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Objectives / Scope Re-wheeling compressors to match late-life field conditions gives significant benefits in operational efficiency and carbon reduction. But changing the compressor wheels and increasing shaft speeds also introduces a risk in terms of the rotor-dynamic stability of the system. API assessments use deterministic methods to assess the design change, but give less information in terms of the key risks and how to control them. This paper outlines new methods for assessing rotor dynamic risks to compressors during re-wheeling and their value over traditional methods. Methods New methods were developed to extend beyond the API requirements in order to assess and manage the rotor-dynamic risk as part of a peer review process of re-wheeling a compressor train. A combination of sensitivity studies on key parameters and Self Organizing Maps (SOMs - a machine learning technique) was used to identify the factors which present the greatest risk to the re-wheeling, and a Monte Carlo analysis was used to identify the change in risk of rotor-dynamic problems when compared with the existing machine. Results The Monte Carlo analysis used random distributions of factors on key input parameters, and the same factors were applied to the existing and re-wheeled designs. It identified that although the re-wheeled design was nominally more stable than the existing design according to the API analysis, it actually presented a greater risk of instability. This is because the distribution of resulting stability values had a higher mean but a greater spread than the existing machine when subject to uncertainty in input parameters. Since the existing machine is free from dynamics problems, the parameter combinations which resulted in an unstable existing machine could be discounted, but the resulting subset of factors when applied to the re-wheeled design still gave some unstable cases. Therefore, the fact that the existing machine is free from dynamics problems does not in itself discount the possibility of problems following the re-wheel. SOMs were used to identify the components which posed the greatest risk to the re-wheeled design. This highlighted that low stiffness in two particular bearings along the high speed shaft would pose the greatest risk to shaft stability, meaning that close attention can be paid by the operators and OEMs to this to manage the risks as the re-wheel progresses. Novel Information This work shows that probabilistic and machine learning techniques have significant value in managing risks during compressor re-wheeling, highlighting risks which would not be identified using standard deterministic methods and focusing attention on the aspects which are most important to manage them.

In this document we assess the potential of RF separation to increase the kaon content of the high-intensity secondary beam to the NA62 experiment in the CERN North Area. The present mixed beam provides a nominal kaon flux of 45 MHz out of a total beam intensity of about 750 MHz. The experiment wishes to increase the kaon flux, without too much increasing the overall beam intensity which might otherwise exceed the rate capability of several detectors and cause increased backgrounds to the detector. To avoid a strong reduction of the kaon flux by decays in a very long beam line, a compact RF separator approach is evaluated as a baseline. Alternative possibilities are briefly described at the end of the document. It turns out that the performance will not meet expectations and that therefore conventional methods to increase the beam intensity will have to be pursued. With the relatively short length proposed, the K phase difference is too small to separate the kaons from the large emittance mixed beam. A larger phase difference would imply more length and hence a drastic reduction of the kaon component due to decays.

BACKGROUND Early childhood caries (ECC), is a common disorder characterized by the presence of one or more decayed (non-cavitated or cavitated lesions), missing (due to caries), or filled tooth surfaces of primary teeth in children 71 months or younger. South Africa has a diverse population in terms of culture, education, income and occupation. This diversity is due to the consequences of historical racial discrimination, poverty, unemployment, lack of accessibility to health services and quality of education. These factors make South Africa unique, and the disease and risk profiles for this country differs from other countries in similar stages of development. For these reasons, it is important to identify what the unique maternal and infant risk factors of ECC are in the South African context. OBJECTIVE The purpose of this study is to determine the risk factors associated with the incidence/prevalence of early childhood caries (ECC) in South Africa in children under the age of 6 years. METHODS All cross-sectional and cohort studies documenting the risk factors associated with the prevalence and incidence of dental disease and severity (decayed, missing and filled scores) will be included. Seven databases will be searched for eligible studies and those included will be based on a pre-specified inclusion criteria. Only studies conducted on South African children who are aged 6 years and younger, where dental caries risk factors are documented will be included. There is no restriction on the time or language of publication. Included articles will be scrutinized for quality by using a risk of bias tool developed by the Joanna Briggs Institute. The results will be presented narratively, and if possible, a meta-analysis will be performed. RESULTS The protocol is registered with PROSPERO, CRD42020216455. The literature search was conducted in November 2020. CONCLUSIONS The results of this study will provide a framework to inform medical and dental personnel to highlight mothers and infants at risk from developing ECC. This framework may be used to prevent non-communicable diseases (NCDs) such as obesity and diabetes incidence in children. CLINICALTRIAL PROSPERO, CRD42020216455

