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Hydrogenic atoms in “superstrong” magnetic fields
PHYSICSLETTERS
Volume 53A, number 5
HYDROGENIC ATOMS IN “SUPERSTRONG”
14 July 1975
MAGNETIC FIELDS
M.L. GLASSER Department of Applied Mathematics. University of Waterloo, Waterloo, Ontario N2L 3GI, Canada
and J.I. KAPLAN School of Science, Indiana-Purdue University, Indianapolis, Indkana 46205, USA Received 30 May 1975 An estimate is made of the magnetic field above which many recent non-relativistic treatments of hydrogenic atoms in strong magnetic fields become invalid. It is argued that this value may be as low as 101oZ Gauss.
Because strong magnetic fields are believed to exist on certain collapsed astronomical objects such as white dwarfs (lo6 - 107Gauss) and neutron stars (1013Gauss) much attention has been directed recently to the energy levels of atoms under such conditions, especially to the hydrogen atom [l-20]. For the most part all these calculations have ignored relativistic electron spin effects and it is our purpose here merely to estimate at what field such effects become important. The results suggest that the non-relativistic treatment used by the above authors should be reconsidered for fields exceeding 101OGauss. To first order, Foldy-Wouthuysen decoupling of the Dirac equation for an electron in a magnetic field [21] leads to the Hamiltonian
Ro’&
pl=BlotBI’,
W=~s&x,,
&ts*Hte$,
(1) gD=--_
eh2V-E 2m2c2
’
%.o
eR
=- -u*(EXTc.), 4m2c2
qf
=-J-J++p,
wheren=p-(e/c)A, H=VXA,& = - V# and we adopt the gauge 4 = Ze/r, A = (-k/y, 0,O). The three parts.of Bc’ correspond to the Darwin, spin-orbit and relativistic mass correction terms in the zero field case, respectively. We shah simply estimate the first order correction to the ground state energy ofsCo due to the termsin91’ AE = (J/olsI’190). To do this we take as $. the variationally determined ground state wave function given by Yafet et al. [l] $u= (2n)-3/4(&,,)-'12 exp{-(x2+r2)/4az
- z2/4a~jIJ),
(2)
where al, u,, have been determined by these authors as functions of the magnetic field up to 1012Z2G. We use as limits of energy and length %= (mZ2e4/21i2) ,
a0 = (fi2/mZe2) ,
and introduce the dimensionless parameters 7=Ccdu/%
7)= (%?/mc2),
E = al/a,, ,
a straightforward, but tedious calculation gives 373
Volume 53A, number 5
PHYSICS LETTERS
14 July 1975
(AEl’??) = 17{(2a)-1/2(a0/u32(ao/u,,) + & (a&~,)~ [S + 2e2 + 2e4 - &, (~~/a,)~(1 + 3 e2)]> - -yn (2n)-ql-
e2>-qq)/a,,>
1+ + (1 - e2)-l/2,2
In
1 -(l
- e2)1/2
1 t (1 - e2)1/2 - Y217C(a0/a1)2+
Now, sinceq-3Z2X
(a,,laO)2(aO/a1>2(2E
10-5,(al/ao)2
+ e2 - 3>> - y3t7@,/aO)2
lo-‘,e
-3,
3(a&,,)2 II+GQ/& I (3)
- +r477(+O)4
the last line exceeds order unity for
HZ 1O”Z Gauss. If AE is compared with the energy of the ground state at this field (roughly 90 Rydberg units), the correction is negligible. However, a more physical criterion is to compare AE to the energy differences among the first few excited states. For the lowest excited state for which a transition is allowed we estimate this difference [9,11,12] as 3-4 Rydbergs. We conclude therefore that the non-relativistic calculation for atoms in this range of magnetic fields must be reconsidered. The first author wishes to thank the Aspen Center for Physics for its hospitality while this work was being completed. This work was supported in part by The National Research Council of Canada under Grant 50, A9344.
