1) The document discusses phonons, which are quantized lattice vibrations in crystals that carry thermal energy. It describes modeling crystal vibrations using a harmonic lattice approach.
2) Normal modes of the lattice vibrations can be described as a set of independent harmonic oscillators. Quantum mechanically, these normal modes are quantized as phonons with discrete energy levels.
3) Phonons can be thought of as quasiparticles that carry momentum and energy in the crystal lattice. Their propagation is described using a phonon field approach rather than independent normal modes.
Dipolar interaction and the Manning formulaIJERA Editor
In this work we want to show that the mathematical model of quantum mechanics, led to its classical approach, is able to reproduce as close macroscopic experimental results captured by the Manning formula, sufficiently verified through their diverse applications in hydraulics. Molecular interaction between the fluid and the wall of the vessel is studied decomposing the Hamiltonian in two parts: free, and interacting. Scaling process is considered from molecular to hydraulic. Participation of the symmetries of Saint-Venant equation in the hydraulic gradient is taken into account. Correlations between different variables are set. The magnitude of scale change is estimated. We conclude that the Compton wavelength induces to the Boussinesq viscosity concept and the characteristic length of the viscous sublayer.
Dipolar interaction and the Manning formulaIJERA Editor
In this work we want to show that the mathematical model of quantum mechanics, led to its classical approach, is able to reproduce as close macroscopic experimental results captured by the Manning formula, sufficiently verified through their diverse applications in hydraulics. Molecular interaction between the fluid and the wall of the vessel is studied decomposing the Hamiltonian in two parts: free, and interacting. Scaling process is considered from molecular to hydraulic. Participation of the symmetries of Saint-Venant equation in the hydraulic gradient is taken into account. Correlations between different variables are set. The magnitude of scale change is estimated. We conclude that the Compton wavelength induces to the Boussinesq viscosity concept and the characteristic length of the viscous sublayer.
Artigo que descreve o trabalho feito com o Chandra nos aglomerados de galáxias de Perseus e Virgo sobre a descoberta de uma turbulência cósmica que impede a formação de novas estrelas.
The quantum electromagnetic flux is studied as operators for magnetic flux quantization in the
Bohr atom. We find that this quantization rule can be found from an elementary analysis of a Bohr electric
oscillator treated like an L-C circuit. The electromagnetic flux quantization agrees for instanton configuration
so there is no radiation in the Bohr atom and we have a stable atomic system .
Artigo que descreve o trabalho feito com o Chandra nos aglomerados de galáxias de Perseus e Virgo sobre a descoberta de uma turbulência cósmica que impede a formação de novas estrelas.
The quantum electromagnetic flux is studied as operators for magnetic flux quantization in the
Bohr atom. We find that this quantization rule can be found from an elementary analysis of a Bohr electric
oscillator treated like an L-C circuit. The electromagnetic flux quantization agrees for instanton configuration
so there is no radiation in the Bohr atom and we have a stable atomic system .
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The Propagation and Power Deposition of Electron Cyclotron Waves in Non-Circu...IJERA Editor
By solving the plasma equilibrium equation, ray equations, and quasi-linear Fokker-Planck equation, the ray
trajectories and power deposition of EC wave has been numerically simulated in non-circular HL-2A tokamak
plasma. The results show that shaping effect and temperature profile has little influence on ECRH, while plasma
density affect propagation and power deposition obviously. when the ordinary mode of EC waves are launched
from the mid-plane and low-field-side, ray trajectories are bended as the parallel refractive index increases and
even recurve to the low-field side when the parallel refractive index reaches to a certain value. Single absorption
decreases with increasing both poloidal and toroidal injection angle, and can be 100% when poloidal injection
angle is 180o and toroidal injection angle is less than 10o.
A numerical wavefront solution for quantum transmission lines with charge discreteness is
proposed for the first time. The nonlinearity of the system becomes deeply related to charge discreteness. The
wavefront velocity is found to depend on the normalized (pseudo) flux variable. Finally we find the dispersion
relation for the normalized flux
0 / .
Is ellipse really a section of cone. The question intrigued me for 20 odd years after leaving high school. Finally got the proof on a cremation ground. Only thereafter I came to know of Dandelin spheres. But this proof uses only bare basics within the scope of high school course of Analytical geometry.
A parallel-polarized uniform plane wave is incident obliquely on a lo.pdfaroraenterprisesmbd
A parallel-polarized uniform plane wave is incident obliquely on a lossless dielectric slab that is
embedded in a free-space medium, as shown in Figure P5-17. Derive expressions for the total
reflection and transmission coefficients in terms of the electrical constitutive parameters,
thickness of the slab, and angle of incidence.
Solution
In this section we look at the power or energy transmitted and reflected at an interface between
two insulators. To do so, we must evaluate the time-averaged power in the incident, reflected,
and transmitted waves which is done by calculating the Poynting vector. The energy current
density toward or away from the interface is then given by the component of the Poynting vector
in the direction normal to the interface. In the second medium, where there is just a single
(refracted) wave, the normal component of S is unambiguously the transmitted power per unit
area. But in the first medium, the total electromagnetic field is the sum of the fields of the
incident and reflected waves. In evaluating E × H, one finds three kinds of terms. There is one
which is the cross-product of the fields in the incident wave, and its normal component gives the
incident power per unit area. A second is the cross-product of the fields in the reflected wave,
giving the reflected power. But there are also two cross-terms involving the electric field of one
of the plane waves and the magnetic field of the other one. It turns out that the time-average of
the normal component of these terms is zero, so that they may be ignored in the present context.