Background: Early childhood caries (ECC) is a common disorder characterized by the presence of one or more decayed (non-cavitated or cavitated lesions), missing (due to caries), or filled tooth surfaces of primary teeth in children 71 months old or younger. South Africa has a diverse population in terms of culture, education, income, and occupation. This diversity is due to the consequences of historical racial discrimination, poverty, unemployment, lack of accessibility to health services, and quality of education. These factors make South Africa unique, and the disease and risk profiles for this country differ from those of other countries at similar stages of development. For these reasons, it is important to identify the unique maternal and infant risk factors for ECC in the South African context. Objective: The purpose of this study is to determine the risk factors associated with the incidence and prevalence of ECC in South Africa in children under the age of 6 years. Methods: All cross-sectional and cohort studies documenting risk factors associated with the prevalence and incidence of dental disease and severity (decayed, missing, and filled scores) will be included. We will search 7 databases for eligible studies, and those included will be based on prespecified inclusion criteria. Only studies conducted with South African children who are aged 6 years and younger in which dental caries risk factors are documented will be included. There is no restriction on the time or language of publication. Included articles will be scrutinized for quality by using a risk of bias tool developed by the Joanna Briggs Institute. The results will be presented narratively, and if possible, a meta-analysis will be performed. Results: The literature search was conducted in November 2020. Conclusions: The results of this study will provide a framework to inform medical and dental personnel to highlight mothers and infants at risk of developing ECC. Trial Registration: PROSPERO International Prospective Register of Systematic Reviews CRD42020216455; https://www.crd.york.ac.uk/prospero/display_record.php?RecordID=216455

Background: Young children are at the highest risk of developing dental caries as they have a lack of autonomy over their diet and oral hygiene practices. Dental caries develops over time due to demineralization of tooth substance (enamel), which results from acid production during sugar metabolism by bacteria. Early onset of dental caries often results in asymptomatic presentation, but if left untreated, it can result in severe pain, infection, and dentoalveolar abscesses. Early childhood caries (ECC) is defined as dental caries in children aged 6 years and younger and is a significant public health problem in South Africa. According to the Global Burden of Disease study, untreated dental caries of primary teeth affects 532 million children. Untreated dental caries has many detrimental effects which can affect the physical development and reduce the quality of life of affected children. Furthermore, long-term untreated dental caries can result in school absenteeism, low BMI, and poor educational outcomes. Objective: The purpose of this study was to determine the prevalence and severity of ECC in South Africa in children under the age of 6 years. Methods: All cross-sectional studies documenting the prevalence and severity of dental disease (decayed, missing, and filled teeth scores) will be included. Various databases will be searched for eligible studies. Only studies conducted on South African children aged 6 years and under will be included. There will be no restriction on the time or language of publication. The quality of all eligible studies will be analyzed by a risk of bias tool developed by the Joanna Briggs Institute. The results will be presented narratively, and if possible, a meta-analysis will be conducted. Results: The protocol is registered with PROSPERO. The literature search was initially conducted in November 2018 and was repeated in November 2020. Conclusions: The results of this study will be used to advise stakeholders of the prevalence and severity of dental disease in children under 6 years of age in South Africa. Trial Registration: PROSPERO CRD42018112161

