- Email: [email protected]
A classical interpretation of the spin-polarization effect of electrons moving in a uniform magnetic field
Volume
27A, number
10
PHYSICS
A CLASSICAL INTERPRETATION OF ELECTRONS MOVING
LETTERS
7 October
OF THE SPIN-POLARIZATION IN A UNIFORM MAGNETIC
1968
EFFECT FIELD
E. STORCK Sektion Physik der Universith’t Miinchen, Germany Received
23 August 1968
The spin-polarization effect of electrons in a magnetic field is explained as the retroaction of a top oscillating on account of an elementary angular and magnetic momentum.
It has been shown by several authors [l-3] that, in the emission of fast electrons moving in a magnetic field, the probability of the quantum transitions accompanied by a change in the orientation of the electron-spin depends essentially on the initial state of polarization: Beginning with the spin in the direction of the field the spinflip is 24 times as probable as vice versa, so that in the state of equilibrium a beam of electrons should be polarized against the direction of the magnetic field at 92 per cent. The velocity T-l of approaching to this limiting value is of the order of magnitude (we use A = c = 1) T-l
= e2m(E/m)2(eB/m2)3
,
where e means the elementary charge, B the magnetic field and E or m energy or mass of the electron. Subsequently with E = 5 GeV and B = = 1.5 x lo4 Oe we have T = 47 sec. Therefore in a modern storage ring this spin-polarization effect should be measurable by all means. After recently [4] having established this effect quantitatively by a rigorous investigation on the basis of quantum electrodynamics, we wish to note here, that it may easily understood already from a classical point of view: as a retroaction in fact of the emission of a top oscillating in the magnetic field on account of its angular momentum +s, .s2 = 1 and its magnetic moment ,u = = -e s/2m. Let us consider for that purpose the balance of energy in the rest system of the top. In consequence of the emission of the magnetic dipole moment oscillating with the freq!ency w= 2~uB= eB/m in the magnetic field B its potential energy decreases by
d(-,u+
of the emission
= - 4 w~(,u&:/~)~
di.
(2) Here we replace the variables B and dl referring to the rest system, by B and dt referring to the laboratory system, by means of
1 dt’ = J1-v2 dt ,
m:=m/E;
(3)
where in our case we have to put v*B = 0 and E = 0. In this way we obtain for the component sB of the vector s in the direction of the magnetic field dSB -= dt
_ fe2,
Em 2 (eB/m2)3(1 ( >
-.si)
.
(4)
This, however, means a turning of s against the direction of the magnetic field, the velocity of which is obviously determined by the same functional expression (1) as in the quantum mechanical case. This consideration was performed at the instigation of Prof. F. Bopp. To Dr. H. J. Meister I am indebted for a discussion.
References E. Storck, Diplomarbeit, MUnchen, April 1963 (unpublished). A. A. Sokolov and I. M. Ternov, Soviet Phys. Doklady 8 (1964) 1203. V. N. Baier and V. M. Katkov, Phys. Letters 24A (1967) 327. E. Storck, Dissertation, Universitlit MUnchen (1968) Z. Naturf., to be published.
651
27A, number
10
PHYSICS
A CLASSICAL INTERPRETATION OF ELECTRONS MOVING
LETTERS
7 October
OF THE SPIN-POLARIZATION IN A UNIFORM MAGNETIC
1968
EFFECT FIELD
E. STORCK Sektion Physik der Universith’t Miinchen, Germany Received
23 August 1968
The spin-polarization effect of electrons in a magnetic field is explained as the retroaction of a top oscillating on account of an elementary angular and magnetic momentum.
It has been shown by several authors [l-3] that, in the emission of fast electrons moving in a magnetic field, the probability of the quantum transitions accompanied by a change in the orientation of the electron-spin depends essentially on the initial state of polarization: Beginning with the spin in the direction of the field the spinflip is 24 times as probable as vice versa, so that in the state of equilibrium a beam of electrons should be polarized against the direction of the magnetic field at 92 per cent. The velocity T-l of approaching to this limiting value is of the order of magnitude (we use A = c = 1) T-l
= e2m(E/m)2(eB/m2)3
,
where e means the elementary charge, B the magnetic field and E or m energy or mass of the electron. Subsequently with E = 5 GeV and B = = 1.5 x lo4 Oe we have T = 47 sec. Therefore in a modern storage ring this spin-polarization effect should be measurable by all means. After recently [4] having established this effect quantitatively by a rigorous investigation on the basis of quantum electrodynamics, we wish to note here, that it may easily understood already from a classical point of view: as a retroaction in fact of the emission of a top oscillating in the magnetic field on account of its angular momentum +s, .s2 = 1 and its magnetic moment ,u = = -e s/2m. Let us consider for that purpose the balance of energy in the rest system of the top. In consequence of the emission of the magnetic dipole moment oscillating with the freq!ency w= 2~uB= eB/m in the magnetic field B its potential energy decreases by
d(-,u+
of the emission
= - 4 w~(,u&:/~)~
di.
(2) Here we replace the variables B and dl referring to the rest system, by B and dt referring to the laboratory system, by means of
1 dt’ = J1-v2 dt ,
m:=m/E;
(3)
where in our case we have to put v*B = 0 and E = 0. In this way we obtain for the component sB of the vector s in the direction of the magnetic field dSB -= dt
_ fe2,
Em 2 (eB/m2)3(1 ( >
-.si)
.
(4)
This, however, means a turning of s against the direction of the magnetic field, the velocity of which is obviously determined by the same functional expression (1) as in the quantum mechanical case. This consideration was performed at the instigation of Prof. F. Bopp. To Dr. H. J. Meister I am indebted for a discussion.
References E. Storck, Diplomarbeit, MUnchen, April 1963 (unpublished). A. A. Sokolov and I. M. Ternov, Soviet Phys. Doklady 8 (1964) 1203. V. N. Baier and V. M. Katkov, Phys. Letters 24A (1967) 327. E. Storck, Dissertation, Universitlit MUnchen (1968) Z. Naturf., to be published.
651