For φ(r), the simulated S(q) shows a maximum located at qM (25.4nm−1), and a shoulder appear on the high-q side of the first peak. Shown in Fig. II.2, the shoulder in the simulated S(q) occurs around 32.5nm−1, which is exactly equal to 2kF. Moreover, the first peak and the shoulder of simulated S(q) agree with those of experimental data above the melting temperature [40], although the values of the simulated S(q) at low q deviate from experimental ones.
In order to investigate the relationship between the shoulder in S(q) and the inter-atomic potential including the repulsive core and Friedel oscillation parts, we examine the variation of the radial distribution functions g(r) and S(q) with the interaction range of the interatomic potential refered to Fig. II.1. First, Fig. II.3 shows the variation of the radial distribution functions gi(r), compared with g(r) of the full-range φ(r). The first peak in all radial distribution functions almost locates at 0.686σ, which is inside
20 40 60
q [nm-1]
0 0.5 1 1.5 2 2.5
S(q)
Exp. data Liquid of Ga
Figure II.2: Comparison of the simulated static structure factor (solid line) with the experimental data of liquid Ga at T=323K (open circles) [40].
1 1.5 2
Figure II.3: The radial distribution functions: g(r) (solid line), g0(r) (dotted line), g1(r) (dashed line), g2(r) (dot-dashed line) and g3(r) (dot-dot-dashed line). g(r) is for the full-range pair potential φ(r), indicated by the thinner dotted line, and gi(r) is for the truncated pair potential φi(r). The second and third shells of the radial distribution functions are enlarged in the inset.
the repulsive core, and the shapes of gi(r) are almost the same, no matter where the interaction distance is truncated. The evident difference is only for tail of radial distribu-tion funcdistribu-tions that indicates the Friedel oscilladistribu-tions of the interatomic potential make the second- and third-shell structures of radial distribution function an outward shift, and this shift enhances the values of radial distribution function at distances around the minimum of the Friedel oscillations. The outward shifts of the second and third shells have little reduction by including the first attractive part referred to Fig. II.3. It means that the Friedel oscillations beyond the first attractive part have a certain effect on the structures of the liquid. Extending the interaction range to the third attractive well of the Friedel oscillations, g3(r) can be completely identical with g(r). The results mildly imply that the shoulder structure may be a result of the interplay between the ledge-shape repulsive core and the Friedel oscillations; however, Matsuda [66] and Canales [67] suggest that the structures of liquid alkal metals are almost indentical by considering the repulsive core and the short-range attractive part of interaction potential.
Secondly, the simulated static structure factors Si(q) with the truncated interatomic potentials φi(r) (i = 0, 1, 2, 3) are shown in Fig. II.4, with S(q) as a reference one. The first peak and the shape around 32.5nm−1 in these static structure factors should be examined particularly. For the S0(q), no shoulder appears on the high-q side of the first peak, which shifts to 25.7nm−1 and has an increase in magnitude. The simulated S1(q) deviates more from a HS fluid than the liquid of pure repulsive core because the first attractive well is included in the interaction range. The first peak of S1(q) is evident
20 25 30 35 40
Figure II.4: The static structure factors: S(q) (solid line), S0(q) (dotted line), S1(q) (dashed line), S2(q) (dot-dashed line) and S3(q) (dot-dot-dashed line). S(q) is for the full-range pair potential φ(r) and Si(q) for the truncated pair potential φi(r) (i = 0, 1, 2, 3) with a cutoff at σi.
to be lower than that of S0(q), but the position has no significant change; however, the values of S1(q) for q between 28nm−1 and 35nm−1 are elevated. For S2(q), the position of the first-peak shifts closer to that of S(q), and the magnitudes in the shoulder region continuously increase to get closer to S(q). To extend the interaction range to including the third attractive well of Friedel oscillations, the magnitude of the first peak is almost close to that of S(q), and a complete shoulder appears on the high-q side of the first peak.
Quite sensitive to the behavior of the static structure factor, the appearance of the shoulder structure can be further identified by the derivative of Si(q) with respect to q, dSi(q)/dq, for q around 32.5nm−1. In general, on the high-q side of the first peak of Si(q), dSi(q)/dq is negative and increases with q. Mathematically, a monotonical increase of dSi(q)/dq with q indicates no appearance of a shoulder in Si(q) in the region investigated.