Background Young children are at the highest risk of developing dental caries as they have a lack of autonomy over their diet and oral hygiene practices. Dental caries develops over time due to demineralization of tooth substance (enamel), which results from acid production during sugar metabolism by bacteria. Early onset of dental caries often results in asymptomatic presentation, but if left untreated, it can result in severe pain, infection, and dentoalveolar abscesses. Early childhood caries (ECC) is defined as dental caries in children aged 6 years and younger and is a significant public health problem in South Africa. According to the Global Burden of Disease study, untreated dental caries of primary teeth affects 532 million children. Untreated dental caries has many detrimental effects which can affect the physical development and reduce the quality of life of affected children. Furthermore, long-term untreated dental caries can result in school absenteeism, low BMI, and poor educational outcomes. Objective The purpose of this study was to determine the prevalence and severity of ECC in South Africa in children under the age of 6 years. Methods All cross-sectional studies documenting the prevalence and severity of dental disease (decayed, missing, and filled teeth scores) will be included. Various databases will be searched for eligible studies. Only studies conducted on South African children aged 6 years and under will be included. There will be no restriction on the time or language of publication. The quality of all eligible studies will be analyzed by a risk of bias tool developed by the Joanna Briggs Institute. The results will be presented narratively, and if possible, a meta-analysis will be conducted. Results The protocol is registered with PROSPERO. The literature search was initially conducted in November 2018 and was repeated in November 2020. Conclusions The results of this study will be used to advise stakeholders of the prevalence and severity of dental disease in children under 6 years of age in South Africa. Trial Registration PROSPERO CRD42018112161; International Registered Report Identifier (IRRID) DERR1-10.2196/25795

BACKGROUND Young children are at the highest risk of developing dental caries as they at high risk for dental caries as they have a lack of autonomy over their diet and oral hygiene practices. Dental caries develops over time due to demineralization of tooth substance (enamel) resulting from acid production of bacterial metabolism from sugars. Early-onset of dental caries often results in asymptomatic presentation, but if left untreated, it can result in severe pain, infection and dentoalveolar abscesses. Early childhood caries (ECC) is defined as dental caries in children aged six years and younger and is a significant public health problem in South Africa. According to the Global Burden of Disease (GBD) study, untreated dental caries of primary teeth affects 532 million children. Untreated dental caries has many detrimental effects which can affect the physical development and reduce the quality of life of affected children. Furthermore, long term untreated dental caries can result in school absenteeism, low BMI and poor educational outcomes. OBJECTIVE The purpose of this study is to determine the prevalence and severity of ECC in South Africa in children under 6 years of age. METHODS All cross-sectional documenting the prevalence and severity of dental disease (decayed, missing and filled teeth scores) will be included. Various databases will be searched for eligible studies, and those included will be done by using prespecified inclusion criteria. Only studies conducted on South African children who are aged 6 years and under, will be included. There will be no restriction on the time nor language of publication. The quality of all eligible studies will be analyzed by a risk of bias tool developed by the Joanna Briggs Institute. The results will be presented narratively, and if possible, a meta-analysis will be conducted. RESULTS The protocol is registered with PROSPERO. The literature search was conducted in November 2018 and was repeated in November 2020. CONCLUSIONS The results of this study will be used to advise stakeholders of the prevalence and severity of dental disease in children less than 6 years of age in South Africa.

This note gives the steps to be followed in order to include the crab cavities module in MAD-X for tracking simulations in SixTrack. Examples of checks that aim to verify that the system is implemented as expected are provided; these checks are important to be done before launching large simulations using, for example, SixDesk.

