A two-phase Hessian approach improves the DFT relaxation of slabs.

A two-phase Hessian approach to DFT slab relaxation of slabs has been implemented and tested. It addresses weaknesses in the modified Broyden and Pfrommer BFGS algorithms specific to relaxing slabs. Complete Hessian and then inverse Hessian matrices with no strain/stress components are first constructed at high force signal-to-noise ratios with no accompanying relaxation. In a second phase the static inverse Hessian is used to relax the slab down to a low force tolerance.

A comparative assessment of waste incinerators in the UK.

The uptake in Europe of Energy from Waste (EfW) incinerator plants has increased rapidly in recent years. In the UK, 25 municipal waste incinerators with energy recovery are now in operation; however, their waste supply chains and business practices vary significantly. With over a hundred more plant developments being considered it is important to establish best business practices for ensuring efficient environmental and operational performance. By reviewing the 25 plants we identify four suitable case study plants to compare technologies (moving grate, fluidised bed and rotary kiln), plant economics and operations. Using data collected from annual reports and through interviews and site visits we provide recommendations for improving the supply chain for waste incinerators and highlight the current issues and challenges faced by the industry. We find that plants using moving grate have a high availability of 87-92%. However, compared to the fluidised bed and rotary kiln, quantities of bottom ash and emissions of hydrogen chloride and carbon monoxide are high. The uptake of integrated recycling practices, combined heat and power, and post incineration non-ferrous metal collections needs to be increased among EfW incinerators in the UK. We conclude that one of the major difficulties encountered by waste facilities is the appropriate selection of technology, capacity, site, waste suppliers and heat consumers. This study will be of particular value to EfW plant developers, government authorities and researchers working within the sector of waste management.

Complementary methods to investigate the development of clogging within a horizontal sub-surface flow tertiary treatment wetland.

A combination of experimental methods was applied at a clogged, horizontal subsurface flow (HSSF) municipal wastewater tertiary treatment wetland (TW) in the UK, to quantify the extent of surface and subsurface clogging which had resulted in undesirable surface flow. The three dimensional hydraulic conductivity profile was determined, using a purpose made device which recreates the constant head permeameter test in-situ. The hydrodynamic pathways were investigated by performing dye tracing tests with Rhodamine WT and a novel multi-channel, data-logging, flow through Fluorimeter which allows synchronous measurements to be taken from a matrix of sampling points. Hydraulic conductivity varied in all planes, with the lowest measurement of 0.1md(-1) corresponding to the surface layer at the inlet, and the maximum measurement of 1550md(-1) located at a 0.4m depth at the outlet. According to dye tracing results, the region where the overland flow ceased received five times the average flow, which then vertically short-circuited below the rhizosphere. The tracer break-through curve obtained from the outlet showed that this preferential flow-path accounted for approximately 80% of the flow overall and arrived 8h before a distinctly separate secondary flow-path. The overall volumetric efficiency of the clogged system was 71% and the hydrology was simulated using a dual-path, dead-zone storage model. It is concluded that uneven inlet distribution, continuous surface loading and high rhizosphere resistance is responsible for the clog formation observed in this system. The average inlet hydraulic conductivity was 2md(-1), suggesting that current European design guidelines, which predict that the system will reach an equilibrium hydraulic conductivity of 86md(-1), do not adequately describe the hydrology of mature systems.

Sustainable construction: composite use of tyres and ash in concrete.

An investigation was carried out to establish the physical, mechanical and chemical characteristics of a non-standard (unprocessed) pulverised fuel ash (PFA) and waste tyres from a former landfill site at the Power Station Hill near Church Village, South Wales, United Kingdom. Investigations are on-going to establish the suitability of the fly ash and/or tyres in road construction (embankment and pavement) and also in concrete to be used in the construction of the proposed highway. This paper reports on concrete-based construction where concrete blends (using various levels of PFA as partial replacement for Portland cement (PC), and shredded waste tyres (chips 15-20mm) as aggregate replacement) were subjected to unconfined compressive strength tests to establish performance, hence, optimising mix designs. Strength development up to 180 days for the concrete made with PC-PFA blends as binders (PC-PFA concrete), with and without aggregate replacement with tyre chips, is reported. The binary PC-PFA concrete does not have good early strength but tends to improve at longer curing periods. The low early strength observed means that PC-PFA concrete cannot be used for structures, hence, only as low to medium strength applications such as blinding, low-strength foundations, crash barriers, noise reduction barriers, cycle paths, footpaths and material for pipe bedding.

