Atomic data for the spectral analysis of magnetic DA white
dwarfs in the SDSS
C. Schimeczek
1
and G. Wunner
Institut für Theoretische Physik 1, Universität Stuttgart, Germany
ABSTRACT
We present binding energies of hydrogen states in magnetic fields B = 0 to
B≈ 5 · 108 T, relevant for the modeling of the atmospheres of magnetic white
dwarfs and neutron stars. We analyzed states with magnetic quantum number
m = 0, . . . , −4 and included states up to a principal quantum number n = 15
for a total of 300 states. Due to a sophisticated data point selection we can
interpolate any energy value for each of these states with an absolute precision of
at least 10−6 Ry for any magnetic field strength. Further, and also as a function of
the magnetic field strength, we present dipole strengths of all transitions between
these states relevant for the Lyman, Balmer, Paschen and Brackett series, with
similarly high precision.
Subject headings: hydrogen, Lyman, Balmer, Paschen, Brackett, magnetic field
1.
Introduction
The observation of over-luminous type Ia supernovae (Scalzo et al. 2010) and their
explanation by explosions of magnetic white dwarfs (MWD) with masses 2.1–2.8 M , significantly above the Chandrasekhar limit (Das & Mukhopadhyay 2013), has high-lighted the
need for an even better understanding of these peculiar stellar remnants which have been
studied intensively over the last decades (Wickramasinghe & Ferrario 2000). High precision
spectroscopy of 521 magnetic white dwarfs in the Sloan Digital Sky Survey (SDSS), with
fields from 1 MG to 733 MG (Kepler 2012), has created a need for more, and more accurate,
data for wavelengths and transition strengths of the hydrogen atom in strong magnetic fields.
It is the purpose of this paper to meet this need. The existing data in the literature are
confined to levels emerging from the field-free states with principal quantum numbers n ≤ 5.
1
Corresponding author
–2–
We will extend the data base to states with principal quantum numbers n ≤ 15. With these
data at hand it should become possible to explain new absorption features in MWD spectra
(Kepler et al. 2013) that cannot be explained with the available atomic data, possibly due
to low magnetic field strengths (Landstreet et al. 2012).
The problem of a hydrogen atom in a magnetic field of arbitrary strength has attracted
the attention of theoretical physicists over several decades (for a review see, e.g., Ruder
et al. (1994)). The specific difficulty lies in the fact that, as the magnetic field increases,
there is a switch-over from the spherical symmetry of the Coulomb field to the cylindrical
symmetry of the magnetic field. In particular in the transition regime the effects of the fields
are of the same order of magnitude, and the Hamiltonian of the atom becomes manifestly
non-integrable in two degrees of freedom. Therefore, to obtain energies and wave functions
one has to resort to numerical methods.
The most comprehensive compilation so far of energies and transition strengths can be
found in Ruder et al. (1994), and in the past these data were used to pin down the field
strengths of MWDs with the help of ”stationary lines”, i.e. wavelengths that go through
a minimum or maximum when the magnetic field is varied. The data also served as input
for powerful codes developed to perform model atmosphere calculations for MWDs (Jordan
1992; Friedrich & Trost 2004; Euchner et al. 2002) and even for such objects with complex
structured surface magnetic fields (Beuermann et al. 2007). The spectra of quite a number
of MWDs have been analyzed so far (Külebi et al. 2009), and absorption features could be
identified and related to the surface magnetic field strengths. The tables in Ruder et al.
(1994) are restricted to the 38 lowest-lying levels, at discrete values of the magnetic field
strength. Other groups calculated energies with very high accuracy (Wang & Hsue 1995;
Kravchenko et al. 1996), but presented results only for a basic selection of states.