Bearing this in mind, we have the following quantities of interest: The time-averaged incident
power per unit area:
P =< S > ·n = c 8 s ² µ |E0| 2 k · n k
The time-averaged transmitted power per unit area:
P 0 =< S 0 > ·n = c 8 s ² 0 µ0 |E 0 0 | 2k 0 · n k 0 d
The reflection coefficient R and the transmission coefficient T are defined as the ratios of the
reflected and transmitted power to the incident power. We may calculate the reflection and
transmission coefficients for the cases of polarization perpendicular and parallel to the plane of
incidence by using the Fresnel equations. If an incident wave has general polarization so that its
fields are linear combinations of these two special cases, then there is once again the possibility
of cross terms in the power involving an electric field with one type of polarization and a
magnetic field with the other type. Fortunately, these turn out to vanish, so that one may treat the
two polarizations individually. For the case of polarization perpendicular to the plane of
incidence, we use the Fresnel equations (52) and (54) for the reflected and transmitted
amplitudes and have
T = q ² 0 µ0 4n 2 cos2 i cos r (n cos i+(µ/µ0) n02n2 sin2 i) 2 q ² µ cosi
Making use of the relations n = ²µ, n 0 = ² 0µ0 , sin r = (n/n0 )sin i, and cosi = 1 sin2 i,
T = 4n(µ/µ0 ) cosi n02 n2 sin2 i [n cosi + (µ/µ0 ) n02 n2 sin2 i] 2 .
By similar means one can write the .
Bound State Solution of the Klein–Gordon Equation for the Modified Screened C...BRNSS Publication Hub
We present solution of the Klein–Gordon equation for the modified screened Coulomb potential (Yukawa) plus inversely quadratic Yukawa potential through formula method. The conventional formula method which constitutes a simple formula for finding bound state solution of any quantum mechanical wave equation, which is simplified to the form; 2122233()()''()'()()0(1)(1)kksAsBscsssskssks−++ψ+ψ+ψ=−−. The bound state energy eigenvalues and its corresponding wave function obtained with its efficiency in spectroscopy.
Key words: Bound state, inversely quadratic Yukawa, Klein–Gordon, modified screened coulomb (Yukawa), quantum wave equation
R,L,C, G parameters of a co-axial & 2-wire transmission line
Field solutions for TE and TM modes for a waveguide
Design and analysis of rectangular waveguide to support TE10 mode
Design and analysis of circular waveguide to support TE11 mode
Investigation of Steady-State Carrier Distribution in CNT Porins in Neuronal ...Kyle Poe
In this work, the carrier distribution of a carbon nanotube inserted into the spinal ganglion neuronal membrane is examined. After primary characterization based on previous work, the nanotube is approximated as a one-dimensional system, and the Poisson and Schrödinger equations are solved using an iterative finite-difference scheme. It was found that carriers aggregate near the center of the tube, with a negative carrier density of ⟨ρn⟩ = 7.89 × 10^13 cm−3 and positive carrier density of ⟨ρp⟩ = 3.85 × 10^13 cm−3. In future work, the erratic behavior of convergence will be investigated.
Similar to Thermodynamics of crystalline states (20)
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
1. Chapter 2
Phonons
Disregarding point symmetry, we can simplify the crystal structure by the space
group [1, 2], representing the thermodynamic state in equilibrium with the sur-
roundings at given values of p and T. In this approach, the restoring forces secure
stability of the lattice, where the masses at lattice points are in harmonic motion.
In this case, we realize that their directional correlations in the lattice are ignored
so that a possible disarrangement in the lattice can cause structural instability.
In this chapter, we discuss a harmonic lattice to deal with basic excitations in
equilibrium structure. Lattice vibrations in periodic structure are in propagation,
specified by frequencies and wavevectors in virtually continuous spectra. Quantum
mechanically, on the other hand, the corresponding phonons signify the dynamical
state in crystals. In strained crystals, as modulated by correlated constituents, low-
frequency excitations dominate over the distorted structure, which is however
thermally unstable as discussed in this chapter.
2.1 Normal Modes in a Simple Crystal
A crystal of chemically identical constituent ions has a rigid periodic structure
in equilibrium with the surroundings, which is characterized by translational
symmetry. Referring to symmetry axes, physical properties can be attributed to the
translational invariance, in consequence of energy and momentum conservations
among constituents.
Constituents are assumed to be bound together by restoring forces in the lattice
structure. Considering a cubic lattice of N3
identical mass particles in a cubic crystal
in sufficiently large size, we can solve the classical equation of motion with nearest-
neighbor interactions. Although such a problem should be solved quantum mechan-
ically, classical solutions provide also a useful approximation. It is noted that the
lattice symmetry is unchanged with the nearest-neighbor interactions, assuring
M. Fujimoto, Thermodynamics of Crystalline States,
DOI 10.1007/978-1-4614-5085-6_2, # Springer Science+Business Media New York 2013
11
2. structural stability in this approach. In the harmonic approximation, we have linear
differential equations, which can be separated into 3N independent equations; this
one-dimensional equation describes normal modes of N constituents in collective
motion along the symmetry axis x, y, or z [3]. Denoting the displacement by a vector
qn from a site n, we write equations of motion for the components qx;n; qy;n and qz;n
independently, that is,
€qx;n ¼ o2
qx;nþ1 þ qx;nÀ1 À 2qx;n
À Á
; €qy;n ¼ o2
qy;nþ1 þ qy;nÀ1 À 2qy;n
À Á
and
€qz;n ¼ o2
qz;nþ1 þ qz;nÀ1 À 2qz;n
À Á
;
where o2
¼ k m= and k and m are the mass of a constituent particle and the force
constant, respectively. As these equations are identical, we write the following
equation for a representative component to name qn for brevity:
€qn ¼ o2
qnþ1 þ qnÀ1 À 2qn
À Á
; (2.1)
which assures lattice stability along any symmetry direction.
Defining the conjugate momentum by pn ¼ m _qn, the Hamiltonian of a harmonic
lattice can be expressed as
H ¼
XN
n¼0
p2
n
2m
þ
mo2
2
qnþ1 À qn
À Á2
þ
mo2
2
qn À qnÀ1ð Þ2
& '
: (2.2)
Each term in the summation represents one-dimensional infinite chain of identi-
cal masses m, as illustrated in Fig. 2.1a.
Normal coordinates and conjugate momenta, Qk and Pk , are defined with the
Fourier expansions
qn ¼
1
ffiffiffiffi
N
p
XkN
k¼0
Qk exp iknað Þ and pk ¼
XkN
k¼0
Pk exp iknað Þ; (2.3)
Fig. 2.1 (a) One-
dimensional monatomic
chain of the lattice constant a.
