Electronic K x rays emitted from muonic atoms: An application of relativistic density-functional theory

概要

Electronic K x rays emitted from muonic atoms: an application of relativistic density
functional theory
X. M. Tong,1, ∗ D. Kato,2, 3 T. Okumura,4, 5 S. Okada,4, 6 and T. Azuma4, †
1

Center for Computational Sciences, University of Tsukuba,
1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
2
National Institute for Fusion Science (NIFS), Toki, Gifu 509-5292, Japan
3
Interdisciplinary Graduate School of Engineering Sciences,
Kyushu University, Kasuga, Fukuoka 816-8580, Japan
4
Atomic, Molecular and Optical Physics Laboratory, RIKEN, Wako 351-0198, Japan
5
Department of Chemistry, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
6
Center for Muon Science and Technology, Chubu University, Kasugai, Aichi 4878501, Japan
We develop a method of the relativistic density functional theory with self-interaction correction,
which is simple and fast yet with reasonable accuracy. Comparison with measured Kα lines and their
hyper-satellites of several atoms, from low-Z to high-Z, reveals that the relativistic local density
approximation is suitable for Kα lines. In contrast, the relativistic local spin density approximation
with self-interaction correction is better for Kαh hyper-satellites. Compared with the non-relativistic
density functional theory, we found that the relativistic effect is significant (about 100 eV) even
for middle Z atoms, like Cu. The screening effects, from inner-shell to outer-shell, and conduction
band, are also discussed. The present work provides all the transition lines of muonic atoms, which
can be used to narrow down the possible transitions by comparing them with the measurements.

I.

INTRODUCTION

Muonic atoms are atoms with an electron replaced by
a negatively charged muon and have been studied extensively [1]. A muon is about 200 times heavier than
an electron, so it moves closer to the nucleus; muonic
atoms thus connect atomic physics and nuclear physics
[2–5]. When a muon is captured to a highly exited state
by an atom, the muonic atom can emit two kinds of x
rays. One is called a muonic x ray due to transitions
between two muon states [6]. The other is called an
electronic x ray due to transitions between two electron
states [7, 8]. X rays emitted from muonic atoms encode
information of the muon state and electron configuration
of exotic atoms. Such information can be decoded by
comparing the measured x-ray energies with the calculated ones. Muonic atoms have also been used to study
the proton size [9], Lamb shift [10], and other higherorder effects [11, 12] as well as a probe of local environments surrounding an exotic atom [13, 14]. The transition energies can be calculated by various methods, from
simple non-relativistic Hartree-Fock-Slater methods [15],
to more complex methods, like the multi-configuration
Dirac-Fock and general matrix elements (MCDFGME)
method [16, 17], which includes the quantum electrodynamics (QED), the Breit interaction, the finite nuclear
size [5], and the vacuum polarization [1, 18]. Due to
the recent development of low-energy muon beams [19]
and new detector technologies [20] the electronic Kα x
rays emitted from muonic atoms can be measured to a

∗
†

tong.xiaomin.ga@u.tsukuba.ac.jp
toshiyuki-azuma@riken.jp

high precision [21]. The most recent reported one was
the measurement of electronic Kα x rays from muonic Fe
atomic ions [8]. These measured Kα x rays are emitted
from different muon states and ionic Fe states. The number of possible transitions is about a quarter million without considering the energy level splitting for the same
electron configuration and different angular momentum
couplings. If we consider all the atomic energy levels, the
number could reach ten million or more. We performed
a simulation based on the non-relativistic density functional theory (DFT) [22] and compared the results with
experiment by shifting the energy systematically. These
shifts are mainly attributed to relativistic effects.
Since similar experiments on other materials and
atoms are being planned, it is necessary to develop a
simple, fast method with reasonable accuracy to calculate all the transition energies. By comparing with the
measured ones, one can narrow down the possible transitions, and then perform a more sophisticated simulation, like the MCDFGME method. For such a goal,
we extended the relativistic density functional theory
with self-interaction correction for atoms [23] to exotic
atoms. This method should, of course, also work for
ordinary atoms. By comparing the measured Kα and
hyper-satellite lines of atoms with ones calculated by various exchange-correlation functionals, we found that the
relativistic local density approximation (LDA) is more
accurate for Kα lines from low to high-Z atoms, while
for the Kαh hyper-satellite, the relativistic local spin density approximation (LSDA) with self-interaction corrections (SIC) works better. Since our goal is to develop a
simple, fast method, we used the simple LDA or LSDA
exchange-correction functional [24, 25], instead of using
a more complicated exchange-correlation functional, like