The Crab Cavities (CCs) may induce emittance growth, driven by phase and amplitude jitter, and lead to a degradation of the High Luminosity LHC (HL-LHC) beam. Furthermore, the tight HL-LHC space constraints call for non-axially symmetric cavity designs that come with higher order non-linearities which can affect the long-term dynamic aperture (DA). In order to study the effect the CCs have on the beam and the engineering challenges to validate the pre-series design, a prototype HL-LHC CC set was installed in the SPS machine in the year-end technical stop (YETS) of 2017-2018, and the first ever CC tests with protons were performed in 2018. This note is a collection of the reports and work done on the emittance growth measurements prior and after the CC installation, as well as simulations and measurements performed to study the CC and SPS non linearities effect. Finally, a summary of the instrumentation observations of the SPS CC experimental measurements is presented.

Over the course of 2018, 7 machine developments were conducted for dedicated crab cavity measurements with each measurement slot consisting of 10 hours of SPS machine time. During this period,many different sets of measurements and studies were performed to improve the understanding of the interaction between protons and crab cavities. This note describes the main results relating to beam dynamics. Additionally, the considerations on crab cavity related instrumentation required for HL-LHC are presented.

The main aim of this document is to have a clearly identified set of beam and machine parameters to be used for numerical simulations and performance assessment. Two scenarios (referring only to the operation at the nominal beam energy of 7 TeV) are discussed: i) Nominal scenario (levelling at a luminosity of 5E34 cm-2s-1) ii) Ultimate scenario (levelling at a luminosity of 7.5E34 cm-2s-1)

In the absence of evidence that a players’ guild formally existed, scholars have tended to assume that the adult acting companies simply copied the guild apprenticeship model as a long-established norm for child labor or that acting company members used their formal guild affiliations to apprentice and train boys as stage players. While some Elizabethan and early Stuart play-boys in the adult London drama companies might have been guild apprentices or been like guild apprentices, others were more like covenant servants or parish apprentices, and while some of these “boys” were children with unbroken voices, others were adults.

The hourglass effect arises due to a coupling between the longitudinal and transverse bunch planes. This coupling will result in a charge density distribution that will vary parabolically through the Interaction Point (IP). Here a method of analytically determining the electric field a particle receives from a charge density distribution which varies parabolically when centred at the IP, is derived for a 2D transverse model of a Gaussian bunch.

Simulations of β$^∗$ luminosity levelling using the strongstrong beam-beam code COMBI are undertaken. Simulations for both flat and round beam profiles are discussed and analysed with respect to the coherent spectra. It is shown that bunches with a round beam profile will have a beambeam parameter that is independent of β$^∗$ over the levelling steps. Flat bunches however will have a beam-beam parameter that is dependent on β$^∗$ over the levelling steps since the beam aspect ratio will change. This will change the tune of the π-mode as the β$^∗$ is levelled, which could lead to a resonance crossing.

An analysis is made of the transmission coefficient for electrons at a sharp interface separating two periodic one-dimensional potentials. Numerical results are presented for the case that the cell potentials are square wells. The transmission coefficient is sensitive to the position of the interface relative to the cells in the two crystals. Results are also presented for two interfaces (a barrier).

The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunneling resistance of the form ρ0 ne bn is assumed for a cluster ofn adjacent insulating sites. In the thermodynamic limit, the mean resistance per site diverges at the critical filling fractionp c =e−b, while the mean square resistivity fluctuations diverge at the lower filling fraction $$p_{c_2 } $$ =p c 2 . Computer simulations of large but finite systems, however, show only a very weak divergence of resistivity atp c and no divergence of the fluctuations at $$p_{c_2 } $$ . For finite lattices, calculation of the resistivity at the critical filling is shown to be simply related to the Petersburg problem. Analytic expressions for the resistivity and resistivity fluctuations are obtained in agreement with the results of computer simulations.

Parts of the linear and quadratic coupling of the electron–phonon system are included in a discussion of optical absorption and emission by an electron that is bound to an impurity in a semiconductor and is in interaction with the lattice vibrations. An adiabatic approach is not used. Comparison is made between an approach based on the Bethe–Sommerfeld expression in the representation in which the Hamiltonian is diagonal and one based on a correlation function approach.

