PHYSICS OF PLASMAS 16, 063302 共2009兲
Hydrogen atoms in Debye plasma environments
S. Paula兲 and Y. K. Ho
Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 106, Taiwan,
Republic of China
共Received 8 December 2008; accepted 20 May 2009; published online 15 June 2009兲
Plasma-screening effects are investigated on hydrogen atoms embedded in weakly coupled plasmas.
In the present context, bound state wave functions are introduced related to the screening Coulomb
potential 共Debye model兲 using the Ritz variation method. The bound energies are derived from an
energy equation, which contains one unknown variational parameter. To calculate the parameter
numerically, fixed-point iteration scheme is used. The calculated energy eigenvalues for various
Debye lengths agree well with the other available theoretical results. The radial wave functions and
radial probability distribution functions are presented for different Debye lengths. The outcomes
show that the plasma affects the embedded hydrogen atom. © 2009 American Institute of Physics.
关DOI: 10.1063/1.3152602兴
I. INTRODUCTION
The hydrogen atom has special significance in quantum
mechanics and quantum field theory as a simple two-body
problem physical system which yielded analytical solution in
closed form. Many surveys have been conducted on the simplest atom, resulting in the vast accumulation of data and
reports that have been systematically arranged and well
documented in literature. Presently, considerable interest has
been cultivated in the study of atomic processes in plasma
environments,1–14 because of the plasma screening effect on
the plasma-embedded atomic systems. The Debye screening
effect played a crucial and significant part in the investigation of plasma environments over the past several decades.
Different theoretical methods have been employed along
with the Debye model to study plasma environments.11–24
Some progress have been made in estimating the influence of
the plasma on atomic structure, but information on scattering
process and on various radiative processes is very limited.
In our present study, we use the Debye model potential.
We apply the Ritz variational method to calculate the nl state
wave functions and their energy eigenvalues of the hydrogen
atom in plasmas. Earlier the method has been used by Jung25
to calculate a few bound 共1s, 2s, and 2p兲 state wave functions and their eigenenergies of hydrogenic ion. He calculated the value of parameter ␣ 关␣ is az = 共a0 / z兲 linked to the
bound state wave functions of free hydrogen atom兴 by neglecting fourth and higher order terms.25 Here, we employ
the fixed-point iteration method to calculate the value of parameter ␣. Our calculated bound state energy eigenvalues are
more precise than that of Jung.25 Finally, we apply this approach to study hydrogen atom immersed in dense plasmas.
The current context presents the theoretical probe of hydrogen atom in the Debye environments. We present bound
state 共1s to 10m兲 energy eigenvalues of hydrogen atom embedded in weakly coupled plasmas for various Debye
lengths. The deviations of radial wave functions and radial
a兲
Electronic mail: [email protected].
1070-664X/2009/16共6兲/063302/7/$25.00
probability distributions of hydrogen atom in plasma background are also presented.
We describe the article as follows. In Sec. II, we present
the formulation of bound state wave functions, the procedure
of the calculation is presented in Sec. III, results are presented in Sec. IV with a short discussion, and some concluding remarks are found in Sec. V.
II. THEORY
The radial Schrödinger equation for hydrogen atom in
dense plasma would be given by
冋 冉
−
册
冊
ប2 d2 l共l + 1兲
Ze2 −r/␭
e D Pnl共r兲 = Enl Pnl , 共1兲
−
2 −
2
2m dr
r
r
where Pnl共r兲 is the radial wave function for the nlth shell.
The numerical solutions26 and higher order perturbation
calculation27 have been evaluated for Eq. 共1兲. Here we shall
consider a simple analytical method to obtain the solution.
Our approach is the same as the procedure of Jung25 but in a
more general way. Jung calculated for the 1s, 2s, and 2p
states only. The solutions for Eq. 共1兲 are assumed to be in the
hydrogenic form with a variation parameter. The trial wave
function is considered as follows:
Pnl共r兲 ⬅ rRnl共r兲
=
冋
1 共n − l − 1兲!
n ␣共n + l兲!