[l] Y. Yafet, R.W. Keyes and E.N. Adams, J. Phys. Chem. Solids 1 (1956) 137. [Z] R.J. Elliott and R. Loudon, J. Phys. Chem. Solids 15 (1960) 196. [3] H. Hasegawa and R.E. Howard, J. Phys. Chem. Solids 21 (1961) 179. [4] S.C. Miller, Phys. Rev. 133 (1964) A1138 [S] R. Cohen, J. Lodenquai and M. Ruderman, Phys. Rev. Lett. 25 (1970) 467. [6] B.B. Kadomtsev, Soviet Phys. JETP 31 (1970) 945. (71 R.O. Mueller, A.R.P. Rau and L. Spruch, Phys. Rev. Lett. 26 (1971) 1136. [8] D. Cabib, E. Fabri and G. Fioro, I1 Nuovo Cimento 10B (1972) 185. [9] J. Callaway, Phys. Lett. 40A (1972) 331. [lo] L.N. Labzowsky and Y.E. Lozovik, Phys. Lett. 40A (1972) 281. [ll] H.C. Praddaude, Phys. Rev. A6 (1972) 1321. [ 121 E. Smith et al., Phys. Rev. D6 (1972) 3700. [13] A. Rajagopal et al., Astroph. J. 177 (1972) 713. [ 141 E. Smith et al., Astrop. J. 179 (1973) 659. [ 151 V.P. Kraimov, J.E.T.P. 64 (1973) 800. [ 161 R. Henry et al., Phys. Rev. D9 (1974) 329. (171 G. Sumelian and R. O’Connell, Astroph. J. (June 15,1974). [ 181 G. Surmelian and R. O’Connell, Astroph. J. (to be published). [ 191 F. Surmelian and R. O’Connell, Astrophysics and Space Science 20 (1973) 85. [20] R.F. O’Connell, Astroph. J. 187 (1974) 275. [21] A. Messiah, Quantum mechanics (North Holland Pub. Co. 1963) Vol. II, p. 946.
374
Volume 53A, number 5
HYDROGENIC ATOMS IN “SUPERSTRONG”
14 July 1975
MAGNETIC FIELDS
M.L. GLASSER Department of Applied Mathematics. University of Waterloo, Waterloo, Ontario N2L 3GI, Canada
and J.I. KAPLAN School of Science, Indiana-Purdue University, Indianapolis, Indkana 46205, USA Received 30 May 1975 An estimate is made of the magnetic field above which many recent non-relativistic treatments of hydrogenic atoms in strong magnetic fields become invalid. It is argued that this value may be as low as 101oZ Gauss.
Because strong magnetic fields are believed to exist on certain collapsed astronomical objects such as white dwarfs (lo6 - 107Gauss) and neutron stars (1013Gauss) much attention has been directed recently to the energy levels of atoms under such conditions, especially to the hydrogen atom [l-20]. For the most part all these calculations have ignored relativistic electron spin effects and it is our purpose here merely to estimate at what field such effects become important. The results suggest that the non-relativistic treatment used by the above authors should be reconsidered for fields exceeding 101OGauss. To first order, Foldy-Wouthuysen decoupling of the Dirac equation for an electron in a magnetic field [21] leads to the Hamiltonian
Ro’&
pl=BlotBI’,
W=~s&x,,
&ts*Hte$,
(1) gD=--_
eh2V-E 2m2c2
’
%.o
eR
=- -u*(EXTc.), 4m2c2
qf
=-J-J++p,
wheren=p-(e/c)A, H=VXA,& = - V# and we adopt the gauge 4 = Ze/r, A = (-k/y, 0,O). The three parts.of Bc’ correspond to the Darwin, spin-orbit and relativistic mass correction terms in the zero field case, respectively. We shah simply estimate the first order correction to the ground state energy ofsCo due to the termsin91’ AE = (J/olsI’190). To do this we take as $. the variationally determined ground state wave function given by Yafet et al. [l] $u= (2n)-3/4(&,,)-'12 exp{-(x2+r2)/4az
