SOME THEORETICAL ASPECTS OF HYDROGEN DIFFUSION IN BCC METALS AT LOW TEMPERATURES

International Journal of Recent advances in Physics (IJRAP) Vol.12, No.1/2, May 2023
DOI:10.14810/ijrap.2023.12202 13
SOME THEORETICAL ASPECTS OF HYDROGEN
DIFFUSION IN BCC METALS AT LOW
TEMPERATURES
Serhii Bobyr, Joakim Odqvist
KTH Royal Institute of Technology, Department of Materials Science and Engineering,
Brinellvägen 23, SE-100 44, Stockholm, Sweden
ABSTRACT
Purpose of the work is to discuss some theoretical aspects of the diffusion of hydrogen atoms in the crystal
lattice of BCC metals at low temperatures using the methods of statistical thermodynamics. The values of
the statistical model calculations of H diffusion coefficients in α-Fe, V, Ta, Nb, K are in good agreement
with the experimental data. The statistical model can also explain deviations from the Arrhenius equation
at temperatures 300-100 K in α-Fe, V, Nb and K. It was suggested that thermally activated fast tunnelling
transition of hydrogen atoms through the potential barrier at a temperature below 300 K provides an
almost free movement of H atoms in the α-Fe and V lattice at these temperatures. The results show that
quantum-statistical effects play a decisive role in the H diffusion in BCC metals at low temperatures. Using
the statistical model allows for the prediction of the diffusion coefficient for H in BCC metals at low
temperatures, where it’s necessary to consider quantum effects.
KEYWORDS
Hydrogen diffusion; BCC metals; Statistical model; Pre-exponential factor; Low temperature; Quantum-
statistical effects
1. INTRODUCTION
The diffusion of hydrogen, H, in metals has been the subject of intense study in recent decades.
This is due to the practical interest in using the metal-hydrogen system for several technological
applications [1, 2]. Diffusion of H in Fe and other metals is very important as it leads to
engineering problems associated with hydrogen embrittlement and degradation of high strength
steels, reactor materials, etc. [3–6].
Hydrogen atoms have an extremely high diffusion mobility in solid metals it is almost the same
as in liquids [7–9]. This high diffusion mobility is thought to be the result of a very low activation
energy due to the quantum nature of hydrogen [9].
In metal lattices, depending on the type of lattice and temperature, hydrogen can occupy two
types of interstitial sites: octahedral and tetrahedral [7–9, 12]. Even in a pure metal, transitions
between interstitials of different types are possible, which entails the ambiguity of the activation
energy of hydrogen diffusion [10–13].
Also, it should be considered that the equilibrium distance between the metal and hydrogen atoms
is much less than between metal atoms [12]. It should be noted that other factors may also lead to
deviations from the Arrhenius law, in particular, the possibility of diffusion jumps of different
lengths and the effect of crystal lattice defects [12, 13].

14
First principles calculations and classical approaches such as molecular mechanics have been
used to study hydrogen diffusion in the case of body-centred cubic, BCC, Fe in several papers
[14-21].
It is assumed that modern ab initio modelling will provide a good description of the diffusion
parameters in Fe-H systems. However, as far as we know, there is no systematic derivation of the
diffusion coefficients in which quantum-statistical effects are considered in a wide temperature
range [13, 22]. The providing of such investigations is important for understanding the possible
mechanisms of hydrogen diffusion in metals.
The purpose of this work is to discuss some theoretical aspects of the diffusion of hydrogen
atoms in the crystal lattice of BCC metals at low temperatures using the methods of statistical
thermodynamics.
2. THEORETICAL METHOD
In the classical limit and using the random-walk model of interstitial diffusion in a BCC lattice,
D0 in the Arrhenius expression for the diffusion parameter D = D0exp (-Ea/RT) can be
represented as [21]:
3
2
1
0 3
1
6
N
j j
N
j j
n
D









(1)
where, n is a geometrical factor for the number of equivalent jump paths, Δ is the jump length,
and νj and νj
+
are the real normal mode frequencies at the initial state and the transition state,
respectively.
Close to the room temperature it is necessary to account for quantum correlations to the diffusion
barrier i.e. the discreteness of H vibrational modes in the metal lattice [21].
Kehr presented another description of H diffusion, assumed, that the lattice vibrations are the
same minimum energy and saddle-point configurations. For BCC metals at room temperature the
limit of hνH >> kBT, where νH is the frequency of the localized H vibrational modes. In this limit
the pre-exponential factor can be represented as [Ref. 21]:
2
0 exp
6
B
B
k T
n E ZPE
D
h k T
 
