• The infared optical properties of Sr2RuO4 and SmTiO3 including an object-oriented resistivity interface /

      Hildebrand, Mylo George.; Department of Physics (Brock University, 1999-05-21)
      The perovskite crystal structure is host to many different materials from insulating to superconducting providing a diverse range of intrinsic character and complexity. A better fundamental description of these materials in terms of their electronic, optical and magnetic properties undoubtedly precedes an effective realization of their application potential. SmTiOa, a distorted perovskite has a strongly localized electronic structure and undergoes an antiferromagnetic transition at 50 K in its nominally stoichiometric form. Sr2Ru04 is a layered perovskite superconductor (ie. Tc % 1 K) bearing the same structure as the high-tem|>erature superconductor La2_xSrrCu04. Polarized reflectance measurements were carried out on both of these materials revealing several interesting features in the far-infrared range of the spectrum. In the case of SmTiOa, although insulating, evidence indicates the presence of a finite background optical conductivity. As the temperature is lowered through the ordering temperature a resonance feature appears to narrow and strengthen near 120 cm~^ A nearby phonon mode appears to also couple to this magnetic transition as revealed by a growing asymmetry in the optica] conductivity. Experiments on a doped sample with a greater itinerant character and lower Neel temperature = 40 K also indicate the presence of this strongly temperature dependent mode even at twice the ordering temperature. Although the mode appears to be sensitive to the magnetic transition it is unclear whether a magnon assignment is appropriate. At very least, evidence suggests an interesting interaction between magnetic and electronic excitations. Although Sr2Ru04 is highly anisotropic it is metallic in three-dimensions at low temperatures and reveals its coherent transport in an inter-plane Drude-like component to the highest temperatures measured (ie. 90 K). An extended Drude analysis is used to probe the frequency dependent scattering character revealing a peak in both the mass enhancement and scattering rate near 80 cm~* and 100 cm~* respectively. All of these experimental observations appear relatively consistent with a Fermi-liquid picture of charge transport. To supplement the optical measurements a resistivity station was set up with an event driven object oriented user interface. The program controls a Keithley Current Source, HP Nano-Voltmeter and Switching Unit as well as a LakeShore Temperature Controller in order to obtain a plot of the Resistivity as a function of temperature. The system allows for resistivity measurements ranging from 4 K to 290 K using an external probe or between 0.4 K to 295 K using a Helium - 3 Cryostat. Several materials of known resistivity have confirmed the system to be robust and capable of measuring metallic samples distinguishing features of several fiQ-cm.
    • Infrared Spectroscopy of (Nb+In) Co-Doped Rutile

      Yee, Susan; Department of Physics
      This work studied rutile TiO2 in pure form and co-doped with In (e􀀀 acceptor) and Nb (e􀀀 donor) at 5% and 10% to explore the effect of co-doping on the infrared active (IR) modes and the complex dielectric response function between 50 and 8000 cm􀀀1 (1.5 - 240 THz, 0.00620 - 0.993 eV). Ceramic pellets of pure, 5% and 10% co-doped TiO2 were prepared using a standard technique. Infrared reflectance (IR) measurements were taken and these data are supplemented with data from the literature to extend the range of frequencies beyond infrared. The dielectric function was determined two ways: (i) by fits of the reflectance to the factorized model of the dielectric function and (ii) by Kramers- Kronig analysis. Co-doping rutile appears to decrease the permittivity at frequencies just below the mode that softens. It is possible that this is due to an increase in porosity resulting from codoping. It appears that the increase in permittivity recently observed elsewhere [1] is not caused by doping induced changes to the phonon modes. The overall effect of co-doping is to make the sample less reflective. The spectrum is composed of three wide, high-reflectance bands. For all levels of co-doping the first band is a mode that softens. The amount of doping does not affect the frequency of the mode that softens. The second and third bands are hard modes. Co-doping appears to introduce four new, impurity, phonon modes that increase in oscillator strength with increasing co-doping level. These modes are centered near w 136, 447, 654 and 793 cm􀀀1 which are close to four, previously observed, Raman active modes in rutile. It is possible that the co-doping process causes the Raman modes to develop a dipole moment and become weakly IR active.
    • Infrared Spectroscopy Of Ba_3+y Co_1+xNb_2O_9 Ceramics