(b) A dispersion curve o vs: k
of the chain lattice.
12 2 Phonons
3. where a is the lattice constant. For each mode of qn and pn, the amplitudes Qk and
Pk are related as
QÀk ¼ Qk
Ã
; PÀk ¼ Pk
Ã
and
XN
n¼0
exp i k À k0
ð Þna ¼ Ndkk0 ; (2.4)
where dkk0 is Kronecker’s delta, that is, dkk0 ¼ 1 for k ¼ k0
, otherwise zero for k ¼ k0
.
Using normal coordinates Qk and Pk, the Hamiltonian can be expressed as
H ¼
1
2m
X2p a=
k¼0
PkPk
Ã
þ QkQk
Ã
m2
o2
sin2 ka
2
'
; (2.5)
from which the equation of motion for Qk is written as
€Qk ¼ Àm2
o2
Qk; (2.6)
where
ok ¼ 2o sin
ka
2
¼ 2
ffiffiffiffi
k
m
r
sin
ka
2
: (2.7)
As indicated by (2.7), the k-mode of coupled oscillators is dispersive, which are
linearly independent from the other modes of k0
6¼ k . H is composed of N
independent harmonic oscillators, each of which is determined by the normal
coordinates Qk and conjugate momenta Pk. Applying Born–von Ka´rman’s boundary
conditions to the periodic structure, k can take discrete values as given by k ¼ 2pn
Na
and n ¼ 0; 1; 2; . . . ; N. Figure 2.1b shows the dispersion relation (2.7) determined
by the characteristic frequency ok:
With initial values of Qkð0Þ and _Qkð0Þ specified at t ¼ 0, the solution of (2.7)
can be given by
QkðtÞ ¼ Qkð0Þ cos okt þ
_Qkð0Þ
ok
sin okt:
Accordingly,
qnðtÞ ¼
1
ffiffiffiffi
N
p
XkN
k¼0
X
n0¼n;nÆ1
qn0ð0Þ cos ka n À n0
ð Þ À oktf g þ
_qn0 ð0Þ
ok
sin ka n À n0
ð Þ À oktf g
!
;
(2.8)
where a n À n0
ð Þ represents distances between sites n and n0
so that we write it as
x ¼ a n À n0
ð Þ in the following. The crystal is assumed as consisting of a
large number of the cubic volume L3
where L ¼ Na, if disregarding surfaces.
2.1 Normal Modes in a Simple Crystal 13
4. The periodic boundary conditions can then be set as qn¼0ðtÞ ¼ qn¼NðtÞ at an
arbitrary time t. At a lattice point x ¼ na between n ¼ 0 and N , (2.8) can be
expressed as
q x; tð Þ ¼
X
k
Ak cos Ækx À oktð Þ þ Bk sin Ækx À oktð Þ½ Š;
where Ak ¼ qkð0Þ
ffiffiffi
N
p and Bk ¼
_qkð0Þ
ok
ffiffiffi
N
p , and x is virtually continuous in the range 0 x
L, if L is taken as sufficiently long. Consisting of waves propagating in Æ x
directions, we can write q x; tð Þ conveniently in complex exponential form, that is,
q x; tð Þ ¼
X
k
Ck exp i Ækx À okt þ jkð Þ; (2.9)
where C2
k ¼ A2
k þ B2
k and tan jk ¼ Bk
Ak
. For a three-dimensional crystal, these one-
dimensional k-modes along the x-axis can be copied to other symmetry axes y and z;
accordingly, there are 3N normal modes in total in a cubic crystal.
2.2 Quantized Normal Modes
The classical equation of motion of a harmonic crystal is separable to 3N indepen-
dent normal propagation modes specified by kn ¼ 2pn
aN along the symmetry axes. In
quantum theory, the normal coordinate Qk and conjugate momentum Pk ¼ Àih @
@Qk
are operators, where h ¼ h
2p and h is the Planck constant. For these normal and
conjugate variables, there are commutation relations:
Qk; Qk0½ Š ¼ 0; Pk; Pk0½ Š ¼ 0 and Pk; Qk0½ Š ¼ ihdkk0 ; (2.10)
and the Hamiltonian operator is
Hk ¼
1
2m
PkP
y
k þ m2
o2
kQkQ
y
k
: (2.11a)
Here, P
y
k and Q
y
k express transposed matrix operators of the complex conjugates Pk
Ã
and Qk
Ã
, respectively.
Denoting the eigenvalues of Hk by ek, we have the equation
HkCk ¼ ekCk: (2.11b)
For real eigenvalues ek , Pk and Qk should be Hermitian operators, which are
characterized by the relations P
y
k ¼ PÀk and Q
y
k ¼ QÀk , respectively. Defining
operators
14 2 Phonons
5. bk ¼
mokQk þ iP
y
k
ffiffiffiffiffiffiffiffiffiffi
2mek
p and b
y
k ¼
mokQ
y
k À iPk
ffiffiffiffiffiffiffiffiffiffi
2mek
p ; (2.12)
we can write the relation
bkb
y
k ¼
1
2mek
m2
o2
kQ
y
k Qk þ P
y
k Pk
þ
iok
2ek
Q
y
k P
y
k À PkQk
¼
Hk
ek
þ
iok
2ek
QÀkPÀk À PkQkð Þ:
From this relation, we can be derive
Hk ¼ hok b
y
k bk þ
1
2
; if ek ¼
1
2
hok: (2.13)
Therefore, Hk are commutable with the operator b
y
k bk, that is,
Hk; b
y
k bk
h i
¼ 0;
and from (2.12)
bk0 ; b
y
k
h i
¼ dk0k; bk0 ; bk½ Š ¼ 0 and b
y
k0 ; b
y
k
h i
¼ 0:
Accordingly, we obtain
Hk; b
y
k
h i
¼ hokb
y
k and bk; Hk½ Š ¼ hokbk:
Combining with (2.11b), we can derive the relations
Hk b
y
k Ck
¼ ek þ hokð Þ b
y
k Ck
and Hk bkCkð Þ ¼ ek À hokð Þ bkCkð Þ;
indicating that b
y
k Ck and bkCk are eigenfunctions for the energies ek þ hok and
ek À hok , respectively. In this context, b
y
k and bk are referred to as creation and
annihilation operators for the energy quantum hok to add and subtract in the energy
ek; hence, we can write
b
y
k bk ¼ 1: (2.14)
Applying the creation operator b
y
k on the ground state function Cknk-times, the
eigenvalue of the wavefunction b
y
k
nk
Ck can be given by nk þ 1
2
À Á
hok, generating a
2.2 Quantized Normal Modes 15
6. state of nkquanta plus 1
2 hok. Considering a quantum hok like a particle, called a
phonon, such an exited state with nk identical phonons is multiply degenerate by
permutation nk! Hence, the normalized wavefunction of nk phonons can be
expressed by 1ffiffiffiffi
nk!