The spectral resolution from the SDSS (York et al. 2000) of roughly 5 Å at wavelengths
of 9200 Å corresponds to an energy resolution of about 5 · 10−5 Rydberg energies for the
hydrogen states. We will present the most complete compilation to date for 300 low-lying
hydrogen states, calculated with a precision of 10−6 Rydberg. We restrict ourselves to a nonrelativistic description as relativistic effects on the energies of bound states of the hydrogen
atom are smaller than the above mentioned spectral energy resolution (Chen & Goldman
1992). Also, we do not account for the motional Stark effect and consider a non-moving atom
here. This is appropriate, as for typical temperatures in magnetic white dwarf atmospheres
the influence of the motional Stark effect on the line spectra can be included via profile factors
and transition wavelength shifts (cf. Faßbinder & Schweizer (1995)). For a description of
the moving hydrogen system in magnetic fields, or the consideration of additional electric
microfields, see, e.g. Potekhin (1994); Vincke et al. (1992).
–3–
2.
Method
We give only a short overview of our method since an exhaustive description has been
given in Schimeczek & Wunner (2014). That paper also explains and provides a link to
the source code of the program used to create the data to be presented in this work. The
Hamiltonian of an electron in the fixed Coulomb potential of a hydrogen core with infinite
mass and in a static magnetic field pointing in z-direction is (cf., e.g., (Ruder et al. 1994))
2
∂
1 ∂2
∂
2
1 ∂
∂2
Ĥ = −
+ 2 2 + 2 − 2iβ
+ β 2 ρ2 + 4βms −
,
(1)
+
2
∂ρ
ρ ∂ρ ρ ∂φ
∂z
∂φ
|r|
where atomic Rydberg units and cylindrical coordinates are used to reflect the symmetry of
the problem. Here, the magnetic field strength β = B/B0 is measured in units of the reference
magnetic field strength B0 ≈ 4.70103 × 105 T. At B0 the quantum length characterizing the
gyration of an electron in a magnetic field, i.e. the Larmor radius, becomes equal to the
Bohr radius, and magnetic and Coulomb field effects are of the same order, and therefore in
competition.
In the presence of a magnetic field the angular momentum l is no longer a good quantum
number. Thus, the conserved quantities that define a symmetry subspace are reduced to the
magnetic quantum number m, the spin projection ms and the z-parity πz . Individual states
in a symmetry subspace are labeled by the excitation number ν. Each state defined by these
high-field quantum numbers corresponds to a state in the weak-field limit with the wellknown quantum numbers n, l, m and ms . The correspondence scheme for these two sets of
quantum numbers was found by Simola & Virtamo (1978), who exploited the non-crossing
rule for energy functions of states in the same symmetry subspace. These quantum number
correspondences can be found in the following way: The z-parity of a state in the weakfield limit is given by πz = (−1)l−m and m and ms are identical in both sets of quantum
numbers. To determine the excitation number ν one applies some simple rules: Even at
high β the states keep their order with respect to n within a symmetry subspace, i.e. states
with higher ν correspond to weak-field states with higher n, whereas states with higher l
correspond to lower ν. Table 1 gives examples for the quantum number correspondences of
several symmetry subspaces.
For the rest of this paper we will use the format (m,πz ,ν) to label a state while we restrict
ourselves to ms = −1/2 and m ≤ 0, since the energy of spin-up states or states with positive
magnetic quantum number m > 0 can be obtained by adding 4β or 4mβ, respectively. For
transitions we use a colon to separate initial and final state. It is convenient to stick to
the simplification of infinite nuclear mass, as the resulting energies E, wave functions ψ and
dipole strengths d are related to the hydrogen atom with finite proton mass via the scaling
–4–
relations (cf. Pavlov-Verevkin & Zhilinskii (1980); Ruder et al. (1994); Becken et al. (1999))
ψ(mp , β, r) =λ−3/2
ψ(mp → ∞, βλ2m , r/λm ) ,
m
2
E(mp , β) =λ−1
m E(mp → ∞, βλm ) − 4β(m + ms )(λm − 1) ,
dqif (mp , β)
=λ2m dqif (mp
→
∞, βλ2m ) ,
(2)
(3)
(4)
with λm = (1 + me /mp ), the electron to proton mass ratio me /mp , and the polarization
index q, which corresponds to the difference of the magnetic quantum numbers of the initial
and the final state.