We report accurate measurements of the broadening of the phonon replica luminescence lines of excitons bound to P, B, Ga, As, and Bi impurities in silicon and formulate a theory of the lineshapes. The broadenings were obtained experimentally from a comparison of the phonon replica lineshapes with the no-phonon lineshape. We explain these broadenings by a new mechanism in which the spread in recombination wave vectors of the localized electron and hole leads to a spread in energy of the photons emitted since the energy of the momentum-conserving phonon is wave vector dependent. We give an expression for the donor luminescence lineshape in terms of the impurity state wave functions and the phonon dispersion. It is not necessary to invoke phonon lifetime broadening to explain our data. We find that the broadening of the TA and TO bound exciton phonon replica lines is considerably smaller than the phonon lifetime broadenings previously invoked to explain the free exciton lineshapes in silicon. We feel that the 'broadening' which has been observed for the free exciton lineshapes can be accounted for by the splitting of the free exciton ground state and the associated nonparabolicity of the free exciton bands.

A mathematical technique previously used for studying absorption and emission from impurity centres in semiconductors is applied to Raman scattering from such centres. The usual adiabatic approximation is not made. The method makes use of the fact that at most one electron can be bound at the impurity site and reduces the problem in a systematic fashion to the evaluation of traces over only phonon states. These traces are evaluated for a particular type of electron–phonon coupling.

The level broadenings for the s-excited states of the P-donor in Si have been obtained from high resolution studies of two-electron luminescence transitions. Combining these studies with standard infrared absorption measurements, we show that the limiting line-broadening mechanism affects the excited s-states more strongly than any of the other states, including the ground state. The experimental results are understood in terms of a model which provides information about the central cell potential and the amplitude of the even-parity donor wavefunctions.

It is pointed out that group-theoretical ideas similar to those used to study bound states of a neutral donor in a many-valley semiconductor can be extended to the problem of screening of a donor ion by delocalized electrons. A transformation from the set of conduction-band Bloch states in Si to a new set of states which transform according to the irreducible representations of the tetrahedral group ${T}_{d}$ is used. This new set of states approximately block diagonalizes tetrahedrally symmetric potentials and simplifies the problem of including both intervalley and umklapp parts. We use the new representation to find the long-wavelength correction due to intervalley and umklapp scattering to the induced potential near a donor ion in an electron gas in Si the RPA. The strength of this correction is estimated by comparison with the experimentally known properties of the P-neutral-donor ground state, using a model potential to represent the effects of intervalley and umklapp scattering on that state.

Special cases of the general result for Raman scattering from an impurity in a semiconductor are discussed. For weak electron–phonon coupling the zero-phonon and one-phonon scattering intensities are derived. For strong electron–phonon coupling a comparison is made between two different approximations that have been previously used.

A model is presented to explain the breadths of the excited impurity levels of phosphorus-doped silicon at zero temperature. The model assumes that the widths of optical transitions involving absorption by the neutral donor and so-called two-electron luminescence from an exciton bound to the neutral donor are the sums of level widths of the initial and final states. The excited s states' level widths are accounted for by assuming that they are governed by the lifetime of the levels for two-phonon decay to the ground state. This decay is brought about by the interaction of the bound electron with the phosphorus ion.

The expression is derived for p-phonon processes accompanying an optically-induced electronic change of state of an electron bound to an impurity site in a semiconductor or insulator. It is shown that a previously-derived line-shape function gives the correct expressions for these processes. The bound electron interacts with the phonons via terms linear and quadratic in the phonon variables.