Introduction of the 5-HT3B subunit alters the functional properties of 5-HT3 receptors native to neuroblastoma cells.

The identification of a second 5-HT(3) (5-HT(3B)) subunit provides an explanation for 5-HT(3) receptor heterogeneity. We investigated whether introduction of recombinant 5-HT(3B) subunits would alter the functional properties of mouse neuroblastoma 5-HT(3) receptors. RT-PCR analysis revealed that NB41A3 cells contain mRNAs encoding 5-HT(3A) and 5-HT(3B) subunits. 5-HT increased intracellular Ca(2+) concentration ([Ca(2+)](i)) and caused the concentration-dependent activation of inward currents recorded at -60 mV. Both actions of 5-HT were antagonized by ondansetron. The 5-HT concentration-response relationship of NB41A3 cells was indistinguishable from that of the related NG108-15 cell line. The selective 5-HT(3)-receptor agonist mCPBG also elevated [Ca(2+)](i) and activated inward currents. 2-M-5HT was less efficacious than 5-HT as an activator of 5-HT(3) receptors in NB41A3 cells and did not significantly increase [Ca(2+)](i). The 5-HT induced increase in [Ca(2+)](i) did not involve caffeine- or thapsigargin-sensitive intracellular Ca(2+) stores. The introduction of the 5-HT(3B) subunit by transient transfection of NB41A3 cells caused 5-HT to become less potent as an activator of 5-HT(3) receptors and altered the kinetics of 5-HT activated currents so that they resembled currents mediated by 5-HT(3AB) receptors. The 5-HT(3B) subunit also abolished the 5-HT induced [Ca(2+)](i) increase seen in untransfected NB41A3 cells. These data are consistent with the hypothesis that NB41A3 cells predominantly express homomeric 5-HT(3A) receptors that become heteromeric 5-HT(3AB) receptors upon introduction of the recombinant 5-HT(3B) subunit.

Alternative transcripts of the GABA(A) receptor epsilon subunit in human and rat.

In both human and rat tissues, complex patterns of transcripts are derived from the genes that encode the gamma-aminobutyric acid (GABA)(A) receptor epsilon subunit. An epsilon subunit transcript (approximately 3.6 kb) is expressed at relatively high levels in regions of the human brain and heart, but is not detected in most other major tissues. The encoded human epsilon subunit (epsilon (h)) confers distinctive properties to receptors into which it assembles. A distinct transcript of the gene (6.2 kb) is expressed abundantly in a variety of human tissues. This alternative transcript (ET2) appears to originate from within the epsilon subunit gene. It is possible that this transcript encodes a truncated subunit (epsilon (hS)), containing all of the transmembrane and intracellular domains. However, a combination of biochemical and electrophysiological analyses does not support this hypothesis. A distinct transcript of the epsilon subunit gene, encoding a large extracellular pro/glx domain, is expressed abundantly in rat and mouse brain. Functional analyses also failed to provide evidence for incorporation of this subunit (epsilon (rL)) into recombinant receptors. However, a shorter rat epsilon subunit (epsilon (r)), which lacks the pro/glx domain, conferred epsilon (h)-like properties to recombinant receptors, providing evidence for a functional rat epsilon subunit. In common with its human orthologue, incorporation of the epsilon (r) subunit into recombinant GABA(A) receptors confers several distinctive properties, including a reduced modulation by the anesthetic propofol and the appearance of spontaneous current.

Application of electron backscatter diffraction (EBSD) to fracture studies of ferritic steels.

The application of electron backscatter diffraction (EBSD) to fracture studies has provided a new method for investigating the crystallography of fracture surfaces. The crystallographic indices of cleavage planes can be measured both directly from the fracture surface and indirectly from metallographic sections perpendicular to the plane of the adjoining fracture surfaces. The results of direct individual cleavage facet plane orientation measurements are presented for carbon-manganese (C-Mn) and low-alloy Mn-Mo-Ni (similar to ASTM A553 type-B). Pressure vessel steel weld metals, obtained from fracture surfaces of Charpy impact test specimens fractured at various test temperatures and for an ultra-low carbon steel (Fe-0.002C- 0.058P) fractured at -196 degrees C by impact. In addition to the direct measurement from the fracture surface, cleavage facet orientation measurements for the ultra-low carbon steel were complemented by the results obtained from the metallographic sections. Fractographic observations revealed that cleavage fracture is accommodated by a microvoid coalescence fracture micromechanism, which was induced by decohesion of second phase particles (inclusions). The correlation between the direct and indirect methodologies shows that the cleavage facet planes are dominated by the [001] plane orientations, and indicated that even when information concerning the full five degrees of freedom is inaccessible, the cleavage facet plane could still be determined. Finally, the advantages and disadvantages of direct orientation measurements from the fracture surface and indirectly by a destructive sectioning technique are discussed.