The wave function ψ is described using a 2D expansion in the (ρ–z)-plane in terms of
B-splines Bµ (de Boor 1972) on a finite element grid as
eimφ X
√
Ψ(r) =
αµν Bµ (ρ)Bν (z).
2π µν
(5)
In comparison with an expansion in terms of spherical harmonics (Zhao & Stancil 2007),
which is a natural choice in the regime of low field strengths, or an expansion in terms of
Landau orbitals (Ruder et al. 1994) in high fields, this expansion proved most efficient for the
description of the wave functions at arbitrary field strengths (Wang & Hsue 1995; Schimeczek
& Wunner 2014). We vary the energy functional corresponding to the Hamiltonian (1) with
respect to the B-spline coefficients αµν and obtain the generalized eigenvalue problem
X
X
αµν Hµνχξ = ε
αµν Sµνχξ ,
(6)
µν
µν
where H and S denote the Hamilton and overlap matrices, respectively. Standard LAPACK
(Anderson et al. 1999) routines are employed to obtain the energy eigenvalues and wave
functions. We take special care to guarantee the precision of our results: By monitoring the
wave functions’ decay behavior at large ρ and z we can select the B-spline basis widths in
such a way that we do account for all important non-zero contributions of the wave function.
Additionally, we increase the number of B-splines until convergence to the desired energy
precision is achieved.
To describe the wave functions and energies of the hydrogen states at arbitrary magnetic
field strengths one needs a smart interpolation scheme. We developed an automated magnetic
field strength variation scheme (AMV) which adapts the number and positions of the field
strength data points individually according to each state’s energy function in dependence on
β. This is superior to equidistant or exponential data point distribution schemes, which do
not account for the rapid changes of the wave functions and energies of states in the vicinity
of anticrossings, whereas the adapted data point scheme allows for an interpolation of the
–5–
energy values at any given field strength with a precision of 10−6 Ry. Typically about 1000
data points are necessary to cover the complete energy function of a state over the full range
β = 0–103 . The adapted distribution of data points, however, also causes a drawback. To
calculate dipole strengths between an initial and final state, one of course needs both states
at the same magnetic field strength. However, our AMV algorithm chooses the magnetic field
strengths individually for each state, and the situation where data for initial and final state
exist at the same B, is a rare exception. A conceptually simple remedy for this situation
would be to calculate all states at the same field strengths, but this would lead to a very
large overhead of calculations. Instead, we make use of the fact that the wave functions
vary continuously with β, even at avoided crossings, and further exploit that a change in the
wave function of a state comes along with a change of the state’s energy value. Under this
assumption the field strength selection of the AMV is also appropriate for the calculation of
dipole strengths, up to quite high magnetic field strengths. We apply a linear interpolation
0.7
exact
interpolated
lower
higher
0.6
∆d1
d
0.5
0.4
∆β1
0.3
∆d1 = 1.3 · 10−3
∆d2 = 5 · 10−6
∆β1 = 0.3
∆β2 = 2.2 · 10−3
0.2
0.1
∆d2
0
1
2
∆β2
3
4
5
6
7
8
9
β
Fig. 1.— (Color online) Dipole strength d of the transition (0,+,1):(0,−,1) in dependence on
β. The interpolated results (blue) deviate only in the seventh digit (second inset) from the
values without interpolation (red). Also shown are the preliminary dipole strengths obtained
with wave functions corresponding to lower (green) and higher (cyan) field strengths. Lines
serve as a guide to the eye.
scheme for the calculation of a transition’s dipole strength: For each field strength β i , where
the initial state was calculated at, we search the nearest lower β1f and higher β2f field strength,
where results exist for the final state. We then calculate the dipole strengths d for both final
–6–
states with the initial one, and interpolate d to identical field strengths β f = β i . After
handling all field strengths of the initial state, we proceed with the field strengths of the
final state in the same manner and afterwards join the results of both calculations. This
ensures that changes of the wave functions of both states are dealt with adequately.
