- •Foreword
- •Preface
- •Contents
- •1 Introduction to Nonlinear Acoustics
- •1.1 Introduction
- •1.2 Constitutive Equations
- •1.3 Phenomena in Nonlinear Acoustics
- •References
- •2 Nonlinear Acoustic Wave Equations for Sound Propagation in Fluids and in Solids
- •2.1 Nonlinear Acoustic Wave Equations in Fluids
- •2.1.1 The Westervelt Equation [1]
- •2.1.2 The Burgers’ Equation [2]
- •2.1.3 KZK Equation
- •2.1.4 Nonlinear Acoustic Wave Equations for Sound Propagation in Solids
- •References
- •3 Statistical Mechanics Approach to Nonlinear Acoustics
- •3.1 Introduction
- •3.2 Statistical Energy Analysis is Transport Theory
- •3.3 Statistical Energy Analysis
- •3.4 Transport Theory Approach to Phase Transition
- •References
- •4 Curvilinear Spacetime Applied to Nonlinear Acoustics
- •4.1 Introduction and Meaning of Curvilinear Spacetime
- •4.2 Principle of General Covariance
- •4.3 Contravariant and Covariant Four-Vectors
- •4.4 Contravariant Tensors and Covariant Tensors
- •4.5 The Covariant Fundamental Tensor gμν
- •4.6 Equation of Motion of a Material Point in the Gravitational Field
- •4.8 The Euler Equation of Fluids in the Presence of the Gravitational Field
- •4.9 Acoustic Equation of Motion for an Elastic Solid in the Presence of Gravitational Force
- •Reference
- •5 Gauge Invariance Approach to Nonlinear Acoustical Imaging
- •5.1 Introduction
- •5.3 Illustration by a Unidirectional Example
- •5.4 Quantization of the Gauge Theory
- •5.5 Coupling of Elastic Deformation with Spin Currents
- •References
- •6.1 Introduction
- •6.2 The Thermodynamic Method
- •6.2.1 Theory
- •6.2.2 Experiment
- •6.3 The Finite Amplitude Method
- •6.3.1 The Wave Shape Method
- •6.3.2 Second Harmonic Measuements
- •6.3.3 Measurement from the Fundamental Component
- •6.4 B/A Nonlinear Parameter Acoustical Imaging
- •6.4.1 Theory
- •6.4.2 Simulation
- •6.4.3 Experiment [17]
- •6.4.4 Image Reconstruction with Computed Tomography
- •References
- •7 Ultrasound Harmonic Imaging
- •7.1 Theory of Ultrasound Harmonic Imaging
- •7.2 Methods Used to Isolate the Second Harmonic Signal Component
- •7.3 Advantages of Harmonic Imaging
- •7.4 Disadvantages of Harmonic Imaging
- •7.5 Experimental Techniques in Nonlinear Acoustics
- •7.6 Application of Ultrasound Harmonic Imaging to Tissue Imaging
- •7.7 Applications of Ultrasonic Harmonic Imaging to Nondestructive Testing
- •7.8 Application of Ultrasound Harmonic Imaging to Underwater Acoustics
- •References
- •8 Application of Chaos Theory to Acoustical Imaging
- •8.1 Nonlinear Problem Encountered in Diffraction Tomography
- •8.4 The Link Between Chaos and Fractals
- •8.5 The Fractal Nature of Breast Cancer
- •8.6 Types of Fractals
- •8.6.1 Nonrandom Fractals
- •8.6.2 Random Fractals
- •8.7 Fractal Approximations
- •8.8 Diffusion Limited Aggregation
- •8.9 Growth Site Probability Distribution
- •8.10 Approximating of the Scattered Field Using GSPD
- •8.11 Discrete Helmholtz Wave Equation
- •8.12 Kaczmarz Algorithm
- •8.14 Applying GSPD into Kaczmarz Algorithm
- •8.15 Fractal Algorithm using Frequency Domain Interpretation
- •8.16 Derivation of Fractal Algorithm’s Final Equation Using Frequency Domain Interpolation
- •8.17 Simulation Results
- •8.18 Comparison Between Born and Fractal Approximation
- •References
- •9.1 Introduction
- •9.2 Mechanisms of Harmonic Generation Via Contact Acoustic Nonlinearity (CAN)
- •9.2.1 Clapping Mechanism
- •9.2.2 Nonlinear Friction Mechanism
- •9.3 Nonlinear Resonance Modes
- •9.4 Experimental Studies on Nonclassical CAN Spectra
- •9.4.1 CAN Application for Nonlinear Acoustical Imaging and NDE
- •9.5 Conclusions
- •References
- •10.1 Introduction
- •10.2 Principles of Modulation Acoustic Method
- •10.3 The Modulation Mode of Method of Crack Location
- •10.4 Experimental Procedure of the Modulation Method for NDT
- •10.5 Experimental Procedures for the Modulation Mode System
- •10.6 Conclusions
- •References
- •11.1 Introduction
5.3 Illustration by a Unidirectional Example |
33 |
5.3 Illustration by a Unidirectional Example
For further simplification, one takes the case of a one-dimensional space. Then the Lagrangian density will become
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∂ ψ |
∂ ψ |
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L = iψ + |
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− ψ + g0R |
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∂ t |
∂ x |
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[(∂x − g Rx ∂x )ψ +][(∂x − gRx ∂x )ψ] |
(5.14) |
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2m |
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Here R is the off-diagonal component of the tensor Rμν , with R = R01 = R10 and Rx = R11. Then the Lagrangian density of the elasticity becomes
L G F |
= − |
1 |
(∂0 Rx )2 |
+ |
(∂x R)2 |
− |
(∂0 R)2 |
− |
2(∂x R)(∂0Rx ) |
(5.15) |
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yielding the Hamiltonian density of the gauge field as
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HG F = |
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+ |
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(5.16) |
2 cs2 ∂ t |
∂ x |
cs2 |
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R is identified with the phonon field φ, with longitudinal polarization for the onedimensional line. The Hamiltonian density can be rewritten as