Evidence for the formation of functionally distinct alphabetagammaepsilon GABA(A) receptors.

1. We transiently introduced the human GABA(A) receptor epsilon subunit cDNA into a human embryonic kidney (HEK) cell line stably expressing alpha1beta3gamma2 receptors (WSS-1 cells) to establish whether the subunit competes with the gamma2 subunit for assembly into receptors. GABA-evoked currents were recorded using the patch-clamp technique from cells transfected with cDNA encoding green fluorescent protein (GFP) alone or in combination with the epsilon subunit cDNA. 2. The epsilon subunit did not change the potency of GABA: the GABA EC(50) was 34 +/- 6 microM in control WSS-1 cells and 37 +/- 6 microM in cells expressing the epsilon subunit. The introduction of the epsilon subunit reduced the peak current amplitude activated by GABA (1 mM) from 1.8 +/- 0.2 nA in control cells to 0.9 +/- 0.2 nA in cells expressing the epsilon subunit (P < 0.05). 3. The epsilon subunit caused the appearance of leak currents recorded in the absence of GABA. Outside-out patches excised from epsilon subunit-containing WSS-1 cells exhibited spontaneously opening GABA(A) channels not seen in patches excised from control GFP-expressing WSS-1 cells. Introduction of the epsilon subunit did not alter the GABA-evoked single-channel cord conductance. 4. The anaesthetic 2,6-diisopropylphenol (propofol, 3 microM) and the benzodiazepine flunitrazepam (1 microM) potentiated GABA-evoked currents recorded from control cells labelled with GFP. The epsilon subunit reduced potentiation by both agents 48-96 h after transfection. 5. The introduction of the epsilon subunit had no effect on the ability of propofol (3-30 microM) relative to GABA (1 mM) to activate GABA(A) receptors in WSS-1 cells. High concentrations of propofol (> or = 100 microM) produced a more marked desensitization of GABA(A) receptor activity in WSS-1 cells transfected with cDNA for the epsilon subunit than in control cells. 6. There was no difference in the potency of Zn(2+) as an inhibitor of currents recorded from control cells (IC(50) = 165 +/- 34 microM) or cells expressing the epsilon subunit (IC(50) = 179 +/- 11 microM). 7. GABA-activated currents recorded both from control cells and cells expressing the epsilon subunit reversed in sign at the Cl- equilibrium potential and exhibited outward rectification. 8. The introduction of the epsilon subunit changes the functional properties of GABA(A) receptors in WSS-1 cells. The resulting receptors have a unique combination of properties indicative of the co-assembly of alpha, beta, gamma and epsilon subunits.

Combined application of electron backscatter diffraction and stereo-photogrammetry in fractography studies.

The main aim of this paper is to report on recent experimental developments that have succeeded in combining electron back-scatter diffraction (EBSD) with stereo-photogrammetry, compared with two other methods for study of fracture surfaces, namely visual fractography analysis in the scanning electron microscope (SEM) and EBSD directly from facets. These approaches will be illustrated with data relating to the cleavage plane orientation analysis in a ferritic and C-Mn steel. It is demonstrated that the combined use of EBSD and stereo-photogrammetry represents a significant advance in the methodology for facet crystallography analysis. The results of point counting from fractograph characterization determined that the proportions of intergranular fracture in C-Mn and ferritic steels were 10.4% and 9.4%, respectively. The crystallographic orientation was determined directly from the fracture surface of a ferritic steel sample and produced an orientation distribution with a clear trend towards the [001] plane. A stereo-photogrammetry technique was validated using the known geometry of a Vickers hardness indent. The technique was then successfully employed to measure the macroscopic orientation of individual cleavage facets in the same reference frame as the EBSD measurements. Correlating the results of these measurements indicated that the actual crystallographic orientation of every cleavage facet identified in the steel specimens is [001].

MRI evaluation of lumbar spine flexion and extension in asymptomatic individuals.

