtained unless n can be estimated independently.{ It is, however, possible to obtain
l directly by measuring the conductivity under conditions where the fre e path may be
compared with some other characteristic length in the metal. E ects of this type have
attracted much attention in the last few years, and it is with them that we shall
henceforth be concerned.
The most obvious method is to us e a thin ®lm or wire and to arrange that the free
path is comparable in magnitude with the thickness or diameter of the specimen; the
arti®cial limitation of the free path by the boundaries of the specimen causes an
increase in the resistivity above its value in the bulk metal, and this may be used to
deduce the ratio of free path to thickness or diameter. This topic i s reviewed in
section 2.
In section 3 we consider the more complicated e ects which occur when a thin
specimen is placed in a magnetic ®eld. The ordinary bulk magneto-resistance e ect
depends in a complicated way on the binding of the electrons in the lattic e and is zero
E. H. Sondheimer502
{ It should be remarked that the value of l de®n ed by Mott and Jones and tabulated on p. 268 of
their book is twice the free path de®ned here; the latter is the physically relevant quantity.

for quasi-free electrons; in thin specimens, however, where boundary scattering of
electrons is important, the alteration of the electron trajectories in a magnetic ®eld in
general leads to a non-zero change of resistance even if the electrons are regarded as
free. These curious `geometrical’ e ects, being classical in nature, are entirely
di erent from the bulk e ect and are essentially simpler to understand. The details
of the phenomena vary with the shape of the specimen and with the relative
con®gurations of specimen, curren t and magnetic ®eld; their analysis in all cases
involves a new quantity with the dimensions of length, namely the radius
r
0
ˆ m·vvc
=°
H of a free-electron orbit in a magnetic ®eld H. The experiments therefore
give, in addition to the free path, a direct estimate of the momentum m
·
vv of the
electrons at the surface of the Fermi distribution, and hence of the number of free
electrons according to equation (1a). The method is, however, severely restricted in
practice by the disturbing e ect of the bulk magneto-resistance phenomenon.
In section 4, ®nally, we consider the so-called anomalous skin e ect in metals.
This is a more sophisticated side e ect in which the free path is compared, not with
the physical dimensions of the specimen, but w ith the distance to which a high-
frequency electric ®eld penetrates into the metal. Experiments on the high-frequency
skin resistance of metals allow values of l
=¯
to be deduced, where
¯
is the classical
skin penetration depth, an d sinc e
¯
depends only on the frequency and the d.c.
electrical conductivity, the free path can again be obtained directly from experi-
mental magnitudes.
1.4. These phenomena present interesting problems from both experimental and
theoretical points of view. Let us consider the orders of magnitude of the various
characteristic lengths. The free path in metals at room temperatures is of the order of
10
¡5
cm or less, but in a pure metal at liquid-helium temperatures the high values of
the conductivity indicate that it may be as large as 10
¡2
cm. It is clear, therefore, that
extremely thin ®lms would be required in experiments on the size e ect at normal
temperatures; such ®lms are di cult to prepare and usually show subsidiary resistive
e ects, often tim e dependent, which tend to obscure the pure geometrical limitation
of the free path with which we are alone concerned. In order to obtain results which
are free from ambiguity, it is therefore essential to carry ou t the experiments at very
low temperatures where relatively large specimens (of thickness 10
¡3
cm) may be
used. The necessity for using specimens of this order of thickness becomes even
greater when we consider the e ects which take place in a magnetic ®eld. These
e ects show up when the orbit radiu s r
0
is comparable with the thickness of the
specimen. Since ·vv is of the order of 10
8
cm/sec, a magnetic ®eld of reasonable
magnitude (several kilogauss) corresponds to r
0
10
¡3
cm; and since r
0
is inversely
proportional to H, impracticably large magnetic ®elds would be require d in the case
of ®lms much thinner than this. Finally, in the anomalous skin e ect l must be large
compared with
¯
; the ratio l
=¯
is proportional to l
3=2
and to the squar e root of the
frequency, so that both very low temperatures and high frequencies are required. For
a pure metal at liquid-helium temperatures and for microwave frequenci es, l
=¯
is of
the order of 100.
One may say, therefore, that the size e ects are essentially low-temperature
phenomena. Since the free path at low temperatures varies from specimen to
specimen and is not a characteristic property of the metal, it is usual to measure
(or estim ate in some way) the value of the bulk conductivity
¼
0
which corresponds to
the free path l, and to express the results of the experiments in terms of the ratio
¼
0
=
l.
Mean free path of electrons in metals 503

Frequently asked questions

Q1. What are the contributions mentioned in the paper "The mean free path of electrons in metals" ?