   
  
  (2)
Where ΔE is the energy difference between the saddle point and minimum energy configuration
on the potential energy surface, ΔZPE is the difference in zero-point energies. Thus, the
exponential term (ΔE+ΔZPE) in Eq. 2 is a ZPE-corrected diffusion barrier. Moreover, we can see
that the pre-exponential factor in Eq. 2 is temperature dependent to the first degree.
In [14] it is proposed to use statistical model expressions for the pre-exponential coefficient. This
approach, based on the first principles of statistical thermodynamics, makes it possible to
adequately consider the statistical distribution of impurity atoms according to their energy.
In this case the diffusion coefficient for impurity atoms in metals is equal to [14]:

15
0
2 2
-W /kT
3
2
e
mk T d
D
h


, W0 = U – μ0. (3)
U is the potential barrier height of the diffusion process,
μ0 is the chemical potential of impurity atoms,
and pre-exponential factor after transformations is [14]:


N
h
d
T
k
m
N
D a
e
3
2
2
2
0
2

, (4)
where d – distance between two nearest identical position sites of impurity particles;
N is the Avogadro number (mol-1
);
k (kB) – Boltzmann constant;
h – Planck constant;
Ne is the atomic number of the diffusive impurity atoms (1 for H);
ma is the atomic mass unit (H).
ρ is the metal density.
Expression (4) of the developed statistical model significantly differs from the expressions used
in simple molecular dynamics [14 - 21]. The interplanar distance d enters this expression to the
first degree. But this formula assumes the dependence of D0 on the square of the atomic number
of the diffusing element and the square of the temperature.
If we compare expression (4) with the classical equation for the diffusion coefficient [11]:
D = KΔ2
ν, (5)
where Δ is the jump length of an atom during diffusion,
ν is the jump frequency, K – constant (usually 1/6),
and in Eq. (5) we put d = Δ, then for the jump frequency we can express:
0
2 2 2
-W /kT
3
2
e
e a
N m k T
KN h




 . (6)
Eqs. (4) and (6) can be used in molecular dynamics for calculation of diffusion parameters.
In the developed statistical model (SM) pre-exponential factor (D0) of hydrogen can be calculate
by [14]:
2 2
0 0.114 /
D T m s


  
, (7)
where Δ is in meters, ρ is in kg/m3
.

16
3. RESULTS
3.1. The Hydrogen Diffusion in A-Iron
Fig. 1 shows the two types of interstitial sites in the BCC lattice, octahedral O (a) and tetrahedral
T (b). By using the experimental lattice parameter of BCC Fe the “radius” of the T-sites and O-
sites were found to be 0.36 Ǻ and 0.19 Ǻ respectively. H has a covalent radius of 0.37 Ǻ, and it
has been suggested, that H fit better in T-sites, than in O-sites [22]. This simple assumption has
been confirmed by DFT calculations [13, 21].
Let’s consider the movement of a hydrogen atom between tetrahedral sites; the most probable
trajectory for the migration of a hydrogen atom from one T-site into the nearest-neighbor T-site is
via a S-site how (Fig. 1, c). In [21] it was found that H does not utilize a straight – line trajectory,
but rather hops from one T-site to a nearby T-site by a curved path distorted towards an O-site.
The S-site is set like a saddle point along the minimum energy path (MEP) for direct hopping of a
H atom between two neighboring T-site [13]. The length of a single jump of a H atom from S-site
is more than one fourth of a lattice parameter. But at temperatures below 300 K, the migration of
a hydrogen atom from one T-site into the other T-site via an O-site becomes possible due to
changed potential barrier [13].
Fig. 1. Positions of O-sites (a) and T-sites (b) of the BCC lattice.
The S-sites are the saddle point along the MEP [13, 21] for direct hopping of a H atom between
two neighboring T-sites (paths 1-2 and 3-4 respectively) (c).
In several experimental studies, the Ea values were practically estimated by the slope of the lnD
versus 1/T plot between 290 K and 1000 K [23-25]. Typical Ea values range from 0.035 eV to
0.142 eV and D0 from 3.35×10-8
m2
s-1
to 2.2×10-7
m2
s-1
for H diffusion in BCC Fe, as compiled
from ten research groups for temperatures as low as 233 K [23].
It was shown in [13] that the diffusion of hydrogen in α-iron deviates from the Arrhenius
equation, which is an additional indication of the dependence of diffusion coefficients on

