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Spin rearrangement of atoms in strong magnetic fields
Volume 40A, number 4
PHYSICS LEUERS
31 July 1972
SPIN REARRANGEMENT OF ATOMS IN STRONG MAGNETIC FIELDS LN. LABZOWSKY and Yu.E. LOZOVIK Institute of Spectroscopy, Academy of Sciences of the USSR, Moscow, USSR
Received 12 June 1972
The process of spin rearrangement of the ground state of an atom in a strong magnetic field is discussed. This rearrangement leads to increasing the spin of an atom. Estimates are made for the helium atom and for a heavy atom in the Thomas-Fermi approximation.
The possibility of the existence of superstrong magnetic fields in astrophysics [1—2] attracts one’s attention to the study of the properties of matter in such fields. Recently [3—5] the electron density distribution in a heavy neutral atom was derived as a function of the fieldH for superstrong fields H~~Z4 ‘i’. In a field H>>Z~the atom in its ground state looks as follows: all the electrons occupy the lowest Landau level and have spins directed opposite to the magnetic field, We shall study weaker fields 1
2e3c/h3 = 2.35X i09 gauss forH, and We employ the units m atomic units for all other quantities.
In the process of the spir rearrangement of the many-electron atom the spin S of an atom increases step by step as H increases; this rearrangement begins from the outer shells and goes to the inner shells. The electrons from the occupied states after transition to the unoccupied states flip their spins opposite to the field. Consequently, the electron density distribution n+(r) for the electrons with the spins oriented in the direction of the field, is more compact, than the distribution n(r) for the electrons with the opposite spins. The evaluation of the distributions n~(r)for the Thomas-Fermi atom gives a finite radius for n+(r) and infinite radius for n(r) in accordance with the quanti. tative consideration above. Since the open shells in the absence of a magnetic field have the maximal spin after Hund’s rule, the spin rearrangement for this shells is similar to the rearrangement of the closed shells: in both cases the electron configuration changes in the process of the rearrangement. We consider now the Thomas-Fermi model for a heavy atom. We derive the condition of the validity of the usual 3-dimensional Thomas-Fermi model to our problem from the inequality TH v/H>)’d, where i’H is the Larmor radius and d is the mean distance between electrons in a heavy atom. Since v ~F Z~and d —Z~ the Fermi momentum), we obtainH~Z~.On the other hand, we must require H ~ I, because only the outher shell rearranges in a field H 1, and the Thomas-Fermi model is not applicable in the last Thus consider the region 1 <
‘~
tions n~(r)by the variation
of the functional
281
Volume 40A, number 4
PHYSICS LETTERS
E[n÷,n] =K f(nt + n~)dr—ZIrn—dr J r
tegral we put n
+~f~)ncr’)dTdTP+~Hf(n+~~n )dr Ir—r under the condition
f n dr
=
Z, where ~
=
(1)
2
~7rn~
— = 2~TI’Nurnercical integration gives us an universal functionf(h)~S/Z, h =HZ~, which increases monotonically as h increases and gives the ri~tlimits S l(H~ I) andS ~Z(H~Z~). The
details of the calculations, as well as the astrophysical applications, will be published elsewhere.
+ n,
and K =(3/l01r2)(67r2)+. After the variation the energy becomes a function of the radii r~of the two distributions. Minimizing this function, we obtain n~(r~)=0. Repeating Jensen’s arguments [7], we can prove that r = 00, but r÷ is finite. The Euler equations for our variational problem are: 5
V+~H+ X = 0
(2)
V-- ~H+X=0
(3)
where V is the potential, X is the Lagrangian multiplier. Requiring, that V(r) = 0, we find X H/2. Then the value of r~may be found from the condition V(r÷) =H. Approximately we may use the condition VTJ(r÷)=H, where VTF is the usual Thomas-Fermi potential. We evaluateS with the aid of the formula:
We thank Prof. V.M. Agranovich, Prof. V.L. Ginzburg, and Prof. B.B. Kadonitsev for the valuable discussions. Some helpful comments were made by Prof. ~ilic. Conversations with M.A. Mechtiev and Dr. V.1. Petviaschwii were also stimulating. References (1] V.L. Ginzburg, Doklady 156 (1964)1, 43; Uspekhi 103 (1971) 393. 21 i.E. Gunn and J.P. Ostriker, Nature 221 (1969) 454; P. Goldreich and W. Julian, Astrophys. J. 157 (1969) 869. 131 B.B. Kadomtsev, Zh. Eksp. Teor. Liz. 58(1970)1765; B. B. Kadomtsev and VS. Zh. Eksp. i Teor. Hz., Pis’ma 13 (1971) 61 Kudrjavtscv, Zh Eksp iTeor Hz 62 (1972) 144
141 R.O. Mueller. A.R.P. Rau and L. Spruch, Phys. Rev. Lctt.
151 r~
S
=f
(n
—
n~)dr+
The difference n
f
n dT.
(4)
—
n.j. in the first integral may be cal-
culated by the perturbation theory; in the second in-
282
26 (1971) 1139. R. Cohen, 1. Lodenquai and M. Ruderman, Phys. Rev. I.ett. 25 (1970) 467.