Figure 1 shows the interpolated results (solid blue line) and the intermediate results
needed for interpolation. Green points correspond to results from β f < β i and cyan ones
to β f > β i . For comparison we also calculated both states at the same field strengths to
obtain dipole strengths without the need of any interpolation (red crosses), which are labeled
“exact”. The excellent agreement of both the interpolated and non-interpolated values of
the dipole strength proves that this method works well and that it is possible to retrieve
accurate dipole strengths using the AMV for data point distribution.
3.
Results
In Fig. 2 we show the energies of the 30 lowest states with m = −1 and πz = +1 as
a function of the magnetic field strength. This comprises all states emerging from the field
free levels with principal quantum numbers up to n = 10. The ground state in the symmetry
subspace is a tightly-bound (tb) state, the binding energy of which diverges logarithmically
in the limit B → ∞ (Avron et al. 1981), whereas the other states are hydrogen-like (hl) and
form a Rydberg series again in the same limit (Loudon 1959).
The Figure demonstrates that in the transition regime B ≈ 10 − 100 MG excited states
originating from levels with n ≥ 6 undergo numerous close avoided crossings. This is even
more evident in Figure 3 where only the transition regime and highly excited levels are
shown. The reason for the avoided crossings is again that for these states in that regime
the switch-over from Coulomb field to magnetic field dominance occurs. (We note that for
Rydberg states n ∼ 40 this behavior can even be seen in laboratory magnetic fields, cf.
(Friedrich & Wintgen 1989)).
In Fig. 4 we show the magnetic field strength versus the transition wavelengths of several
lines from the Balmer series. The range of the magnetic field strength is chosen to cover
typical field strengths of MWDs (Vanlandingham et al. 2005). At low B the lines split into
Zeeman triplets corresponding to the three possible quantum number differences between
initial and final state q := ∆m = 0, ±1 in the dipole approximation. The wavelengths of
transitions that correspond to q = +1 transitions to states with positive m rapidly drop
to zero for high magnetic fields, whereas those of transitions with q = −1 from states with
m > 0 diverge λ → ∞, at intermediate magnetic fields. Both effects are caused by the linear
–7–
−3
−4
log β
−2
−3
−1
0
log −E [Ry]
−2
−1
0
−1
0
1
2
3
4
log B [MG]
Fig. 2.— Energy E as a function of the magnetic field strength B for the lowest 30 states in
the symmetry subspace m = −1, πz = +1. Colors are used to distinguish different energy
levels (color online). Minor B-axis ticks are placed at 2, 5 and 8 times the corresponding
major tick.
–8–
ν
0+
0−
1+
1−
2+
2−
1
2
3
4
5
6
7
8
9
10
1s0
2s0
3d0
3s0
4d0
4s0
5g0
5d0
5s0
6g0
2p0
3p0
4f0
4p0
5f0
5p0
6h0
6f0
6p0
7h0
2p1
3p1
4f1
4p1
5f1
5p1
6h1
6f1
6p1
7h1
3d1
4d1
5g1
5d1
6g1
6d1
7i1
7g1
7d1
8i1
3d2
4d2
5g2
5d2
6g2
6d2
7i2
7g2
7d2
8i2
4f2
5f2
6h2
6f2
7h2
7f2
8j2
8h2
8f2
9j2
Table 1: Correspondences of high-field and weak-field quantum numbers. The first row and
column give the high-field quantum numbers in the format m πz and ν, whereas the body of
the table contains corresponding weak-field states in the common format nlm .
1
2
−E [mRy]
3
5
7
10
20
30
10
20
30
B [MG]
50
70
100
Fig. 3.— Detail of Figure 2 showing numerous avoided crossings of levels emerging from
field free states principal quantum numbers n ≥ 6 (color online).