p b
y
k
nk
Ck. The total lattice energy in an excited state of n1; n2;
::::: phonons in the normal modes 1, 2,. . ... can be expressed by
U n1; n2; :::::ð Þ ¼ Uo þ
X
k
nkhok; (2.15a)
where Uo ¼
P
k
hok
2 is the total zero-point energy. The corresponding wavefunction
can be written as
C n1; n2; :::::ð Þ ¼
b
y
1
n1
b
y
2
n2
:::::
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1! n2! :::::
p C1C2:::::ð Þ; (2.15b)
which describes a state of n1; n2; :::: phonons of energies n1hok1
; n2hok2
; :::::. The
total number N ¼n1 þ n2 þ ::::: cannot be evaluated by the dynamical theory;
however, we can determine the value in thermodynamics, as related to the level
of thermal excitation at a given temperature.
2.3 Phonon Field and Momentum
In a one-dimensional chain of identical mass particles, the displacement mode qk is
independent from each other’s modes, and hence representing normal modes in a
three-dimensional crystal. However, this model is only approximate, in that these
normal modes arise from the one-dimension harmonic chain model, where mutual
interactions between different normal modes are prohibited. For propagation in
arbitrary direction, the vibrating field offers more appropriate approach than the
normal modes, where quantized phonons move in any direction like free particles in
the field space.
Setting rectangular coordinates x; y; z along the symmetry axes of an orthorhom-
bic crystal in classical theory, the lattice vibrations are described by a set of
equations
px;n1
2
2m
þ
k
2
qx;n1
À qx;n1þ1
À Á2
þ qx;n1
À qx;n1À1
À Á2
n o
¼ ex;n1
;
py;n2
2
2m
þ
k
2
qy;n2
À qy;n2þ1
À Á2
þ qy;n2
À qy;n2À1
À Á2
n o
¼ ey;n2
;
and
pz;n3
2
2m
þ
k
2
qz;n3
À qz;n3þ1
À Á2
þ qz;n3
À qz;n3À1
À Á2
n o
¼ ez;n3
;
(2.16)
16 2 Phonons
7. whereex;n1
þ ey;n2
þ ez;n3
¼ en1n2n3
is the total propagation energy along the direction
specified by the vector q n1; n2; n3ð Þ and k is the force constant.
The variables qx;n1
; qy;n2
; qz;n3
in (2.16) are components of a classical vector
q n1; n2; n3ð Þ , which can be interpreted quantum theoretically as probability
amplitudes of components of the vector q in the vibration field. We can therefore
write the wavefunction of the displacement field as C n1; n2; n3ð Þ ¼ qx;n1
qy;n2
qz;n3
,
for which these classical components are written as
q x; tð Þ ¼
X
kx
Ck;x exp i Ækxx À okx
t þ jkx
À Á
;
q y; tð Þ ¼
X
ky
Cky
exp i Ækyy À oky
t þ jky
;
q z; tð Þ ¼
X
kz
Cky
exp i Ækzz À okz
t þ jkz
À Á
;
and hence, we have
C n1; n2; n3ð Þ ¼
X
k
Ak exp i Æk:r À
n1ex;n1
þ n2ey;n2
þ n3ez;n3
h
t þ jk
:
Here, Ak ¼ Ckx
Cky
Ckz
; jk ¼ jkx
þ jky
þ jkz
, and k ¼ kx; ky; kz
À Á
are the ampli-
tude, phase constant, and wavevector of C n1; n2; n3ð Þ , respectively. Further
writing
n1ex;n1
þ n2ey;n2
þ n3ez;n3
h
¼ ok n1; n2; n3ð Þ ¼ ok; (2.17a)
the field propagating along the direction of a vector k can be expressed as
C k; okð Þ ¼ Ak exp i Æk:r À okt þ jkð Þ; (2.17b)
representing a phonon of energy hok and momentum Æ hk. For a small kj j, the
propagation in a cubic lattice can be characterized by a constant speed v of propaga-
tion determined by ok ¼ v kj j, indicating no dispersion in this approximation.
The phonon propagation can be described by the vector k, composing a recipro-
cal lattice space, as illustrated in two dimensions in Fig. 2.2 by
kx ¼
2pn1
L
; ky ¼
2pn2
L
and kz ¼
2pn3
L
;
2.3 Phonon Field and Momentum 17
8. in a cubic crystal, where aN ¼ L. A set of integers n1; n2; n3ð Þ determines an
energy and momentum of a phonon propagating in the direction of k, where ok
¼ ok n1; n2; n3ð Þ and kj j ¼ 2p
L
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n2
x þ n2
y þ n2
z
q
. In the reciprocal space, all points on a
spherical surface of radius kj jcorrespond to the same energy hok , representing a
sphere of a constant radius kj j and energy ek. Quantum mechanically, we can write
the phonon momentum as p ¼ hk to supplement the energy hok, characterizing a
phonon particle.
2.4 Thermal Equilibrium
In thermodynamics, a crystal must always be in thermal contact with the surround-
ings. At constant external pressure p, the quantized vibration field can be in
equilibrium with the surroundings at a given temperature T. A large number of
phonons are in collision-free motion, traveling in all directions through the lattice,
colliding with surfaces to exchange their energy and momentum with the surround-
ings. Assuming the crystal volume as unchanged, the average of phonon energies
can be calculated with the Boltzmann probability at T.
