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Puska, M. J.; Nieminen, R. M.
Theory of hydrogen and helium impurities in metals
Published in:
Physical Review B
DOI:
10.1103/PhysRevB.29.5382
Published: 15/05/1984
Document Version
Publisher's PDF, also known as Version of record
Please cite the original version:
Puska, M. J., & Nieminen, R. M. (1984). Theory of hydrogen and helium impurities in metals. Physical Review B,
29(10), 5382-5397. https://doi.org/10.1103/PhysRevB.29.5382
PHYSICAL
REVIE%
8
VOLUME
29,
NUMBER
10
Theory
of
hydrogen
and helium
impurities
in
metals
15 MAY 1984
M.
J.Puska
Laboratory
of
Physics,
Helsinki
University
of
Technology,
SF
02-150
Espoo,
Finland
R.
M. Nieminen
Department
of
Physics,
Uniuersity
of
Jyuaskyla,
SF
4010-0
Jyuaskyla,
Finland
(Received
10 November 1983)
A
powerful
computational
scheme is
presented
for calculating
the
static
properties
of
light
inter-
stitials in metallic
hosts.
The
method entails
{i)
the
construction
of the
potential-energy
field
using
the
quasiatorn
concept,
(ii)
the
wave-mechanical
solution of the
impurity
distribution
(
zero-point
motion"
),
(iii)
calculation
of the
forces exerted on the
adjacent
host atoms and
their
displacements,
and
(iv)
iteration to self-consistency.
We
investigate
self-trapping phenomena
in bcc and fcc metals
in
detail,
and calculate both the
ground
and
low-lying
excited states.
Implications
of the
wave-
mechanical
or
band
pictuxe
to
diffusion
mechanisms
and
inelastic
scattering experiments
are
dis-
cussed.
Impurities
treated
are
p,
H, D, T,
and
He,
and
particular
attention is
paid
to
isotope
ef-
fects
among
the
hydrogenic
impurities.
It is
argued
that
especially
for
p+
and
H the
quantum
na-
ture of
the
impurity
is
crucial. The calculated results are in
agreement
with
a
wealth
of experimen-
tal
data.
I. INTRODUCTION
The behavior
of
light
impurities
(notably
helium,
the
hydrogen
isotopes,
and
positive
muons) remains a
subject
of
considerable
interest
in
metal
physics.
It
is a compli-
cated
subject
for
two
reasons.
Firstly,
the
impurity
pro-
vides
a
large
perturbation
for the
host-metal
electrons.
This leads to a
strong
electronic interaction
which
may
be
basically repulsive
in
nature,
as
for
inert
gases,
or
a
more
involved
"hybridization"
or
"bonding-antibonding"
in-
teraction
for
chemically active
species
such as
hydrogen.
Secondly,
the
Impurity
couples
strongly
to
the
host-Ion
coordinates,
which leads to
polaron-type
lattice
distor-
tions.
The
wave
mechanics
of
the
degrees
of freedom of
the light-mass
impurity
is
important
in
this context. Both
of
these
aspects
must enter
into
any
satisfactory
descrip-
tion of
impurity-related
phenomena:
ground-state
elec-
tronic
structure,
energetics
and site
assignment,
lattice
dis-
placements
in the
self-trapped
state,
local-mode
excita-
tions,
diffusion-related
effects,
etc.
The treatment
of
the electronic structure
of
impurities
in solids has shown remarkable recent
progress.
Tech-
niques
for self-consistent
ab
initio calculations
have been
developed
with
increasing computational
power
and
accu-
racy.
However, even
within
the local-density
approxima-
tion
for
exchange and correlation,
such
calculations are
still
time
consuming
and
costly,
and
only
feasible
mainly
for
relativdy
simple,
high-symmetry
situations.
However,
the
progress
in this
area has
made
it
possible
to create
simpler
schemes more
easily adaptable
to
complicated
low-symmetry
situations.
A
particularly interesting
ap-
proach
is
the
effective-medium or
quasiatom theory
pro-
posed
independently
by
Stott
and Zaremba
and Ngrskov
and
Lang.