2
the general gradient approximation (GGA) [26] or metaGGA [27] with tunable parameters. The present method
works for both muonic and electronic x rays.
Since our goal is to provide the electronic K x-ray energies for all charge states in order to identify possible transitions measured in the experiment, we do not present
the muonic x-ray nor the transition rates. The transition rates can be calculated [28] once the transitions are
identified or narrowed down.
We also studied electron screening effects from different
orbits and concluded that the outer screening (M -shell,
N -shell, and conduction band in the solid) are not important. This justifies our use of the isolated muonic Fe
data to compare with the measurements from an Fe foil.
II.

THEORETICAL METHOD

The descriptions of density functional theory with
self-interaction correction for atoms can be found in
Refs. [22, 23]. Here we only present the working equations for exotic atoms. In the relativistic density functional theory, the Dirac equation of the electrons in an
exotic atom is written as (atomic units ~ = me = e = 1
are used unless stated otherwise)


cα · p + βc2 + vef f,σ (r) ψiσ (r) = iσ ψ(r).
(1)
The Dirac equation of the muon in the exotic atom is
written as


cα · pµ + mµ βc2 + vµ (rµ ) ψµ (rµ ) = µ ψµ (rµ ). (2)
Here mµ is the muon mass, c is the light velocity in vacuum, and r, rµ are the electron and muon coordinates,
respectively. The electron moves in an optimized effective potential [29], plus the electron-muon interaction as
Z
ρµ (rµ )
drµ ,
(3)
vef f,σ (r) = V OEP (r) +
|r − rµ |
and the muon moves in an effective potential formed by
the nucleus and the electrons as
Z
ρe (r)
vµ (rµ ) = VN (rµ ) +
dr.
(4)
|r − rµ |
Here, V OEP is the optimized effective potential [29, 30],
which can be obtained as detailed in Refs. [23, 31] and the
last term in Eq. (3) represents the muon-electron interaction. VN is the muon-nucleus interaction, and the last
term in Eq. (4) is the electron-muon interaction. ρe , ρµ
are the electron and muon charge densities, respectively.
Eqs. (1)-(4) are solved iteratively (self-consistently) until
the effective potentials reach convergence. The total energy for a given electron configuration and muon state is
obtained, and the emitted x-ray energy is calculated as
the total energy difference between the transition initial
and final states.
We used the pseudo-spectra grid [32, 33] to discretize
the space for the electron and muon radial wave functions separately due to the mass difference between muon

and electron. If we replace the Dirac equations with the
Schr¨odinger equations, we get the non-relativistic version
[22] of the DFT method for exotic atoms.

III.

RESULTS AND DISCUSSION

Within the DFT method, there are many exchangecorrelation functionals (several tens or hundreds), from
LSDA (LDA) to meta-GGA [34]. Even for the simplest
LDA, one can still consider the self-interaction correction (SIC). Our goal is to find a simple, fast, yet reliable
method to estimate electronic K x-ray energies and its
hyper-satellite energies for muonic atoms. By setting the
muon density to zero, the method should also work for
characteristic K x rays emitted from an atom with one
1s vacancy, which has been studied extensively [35]; this
data, from both theory and experiment, are tabulated in
Ref. [36]. Data also exists [37, 38] for K x-ray hypersatellites emitted from hollow atoms with two 1s vacancies. Therefore, we first compare the characteristic x-ray
energies calculated by DFT methods with LSDA or LDA
with or without SIC to find which exchange-correlation
functional is better for K x rays of atoms. Here we labeled the relativistic DFT with local spin density approximation with self-interaction correction as R-LSDA/SIC
and without SIC as R-LSDA/nonSIC and DFT with local density approximation with SIC as R-LDA/SIC and
without SIC as R-LDA/nonSIC. To study the relativistic
effects, we also present the corresponding non-relativistic
ones labeled as NR-LSDA or NR-LDA. First, we compare
our results with other existing data for the case of ordinary atoms to check the validity of the present method.