      Ibrahim, Samia; Department of Physics
      The dielectric properties of ceramic Ba3+y Co1+xNb2O9 where x= 0,-0.07,0.03 and y = 0, -0.03 were characterized because it might used for a wide range of applications including wireless communication used in mobile communication, ultra high speed local area networks, intelligent transport system and satellite communications. Room temperature optical re ectivity measurements of ceramics sintered at di erent temperatures between 1200 C to 1500 C were made covering the spectral range between 70-8000 cm􀀀1. The Lorentz model was used to t the re ectance data to make extrapolations for Kramers Kronig (K-K) analysis and to estimate the microwave properties. K -K analysis was applied to extract the other optical response functions from the re ectance data ( optical conductivity and dielectric permittivity). Powder X-Ray di raction measurements were done with 2 in the range between 10 to 80 degree for all samples. Most samples exhibit some degree of 1:2 ordering which appears as small superlattice peaks at 17.6 and 12 degree. All samples exhibit a small amount of impurity phases. The main purpose of this work is to study the e ect of the density, 1:2 ( Co:Nb) ordering and concentration of Cobalt on the dielectric properties. It was shown that density has a clear e ect on the dielectric properties. For example 1 (50 cm􀀀1) decreased if the densjhfnadfity decreased. On the other hand the change of the concentration of Cobalt does not have any real e ect on the dielectric properties. 1:2 order also has an e ect on the dielectric properties. It was observed that the scattering rate of the optical phonon was smaller in sample exhibiting more 1:2 order.
    • Infrared Spectroscopy of Gadolinium

      Obied, Laila; Department of Physics (2013-04-10)
      Measurements of the optical reflectivity of the normal incident light along c-axis [0001] have been made on a Gadolinium single crystal, for temperatures between 50 K and room temperature just above the Curie temperature of Gd, which is 293 K. And covering the spectrum range between 100 -11000 cm-I . This work is the first study of Gd in the far infrared range. In fact it fills the gap below 0.2 eV which has never been measured before. Extreme attention was paid to the fact that Gadolinium is a very reactive metal with air. Thus, the sample was mechanically polished and carefully handled during the measurement. However, temperature dependent optical measurements have been made in the same frequency range for a sample of Gd2O3. For comparison, both samples of Gd and Gd2O3 were examined by X-Ray diffraction. XRD analysis showed that the sample was pure gadolinium and the oxide layer either does not exist, or is very thin. Furthermore, this fact was supported by the absence of any of Gd2O3 features in the Gd sample reflectivity. Kramers Kronig analysis was applied to extract the optical functions from the reflectance data. The optical conductivity shows a strong temperature dependence feature in the mid-infrared. This feature disappears completely at room temperature which supports a magnetic origin.
    • Infrared spectroscopy of Mg-doped SrRuO3 thin films /

      Eftekhari, Fatemeh.; Department of Physics (Brock University, 2006-06-01)
      The reflectance of thin films of magnesium doped SrRu03(Mg-SR0) produced by pulsed laser deposition on SrTiOa (100) substrates has been measured at room temperature between 100 and 7500 cm~^. The films were chosen to have wide range of thickness, stoichiometry and electrical properties. As the films were very thin (less than 300 nm), and some were insulating the reflectance data shows structures due to both the film and the substrate. Hence, the data was analyzed using Kramers-Kronig constrained variational fitting (VDF) method to extract the real optical conductivity of the Mg-SRO films. Although the VDF technique is flexible enough to fit all features of the reflectance spectra, it seems that VDF could not eliminate the substrate's contribution from fllm conductivity results. Also the comparison of the two different programs implementing VDF fltting shows that this technique has a uniqueness problem. The optical properties are discussed in light of the measured structural and transport properties of the fllms which vary with preparation conditions and can be correlated with differences in stoichiometry. This investigation was aimed at checking the VDF technique and also getting answer to the question whether Mg^"*" substitutes in to Ru or Sr site. Analysis of our data suggests that Mg^+ goes to Ru site.
    • Interaction between vortices and impurities in a d-wave superconductor /