In it one
formulates
the
electronic interaction
between the
impurity
and the host
in
terms
of
the
ground-state
electron
density
in the
unperturbed
host
and
some
coupling properties
of the inserted
impurity
atom.
The
latter can
be
calculated once
and
for
all for
any
given
atom,
whereafter the electronic interaction terms are
nu-
merically
easy
to evaluate
for
any
system
where a
reason-
able
description of
the host electron
density
can
be
found.
As will
bc dlscusscd
below,
thc
pcrturbatlve
interaction
terms can be calculated
relatively
accurately
for inert
atoms,
but the
situation
is less
satisfactory
for
hydrogen.
For
the latter
no
systematic
expansion
of the interaction
exists
and the
approach
is
partly
guided
by
physical
intui-
tion. It
must
thus
be
remembered that while
the
potential-energy
surfaces
thus obtained are
qualitatively
correct, quantitative accuracy
in
the interaction
energies is
significant
only
at the
level of
about 0.2
—
0.
5 eV.
Nevertheless,
the
simplicity
of the
scheme makes further
work in
improving
the
description
very
much
worthwhile:
It
makes it
possible
to
investigate
the trends in
large
classes of
complicated systems
where
more
rigorous ap-
proaches
are
at
present
all
but
impossible.
The
applica-
tions discussed so far include
hydrogen
heats
of
solution
ln
transition
metals,
hydI
ogcn
at
nlckcl
and other
transition-metal
surfaces,
as well as
hydrogen
at
defects
in
nickel,
trapping
of
helium
in
metal vacancies,
and
helium
surface
scattering.
'
Very
recently,
Daw
and
co-
workers"'
have
applied
a
related
semiempirical
scheme
in
their
treatment of
hydrogen
embrittlement
and hydro-
gen trapping
to interstitial
impurities.
The
other
aspect
of
light
impurities
interacting
with
solids has
received much
less
attention
in the
form of
quantitative
calculations.
A
straightforward
route to
fol-
low
would be
to
describe
the
impurity-host
and
host-host
interactions
by
(semi-)
empirical
pair
potentials,
and
search
for
equilibrium
relaxations
using
either computer-
simulation
or lattice
—
response-function
techniques.
'
However,
a
pair-potential
description
is
of
limited use for
mcta11ic
systems,
where
vo1umc-
of
density-depcndeIlt
tcrills 111 tlic total
cilcl"gy
al'c
1mportant.
M01'covcl',
cvc11
if
satisfactory
pair
potentials
could
be
defined,
these
1984
The
American
Physical Society
5383
methods
completely
ignore
the
zero-point
motion
of the
impurity,
which
can be
important.
An
extreme
example
is
the case of a
positron,
where the
quantum
kinetic
ener-
gy
dominates and dictates
the basic features of
diffusion
and
trapping
at
defects.
In
important
recent publica-
tions,
'
Sugimoto
and
Fukai
have considered
the
quantum-mechanical
nature
of
hydrogenic
impurities
in
some
bcc
transition metals. Their
approach
is
based
on
empirical metal-hydrogen
pair
potentials,
but
includes
a
proper
account
of the
zero-point
effects,
which can be
substantial.
In another recent
publication,
Casella'
has
suggested
that
the
hydrogen
states excited
in neutron
scattering
are
wave-mechanical
band
states,
and shows
this to be in accord with
experimental findings.
Further
evidence
for
the
quantum
nature
is
provided
by,
e.
g.
,
the
observation'
of
a
tunnel-split
oscillator
ground
state
for
hydrogen
in
oxygen-doped
Nb.
In
this
paper
we
seek
to
combine
the electronic,
lattice-
relaxation,
and zero-point
effects
into
a
unified
theory
of
light
impurities
in metals.
%'e
describe the electronic
in-
teraction in
terms of
the
effective-medium
theory,
and
calculate
a
potential-energy
field
of an
impurity
in a
given
host-ion
configuration.
In
practice,
this is facilitated
by
constructing
the host electron
density
as a
superposition
of atomic densities.