In equilibrium, the total energy of a crystal can be expressed asU þ Us, whereUs
is the contribution from the heat reservoir and Urepresents the energy of a stable
crystal. In this case, the total energy U þ Us should be stationary with any
thermodynamic variation around equilibrium. Using probabilities w and ws for
keeping the crystal in equilibrium with the surroundings, the product wws should
be calculated as maximizing U þ Us to determine the most probable value. Setting
this variation problem as
2πny
ky =
a
2πnx
kx =
a
dk
k
4
3
2
1
0
1 2 3 4
Fig. 2.2 Two-dimensional
reciprocal lattice. A lattice
point is indicated by kx; ky
À Á
.
Two quarter-circles of radii k
and k + dk show surfaces of
constant ek and ekþdk for small
kj j in kxky plane.
18 2 Phonons
9. d wwsð Þ ¼ 0 and d U þ Usð Þ ¼ 0
for arbitrary variations dw and dws, these variations can be calculated as
wsdw þ wdws ¼ 0 and dU þ dUs ¼ 0;
respectively. We can therefore write
d ln wð Þ
dU
¼
d ln wsð Þ
dUs
;
which is a common quantity between U and Us. Writing it as equal to b ¼ 1
kBT , we
can relate b to the conventional absolute temperature T. Therefore,
d ln wð Þ
dU
¼ b ¼
1
kBT
or w ¼ wo exp À
U
kBT
: (2.18)
Here, w is called the Boltzmann probability; wo is the integration constant that can
be determined by assuming U ¼ 0 at T ¼ 0 K, where kB is the Boltzmann constant.
Quantum theoretically, however, T ¼ 0 is fundamentally unreachable, as stated in
the third law of thermodynamics. Accordingly, we write U ¼ 0 þ Uo at 0 K, where
Uo ¼ 1
2 Nhoo is the zero-point energy.
Although we considered only vibrations so far, physical properties of a crystal are
also contributed by other variables located at lattice points or at interstitial sites.
Though primarily independent of lattice vibrations, these variables can interact with
the lattice via phonon scatterings. If accessed by random collisions of phonons,
energiesei of these variables are statistically available with Boltzmann’s probabilities
wi, so we can write equations
U ¼
X
i
ei and w ¼ Piwi;
where
wi ¼ wo exp À
ei
kBT
and
X
i
wi ¼ 1; (2.19)
as these wi are for exclusive events. In this case, the function Z ¼
P
i
exp À ei
kBT
,
called the partition function, is useful for statistical calculation. In such a system
as called microcanonical ensemble, thermal properties can be calculated directly
with Z.
The Boltzmann statistics is a valid assumption for a dynamical system under the
ergodic hypothesis. Despite of the absence of rigorous proof, the Boltzmann statis-
tics can usually be applied to phonon gas in a fixed volume. Thermodynamically,
2.4 Thermal Equilibrium 19
10. however, it is only valid for isothermal processes, because the volume is not always
constant in adiabatic processes of internal origin. The anharmonic lattice cannot be
ergodic in strict sense, whereas the harmonicity is essentially required for stable
crystals at constant volume and pressure.
2.5 Specific Heat of a Monatomic Crystal
The specific heat at a constant volume CV ¼ @U
@T
À Á
V
is a quantity measurable with
varying temperature under a constant external pressure p. The phonon theory is
adequate for simple monatomic crystals, if characterized with no structural
changes.
For such a crystal, the specific heat and internal energy are given by quantized
phonon energies ek ¼ nk þ 1
2
À Á
hok , for which the wavevector k is distributed
virtually in all directions in the reciprocal space. Assuming 3N phonons in total,
the energies ek are degenerate with the density of k-states written as gðkÞ, which is a
large number as estimated from a spherical volume of radius kj j in the reciprocal
space. In this case, the partition function can be expressed as
Zk ¼ gðkÞ exp À
ek
kBT
¼ gðkÞ exp À
hok
2kBT
X1
nk¼0
exp À
nkhok
kBT
;
where the infinite series on the right converges, if hok
kBT 1. In fact, this condition is
satisfied at any practical temperature T lower than the melting point so that Zk is
expressed as
Zk ¼ gðkÞ
exp À
hok
2kBT
1 À exp À
hok
kBT
:
The total partition function is given by the product Z ¼ PkZk , so that ln Z ¼P
k ln Zk; the free energy can therefore be calculated as the sum of ln Zk, namely,
F ¼ kBT
P
k
ln Zk, where
ln Zk ¼ À
hok
2
þ kBT ln gðkÞ À kBT ln 1 À exp À
hok
kBT
'
:
By definition, we have the relation F ¼ U À TS ¼ U þ T @F
@T
À Á
V
from which we
can derive the formula U ¼ ÀT2 @
@T
F
T
À Á
. Using the above ln Zk, we can show that the
internal energy is given by
U ¼ Uo þ
X
k
hok
exp hok
kBT À 1
and Uo ¼
1
2
X
k
hok: (2.20)
20 2 Phonons
11. The specific heat at constant volume can then be expressed as
CV ¼
@U
@T
V
¼ kB
X
k
hok
kBT
2
exp
hok
kBT
exp
hok
kBT
À 1
2
: (2.21)
To calculate CV with (2.21), we need to evaluate the summation with the number
of phonon states on energy surface ek ¼ h ok þ 1
2
À Á
in the reciprocal space. In
anisotropic crystals, such a surface is not spherical, but a closed surface, as
shown in Fig. 2.3a. In this case, the summation in (2.21) can be replaced by a
volume integral over the closed surface, whose volume element is written as d3
k
¼ dk:dS ¼ dk? dSj j. Here, dk?is the component of k perpendicular to the surface
element dSj j ¼ dS, as illustrated two-dimensionally in Fig. 2.3b. We can write
dok ¼
dek
h
¼
1
h
gradke kð Þj jdk?;
where 1
h gradke kð Þj j ¼ vg represents the group velocity for propagation, and hence
dk? ¼ dok
vg
. Using these notations, we can reexpress (2.21) as
CV ¼ kB
ð
ok
hok
kBT
2
exp
hok
kBT
exp
hok
kBT
À 1
2
D okð Þdok; (2.22a)
where
a b
n
kx
dS
dS
0
kz
kx
ky
ky
Fig. 2.3 (a) A typical constant-energy surface in three-dimensional reciprocal space, wheredSis a
differential area on the surface. (b) The two-dimensional view in the kxky-plane.