      Farhoodfar, Avid.; Department of Physics (Brock University, 2005-06-15)
      High temperature superconductors were discovered in 1986, but despite considerable research efforts, both experimental and theoretical, these materials remain poorly understood. Because their electronic structure is both inhomogeneous and highly correlated, a full understanding will require knowledge of quasiparticle properties both in real space and momentum space. In this thesis, we will present a theoretical analysis of the scanning tunneling microscopy (STM) data in BSCCO. We introduce the Bogoliubov-De Gennes Hamiltonian and solve it numerically on a two-dimensional 20 x 20 lattice under a magnetic field perpendicular to the surface. We consider a vortex at the center of our model. We introduce a Zn impurity in our lattice as a microscopic probe of the physical properties of BSCCO. By direct numerical diagonalization of the lattice BogoliubovDe Gennes Hamiltonian for different positions of the impurity, we can calculate the interaction between the vortex and the impurity in a d-wave superconductor.
    • Investigation into the Magnetoelectric Effect and Magnetic Properties of Iron-doped Cobalt Molybdate

      Pula, Mathew; Department of Physics
      The plausibility of revealing linear magnetoelectric coupling is investigated in the cobalt molybdate (Co2Mo3O8). Recently, Wang et al.[Scientific Reports. 2015;Vol. 5:Article 12268] showed that iron molybdate(Fe2Mo3O8) can be induced into a ferrimagnetic state from a nominal antiferromagnetic state via application of a magnetic field. As such, it may be possible that cobalt molybdate exhibits a similar effect intrinsically or with addition of iron dopant. Single crystals of the hexagonal molybdate (Co(1-x)Fe(x))2Mo3O8 (x=0, 0.25, 0.5, 0.75, 1) were synthesized via chemical vapour transport. The magnetic properties were investigated along the polar axis and in the basal plane. Despite doping with iron, no metamagnetic phase transition was present in (Co(1-x)Fe(x))2Mo3O8 (x=0.25, 0.5, 0.75). Low field measurements of the susceptibility reveal the presence of an anisotropic ferromagnetic-like moment, which is suppressed at moderate fields. This is believed to be a product of an exchange-bias-like phenomena, which is not fully understood. The magnetocapacitance was measured along the c-axis for x=(0.25, 0.5). Co1.5Fe0.5Mo3O8 exhibits the conventional magnetodielectric effect, with a proportionality constant of 5.1(0.3) x10^(-14) Oe^(-2) at 40K, while the capacitance of Co1Fe1Mo3O8 shows linear dependency on H, with slope -6.99(0.07) x10^(-9) Oe^(-1) at 49K.
    • Kr-Ar laser Raman spectrometer for low temperature measurements /

      Wardlaw, Graeme M.; Department of Physics (Brock University, 2004-06-29)
      A Czerny Mount double monochromator is used to measure Raman scattered radiation near 90" from a crystalline, Silicon sample. Incident light is provided by a mixed gas Kr-Ar laser, operating at 5145 A. The double monochromator is calibrated to true wavelength by comparison of Kr and Ar emission Une positions (A) to grating position (A) display [1]. The relationship was found to be hnear and can be described by, y = 1.219873a; - 1209.32, (1) where y is true wavelength (A) and xis grating position display (A). The Raman emission spectra are collected via C"*""*" encoded software, which displays a mV signal from a Photodetector and allows stepping control of the gratings via an A/D interface. [2] The software collection parameters, detector temperature and optics are optimised to yield the best quality spectra. The inclusion of a cryostat allows for temperatmre dependent capabihty ranging from 4 K to w 350 K. Silicon Stokes temperatm-e dependent Raman spectra, generally show agreement with Uterature results [3] in their frequency haxdening, FWHM reduction and intensity increase as temperature is reduced. Tests reveal that a re-alignment of the double monochromator is necessary before spectral resolution can approach literature standard. This has not yet been carried out due to time constraints.
    • La1-xSrxMnO3 as a candidate for a room temperature pressure sensor /