We point out the existence of experimental results for lightly doped n-type GaAs and n-type InP in which the conductivity \ensuremath{\sigma} is with s close to (1/4, indicating variable-range hopping. We show that the experimental range of temperatures T, 1\ensuremath{\lesssim}T\ensuremath{\lesssim}7 K, in which this behavior holds, is well in excess of the temperature range predicted for these materials by established theories of variable-range hopping, and that the experimental ${T}_{0}$ values are too small to be understood on the basis of these theories. As a first step toward understanding these experimental results, we consider a two-dimensional lightly doped semiconductor. We choose a flat density of states with width \ensuremath{\Delta}\ensuremath{\varepsilon}. We model the semiconductor as a Miller-Abrahams-type resistor network; we use the full form of the resistance and do not take the low-temperature asymptotic form because we are interested in temperatures for which kT is of order \ensuremath{\Delta}\ensuremath{\varepsilon}.We use percolation theory to obtain a set of equations which specify the overall resistance R(t) of the network as a function of t=kT/\ensuremath{\Delta}\ensuremath{\varepsilon}. This theoretical result for R(t) depends on the form chosen for the temperature dependence of the critical number ${B}_{c}$(t) of bonds (resistors) per site required for percolation. We find that, for a flat density of states and two dimensions, there is no extensive temperature range in which with ${T}_{0}$ substantially smaller than the theoretical ${T}_{0}$ for variable range hopping, and with s close to the two-dimensional value of (1/3. Instead, we find a wide temperature range for which \ensuremath{\sigma}=${\ensuremath{\sigma}}_{0}$${e}^{\mathrm{\ensuremath{-}}{\ensuremath{\varepsilon}}_{3}/\mathrm{kT}}$ with ${\ensuremath{\varepsilon}}_{3}$=0.28 \ensuremath{\Delta}\ensuremath{\varepsilon}. This value of ${\ensuremath{\varepsilon}}_{3}$ is considerably smaller than both the value found by Hayden and Butcher and the result reported by Skal, Shklovskii, and Efros. We believe that the difference between our result and those of these other workers may be at- tributed to their use of the low-temperature form of the resistance (kT\ensuremath{\ll}\ensuremath{\Delta}\ensuremath{\varepsilon}) in a temperature range in which kT is of order \ensuremath{\Delta}\ensuremath{\varepsilon}. Our theory also provides us with the temperature dependences of the typical hopping distance and the typical energy associated with a hop in the temperature range where kT is of order \ensuremath{\Delta}\ensuremath{\varepsilon}.

Heat treatment of silicon doped with phosphorus is found to produce marked changes in the photoluminescence and the electron paramagnetic resonance for samples containing impurity concentrations above the critical concentration [Formula: see text] for the semiconductor–metal transition. A possible explanation is suggested in terms of a heat-induced defect which is populated only when the phosphorus-doped electrons are delocalized. Clusters of Si-interstitials are considered. At concentrations greater than about 2.0 × 1019 P/cm3 the heat-induced changes are exceedingly small.

We present the results of a numerical simulation of a two-dimensional lightly doped compensated semiconductor. We choose a flat density of states with width Δε. We model the semiconductor as a Miller and Abrahams type resistor network; we use the full form of the resistance and do not take the low-temperature asymptotic form because we carry out the simulation at temperatures for which kT is of order Δε. We find that there is a wide temperature range for which [Formula: see text] with ε3 = 0.28Δε. This value of ε3 is considerably smaller than values found by others. We believe that the difference between our result and those of other workers may be attributed to their use of the low-temperature form of the resistance [Formula: see text] in a temperature range in which kT is of order Δε.

The resistivity of a one-dimensional lattice consisting of randomly distributed conducting and insulating sites is considered. Tunnelling resistance of the form R0iebi is assumed for a cluster of i adjacent insulating sites. Three different ensembles are considered and compared. In the first ensemble, the number of insulating "atoms" is fixed and distributed in a linear chain; in the second one, there exists a fixed probability p of having an insulator "atom" occupying a site in a linear chain; and finally in the third one, a line is bent into a circle and the probability p is considered. It is observed that in the thermodynamic limit, the average ensemble resistance per site diverges at the critical filling fraction pc = e−b, while the variance of the resistance diverges at the lower filling fraction [Formula: see text]. Computer simulations of large but finite systems, however, exhibit a much weaker divergence of the resistance per site at pc and no divergence of the variance at pc1.