2.5 Specific Heat of a Monatomic Crystal 21
12. D okð Þ ¼
L
2p
3 þ
S
dS
vg
(2.22b)
is the density of phonon states on the surface S.
Tedious numerical calculations performed in early studies on representative
crystals resulted in such curves as shown in Fig. 2.4a, for example, of a diamond
crystal. However, in relation with dispersive longitudinal and transversal modes,
the analysis was extremely difficult to obtain satisfactory comparison with experi-
mental results. On the other hand, Einstein and Debye simplified the functionD okð Þ
independently, although somewhat oversimplified for practice crystals. Neverthe-
less, their models are proven to be adequate in many applications to obtain useful
formula for Uand CV for simple crystals [4].
2.6 Approximate Models
2.6.1 Einstein’s Model
At elevated temperatures T, we can assume that thermal properties of a crystal are
dominated by n phonons of energy hoo. Einstein proposed that the dominant mode
at a high temperature is of a single frequency oo, disregarding all other modes in the
vibration spectrum. In this model, using the expression (2.22b) simplified as D ooð Þ
¼ 1, we can express the specific heat (2.22a) and the internal energy as
6
ω
a b
4
2
0 01 2 2 4 6 8
ωD
ω
2π
D (ω)
trans.
trans.
long.
long.
3 2π
k
Fig. 2.4 (a) Examples of practical dispersion curves. Longitudinal and transverse dispersions are
shown by solid and broken curves, respectively. (b) The solid curve shows an example of an
observed density function, being compared with the broken curve of Debye model.
22 2 Phonons
13. CV ¼ 3NkB
x2
exp x
exp x À 1ð Þ2
and U ¼ 3NkB
1
2
x þ
x
exp x À 1
; (2.23)
respectively, where x ¼ YE
T ; the parameter YE ¼ hoo
kB
is known as the Einstein
temperature. It is noted that in the limit x ! 0, we obtain CV ! 3NkB.
At high temperatures, U can be attributed to constituent masses, vibrating
independently in degrees of freedom 2; hence, the corresponding thermal energy
is 2 Â 1
2 kBT, and
U ¼ 3NkBT and CV ¼ 3NkB: (2.24)
This is known as the Dulong–Petit law, which is consistent with Einstein’s
model in the limit of T ! 1.
2.6.2 Debye’s Model
At lower temperatures, longitudinal vibrations at low frequencies are dominant
modes, which are characterized approximately by a nondispersive relation o ¼ vgk.
The speed vg is assumed as constant on a nearly spherical surface for constant
energy in the k-space. Letting vg ¼ v, for brevity, (2.22b) can be expressed as
D oð Þ ¼
L
2p
3
4po2
v3
: (2.25)
Debye assumed that with increasing frequency, the density D oð Þ should be
terminated at a frequency o ¼ oD, called Debye cutoff frequency, as shown by the
broken curve in Fig. 2.4b. In this case, the density function D oð Þ / o2
can be
normalized as
RoD
0
D oð Þdo ¼ 3N, so that (2.25) can be replaced by
D oð Þ ¼
9N
o3
D
o2
: (2.26)
Therefore, in the Debye model, we have
U ¼ 3NkBT
ðoD
0
ho
2
þ
ho
exp ho
kBT À 1
0
@
1
A 3o2
do
o3
D
and
2.6 Approximate Models 23
14. CV ¼ 3NkB
ðoD
0
exp
ho
kBT
exp
ho
kBT
À 1
2
ho
kBT
2
3o2
do
o3
D
:
Defining Debye temperature hoD
kBT ¼ YD and ho
kBT ¼ x, similar to Einstein’s model,
these expressions can be simplified as
U ¼
9
8
NkBYD þ 9NkBT
T
YD
3 ð
YD
T
0
x3
exp x À 1
dx
and
CV ¼ 9NkB
T
YD
3 ð
YD
T
0
x4
exp x
exp x À 1ð Þ2
dx:
Introducing the function defined by
Z
YD
T
¼ 3
T
YD
3 ð
YD
T
0
x3
dx
exp x À 1
; (2.27)
known as the Debye function, the expression CV ¼ 3NkBZ YD
T
À Á
describes
temperature-dependent CV for TYD. In the limit of YD
T ! 1, however, these U
and CV are dominated by the integral
ð1
0
x3
dx
exp x À 1
¼
p4
15
;
and hence the formula
U ¼
9
8
NkBYD þ 9NkBT
T
YD
3
p4
15
and
CV ¼ 9NkB
T
YD
3
p4
15
: (2.28)
24 2 Phonons
15. can be used at lower temperatures than YD. In the Debye model, we have thus the
approximate relation CV / T3
for TYD, which is known as Debye T3
-law.
Figure 2.5 shows a comparison of observed values of CV from representative
monatomic crystals with the Debye and Dulong–Petit laws, valid at low and high
temperatures, respectively, showing reasonable agreements.
2.7 Phonon Statistics Part 1
Quantizing the lattice vibration field, we consider a gas of phonons hok; hkð Þ. A
large number of phonons exist in excited lattice states, behaving like classical
particles. On the other hand, phonons are correlated at high densities, owing to
their quantum nature of unidentifiable particles. Although dynamically unspecified,
the total number of phonons is thermodynamically determined by the surface
boundaries at T, where phonon energies are exchanged with heat from the sur-
roundings. In equilibrium, the number of photons on each k-state can be either one
of n ¼ 1; 2; . . . ; 3N. Therefore, the Gibbs function can be expressed by G p; T; nð Þ,
but the entropy fluctuates with varying n in the crystal. Such fluctuations can be
described in terms of a thermodynamic probability g p; T; nð Þ, so that we consider
that two phonon states, 1 and 2, can be characterized by probabilities g p; T; n1ð Þ and
g p; T; n2ð Þ in an exclusive event, in contrast to the Boltzmann statistics for indepen-
dent particles.