The
Schrodinger
equation
for the
impurity-mass
coordinate is then
solved
numerically.
The
forces
exerted
on
the host atoms
by
the
impurity
are
cal-
culated,
and the
ensuing
lattice relaxations
are evaluated
by
Green-function
techniques.
A new
electron-density
map
and
potential-energy
field are
then
constructed,
and
the
process
is
iterated
to
self-consistency.
The
main
difference
with the Sugimoto-Fukai approach' is the
potential-energy
construction, which we
carry
out without
any
adjustable
parameters.
We
present
a number of
appli-
cations
for
both
bcc
and
fcc
metals
concerning
the
self-
trapping
of
helium,
hydrogen
isotopes,
and muons
in the
interstitial
regions.
We also
discuss
the excited-state
characteristics
and the diffusion mechanisms. While the
main
emphasis
in
this
paper
is
on
the
laying
out of
the
theoretical
principles
and
assessing
the
potential
and the
deficiencies of
the method,
we
also
compare
our results
with existing
experimental
information. The overall
per-
formance and
predictive
power
of
the
method are
deemed
good,
but
directions
for further
improvements
are
pointed
out.
II. FORMULATION
We
first
briefly
summarize
the main
formulas
given
by
Sugimoto
and
Fukai'
since
they
also form our
starting
point.
In the adiabatic
approximation,
the total
energy
of
the combined
light-impurity
—
metal
system
is
E([u(R)
j,
a)=EL(f
u(R)j)+E
([u(R)j),
where
j
u(R
)
j
is
the
set of
the
host-atom
displacements
at lattice
positions
R,
EL
is
the (lattice)
energy
of the host
atoms
in
the
presence
of the
impurity,
and E is the
im-
purity energy
eigenvalue. Fundamentally,
all the
energies
in
Eq.
(I)
are
of
electronic
origin,
but we assume in the
spirit
of the Born-Gppenhecmer
approximation
that
they
can
be
reduced to a
simpler
representation
in terms
of
well-defined
coupling
constants. The
key
equation
is
then
the
Schrodinger
equation
for the
impurity
coordinate,
where
Eo
is
the
energy
of
the
undeformed
lattice,
and
F(R
)
is
the force
acting
on
the host atom
at
R,
F~(R
)
=—
BEL
x,
y
Bu;(R
)
.
0
(4)
The
subscript
0
denotes
that
the
derivative
is evaluated at
the
equilibrium
position
in
the
undeformed
crystal.
Con-
sequently,
the
forces vanish in the
perfect
lattice.
P(R', R
)
is
the
dynamic
matrix,
QJ(R',
R)=
Q
2E
au,
(R
)au,
(R
)
The
impurity
(defect)
brings
in two features. The
dynam-
ic matrix
changes
to
and the forces cease
to
vanish. The
equilibrium
configu-
ration
corresponds to
the variational minimum
of
Eq.
(3)
with
respect
to the
displacements u(R
),
whereby'
'
F(R
)
=
gP*(R,
R')u
{R')
.
The inverse of
Eq.
(7)
reads
u(R)
=
g
G*(R,
R')F
(R
),
where
G*
is the static Green function for the
deformed
lattice.
'
GJ(R,
R')
gives
the
displacement
in
the
direc-
tion
i
for
the
atom
at
R
when
a
unit force
in
the direction
j
is
exerted on the atom
at
R'.
Combining
(6)
and
{8),
one
has (in
matrix
notation)
u=6(F+5P
u
)
=GF*,
where G is
the perfect-lattice
Green function
(G=P
'),
and
F'
is the
Kanzaki
force exerted
by
the defect on a
lattice
atom
in
the relaxed
position.
The
lattice-
deformation
energy
is now
expressed
simply
as
where
M
is the
impurity
mass and
V(
r
)
is
the
potential-energy
field
"experienced"
by
the
impurity
atom.
V(r
)
arises from the
"embedding"
interaction
be-
tween the
impurity
and
host,
and
will
be
developed
below
in terms
of
the
effective-medium
theory.