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1 1 ∂ φ 2 |
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HF G = |
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+ |
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(5.17) |
2 cs2 ∂ t |
∂ x |
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producing the linear acoustic wave equation with wave velocity cs …
5.4 Quantization of the Gauge Theory
The above treatment is classical gauge theory for electron–phonon interaction. To quantize the gauge theory, the phonon variables will be introduced. The phonon field, the displacement operator will be expanded in terms of the creation and annihilation operators as aq+ and aq as follows [4]:
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1 |
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i (q x |
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Wq t ) |
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i (q x |
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Wq t ) |
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φ (x, t) |
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a |
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e |
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a+e− |
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(5.18) |
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2Lρ Wq |
q |
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34 |
5 Gauge Invariance Approach to Nonlinear Acoustical Imaging |
where ρ = mass density, Wq = energy dispersion relation, L = size of the system, and φ(x, t) = displacement operator.
The second quantized Hamiltonian can be written as
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F G = ∫ |
ds H |
F G = |
W a+a |
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(5.19) |
H |
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q q |
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q
When the electron field is second-quantized, the standard electron–phonon interaction can be obtained. Expanding the electron field in plane waves, one has.
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ei k x ckσ |
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e−i k x c+ |
(5.20) |
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√L |
kσ |
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k,σ |
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where (ckσ , ck+σ ) are fermion destruction and creation operators of particles with wave number k and spin σ.
5.5 Coupling of Elastic Deformation with Spin Currents
The gauge invariance of electron–phonon interaction will also produce a coupling between the electron spin and the strain field of the crystal lattice via the spin = orbit interaction. The above coupling is of fundamental origin. The consideration of spin–orbit interaction will lead to the electronic Hamiltonian density [5]:
H |
S O = − |
i μB |
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ψ +.σ |
× |
Eψ ψ +σ |
× |
E. |
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(5.21) |
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4m |
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× |
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where μB = Bohr magnetor, and E = electric field.
Such effect is known as spin–orbit coupling. In order to preserve the gauge invariance of the theory, one has to replace the ordinary derivative by the covariant
derivative D, producing |
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i μB |
( ψ +.σ × Eψ−ψ +σ × E. ψ) |
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HS O |
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4m |
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μB |
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gεi j k E j Rkl [∂l (ψ +)σi − ψ +σi σl ψ] |
(5.22) |
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where repeated indices are to be summed over and g is the coupling constant defined previously. In the extra term appearing in (5.22), the electron spin is coupled to the space-like elastic field Rkl and to the electric field of the strained lattice.
In this treatment of gauge invariance approach to electron–phonon interaction, the phonons are related to a gauge field associated to local symmetry property as
5.5 Coupling of Elastic Deformation with Spin Currents |
35 |
gauge bosons. This is in contrast to the general understanding of phonons as Goldstone bosons, associated with the spontaneous breaking of a global symmetry. In our treatment of the gauge invariance approach to electron–phonon interaction, we are dealing with local gauge invariance of non-Abelian group.
References
1.Gan, Woon Siong. 2007. Gauge Invariance Approach to Acoustic Fields, Acoustical Imaging. In vol. 29, ed. I. Akiyama, 389–394. Springer.
2.Yang, C.N., and R.L. Mills. 1954. Conservation of isotopic spin and isotopic gauge invariance.
Physical Review 96 (1): 191–195.
3.Ryder, L.H. 1991. Quantum Field Theory, 2nd edn. Cambridge University Press.
4.Kittel, C. 1963. Quantum Theory of Solids. New York: John Wiley.
5.Dartora, C.A., and G.G. Cabrera. 2008. The electron–phonon interaction from fundamental local gauge symmetries in solids. Phys. Rev. B 78: 012403.