册冉 冊
1/2
2r
n␣
l+1
2l+1
Ln−l−1
冉 冊
2r −r/n␣
,
e
n␣
共2兲
where ␣ is the variational parameter and
␣ → az
for
␭D → ⬁,
共3兲
az = a0 / z, a0 is Bohr radius, and ␭D → ⬁ indicates plasma free
2l+1
situation. Ln−l−1
is the usual Laguerre polynomial. Substituting the expression of trial wave function into the Schrödinger
equation, we get
16, 063302-1
© 2009 American Institute of Physics
063302-2
Phys. Plasmas 16, 063302 共2009兲
S. Paul and Y. K. Ho
具Enl典 = −
冋
2
1
ប2
2 2 −
2m n ␣
␣
− Ze2
冕
⬁
0
冕
⬁
0
1
Pnl共r兲 Pnl共r兲dr
r
册
1
Pnl共r兲 e−r/␭D Pnl共r兲dr.
r
共4兲
After evaluating the integrations and simplification 共for more
see the Appendix兲, we have
ប2 1
Ze2
2 2 − 2 .
2m n ␣
n␣
具Enl典 =
冉 冊冉
n−l−1
⫻
兺
k=0
冉
1
n␣
1+
2␭D
n+l
n−l−1
k
k
冊
冊冉 冊
IV. RESULTS AND DISCUSSION
2n
n␣
2␭D
2k
共5兲
,
which is the expectation value of the nl state energy of hydrogen atom in plasmas. Equation 共5兲 shows that the energy
level of hydrogen atom in plasmas depends on both n 共the
principal quantum number兲 and l 共the orbital quantum number兲. Parameter ␣ is obtained from the minimization condition of 具Enl典, i.e., 共⳵ / ⳵␣兲具Enl典 = 0, which gives
␣
az =
n␣
1+
2␭D
冉
冤
冊 冉
2n
n−l−1
兺
k=0
+
where
Ak =
共1 − 2k兲Ak
冉 冊冉
冉 冊
␣
␭D
n␣
1+
2␭D
n2
冉 冊
n+l
n−l−1
k
k
n␣
2␭D
冊
n−l−1
冊
2k
兺
k=0
冥
Ak
冉 冊
n␣
2␭D
2k
共6兲
,
共7兲
.
It follows from Eqs. 共2兲 and 共6兲 that the radial wave
functions Rnl共r兲 depend on ␣ as well as the Debye length ␭D.
The radial probability densities are denoted by
Pnl共r兲 = r2兩Rnl共r兲兩2 ,
共8兲
where Pnl共r兲 are functions of n, l, and ␭D.
III. CALCULATION
To calculate the values of ␣ for various Debye lengths,
we rewrite Eq. 共6兲 as
冉
n␣
␣ = az 1 +
2␭D
冊
2n
冤冉
n−l−1
+
兺
k=0
共1 − 2k兲Ak
冉 冊
␣
␭D
n␣
1+
2␭D
n2
冉 冊
n␣
2␭D
2k
n−l−1
冊兺
k=0
冥
Ak
point ␣0 = az 关by the help of condition 共3兲兴. After calculating
the value of ␣ for a particular element of the set 兵n , l , ␭D其 say
共n1 , l1 , ␭D1兲, we calculate the energy level En1l1 from Eq. 共5兲
for ␭D = ␭D1. Aside from the energy level we also calculated
radial wave functions and probability distributions of electron for different Debye lengths.