The
practical
details
of the numerical
solution of
Eq.
(2)
are
discussed
in
Appendix
A.
In the harmonic
solid
approximation,
the lattice
energy
EL
is
EL
—
—
Eo
—
gF(R) u(R)+
—,
g
u(R')P(R',
R)u
(R),
5384
M.
J. PUSKA AND R. M. NIEMINEN
29
KEI
=El
E—
p
—
—
—,
g
F"
(R
).
u(R
)
.
(10)
F*«)=
—
fd
IP
(
)I'
~
IR+
«)—
&
I
R+u(R)
—
r
I
Finally,
the
Kanzaki
forces can be obtained
from the
im-
purity
distribution
I
f
I
and
the
potential-energy
field
V. The latter is
a
function of
the
atomic
positions.
For
an atom
at
the relaxed
position
R+
u(R
),
In other
words,
Eq.
(12)
contains
the
Brillouin-zone
(BZ)
integration
over
the inverse
squares
of
the
phonon
modes
co
(q
)
multiplied
by
the
corresponding polarization
vec-
tors
e;
(q
).
The discrete-lattice
Green
functions are
numerically
available
for
most
metals
of
interest.
'
'
'
In
cubic
crystals
it is
straightforward to
also derive the
following
elastic
continuum
approximations:
AJ
(
q
)
=
~
g
c
kjlgk jl
k,
l
(11)
with
The
Green
function for the ideal discrete lattice
can
be
written
as'
—
iq.
R
G&(R)=
3
g
f
dq e;
(q
)e& (q
)
(2n.
)
Bz
~~(q
)
(12)
where
0 is
the unit-cell
volume,
e;
(
q
)
(cr
=
1,
2,
3)
are the
eigenvectors,
and
co are the
corresponding
eigenvalues
of
the matrix
P
'(q
)
when
y(q
)=
gy(R)e
(13)
R
Cikjl
c125ik5jl +c44(5ij
5kl
+
5ii5jk
)
+(cl1
—
c12
2c44)5ijkl
.
(15)
Gj(R
)
=
f
y;j(y)dp,
(16)
where
Above
cii,
c12,
and
c44
are the
(Voigt)
elastic
constants,
j
=k
=I,
or
5;jkl
=0
ot
lows
for
the
elastic
Green
function
that
LJ
y;,
(y)=
c44
+
dKj
KiKj(C44+Cii
)
2
44+C12
(c
4+4d
K)(c
4+4d
K)
j1+
g
2
&k
C44
+
dKk
d
=
c
1 1
—
c
iq
—
2c44,
Ki
=
cosy sing+
sinter
cos8
cosp,
K2
—
—
—
cosy
cosp+
sing
cos8
sing,
Ks
—
—
sing
sin8, (17)
and
8
and
P
are
the
polar
angles
of R.
The
integral
in
Eq.
(16)
can be
carried
out
analytically
only
in a few main
symmetry
directions,
but
is
easily
obtained
numerically
in
a
general
direction. The Green function
(16) diverges
at
the
origin,
but the
611(0)
component
needed
in the
calcu-
lations
can be
evaluated in
the
Debye model,
in
which this
divergence
does
not
exist.
'
The
final form in
the elastic
continuum model
reads as
(cii+2c44)
'VD
Gii(0)=
2~'c„(c»
)'"
where
qD
is the
Debye
wave vector. The
agreement
be-
tween
the
elastic and
discrete-lattice
Green functions
is
normally
better
than
—
10%.
Larger
deviations
occur for
metals with
long-range
elastic
coupling,
of which
Nb is a
typical
example.
In
practice,
only
few
(in
our
calculations,
two)
nearest-
neighbor
atom shells around the
impurity
are
allowed
to
relax.
This
limitation
and the
use of
symmetry
reduces
the
dimension
of the Green-function
matrix
consider-
ably.
'
The
expressions
for
the
resulting
reduced Green
functions for
the
tetrahedral
and
octahedral interstices
in
both
bcc and fcc lattices
are
given
in
Appendix
B.