      Teymoori, Roshanak.; Department of Physics (Brock University, 2003-05-21)
      Perovskite manganite compounds, Lai-xDxMnOs (D-divalent alkaline earth Ca, Sr or Ba), whose electrical and magnetic properties were first investigated nearly a half century ago, have attracted a great deal of attention due to their rich phase diagram. From the point of view of designing a future application, the strong pressure dependence of the resistivity and the accompanying effects in thin films have potential for application in pressure sensing and electronic devices. In this study we report our experimental investigations of pressure dependence of the resistivity of Lao.siSvo^iQMnOs and Lai-xSvxMnOs (LSMO) epitaxial films with x= 0.15, 0.20, 0.25, 0.30, 0.35, on SrTiOs substrates.
    • Low frequency Raman scattering in amorphous materials: fused quartz, "pyrex" boro-silicate glass and soda-lime silicate glass

      Elozi, Khaled.; Department of Physics (Brock University, 1991-07-09)
      Raman scattering in the region 20 to 100 cm -1 for fused quartz, "pyrex" boro-silicate glass, and soft soda-lime silicate glass was investigated. The Raman spectra for the fused quartz and the pyrex glass were obtained at room temperature using the 488 nm exciting line of a Coherent Radiation argon-ion laser at powers up to 550 mW. For the soft soda-lime glass the 514.5 nm exciting line at powers up to 660 mW was used because of a weak fluorescence which masked the Stokes Raman spectrum. In addition it is demonstrated that the low-frequency Raman coupling constant can be described by a model proposed by Martin and Brenig (MB). By fitting the predicted spectra based on the model with a Gaussian, Poisson, and Lorentzian forms of the correlation function, the structural correlation radius (SCR) was determined for each glass. It was found that to achieve the best possible fit· from each of the three correlation functions a value of the SCR between 0.80 and 0.90 nm was required for both quartz and pyrex glass but for the soft soda-lime silicate glass the required value of the SCR. was between 0.50 and 0.60 nm .. Our results support the claim of Malinovsky and Sokolov (1986) that the MB model based on a Poisson correlation function provides a universal fit to the experimental VH (vertical and horizontal polarizations) spectrum for any glass regardless of its chemical composition. The only deficiency of the MB model is its failure to fit the experimental depolarization spectra.
    • Magnetic and Dielectric Properties of Cu3-xNixWO6 and Cu3W1-xMoxO6

      Dan, Xu; Department of Physics
      Cu3WO6 is a compound with an interesting coordination chemistry for both Cu and W. In this research, all samples were made by using the standard Solid Phase Reaction method with and without any doping. Using Powder x-ray diffraction and Rietveld analysis, we did not observe any distortion of the cubic crystal. Ni substitution for Cu, and Mo substitution for W, will act as a negative pressure on the lattice parameter of Cu3WO6. Magnetization measurements of Cu3-xNixWO6 indicate that all compounds undergo an antiferromagnetic phase transition at the Néel temperature. However, a significant change was observed in Néel temperature with Ni’s concentration. All compounds show Curie–Weiss antiferromagnetic behavior at high temperatures. The value of the 𝛍eff is close to the theoretical calculation in Cu3WO6. And the magnitude of 𝛍eff (exp) increases with Ni’s concentration. A spin-singlet ground state with energy gap at low temperatures was observed for all compounds. The energy gap 𝚫 is decreasing with the increasing concentration of Ni. The dielectric permittivity as a function of temperature and different frequency from1 kHz to 20 kHz for all samples, were investigated. A peak in dielectric loss ε'' appears between 150 K to 260 K in all samples of Cu3-xNixWO6. The peak position has a linear relationship with log10(𝑓) as a function of temperaure. The doping of Ni causes a gradual shift in the peak position. The activation energy Ea is decreasing with the increasing of Ni’s concentration.
    • Magnetic and high pressure studies in the YPd5B3C3 system

