Quantum mechanical models of metal surfaces and nanoparticles

SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY Wolfgang Gräfe Quantum Mechanical Models of Metal Surfaces and Nanoparticles 123 SpringerBriefs in Applied Sciences and Technology www.pdfgrip.com More information about this series at http://www.springer.com/series/8884 www.pdfgrip.com Wolfgang Gräfe Quantum Mechanical Models of Metal Surfaces and Nanoparticles 123 www.pdfgrip.com Wolfgang Gräfe Berlin Germany ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-3-319-19763-0 ISBN 978-3-319-19764-7 (eBook) DOI 10.1007/978-3-319-19764-7 Library of Congress Control Number: 2015941358 Springer Cham Heidelberg New York Dordrecht London © The Author(s) 2015 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) www.pdfgrip.com Preface In this book I consider two simple quantum mechanical models of metal surfaces It is the aim to give an ostensive picture of the forces acting in a metal surface and to deduce analytical formulae for the description of their physical properties The starting points of my approach to the surface physics were strength and fatigue limit As the cause of these features I consider a near-surface stress with the dimension of a force per area In this book I explain the relation between the near-surface stress and the familiar surface parameters In order to make the understanding of my theory easier I have applied the concept of the separation of the three-dimensional body into three one-dimensional subsystems This book has been written for experts and newcomers in the field of surface physics Wolfgang Gräfe v www.pdfgrip.com Acknowledgments Without the patience and without the care of my wife Herta I would not have accomplished this book vii www.pdfgrip.com Contents Introduction 1.1 Electrocapillarity of Liquids 1.2 Surface Free Energy and Surface Stress of Solids 1.3 The Estance or the Surface Stress-Charge Coefficient 1.4 Experimental Data in the Literature 1.5 State of the Theoretical Knowledge 1.6 The Aim of the Following Text References 1 3 6 The Model of Kronig and Penney 2.1 The Density of the Electron Energy Levels n(E) 2.2 Remarks Reference 11 13 13 Tamm’s Electronic Surface States Reference 15 18 The Extension of the Kronig–Penney Model by Binding Forces 19 The Separation of the Semi-infinite Model and the Calculation of the Surface Parameters for the Three-Dimensional body at T = K (Regula Falsi of Surface Theory) 5.1 The Separability of the Chemical Potential 5.2 The Separability of the Fermi Distribution Function 5.3 The Calculation of Surface Energy, Surface Stress, and Surface Charge at T = K (Regula Falsi of Surface Theory) References 25 27 29 30 34 ix www.pdfgrip.com x Contents The Surface Parameters for the Semi-infinite Three-Dimensional Body at Arbitrary Temperature 6.1 Discussion References The Surface Free Energy φ and the Point of Zero Charge Determined for the Semi-infinite Model 7.1 Electron Transitions from the Bulk into the Surface and the Contribution to the Surface Free Energy φTr 7.2 The Point of Zero Charge (PZC) and the Fermi Level Shift 7.3 The Contribution of the Electrostatic Repulsion Between the Electrons in the Surface Bands to the Surface Energy Reference 35 40 41 43 43 47 48 51 53 53 54 56 57 58 60 64 65 68 Surface Stress-Charge Coefficient (Estance) 69 10 Regard to the Spin in the Foregoing Texts 73 11 Detailed Calculation of the Convolution Integrals References 75 80 12 Comparison of the Results for the Semi-infinite and the Limited Body 