Monte Carlo Simulation of the Stern Gerlach Experiment

1. Page 1/28 Mar 2008 SGE A Monte-Carlo Simulation of the Stern-Gerlach Experiment Dr. Ahmet BNGL Gaziantep niversitesi Fizik Mhendislii Blm Nisan 2008

2. Page 2/28 Mar 2008 SGE Content Stern-Gerlach Experiment (SGE) Electron spin Monte-Carlo Simulation You can find the slides of this seminar and computer programs at: http://www1.gantep.edu.tr/~bingul/seminar/ spin

3. Page 3/28 Mar 2008 SGE The Stern-Gerlach Experiment The Stern-Gerlach Experiment (SGE) is performed in 1921, to see if electron has an intrinsic magnetic moment. A beam of hot (neutral) Silver ( 47 Ag) atoms was used. The beam is passed through an inhomogeneous magnetic field along z axis. This field would interact with the magnetic dipole moment of the atom, if any, and deflect it. Finally, the beam strike s a photographic plate to measure, if any, deflection.

4. Page 4/28 Mar 2008 SGE The Stern-Gerlach Experiment Why Neutral Silver atom? No Lorentz force ( F = q v x B ) acts on a neutral atom, since the total charge (q) of the atom is zero . Only the magnetic moment of the atom interacts with the external magnetic field. Electronic configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 4p 6 4d 10 5s 1 So, a neutral Ag atom has zero total orbital momentum. Therefore, if the electron at 5s orbital has a magnetic moment, one can measure it. Why inhomogenous magnetic Field? In a homogeneous field, each magnetic moment experience only a torque and no deflecting force. An inhomogeneous field produces a deflecting force on any magnetic moments that are present in the beam.

5. Page 5/28 Mar 2008 SGE The Stern-Gerlach Experiment In the experiment, they saw a deflection on the photographic plate. Since atom has zero total magnetic moment, the magnetic interaction producing the deflection should come from another type of magnetic field . That is to say: electrons (at 5s orbital) acted like a bar magnet. If the electrons were like ordinary magnets with random orientations , they would show a continues distribution of pats. The photographic plate in the SGE would have shown a continues distribution of impact positions. However, in the experiment, it was found that the beam pattern on the photographic plate had split into two distinct parts . Atoms were deflected either up or down by a constant amount, in roughly equal numbers. Apparently, z component of the electrons spin is quantized.

6. Page 6/28 Mar 2008 SGE The Stern-Gerlach Experiment A plaque at the Frankfurt institute commemorating the experiment

7. Page 7/28 Mar 2008 SGE Electron Spin 1925: S.A Goutsmit and G.E. Uhlenbeck suggested that an electron has an intrinsic angular momentum (i.e. magnetic moment) called its spin . The extra magnetic moment s associated with angular momentum S accounts for the deflection in SGE. Two equally spaced lined observed in SGE shows that electron has two orientations with respect to magnetic field.

8. Page 8/28 Mar 2008 SGE Electron Spin Orbital motion of electrons, is specified by the quantum number l . Along the magnetic field, l can have 2 l +1 discrete values.

9. Page 9/28 Mar 2008 SGE Electron Spin We have two orientations: 2 = 2 s +1 s = 1/2 The component S z along z axis: Similar to orbital angular momentum L , the spin vector S is quantized both in magnitude and direction, and can be specified by spin quantum number s .

10. Page 10/28 Mar 2008 SGE Electron Spin It is found that intrinsic magnetic moment ( s ) and angular momentum ( S ) vectors are proportional to each other: where g s is called gyromagnetic ratio. For the electron, g s = 2.0023 . The properties of electron spin were first explained by Dirac (1928), by combining quantum mechanics with theory of relativity.

11. Page 11/28 Mar 2008 SGE Monte-Carlo Simulation Experimental Set-up :

12. Page 12/28 Mar 2008 SGE Monte-Carlo Simulation Initial velocity v of each atom is selected randomly from the Maxwell-Boltzman distribution function: around peak value of the velocity: Ag atoms and their velocities : Note that: Components of the velocity at (x 0 , 0, z 0 ) are assumed to be: v y0 = v , and v x0 = v z0 = 0. Temperature of the oven is chosen as T = 2000 K. Mass of an Ag atom is m =1.8 x 10 25 kg.