A.

K x ray and its hyper-satellite of atoms

Table I lists the reported Kα1,2 , Kβ1 energies for the
low-Z atom Ar (Z = 18), the high-Z atom Rn (Z = 86),
and the spin-polarized atom Eu (Z = 63) and the energy
differences between experiments and calculations with
the relativistic and non-relativistic DFT methods using various exchange-correlation functionals. The hypersatellites Kαh1,2 from Cu [38] and Pb atoms [37] are also
listed. For the theoretical results, we only present the energy difference with the measured one, which is defined
as ∆ = Eexp − Ethe . Overall, the results of the relativistic simulations are better than the non-relativistic
ones: the maximum error for the relativistic simulation
is less than 0.5%, while the non-relativistic results are
in reasonable agreement with the measured ones only
for low-Z atoms, like Ar, as we expected. For high-Z
atoms, like Rn, the relativistic effect is about 10%. Note
that for the non-relativistic simulation, we cannot distinguish Kα1 and Kα2 . Therefore, a relativistic calculation
is needed even for a low-Z atom, like Ar. For high-Z
atoms, the relativistic effects cannot be ignored. For the
non-relativistic simulation, the results are not so sensitive

3
TABLE I. Transition energies of Kα1,2 , Kβ1 for Ar, Rn and Eu atoms and Kαh1,2 hyper-satellites for Cu and Pd atoms from
experiments. The energy differences between the experiments and the simulations are also presented. All the energies are in
eV. The bold font highlights the best performing method for a given case.

Element
Ara (Z = 18)

R-LSDA
SIC
non-SIC
2.29
17.80
2.55
18.04
0.62
18.84

R-LDA
SIC
non-SIC
-15.85
-0.26
-15.58
0.02
-18.53
-0.22

NR-LSDA
SIC
non-SIC
14.33
28.09
12.22
25.98
11.94
28.41

NR-LDA
SIC
non-SIC
-4.05
9.76
-6.16
7.65
-7.45
9.08

Kα1
Kα2
Kβ1

Eexpt.
2957.68
2955.57
3190.49

Rna (Z = 86)

Kα1
Kα2
Kβ1

83783
81066
94867

-315
-282
-319

11
37
31

-363
-328
-371

-36
-10
-21

10508
7791
10503

10580
7863
10595

10430
7713
10420

10502
7785
10512

Eua (Z = 63)

Kα1
Kα2
Kβ1

41542.63
40902.33
47038.40

-96.17
-83.57
-103.60

45.43
55.73
53.30

-142.67
-129.67
-151.60

-0.77
10.03
5.50

2588.43
1948.13
2568.30

2640.53
2000.23
2634.50

2530.23
1889.93
2504.80

2582.33
1942.03
2571.10

Cub (Z = 29)

Kαh1
Kαh2

8352.60
8329.10

-2.35
-1.85

29.87
30.31

26.28
26.57

58.39
58.51

107.82
84.32

131.62
108.12

137.48
113.98

161.22
137.72

Pbc (Z = 82)

Kαh1

76250±60

-133

157

80

210

8876

8945

8953

9022

a
b
c

Ref. [36]
Ref. [38]
Ref. [37]

to SIC or non-SIC for high-Z atoms because the relativistic effect is the major error source. For low-Z atoms, the
results of LSDA/SIC are better for ionization potentials,
as reported in the previous work [23]. Indeed, the NRLDA/SIC results for Ar atoms are the best among the
non-relativistic simulations. Thus, we may think that
the results of the relativistic SIC should also be better
than the non-SIC ones. To our surprise, the K x-ray
energies calculated with R-LDA without SIC are better
than the ones with SIC. For all of the K x rays in the
table, the maximum relative error of R-LDA/nonSIC is
less than 0.05%. For a very heavy atom, like Rn, the RLDA/nonSIC and R-LSDA/nonSIC results are comparable. A possible reason is that there are two 1s electrons in
the transition final state, thereby, the spin average is better than the self-interaction correction, or there are some
cancellations between the relativistic effect and the selfinteraction correction. The original motivation for introducing SIC [39] is to correct for the long range Coulomb
tail, which significantly improves the orbital energies [22]
while only moderately improving the total energies [31].
We use the total energy difference to calculate the transition energies, not the orbital energies. Also, the SIC
corresponds to a single-electron correction, and its relative contribution is smaller for many-electron atoms or
high-Z atoms. This could be another possible reason that
R-LDA/nonSIC is better for the K x rays. For a given
transition, the number of 1s electrons in the final state