17
temperature. In [13] the centroid path-integral molecular dynamics method (CMD) was used to
change free energy profiles for H migration at various temperatures [see Fig. 5 in Ref. 13].
According to these data, at temperatures below 300 K, hydrogen diffusion can occur both through
S-sites and O-sites with similar potential barrier values. We used this important theoretical result
in the present work.
The results of the molecular dynamics and SM calculations are compared with the experimental
data found in [11, 13, 24, 25], see Figure 2, Table 1.
We can see that the present SM approach is valid for predicting the values of the diffusion
coefficients below the temperature of 300 K. For the calculations we used the potential by
Kimizuka, Mori and Ogata [13]. Estimated D values show a good agreement with classical MD
results, see Fig 2.
SM calculations with temperature-dependent potentials (TDP) by Kimizuka et al have given a
plausible description of H diffusion in Fe below 300 K and show a good agreement for the
temperature 500 K. This result shows the necessity of considering the increase of the potential
barrier at high temperatures.
Table 1. Calculated values for an activation energy (Ea) and a pre-exponential factor (D0) and of H
diffusivity in α-Fe for the temperatures below 300 K, together with experimental values.
Diffusion
parameters
Ea
(eV)
D0×10−8
(m2
/s)
D×10−8
(m2
/s)
T= 293 K
D×10−8
(m2
/s)
T= 250 K
D×10−8
(m2
/s)
T= 100 K
Calc. (MD)
(Ref. 13)
0.037 3.23 0.80 0.62 0.053
Calc. (CMD)
(Ref. 13)
TDP**
0.075*
3.23 0.83 0.70 0.45
Calc. (SM) TDP** 3.68* 0.31 0.20 0.05
Calc. (SM) 0.0
0.03*
2.61* 0.89 0.65 0.105
Expt. (Ref. 11) 0.044 4.15 0.1-1.0 0.06 -
Expt. (Ref. 24)
(290 -1040 K)
0.040 4.20 0.87 - -
Expt. (Ref. 25)
(290 -1040 K)
0.035 3.35 0.81 - -
*
At 500 K. ** temperature-dependent potentials by Kimizuka et al.

18
Fig 2. Diffusion coefficients of H in α-Fe in the temperature range 100-500 K.
The solid triangles and open squares represent the CMD and classical MD results for the potential
used by Kimizuka, Mori and Ogata, respectively [13]. The crosses represent the classical MD
results for the potential of Wen et al. [13]. The stars represent the SM results for a potential of
0.037 eV. The oval symbols represent the SM results for temperature-dependent potentials (TDP)
by Kimizuka, Mori and Ogata [13]. Diamonds represent the SM results with a potential equal
zero. Black and colored circles show experimental results on hydrogen diffusion from [11] and
[13] respectively.
We also performed calculations using Eq. (7) with zero potential. In doing so, we account for the
fact, as was done in [13], that at low temperatures H migration along not only the S-sites but also
O-sites becomes possible. For comparison we estimated D values based on SM approximation by
Eq. (7) in the temperature range 100 – 293 K; the obtained values of D show a good agreement
with experimental and CMD calculated data, see Table 1. At a temperature of 500 K the
calculated value is more than the experimental values, which also indicates the necessity of
considering the increase in the potential barrier at this temperature. As shown in our calculations,
for full agreement with experimental values the calculated value for an activation energy Ea must
be 0.0022 eV at 293 K and 0.03 eV at 500 K. Last value is close to the potential of Kimizuka et
al. and corresponds to their approach with temperature-dependent potentials [13].
Our approach based on first principles of statistical thermodynamics makes it possible to
coherently describe the temperature dependence of the diffusion coefficient H in α-Fe for
temperatures below 500 K. However, experimental data on the diffusion of hydrogen in α-Fe at
low temperatures near 100 K are clearly insufficient. Further experimental studies of this process
followed by a theoretical description are therefore of high interest.
3.2. Hydrogen Diffusion in BCC Low Temperatures
In [27] it was found that H in BCC metals, V, Nb and Ta, had a preference to occupy T-sites, but
O-sites in FCC-Pd, which could be correlated qualitatively with protonic electrostatic potential in
pure bulk metals.