–9–
4
log B [MG]
3
2
1
0
3500
4500
5500
λ [Å]
6500
7500
Fig. 4.— Magnetic field strength as a function of the wavelength of the first 325 transitions
in the Balmer series, which emerge from the field-free Balmer transitions up to principal
quantum numbers n = 10.
energy shift of 4mβ for all states with positive magnetic quantum numbers in relation to
their counterparts with negative m. One can also recognize stationary line components (see,
e.g. Angel (1978); Wunner et al. (1985)) of several transitions in the intermediate regime of
the magnetic field strength B ≈ 102 − 103 MG.
In Tabs. 2 – 5 we compare energies and dipole strengths of important tb-tb transitions
with results taken from Baye et al. (2008), who used a Lagrange-mesh method and spherical or semi-parabolic coordinates. We show 10 tb-tb transitions from (0,+,1):(−1,+,1) to
(−9,+,1):(−10,+,1) in Tabs. 2 and 3 and find that our energy precision is as high as we
expected: 7 digits. Some energy values of Baye et al. (2008) at β = 500 differ from ours
in Tab. 3. It must be noted, however, that the precision of the results given by Baye et al.
(2008) rapidly decreases at high field strengths. Besides those outliers, our dipole strengths
agree with the results of Baye et al. (2008) by at least 8 digits. This is positively surprising,
as we rely only on the energy values to check the convergence of our calculations.
We also compare with the results of Baye et al. (2008) in Tabs. 4 and 5 and, in the latter
one, additionally with values of Forster et al. (1984), who published the most complete data
set for hydrogen transitions in the literature so far. Their data include finite nuclear mass
corrections except for dipole strengths, thus we restrict ourselves to the comparison of dipole
– 10 –
strengths with Forster et al. (1984). They did not present data for states with principal
quantum number n > 3, thus no results of Forster et al. (1984) are shown in Tab. 5. The
transitions listed in both tables are of the type tb-hl or hl-hl and are part of the Lyman,
Balmer, Paschen, or Brackett series.
As found previously, our energy values deviate only in the case of high magnetic fields,
i.e. β = (50, 500) from the energies given in the literature. Note that for most states Baye
et al. (2008) presented no results at β = 500, presumably due to convergence problems.
The dipole strengths do not agree as perfectly as in the previous tables, but we still find
most states to have 7-8 significant digits in common with the values from Baye et al. (2008).
Deviations are larger in rare cases only and are limited to high field strengths again, where
the results of Baye et al. (2008) suffer from a reduced energy precision as well. Our results
agree with Forster et al. (1984) in the first three significant digits. The reduced accuracy is
due to the fact that their calculations were performed using truncated spherical or Landau
expansions, and therefore are only approximate. All comparisons demonstrate that we have
reached an excellent accuracy, not only for the energy values but also for the dipole strengths
of the transitions.
An impressive demonstration for both the efficiency of our dipole strength calculation
scheme and the AMV data point selection algorithm can be seen in Fig. 5, which shows the
dipole strength of the transition from the ground state to a highly excited state with many
avoided crossings. There, even the smallest features of the dipole strength function d(β) can
be resolved, as is shown by the insets.
3.1.
Improved spectral plots
The high line density at lower wavelengths in the B versus λ plot of Fig. 4 can hamper
the identification of possible absorption features. For the higher Paschen and Brackett
series, numerous lines that differ only marginally in wavelength exist, and this problem
is aggravated. The combination of both wavelengths and line strengths into a single plot
improves clarity and allows for a fast identification of the important line features at any B.
