An- Najah university- Nablus -Palestine Zeeman effcet Madleen albalshi

Outline Introduction Discussion A. Zeeman splitting mathematically B. Relation between energy and applied field C. Weak field Zeeman effect 1. Normal Zeeman splitting 2. Anomalous Zeeman splitting D. Strong field Zeeman effect E. Important F. Solved problem Conclusions References Introduction Discussion A. Zeeman splitting mathematically B. Relation between energy and applied field strength C. Weak field Zeeman effect 1. Normal Zeeman splitting 2. Anomalous Zeeman splitting D. Strong field Zeeman effect E. Important F. Solved problem Conclusions References Outline

Introduction Zeeman effect was first predicted by Lorentz in 1895 as a part of his classic theory of the electron , and experimentally confirmed some years later by Zeeman . It is defined as interaction of atoms with external magnetic field which result in splitting of energy levels with different angular momenta.

Example of nuclei with these properties Nuclei with an odd mass or odd atomic number have ″nuclear spin″. Which behave in a similar fashion to a simple, tiny bar magnet.  Discussion Example of nuclei with these properties 1H 13C 19F 31P

In the absence of a magnetic field, these are randomly Oriented but when a field is applied they line up parallel to the applied field, either spin aligned or spin opposed. The  more highly populated state is the lower energy spin aligned situation.  Two schematic  representations of These arrangements are shown: The difference in energy between these tow spin state is the Zeeman splitting.  

A. Zeeman splitting mathematically: Difference in energy between tow spin state gyriomagnetic × planks × applied field = ratio constant strength 2 π ∆E = g × ћ × B0 ∆E α B0 A. Zeeman splitting mathematically:

B.Relation between energy and applied field strength Energy difference between two Spin state increase as applied field strength increase The number of align nuclei increase as the strength of the applied field increase

the population of nuclei in upper and lower energy states is given as: N upper = е-∆E/KT N lower where k is the Boltzmann constant T is the absolute temperature (˚K)

These interaction between atoms and external field can be classified into two regimes: Weak fields: Zeeman effect, either normal or anomalous. Strong fields: Paschen-Back effect.

The good quantum numbers C. Weak field Zeeman effect If the strength of the external magnetic field is much less than the internal field the fine structure dominate. The good quantum numbers In first-order perturbation theory, the Zeeman correction to the energy is found by the following: n, l, j, mj

fine structure results from a magnetic interaction between the orbital magnetic moment and spin magnetic moment of the electron, called spin-orbit coupling.

spin-orbit coupling An electron revolving about a nucleus finds itself in a magnetic field produced by the nucleus which is circling about it in its own frame of reference. This magnetic field then acts upon the electron’s own spin magnetic moment to produce substates in terms of energy

But Since the Lande g-factor (gJ)

Where μB is the bohr magneton and it is given by: Thus, Where μB is the bohr magneton and it is given by: E total = E bohr + E fine-structure + E Zeeman correction

Fine structure constant (е2/4πϵ0ћc) Example: Find the total energy for the ground state. For the ground state: n=1, l=0, j=1/2, thus gj=2 splits into tow level E total = + + Fine structure constant (е2/4πϵ0ћc) E =

How to calculate j? J |l-s | ≤ j ≤ l+s For s orbit, l=0 For p orbit, l=1 1/2 │ 1-1/2│ ≤ j ≤ 1+1/2 j 3/2 example: find the values of j for d orbit? For d orbit, l=2 3/2 │ 2-1/2│ ≤ j ≤ 2+1/2 j 5/2

1. Normal Zeeman effect: agrees with the classical theory of Lorentz Observed in atoms with no spin such as: Filled shells, Even number of electrons can produce S = 0 state, all ground states of Group II (divalent atoms) have ns2 configurations. Interaction energy between magnetic moment and a uniform magnetic field is: ∆E = -μ. B

If B is only on the z-direction ∆E = -μzBz =μBBzml orbital magnetic quantum number For a transition to occur Selections rules must satisfied ∆m l ±1

For a transition between two Zeeman-split atomic levels, allowed transition energies are: hʋ= hʋ0 + μBBz ∆ml= +1 hʋ= hʋ0 ∆ml= 0 hʋ= hʋ0 - μBBz ∆ml= -1

2. Anomalous Zeeman effect: Discovered by Thomas Preston in Dublin in 1897. It is purely quantum mechanical, and depends on electron spin which mean it is occurs in atoms with non-zero spin => atoms with odd number of electrons. Interaction energy is: ∆E = -μzBz = g jμBBzmj The selection rules ∆j = 0,±1 ∆mj = 0,±1

The good quantum number D. Strong field Zeeman effect If the strength of the external magnetic field is much greater than the internal field the zeeman effect dominate. The good quantum number The Zeeman Hamiltonian is: n, l, ml, ms

and the unperturbed energies are: In first order perturbation theory, the fine-structure correction to these levels is: Hf-s

E total = E bohr + E fine-structure correction+ E Zeeman effect E total =

E. Important: Zeeman effect is needed for many aspects of modern research in atomic physics. Zeeman effect played an important historical role in the development of quantum mechanics, leading to the discovery of : spin the g-factor of the electron Thomas precession.

