Fun with Formulas !

Fun with Formulas !Werner Joho

Paul Scherrer Institute (PSI) CH5232 Villigen, Switzerland4.July 2003(updated 1.10.2010)

IntroductionFormulas can be fun. They often can be made to look simple, transparent and thus beautiful (in the spirit of Einstein and Chandrasekhar). This can be achieved with some simple rules and a few tricks of the trade. This paper is a sample of some simple formulas, collected during my career as a physicist.The following material was presented (but not published) at a seminar talk given at the CERN Accelerator School on Synchrotron Radiation and Free Electron Lasers,Brunnen, Switzerland, 2-9 July 2003

(this file is available on the WEB with google: JOHO PSI)

Content philosophy for formulas capital growth new interpretation of Ohms law logarithmic derivatives the relativistic equations of Einstein the magic triangle formed by the logarithmic derivatives of the relativistic parameters Alternative Gradient Focusing, constructed by hand binomial curves everywhere, approximation of a variety of functions, like beam profiles, the fringe field of magnets, the flux and brightness of synchrotron radiation etc. simple representation of phase space ellipses how to win money with statistics ! design of beautiful tables with a Hamiltonian

philosophy for formulas simplify formulas, they should look beautiful formula should indicate the proper dimensions use units of 1'000 (cm should not exist in formulas!) choose right scales for plots (e.g. logarithmic)example: c = 3 108 m/s ?? better is:c = 0.3109 m/s or 300 m/ms or 0.3 mm/ps !!for comparison of electric forces: q (kV/mm) with magnetic forces: qvB = qcBc = 300 (kV/mm)/T !!

m0 = 4p 10-7 Vs/Am ?? better is:m0 = 0.4p mH/m = 0.4p T/(kA/mm)

how to avoid akward numbers in electrodynamics

electron mass: m = 9.1110-31 kg (?) => forget it !

use mc2 ( eUo , Uo = .511 MV

(o = 8.85410-12 As/Vm (?) => forget it !

use (o (o = 1/c2 with (o = 0.4 ( (H/m

introduce impedance Zo :

(( 29.9792458 ()

Alfven current IA (used in space charge calculations):

IA = 4((omc3/e (?) => forget it ! (similarly for "perveance")

use instead Ohms-law : IA = Uo/Zo = 511 kV/30( = 17 kA

charged particle in a magnetic field Bo

=> Larmor-frequency

electron:

= 1.761011 C/kg (?) => forget it !

use

=

= 28 GHz/T (15.25 MHz/T for protons)

how to avoid akward numbers in electro-dynamics

Zo ( EMBED Equation.3 = EMBED Equation.3 = 30 (

(o = EMBED Equation.3

_1118511451.unknown

_1118512706.unknown

_1118512821.unknown

_1118514183.unknown

_1118512633.unknown

_1118511324.unknown

use logarithmic derivatives !

Einstein triangle

in relativistic equations use dimensionless quantities for

velocity, energy and momentum on a democratic basis!

=> Einstein triangle and magic triangles for logarithmic derivatives

velocity:

v = ( c

total energy:E = ( E0 , E0 = mc2 = eU0 (0.511 MeV for electron)

( = (1-(2)-1/2 ,

momentum:p =

E0/c =

mc

Pythagoras- connection: E2 = E02 + (pc)2

give the Einstein triangle and derivatives

1) highly relativistic case

( >> 1 , ( ( 1 => use angle (

( ( cos( ( 1 (2

1/( ( sin( ( (

( (( = 1/tan( ( 1/(

a) race to the moon between electron and photon: electron looses by

(L = (1 - () L (

L

SLS: 2.4 GeV, (L = 8 m

ESRF: 6 GeV, (L = 1.4 m

LEP II: 100 GeV, (L = 5 mm

b) race over one undulator period (u : if electron is just one wavelength ( behind photon (slippage) => positive interference

K = 0.0934 B[T] (u [mm]

detour due to slalom in B-field !

2) mildly relativistic case: use angle (

the following table gives some easy reference values for

quick estimates or for calculations with a pocket calculator!

(

( = sin(

cos(

(= 1/cos(

p = (( = tan(

300

0.5

0.87

1.15

0.58

36.90

0.6

0.8

1.25

0.75

450

0.71

0.71

1.41

1

53.10

0.8

0.6

1.67

1.33

600

0.87

0.5

2

1.71

Einstein triangle

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

(L = ( = EMBED Equation.3 (u (1 + K2)

_1118764697.unknown

_1118765678.unknown

_1118766891.unknown

_1119003940.unknown

_1118765979.unknown

_1118764857.unknown

_1118473171.unknown

_1118764630.unknown

_1118431102.unknown

Magic Triangles (W.Joho) with logarithmic derivatives of relativistic parameter

multiplication factors form inverse triangle

trigonometric functions for relativistic formula !