- z2/4a~jIJ),
(2)
where al, u,, have been determined by these authors as functions of the magnetic field up to 1012Z2G. We use as limits of energy and length %= (mZ2e4/21i2) ,
a0 = (fi2/mZe2) ,
and introduce the dimensionless parameters 7=Ccdu/%
7)= (%?/mc2),
E = al/a,, ,
a straightforward, but tedious calculation gives 373
Volume 53A, number 5
PHYSICS LETTERS
14 July 1975
(AEl’??) = 17{(2a)-1/2(a0/u32(ao/u,,) + & (a&~,)~ [S + 2e2 + 2e4 - &, (~~/a,)~(1 + 3 e2)]> - -yn (2n)-ql-
e2>-qq)/a,,>
1+ + (1 - e2)-l/2,2
In
1 -(l
- e2)1/2
1 t (1 - e2)1/2 - Y217C(a0/a1)2+
Now, sinceq-3Z2X
(a,,laO)2(aO/a1>2(2E
10-5,(al/ao)2
+ e2 - 3>> - y3t7@,/aO)2
lo-‘,e
-3,
3(a&,,)2 II+GQ/& I (3)
- +r477(+O)4
the last line exceeds order unity for
HZ 1O”Z Gauss. If AE is compared with the energy of the ground state at this field (roughly 90 Rydberg units), the correction is negligible. However, a more physical criterion is to compare AE to the energy differences among the first few excited states. For the lowest excited state for which a transition is allowed we estimate this difference [9,11,12] as 3-4 Rydbergs. We conclude therefore that the non-relativistic calculation for atoms in this range of magnetic fields must be reconsidered. The first author wishes to thank the Aspen Center for Physics for its hospitality while this work was being completed. This work was supported in part by The National Research Council of Canada under Grant 50, A9344.
[l] Y. Yafet, R.W. Keyes and E.N. Adams, J. Phys. Chem. Solids 1 (1956) 137. [Z] R.J. Elliott and R. Loudon, J. Phys. Chem. Solids 15 (1960) 196. [3] H. Hasegawa and R.E. Howard, J. Phys. Chem. Solids 21 (1961) 179. [4] S.C. Miller, Phys. Rev. 133 (1964) A1138 [S] R. Cohen, J. Lodenquai and M. Ruderman, Phys. Rev. Lett. 25 (1970) 467. [6] B.B. Kadomtsev, Soviet Phys. JETP 31 (1970) 945. (71 R.O. Mueller, A.R.P. Rau and L. Spruch, Phys. Rev. Lett. 26 (1971) 1136. [8] D. Cabib, E. Fabri and G. Fioro, I1 Nuovo Cimento 10B (1972) 185. [9] J. Callaway, Phys. Lett. 40A (1972) 331. [lo] L.N. Labzowsky and Y.E. Lozovik, Phys. Lett. 40A (1972) 281. [ll] H.C. Praddaude, Phys. Rev. A6 (1972) 1321. [ 121 E. Smith et al., Phys. Rev. D6 (1972) 3700. [13] A. Rajagopal et al., Astroph. J. 177 (1972) 713. [ 141 E. Smith et al., Astrop. J. 179 (1973) 659. [ 151 V.P. Kraimov, J.E.T.P. 64 (1973) 800. [ 161 R. Henry et al., Phys. Rev. D9 (1974) 329. (171 G. Sumelian and R. O’Connell, Astroph. J. (June 15,1974). [ 181 G. Surmelian and R. O’Connell, Astroph. J. (to be published). [ 191 F. Surmelian and R. O’Connell, Astrophysics and Space Science 20 (1973) 85. [20] R.F. O’Connell, Astroph. J. 187 (1974) 275. [21] A. Messiah, Quantum mechanics (North Holland Pub. Co. 1963) Vol. II, p. 946.
374