12.1 The Semi-infinite Body 12.1.1 Surface States 12.1.2 Density Distribution of the Energy Levels 12.1.3 Remark 12.1.4 Surface Free Energy 81 82 82 82 82 82 A Model with a Limited Number of Potential Wells 8.1 Modeling a Nanoparticle and a Solid Surface 8.2 The Energy of the Electrons in the Bulk and in the Surface Bands 8.3 Calculation of the Surface Parameters 8.3.1 The Surface Energy uESB of the Electrons in a Surface Band of a Nanocube 8.3.2 Calculation of the Surface Free Energy u in a Nanocube 8.3.3 Calculation of the Surface Stress for a Nanocube and a Plate-like Body 8.4 Remarks 8.5 The Surface Charge Densities and the Point of Zero Charge in a Nanocube Reference www.pdfgrip.com Contents xi 12.2 The Limited Body 12.2.1 Surface States 12.2.2 Density Distribution of the Energy Levels 12.2.3 Surface Free Energy of a Nanocube 12.2.4 Surface Free Energy of a Plate-like Body 12.3 Summary 83 83 83 84 84 84 13 Calculation of Surface Stress and Herring’s Formula 13.1 Conclusions References 85 89 89 14 Miscellaneous and Open Questions 14.1 The Scientific Ambition of this Book 14.2 Own Results 14.2.1 Semi-infinitely Extended Body at 300 K 14.2.2 Nanocube of 10 × 10 × 10 Potential Wells at K 14.3 Support for the Presented Theory 14.4 Fatigue Limit 14.5 Surface Stress and Young’s Modulus 14.6 Electrocapillarity 14.7 The Minimum of the Surface Free Energy 14.8 Fermi Level/Chemcal Potential 14.9 The Influence of the Number of Atoms on the Results References 91 91 92 92 92 93 94 95 95 96 96 97 97 Index 99 www.pdfgrip.com Surface Stress-Charge Coefcient (Estance) 1ii fị ẳ 71 drii drii df ¼ ¼ dq df dq M @E ị ci X i l00 ẳ  l00 @bi 00 Â > > > < > > > : e M 00 X l00 e ES ỵEi þEj Àf k kT 1þe e ES þEi þEj Àf k kT ES ỵEi ỵEj f k kT 2 kT 9À1 > > > = ð9:4Þ  2 S E ỵEi ỵEj f kT > > k > ; ỵ e kT (Deviating from Eq (8.10), here we carry out the summations of each degenerated state indexed by μ’’ instead of a summation using the degenerations nμ’ See comment to the Eq (8.6).) The estance dσii/dq calculated with Eq (9.4) shows a distinct dependence on ζ These results are also presented in Fig 9.1 www.pdfgrip.com Chapter 10 Regard to the Spin in the Foregoing Texts Abstract Now we consider the influence of the spin on the foregoing results For this, we complete the wave functions for the three-dimensional space ψ(r) by the eigenfunctions of the spin operator + and wrị ! vặ wrị: Keyword Spin Now we consider the influence of the spin on the foregoing results For this, we complete the wave functions for the three-dimensional space ψ(r) by the eigenfunctions of the spin operator + and wrị ! vặ wrị: 10:1ị If the spin is an invariant, this separated ansatz for the wave function is the correct solution of the Schrödinger equation Furthermore, we separate the three space coordinates in the wave function and obtain vặ wx1 ịwx2 ịwx3 ị: 10:2ị The numerical corrections necessary for the regard to the spin in the foregoing outcomes consist only in a multiplication of the final results for the surface energy φ, the surface stress σii, the near-surface stress sii, and the surface charge density q by the factor © The Author(s) 2015 W Gräfe, Quantum Mechanical Models of Metal Surfaces and Nanoparticles, SpringerBriefs in Applied Sciences and Technology, DOI 10.1007/978-3-319-19764-7_10 www.pdfgrip.com 73 Chapter 11 Detailed Calculation of the Convolution Integrals Abstract The convolution integrals for calculation of the density of electron states in the surface layer ni+j(Ei+j) are considered in detail The