Flexion and extension movements or positions have been advocated in the treatment of various forms of low back dysfunction due to the potential pain relieving effects attributed to displacements of the intervertebral disc (IVD). Objective in vivo determination of the segmental behaviour of the disc to contrasting positions has until recently been difficult. Magnetic resonance imaging (MRI) was used in this study to evaluate the influence of sagittal plane positions on lumbar IVD height and nucleus displacement in a small asymptomatic population.T2-weighted sagittal plane images from L1 to S1 were obtained from 10 subjects (mean age: 30+/-5 years) positioned supine in lumbar flexion, followed by extension. Changes in disc height and localization of nucleus position (determined by peak MRI signal intensity) between the two positions were calculated. Discs were classified for degenerative changes using a semi-quantitative grading scale. The mean range of lumbar sagittal movement achieved in the MRI was 44 degrees (range: 22-77 degrees ). Between flexion and extension, a significant increase in measured anterior disc height of 1.1 mm (P<0.0001) and anterior displacement of the nucleus of 6.7% (P<0.0001) was observed. Despite the anterior displacement of the nucleus in extension observed in the pooled analysis, 30% of discs did not follow this trend. Nucleus degeneration was observed in at least one disc in nine subjects and in 26% of all discs examined. Lumbar spine position was found to be associated with small measured changes in anterior disc height and nucleus position, however, this response was variable within and between individuals. The theoretical concept of a stereotypical effect of spinal position on the lumbar IVD is challenged by these initial data. Since the health of the disc is often unknown in clinical practice, manual therapy treatment for lumbar spine pain should be based on the symptomatic response to movement and position rather than biomechanical theory.

Evidence for expression of heteromeric serotonin 5-HT(3) receptors in rodents.

The gene and cDNAs that encode a novel subunit of rodent serotonin 5-HT(3) receptors were isolated from mouse and rat tissues. Each of the new rodent subunits shares 40% amino acid identity with the rat 5-HT(3A) subunit and 73% identity with the human 5-HT(3B) subunit. Despite a relatively low level of structural conservation, sequence analysis and functional studies suggest that the new rodent subunits are orthologues of the human 5-HT(3B) subunit. In common with homologous human receptors, rat heteromeric 5-HT(3) receptors displayed a substantially larger single-channel conductance than homomeric 5-HT(3A) receptors. In addition, the rat heteromeric receptors were less sensitive to antagonism by tubocurarine. However, in contrast to human heteromeric receptors, those of the rat displayed pronounced inward rectification of both the whole-cell and single-channel current amplitudes. Transcripts of the mouse 5-HT(3A) and 5-HT(3B) subunits are coexpressed in several cell lines that possess endogenous 5-HT(3) receptors. In addition, treatment of rat PC12 cells with nerve growth factor induced expression of both subunit mRNAs, with a similar time course for accumulation of each transcript. The combination of functional data and expression patterns is consistent with the existence of heteromeric 5-HT(3) receptors in rodent neurons.

The influence of an endogenous beta3 subunit on recombinant GABA(A) receptor assembly and pharmacology in WSS-1 cells and transiently transfected HEK293 cells.

Cell lines are commonly used for studying recombinant heterooligomeric ion channels with defined subunit composition. Such studies often ignore the contribution of endogenous proteins in the assembly of mature channels. We examined whether an endogenous subunit was required for the functional expression of gamma-aminobutyric acid type A (GABA(A)) receptors in WSS-1 cells, HEK293 cells stably expressing recombinant alpha1 and gamma2 subunits. Our pharmacological and RT-PCR analyses of GABA(A) receptors and their mRNAs in WSS-1 cells confirm the presence of alpha1 and gamma2 subunits and suggest the existence of an endogenous beta3 subunit. Whole-cell GABA-evoked currents recorded from untransfected WSS-1 cells were blocked by bicuculline methiodide and enhanced by anesthetics and anticonvulsants including the subunit-selective compounds diazepam and loreclezole. These data suggest that, in addition to the gamma2 subunit, WSS-1 cell receptors also contain beta2/3 subunits. RT-PCR revealed that WSS-1 cells and parental HEK293 cells contain beta3 mRNA. We examined the contribution of the beta3 subunit in the function of receptors formed by expression of alpha1 and gamma2S subunits. Untransfected HEK293 cells were unresponsive to GABA. Cells transfected with alpha1 and gamma2S cDNAs displayed small diazepam and loreclezole responsive GABA-activated currents. By contrast, the expression of alpha1 and gamma2S cDNAs in the neuroblastoma NB41A3 cell line, that lacks beta subunit mRNAs, failed to produce functional receptors. These data reaffirm that alpha1 and gamma2S subunits alone do not form functional GABA(A) receptors and that receptors of WSS-1 cells contain alpha1, beta3 and gamma2S subunits.

The 5-HT3B subunit is a major determinant of serotonin-receptor function.

The neurotransmitter serotonin (5-hydroxytryptamine or 5-HT) mediates rapid excitatory responses through ligand-gated channels (5-HT3 receptors). Recombinant expression of the only identified receptor subunit (5-HT3A) yields functional 5-HT3 receptors. However, the conductance of these homomeric receptors (sub-picosiemens) is too small to be resolved directly, and contrasts with a robust channel conductance displayed by neuronal 5-HT3 receptors (9-17 pS). Neuronal 5-HT3 receptors also display a permeability to calcium ions and a current-voltage relationship that differ from those of homomeric receptors. Here we describe a new class of 5-HT3-receptor subunit (5-HT3B). Transcripts of this subunit are co-expressed with the 5-HT3A subunit in the amygdala, caudate and hippocampus. Heteromeric assemblies of 5-HT3A and 5-HT3B subunits display a large single-channel conductance (16 pS), low permeability to calcium ions, and a current-voltage relationship which resembles that of characterized neuronal 5-HT3 channels. The heteromeric receptors also display distinctive pharmacological properties. Surprisingly, the M2 region of the 5-HT3B subunit lacks any of the structural features that are known to promote the conductance of related receptors. In addition to providing a new target for therapeutic agents, the 5-HT3B subunit will be a valuable resource for defining the molecular mechanisms of ion-channel function.

A comparison of barley isolated microspore and anther culture and the influence of cell culture density.

Comparisons were made between the efficiency of barley plant regeneration from anther culture (AC) and isolated microspore culture (IMC) for the European winter cultivar `Igri' and the spring F1 Australian breeder's hybrid Amagi Nijo×WI2585. In both cases, IMC produced a higher number of green regenerant plantlets per anther than AC. For `Igri' there was a 100- to 200-fold improvement and for Amagi Nijo×WI2585 there was a five- to ninefold improvement of IMC over AC. To improve the consistency and reliability of the IMC method, we investigated several parameters, including maltose concentration, subculture protocol, microspore plating density and colony plating density. Subculturing during the liquid culture phase produced no significant improvement in the number of microspores developing into colonies. The optimal concentration of maltose in the liquid induction medium was found to be 90 g l-1. Both microspore plating density and colony plating density were found to influence plant regeneration. Microspores produced the highest numbers of colonies when plated at densities greater than 5×104 ml-1, and colonies produced optimal numbers of green plantlets when plated at 12.5-25 colonies/cm2.

Modulation by general anaesthetics of rat GABAA receptors comprised of alpha 1 beta 3 and beta 3 subunits expressed in human embryonic kidney 293 cells.