However, once a shoulder appears in Si(q), the behavior of dSi(q)/dq is distorted to show some extremes, instead of increasing monotonically. Shown in the inset of Fig. II.4 are the numerical results of dSi(q)/dq for q in the shoulder region. Both dS0(q)/dq and dS1(q)/dq increase monotonically, indicating that no shoulder appears on the high-q side of the first peak of S0(q) or S1(q). dS2(q)/dq shows very weak extremes, signaling the emergence of a weak shoulder in S2(q). Manifested by the clearly observed extremes in dS3(q)/dq, the structures in the liquid simulated with φ3(r) are developed well enough to produce a shoulder on the high-q side of the first peak of S3(q). Our results clearly indicate that the liquid structures are mainly determined by the repulsive core and the long-range Friedel oscillations induced by the conduction electrons, and the effective interaction range must
Pair potential N1201 N1211 N1301 N1311 N1421 N1422 N1431
φ(r) 0.189 0.045 0.134 0.252 0.042 0.092 0.065 φ0(r) 0.164 0.047 0.118 0.252 0.050 0.101 0.077 φ1(r) 0.177 0.046 0.124 0.254 0.047 0.096 0.071 φ2(r) 0.190 0.046 0.132 0.251 0.042 0.091 0.065 φ3(r) 0.189 0.045 0.135 0.251 0.042 0.092 0.064
Table II.1: The averaged fractions of atomic bonded pairs, Nijkl [63], in the liquids simu-lated with the full-range pair potential φ(r) and the truncated φi(r)
include at least the first three attractive wells.
To manifest further the modulation on local structures by Friedel oscillations, we calculate the numbers of atomic bonded pairs (ABPs) [62] in the liquids for the pair potentials at different truncations. In the previous studies, it has been shown numerically that as the system simulated with the pair potential generated by the GEINMP theory is quenched from the liquid phase into the amorphous solids or the β-phase crystal, the 1201-type atomic bounded pairs (ABPs), clusters of four atoms formed by a root pair and two neighboring atoms, become predominated [63]. Also, some large clusters formed by more 1201-type ABPs may produce a high-q shoulder in S(q) [59]. In Table II.1, our calculated results for the APBs show that by truncating the pair potential φ(r) at σ0, the numbers of the 1201- and 1301-type ABPs decrease significantly but those of the 1421-, 1422- and 1431-type increase. With the cutoff at σ1, the numbers of those ABPs mentioned above are still different from those of the full-range φ(r), although the differences are reduced.
As the cutoff is extended up to σ2, the numbers of the ABPs are almost the same as those
of φ(r). Thus, our present analysis on the ABPs suggests that the Friedel oscillations within the intermediate region, up to the third attractive wells, cause a modulation on the local structures determined by the repulsive core and such a modulation favors the emergence of some solid-like clusters which cause a shoulder on the high-q side of the first peak in S(q).
To investigate the anomaly in collective dynamics, we have calculated the dynamic structure factor SM D(q, ω), a time Fourier transform of the intermediate scattering func-tion of Ga interatomic potential. These intermediate scattering funcfunc-tions are obtained by simulations for wavevectors chosen to be the reciprocal lattice points of the simulated box. In order to fit the experimental data, SM D(q, ω) has to be modified to satisfy the detailed balance condition and convoluted with the instrumental resolution function R(ω) [11]:
INth(q, ω) =
Z ¯hω0/KT
1 − exp(−¯hω0/KT )SM D(q, ω0)R(ω − ω0)dω0, (II.9)
and the calculated dynamics structure factors INth(q, ω) are directly compared with the experimental data of IXS. In hydrodynamic regime, a comparison between the best fitting for selected q and the experimental spectrum is reported in Fig. II.5. Only for q = 3.25nm−1, the central peak of INth(q, ω) is disappeared because of the effect of simulated system size; however, the two Brillouin peaks can be fitted well. For several q values in the kinetic regime, from the first peak to the second minimum in S(q), the dynamic structure factors calculated by our simulation are also in good agreement with the experimental
-20 0 20
Figure II.5: Dynamic structure factor at the indicated wavevectors. The IXS spectra at 315K are shown as open circles, and the simulated dynamic structure factors at 323K are shown as solid lines. The width of the instrument resolution function is about 3.0meV .
data, and the comparison is reported in Fig. II.6.
Since attention is focused on S(q, ω), it is more convenient to examine the CL(q, ω), which is defined in Eq. II.6. In the same manner as the experimental inelastic x-ray data [68, 69], we analyze the longitudinal current spectra which is fitted by the simple damped harmonic oscillator (DHO) function
CL(q, ω) = A γLω
(ω2− ωL2)2+ (γLω)2, (II.10)
where A is a fitting; on the other hand, ωL and γL is energy position shift and width of spectra, respectively. For several q values between q = 3.25nm−1 and q = 15.0nm−1, the longitudinal current spectra with the DHO fitting lines are shown in Fig. II.7.