numerical values of this density function have been determined For differentiation of surface energy with respect to the strain ε, according to Herring’s equation the extended Leibniz rule has been applied The area of integration for the practical accomplishment of the convolution procedure is depicted Keyword Convolution integral For the sake of shortness, in the Chaps and the quantities and +∞ have been used as the bounds of integration in the convolution integrals of the type Zỵ1 n1ỵ2 E1ỵ2 ị ẳ n1 E1 ịn2 E1ỵ2 E1 ịdE1 : ð11:1Þ The correct form of this convolution integrals for integration over the lower allowed energy band from EB until ET is T E Z1 eị n1ỵ2 E1ỵ2 ị ẳ n1 E1 ịn2 E1ỵ2 E1 ịdE1 : 11:2ị E1B ðeÞ The graph of the density function n1+2(E1+2) resulting from Eq (11.2) is depicted in Fig 11.1 Formally, we have for the surface energy of the electrons in the surface band uESB ẳ L1 L2 T E Z eị p E S ỵ E1ỵ2 f E S ỵ E1ỵ2 n1ỵ2 E1ỵ2 ịdE1ỵ2 : 11:3ị E B eị â The Author(s) 2015 W Grọfe, Quantum Mechanical Models of Metal Surfaces and Nanoparticles, SpringerBriefs in Applied Sciences and Technology, DOI 10.1007/978-3-319-19764-7_11 www.pdfgrip.com 75 76 11 Detailed Calculation of the Convolution Integrals Fig 11.1 The graph of the density n1+2(E1+2) versus the energy E1+2 according to Eq (11.2) For differentiation of Eq (11.3) with respect to the strain ε, we have to apply the extended Leibniz rule [1] d de xẳbeị Z Zbeị f x; eịdx ẳ xẳaeị @f x; eị dx ỵ f beị; eịb0 eị f aeị; eịa0 eị: 11:4ị @e xẳaeị By Herrings equation, we obtain for the surface stress rESB @uESB @ ẳ uESB ỵ ẳ L1 L2 @e11 e11 T T EZ ỵE2 pE S ỵ E1ỵ2 fịE S þ E1þ2 Þn1þ2 ðE1þ2 ÞdE1þ2 E1B þE2B T T > EZ1 ỵE2   < @E1ỵ2 @n1ỵ2 E1ỵ2 ị ẳ pE S ỵ E1ỵ2 fị n1ỵ2 E1ỵ2 ị ỵ E S ỵ E1ỵ2 ị dE1ỵ2 > L1 L2 : @e11 @e11 E1B ỵE2B @E1T ỵ E2T ị @e11 ' B B S B B S B B B B @E1 ỵ E2 ị pE ỵ E1 ỵ E2 fịE ỵ E1 ỵ E2 ịn1ỵ2 E1 ỵ E2 ị : @e11 ỵ pE S ỵ E1T ỵ E2T fịE S ỵ E1T ỵ E2T ịn1ỵ2 E1T þ E2T Þ ð11:5Þ According to Fig 11.1 it is B ẳ n1ỵ2 E1B ỵ E2B ị ẳ n1ỵ2 E1ỵ2 and T ẳ n1ỵ2 E1T ỵ E2T ẳ 0: n1ỵ2 E1ỵ2 11:6ị www.pdfgrip.com 11 Detailed Calculation of the Convolution Integrals 77 Fig 11.2 The area of integration for the convolution integral belonging to the lower energy band of a two-dimensional system; The dashed line is the graph for the function E1 + E2 = E1+2 Therefore, we obtain ESB rESB ỵ 11 ẳu ẳ L1 L2 @uESB e11 E1T ỵE2T Z E1B þE2B   À Á @n1þ2 ðE1þ2 Þ À Á @E1ỵ2 p ES ỵ E1ỵ2 f n1ỵ2 E1ỵ2 ị þ E S þ E1þ2 dE1þ2 : @e11 @e11 ð11:7Þ Figure 11.2 shows the area of integration for the practical accomplishment of the convolution procedure according to Eqs (11.3) and (11.7) For the sake of simplicity, we suppose E1B ¼ E2B E1T ẳ E2T : and 11:8ị In Fig 11.2, the dashed line is the function E1 + E2 = E1+2 The energy E1+2 rises along the diagonal in the square, from EB1 + EB2 to ET1 + ET2 The practical calculations of the surface energy and the surface stress have been carried out with the following formulae u ESB B T > EZ1 ỵE1 < ẳ p E S ỵ E1ỵ2 f L1 L2 > : E1ỵ2 E1B E1B ỵE2B E1T ỵE2T Z þ E1B þE1T À Á p E S þ E1þ2 f Z E1T E1ỵ2 E1T Z ES þ E1þ2 n1 ðE1 Þn2 ðE1þ2 À E1 ÞdE1 dE1þ2 E1B > = ES ỵ E1ỵ2 n1 E1 ịn2 E1ỵ2 E1 ịdE1 dE1ỵ2 : > ; ð11:9Þ www.pdfgrip.com 78 11 Detailed Calculation of the Convolution Integrals The application of Herring’s formula, Eq (1.3), to the Eq (11.9) yields @uESB ESB rESB ỵ 11 ẳ u e 11 B T EZ ỵE1 > < @ ẳ p ES ỵ E1ỵ2 f L1 L2 > @e 11 : B B B E1ỵ2 Z E1 E1 ỵE2 ỵ p ES ỵ E1B ỵ E1T f Z E1T E1B ZE1 E S ỵ E1ỵ2 n1 E1 ịn2 E1ỵ2 E1 ịdE1 dE1ỵ2 E1B @E1B ỵ E1T ị ES ỵ E1B ỵ E1T n1 E1 ịn2 E1B ỵ E1T E1 dE1 @eii B p E ỵ S E1B ỵ E2B f E1B @ þ @e11 E1T þE2T Z À Á À Á @ðE1B þ E2B Þ ES þ E1B þ E2B n1 ðE1 ịn2 E1B ỵ E2B E1 dE1 @eii ZE1 T p E ỵ E1ỵ2 f S E1B ỵE1T ES ỵ E1ỵ2 n1 E1 ịn2 E1ỵ2 E1 ịdE1 dE1ỵ2 E1ỵ2 E1T ỵ p ES ỵ E1T ỵ E2T f Z E1T E1T ZE1 T p ES ỵ E1B ỵ E1T À f Á E1B À Á À Á @ðE1T þ E2T Þ E S þ E1T þ E2T n1 E1 ịn2 E1T ỵ E2T E1 dE1 @eii > B T = @E ỵ E ị 1 : ES ỵ E1B ỵ E1 n1 E1 ịn2 E1B þ E1T À E1 dE1 > @eii ; Á T À Á ð11:10Þ As discussed in Chaps and 8, we differentiate only the eigenvalues E and the densities n(E,ε) with respect to ε The third and seventh term counterbalance each other rESB 11 B T B EZ E1ỵ2 ỵE1 > Z E1 < @E1 S ẳ p E ỵ E1ỵ2 f n1 E1 ịn2 E1ỵ2 E1 ị > L1 L2 : @e11 E1B ỵE2B E1B @n1 E1 ị ỵ ES ỵ E1ỵ2 n2 E1ỵ2 E1 ịdE1 @e11 À Á À S Á À Á À B @ E1ỵ2 E1B B ỵ E ỵ E1ỵ2 n1 E1ỵ2 E1 n2 E1 @e B ! 11 À S Á À BÁ À Á @ E1 E ỵ E1ỵ2 n1 E1 n2 E1ỵ2 E1B dE1ỵ2 @e11 T T T EZ ỵE2 ZE1 S @E1 ỵ p E ỵ E1ỵ2 f n1 E1 ịn2 E1ỵ2 E1 ị @e11 E1B ỵE1T E1ỵ2 E1T www.pdfgrip.com 11 Detailed Calculation of the Convolution Integrals 79 @n1 E1 ị ỵ E S ỵ E1ỵ2 n2 E1ỵ2 E1 ịdE1 @e11 @E1T ỵ E S ỵ E1ỵ2 n1 E1T n2 E1ỵ2 E1T @e11 ! ' S @ E1ỵ2 E1T E ỵ E1ỵ2 n1 E1ỵ2 E1T n2 E1T dE1ỵ2 : @e11 11:11ị In the case that n1(E1) = n2(E1), the members in the third and in the last row as well as in the fourth and the seventh row in Eq (11.11) vanish for the following reasons: The energy level EB corresponds to kc = but, the smallest value of k is k¼ 2p : L ð11:12Þ Strongly speaking, the energy level EB is a limit which cannot be reached by the allowed energy levels Therefore, we have n(EB) = According to the extended Leibniz rule Eq (11.4), it follows for the derivation of the total number of energy levels Nel with respect to @N @ ẳ @eii @eii T E Z eị el nðEi ÞdEi E B ðeÞ T E Z ðeÞ ¼ E B ðeÞ ð11:13Þ À Á @E À Á @E @nEi ị dEi ỵ n E T eị n E B ðeÞ : @eii @eii @eii T B In the infinitely extended body, the number of the energy levels between EB(ε) and ET(ε) will not be changed due to a dilatation of the solid That means @ @eii T E Z eị nEi ịdEi ẳ : 11:14ị E B ðeÞ Furthermore, also the sum of all changes of n in the energy range between EB(ε) and ET(ε) is zero, that is T E Z ðeÞ E B eị @nEi ị dEi ẳ : @eii www.pdfgrip.com 11:15ị 80 11 Detailed Calculation of the Convolution Integrals For this reason, it follows À Á @E T À Á @E B n ET eị; e ẳ n E B eị; e : @eii @eii ð11:16Þ We see from Fig 4.2 and Eq (4.2) that the quantities @ET =@eii and @EB =@eii are not zero Because of nðE B ðeÞ; eÞ ¼ and @E=@e 6¼ 0; nðE T ðeÞ; eÞ ¼ and we obtain rESB 11 B T B EZ E1ỵ2 ỵE1 > Z E1 < S @E1 ẳ p E ỵ E1ỵ2 f n1 E1 ịn2 E1ỵ2 E1 ịdE1 L1 L2 > @e 11 : E1B ỵE2B B E1ỵ2 Z E1 ỵ E1B E1T ỵE2T Z ỵ E1B ỵE1T Z E1T ỵ E1ỵ2 E1T E1B @n1 E1 ị ES ỵ E1ỵ2 n2 E1ỵ2 E1 ịdE1 5dE1ỵ2 @e11 p E S ỵ E1ỵ2 f ZE1 T E1ỵ2 E1T @E1 n1 E1 ịn2 E1ỵ2 E1 ịdE1 @e11 > = À S Á @n1 ðE1 Þ E þ E1þ2 n2 ðE1þ2 À E1 ÞdE15dE1þ2 : > @e11 ; ð11:17Þ Also this result corresponds to the Eq (6.5) Remark Haiss [2] considered the derivation of the surface energy with respect to the surface strain at εik = References Rothe R (1954) Höhere Mathematik für Mathematiker, Physiker, Ingenieure, Teil II, B Teubner Verlagsgesellschaft Leipzig, pp S 143 Haiss W (2001) Surface stress on clean and adsorbate-covered solids Rep Prog Phys 64: 591–648 www.pdfgrip.com Chapter 12 Comparison of the Results for the Semi-infinite and the Limited Body Abstract The results for the semi-infinite and the limited bodies are compared The findings obtained with the different models for the offspring surface