1. Radioligand binding and patch-clamp techniques were used to study the actions of gamma-aminobutyric acid (GABA) and the general anaesthetics propofol (2,6-diisopropylphenol), pentobarbitone and 5 alpha-pregnan-3 alpha-ol-20-one on rat alpha 1 and beta 3 GABAA receptor subunits, expressed either alone or in combination. 2. Membranes from HEK293 cells after transfection with alpha 1 cDNA did not bind significant levels of [35S]-tert-butyl bicyclophosphorothionate ([35S]-TBPS) (< 0.03 pmol mg-1 protein). GABA (100 microM) applied to whole-cells transfected with alpha 1 cDNA and clamped at -60 mV, also failed to activate discernible currents. 3. The membranes of cells expressing beta 3 cDNAs bound [35S]-TBPS (approximately 1 pmol mg-1 protein). However, the binding was not influenced by GABA (10 nM-100 microM). Neither GABA (100 microM) nor picrotoxin (10 microM) affected currents recorded from cells expressing beta 3 cDNA, suggesting that beta 3 subunits do not form functional GABAA receptors or spontaneously active ion channels. 4. GABA (10 nM-100 microM) modulated [35S]-TBPS binding to the membranes of cells transfected with both alpha 1 and beta 3 cDNAs. GABA (0.1 microM-1 mM) also dose-dependently activated inward currents with an EC50 of 9 microM recorded from cells transfected with alpha 1 and beta 3 cDNAs, clamped at -60 mV. 5. Propofol (10 nM-100 microM), pentobarbitone (10 nM-100 microM) and 5 alpha-pregnan-3 alpha-ol-20-one (1 nM-30 microM) modulated [35S]-TBPS binding to the membranes of cells expressing either alpha 1 beta 3 or beta 3 receptors. Propofol (100 microM), pentobarbitone (1 mM) and 5 alpha-pregnan-3 alpha-ol-20-one (10 microM) also activated currents recorded from cells expressing alpha 1 beta 3 receptors. 6. Propofol (1 microM-1 mM) and pentobarbitone (1 mM) both activated currents recorded from cells expressing beta 3 homomers. In contrast, application of 5 alpha-pregnan-3 alpha-ol-20-one (10 microM) failed to activate detectable currents. 7. Propofol (100 microM)-activated currents recorded from cells expressing either alpha 1 beta 3 or beta 3 receptors reversed at the Cl- equilibrium potential and were inhibited to 34 +/- 13% and 39 +/- 10% of control, respectively, by picrotoxin (10 microM). 5 alpha-Pregnan-3 alpha-ol-20-one (100 nM) enhanced propofol (100 microM)-evoked currents mediated by alpha 1 beta 3 receptors to 1101 +/- 299% of control. In contrast, even at high concentration 5 alpha-pregnan-3 alpha-ol-20-one (10 microM) caused only a modest facilitation (to 128 +/- 12% of control) of propofol (100 microM)-evoked currents mediated by beta 3 homomers. 8. Propofol (3-100 microM) activated alpha 1 beta 3 and beta 3 receptors in a concentration-dependent manner. For both receptor combinations, higher concentrations of propofol (300 microM and 1 mM) caused a decline in current amplitude. This inhibition of receptor function reversed rapidly during washout resulting in a "surge' current on cessation of propofol (300 microM and 1 mM) application. Surge currents were also evident following pentobarbitone (1 mM) application to cells expressing either receptor combination. By contrast, this phenomenon was not apparent following applications of 5 alpha-pregnan-3 alpha-ol-20-one (10 microM) to cells expressing alpha 1 beta 3 receptors. 9. These observations demonstrate that rat beta 3 subunits form homomeric receptors that are not spontaneously active, are insensitive to GABA and can be activated by some general anaesthetics. Taken together, these data also suggest similar sites on GABAA receptors for propofol and barbiturates, and a separate site for the anaesthetic steroids.

Insensitivity to anaesthetic agents conferred by a class of GABA(A) receptor subunit.

A common feature of general anaesthetic agents is their ability to potentiate neuronal inhibition through GABA(A) (gamma-aminobutyric acid) receptors. At concentrations relevant to clinical anaesthesia, these agents cause a dramatic stimulation of the chloride currents that are evoked by the binding of the natural ligand, GABA. Although there is widespread evidence that the sensitivity of GABA(A) receptors to anaesthetic agents is heterogeneous, the structural basis of these differences is largely unknown. Variations in subunit composition can have profound effects on the sensitivity of GABA(A) receptors to modulatory agents such as benzodiazepines. However, strict subunit specificity has not been demonstrated for the potentiating effects of anaesthetic agents. Here we describe a new class of human GABA(A) receptor subunit (epsilon) that can assemble with alpha- and beta-subunits and confer an insensitivity to the potentiating effects of intravenous anaesthetic agents. The epsilon-subunit also abolishes the normal outward rectification of recombinant receptors in which it assembles. The expression pattern of this subunit in the brain suggests a new target for manipulation of neuronal pathways within the basal ganglia.

Light-trapping lenses for solar cells.

A new type of lens is described that, when used as a secondary concentrator together with a primary Fresnel lens to illuminate a silicon solar cell, would tend to return escaping light to the cell and therefore enhance the light trapping caused primarily by internal reflection within the silicon. In the ideal case of a perfect mirror at the back surface of the cell, it is calculated that, with a lens with a refractive index of 1.5, the cell could be reduced in thickness by a factor of 3 and still absorb as much light. The thinner cell would permit higher concentration and efficiency. Uniformity of illumination would also be improved by the lens. There are no metallic reflectors used; instead the lens traps light by total internal reflection. Its geometry and properties are presented in terms of the refractive index of the lens material.