Shown in Fig. II.8, the domainant q dependence of the shift ωL is linear below q = 10.0nm−1, which confirms our results with those observation in the inelastic x-ray measurements [11]. The sound velocity of our system is estimated about 2850ms−1, which is slightly lower than the value 3000ms−1 observed in the IXS experiment at 315K. How-ever, the estimated value of sound velocity is almost the same as the value (2800ms−1) as deduced by ultrasonic measurements [11]. Also, the interaction range has very little effect on the sound velocity of system because the sound velocity for each simulated inter-action potential is around 2850 ± 20ms−1. Shown in Fig. II.9, the damping factor γL/q of neutron data is compared with the data obtained by fitting our MD simulation data (Eq.
II.10). The two sets of data are once again in good agreement with each other and both are consistent with that γL/q, is approximated to be a q-independent constant between
-20 0 20
Figure II.6: Comparison between the simulated dynamic structure factors at T = 323K (solid lines) and the IXS experimental data of liquid Ga at T = 315K (circles) [47].
0 10 20 30
Figure II.7: The simulated longitudinal current spectra at 323K (open circles), and the fitting function of the damped harmonic oscillator described in the text (dashed lines).
0 4 8 12 16
q(nm-1)
0 5 10 15 20 25
ω L (meV)
φ(r) φ0(r) φ1(r) φ2(r) φ3(r) IXS
Figure II.8: Dispersion relation obtaine by fitting the simulated longitudinal current spec-tra with damped harmonic oscillator model. The IXS data at 315K from [11] are shown as solid circles. The solid lines are the linear fit to the data of q less than the available data of the IXS experiment. The estimated value of sound velocity for each pair potential is Cf ull = 2847.3m/s, C1min = 2829.7m/s, C1max = 2843.4m/s, C2max = 2875.6m/s, and C3max = 2870.0m/s, respectively.
0 0.4 0.8 1.2 1.6
q(A-1)
0 4 8 12 16
γ L(q)/q(meV A) φ(r)
φ0(r) φ1(r) φ2(r) φ3(r) INS
o o
Figure II.9: The damping factor γL/q versus q. Neutron data at 315K from [11] are presented as solid circles, and MD simulated data with various interaction range are presented as open symbols.
q = 0.2nm−1 and q = 0.8nm−1. On the other hand, the γL/q values of various interaction ranges are almost the same and that means the interaction range also can not affect the linewidth of longitudinal current spectrum. It is worth pointing out that the relaxation time of the longitudinal current function is linear with the q value in low-q region, but the interaction range of pair potential has no effect on the relaxation time.
As wavevectors larger than 15nm−1, SM D(q, ω) can be fitted with a single Lorentzian . We define Z(q) for each q as the half width at half maximum (HWHM) of the Lorentzian.
Shown in Fig. II.10, the Z(q) data obtained by our simulations are compared with the experimental data from IXS [47] and QENS [48] technique. The linewidth Z(q) of both experiments and simulation have a minimum, the de Gennes narrowing, which occurs very close to qM the location of the maximum of S(q). In experiments, an anomaly, which is a shoulder, is observed on the high-q side of the de Gennes narrowing. In our simulation, the linewidth Z(q) agrees well with the general features of both the experimental data, especially the shoulder around 32.5nm−1. Considering S(q) and Z(q) both generated by our simulations with φ(r), we confirm that the anomalies in these two functions occur at the same position.
Fig. II.11 shows that both the de Gennes narrowing and the shoulder of the linewidth Z(q) are predicted quite well by the revised Enskog theory, in which σHS must be the position of the first peak of g(r) (0.686σ) [58] and the reduced density equals ρσ30 = 3.305. While the reduced density is equal to 3.305, the packing fraction η is estimated
20 30 40 50
q [nm-1]
0 1 2 3
Linewidth [meV]
IXS QENS Simulation S(q) of φ(r)
S(q)
2
1
0
Figure II.10: The spectral linewidth (HWHM) of dynamic structure factors. The open circles and squares are the experimental data of IXS [47] and QENS [48] at T = 315K, respectively. The solid circles are the simulated results with φ(r) at T = 323K. The dashed line is the simulated S(q), with a scale referred to the right axis.
