Exercise:
There are known isotopes of Krypton a noble gas isotopeKr with atomic mass numbers from through . The six most stable isotopes are the ones reported in the table below with their relative abundance in nature. center tabular |p.cm|p.cm| hline Isotope & Abundance hline hline isotopeKr & .% hline isotopeKr & .% hline isotopeKr & .% hline isotopeKr & .% hline isotopeKr & .% hline isotopeKr & .% hline tabular center We want to use a mass spectrometer to confirm the relative abundance in a sample at our disposal. The spectrometer we use corresponds to a simplified version of the Thoon mass spectrometer. There is a source where Krypton ions are generated. The ionization process removes one electron from the atoms; then the ions are accelerated along the x-direction via a parallel plate capacitor with an acceleration potential U_ towards the spectrometer. After the acceleration step the ions enter in the spectrometer where a magnetic field vecB -Bvece_z deflects their trajectory. The ions are then detected via a micro-channel plate detector a detector consisting of an array of adjacent particle detectors with a glqq pixelgrqq size Delta y. The detector is placed at a distance D from the entrance po of the spectrometer and oriented perpicular to the x-direction in the yz-plane. abcliste abc Asing that the initial velocity of the ions is negligible find the velocity of the ions after the linear acceleration stage. abc Describe the trajectory the ions take after entering the mass spectrometer. Asing there is no detector what is the maximum displacement s projected along the x-direction? What does this imply for the distance D at which the detector is placed? abc Asing that the condition you find in b holds show that the lateral deflection Delta y of the ions on the detector of the mass spectrometer is Delta y sleft -sqrt-fracD^s^right abc Simplify the expression found in c to find the deflection as a function of the mass m the charge q of the ions the acceleration voltage U_ as well as the distance D in the limit that D ll s. H: You may use the Taylor-Expansion sqrt pm x^ approx pm fracx^ for x ll . abc Let’s ase the acceleration potential is U_ kilovolt magnetic field strength is B tesla the detector is placed at a distance D centimeter. What is the maximum value of Delta y that will allow all different isotopes to be resolved? H: in order to resolve two different isotopes they should hit two different pixels of the detector. Also ase that the neutron and proton mass is the same with a value m .kilogram which is much bigger than the electron mass which can therefore be neglected. abcliste

Solution:
abcliste abc The velocity after the acceleration step can be computed from energy conservation where at the ning an ion of charge q and mass m has only electric potential which is fully converted to kinetic energy. Thus qU_ fracmv^ &Rightarrow v sqrtfracqU_m the velocity pos in x direction. abc In the mass spectrometer the charged ions are deflected due to the Lorentz force acting on them. It is: vecF_L qvecvtimes vecB qpmatrix v pmatrixtimes pmatrix -B pmatrix pmatrix qvB pmatrix qvBvece_y F_Lvece_y If undisturbed the ions will move on a circular trajectory with radius R the so-called Larmor radius which can be computed from the centripetal force F_Z F_L F_Z qvB mfracv^R R fracmvqB The maximum displacement of the ions in x-direction equals this radius R. If R is smaller than D the ions will never reach the detector hence we need D R s for the spectrometer to work as ed. abc We can write: Delta y R - Rcos phi Rleft -sqrt-sin^phiright Rleft -sqrt-fracDelta x^R^right In part b we saw that Rs and we can also use tha fact that the detector is placed at Delta x D to obtain Delta y Rleft -sqrt-fracD^s^right abc If D ll R i.e. when the detector is placed close to the input of the mass spectrometer with respect to the radius R we can approximate the squarroot as sqrt-fracD^R^ approx -fracD^R^ to get Delta y approx Rleft - left -fracD^R^rightright fracD^R^ Next we replace the radius with the expression found in b given in the such that Delta y &approx fracD^R fracqBD^mv fracqBD^msqrtfracmqU_ sqrtfracqmfracBD^sqrtU_ where we roduced the velocity from obtained in a. abc It’s easiest if we first compute the mass for of the isotopes as m Am_p where we ase that protons and neutrons have the same mass m_p and electrons can be neglected in the mass calculation. The masses are reported in the table below. Then is used to compute the deflections with the given values. Note that the charge of each isotope as it is once ionized is +e .coulomb. center tabular |p.cm|p.cm|p.cm|p.cm|p.cm| hline Isotope & Abundance & m kg & delta y micrometer & Spacing micrometer hline hline isotopeKr & .% & . ^- & & fracna hline isotopeKr & .% & . ^- & & hline isotopeKr & .% & . ^- & & hline isotopeKr & .% & . ^- & & hline isotopeKr & .% & . ^- & & hline isotopeKr & .% & . ^- & & hline tabular center In the last column we computed the spacing to the isotope above. We can see that the minimum spacing is around micrometer which implies that we need detectors with a pixel size L smaller than this value e.g. around micrometer. abcliste