冉 冊
n␣
2␭D
2k
−1
共9兲
and apply the fixed point iteration method with the initial
The results of applying the very simple numerical technique 共fixed-point iteration兲 and Ritz variational method to
the current problem “hydrogen atom in plasma background”
are excellent. The bound energies of the states 1s through
10m are tabulated in Table I. It will be seen that in the weak
screening regions, the various l levels for a given principal
quantum number lie fairly close together. As the levels approach the continuum, the energy of different l levels varies
significantly with ␭D. The agreement between our results and
available calculated data of Rogers et al.26 共not listed in
Table I兲 is excellent. Rogers et al.26 worked out energy eigenvalues by using a linear difference method and large scale
numerical computation. Earlier, a number of authors employed a variety of techniques to calculate the bound state
energies of an electron in the screened field of a proton.28–34
Harris used first-order perturbation calculation and k-order
k = 共1 , 2 , 3兲 variational calculation varying Z,28 Z is the
nuclear charge. The computed results agree well for the 1s
state, slightly differ for the 2s state, and disagree for the
higher principal quantum number compared with our calculated results. First-order perturbation theory was also applied
by Smith29 to evaluate bound state energy eigenvalues for the
Debye potential. The results match up to three decimal
places with the present results. Lam calculated energies of
1s, 2s, 3s, and 4s states by using perturbation calculation and
one-parameter variation calculation.30 He proposed a simple
variational wave function for the ground state.31 The calculated results tally well with our results unless ␭D is too small.
Dunlap and Armstrong32 formed a basis for an irreducible
unitary representation of SO共2, 1兲. They calculated 1s, 2s,
3s, and 4s states energy eigenvalues for screened Coulomb
potential by using the basis. The computed 1s state results
agree excellent with the present results, but the agreement of
remaining results is not so good. Nauenberg33 described a
procedure to obtain a sequence of wave functions to manage
the problem. Bessis et al.34 applied perturbation calculation
on the basis of Hulthén functions. They computed energy
eigenvalues of 1s, 2s, 3s, 4s, 5s, 2p, 3p, 3d, 4p, and 4d
states. Their calculated results agree nicely with out results
except when ␭D is small. In our previous work, we calculated
energy eigenvalues of a few states 共1s to 4f兲 by using pseudostate method with a new basis set,35 the agreement between present results and those is good.
In Figs. 1–6, we present the radial wave functions of
