Basic Mathematics and Units

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2 Basic Mathematics and Units Rende Steerenberg BE/OP

3 Contents Vectors & Matrices Differential Equations Some Units we use 3

4 Vectors & Matrices Differential Equations Some Units we use 4

5 Scalars & Vectors Scalar, a single quantity or value Vector, (origin,) length, direction 5

6 Coordinate systems A vector has or more quantities associated with it (,y) Cartesian coordinates (r, ) Polar coordinates y r r r is the length of the vector θ gives the direction of the vector 6

7 Vector Cross Product a and b are two vectors in the in a plane separated by angle θ θ a a b b The cross product a b is defined by: Direction: a b is perpendicular (normal) on the plane through a and b The length of a b is the surface of the parallelogram formed by a and b 7

Cross Product & Magnetic Field The Lorentz force is a pure magnetic field B field Force Direction of current The reason why our particles move around

8 Cross Product & Magnetic Field The Lorentz force is a pure magnetic field B field Force Direction of current The reason why our particles move around our circular machines under the influence of the magnetic fields Electro Magnetic Theory by Werner Herr Transverse Beam Dynamics by Bernhard Holzer 8

9 Lorentz Force in Action Velocity v Magnetic Field charge = -e Particle with mass m charge = e charge = e The larger the energy of the beam the larger the radius of curvature 9

10 A Rotating Coordinate System Horizontal Vertical It travels on the central orbit Longitudinal y s or z 10

11 Magnetic Rigidity As a formula this is: Like for a stone attached to a rotating rope F = evb = mv r Radius of curvature mv p Which can be written as: B r = = Momentum e e p=mv Bρ is called the magnetic rigidity, and if we put in all the correct units we get: Bρ = p [KG m] = p [T m] (if p is in [GeV/c]) 11

12 Moving a Point in a Coordinate System To move from one point (A) to any other point (B) one needs control of both Length and Direction. Rather clumsy! B ( new, y ) new y A ( old, y ) old Is there a more efficient way of doing this? y new new = = a c old old by dy old old equations needed!!! 1

13 Matrices & Vectors So, we have: y new new = = a c old old by dy old old Lets write this as one equation: Rows A and B are Vectors or Matrices A and B have rows and 1 column M is a Matri and has rows and columns Columns 13

Matri Multiplications This implies: y new new =

i = ae bg, j = af bh, k = ce dg, l = = a c y b e

14 Matri Multiplications This implies: y new new = = a c old old by dy i k This matri multiplication results in: old old j l Equals This defines the rules for matri multiplication i = ae bg, j = af bh, k = ce dg, l = = a c y b e d g cf new new dh f h = a c b d y old old Is this really simpler? 14

15 Moving a Point & Matrices Lets apply what we just learned and move a point around: y y1 1 y 3 1 y3 3 M1 transforms 1 to M transforms to 3 This defines M3=MM1 15

16 Matrices & Accelerators We use matrices to describe the various magnetic elements in our accelerator. The and y co-ordinates are the position and angle of each individual particle. If we know the position and angle of any particle at one point, then to calculate its position and angle at another point we multiply all the matrices describing the magnetic elements between the two points to give a single matri Now we are able to calculate the final co-ordinates for any initial pair of particle co-ordinates, provided all the element matrices are known. 16

17 The Unit Matri There is a special matri that when multiplied with an initial point will result in the same final point. y new new = y old old The result is : X new = X old Y new = Y old The Unit matri has no effect on and y 17

18 Going Backwards What about going back from a final point to the corresponding initial point? y new new = a c b d y old old or B = MA For the reverse we need another matri M -1 such that The combination of M and M -1 does have no effect M -1 is the inverse or reciprocal matri of M. 18

19 Calculating the Inverse Matri If we have a matri: é M = ê a b ë c d ù ú û Then the inverse matri is calculated by: M -1 = 1 (ad - bc) d -b -c a The term (ad bc) is called the determinate, which is just a number (scalar). é ê ë ù ú û 19

20 Eample: Drift Space Matri A drift space contains no magnetic field. A drift space has length L. 0 = 1 + L. 1 L 1 1 ' 0 ' ' L = = 1 small = ' ' 1 1 L }

21 A Practical Eample Changing the current in two sets of quadrupole magnets (F & D) changes the horizontal and vertical tunes (Q h & Q v ). This can be epressed by the following matri relationship: Q Q h v = a c b d I I F D or Q = M I Change I F then I D and measure the changes in Q h and Q v Calculate the matri M Calculate the inverse matri M -1 Use now M -1 to calculate the current changes (ΔI F and ΔI D ) needed for any required change in tune (ΔQ h and ΔQ v ). 1

22 Vectors & Matrices Differential Equations Some Units we use

23 The Pendulum Lets use a pendulum as eample The length of the pendulum is L It has a mass m attached to it It moves back and forth under the influence of gravity M L Lets try to find an equation that describes the motion of the mass m makes. This will result in a Differential Equation 3