state and the surface free energy are similar but, not identical Keyword Surface data from different models In the foregoing chapters, the allowed energy levels were calculated for a Meander-like potential run in the bulk as it is symbolized in Fig 3.1 The width of the potential wells a and the width of the potential barriers b in the bulk are a = b = × 10−10 m The height of the barriers in the bulk amounts to Ui = 0.8·10−18 J (5 eV) and for the height of the potential step at the surface US it has been assumed the value 1.6 × 10−18 J (10 eV) The precondition for the application of the two models considered in this booklet is the separability of the potential energy of the body If the system is separable, the chemical potential of the three-dimensional system ζ is the sum of the chemical potentials for the subsystems ζi The Fermi function for the three-dimensional system can be written in the form p E f ị ẳ 1ỵe E1 ỵE2 ỵE3 f1 ỵf2 ỵf3 ị kT 5:19ị and the Fermi distribution function for the ith partial system is pEi fi ị ẳ 1ỵe Ei fi kT ð5:20Þ On the condition of Eq (5.21) even the Eq pE fị ẳ pE1 f1 ịpE2 f2 ÞpðE3 À f3 Þ ð5:23Þ holds © The Author(s) 2015 W Gräfe, Quantum Mechanical Models of Metal Surfaces and Nanoparticles, SpringerBriefs in Applied Sciences and Technology, DOI 10.1007/978-3-319-19764-7_12 www.pdfgrip.com 81 82 12 Comparison of the Results for the Semi-infinite … 12.1 The Semi-infinite Body 12.1.1 Surface States In the one-dimensional model for the semi-infinitely extended solids, an offspring surface state appears closely above the lower energy band if the surface energy step US exceeds a certain level For the lattice parameters listed above, this threshold amounts to nearly 1.3 × 10−18 J In the case mentioned above (US = 1.6 × 10−18 J), −18 the energy of the offspring surface state is ESo J k = 0.506 × 10 For the same parameters, the additional surface state appears at ESa = k 0.678 × 10−18 J 12.1.2 Density Distribution of the Energy Levels For a one-dimensional system, the density distribution of the energy levels is given in Eq (2.9) and presented in Fig 2.4 The calculation of the density distribution of the energy levels for a two-dimensional system has been carried out by a convolution integral in Chap 12.1.3 Remark The calculation of the density distribution for a three-dimensional body should be principally performable by a twofold convolution with a formula of the kind in Eq (12.1) Z1 Z1 n1 E1 ịn2 E1ỵ2 E1 ịdE1 n3 E E1ỵ2 ịdE1ỵ2 12:1ị 12.1.4 Surface Free Energy The contribution to the maximum surface free energy φTR of a semi-infinite body caused by the electron transitions from the upmost energy level in the bulk ETk into the surface state ESk amounts to www.pdfgrip.com 12.1 The Semi-infinite Body 83 À Á ð0:506 Á 10À18 À 0:502 Á 10À18 ÞJ J max uTR ẳ EkS EkT ị ẳ ¼ 0:025 2 À10 c m ð4 Á 10 mÞ ð12:2Þ 12.2 The Limited Body Now, we consider one-dimensional models with only ten potential wells Originally it was the aim to demonstrate a completely different procedure for an easy and principally correct numerical calculation of the surface parameters in a three-dimensional model With the model of a limited solid also the calculation of the wave functions was aspired The deduction of the surface free energy proved to be particularly descriptive 12.2.1 Surface States With the limited model, it is not possible to demonstrate in a convincing manner the existence of a lower surface state, which is an offspring of the “lower energy band” Instead of the surface