Statistical Physics of Nanoparticles in the Gas Phase

420 Pages·2018·12.493 MB·English

by  Klavs Hansen

Springer Series on Atomic, Optical, and Plasma Physics 73
Klavs Hansen
Statistical Physics
of Nanoparticles
in the Gas Phase

Second Edition
Springer Series on Atomic, Optical,
and Plasma Physics
Volume 73
Editor-in-chief
GordonW.F.Drake,DepartmentofPhysics,UniversityofWindsor,Windsor,ON,
Canada
Series editors
JamesBabb,Harvard-SmithsonianCenterforAstrophysics,Cambridge,MA,USA
Andre D. Bandrauk, Faculté des Sciences, Université de Sherbrooke, Sherbrooke,
QC, Canada
Klaus Bartschat, Department of Physics and Astronomy, Drake University,
Des Moines, IA, USA
Philip George Burke, School of Mathematics and Physics, Queen’s University,
Belfast, UK
Robert N. Compton, Knoxville, TN, USA
Tom Gallagher, University of Virginia, Charlottesville, VA, USA
Charles J. Joachain, Faculty of Science, Université Libre Bruxelles, Bruxelles,
Belgium
Peter Lambropoulos, FORTH, IESL, University of Crete, Iraklion, Crete, Greece
Gerd Leuchs, Institut für Theoretische Physik I, Universität Erlangen-Nürnberg,
Erlangen, Germany
PierreMeystre, OpticalSciences Center, Universityof Arizona, Tucson, AZ,USA
The Springer Series on Atomic, Optical, and Plasma Physics covers in a
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fundamental ideas, methods, techniques, and results in the field.
More information about this series at http://www.springer.com/series/411
Klavs Hansen
Statistical Physics
of Nanoparticles
in the Gas Phase
Second Edition
123
KlavsHansen
Schoolof Science
Tianjin University
Tianjin, China
ISSN 1615-5653 ISSN 2197-6791 (electronic)
SpringerSeries onAtomic, Optical, andPlasma Physics
ISBN978-3-319-90061-2 ISBN978-3-319-90062-9 (eBook)
https://doi.org/10.1007/978-3-319-90062-9
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2ndedition:©SpringerInternationalPublishingAG,partofSpringerNature2018
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To Stefan, Simon, Rasmus and Jakob
Preface to the Second Edition
Several changes have been made to the text in this second edition relative to the
first,andnochapterhasbeenleftuntouchedintheprocess.Thechangesareoftwo
types. One is an expansion of the scope of the material covered. The other is the
introduction of new developments of the field. The revisions have resulted in two
new chapters and several new sections. In addition to these changes, uncountably
many minor changes have been made in figures and in the text to improve read-
ability.Apartofthepreparationofthiseditionwasthecorrectionoftheerrorsand
(mostly)misprintsspottedinthefirstedition.Duringthisprocess,itbecameclearto
me that while two errors may cancel each other, mistakes never do. Happily, most
mistakes were of minor importance.
The idea behind the book remains unchanged: To provide a guided tour of a
number of interesting phenomena in the field. A reader who compares this edition
with the first will realize that occasionally the presentation has changed radically.
Butalthoughtheflavormayhavechangedhereandthere,hopefullythenutritional
value still makes it worth for the reader to consume the dish.
I want to thank Takeshi Furukawa, Bernd von Issendorff, Vitali Zhaunerchyk,
andMathiasWeberforprovidingillustrativeexperimentaldata;PetrSlavícĕkforan
educational conversation on holy water at an Erice Workshop; and Hanna
Vehkamäki for enlightening me on the first nucleation theorem.
Corrections,suggestions,anderrorreportsarewelcomeandcanbemailedtome
at [email protected].
Tianjin, China Klavs Hansen
vii
Preface to the First Edition
This book is a developed version of lecture notes that were prepared for graduate
students at the University of Gothenburg and Chalmers University of Technology.
It aims to fulfill two needs. First of all, it should summarize some of the relevant
literatureandprovideacollectionofresultsforanybodywhoworkswithstatistical
aspects of nanoparticles. Thermal processes are ubiquitous and a proper under-
standing of the field is necessary for a complete description of the physics of
nanoparticles. It is therefore important to have collected a number of results that
refer specifically to small particles and the special features they exhibit.
Butamerecollectionofliteratureresultswouldlimittheusefulnessofthebook.
A textbook which only reviewed literature results would run a serious risk of
degenerating into a supermarket for shopping for the equation which seemed to fit
some particular data set. The other purpose is therefore to derive results from
scratch in a manner that allows the reader to follow the important steps in the
derivation and to gain an understanding of the applicability and the limitations
of the equations, both from the literature and of the homegrown variety. The term
‘homegrown’ does not carry any derogatory meaning. As with food, homemade
maybeworseorbetterthandishessoldready-made.Andaswithfood,oneusually
wants to know the ingredients before the dish is consumed. One important