19
Despite a significant number of experimental studies of hydrogen diffusion in metals (see e.g.
[26–35]), there are very few reliable data, especially on the hydrogen distribution at temperatures
below room temperature. Experimental data on quantum diffusion of hydrogen have been
obtained for Nb and Ta [28, 29] and in Na and K for deuterium [30, 31].
Theory of hydrogen diffusion in metals has been developed by various models, namely: the small
polaron model of Flynn and Stoneham [34, 36, 37], semiclassical transition state theory [37],
Feynman path-integral molecular dynamics model [13, 38], density functional theory (DFT) [35,
38] centroid and ring polymer molecular dynamics, [39], statistical thermodynamics [40, 41], and
others. In quantum theories of diffusion, the interpretation of bends on the D(T) dependences in
Ta and Nb [34, 35] was successful. Ab initio DFT calculations of diffusion activation energies for
both migration mechanisms were performed, which showed excellent agreement with
experimental data for H in Nb and Ta [40].
Experimental results on the temperature dependencies of the diffusion coefficients D(T) for
deuterium in K were presented in [26]. At temperatures from 250 to 120 K, the mechanism of
above-barrier motions of atoms dominates in the process of mass transfer, while below 120 K,
this is the tunneling mechanism.
In work [41] was presented the statistical model of H diffusion in FCC metals at a high
temperature (900 – 2000 K). This thermodynamics model makes it possible to explain the high
values of the pre-exponential factor of H diffusion coefficients (~10–7
m2
s–1
− 10–6
m2
s–1
) in FCC
metals (γ-Fe, Ni, Pd, Al, Ag) at high temperature due to the coefficient T2
in the diffusion
equations.
In the present work we propose and develop a statistical model for describing of H diffusion in
BCC metals at low temperatures. The values of a pre-exponential factor for Nb, Ta, V were found
from first principles, see Eq. 7.
The calculated activation energy, pre-exponential factor and experimental values from the [11]
and [26] are presented in Table 2 and in the Fig. 3.
Table 2. Calculated values for an activation energy (Ea), pre-exponential factor (D0) and H diffusivity D in
the considered metals at the temperature below 300 K, together with data from [11] and [26].
Metals K (De)* Ta Nb V
Temperature K 100 200 200 300 200 300 200 300
Calc. (Ref. 11)
Ea (eV)
0.0024 0.16 0.04 0.14 0.068 0.106 0.045 0.045
Calc. (Ref. 11)
D0 10-8
(m2
s–1
)
3.28
×10-8
1.24
×10-4
0.02 4.4 0.9 5.0 0.31 0.31
Expt. (Ref. 11)
D 10-8
(m2
/s)
2.0×
10-8
1.0×
10-5
0.002 0.02 0.17 0.8 1.7 4.0
Calc. (SM)
D0 10-8
(m2
/s)
0.31
×10-4
1.25
×10-4
0.005 4.0 2.2 5.0 1,76 4.37
Calc. (SM)
Ea (eV)
0.135 0.016 0.077 0.044 0.0023
*Deuterium diffusion in K at the data in [26].