plays an important role.
To check this scenario, we studied the K x-ray hypersatellites (the final state having only one 1s electron)
where the local-spin density approximation with selfinteraction correction plays a crucial role. In the RLSDA/SIC, we assume that the electron spin-flip is forbidden and the transition happens only among the same
spin states. Indeed, we found that the R-LSDA/SIC is
better for the K x-ray hyper-satellites of Cu. For heavy
atoms, like Pb (Z = 82), both R-LSDA/SIC and RLDA/SIC are better than the others. For such a heavy
atom, the Breit interaction [40], which is about several
tens of eV [41, 42], is ignored in the present simulation.
To confirm the reliability of the R-LSDA/SIC, Table II
lists the Kαh hyper-satellite energies of the 3d transition
metals reported in Ref. [38] and the values of the relativistic multi-configuration Dirac-Fock (RMCDF) methods
[41, 44, 45] and the active space approximation (ASA)
methods [43]. To focus on the reliability of the simulations, we present the experimental results along with the
differences between the measured ones and simulations
as ∆ = Eexp − Ethe .
For Kαh2 x rays, our results are comparable with other
simulations, and the errors are less than 5 eV for all elements in Table II. For Kαh1 x rays, the results of Ref. [43]
are better than ours, but the largest error of our results is
still less than 10 eV. Overall, our results are comparable
with RMCDF-type simulations, but numerically much

4
TABLE II. Kαh1,2 hyper-satellite energies of the 3d transition metals reported in Ref. [38]. The energy differences between the experiment and present simulations (∆p ) with RLDA/SIC and the differences between the experiment and the
results of RMCDF (∆R ) [38] or ASA (∆A ) [43] methods are
also presented. All energies are in eV.
Element
V
Cr
Mn
Fe
Co
Ni
Cu
Zn

Kαh2
5176.6
5649.2
6143.4
6659.7
7194.4
7752.3
8329.1
8929.5

∆p
0.89
0.56
1.00
2.53
1.10
1.43
-1.85
-1.17

∆R
1.50
0.70
2.80
4.00
3.10
3.00
-0.20
0.60

∆A
0.8
2.7

0.6

Kαh1
5191.7
5665.1
6160.9
6678.8
7214.9
7774.1
8352.6
8955.8

∆p
7.27
5.96
5.92
6.69
4.00
2.62
-2.35
-2.65

∆A

µ-state
1s
5s
10s
a

0.9

Present
0.0
13.4
119.8

NR-LDAa
0.0
12.5
114.9

NR-HF[15]
0.0
11.9
112.9

RMCDF
-0.6
12.6
118.3

Note that the results are slightly different from the values in
the Supplement of Ref. [8] due to that values were calculated
with NR-LSDA/SIC.

2.9

0.4

simpler and faster. The present DFT method is equivalent to the level-average of RMCDF without calculating
all of the energy levels for the same electron configuration and different total angular momenta. We need to
calculate three total energies for the Kαh1 , Kαh2 transition
energies: one initial and two final states. It took less than
a minute with a desktop computer. The present method
is highly parallelized for use with a many-core computer
or even a supercomputer. Thus we can estimate the K
x-ray energy quickly.
Note that the above numerical results depend weakly
on the simulation parameters, such as the box size and
the number of grids. The results may change within a
few eV for heavy atoms if we reduced the box size and
the number of grids by half. In the present simulation,
we did not tune these parameters and used a box size
of Rmax = 20 a.u. with the number of grids N = 320.
We found that the R-LDA/nonSIC is better for K x rays
and R-LSDA/SIC works better for K h hyper-satellites.
Therefore, in the following discussion, all the results are
obtained with the R-LDA/nonSIC for muonic atoms or
ions since only the electronic Kα transitions of muonic
Fe were reported in the recent experiment [8]. Of course,
our method should also work for Kαh hyper-satellites.
B.