hydrogen atom for various Debye lengths 共as indicated in the
figures兲 along with that of for free hydrogen atom. The peaks
reduce with the decreasing of Debye length and maximum
deeps appear for plasma free situation.
063302-3
Phys. Plasmas 16, 063302 共2009兲
Hydrogen atoms in Debye plasma environments
TABLE I. Energy levels of nl states for H in the Debye screening plasmas in Rydberg units.
nl \ ␭D
200
150
100
1s
2s
⫺0.990 037
⫺0.240 148
⫺0.986 733
⫺0.236 929
⫺0.980 149
⫺0.230 587
70
⫺0.971 732
⫺0.222 615
50
40
30
20
10
⫺0.960 592
⫺0.212 297
⫺0.950 922
⫺0.203 553
⫺0.934 964
⫺0.189 547
⫺0.903 632
⫺0.163 558
⫺0.814 103
⫺0.100 058
2p
⫺0.240 124
⫺0.236 886
⫺0.230 490
⫺0.222 421
⫺0.211 926
⫺0.202 983
⫺0.188 563
⫺0.161 461
⫺0.092 845
3s
3p
3d
⫺0.101 440
⫺0.101 416
⫺0.101 369
⫺0.098 359
⫺0.098 317
⫺0.098 233
⫺0.092 398
⫺0.092 306
⫺0.092 123
⫺0.085 118
⫺0.084 936
⫺0.084 574
⫺0.076 048
⫺0.075 706
⫺0.075 026
⫺0.068 676
⫺0.068 162
⫺0.067 138
⫺0.057 492
⫺0.056 628
⫺0.054 912
⫺0.038 912
⫺0.037 165
⫺0.033 727
⫺0.008 491
⫺0.003 721
4s
4p
⫺0.053 075
⫺0.053 052
⫺0.050 176
⫺0.050 136
⫺0.044 715
⫺0.044 629
⫺0.038 318
⫺0.038 150
⫺0.030 798
⫺0.030 491
⫺0.025 092
⫺0.024 640
⫺0.017 246
⫺0.016 513
⫺0.007 069
⫺0.005 732
⫺0.000 950
⫺0.003 143
4d
⫺0.053 006
⫺0.050 055
⫺0.044 456
⫺0.037 816
⫺0.029 880
⫺0.023 740
⫺0.015 065
4f
⫺0.052 936
⫺0.049 934
⫺0.044 196
⫺0.037 317
5s
5p
5d
⫺0.030 880
⫺0.030 857
⫺0.030 813
⫺0.028 201
⫺0.028 163
⫺0.028 086
⫺0.023 333
⫺0.023 252
⫺0.023 091
⫺0.017 957
⫺0.017 805
⫺0.017 501
⫺0.028 971
⫺0.012 185
⫺0.022 407
⫺0.008 316
⫺0.012 936
⫺0.004 039
⫺0.011 914
⫺0.011 377
⫺0.007 929
⫺0.007 165
⫺0.003 462
⫺0.002 335
⫺0.000 713
5f
⫺0.030 746
⫺0.027 971
⫺0.022 850
⫺0.017 048
⫺0.010 581
⫺0.006 039
5g
⫺0.030 657
⫺0.027 819
⫺0.022 530
⫺0.016 449
⫺0.009 536
⫺0.004 574
6s
6p
6d
⫺0.019 014
⫺0.018 992
⫺0.018 950
⫺0.016 587
⫺0.016 551
⫺0.016 478
⫺0.012 381
⫺0.012 307
⫺0.012 158
⫺0.008 122
⫺0.007 985
⫺0.007 713
⫺0.004 209
⫺0.003 979
⫺0.003 524
⫺0.002 222
⫺0.001 919
⫺0.001 328
6f
6g
⫺0.018 886
⫺0.018 801
⫺0.016 370
⫺0.016 226
⫺0.011 936
⫺0.011 641
⫺0.007 309
⫺0.006 776
⫺0.002 854
⫺0.001 981
⫺0.000 476
6h
7s
7p
7d
7f
⫺0.018 695
⫺0.012 045
⫺0.012 025
⫺0.011 984
⫺0.011 924
⫺0.016 047
⫺0.009 895
⫺0.009 861
⫺0.009 793
⫺0.009 691
⫺0.011 276
⫺0.006 403
⫺0.006 334
⫺0.006 199
⫺0.005 996
⫺0.006 119
⫺0.003 316
⫺0.003 197
⫺0.002 960
⫺0.002 609
⫺0.000 921
⫺0.001 252
⫺0.001 077
⫺0.000 736
⫺0.000 252
7g
7h
⫺0.011 843
⫺0.011 742
⫺0.009 557
⫺0.009 389
⫺0.005 728
⫺0.005 397
⫺0.002 149
⫺0.001 586
7i
8s
8p
⫺0.011 622
⫺0.007 702
⫺0.007 683
⫺0.009 189
⫺0.005 846
⫺0.005 814
⫺0.005 004
⫺0.003 102
⫺0.003 041
⫺0.000 926
⫺0.001 197
⫺0.001 100
8d