20
The SM calculations results show a good agreement with experimental data, especially in case of
H diffusion in Nb and V (Fig. 3). The statistical model allows for the prediction of the diffusion
coefficients for H in Nb and V at low temperatures. For Ta, as in the classical description, we
had to use two diffusion SM equations with different activation energies, which indicates a
change in the diffusion parameters of H in Ta near the Debye temperature.
Fig. 3. Temperature dependences of diffusion coefficient for H in Nb, Ta, V and deuterium De in K.
Circles demonstrate experimental results on hydrogen diffusion [11] and deuterium diffusion in
K [26]. The star symbols and lines represent the SM results.
The calculated values of the SM-preexponential factor (D0) for H diffusion in Nb, Ta, V at the
high temperature is in good agreement with the known data (Table 3). But the calculated values
of an activation energy are significantly different from those accepted in [11] values, especially in
case of H diffusion in V. This value 0.0023 eV is almost equal to the calculated value of an
activation energy of deuterium diffusion in K at a temperature below 100 K - 0.0024 eV. Such
low activation energy values indicate quantum mechanism of deuterium diffusion in K and H
diffusion in V.
4. DISCUSSION
As was found in [26] for the systems Nb–H, Ta–H, and K–D, the bend temperatures Tb are 250,
229, and 120 K and Debye temperatures 260, 225, and 100 K, respectively.
Using the statistical model, we will discuss this issue of relating the quantum mechanical motion
of hydrogen in metals to the Debye temperature of these metals. Let's present the calculation data
of Table 2 in activation energy Ea of H diffusion in BCC metals versus – temperature diagram
indicating the Debye temperature of these metals and scale of energy Eh=kBT. This energy makes
it possible to estimate the average energy of a hydrogen atom at a certain temperature (Fig. 4).
From the above data, there is an inverse relationship between the activation energy Ea of H in
BCC metals and the Debye temperature of these metals. The higher the Debye temperature for a
metal, the lower the energy barrier that hydrogen needs to overcome at a temperature of 300 K.

21
Similar results were obtained earlier in a different form in [26]. The mechanism of this
connection requires a separate theoretical study.
Fig. 4. Correlation of diffusion activation energy Ea of Н in BCC metals at the 300 K with energy Eh and
the Debye temperature points of these metals.
The black squares and diamonds symbols represent the SM results and the classical results for
activation energy respectively and the Debye temperature points of BCC metals on the
temperature scale. For α-Fe, potential by Kimizuka et al. was used.
When using the Arrhenius equation for diffusion calculations, the calculated values of the
activation energy of hydrogen are greater than the values of the statistical model, i.e. these
parameters are dependent on the experimental data processing model. The true values of the
energy barrier for hydrogen diffusion can be calculated from first principles [17, 18, 21, 37, 38].
The transition from the theoretical values of the energy barrier to the model values of the
activation energy is associated with considering additional influencing factors, primarily physical
parameters: - zero-point energy [21], decay of local deformation near the hydrogen atom [40],
temperature-dependent potentials [13], etc. The presentation of the results is greatly influenced
by the parameters of the diffusion models used for calculations - the Arrhenius equation itself
[10, 11], the Einstein formula of molecular dynamics [9, 20], the Kerr diffusion equation [21], the
operator expression of quantum statistical mechanics [40], and T2
-diffusion equation of statistical
thermodynamics [41].
In view of the foregoing, let us analyze the data presented in Fig. 4. At a temperature of 300 K,
Eh-energy has a value of ~0.026 eV, which is very close to the apparent values of the diffusion
activation energy of H determined from the Arrhenius equation in iron (0.037 eV) and vanadium
(0.044 eV). The ab initio determined value of the energy barrier at this temperature is U300 ≈ 0.05
eV for iron, according to the data of the work [13]. Thus, at a given ratio of Eh and U300, there are
all conditions for a thermally activated fast tunneling transition of hydrogen atoms through the
potential barrier of iron atoms at this temperature [13, 40, 42].