state we find only an enhancement of the electron concentration near the surface 12.2.2 Density Distribution of the Energy Levels The determination of the distribution of the energy levels has been accomplished by a simple calculation of the triples Ek + Ei + Ej and their sequencing according to their magnitude This can be realized easily The positions of the energy levels for different numbers of the potential wells are presented in Fig 12.1 We can see from Fig 12.1 that for the given values of the potential maxima in the bulk as well as for the potential steps at the surfaces and for a given number N of potential wells only N − energy levels fall into the allowed energy band for the bulk of an infinite body In the model of the limited body, the taking of the remaining energy level as an offspring surface state or as a surface resonance is an arbitrary decision However, if each potential well contributes one occupied energy level to the solid, the surface is uncharged if the “lower surface” state is occupied www.pdfgrip.com 84 12 Comparison of the Results for the Semi-infinite … Fig 12.1 The positions of the energy levels for different numbers of potential wells In the one-dimensional model with ten potential wells, the energy eigenvalue of the additional surface state amounts to 0.8 × 10−18 J 12.2.3 Surface Free Energy of a Nanocube For a nanocube with 10 × 10 × 10 potential wells, the dependence of the surface free energy on the chemical potential is depicted in Fig 8.8 The maximum value amounts to 0.116 J/m2 12.2.4 Surface Free Energy of a Plate-like Body The maximum value of surface free energy calculated for a section with 10 × 10 × 10 potential wells of a plate-like body considered in Chap also adds up to 0.116 J/m2 12.3 Summary The results obtained with the different models for the offspring surface state and the surface free energy are similar but, not identical If in the case of a limited body and empty surface states the chemical potential is sufficiently decreased, the situation can arise that the surface free energy becomes negative This may be deduced from the lowest graph in Fig 8.4 The lowest thin line symbolizes the bottom of the lowest allowed energy band for the model of Kronig and Penney The next higher line is the top of the allowed energy band and the highest line gives the position of the offspring surface state The rightmost energy levels have been calculated for 100 potential wells www.pdfgrip.com Chapter 13 Calculation of Surface Stress and Herring’s Formula Abstract The calculation of the surface stress with the Herring’s Eq is analyzed in depth Without an additional assumption, it is not possible to apply the Herring’s equation to the formulae for the surface energy and the surface stress which were deduced in Chap for the limited body Let us consider the Eq (8.2) uESB ẳ 1 X S 2ị Ek ỵ Eil ỵ Ejl ịl0 nl0 Li Lj l0 1ỵe ị E S ỵE l ỵE l Àf i j l0 k kT : ð8:2Þ For the derivation of this formula with respect to the strain εii, we have introduced in Chap an additional restriction for the calculation of the surface stress; it has to be executed the partial derivation of Eq (8.2) A more detailed analysis of this problem shows that no derivation of the Fermi function with respect to the strain εii can appear in the formula for surface stress Keyword Applicability of Herring's equation The formula of Herring, Eq (1.3), has been deduced by the consideration of an idealized reversible cyclic process This is described in the paper of Haiss [1] Therefore, Herring’s