8f
⫺0.007 645
⫺0.007 587
⫺0.005 751
⫺0.005 657
⫺0.002 919
⫺0.002 737
⫺0.000 907
⫺0.000 626
8g
8h
⫺0.007 511
⫺0.007 416
⫺0.005 532
⫺0.005 376
⫺0.002 498
⫺0.002 203
⫺0.000 264
8i
⫺0.007 302
⫺0.005 190
⫺0.001 854
8k
9s
9p
9d
9f
9g
9h
9i
9k
9l
10s
10p
10d
10f
10g
10h
10i
10k
10l
10m
⫺0.007 170
⫺0.004 895
⫺0.004 877
⫺0.004 841
⫺0.004 787
⫺0.004 715
⫺0.004 626
⫺0.004 519
⫺0.004 396
⫺0.004 255
⫺0.003 050
⫺0.003 033
⫺0.002 999
⫺0.002 948
⫺0.002 881
⫺0.002 798
⫺0.002 698
⫺0.002 583
⫺0.002 453
⫺0.002 307
⫺0.004 976
⫺0.003 348
⫺0.003 319
⫺0.003 260
⫺0.003 173
⫺0.003 058
⫺0.002 915
⫺0.002 744
⫺0.002 547
⫺0.002 325
⫺0.001 820
⫺0.001 793
⫺0.001 740
⫺0.001 661
⫺0.001 556
⫺0.001 426
⫺0.001 272
⫺0.001 094
⫺0.000 894
⫺0.000 672
⫺0.001 454
⫺0.001 368
⫺0.001 315
⫺0.001 209
⫺0.001 052
⫺0.000 846
⫺0.000 594
⫺0.000 298
⫺0.000 581
⫺0.000 538
⫺0.000 454
⫺0.000 331
⫺0.000 174
⫺0.001 199
⫺0.000 206
⫺0.000 518
⫺0.000 943
⫺0.000 606
⫺0.000 033
⫺0.000 354
⫺0.000 048
⫺0.000 055
⫺0.000 664
⫺0.000 480
⫺0.000 152
⫺0.000 334
⫺0.000 155
⫺0.000 131
⫺0.000 015
⫺0.000 459
⫺0.000 350
⫺0.000 153
⫺0.000 272
⫺0.000 164
⫺0.000 142
⫺0.000 041
⫺0.000 055
⫺0.000 003
⫺0.000 471
⫺0.000 403
⫺0.000 273
⫺0.000 097
⫺0.000 210
⫺0.000 143
⫺0.000 029
⫺0.000 128
⫺0.000 062
⫺0.000 067
⫺0.000 008
⫺0.000 025
⫺0.000 228
⫺0.000 183
⫺0.000 101
⫺0.000 000 5
⫺0.000 108
⫺0.000 064
⫺0.000 066
⫺0.000 024
⫺0.000 034
⫺0.000 012
063302-4
S. Paul and Y. K. Ho
FIG. 1. 共Color online兲 Radial wave functions of the 1s state for various
Debye lengths.
FIG. 2. 共Color online兲 Same as Fig. 1 but for the 2s state.
FIG. 3. 共Color online兲 Same as Fig. 1 but for the 2p state.
Phys. Plasmas 16, 063302 共2009兲
FIG. 4. 共Color online兲 Same as Fig. 1 but for the 3s state.
FIG. 5. 共Color online兲 Same as Fig. 1 but for the 3p state.
FIG. 6. 共Color online兲 Same as Fig. 5 but for the 3d state.
063302-5
Phys. Plasmas 16, 063302 共2009兲
Hydrogen atoms in Debye plasma environments
FIG. 7. 共Color online兲 Radial probability distribution functions of the 1s
state for various Debye lengths.
FIG. 9. 共Color online兲 Same as Fig. 7 but for the 2p state.
equation satisfies an infinite boundary condition. In Debye
plasma, beyond the Debye length, the electric field will vanish due to the Debye screening.
The first six radial distribution functions plotted against
distance from the nucleus are shown, for different Debye
lengths, in Figs. 7–12. Orbitals 1s, 2p, and 3d have only one
maximum, indicating no node point. All the rest have one or
more minima, which corresponds to nodal spheres. The most
probable 共highest peak兲 distance of the electron from the
nucleus increases in going from higher Debye length to
lower Debye length.
