# Requirements

This section contains the requirements with regard to the content of the degree programme.All requirements regarding formal aspects of your application are listed in the “Application” section.

The Master’s degree programme in Materials Science and Engineering requires a Bachelor’s degree in a field of natural sciences or techniques.

Basics in the following keywords must be known for theory and practice.

Here you can print out a checklist.

## Materials Science

- Models of atoms, Schrödinger equation and principles of quantum theory, bonding, concept of potential wells
- Perfect crystals: formal description, important lattices
- Real crystals: defects in crystals, types of defects, interaction of defects
- Thermodynamic
equilibrium: types of equilibrium, 1
^{st}and 2^{nd}laws of thermodynamics, statistical interpretation of the entropy, applications to point defects - Kinetics: reaction rates and the Boltzmann factor, phenomenological diffusion and atomic mechanisms
- Mechanical properties I: stress and strain, phenomenological description of deformation, brittle behaviour and fracture
- Mechanical properties II: mechanisms of plastic deformation, dislocation movement and multiplication, theory of yield strength
- Structural and mechanical properties of amorphous materials and polymers
- General structural properties, modulus of elasticity, viscose and inelastic behaviour, deformation and fracture
- Ageing and failure of materials: mechanisms in general, fatigue and creep, corrosion, electromigration and special mechanisms
- Electronic properties in general, electronic properties and materials science.
- Electrons in crystals: classical theory, the Hall effect, quantum descriptions, the free electron gas model, density of states, Fermi distribution and Fermi energy, properties of the free electron gas
- Diffraction in crystals: basic consideration of diffraction in crystals, Bragg’s law, reciprocal lattices and the Ewald Construction, intensity of the diffracted waves
- Electrons in a periodic potential: free electron gas plus Bragg’s law, the Brillouin construction of diffraction, band structures and electronic properties, band-band transitions and standard representation of semiconductor band structures
- Semiconductors: intrinsic conductivity, conductivity as a function of temperature, the concept of holes, doping and extrinsic conductivity
- Semiconductor contacts: surface states, space-charge regions, p-n-junctions, currents and current-voltage characteristics, recombination and diffusion currents

## Mathematics

### Calculus

- Real and complex numbers, complete induction, sets, functions. Series of real numbers, convergence, Cauchy series
- Steadiness, theorems on continuous functions, polynoms, nulls, rational functions
- Inverse functions: exponential functions and logarithms, trigonometric functions, hyperbola functions
- Differential calculus: characteristics of differentiable functions and differentiation rules, derivatives of basic functions, median theorem, extrema, Taylor’s Theorem and l'Hôpital'srule
- Integral calculus: antiderivatives, indefinite integrals, substitutional rules, partial integration, factoring of polynomials, Riemann integrals, examples of continuous and monotone functions, main clause of differential and integral calculus.
- Indefinite integrals: gamma functions, Stirling’s formula
- Infinite series: convergence criterion, power series, monotonous convergence, differentiation and integration of individual terms, examples of Taylor series’, criteria of convergence with respect to Fourier series’
- Arc length, curvature, convergence criteria; power series, uniform convergence, differentiation and integration by segments, examples of Taylor series’
- Fourier series: questions of convergence, Bessel's inequality
- Taylor's Theorem
- Extrema of functions in several variables, the method of least squares, Lagrange multipliers
- Integration in Rn: integral over domains, iterated integrals (Fubini), volume, substitution rule: polar and spherical coordinates

### Algebra

- Euclidean vector spaces in R2, R3: vectors, scalar products, matrices, linear maps in R2, vector products in R3
- Analytic geometry in R2, R3 vector spaces: linear independence, basis, dimensions, linear maps and matrices, ranks
- Linear systems of equations: solvability, the algorithm of Gaussian elimination, L-R-factorization, inverse matrices, Cramer's rule
- Eigenvalues and eigenvectors, characteristic polynomials, scalar products and norms, Schwarz Inequality
- Legendre polynomials, orthogonal and unitary maps
- Linear transformations: eigenvalues and eigenvectors of symmetric and orthogonal matrices, quadratic forms
- Some topology in Rn: open, closed, tangent planes, directional derivatives, special partial derivatives, gradients, direction of maximum slope

## Chemistry

### A. Inorganic Chemistry

• The periodic system of elements, names, periodic properties, electron configuration

• Usage of the elements and their compounds

• Atomic structures, crystals, amorphous materials

• Chemical bond types: covalent, ionic, metallic, van-der-Waals

• The reactivity of chemical elements, redox potential, oxidation, reduction

• Simple inorganic synthesis

• Acids and bases, strength, order

• pH values, neutralization, titration

• Chemical equilibrium

• Indicators

• Basic chemical analysis

### Organic Chemistry

- Alkanes, alkenes, alkynes
- Alcohols, ketones, aldehydes, acids
- Cyclic molecules, hetero structures, amoatics
- Nucleophiles, electrophile substitution and addition
- Common reactions
- Polymerisation, reactions, modes, usage
- Nomenclature
- Basics in stereo chemistry, chirality, stereo selectivity
- Basics in natural products chemistry

### Physical
Chemistry

- Ideal and real gases
- Fundamentals of the kinetic theory of gases
- Thermodynamic state variables
- Laws of thermodynamics
- Chemical transformations
- Thermochemistry
- Phase transition and equilibrium
- Multicomponent systems
- Phase diagrams
- Chemical equilibriums
- Fundamentals of equilibrium electrochemistry

## Physics

- Fundamentals of physics with a special emphasis on classical mechanics, electricity, optics and acoustics
- Introduction into physics of atoms, molecules, nuclei, elementary particles
- Basics of solid state physics