As oscillator strengths f obey sum rules and can be converted to absorption cross sections
(see, e.g. Seaton (1987)) we use these quantities to represent the line strengths. Dark colors
represent high values of f , or strong transitions, and light colors correspond to low values
of f , or weak transitions, respectively. In Fig. 6 we present the Balmer series in the same
manner as in the previous picture, and in the same parameter ranges as in Fig. 4. The
bunch of higher excited lines below 4000 Å and between 10 MG and 100 MG are clearly
weaker than the Hα and Hβ lines (beginning near 6500 Å and 4800 Å), which dominate in
– 11 –
∆E
transition
(0,+,1):(−1,+,1)
(−1,+,1):(−2,+,1)
(−2,+,1):(−3,+,1)
(−3,+,1):(−4,+,1)
(−4,+,1):(−5,+,1)
d
β
this work
Baye et al. (2008)
this work
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.7491437
1.2447496
2.3100871
4.0480043
0.2070981
0.4344152
0.8931868
1.6666208
0.1061581
0.2408911
0.5163217
0.9934801
0.0675620
0.1594855
0.3503001
0.6877212
0.0479903
0.1160647
0.2592639
0.5166204
0.7491436766184
1.244749643740
2.3100871420
4.048004334
0.2070980665482
0.4344151326806
0.89318683468
1.666620774
0.1061581093086
0.2408911847390
0.51632165674
0.99348012
0.0675619985856
0.1594854970340
0.350300169462
0.68772108
0.0479903346252
0.1160646343654
0.259263875392
0.51662082
0.501548538
0.0858207408
0.0095906859
0.0009847157
1.6257462598
0.1900476417
0.0197197002
0.0019901842
2.6937746178
0.2918350349
0.0297727575
0.0029923515
3.7339389430
0.3928736119
0.0398031538
0.0039935558
4.7610248616
0.4935743088
0.0498234242
0.0049943386
Baye et al. (2008)
0.5015485403
0.085820740633
0.009590685913
0.000984715683
1.625746259709
0.190047641716
0.019719700164
0.001990184250
2.693774614343
0.291835034653
0.029772757547
0.002992351487
3.733938912828
0.392873611786
0.039803153811
0.003993555797
4.761024874987
0.493574308548
0.049823424158
0.004994338592
Table 2: Energies ∆E and dipole strengths d in Rydberg atomic units for tightly-bound
transitions at high β compared with recent results in the literature.
– 12 –
∆E
transition
(−5,+,1):(−6,+,1)
(−6,+,1):(−7,+,1)
(−7,+,1):(−8,+,1)
(−8,+,1):(−9,+,1)
(−9,+,1):(−10,+,1)
d
β
this work
Baye et al. (2008)
this work
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.0364341
0.0895997
0.2027038
0.4086942
0.0289251
0.0720204
0.1645873
0.3350801
0.0237135
0.0596180
0.1373856
0.2820181
0.0199174
0.0504685
0.1171277
0.2421633
0.0170485
0.0434824
0.1015351
0.2112586
0.0364340808404
0.0895997130168
0.20270375298
0.40869364
0.0289251519534
0.0720203995592
0.16458736470
0.3350814
0.0237134872722
0.059617997502
0.1373855808
0.2820170
0.0199174080016
0.050468516186
0.1171276474
0.2421658
0.0170485088226
0.043482362916
0.1015351488
0.2112566
5.7808773687
0.594089110
0.0598381731
0.0059948959
6.7962493128
0.6944887016
0.0698495286
0.0069953169
7.8086200700
0.7948109926
0.0798586247
0.0079956487
8.8188637127
0.8950783904
0.0898661273
0.0089959183
9.8275337063
0.9953051121
0.0998724566
0.0099961429
Baye et al. (2008)
5.780877372661
0.594089109541
0.059838173063
0.00599489585
6.796249321741
0.694488701441
0.069849528609
0.00699531692
7.80862013370
0.794810992722
0.079858624721
0.00799564864
8.81886369967
0.895078390800
0.089866127402
0.0089959184
9.8275336917
0.995305112192
0.099872456665
0.0099961429
Table 3: Energies ∆E and dipole strengths d in Rydberg atomic units for tightly-bound
transitions at high β compared with recent results.
– 13 –
−3
−4
log d
−5
−6
−7
−8
−9
−10
−3
−2
−1
0
log B [MG]
1
2
3
Fig. 5.— Dipole strength d of the transition (0,+,1):(0,−,30) as a function of B. The
complex structure of the graph is caused by the upper state which passes through several
avoided crossings. Insets show a detailed view of the first, very small-banded anticrossings,
including the data points (color online).