formula should be of fundamental importance and each result for the surface stress should be in accordance with this formula The Herring’s Eq relates the surface free energy to the surface stress, but in Eq (7.5) we have seen that Herring’s Eq is also applicable to the surface energy caused by the electrons in the surface bands φESB For the surface energy φESB, we write without any doubt uESB fị ẳ Li Lj Z1 EnS E ịpE fÞdE : ð13:1Þ The term nS(E) means the density of the electron states in the surface bands © The Author(s) 2015 W Gräfe, Quantum Mechanical Models of Metal Surfaces and Nanoparticles, SpringerBriefs in Applied Sciences and Technology, DOI 10.1007/978-3-319-19764-7_13 www.pdfgrip.com 85 86 13 Calculation of Surface Stress and Herring’s Formula By an expansion of the body in the xi-direction, a surface stress σii is generated by the electrons which are located in the near-surface layer In analogy to the calculations in Chap 4, we make for this surface stress the ansatz FS ci rii ẳ i fị ẳ Lj Li Lj Z1 fi ðE ÞnS ðE ÞpðE À fÞdE : ð13:2Þ The computation of this formula has been accomplished without any recourse to the surface energy and to the formula of Herring in Eq (1.3) Applying Herring’s formula to Eq (13.1), we have to keep in mind the Leibniz rule and the fact that E is an integration variable and does not depend on strain ε In the case the density nS depends on the strain εii, the Eq (13.2) is not the derivative of Eq (13.1) and the both Eqs not comply with Herring’s formula Furthermore, without an additional assumption it is not possible to apply the Herring’s equation to the formulae for the surface energy and the surface stress which were deduced for the limited body in Chap Let us consider the Eq (8.2) uESB ẳ 1 X S 2ị Ek ỵ Eil þ Ejl Þl0 nl0 Li Lj l0 1þe ES ỵE l ỵE l ịl0 f i j k kT : ð8:2Þ In Chap 8, we have introduced an additional restriction for the calculation of the surface stress It has to be executed the partial derivation of Eq (8.2) with respect to the strain εii This has been accomplished in a non-stringent manner By observation of this additional restriction, we have obtained the formula  M 00  < = l X 1 @Ei rESB ẳ : 8:7ị E S ỵE l ỵE l ị 00 Àf ; Li Lj :l00 ¼1 @eii l00 i j l k kT 1ỵe One would intuitively write Eq (8.7) because the formula F = dE/dx holds only for potential energies and not for the energies in general The energy in the Fermi distribution is unspecified Let us consider the transition from an infinitely extended body to a limited one The energy of an infinitely extended chain of atoms is ZET Ei1D fi ị ẳ Eie g; bi ịni g; bi ị EB gfi ỵ e kT dg : ð13:3Þ The symbol Eei means the energy eigenvalues and “1D” marks the onedimensional body www.pdfgrip.com 13 Calculation of Surface Stress and Herring’s Formula 87 In this formula, we substitute the density of the energy levels in an allowed energy band n(E) by a sum of Dirac’s delta functions ni ðg; bi Þ ! N X À Á d g Ei;m bi ị 13:4ị mẳ1 and have Ei1D fi ị Zỵ1 X N ẳ e Ei;m bi ịd g Ei;m bi ị mẳ1 gfi ỵ e kT dg : 13:5ị For an approximation of the δ-function, we use a sequence of normal distributions À Á ðgÀEi;m ðbi ÞÞ d g Ei;m bi ị; o ẳ p e 2o2 2po ð13:6Þ with the parameter ο, the Greek letter “omicron,” and obtain for the energy Ei1D fi ị ẳ lim o!0 Zỵ1 X N gEi;m bi ịị 1 e Ei;m ðbi Þ pffiffiffiffiffiffi eÀ 2o2 gÀfi dg : 2po ỵ e kT mẳ1 13:7ị The quantity ο2 is the variance Taking notice of the Leibniz rule, it follows for the approximation of the force in the one-dimensional body ỵ1