      Murdoch, Jim.; Department of Physics (Brock University, 1995-07-09)
      The macroscopic properties of the superconducting phase in the multiphase compound YPd5B3 C.3 have been investigated. The onset of superconductivity was observed at 22.6 K, zero resistance at 21.2 K, the lower critical field Hel at 5 K was determined to be Hel (5) rv 310 Gauss and the compound was found to be an extreme type-II superconductor with the upper critical field in excess of 55000 Gauss at 15 K. From the upper and lower critical field values obtained, several important parameters of the superconducting state were determined at T = 15 K. The Ginzburg-Landau paramater was determined to be ~ > 9 corresponding to a coherence length ~ rv 80A and magnetic penetration depth of 800A. In addition measurements of the superconducting transition temperature Te(P) under purely hydrostatically applied pressure have been carried out. Te(P) of YPd5B3 C.3 decreases linearly with dTe/dP rv -8.814 X 10-5 J</bar. The discussion of Te(P) will focus on the influence pressure has on the phonon spectrum and the density of states near the Fermi level.
    • Magnetic and transport properties of (Ba1-xKx)Fe2As2

      Reuvekamp, Patrick.; Department of Physics (Brock University, 2009-01-28)
      Single crystals of (Bal - xKx)Fe2As2 were prepared using the Sn flux method. Two heating methods were used to prepare the single crystals: the slow heating and rapid heating methods. It was found that the single crystals grown using the slow heating method were not superconducting due to a significant loss of potassium. When the rapid heating method was used, the single crystals were observed to be superconducting with the desired potassium concentration. The energy dispersive X-ray spectroscopy analysis indicated the presence of multiple phases in the single crystals. Using single crystal X-ray diffraction, the crystal structure of the single crystals was found to be 14/mmm tetragonal at room temperature. The magnetic measurements on the single crystals indicated the presence of multiple phases and magnetic impurities.
    • Magnetic properties of the Biâ Srâ CaCuâ Oâ single crystal

      Abdussalam, Giamal F.; Department of Physics (Brock University, 1991-07-09)
      The Bi2Sr2CaCu20g single crystal with a superconducting transition temperature equal to 90 ± 2 K was prepared. The irreversibility line of the single crystal for a mgnetic field direction along the c-axis and T* in the ab-plane was determined. The reduced temperature (l - T ) is proportional to H 1.1 for fields below 004 T and proportional to HO.09 for fields above 0.4 T. The zero temperature upper critical field Hc2(0) and coherence length ~ (0) were determined from the magnetization meaurements to be H-lC2=35.9T , H//C2=31.2T, ~c(0)=35.0 A, and ~ab(0)=32.5A,and from the magnetoresistance measurements to be H-lc2 = 134.6T , H//C2=55.5T '~c(0)=38.1 A, and ~ab(0)=2404 A for both directions of the applied magnetic field. The results obtained for Hc2(0) and ~(O) are not reliable due to the rounding that the single crystal exhibits in the magnetization and magnetoresistance curves. The magnetization relaxation of the single crystal was investigated, and was found to be logarithmic in time, and the relaxation rate increases with temperature up to 50 -60 K, then decreases at higher temperatures.
    • The measured variation of the Debye-Waller factor of aluminum from 295K to 815K by using the energy dispersive x-ray diffraction technique

      Nguyen, Son-Ha.; Department of Physics (Brock University, 1992-07-09)
      The Energy Dispersive X-ray Diffraction System at Brock University has been used to measure the intensities of the diffraction lines of aluminum powder sample as a function of temperature. At first, intensity measurements at high temperature were not reproducible. After some modifications have been made, we were able to measure the intensities of the diffraction lines to 815K, with good accuracy and reproducibility. Therefore the changes of the Debye-Waller factor from room temperature up to 815K for aluminum were determined with precision. Our results are in good agreement with those previously published.
    • Molecular dynamics calculation of mean square displacement in alkali metals and rare gas solids and comparison with lattice dynamics