161 HA. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer-Verlag. Berlin, 1957).
r~
0
31 July 1972
171
P. Gonibas, Die statistischc Theorie des Atoms und jhre Anwendungen (Wien, 1949).
PHYSICS LEUERS
31 July 1972
SPIN REARRANGEMENT OF ATOMS IN STRONG MAGNETIC FIELDS LN. LABZOWSKY and Yu.E. LOZOVIK Institute of Spectroscopy, Academy of Sciences of the USSR, Moscow, USSR
Received 12 June 1972
The process of spin rearrangement of the ground state of an atom in a strong magnetic field is discussed. This rearrangement leads to increasing the spin of an atom. Estimates are made for the helium atom and for a heavy atom in the Thomas-Fermi approximation.
The possibility of the existence of superstrong magnetic fields in astrophysics [1—2] attracts one’s attention to the study of the properties of matter in such fields. Recently [3—5] the electron density distribution in a heavy neutral atom was derived as a function of the fieldH for superstrong fields H~~Z4 ‘i’. In a field H>>Z~the atom in its ground state looks as follows: all the electrons occupy the lowest Landau level and have spins directed opposite to the magnetic field, We shall study weaker fields 1
2e3c/h3 = 2.35X i09 gauss forH, and We employ the units m atomic units for all other quantities.
In the process of the spir rearrangement of the many-electron atom the spin S of an atom increases step by step as H increases; this rearrangement begins from the outer shells and goes to the inner shells. The electrons from the occupied states after transition to the unoccupied states flip their spins opposite to the field. Consequently, the electron density distribution n+(r) for the electrons with the spins oriented in the direction of the field, is more compact, than the distribution n(r) for the electrons with the opposite spins. The evaluation of the distributions n~(r)for the Thomas-Fermi atom gives a finite radius for n+(r) and infinite radius for n(r) in accordance with the quanti. tative consideration above. Since the open shells in the absence of a magnetic field have the maximal spin after Hund’s rule, the spin rearrangement for this shells is similar to the rearrangement of the closed shells: in both cases the electron configuration changes in the process of the rearrangement. We consider now the Thomas-Fermi model for a heavy atom. We derive the condition of the validity of the usual 3-dimensional Thomas-Fermi model to our problem from the inequality TH v/H>)’d, where i’H is the Larmor radius and d is the mean distance between electrons in a heavy atom. Since v ~F Z~and d —Z~ the Fermi momentum), we obtainH~Z~.On the other hand, we must require H ~ I, because only the outher shell rearranges in a field H 1, and the Thomas-Fermi model is not applicable in the last Thus consider the region 1 <
‘~
tions n~(r)by the variation
of the functional
281
Volume 40A, number 4
PHYSICS LETTERS
E[n÷,n] =K f(nt + n~)dr—ZIrn—dr J r
tegral we put n
+~f~)ncr’)dTdTP+~Hf(n+~~n )dr Ir—r under the condition
f n dr
=
Z, where ~
=
(1)
2
~7rn~
— = 2~TI’Nurnercical integration gives us an universal functionf(h)~S/Z, h =HZ~, which increases monotonically as h increases and gives the ri~tlimits S l(H~ I) andS ~Z(H~Z~). The
details of the calculations, as well as the astrophysical applications, will be published elsewhere.
+ n,
and K =(3/l01r2)(67r2)+. After the variation the energy becomes a function of the radii r~of the two distributions. Minimizing this function, we obtain n~(r~)=0. Repeating Jensen’s arguments [7], we can prove that r = 00, but r÷ is finite. The Euler equations for our variational problem are: 5
V+~H+ X = 0
(2)
V-- ~H+X=0
(3)
where V is the potential, X is the Lagrangian multiplier. Requiring, that V(r) = 0, we find X H/2. Then the value of r~may be found from the condition V(r÷) =H. Approximately we may use the condition VTJ(r÷)=H, where VTF is the usual Thomas-Fermi potential. We evaluateS with the aid of the formula:
We thank Prof. V.M. Agranovich, Prof. V.L. Ginzburg, and Prof. B.B. Kadonitsev for the valuable discussions. Some helpful comments were made by Prof. ~ilic. Conversations with M.A. Mechtiev and Dr. V.1. Petviaschwii were also stimulating. References (1] V.L. Ginzburg, Doklady 156 (1964)1, 43; Uspekhi 103 (1971) 393. 21 i.E. Gunn and J.P. Ostriker, Nature 221 (1969) 454; P. Goldreich and W. Julian, Astrophys. J. 157 (1969) 869. 131 B.B. Kadomtsev, Zh. Eksp. Teor. Liz. 58(1970)1765; B. B. Kadomtsev and VS. Zh. Eksp. i Teor. Hz., Pis’ma 13 (1971) 61 Kudrjavtscv, Zh Eksp iTeor Hz 62 (1972) 144
141 R.O. Mueller. A.R.P. Rau and L. Spruch, Phys. Rev. Lctt.
151 r~
S
=f
(n
—
n~)dr+
The difference n
f
n dT.
(4)
—
n.j. in the first integral may be cal-
culated by the perturbation theory; in the second in-
282
26 (1971) 1139. R. Cohen, 1. Lodenquai and M. Ruderman, Phys. Rev. I.ett. 25 (1970) 467.
161 HA. Bethe and E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer-Verlag. Berlin, 1957).
r~
0
31 July 1972
171
P. Gonibas, Die statistischc Theorie des Atoms und jhre Anwendungen (Wien, 1949).