TABLE III. Energy shifts (in eV) of Kα lines of µFe from the
Kα of Mn atoms calculated with R-LDA/non-SIC.

Energy shifts of electronic K x ray of muonic Fe

The muon is initially captured in highly excited states
peaked at a principal quantum number of about n ≈
√
mµ ≈ 14 for low incident energies close to the threshold; the peak moves to higher n as the incident energy
increases [46, 47]. Then the muon gradually cascades
into lower excited states through Auger decay by removing electrons from Fe atomic ions or through radiative
decay by losing energy via emitting a photon. The cascade process has been studied in Refs. [48, 49]. During
the cascading, various charged hollow ions are formed.
We first analyze the transition energy shifts of K x rays

with different muon states for muonic Fe (µFe) with only
one K-vacancy. There is no high-precision experimental
data for muon state-specified electronic K x rays. Thus
we use the MCDFGME [8] data as references. The energy shifts are almost the same for Kα1,2 , and we do
not distinguish the two lines. Table III lists the Kα energy shifts of muonic Fe atoms with respect to the Kα
line of Mn atoms. As the muon state nµ becomes lower,
the screening effect of the muon on the nucleus becomes
more significant, and at nµ = 1, Kα lines of µFe becomes
identical to Kα of Mn atoms. Overall, our results are in
reasonable agreement with the non-relativistic HartreeFock results [15], although the Kα energies may differ due
to relativistic effects. Our results are also in agreement
with MCDFGME [8], in which QED and other higherorder effects are considered. The discrepancies between
the present simulations and MCDFGME ones are less
than 2 eV.

C.

Electronic Kα x rays from muonic Fe ions

There are various type of µFe ions with different charge
states formed during the cascade process. In the solid
state, the vacancies of the hollow ions can be refilled by
electrons in the conduction band. We need to consider
all muon states from n = 1 to n = 12, which is 144 states
if all possible j states are considered. The probability
for a muon captured into a high-n (> 12) state with a
K-vacancy is negligibly small. For spin-averaged simulations with LDA, the number of electron configurations
of the L-shell with the electron number varying from 0
to 8 is 45, which is the same for the M -shell. Therefore,
the total transition lines are 144 × 45 × 45 = 291, 600,
roughly a quarter million. If we consider all energy levels
with the same electron configuration and different total
angular momenta, the number can easily reach for ten
million or even more. This is our motivation for a simple, fast method to estimate Kα energies of muonic atoms
and ions.
Since the experiment was performed with an Fe foil,
we need to check if the conduction band electrons affect
the x-ray energies emitted from hollow muonic Fe ions
produced in the metal. We modified the method employed to study x ray emission from hollow nitrogen ions

5

12

µ Fe

10

10

8

8
nµ

nµ

12

6

6
4

4
R-LDA with
NR-LDA with
NR-LDA with

2
0

µ Fe

5850

5900

5950

[Ar]3d 6 4s2
[Ar]3d 6 4s2
[Ar]3d 0 4s0
6000

6050

nL = 8, nM = 8
nL = 8, nM = 0
nL = 1, nM = 8

2
0

6100

5900

6000

Kα Energy (eV)

FIG. 1. Kα energies of µFe as a function of muonic states
(nµ , lµ , or jµ ) calculated by the LDA/nonSIC methods. For
the relativistic simulation, only Kα1 lines are presented.