      Heiser, Gernot A.; Department of Physics (Brock University, 1984-07-09)
      Molec ul ar dynamics calculations of the mean sq ua re displacement have been carried out for the alkali metals Na, K and Cs and for an fcc nearest neighbour Lennard-Jones model applicable to rare gas solids. The computations for the alkalis were done for several temperatures for temperature vol ume a swell as for the the ze r 0 pressure ze ro zero pressure volume corresponding to each temperature. In the fcc case, results were obtained for a wide range of both the temperature and density. Lattice dynamics calculations of the harmonic and the lowe s t order anharmonic (cubic and quartic) contributions to the mean square displacement were performed for the same potential models as in the molecular dynamics calculations. The Brillouin zone sums arising in the harmonic and the quartic terms were computed for very large numbers of points in q-space, and were extrapolated to obtain results ful converged with respect to the number of points in the Brillouin zone.An excellent agreement between the lattice dynamics results was observed molecular dynamics and in the case of all the alkali metals, e~ept for the zero pressure case of CSt where the difference is about 15 % near the melting temperature. It was concluded that for the alkalis, the lowest order perturbation theory works well even at temperat ures close to the melting temperat ure. For the fcc nearest neighbour model it was found that the number of particles (256) used for the molecular dynamics calculations, produces a result which is somewhere between 10 and 20 % smaller than the value converged with respect to the number of particles. However, the general temperature dependence of the mean square displacement is the same in molecular dynamics and lattice dynamics for all temperatures at the highest densities examined, while at higher volumes and high temperatures the results diverge. This indicates the importance of the higher order (eg. ~* ) perturbation theory contributions in these cases.
    • Monte Carlo study of the XY-model on quasi-periodic lattices

      Reid, R. William.; Department of Physics (Brock University, 1996-07-09)
      Monte Carlo Simulations were carried out using a nearest neighbour ferromagnetic XYmodel, on both 2-D and 3-D quasi-periodic lattices. In the case of 2-D, both the unfrustrated and frustrated XV-model were studied. For the unfrustrated 2-D XV-model, we have examined the magnetization, specific heat, linear susceptibility, helicity modulus and the derivative of the helicity modulus with respect to inverse temperature. The behaviour of all these quatities point to a Kosterlitz-Thouless transition occuring in temperature range Te == (1.0 -1.05) JlkB and with critical exponents that are consistent with previous results (obtained for crystalline lattices) . However, in the frustrated case, analysis of the spin glass susceptibility and EdwardsAnderson order parameter, in addition to the magnetization, specific heat and linear susceptibility, support a spin glass transition. In the case where the 'thin' rhombus is fully frustrated, a freezing transition occurs at Tf == 0.137 JlkB , which contradicts previous work suggesting the critical dimension of spin glasses to be de > 2 . In the 3-D systems, examination of the magnetization, specific heat and linear susceptibility reveal a conventional second order phase transition. Through a cumulant analysis and finite size scaling, a critical temperature of Te == (2.292 ± 0.003) JI kB and critical exponents of 0:' == 0.03 ± 0.03, f3 == 0.30 ± 0.01 and I == 1.31 ± 0.02 have been obtained.
    • NMR characterization of chlorhexidine in lipid-based formulations /

      Trsková, Zuzana.; Department of Physics (Brock University, 2004-07-14)
      A mixture of Chlorhexidine digluconate (CHG) with glycerophospholipid 1,2-dimyristoyl- <^54-glycero-3-phospocholine (DMPC-rf54) was analysed using ^H nuclear magnetic resonance. To analyze powder spectra, the de-Pake-ing technique was used. The method is able to extract simultaneously both the orientation distribution function and the anisotropy distribution function. The spectral moments, average order parameter profiles, and longitudinal and transverse relaxation times were used to explore the structural phase behaviour of various DMPC/CHG mixtures in the temperature range 5-60°C.
    • Numerical Solutions of Laplace's Equation for Various Physical Situations