Leitmotifofthederivationspresentedhereistomakeitcleartothereaderwhatare
physical assumptions and what are mathematical approximations. A calculation of
somephysicalquantitymayatthesametimebeextremelyprecise,lessaccurateand
completely unreliable. One should not confuse precision in the numerical estimate
of the consequences of a model with the accuracy with which the model has been
derived. And one should under no circumstances confuse the accuracy of a model
calculation with its reliability.
Withthesetwopurposesinmind,thematerialinthisbookisoftenpresentedin
morethanonewayandsomeofitisredundantfromastrictlylogicalpointofview.
Occasionally models are elaborated beyond the applicability to a real physical
system. The purpose of this is mainly to explore the limits of the approximations
and demonstrate the power of the methods. On the other hand, some problems are
presented and solved with a degree of simplification which is not on par with the
ix
x PrefacetotheFirstEdition
best available experimental data of specific systems. Readers should be equipped
with the necessary tools to improve on the text and to provide their own solutions
for specific systems. The text also contains subjects and subsections that are
intendedtoprovideexamplesandillustrations.Itrustthatthereaderwillbeableto
distinguish between fundamental results, applications and illustrative examples.
Thebookhasastrongemphasisonmicrocanonicalphysics.Supportedparticles
are veryrelevant for applications ofnanotechnology butfundamental properties of
nanosystems are best studied free of the disturbing and frequently uncontrolled
influence of a substrate. A large number of experiments have been and more will
continuetobeperformedinmolecularbeams,iontrapsandstoragerings,forwhich
the microcanonical description is the appropriate one. But since microcanonical
propertiescanbeconvertedtocanonicalbyasimpleintegration,alsoworkerswho
equilibrate their particles to a external heat bath will find useful results here.
Booksonstatisticalphysicsarefullofequations.Remarkably,onecangetaway
with very little advanced mathematics and yet describe a very wide range of
observable phenomena. The number of equations is therefore not a measure of the
level of mathematical sophistication the student is required to master. They are
simplytheretoshowhowonegetsfrompointatopointb,andtoshowwhatpoint
b looks like. After all, equations are economically expressed figures.
This book also contains material which is covered in most basic courses on
statistical physics.Experiencehasshownthat these skills often needtobebrushed
up and that some recapitulation of subjects is necessary in practice. When the
present text falls short of this target, the reader can consult the additional reading
list in Appendix A.
I have received a number of suggestions for the contents and corrections to the
text that have helped immensely, from O. Echt (Chapters 1, 3, 7), V. V. Kresin
(Chapter12),and fromstudents who havetaken mycourse.Butas thesole author
of this manuscript I have no other to blame for its shortcomings.
Perfectionismisonlythedesiretospendtimeadmiringyour(almost)completed
work, and it is time to publish this volume. Readers will hopefully report sug-
gestions, misprints, miscalculations and plain errors to me at klavs.hansen@-
physics.gu.se.
Gothenburg, Sweden Klavs Hansen
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Thermodynamic Concepts . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The Microcanonical Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 The Level Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Temperature and Boltzmann Factor . . . . . . . . . . . . . . . . . . . . . 14
1.6 The Canonical Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Mean Values in the Canonical Ensemble . . . . . . . . . . . . . . . . . 23
1.8 The Grand Canonical Ensemble. . . . . . . . . . . . . . . . . . . . . . . . 24
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 The Relation Between Classical and Quantum Statistics. . . . . . . . . 31
2.1 Fermi and Bose Statistics of Independent Particles . . . . . . . . . . 34
2.2 Classical Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 A Few Elementary and Useful Results from Classical
Statistical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Semiclassical Calculations of Spectra. . . . . . . . . . . . . . . . . . . . 41
2.5 Quantum Corrections to Interatomic Potentials . . . . . . . . . . . . . 42
2.6 Classical Limits, Example 1: The Harmonic Oscillator . . . . . . . 45
2.7 Classical Limits, Example 2: A Free Particle . . . . . . . . . . . . . . 47
2.8 Classical Limits, Example 3: A Particle in the Earth
Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Microcanonical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Finite Size Heat Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Level Densities and Canonical Partition Functions . . . . . . . . . . 63
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
xi