(0:20) There you are, you've got it with you. (0:26) It's about money, it's about a lot of money. (0:29) Don't worry, we'll look into it very, very carefully.

(0:33) Come along. (0:36) Martin, you once told me about an exciting story about an ominous vintage racing car. (0:43) What exactly was it again? (0:44) Exactly, this vintage racing car was about whether it really was this legendary vintage racing car (0:49) or whether it was perhaps just a very well-made copy.

(0:53) And of course there is a lot of money involved. (0:56) And this is where spark emission spectroscopy comes into play. (0:59) The spark emission spectrometer can be used to analyse the elemental composition of metallic samples. (1:04) analyse super precisely.

(1:06) From lithium with the atomic number 3 to uranium with the atomic number 92. (1:10) But how does it all work? (1:13) A strong electrical source causes a spark discharge. (1:17) The sample material vaporises and the released atoms are stimulated to emit radiation.

(1:24) A small part of the sample is discharged by an electrode, (1:27) an electrical voltage source, is heated to several thousand degrees Celsius. (1:32) When the material vaporises on the surface, the atoms are excited. (1:38) This means that electrons are lifted into the outer electron shell (1:41) and then the electrons fall back into the actual shell and thus emit light.

(1:47) The arrangement of the spectral elements, both the wavelength and their colours, (1:53) as well as their intensity ratio is characteristic for each chemical element. (1:59) As can be clearly seen here in the graphic. (2:01) This is therefore comparable to a fingerprint, which is always unique.

(2:07) And so we also use the spark emission spectrometer like a detective to collect evidence. (2:13) These wavelengths in the range from 120 to 770 nanometres are detected and analysed by the spectrometer's optical system. (2:21) To ensure optimum resolution of difficult analysis lines, (2:25) two optical systems are used in our spectrometer.

(2:29) One measures precise wavelengths from 120 to 240 nanometres, the other wavelengths from 210 to 770 nanometres. (2:39) The measured wavelengths are characteristic of the elements contained in the sample (2:44) and the intensity of the radiation for the concentration of the corresponding element in the sample. (2:50) And as complicated as the whole thing sounds, the measurement itself takes less than a minute (2:55) and the exact composition of the analysed sample appears on the screen.

(3:00) Here are the results of such an analysis. (3:04) We can see how precisely the individual ingredients are analysed. (3:08) Larger quantities are in addition to manganese, which is a typical iron companion, (3:12) still contain chromium 17%, nickel 11% and molybdenum 2%.

(3:17) In addition, the sample contains less than 0.02% carbon. (3:21) It is therefore an X2CrNiMo1711II or 1.4406. (3:31) The trick is therefore to break down the light generated in this way into the individual wavelengths, (3:36) to then have them analysed by an analysis software. (3:40) But Peter will explain to you how it works.

(3:44) You can imagine the problem we are facing mathematically as follows. (3:47) You mix different colours together and ask yourself at the end, (3:51) it is possible to break these colours back down into their basic components. (3:56) Sounds difficult? Not with the help of an orthogonal base.

(4:00) In 1812, the mathematician Joseph Fourier submitted to the French Academy of Sciences (4:05) a work on the subject of heat. (4:07) He described the propagation of heat in solids using a partial differential equation. (4:13) And he even solved this equation.

(4:14) However, the most impressive part of his work was not the equation itself and its solution, (4:20) but its solution method. (4:22) Some solutions to the equation are exponentially decaying oscillations over time. (4:28) Since the heat conduction equation is linear and homogeneous, (4:31) the sum of an infinite number of such oscillations also remains a solution.

(4:37) If we now look at the state of these superimposed oscillations at time t equal to 0, (4:42) then the naïve thought quickly arises that every initial profile (4:47) can be expressed as a superposition of an infinite number of oscillations. (4:50) The so-called Fourier series. (4:53) Under certain conditions, this is even true.

(4:55) But what do the propagation of heat in solids have to do with (4:59) with mixing colours in a table box and a vintage racing car? (5:05) The basis functions of the Fourier series are with respect to a suitable scalar product (5:10) are orthogonal to each other and form an orthonormal system with a suitable normalisation. (5:16) Since the basis function 1 has a different normalisation than the other basis functions, (5:21) we pull them out of the sum and redefine them. (5:25) Through some transformations, consisting of splitting, inserting intelligent zeros (5:29) and the Euler formula for complex numbers, the Fourier series (5:33) into a complex representation.

(5:37) In geometric terms, this complex representation is a sum of vectors, (5:42) each pointing to a point on a circle. (5:46) They rotate at a speed of N times Omega. (5:49) Simply put, this indicates which signal is involved.

(5:52) And the radii of the circles Cn indicate the corresponding intensity. (5:58) Thanks to the orthogonality of the basis functions, we can now use this colour mixture (6:02) into its basic components. (6:04) We multiply by a suitable factor, (6:07) integrate over the entire period and adjust according to intensity.

(6:11) This not only allows you to determine which and how many colours are contained in this colour mixture, (6:17) but also which frequencies in an audio recording are disturbing noise, (6:22) which frequencies in a car overlap with the natural frequencies of the body, (6:27) which frequencies must be counteracted when constructing buildings in an earthquake region, (6:32) but also which and how many of the substances are contained in the alloy of a classic racing car. (6:57) And what was the actual outcome of the story? (7:00) Yes, so we analysed the steel and it came out, (7:03) It was a steel with a particularly high purity in terms of phosphorus and sulphur content. (7:08) Well, but it has only been possible to produce steel of this quality since the 1970s.

(7:13) So, it was a nice copy, but it was a copy.