15 20 25 30 35 40
Figure II.11: The linewidth HWHM of S(q, ω) as a function of q for the simulated liquid of Ga at T = 323K. The circles are the results of the liquid simulated with φ(r) at T = 323K. The dotted line is the simulated S(q), the dashed line is the prediction of the revised Enskog theory Eq. II.4, the dotted-dashed line is the prediction of the revised Enskog theory Eq. II.4 without d(q), and the dotted-dotted-dashed line is the d(q) function only. The values of σHS are 0.686σ in the upper panel and 0.664σ in the lower one.
evaluated by Eq. II.5 is 1.66 × 10−5cm2s−1, which is close to the self-diffusion coefficient Ds = 1.77 × 10−5cm2s−1 obtained from the velocity autocorrelation function. On the other hand, as σhs is chosen to be 0.664σ obtained by the E-MCRS theory [33], which accurately predicts the Helmholtz free energy and entropy of the liquid, DE evaluated by Eq. II.5 is 1.86 × 10−5cm2s−1. The results are not as good as those fitted by 0.686σ:
the de Gennes narrowing in the range between q = 17.5nm−1 and q = 30.0nm−1 are still described well by revised Enskog theory, but the shoulder is gradually disappeared. Also, shown in Fig. II.11, by comparing the predictions with and without dynamic factor d(q) in Eq. II.4, we find that the d(q), associated with the cage diffusion, cause a quite good fitting between the prediction function and simulated Z(q) for the wavevectors around the shoulder. This information indicates that the mechanism for the occurrence of the shoulder in Z(q) should be related to the cage diffusion. For the single-particle dynamics, the cage diffusion in a liquid usually depicts that a particle is confined in a cage which is composed of its neighbours; therefore, this particle collides with its neighbours in a short timescale but diffuses out of the cage in a longer timescale. Alternatively, from the viewpoint of collective dynamics, the cage diffusion can be considered as the relaxation of the cage structure, with the relaxtion time related to the stability of the cage.
Shown in Fig. II.12, to investigate the role of the Friedel oscillations on anomaly in Z(q), we calculate the linewidths Zi(q) with the pair potentials φi(r) for i = 0, 1, 2, 3.
Both Z0(q) and Z1(q) have a minimum nearby qM, but they are monotonically ascendent beyond qM up to 40nm−1, which is close to the first minimum of S(q). This behavior is
20 25 30 35 40
q [nm-1]
0 1 2 3
Linewidth [meV]
Z0(r) Z1(r) Z2(r) Z3(r) Z(r)
Figure II.12: The spectral linewidth Z(q) for the full-range φ(r) and Zi(q) for φi(r) with a finite range. All symbols are the simulated results.
similar as those of the LJ liquids and liquid alkali metals.
As the range of interatomic potential is extended up to the second maximum of the Friedel oscillations, a shoulder clearly appears on the high-q side of the de Gennes narrow-ing; furthermore, this shoulder is almost developed as well as the one in Z(q) as the range of pair potential is extended up to the third maximum of the Friedel oscillations. In our model, the interaction range of pair potential to produce an anomaly in Z(q) should be farther than the second attractive well; however, this range is shorter than that causing the shoulder in S(q). Fig. II.13 shows that the comparison between the simulated Zi(q)
20 25 30 35 40
Figure II.13: Comparison between the simulated results (the symbols) and the predictions of the revised Enskog theory with the HS diameter σhschosen to be the first-peak position of g(r), DE evaluated by Eq. II.5 with the chosen σhs and Shs(q) replaced by S(q) or Si(q). Each linewidth function for a pair potential is shifted upward 0.3 meV from the lower one.
and the predictions of the revised Enskog theory with the same values of σhs and DE, and the Shs(q) in Eq. II.4 replaced by the corresponding S(q) presented in Fig. II.4. Arising from the dynamic factor d(q), a shoulder is indeed produced in the linewidth function ZE(q), no matter what the range of the pair potential is. The results are consistent with that of Fig. II.13. The shoulder in Z2(q), Z3(q) and Z(q) is in good agreement between the prediction of the theory and the simulation result for q from 22.5nm−1 to 35.0nm−1, but the prediction of the theory for q between 30nm−1 and 35nm−1 is deviated from the
Z0(q) and Z1(q), having no shoulder. This is clearly present that the dynamic factor d(q), the characteristic of cage factor, is the essential factor to make the revised Enskog theory a successful prediction for the shoulder in Z(q) [47, 58]. Thus, the shoulder in Z(q) is certainly caused by cage diffusion.