Equations
(2), (9),
and
(11)
constitute a set of
equations
which has to
be
solved
self-consistently. What
remains
is
a
description
of the interaction
energy
V. We utilize
the
quasiatom concept.
First,
we assume that
a
satisfacto-
ry
approximation
for the electron
density
np(r
)
in
the
host (without
the
impurity,
but
with atomic
relaxations)
is
np=
1
r
np
1
1 1
This
implies
a
self-consistency
requirement for
np
and
P,
(
r
).
In
practice,
this is
not
very
severe since the
atom-
induced Coulomb
potential
P,
is
a
weak function
of the
(jellium)
density
where it is
calculated.
For further
development,
it
is
important
that
the
quasiatom
is
neutral
inside
R„and
also that
R,
is smaller
than
any
impurity-
host
nuclear
separation.
Because of
long-range
Friedel
os-
cillations, this,
in
practice,
necessitates
a
renormalization
of the
positive charge
(see
Sec.
III
A).
With
these
assumptions,
the
interaction
energy
V
be-
tween
the
impurity
at
r
=
0 and
the host
[electron density
np(
r
)]
can
be
written
as a
perturbation
series,
b,
Eh,
(np)
—
a,
np+5+he;,
(20)
where
&Eh, (np)
is the immersion
energy
of
an atom
available. We further
suppose
that the
impurity
nucleus
is
placed
at the
position
r
=0.
The electronic
perturba-
tion
due
to
the
impurity
atom
is,
owing
to
metallic
screen-
ing,
localized in
space.
We
assume that the
impurity-
induced effects are confined
within a
sphere
of
cutoff
ra-
dius
R,
.
Inside this
sphere
the
charge
density
p,
(r
)
and
Coulomb
potential
P,
(r
)
are taken
equal
to
those for
an
atom embedded
in
a
homogeneous electron
gas.
The
density
np
where
the
embedding
is
imagined
to
take
place
is
chosen
by
using
P,
(r
)
as
the
sampling
function
for the
unperturbed host,
THEORY OF HYDROGEN AND HELIUM
IMPURITIES
IN
METALS
into
a
homogeneous
electron
gas
of
density
no,
and
a.
=
—
J„dry.
(r)=
—
f,
f„dr
dr
p,
(r
)
0
4 0
r
—
r'
(21)
host-imp
,
host
yell-ImP
(22)
The
feasibility
of the
effective-medium
theory
as
out-
lined
above
demands that
(i)
the
intrinsic
dependence
on
the
cutoff radius
R,
in
Eq.
(20)
should
be small
for
reasonable values
of
R„and
(ii)
one
should
be able
to
esti-
mate the
one-electron
term
(22)
without
excessive effort
(that
is,
certainly
short of
performing
the full
calculation
of
the
energy
spectrum
in
the combined
host-impurity
system).
We
shall examine these
questions
for
hydrogen
and helium in detail below.
In
all the
applications
described below we
have
simply
constructed the host
electron
density
by
superimposing
free-atom
(at)
densities,
i.e.
,
no(r
)=
gn«(r
R;),
—
where
the
sum
goes
over
the
atomic
positions.
For
thc
present
purposes,
this
approximation
is
accurate
enough:
Comparison
with available
self-consistent
calculations25
shows
that
the differences in
the
interstitial
regions
are
less than
about 10%
for
close-packed
metallic elements.
For
vacancies and other lattice
defects,
differences
may
become
larger
but the
qualitative
features are not
changed.
III.
HYDROGEN
IN METALS
A.
Hydrogen
effective-medium
potential
In
the
case of
hydrogen
the
covalent term
5+,
.
he;
of
the
effective-medium
potential
is
important because the
hydrogen
1s
level
interacts
strongly
with
the host
valence
bands.
Therefore
the situation
is
completely
different
from
that for
helium, where
the
tightly
bound 1s
level
is
nearly
inert,
also in
a metallic environment
(see
Sec.
IV).