2) mildly relativistic case: use angle (

the following table gives some easy reference values for

quick estimates or for calculations with a pocket calculator!

(with use of Pythagoras triangle with sides 3,4,5)

(

( = sin(

cos(

(= 1/cos(

p = (( = tan(

300

0.5

0.87

1.15

0.58

36.90

0.6

0.8

1.25

0.75

450

0.71

0.71

1.41

1

53.10

0.8

0.6

1.67

1.33

600

0.87

0.5

2

1.71

highly relativistic case

highly relativistic case

( >> 1 , ( ( 1 => use angle (

( ( cos( ( 1 (2 / 2

1/( ( sin( ( (

( (( = 1/tan( ( 1/(

a) race to the moon between electron and photon: electron "looses" by

(L = (1 - () L (

L

SLS: 2.4 GeV, (L = 8 m

ESRF: 6 GeV, (L = 1.4 m

LEP II: 100 GeV, (L = 5 mm

b) race over one undulator period (u : if electron is just one or n wavelengths ( behind photon (slippage) => positive interference

K = 0.0934 B[T] (u [mm]

detour due to slalom in B-field !

EMBED Equation.3

EMBED Equation.3

(L = ( = EMBED Equation.3 (u (1 + K2 / 2)

_1118473171.unknown

_1118765979.unknown

_1287400419.unknown

_1118765678.unknown

_1118431102.unknown

Undulator RadiationTESLAproduced by an electron beam of energy E = gmc2

AG-focusing simple example of alternative gradient focusing:

FODO-lattice with thin lenses (focal length f)if L = 2f => construction is possible by hand !it takes 6 periods to get a 3600-oscillationi.e. the phase advance/period is = 600for L = 2f => = 600 (graphic example) for L = 4f => = 1800 (instability !)exact solution with transfer matrices gives

magnetic fringe field with binomial

Diagramm1

1

0.9999999667

0.9999957334

0.9999271106

0.9994544624

0.9974093148

0.9908392374

0.9739653749

0.9385101073

0.8778347684

0.793700526

0.6973577481

0.6020201164

0.5160898112

0.4425012051

0.3809660936

0.3299325428

0.2876075066

0.252358926

0.2228252474

0.1979110744

0.1767475649

0.158647976

0.1430684362

0.1295761719

0.1178246951

0.1075347247

0.0984795879

0.0904740456

0.0833657075

0.077028403

0.0713570312

0.0662635336

0.0616737245

0.0575247801

0.0537632356

0.0503433787

0.0472259528

0.0443771039

0.0417675212

0.0393717318

N=7, S=-3

u

B

magnet edge

short range

Binomial Curve3/28/03Binom_curves.xls

short range:

y=A*((1-(X/XL)^^N)^^1/Sy(1)=0.5y(XL)=0

A1.0000

N4.0000

S0.2500

Einfgen-Name-Definieren

1/S4.0000

XL1.5834

xy=binom

01.0000

0.10.9999

0.20.9990

0.30.9949

0.40.9838

0.50.9608

0.60.9200

0.70.8557

0.80.7637

0.90.6434

10.5000

1.10.3462

1.20.2016

1.30.0886

1.350.0494

1.40.0228

1.50.0014

1.580.0000

short range

1

N=4, S=0.25

XL=1.583

X

Y

Binomial

long range

Binomial Curve6/5/03Binom_curves.xls

long range

y=A/(1+(X/XL)^^N)^^1/Ty(1)=0.5y(XL)=0

A1.0

N7.0

T3.0000

Einfgen-Name-Definieren

1/T0.3333

XL0.7573

xy=binom

01.0000

0.11.0000

0.21.0000

0.30.9995

0.40.9962

0.50.9824

0.60.9421

0.70.8592

0.80.7400

0.90.6127

10.5000

1.10.4088

1.20.3372

1.30.2813

1.40.2373

1.50.2024

1.60.1743

1.70.1514

1.80.1325

1.90.1169

20.1037

2.10.0925

2.20.0830

2.30.0749

2.40.0678

2.50.0616

2.60.0562

2.70.0515

2.80.0473

2.90.0436

30.0403

long range

N=7, S=-3

u

y

magnet edge

short range (2)

Binomial Curve3/28/03Binom_curves.xls

short range:

y=A*((1-(X/XL)^^N)^^1/Sy(1)=0.5y(XL)=0

A1.0000

N20.0000

S0.0500

Einfgen-Name-Definieren

1/S20.0000

XL1.1841

xy=binom

01.0000

0.031.0000

0.061.0000

0.091.0000

0.121.0000

0.151.0000

0.181.0000

0.211.0000

0.241.0000

0.271.0000

0.31.0000

0.331.0000