13. Page 13/28 Mar 2008 SGE Monte-Carlo Simulation The Slit : Initial position (x 0 , 0, y 0 ), of each atom is seleled randomly from a uniform distribution. That means: the values of x 0 and z 0 are populated randomly in the range of [Xmax, Zmax], and at that point, each atom has the velocity (0, v , 0).

14. Page 14/28 Mar 2008 SGE Monte-Carlo Simulation The Magnetic Field : In the simulation, for the field gradient (dB/dz) along z axis, we assumed the following 3-case: uniform magnetic field : constant gradient : field gradient is modulated by a Gaussian i.e. We also assumed that along beam axis:

15. Page 15/28 Mar 2008 SGE Monte-Carlo Simulation Potential Energy of an electron: Componets of the force: Equations of motion : Consequently we have,

16. Page 16/28 Mar 2008 SGE Monte-Carlo Simulation Differential equations and their solutions: since v 0x = 0 since v 0y = v and y 0 = 0 since v 0z = 0 Equations of motion : So the final positions on the photographic plate in terms of v , L and D : Here x 0 and z 0 are the initial positions at y = 0.

17. Page 17/28 Mar 2008 SGE Monte-Carlo Simulation Spin vector components: S = ( S x , S y , S z ) In spherical coordinates: S x = | S | sin( ) cos( ) S y = | S | sin( ) sin( ) S z = | S | cos( ) w here the magnitude of the spin vector is: Quantum Effect :

18. Page 18/28 Mar 2008 SGE Monte-Carlo Simulation Angle can be selected as: where R is random number in the range (0,1). However, angle can be selected as follows: if S z is not quantized, cos will have uniform random values: else if S z is quantized, cos will have only two random values: Quantum Effect :

19. Page 19/28 Mar 2008 SGE Monte-Carlo Simulation Geometric assumptions in the simulation: L = 100 cm and D = 10 cm X max = 5 cm and Z max = 0.5 cm

20. Page 20/28 Mar 2008 SGE Monte-Carlo Simulation Physical assumptions in the simulation: N = 10,000 or N = 100,000 Ag atoms are selected. Velocity ( v ) of the Ag atoms is selected from MaxwellBoltzman distribution function around peak velocity. The temperature of the Ag source is takes as T = 2000 K. (For the silver atom: Melting point T = 1235 K ; Boiling point 2435 K) Field gradient along z axis is assumed to be: uniform magnetic field constant field gradient along z axis field gradient is modulated by a Gaussian z component of the spin ( S z ) is either quantized according to quantum theory such that cos = 1/sqrt(3) or cos is not quantized and assumed that it h as random orientation .

21. Page 21/28 Mar 2008 SGE Results Hereafter slides, you will see some examples of simulated distributions that are observed on the photographic plate. Each red point represents a single Ag atom. You can find the source codes of the simulation implemented in Fortran 90, ANSI C and ROOT programming languages at: http://www1.gantep.edu.tr/~bingul/seminar/spin

22. Page 22/28 Mar 2008 SGE Results N = 10,000 N = 100,000 dB/dz = 0

23. Page 23/28 Mar 2008 SGE Results N = 10,000 N = 100,000 dB/dz = 0

24. Page 24/28 Mar 2008 SGE Results N = 10,000 N = 100,000 dB/dz = constant > 0

25. Page 25/28 Mar 2008 SGE Results N = 10,000 N = 100,000 dB/dz = constant > 0

26. Page 26/28 Mar 2008 SGE Results dB/dz = constant * exp(kx 2 ) N = 10,000 N = 100,000

27. Page 27/28 Mar 2008 SGE Results N = 10,000 N = 100,000 dB/dz = constant * exp(kx 2 )

28. Page 28/28 Mar 2008 SGE End of Seminar Thanks. April 2008