– 14 –
4
5
4
3
2
2
− log(f /fmax )
log B [MG]
3
1
1
0
3500
0
4500
5500
λ [Å]
6500
7500
Fig. 6.— (Color online) Wavelengths of the Balmer series as in Fig. 4 but color coded by
their respective oscillator strengths.
– 15 –
4
5
3
4
2
3
1
2
0
1
−1
− log(f /fmax )
log B [MG]
oscillator strength over the entire range of β. Note that the dense lines at λ . 4000 Å are
weak and do not contribute significantly to the spectrum. The stationary components of
the Lyman-α and -β lines, however, have a high dipole strength over the full range of the
magnetic field strength.
0
0
10000
20000
30000
40000
50000
λ [Å]
Fig. 7.— (Color online) Wavelengths of the Lyman, Balmer, Paschen and Brackett series
including thermal weighting for T = 20, 000 K. The line colors correspond to the oscillator
strengths.
An overview of the strongest transitions from the Lyman, Balmer, Paschen and Brackett series is displayed in Fig. 7. The combination of the series into a single plot suggests
thermal weighting for the initial states, which is why we multiply the oscillator strengths by
a Boltzmann weight
w = e−(Ei −Eg )/(kB T) ,
(7)
corresponding to the energies Ei and Eg of the initial and ground state. We selected a typical
temperature of 20, 000 K, which is a reasonable choice for MWDs (cf. Kepler et al. (2013)).
Despite this Boltzmann factor, even the lines from the Brackett series stay clearly visible
and quite intense up to intermediate field strengths, due to the rather high thermal energy
– 16 –
compared to the excitation energy of the initial states. At larger field strengths, the effects of
thermal excitation become visible: all transitions but those from the ground state (Lyman
series) fade out, which is caused by the well-known logarithmic divergence of the ground
state energy at large B.
4.
Conclusion
We presented energies, dipole strengths and oscillator strengths of hydrogen transitions
important to the modeling of MWD spectra. Extensive comparison with the literature proves
that we have achieved a six-digit accuracy of the calculated transition energies and dipole
strengths for all investigated magnetic field strengths. We calculated transition strengths for
the Lyman, Balmer, Paschen and Brackett series for states up to principal quantum numbers
n = 15 over the full range of the magnetic field strength B = 0 – 103 T, relevant for magnetic
white dwarfs or neutron stars. Further, we combined the energy and transition strength
data to find a compact representation of the line data that is useful for the identification of
spectral features.
The associated computer program is available via the URL given in Schimeczek &
Wunner (2014) so that every astronomer in the field of magnetic DA white dwarfs can
calculate the data required for the interpretation of his or her data by him- or herself.
Further, all data calculated so far can be downloaded from our website at http://itp1.unistuttgart.de/institut/mitarbeiter/schimeczek/h2db/index.php.
The authors would like to thank John Landstreet for helpful comments on this article.
This work was supported by bwGRiD (http://www.bw-grid.de/), a member of the German
D-Grid initiative.
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This preprint was prepared with the AAS LATEX macros v5.2.