At constant p, the equilibrium between the crystal and reservoir can therefore be
specified by minimizing the total probability g p; T; nð Þ ¼ g p; T; n1ð Þ þ g p; T; n2ð Þ,
considering such binary correlations dominant under n ¼ n1 þ n2 ¼ constant,
leaving all other niði 6¼ 1; 2Þ as unchanged. Applying the variation principle for
small arbitrary variations dn1 ¼ Àdn2, we can minimize g p; T; nð Þ to obtain
Cv
T
T3
-law
3R
0.6
0.4
0.2
0 .2 .4 .6 .8 1.0 1.2
Al
Cu
Ag
Pb 95K
215K
309K
396K
Dulong-Petit
ΘD
Fig. 2.5 Observed specific
heat CV 3R= against T YD= for
representative metals. R is the
molar gas constant. In the
bottom-right corner, values of
D for these metals are shown.
The T3
-law and Dulong–Petit
limits are indicated to
compare with experimental
results.
2.7 Phonon Statistics Part 1 25
16. dgð Þp;T ¼
@g1
@n1
p;T
dn1 þ
@g2
@n2
p;T
dn2 ¼ 0;
from which we derive the relation
@g1
@n1
p;T
¼
@g2
@n2
p;T
:
This is a common quantity between g1 and g2, which is known as the chemical
potential. Therefore, we have equal chemical potentials m1 ¼ m2 in equilibrium
against phonon exchange. Writing the common potential as m, a variation of the
Gibbs potential G for an open system at equilibrium can be expressed for an
arbitrary variation d n as
dG ¼ dU À TdS þ pdV À mdn; (2.29)
where dn represents a macroscopic variation in the number of phonons n.
Consider a simple crystal, whose two thermodynamic states are specified by the
internal energy and phonon number, Uo; Noð Þ and Uo À e; No À nð Þ , which
are signified by probabilities go and g, as related to their entropies S Uo; Noð Þ and
S Uo À e; No À nð Þ, respectively. Writing the corresponding Boltzmann relations,
we have
go ¼ exp
S Uo; Noð Þ
kB
and g ¼ exp
S Uo À e; No À nð Þ
kB
:
Hence,
g
go
¼
exp S Uo À e; No À nð Þ=kBf g
exp S Uo; Noð Þ=kBf g
¼ exp
DS
kB
;
where
DS ¼ S Uo À e; No À nð Þ À S Uo; Noð Þ ¼ À
@S
@Uo
No
e À
@S
@No
Uo
n:
Using (2.29), we obtain the relations
@S
@Uo
No
¼
1
T
and
@S
@No
Uo
¼ À
m
T
so that
26 2 Phonons
17. g ¼ go exp
mn À e
kBT
: (2.30)
For phonons, the energy e is determined by any wavevector k, where kj j ¼ 1; 2;
::::; 3N, and N can take any integral number. The expression (2.30) is the Gibbs
factor, whereas for classical particles, we use the Boltzmann factor instead. These
factors are essential in statistics for open and closed systems, respectively. For
phonons, it is convenient to use the notation l ¼ exp m
kBT , with which (2.30) can be
written as g ¼ goln
exp À e
kBT
. The factor l here implies a probability for the
energy level e to accommodate one phonon adiabatically [5], whereas the conven-
tional Boltzmann factor exp À e
kBT
is an isothermal probability of e at T. Origi-
nally, the chemical potential m was defined for an adiabatic equilibrium with an
external chemical agent; however, for phonons l is temperature dependent as
defined by l ¼ exp m
kBT . Here, the chemical potential is determined as m ¼ À @G
@n
À Á
p;T
from (2.29), which is clearly related with the internal energy due to phonon
correlations in a crystal.
For phonon statistics, the energy levels are en ¼ nho, and the Gibbs factor is
determined by e ¼ ho and n. The partition function can therefore be expressed as
ZN ¼
XN
n¼0
ln
exp À
ne
kBT
¼
XN
n¼0
l exp À
e
kBT
'n
:
Consideringl exp À e
kBT
1, the sum ofthe infinite series evaluated for N ! 1is
Z ¼
1
1 À l exp À
e
kBT
:
With this so-called grand partition function, the average number of phonons can
be expressed as
nh i ¼ l
@ ln Z
@l
¼
1
1
l
exp À
e
kBT
À 1
¼
1
exp
e À m
kBT
À 1
: (2.31)
This is known as the Bose–Einstein distribution. It is noted that the energy e is
basically dependent on temperature, whereas the chemical potential is small
and temperature independent. Further, at elevated temperatures, we consider that
for e m; (2.31) is approximated as nh i % exp mÀe
kBT % exp Àe
kBT , which is the
Boltzmann factor. However, there should be a critical temperature Tc for nh i ¼ 1
to be determined by e Tcð Þ ¼ m, which may be considered for phonon condensation.
2.7 Phonon Statistics Part 1 27
18. So far, phonon gas was specifically discussed, but the Bose–Einstein statistics
(2.31) can be applied to all other identical particles characterized by even parity;
particles obeying the Bose–Einstein statistics are generally characterized by even
parity and called Bosons. Particles with odd parity will be discussed in Chap. 11 for
electrons.
2.8 Compressibility of a Crystal
In the foregoing, we discussed a crystal under a constant volume condition. On the
other hand, under constant temperature, the Helmholtz free energy can vary with a
volume change DV, if the crystal is compressed by
DF ¼
@F
@V
T
DV;
where p ¼ À @F
@V
À Á
T
is the pressure on the phonon gas in a crystal. At a given
temperature, such a change DFmust be offset by the external work À pDV by
applying a pressure p, which is adiabatic to the crystal.