The derivation of the Stokes' shift has been performed without making the adiabatic approximation. A canonical transformation is used to show the relationship between that method and the more usual one in which the adiabatic approximation is made.

The theory of the phonon broadening of impurity spectral lines is discussed within the framework of a Kubo-type formulation of the adiabatic dielectric susceptibility and the subsequent calculation of this using double-time Green's functions. The interaction of the bound electron (or hole) with the lattice vibrations is assumed to be weak and is treated as a small perturbation of the independent systems of electron and vibrating lattice; the use of the adiabatic approximation is thereby avoided. The cases in which the bound carrier states are degenerate and nondegenerate are discussed separately. The results are compared with previous work in the field.

The drift velocity of electrons in n-type germanium has been measured as a function of applied electric field at lattice temperatures of 77° K and 295° K. Three directions of applied field were used, viz. (100), (110), and (111) crystal directions. The range of field strength was from 500 v/cm to 75 kv/cm. A longitudinal anisotropy was observed at 77° K but not at 295° K. All specimens showed saturation of the drift velocity at high fields. At 77° K, all (100) specimens exhibited a breakdown effect, the cause of which is not known. The results are analyzed on the basis of an extension of Stratton's theory to the case of a many-valley semiconductor.

The general theory of the phonon broadening of impurity spectral lines discussed in an earlier paper is applied to shallow impurity levels in silicon. With the use of a modified hydrogenic model and a deformation potential description of the electron–phonon interaction, expressions are obtained for typical contributions to the half-widths. Some numerical estimations are made for both acceptor and donor cases and are compared with experiment.

The theory of the phonon broadening of optical absorption lines associated with impurities in semiconductors is discussed, using the diagram technique for the evaluation of an appropriate two-particle temperature Green's function. The single-particle propagator for an electron moving in the impurity potential and simultaneously in interaction with the phonons is calculated and used. The interaction with the phonons is treated as weak. The line-shape function is found, and a comparison is made between this method of calculation and a previous one based on the decoupling of an infinite hierarchy of equations for double-time temperature-dependent Green's functions.

For a monovalent donor impurity in a semiconductor the number of electrons that can be bound to an impurity site is either zero or one. The one bound electron can have either direction of spin. This means that the usual Fermi–Dirac distribution function is not applied to such impurity states. A new derivation of the distribution function is presented in terms of annihilation and creation operators for the case of no interaction with the phonons. With the use of double-time temperature-dependent Green's functions, both the electron and the phonon distribution functions are derived when there is interaction between the bound electron and phonons.

A new method of deriving the Stokes' shift in impurity spectral lines is presented. The adiabatic approximation is not used. The method makes use of the fact that at most one electron can be bound at the impurity site and reduces the problem in a systematic fashion to the evaluation of traces over only phonon states. A new method is given of evaluating these traces for a particular electron–phonon system which has the chief features required to discuss the Stokes' shift.

A calculation is made of the temperature dependence of the energy levels of shallow donor impurities in silicon. This temperature dependence arises from the electron–phonon interaction and we consider mixing only of the {1s}, {2s), and {2p0} electronic states. A comparison is made with experiment for the case of phosphorus-doped silicon.

It is shown that the spin [Formula: see text] Ising model can be formulated as a spinless fermion many-body problem and that the Green's function technique can be applied to it. The hierarchy of Green's function equations of motion terminates at the (q + 1)-particle Green's function, where q is the coordination number. This finite number of equations yields Fisher's transformation of correlations. The technique discussed in this paper can be used to obtain exact results for the one-dimensional Ising model.