      Vendromin, Colin; Department of Physics
      There are two projects in this thesis. In the first project, a general method is introduced to numerically calculate the resistance of truncated resistors in cylindrical coordinates, with non-constant cross-sectional area. The problem of finding the resistance of a truncated conical resistor is given in some introductory textbooks as a simple problem. The textbook method is flawed however, and leads to the wrong answer. The textbook method assumes that the electric potential distribution inside the truncated cone is approximately equivalent to a cylindrical resistor. This assumption ignores the constricting affect that the boundary of the truncated conical resistor has on the electric potential inside. The deformation of the electric field is not accounted for by excess charge or changing magnetic fields, instead it is the result of a derivative operation called the shear of the field. Numerical solutions for the resistance of truncated conical, ellipsoidal, and hyperboloidal resistors are presented as a function of a/b, where a is the radius of the smallest cross-sectional area and b is radius of the largest. It was found that the textbook solution always underestimates the numerical value of the resistance. In the second project, dielectric breakdown clusters were grown with a stochastic two dimensional Dielectric Breakdown Model (DBM) on a honeycomb, square, and triangle lattice, as well as on a random distribution of nodes. On the regular lattices the number of nearest neighbours was a constant at all lattice sites. For a random distribution of nodes there was variation in the number of nearest neighbours at different nodes. Some percentage of the nodes were isolated from the rest of the distribution, because they had 0 nearest neighbours. Distributions of nodes in which many of the nodes had 0 nearest neighbours indicated a medium with high density fluctuations. The motivation for this work was to study the relationship between the fractal dimension of the dielectric breakdown clusters and the number of nearest neighbours, and the density variation of the medium. The singularity spectra were calculated for the clusters, as well as their fractal dimension using box counting, and sandbox methods. It was found that the dielectric breakdown model produces monofractal clusters. As such, the dimension of the clusters can be represented by a single fractal dimension. In the DBM, the probability of a perimeter site connecting to the cluster is proportional to the strength of the local electric field raised to an exponent. If the exponent is a large positive number then perimeter sites which feel a stronger electric field are more likely to connect to the cluster. Increasing the exponent produces clusters which resemble lightning, with a fractal dimension lower than the dimension of the lattice. Similarly increasing the percentage of isolated nodes decreases the fractal dimension.
    • On the anharmonic, multi-phonon, Debye-Waller contributions to the phonon-limited resistivity of metals : applications to Na and K

      VanderSchans, Martinus Bartus.; Department of Physics (Brock University, 1979-07-09)
      The anharmonic, multi-phonon (MP), and Oebye-Waller factor (OW) contributions to the phonon limited resistivity (;0) of metals derived by Shukla and Muller (1979) by the doubletime temperature dependent Green function method have been numerically evaluated for Na and K in the high temperature limit. The anharmonic contributions arise from the cubic and quartic shift of phonons (CS, QS), and phonon width (W) and the interference term (1). The QS, MP and OW contributions to I' are also derived by the matrix element method and the results are in agreement with those of Shukla and Muller (1979). In the high temperature limit, the contributions to;O from each of the above mentioned terms are of the type BT2 For numerical calculations suitable expressions are derived for the anharmonic contributions to ~ in terms of the third and fourth rank tensors obtained by the Ewald procedure. The numerical calculation of the contributions to;O from the OW, MP term and the QS have been done exactly and from the CS, Wand I terms only approximately in the partial and total Einstein approximations (PEA, TEA), using a first principle approach (Shukla and Taylor (1976)). The results obtained indicate that there is a strong pairwise cancellation between the: OW and MP terms, the QS and CS and the Wand I terms. The sum total of these contributions to;O for Na and K amounts to 4 to 11% and 2 to 7%, respectively, in the PEA while in the TEA they amount to 3 to 7% and 1 to 4%, respectively, in the temperature range.