Low Energy Phonons in Amorphous Materials

  • Roger Maynard


The Resonant interaction: All the properties(1) observed so far on glasses and polymers are conceptually strongly akin to the problem of the interaction between the electronic spins of paramagnetic impurities and phonons. More precisely the understanding of the low temperature properties requires the study of the propagation of phonons in a resonant medium characterized by a broad distribution of the energy splittings of the resonant spins. Therefore, the natural starting point is the theory of Jacobsen and Stevens(2) for the coupling of phonons and spins. Here, the spins are the configurational defects pictured by a double well potential proposed first by Anderson, Halperin, Varma(3) and Phillips(4). Retaining only the double degrees of}freedom, these defects can be represen-ted by a fictive 1/2 spin \(\mathop {S\alpha }\limits^ \to\) located at site \(\mathop {Ra}\limits^ \to\), with an energy splitting Eα. The coupled Hamiltonian
$${H_o} = \sum\limits_k {\forall {\omega _k}} a_k^ + {a_k} + \sum\limits_\alpha {E\alpha S\mathop \alpha \limits^z }$$
where ω=vk is the frequency of a phonon k of a unique acoustic branch we consider for simplicity. The interaction Hamiltonian is a quadratic form of the spin and phonon variables:
$$Hint = \sum\limits_\alpha {B_x^\alpha } \eta \left( {\mathop {R\alpha }\limits^ \to } \right)S_\alpha ^x\eta \left( {\mathop r\limits^ \to } \right) = \frac{1}{{\sqrt v }}\sum\limits_k {{{\mathop {2\rho v}\limits^{\left( {\frac{{\forall \omega k}}{2}} \right)} }^{1/2}}} \left( {{a_k} + a_{ - k}^ + } \right){e^{i{{\vec k}_{\vec \Gamma }}}}$$
ρ the mass density, \(\eta \left( {\vec r} \right)\) is the local strain operator (only the isotropic local dilatation is considered for simplicity) and S α x is the x-component of the Pauli matrix describing the spin flip process.


Sound Velocity Energy Splitting Resonant Medium Natural Starting Point Local Dilatation 
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Copyright information

© Plenum Press, New York 1976

Authors and Affiliations

  • Roger Maynard
    • 1
  1. 1.Centre de Recherches sur les Très Basses TempératuresCNRSGrenoble-CédexFrance

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