– 19 –
∆E
transition
(0,+,1):(0,−,1)
(0,−,1):(0,+,2)
(−1,+,1):(0,+,2)
(−1,−,1):(−1,+,1)
(−1,−,1):(0,−,1)
(−1,+,2):(0,+,2)
(−1,+,2):(0,+,1)
(−1,+,2):(−2,+,1)
(−1,+,2):(−1,−,1)
β
this work
0.5
5
50
500
0.5
5
50
500
0.5
5
50
500
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
1.1423246
2.7302946
6.6523729
14.3398565
0.1990752
0.3473961
0.4148724
0.3932751
0.5922561
1.8329411
4.7571582
10.6851273
0.5000594
1.5729323
4.3837790
0.1068785
0.0873873
0.0414932
0.0699795
0.0533006
0.0428578
1.4113793
3.1309913
7.1101031
0.4551375
1.4518265
3.9068292
0.1621762
0.3133094
0.4162370
Baye et al. (2008)
1.1423245615790
2.7302946308172
6.6523729439898
14.3398570529642
0.1990752666196
0.347396038526
0.41487238846
0.393318
0.5922561515806
1.83294102560
4.7571581900
10.685170
0.50005938912638
1.5729323040730
4.3837790458
0.1068785041660
0.087387317002
0.0414932442
0.0699794756
0.053300668730
0.04285778
1.4113793038
3.13099133810
7.11010312
0.4551375606
1.45182656162962
3.90682914
0.16217623808070
0.31330939024004
0.416237028
d
this work
Baye et al. (2008)
0.5901316089
0.2246670349
0.0621460120
0.0158531753
7.9694837072
4.3715764996
3.3171608053
3.0554854835
0.0854715825
0.0012965569
0.0000332168
0.0000009848
1.7275501912
0.4772969856
0.1222156237
0.8703732787
0.0981570787
0.0099813979
0.8189677854
0.0959952311
0.0098677543
0.0082376074
0.0005187584
0.0000207508
0.0294804058
0.0007164006
0.0000211632
10.6799990883
5.0827007631
3.4851901912
0.59013161498
0.2246670334
0.0621460109
0.01585325
7.9694836333
4.3715765681
3.31716083
3.055609
0.08547157589
0.0012965574214
0.0000332168265
0.0000009853305
1.727550194
0.477296986
0.122215623
0.8703732714
0.098157077652
0.009981397847
0.818967796
0.0959952296538
0.009867754
0.00823760725
0.0005187584226
0.00002075083
0.02948040627
0.0007164006011
0.000021163182
10.679999056
5.0827007537
3.4851901
Forster et al. (1984)
0.5902
0.2252
0.06217
0.01585
7.944
4.371
3.317
3.055
0.08129
0.001290
0.00003318
0.0000009846
1.728
0.4773
0.1222
0.8704
0.09816
0.009981
0.8190
0.09601
0.009868
0.008238
0.0005169
0.00002073
0.02948
0.0007161
0.00002116
10.68
5.083
3.485
Table 4: Energies ∆E and dipole strengths d in Rydberg atomic units for transitions at high
β.
– 20 –
∆E
d
transition
β
this work
Baye et al. (2008)
this work
(−2,−,1):(−2,+,1)
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
0.5
5
50
0.3461527
1.1915152
3.5208175
0.0531914
0.0529981
0.0302253
0.0278627
0.0260014
0.0222445
0.6900983
1.9122431
4.8222605
0.3768421
1.2369368
3.4127520
0.1368475
0.2863127
0.4082562
0.34615270934706
1.191515217472
3.5208174680
0.0531913867690
0.0529980460832
0.0302252578
0.0278626883564
0.0260014316
0.0222458
0.6900983155632
1.9122431258
4.8222618
0.3768421397066
1.2369368084
3.4127532
0.13684753966790
0.286312774618
0.40825624
2.6261074153
0.6735775630
0.1694470246
1.8734114524
0.1980555125
0.0199774636
1.8893737464
0.1973477142
0.0199126876
0.0078113897
0.0004166955
0.0000161847
0.0171736445
0.0005168126
0.0000164132
12.9601817835
5.6785287950
3.6318264550
(−2,−,1):(−1,−,1)
(−2,+,2):(−1,+,2)
(−2,+,2):(−1,+,1)
(−2,+,2):(−3,+,1)
(−2,+,2):(−2,−,1)
Baye et al. (2008)
2.6261074103
0.673577564
0.169447025
1.8734114506
0.198055512058
0.019977463603
1.8893737425
0.197347713627
0.019912691
0.00781138940
0.0004166954691
0.00001618473
0.01717364
0.0005168126918
0.00001641326
12.960181745
5.678528757859
3.631829
Table 5: Energies ∆E and dipole strengths d in Rydberg atomic units for transitions at high
β compared with recent results.