In this paper, it was shown that the number of free electrons in a metal is of the same order as the number number of atoms, however, the precise number depends on the detailed conduction bands of the energy bands and need not be a simple multiple or submultiple of the numbers of atoms. 

Q2. What is the effect of the process of drawing wires of these metals?

The process of drawing wires of these metals produces an amorphous layer which is not easily removed by annealing, but which, in Chambers’ experiments, was removed by electrolytic polishing. 

Q3. What is the case of a thin wire in a longitudinal magnetic ®eld?

When the electric and magnetic ®elds are parallel, the magnetic force on the electrons is always perpendicular to the electric force; the authors can then regard the electric ®eld alone as producing a drift current in the usual way, and the magnetic ®eld simply as modifying the electronic trajectories. 

Q4. What is the eective conductivity of a conductor?

The e ective conductivity ¼ is obtained by integrating the current density over the cross-sectional area S of the conductor, and the ratio of ¼ to the bulk conductivity ¼0 may be written in the compact form¼ 

Q5. What is the simplest way to determine the distribution function of the electrons?

The problem is essentially a one-dimensional one, and the distribution function of the electrons may be written in the formf ˆ f0 ‡ f1…v; z†; …8†where the function f1 which has to be determined depends on the space variables only through z. 

Q6. What was the main purpose of the modern electron theory of metals?

The foundations of the modern electron theory of metals were laid at the beginning of the present century, when the existence of a gas of free electrons was postulated by Drude in order to explain the conducting properties of metals; the behaviour of the electrons was subsequently analysed by Lorentz by means of the statistical methods of the dynamical theory of gases. 

Q7. What is the conductivity of thin wires in a longitudinal magnetic ®eld?

The conductivity of thin wires in a longitudinal magnetic ®eld MacDonald, at the time of his discovery of the e ect, correctly interpreted its physical origin, and in particular he explained the simple decrease in resistance which occurs in a wire in a longitudinal ®eld as being due to the lessened infuence of scattering at the walls of the wire when the electrons are forced to pursue spiral paths around the lines of force of the magnetic ®eld. 

Q8. What is the distribution function of the electrons leaving each surface?

The distribution function of the electrons leaving each surface must then be independent of direction; equation (10) shows that this can only be satis®ed if the authors choose F…v† so that f1…v; 0† ˆ 0 for all v such that vz > 0 (that is, for electrons moving away from the surface z ˆ 0), and f1…v; a† ˆ 0 for all v such that vz < 0: 

Q9. What is the obvious method to determine the free path in a metal?

The most obvious method is to use a thin ®lm or wire and to arrange that the free path is comparable in magnitude with the thickness or diameter of the specimen; the arti®cial limitation of the free path by the boundaries of the specimen causes an increase in the resistivity above its value in the bulk metal, and this may be used to deduce the ratio of free path to thickness or diameter. 

Q10. What is the definition of the term sizee ect?

This term takes into account the non-uniform distribution in space of the conduction electrons which is characteristic of the sizee ect phenomena. 

Q11. What is the limit of zero magnetic ®eld?

In the limit of zero magnetic ®eld … ˆ 0†, equation (54) for the conductivity reduces to equation (16), and (55) gives a corresponding expression for the Hall coe cient of a thin ®lm in a vanishingly small magnetic ®eld. 

Q12. What is the advantage of using a magnetic ®eld?

The advantage of using a magnetic ®eld is that all the required information can be obtained from one specimen only, and it is not necessary to assume, as in zero-®eld experiments, that the free path in a thin specimen is the same as in a bulk specimen. 

Q13. What are the probable errors for copper, silver, gold, and aluminium?

The values for copper, silver, gold, and tin were obtained by Chambers,{ and are subject to probable errors of about 5%; the values for mercury and aluminium have been deduced from Pippard’s measurements and have larger probable errors. 

Q14. Why does the departure from theory in high ®elds interfere with the comparison between theory and?

The departure from theory in high ®elds is, in fact, due to the onset of the bulk e ect, but this does not interfere seriously with the comparison between theory and experiment.