Zeeman effect is used in astrophysics, it is used to show the variation of magnetic field on the sun. The Zeeman effect is utilized in many Laser cooling applications such as the Zeeman slower.

Solved problems: Determine the Zeeman splitting for the states |1s> and |2p> of the hydrogen atom in a magnetic field of 10 Tesla. Sol: 10 Tesla strong field Zeeman effect E n,m,ms = (-13.6/ n2)eV + µBBext (ml + 2ms) µB = eћ/(2m) = 9.3*10−24 J/T µBBext = 9.3*10−24 J/T· 10 T = 9.3*10−23 J = 9.3*10−23·(1019/1.6) eV = 5.8*10−4 eV

For the state |1s> = |n = 1, l = 0, ml = 0> one has s =1/2 and consequently ms = −1/2,1/2. It follows that ∆ E1,0,−1/2= - µBBext = -5.8 *10−4 eV ∆ E1,0,1/2 = µBBext = 5.8 *10−4 eV For the state |2p> = |n = 2, l = 1, ml = -1,0,1> one has s =1/2 and then ms = −1/2,1/2. It follows that: ∆E 2,-1,−1/2= -2 µBBext = -11.6 *10−4 eV ∆ E2,0,-1/2 = - µBBext = - 5.8 *10−4 eV ∆E 2,1,−1/2= ∆E 2,−1,1/2= 0

∆E2,0,1/2= µBBext = 5.8 *10−4 eV ∆E2,1,1/2= 2µBBext = 11.6 *10−4 eV Calculate the Zeeman splitting for the state |2p> of the hydrogen atom in a magnetic field of 5 Gauss. Sol: 5 gauss weak field En,l,j,mj = (-13.6/n2) eV + µB Bext gl,j mj µB = eћ/(2m) = 9.3*10−24 J/T

gl,j = 1 + j(j + 1) − l(l + 1) + 3/4 2j(j + 1) For the state |2p> = |n = 2, l = 1, ml = −1, 0, 1> one has s =1/2 and then ms = −1/2, ½. In addition, j = l ±1/2 =1/2,3/2, from which mj = −3/2, −1/2,1/2,3/2 In the case j = 1/2 the Land´e factor reads: g1,1/2= 1 +3/4 − 2 + 3/4 = 1 − 1/3 =2/3 3/2 In the case j = 3/2 the Land´e factor reads: g1,3/2= 1 +15/4 − 2 + 3/4 = 1 + 1/3 = 4/3 15/2

µBB = 9.3*10-24 J/T· 5 *10-4 T = 4.6*10-27 J = 4.6*10-27·(1019/1.6) eV = 2.9 *10-8 eV For j =1/2 one has mj = −1/2,1/2 and then ∆E1,1/2,−1/2 = -(1/3) µBB = -0.97 *10-8 eV ∆E1,1/2,1/2= (1/3) µBB = 0.97 *10-8 eV . For j =3/2 one has instead mj = −3/2, −1/2, 1/2, 3/2 thus, ∆E1, 3/2, −3/2 = -2µBB = -5.8 *10-8 eV ∆E1,3/2,−1/2 = - (2/3) µBB = -1.93 *10-8 eV ∆E1,3/2,1/2 =(2/3) µBB = 1.93 *10-8 eV ∆E1,3/2,3/2 = 2µBB = 5.8 *10-8 eV

Conclusions S=0 S≠0 n, l, j, mj n, l, ml, ms Zeeman effect Week field Normal effect Anomalous effect Strong field n, l, j, mj n, l, ml, ms S=0 S≠0

References David Jeffery Griffiths, Introduction to Quantum Mechanics, 2nd ed. (Pearson Prentice Hall, USA, 2005), PP. 277-280. Springer, Theoretical and Quantum Mechanics: Fundamentals for Chemists, (Springer, Netherlands, 2006), P.456. William H. Brown, Organic Chemistry, 7th ed. (Cengage Learning, 2013), P.513. Nouredine Zettili, Quantum Mechanics: Concepts and Applications, 2nd ed. (John Wiley & Sons, United kingdom, 2009), P.505. Fouad G. Major, The Quantum Beat: Principles and Applications of Atomic Clocks, 2nd ed. (Springer, USA, 2007), P.349.  

Hermann Haken, Hans Christoph Wolf, The Physics of Atoms and Quanta: Introduction to Experiments and Theory, 7th ed. (Springer, Germany, 2006 ), P.191. A. M. Stoneham, Theory of Defects in Solids: Electronic Structure of Defects in Insulators and Semiconductors, (Oxford University Press, New York, 2001), P.646. S. S. Hasan, Robert J. Rutten, Magnetic Coupling between the Interior and Atmosphere of the Sun, (Springer, London, 2010), P.122.