In
transition
metals the
hybridization
of
the
hydrogen
1s
and host
d
levels
occurs,
and the
covalent
contribution
can
approximately
be
expressed as
a
sum of two
terms,
is
proportional
to the
mean
quasiatomic
Coulomb poten-
tial. The last term in
Eq.
(20)
denotes the
change
in the
one-electron
eigenvalue
differences
when the
impurity-
jellium
system
is
replaced
by
the
impurity
—
true-host
sys-
tem,
5+he;=
ge;
5+he;=
J„[no(r
)
—
no]EV,
(r
)dr
—
2(1
—
f)20
',
g,
(24)
1
i-r
—
R,
i'
where
6
V,
(r)
is
the
total effective
potential
change
due
to
the
hydrogen.
The
second term
is
derived in
the
atomic-
sphere
approximation
(ASA). In
Eq.
(24),
f
is
the
rela-
tive
filling
of
host
d
level,
C~
—
V
is
the
separation
be-
tween the
d-band
center
of
gravity
and
the metal effective
potential
at the
hydrogen
site,
6,
and
b,
~
are
functions
of
the
potential
and atomic radii in
ASA,
and
finally
the
sum in the
equation
goes
over all
host lattice
sites.
As
suggested
by
N@rskov,
'
we have chosen
the
cutoff
rad|us
R,
=2.
5ao
for
hydrogen-induced
density
and
po-
tential.
The induced
density
is calculated
by
embedding
hydrogen
in
a uniform electron
gas
with
a
density
param-
eter
r,
=3ao.
We
also
follow the
recommendation
of
Ref.
7 and renormalize
the
charge
inside the
sphere
determined
by
this
radius
by
adding
a uniform
neutralizing
back-
ground charge
dlstrlbutlon.
Also,
as
recommended,
kVg
is
rigidly
shifted
so
that
it vanishes
at
the
surface of the
renormalization
sphere.
When
these
instructions
are
fol-
lowed,
the
effective-medium
potential
is
relatively
insensi-
tive
to
reasonable variations in
the
cutoff
radius
R,
. On
the
other
hand,
a different
procedure,
for
example,
the
re-
normalization
of the
hydrogen
nuclear
charge,
leads
to a
severe
dependence
on the cutoff
radius.
In
the actual calculations
we have omitted
the small
second
(hybridization)
term
in
the covalent
contribution
[Eq.
(24)]
and used
N@rskov's
rough
estimate
for
the first
term
no
r
—
no
hV~
r
r=
—
~„&go,
(25)
where
a„=31
cV
ao.
These
approximations
are
made
be-
cause
we want
to
keep
the construction
of the
potential
as
simple
as
possible,
and the determination
of
the hybridiza-
tion term
appears
somewhat
arbitrary
against
the
back-
ground
of
approximations
made
in
calculating the
leading
contributions
to the
potential.
For
the same
reason,
we
have
not included the
recently
suggested
correction
which takes into account
the
orthogonality
repulsion
be-
tween
hydrogen
and
metal-ion
cores.
However',
these
omissions are not
severe,
because, as
emphas1zed
by
Ngrskov,
the
hybridization and
repulsion
terms
are
only
small corrections to
the
potential
energy,
introduced in
or-
der
to
get
the
energy
minima
to
agree
with
such experi-
mental results
as
the
heat
of solution or
the
binding
ener-
gy
of
hydrogen
chemisorbed
on metal
surfaces.
The
lead-
ing
contnbutions, which
we
keep,
determine,
in
turn,
the
trends
and
the
spatial
variation
of
the
potential,
which
are
essential in
our
application. Thus,
we calculate the
poten-
tial for
hydrogen
in the
host from the
sampled density
[Eq.
(19)]
corresponding
to
a
given
point using
the
follow-
ing
interpolation formulas
(no
in
ao
and
~h,
in
eV):
V(
r
)=&Rh,
[no(r
)]
cf,
„no(r )
a„no(—
r
),
—
Pl
0
130noln
EEh,
(no)=
~
0.004
4
398(no
—
0.
0.
127)
+150no
—
2.
81,
0.
0127&no
.