It is realized that volume-dependent energies need to be included in the free
energy of a crystal in order to deal with the pressure from outside. Considering an
additional energy Uo ¼ UoðVÞ, the free energy can be expressed by
F ¼ Uo þ 9NkBT
T
YD
3 ð
YD
T
0
x
2
þ ln 1 À exp Àxð Þf g
!
x2
dx
¼ Uo þ 3NkBTZ
YD
T
; (2.32)
where Z YD
T
À Á
is the Debye function defined in (2.27), for which we have the relation
@Z
@ ln YD
T
¼ À
@Z
@ ln T
V
¼ À
T
YD
@Z
@T
: (2.33)
Writing z ¼ z T; Vð Þ ¼ TZ YD
T
À Á
for convenience, we obtain
@z
@V
T
¼ À
g
V
@z
@ ln YD
T
¼
gT
V
@Z
@ ln T
V
;
where the factor
28 2 Phonons
19. g ¼ À
d ln Y
d ln V
is known as Gru¨neisen’s constant. Using (2.30), the above relation can be
reexpressed as
@z
@V
T
¼
g
V
T
@z
@T
V
À z
'
:
From (2.29), we have NkBz T; Vð Þ ¼ F À Uo; therefore, this can be written as
@ F À Uoð Þ
@V
'
T
¼
g
V
T
@ F À Uoð Þ
@T
V
À F þ Uo
'
: (2.34)
Noting Uo ¼ UoðVÞ, the derivative in the first term of the right side is equal to
T @F
@T
À Á
V
¼ ÀTS; hence, the quantity in the curly brackets is À U þ Uo ¼ Uvib that
represents the energy of lattice vibration. From (2.31), we can derive the expression
for pressure in a crystal, that is,
p ¼ À
dUo
dV
þ
gUvib
V
; (2.35)
which is known as Mie–Gru¨neisen’s equation of state.
The compressibility is defined as
k ¼ À
1
V
@V
@p
T
; (2.36)
which can be obtained for a crystal by using (2.32). Writing (2.32) as pV ¼ ÀV dUo
dV
þgUvib and differentiating it, we can derive
p þ V
@p
@V
T
¼ À
dUo
dV
À V
d2
Uo
dV2
þ g
@Uvib
@V
T
:
Since the atmospheric pressure is negligible compared with those in a crystal, we
may omit p, and also from (2.31)
@Uvib
@V
T
¼
g
V
T
@Uvib
@T
V
À Uvib
'
in the above expression. Thus, the compressibility can be obtained from
2.8 Compressibility of a Crystal 29
20. 1
k
¼ ÀV
@p
@V
T
¼
dUo
dV
þ V
d2
Uo
dV2
À
g2
V
TCV À Uoð Þ; (2.37)
where CV ¼ @Uvib
@T
À Á
V
is the specific heat of lattice vibrations.
If p ¼ 0, the volume of a crystal is constant, that is, V ¼ Vo and dUo
dV ¼ 0, besides
Uvib ¼ const: of V. Therefore, we can write 1
ko
¼ Vo
d2
Uo
dV2
V¼Vo
, meaning a
hypothetical compressibility ko in equilibrium at p ¼ 0. Then with (2.34) the
volume expansion can be defined as
b ¼
V À Vo
Vo
¼
kogUvib
V
: (2.38)
Further, using (2.32)
@p
@T
V
¼
g
V
@Uvib
@T
V
¼
gCV
V
;
which can also be written as
@p
@T
V
¼
1
V
@V
@T
p
À
1
V
@V
@p
T
;
and hence we have the relation among g; k, and b, that is, g ¼ À V
CV
b
k .
Such constants as Y; k; b, and g are related with each other and are significant
parameters to characterize the nature of crystals. Table 2.1 shows measured values of
YDby thermal and elastic experiments on some representative monatomic crystals.
Exercise 2
1. It is important that the number of phonons in crystals can be left as arbitrary,
which is thermodynamically significant for Boson particles. Sound wave propa-
gation at low values of k and o can be interpreted for transporting phonons, which
is a typical example of low-level excitations, regardless of temperature. Discuss
why undetermined number of particles is significant in Boson statistics. Can there
be any other Boson systems where the number of particles if a fixed constant?
Table 2.1 Debye
temperatures YD determined
by thermal and elastic
experimentsa
Fe Al Cu Pb Ag
Thermal 453 398 315 88 215
Elasticb
461 402 332 73 214
a
Data: from Ref. [3]
b
Calculated with elastic data at room temperature
30 2 Phonons
21. 2. Einstein’s model for the specific heat is consistent with assuming crystals as a
uniform medium. Is it a valid assumption that elastic properties can be attributed
to each unit cell? What about a case of nonuniform crystal? At sufficiently high
temperatures, a crystal can be considered as uniform. Why? Discuss the validity
of Einstein’s model at high temperatures.
3. Compare the average number of phonons nh i calculated from (2.26) with that
expressed by (2.31). Notice the difference between them depends on the chemi-
cal potential: either m ¼ 0 or m 6¼ 0: Discuss the role of a chemical potential in
making these two cases different.
4. The wavefunction of a phonon is expressed by (2.17b). Therefore in a system of
many phonons, phonon wavefunctions should be substantially overlapped in the
crystal space. This is the fundamental reason why phonons are unidentifiable
particles; hence, the phonon system in crystals can be regarded as condensed
liquid. For Boson particles 4
He, discuss if helium-4 gas can be condensed to a
liquid phase at 4.2K.
5. Are the hydrostatic pressure p and compressibility discussed in Sect. 2.8 ade-
quate for anisotropic crystals? Comment on these thermodynamic theories
applied to anisotropic crystals.
References
1. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1964)
2. R.S. Knox, A. Gold, Symmetry in the Solid State (Benjamin, New York, 1964)
3. C. Kittel, Quantum Theory of Solids, (John Wiley, New York, 1963)
4. C. Kittel, Introduction to Solid State Physics, 6th edn. (Wiley, New York, 1986)
5. C. Kittel, H. Kroemer, Thermal Physics (Freeman, San Francisco, 1980)
References 31