22
In the statistical model of diffusion, this situation becomes even more obvious, since the
calculated value is Ea = 0.0022 eV, which indicates an almost free movement of H atoms in the
iron lattice at this temperature. This can explain the very low solubility of H in α-Fe and the
extremely high diffusion mobility.
A similar situation occurs for the diffusion of hydrogen in vanadium, for which Ea = 0.0023 eV.
Therefore, we assume that the patterns of hydrogen diffusion in vanadium manifest themselves in
the same way as in iron. However, this assumption requires additional both experimental and
theoretical studies for its verification.
Other studied BCC metals have at room temperature a significant potential barrier for H
diffusion, which indicates its diffusion over the barrier.
As the diffusion temperature rises to 500 K, i.e. above the Debye temperature for iron in the
statistical model, it is necessary to use the energy value Ea = 0.03 eV is close to the CMD model
value (0.037 eV). At the same time, the value of the energy barrier is calculated from first
principles U500 ≈ 0.075 eV [13]. Considering the difference in zero-point energy (-0.046 eV) [21],
the ZPE-corrected diffusion barrier becomes equal to 0.029 eV corresponding to the value Ea in
the statistical model.
It can be assumed that above the Debye temperature the H vibrational modes become more than
the maximal frequency of Fe lattice vibrations; this decoupling and the increasing interaction of
H atoms with phonons leads to an increase in the potential barrier of H diffusion.
Other BCC metals – Nb, Ta and K also show an increasing potential of H diffusion above the
Debye temperature. Therefore, the question of transition from tunneling to over-barrier H
diffusion at a temperature close to the Debye temperature remains open for further research.
When the diffusion temperature drops below 300 K, the energy decreases, but the magnitude of
the potential barrier decreases accordingly, supporting the process of quantum diffusion of H and
an almost free movement of H atoms in the iron lattice. This indicates, different from [26], that
the quantum effects strongly control the mechanism of local diffusion of H in α-Fe at low
temperature [13]. We can assume the same patterns for diffusion of V and K at temperatures
below the Debye temperature.
For Nb and Ta, quantum-statistical effects manifest themselves in a change in the diffusion
mechanism near the Debye temperature in the presence of a significant energy barrier for
hydrogen diffusion behind the over barrier mechanism [34, 40].
5. CONCLUSIONS
It was shown that the most probable trajectory for the migration of a hydrogen atom from one T-
site into the nearest-neighbor T-site going on across a S-site is by a curved path distorted toward
the O-site. The length of a single jump of an H atom during S-site is more than one fourth of a
lattice parameter. But at temperatures below 300 K it is also possible with the migration of a
hydrogen atom from one T-site into the other T-site across an O-site due to temperature-
dependent potentials and changing (decreasing) diffusion path for H.
It was proposed to use the statistical model for a more accurate calculation of the hydrogen
diffusion in BCC metals at low temperatures. The calculations performed are compared with the
known experimental data on the diffusion of hydrogen atoms in α-Fe with a good agreement
between the results.

23
It was suggested using the SM model that thermally activated fast tunneling transition of
hydrogen atoms through the potential barrier at temperatures below 300 K provides an almost
free movement of H atoms in the iron lattice at these temperatures. We may also assume that the
patterns of hydrogen diffusion in vanadium manifest themselves in the same way as in iron.
Our approach makes it possible to coherently describe the temperature dependence of the
diffusion coefficient H in α-Fe at temperatures below 300 K. However, experimental data on the
diffusion of hydrogen in α-Fe at low temperatures are clearly insufficient. Further experimental
studies of this process followed by a theoretical description are expedient.
The results show that quantum-statistical effects play a decisive role in the H diffusion in BCC
metals at low temperatures. Using the statistical model allows for the prediction of the diffusion
coefficient for H in BCC metals at low temperatures, where it’s necessary to consider quantum
effects. Thus, the values of the SM pre-exponential factor of H diffusion in BCC metals are in
good agreement with the experimental data at room temperature.
Undoubtedly, this model does not consider many specific factors affecting the diffusion
parameters of atoms in a metal, such as the interaction of impurity atoms, the presence of
vacancies, influence of the Debye temperature, quantum tunneling at a cryogenic temperature,
etc. Further development of this model and quantum theories of diffusion will allow for
consideration of the influence of these factors.
ACKNOWLEDGEMENTS
This project was funded by the Swedish Foundation for Strategic Research (SSF), contract
UKR22-0070. We want to thank profs. Annika Borgenstam and Liubov Belova for supporting
this project and prof. Andrei Ruban for fruitful discussions and critical remarks.
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25
AUTHORS
DSc Serhii Bobyr is a Leading Researcher in Metal Physics and Materials science with
rich on hands experience in academic and commercial projects participation and
management with several developments patented and implemented in industrial
enterprises.
From 15.09.2022 is a Leading Researcher at КТН - Royal Institute of Technology,
Stockholm, Sweden
2016 – 2022 - Leading Researcher at Z.I. Nekrasov Iron & Steel Institute of the
National Academy of Science of Ukraine (ISI NASU).
2016 -Winner of the prize of the G.V. Kurdyumov name of the National Academy of Sciences of Ukraine.
2011 - DSc. Degree, 2005 - 2016 - Senior Researcher at ISI NASU.
Dr. Joakim Odqvist works as a professor at KTH Royal Institute of Technology,
Stockholm, Sweden. His research interests are phase transformations in metallic alloys
and computational materials design. In 2004 he was awarded a PhD degree in Materials
Science from KTH.

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