Differential Equation L The distance from the

force due to gravity is -M g sinθ (force

24 Differential Equation L The distance from the centre = Lθ (since θ is small) The velocity of mass M is: The acceleration of mass M is: M Newton: Force = mass acceleration Mg Restoring force due to gravity is -M g sinθ (force opposes motion) d ( ) dt g L = 0 θ is small L is constant 4

25 Solving a Differential Equation d ( ) dt d( ) dt g L = 0 Find a solution Try a good guess Differentiate our guess (twice) Differential equation describing the motion of a pendulum at small amplitudes. = Acos( t) d ( ) = A sin( t) and = A cos( t) dt Put this and our guess back in the original Differential equation. g cos( t) cos( t) L = 0 5

26 Solving a Differential Equation Now we have to find the solution for the following equation: g cos( t) cos( t) = L 0 Solving this equation gives: The final solution of our differential equation, describing the motion of a pendulum is as we epected : Oscillation amplitude = Acos g L t Oscillation frequency 6

27 Position & Velocity The differential equation that describes the transverse motion of the particles as they move around our accelerator. d () dt + K ( ) = 0 The solution of this second order describes oscillatory motion For any system, where the restoring force is proportional to the displacement, the solution for the displacement will be of the form: 7

28 Phase Space Plot Plot the velocity as a function of displacement: d dt 0 It is an ellipse. As ωt advances by π it repeats itself. This continues for (ω t + k π), with k=0,±1, ±,..,..etc 0 8

29 Oscillations in Accelerators Under the influence of the magnetic fields the particle oscillate = displacement = angle = d/ds ds d s 9

30 Transverse Phase Space Plot This changes slightly the Phase Space plot X Position Angle ' = d ds φ φ = ωt is called the phase angle X-ais is the horizontal or vertical position (or time). Y-ais is the horizontal or vertical phase angle (or energy). 30

31 Transverse Phase Space Plot We distinguish motion in the Horizontal & Vertical Plane y φ φ y Horizontal Phase Space Vertical Phase Space 31

32 Transverse Emittance To be rigorous we should define the emittance slightly differently. Observe all the particles at a single position on one turn and measure both their position and angle. This will give a large number of points in our phase space plot, each point representing a particle with its co-ordinates,. emittance beam acceptance Symbol: e Epressed in 1s, s,.. Units: mm mrad The emittance is the area of the ellipse, which contains all, or a defined percentage, of the particles. The acceptance is the maimum area of the ellipse, which the emittance can attain without losing particles 3

33 Vectors & Matrices Differential Equations Some Units we use 33

34 Relativity c velocity CPS Einstein: } energy increases E not velocity SPS / LHC = mc PSB Newton: 1 E = mv energy More about Relativity by Werner Herr 34

35 The Units we use for Energy The energy acquired by an electron in a potential of 1 Volts is defined as being 1 ev The unit ev is too small to be used today, we use: 1 KeV = 10 3, MeV = 10 6, GeV = 10 9, TeV =

36 Energy: ev versus Joules The unit most commonly used for Energy is Joules [J] In accelerator and particle physics we talk about ev!? The energy acquired by an electron in a potential of 1 Volt is defined as being 1 ev 1 ev is 1 elementary charge pushed by 1 Volt. 1 ev = Joules 36

37 The Energy in the LHC beam The energy in one LHC beam at high energy is about 30 Million Joules This corresponds to the energy of a TGV engine going at 150 km/h... but then concentrated in the size of a needle 37

Energy versus Momentum E = mc Einstein s formula: which for a mass at rest is: E = m c 0 0 The ratio between the total energy and the rest energy is = E E 0 Then the

38 Energy versus Momentum E = mc Einstein s formula: which for a mass at rest is: E = m c 0 0 The ratio between the total energy and the rest energy is = E E 0 Then the mass of a moving particle is: We can write: Momentum is: = mvc mc p = mv The ratio between the real velocity and the velocity of light is m = m 0 pc = or p = E = E c v c 38

39 Energy versus Momentum Energy Momentum Therefore the units for momentum are: MeV/c, GeV/c, etc. Energy are: MeV, GeV, etc. Attention: when β=1 energy and momentum are equal when β<1 the energy and momentum are not equal 39

40 A Practical Eample (PSB- PS) Kinetic energy at injection E kinetic = 1.4 GeV Proton rest energy E 0 =938.7 MeV The total energy is then: E = E kinetic + E 0 =.34 GeV We know that = E E, which gives γ = We can derive = 1 1, which gives β = E Using p = we get p =.14 GeV/c c In this case: Energy Momentum 40

41 Pure mathematics is, in its way, the poetry of logical ideas. Albert Einstein 41

Basic Mathema,cs. Rende Steerenberg BE/OP. CERN Accelerator School Basic Accelerator Science & Technology at CERN 3 7 February 2014 Chavannes de Bogis

Basic Mathema,cs. Rende Steerenberg BE/OP. CERN Accelerator School Basic Accelerator Science & Technology at CERN 3 7 February 2014 Chavannes de Bogis Basic Mathema,cs Rende Steerenberg BE/OP CERN Accelerator School Basic Accelerator Science & Technolog at CERN 3 7 Februar 014 Chavannes de Bogis Contents Vectors & Matrices Differen,al Equa,ons Some Units