q+

N inside of metal in Ref. [50] to study muonic atomic
ions in a metal. For Nq+ ions in Al metal, the emitted
x-ray energies differ from the corresponding emission in
vacuum. The energy shifts also depend on the charge
state q. Unlike Nq+ hollow atoms, x-ray energies from
µFe hollow ions in bulk are almost the same as the ones
in vacuum. By further analysis, we confirmed that the
screening effect of electrons in the conduction band on
µFe is attributed to an outer screening that does not affect the Kα transition energies. This is different from the
case of Nq+ hollow atoms in bulk, in which the screening
affects the initial state of the transition. Therefore, all of
the following µFe results are calculated for isolated µFe
atomic ions.
Figure 1 shows the electronic Kα (or Kα1 for the relativistic case) energies of µFe ions with different muon
states determined by the relativistic and non-relativistic
LDA simulations. To show the outer screening effect,
we plot the data of µFe ions where all the valence electrons in 3d, 4s orbits are removed for the non-relativistic
LDA simulation. For a given principal quantum number
nµ , there are still many possible jµ or lµ states, which
results in spreading of the energy. Note that the outer
screening effect becomes negligibly small by removing the
eight valence electrons. This screening effect should be
larger than the screening from the electrons in the conduction band. This comparison justifies using µFe data
to compare with the ones measured in solid [8]. We also
see that the relativistic effect shifts the non-relativistic
results systematically, and the shifts weakly depend on
the muon states. The non-relativistic shift of this calculation is 47 eV, and the value of MCDFGME is 50
eV [51]. This justifies the procedure in Ref. [8], where we
used the non-relativistic results and shifted by a constant
value systematically.
Figure 2 shows Kα1 energies of µFe ions with different
muon states determined by the relativistic LDA simulations for changing the number of electrons in M - and

6100

6200

6300

Kα1 Energy (eV)

FIG. 2. Kα1 energies of µFe as a function of muonic states
(nµ , jµ ) with different number of M, L-shell electrons calculated by the relativistic LDA method.

L-shells. Comparing Kα1 with a full 3s2 3p6 and without
3s, 3p electrons, the screening effect of the 3s, 3p can be
as large as 20 eV. This is not negligibly small, and the
electron configuration in the M -shell should be taken into
account for a high-precision simulation. Comparing the
results with the full L-shell electrons and one L-shell electron, we found the energy shifts can be as large as 200
eV. This shows that the L-shell electron is important for
the Kα transition, and the electron configuration in the
L-shell must be considered. The different muon states
also contribute to the Kα energy, and the measured xray energy encodes the information of the muon states
and the electron configurations of µFe ions.
As we mentioned that there are about a quarter million
Kα lines in the simulation, we do not present all of them
but use the data to compare with the measurements. All
the data are available from the corresponding author,
XMT, upon request.

IV.

CONCLUSION

We developed a relativistic density functional theory
to study electronic Kα x-ray energies of muonic atoms.
By comparing the available Kα and its hyper-satellite energies calculated with various exchange-correlation functionals for various atoms, from low to high-Z atoms, we
conclude that (1) while relativistic effects are obviously
important for high-Z atoms, which are known qualitatively without simulation, even for middle-Z atoms relativistic simulations are necessary; (2) the relativistic local density approximation is suitable for Kα while the
relativistic local spin density approximation with selfinteraction correction is suitable for hyper-satellites. We
also studied the screening effect from electrons in the
inner-shell/outer-shell orbits and the conduction band.
We found that the outer screening effect from the conduction band and valence electrons are negligibly small
(about a few eV for muonic Fe ions). In contrast, the

6
inner-shell (L-shell) screening effect is comparable to or
even larger than the energy shifts from different muon
states. Such a simple, fast, reasonably accurate method
can be used to calculate all K x-ray energies, either
muonic x rays or electronic x rays. The number of the
transitions can reach about a quarter million, even million depending on the targets. Comparing the simulations by this method with experiments, one can narrow
down the possible transitions and then study the specific
transitions with a more elaborate method, like MCDFGME with QED and other higher-order effects.

tational Sciences, University of Tsukuba. XMT was
supported by the JSPS KAKENHI (Grant-in-Aid for
Scientific Research (C) 22K03493. TA, TO and SO
were supported by the JSPS KAKENHI (Grant-in-Aid
for Scientific Research on Innovative Areas, Toward
new frontiers: encounter and synergy of state-of-theart astronomical detectors and exotic quantum beams
18H05457 and 18H05458, Grant-in-Aid for Scientific Research (A) 18H03714, and Grant-in-Aid for Young Scientists 20K15238), and the RIKEN Pioneering Projects.
We are grateful to Prof. P. Indelicato for providing MCDFGME data.

ACKNOWLEDGMENTS

This work was supported by the Multidisciplinary
Cooperative Research Program in Center for Compu-

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