Debye plasma has been considered here, the concept of
the Debye screening is valid only in a steady, thermodynamical equilibrium, and linear plasma. Three assumptions are
used to derive the Debye screening potential. 共1兲 The electron density Ne and ion density Ni are in the steady state and
obey the Boltzmann formula. 共2兲 The derivation of charge
density from its quasineutral value is much less than the
quasineutral value, so the Boltzmann formulas of Ne and Ni
can be expended into Taylor’s series, respectively, and only
the linear terms are taken. 共3兲 The potential in the Poisson
Here, we report an approach to determine bound state
wave functions for screening Coulomb potential, based on
the Ritz variation method. Using a very simple numerical
scheme 共fixed-point iteration兲, we calculated eigenenergies
of a large number of bound states and presented some of
them in Table I. Our calculated energy eigenvalues agree
splendidly with the available computed results of Rogers et
al.26 We also presented the deviation of radial wave functions
and radial probability distribution functions for the hydrogen
atom planted in weakly coupled Debye plasmas. Our results
show that different values of Debye lengths, as well as
plasma environments, have a considerable effect on the hydrogen atom embedded in dense plasmas.
FIG. 8. 共Color online兲 Same as Fig. 7 but for the 2s state.
FIG. 10. 共Color online兲 Same as Fig. 7 but for the 3s state.
V. CONCLUSION
063302-6
Phys. Plasmas 16, 063302 共2009兲
S. Paul and Y. K. Ho
determined by the plasma temperature and density. With the
increase in plasma density at a given temperature, the Debye
length decreases, thus the effect from plasma temperature
and density cannot be neglected. Finally, we mention that
interested readers in dynamic screening effects are referred
to the references where such effects were studied for electron
capture processes37,38 and for constructing dynamic screening potential using the plasmas dielectric functions in a calculation of election capture cross sections.39
ACKNOWLEDGMENTS
The authors would like to thankfully acknowledge financial support by the National Science Council of Taiwan,
ROC.
APPENDIX: EVALUATION OF INTEGRAL
冕
FIG. 11. 共Color online兲 Same as Fig. 7 but for the 3p state.
⬁
0
The dynamic motion of the plasma electrons has to be
considered in order to investigate the plasma screening effect
on the hydrogen atom. It can be considered qualitatively by
the introduction of the plasma dielectric function.36 The effects may be very important for high density plasma, but for
low-density plasma the effect can be neglected. The static
plasma screening formula obtained by the Debye–Huckel
model overestimates the plasma screening effects on the hydrogen atom in dense plasma. It is, indeed, necessary to recalculate all the data in dense plasma on the basis of kinetic
plasma theory, which, in particular, permits to account the
collective plasma effects, namely, dynamic screening along
with plasma fluctuations. The static screening result presented here is subject to the condition that the plasma is a
thermodynamically equilibrium plasma and neglect the contributions from ions in plasma since electrons provide more
effective shielding than ions. In the static plasma screening,
we observed that the two-photon transition amplitude is
mainly determined by the Debye length, which in turn is
1
Pnl共r兲 e−r/␭D Pnl共r兲dr.
r
共A1兲
Substituting the expression of Pnl共r兲 in integral 共A1兲, we
have
1 共n − l − 1兲!
n2 ␣共n + l兲!
冕
⬁
0
1
2l+1
2l+1
共kr兲2l+2Ln−l−1
共kr兲Ln−l−1
共kr兲e−kr e−r/␭Ddr.
r
共A2兲
Using the integration formula,40 we get
冉
冉
1
−1
1 k␭D
n 2␣ 1
+1
k␭D
冊
冊
n−l−1
冢 冣
1
共2l+1,0兲
n+l+1 Pn−l−1
2
k 2␭ D
1
2
k 2␭ D
+1
共A3兲
,
−1
共2l+1,0兲
is the Jacobi polynomial. Applying the
where Pn−l−1
expression41 of the Jacobi polynomial, the above expression
reduces to
1
n 2␣
冉
1
n␣
1+
2␭D
n−l−1
冊
2n 兺
k=0
冉 冊冉
n+l
n−l−1
k
k
冊冉 冊
n␣
2␭D
2k
.
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