(D) Chloride process metallurgy

Shelton, R.A.J.

doi:10.1179/095066076790136717

Cited by 7 publications

(17 citation statements)

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“…This is consistent with the interpretations [19,21] that these tails are due to the so-called 'anomalously localised' electronic states, which occur in rare, trapping disorder configurations. Indeed our result (11) can be understood in terms of the 'optimal fluctuation' concept, discussed in this context in [19]. The optimal fluctuation in our case corresponds to having a constant mass m 0 within the sample.…”

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confidence: 71%

“…Here we only quote the results. Formulae (11) and (15) are not affected. The power law in the denominator of (12) changes to 1/Q 1+m0/g 0 .…”

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confidence: 96%

“…It seems natural to undertake the next step and investigate the probability distribution functions for the above systems. To this end, we adopt a simple mathematical technique, based on the recent observation by Shelton and Tsvelik [11] that, at ǫ = 0, the random mass Dirac model can be formulated as a one-particle quantum mechanical problem. Indeed, if we introduce the combinations χ ± = (R ∓ L)/ √ 2 of the chiral components of the electron field, then the Dirac equation becomes…”

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confidence: 99%

“…This formula is valid when the factor in the exponential is large, i.e. ln 2 τ 0 ≫ L. As a simple application to random spin chains, we consider long-time relaxation of the magnetisation M (t) inside a finite segment of length L. Due to (11) we find that M (t) ∼ exp [− ln 2 (t)/8gL]: a very slow decay.…”

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confidence: 99%

“…The power law in the denominator of (12) changes to 1/Q 1+m0/g 0 . Both formulae (16) and (17) Notice that the probability of long time delays, (11), as well as the probability of a small transmission, (15), follows the so-called log-normal law. To our knowledge, the log-normal tails of the distribution functions in onedimensional disordered systems were first obtained in Ref.…”

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confidence: 99%

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Statistical properties of a localization-delocalization transition in one dimension

et al. 1999

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We study a one-dimensional model of disordered electrons (also relevant for random spin chains), which exhibits a delocalisation transition at half-filling. Exact probability distribution functions for the Wigner time and transmission coefficient are calculated. We identify and distinguish those features of probability densities that are due to rare, trapping configurations of the random potential from those which are due to the proximity to the delocalisation transition. PACS No:The Anderson transition in dimensions D < 3 has recently attracted a renewed interest in relation to such systems as random antiferromagnetic spin chains [1] and high mobility Si MOSFET's [2]. Experiments on both systems could not be accounted for in the standard scaling theory of localisation [3].The simplest disordered model known to exhibit metallic behaviour, in spite of being one-dimensional, is the random-hopping model,where t n > 0 are random variables, with n-independent average, t n = t, and c n annihilates a spinless fermion at site n. This 1D model has a single delocalised state at the middle of the band, ǫ = 0. This is an interesting example of the failure of the scaling theory of localisation. Moreover, this model has many common features with a wide class of random spin chains; such as the spin-1/2 random Heisenberg chain H = n J n S n · S n+1 , where J n > 0 are randomly distributed. (The XX version of the latter model is, in fact, exactly equivalent to (1), upon the Jordan-Wigner transformation.) For the random hopping model, a great deal is known about such self-averaging quantities as the total density of states. More recently, some of the correlation functions have also been calculated. However, the behaviour of probability distributions in the proximity of the delocalisation transition is virtually unexplored. It is the purpose of this letter to address this issue.In the continuum limit, model (1) becomes what is known as the random-mass Dirac model (see e.g. [4,5]):where R and L are the chiral components of the electron field operator. The derivation of the continuum limit assumes weak disorder such that t n = t+δt n (the Fermi velocity, v F , associated with t is set to 1). It is the staggered component of the random hopping, δt n → (−1) n m(x), which enters into the continuum theory. In field-theoretic language, this corresponds to a random mass.It was found by Dyson in 1953 [6], that the average electron density of states for a model equivalent to (1) diverges at the middle of the band: ρ(ǫ) ∼ 1/(ǫ| ln ǫ| 3 ). By the Thouless relation [7], such a density of states implies a divergent localisation length λ ǫ ∼ | ln ǫ|. The criticality of the model at half-filling was established by Gogolin and Mel'nikov [8]. In particular, they found that model (2) has a finite conductivity in contrast with the Mott law in the standard localised regime. Calculations of Ref.[9] first indicated that it is not the Thouless length λ ǫ , but rather the length l ǫ ∼ ln 2 ǫ, which is likely to govern the correlation functions. The role ...

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confidence: 71%

“…Here we only quote the results. Formulae (11) and (15) are not affected. The power law in the denominator of (12) changes to 1/Q 1+m0/g 0 .…”

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confidence: 96%

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confidence: 99%

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confidence: 99%

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confidence: 99%