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Enhancement of nuclear tunneling through Coulomb-barriers using molecular cages
Chemical Physics Letters 420 (2006) 241–244 www.elsevier.com/locate/cplett
Enhancement of nuclear tunneling through Coulomb-barriers using molecular cages Dvira Segal
a,*
, Tamar Seideman b, Gershon Kurizki a, Moshe Shapiro
a,c,d
a
b
Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113, USA c Department of Chemistry, University of British Columbia, Vancouver V6T1Z1, Canada d Department of Physics, University of British Columbia, Vancouver V6T1Z1, Canada Received 4 November 2005; in final form 18 December 2005 Available online 24 January 2006
Abstract We present an electrostatic mechanism that gives rise to dramatic enhancements of tunneling through the inter-nuclear Coulomb-barrier. The enhancement is due to an increase in the effective negative electric charge in the mid region between two protons (deutrons) þ belonging to an Hþ 2 ðD2 Þ molecule placed inside a molecular cage, such as a fullerene molecule. This charge increase results in a marked reduction of the outer regions of the Coulomb-barrier, thereby dramatically increasing the nuclear tunneling rates. Coupled with compression of the molecular cage this effect may lead to meaningful enhancements in fusion rates. 2006 Elsevier B.V. All rights reserved.
Fusion of two nuclei, such as that of two deutrons to form a helium nucleus 2Dþ ! He2þ ;
ð1Þ 6
typically requires temperatures in excess of 10 K [1] and/ or high compression. The widespread skepticism regarding the occurrence of fusion at much lower temperatures (‘cold fusion’) stems from the fact that tunneling rates at room temperatures are way too small to give rise to fusion. An exceptional process is a muon catalyzed fusion demonstrated experimentally by Alvarez et al. [2]. In this special case the heavy (negative) muon causes the nuclei to bind closely together, thus enhancing the reaction rate. In general, the estimates of the tunneling rates are based on the following arguments: The rate of the fusion reaction is given by 2
w ¼ CjWðRN Þj ;
ð2Þ
where C is a constant and jW(RN)j2 is the probability–density for two nuclei to be at zero separation, or more pre*
Corresponding author. Fax: +972 8 9344123. E-mail address: [email protected] (D. Segal).
0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.12.076
˚ , where cisely, at the inter-nuclear distance RN 106 A the strong nuclear forces become operative. In order to reach such a small inter-nuclear separation, the system must tunnel through the entire RN < R < Rt repulsive region of the inter-nuclear potential V(R) of Fig. 1, where Rt, satisfying the E = V(Rt) relation, is the smallest classical turning point at energy E. The nuclear probability–density at RN is given in WKBJ approximation as, ( Z 1=2 ) RN 2M 2 jWðRN Þj A exp 2 ðV ðRÞ EÞ dR ; ð3Þ h2 Rt where A is a normalization constant and M is the reduced mass of the nuclei. Using Eqs. (2) and (3), the fusion rate for the ground vibrational energy E = E0 can be written approximately as [3,4], h i 2ME0 exp 1 pð2Rt =AM Þ1=2 ; wC 2 2 ð4Þ 4p h AM where AM = h2/Me2 is the Bohr radius associated with the reduced nuclear mass M. It is clear from Eq. (4) that even a slight decrease in Rt can result in a vast increase in the tunneling rate. Below, we
242
D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
-15.5 V(R) [eV]
V(R) [eV]
10
with R being the inter-nuclear distance. The range of variation of prolate spheroidal coordinates is 1 6 n 6 1, 1 6 g 6 1, and 0 6 / 6 2p. With the above coordinate definitions, the Schro¨dinger equation for an electron in the presence of two fixed positive nuclei is given as, ( h2 4 o 2 o o o ðn 1Þ þ ð1 g2 Þ 2 2 2 on og og 2me R ðn g Þ on # ) n2 g2 o2 4e2 n e2 þ 2 þ Wðn; g; /Þ 2 2 Rðn g2 Þ R ðn 1Þð1 g2 Þ o/
0
-16
0.6
1 1.4 R [Angstrom]
-10
¼ EWðn; g; /Þ; 0
1
2 3 R [Angstrom]
4
5
Fig. 1. Molecular potential energy V(R) curve and ground state vibrational function (inset) of the Hþ 2 system. The smallest classical turning point Rt is marked at the inset.
show that it is possible to induce such a decrease by confinþ ing the Hþ 2 ðD2 Þ molecular ion within a cage of a much larger molecule, such as C20 [5]. By causing a small increase in the electronic charge between the nuclei, the cage provides an effective screening of the Coulomb repulsion and hence lowers the Coulomb repulsion. The more effective screening lowers Rt and greatly increases w. In a more quantitative manner it follows from Eq. (4), that the lowering of Rt by a factor c < 1, where ð1Þ ð0Þ c ¼ Rt =Rt , increases the tunneling rate by 12 1 ð0Þ 1 c2 . f wð1Þ =wð0Þ ¼ exp p 2Rt =AM ð5Þ ð0Þ
Taking Rt for the lowest vibrational state of Dþ 2 to be ˚ , and AM = 2.9 · 1014 m, we see that a 40% change 1 A in Rt, i.e., taking c = 0.6, results in an enhancement factor w(1)/w(0) of 1025. In order to estimate the values attainable for c in practice we have performed detailed calculations of the electronic ground state of Xþ 2 (X = H, D) inside molecular cages [5–7], e.g., small fullerene molecules. It was verified experimentally that C60 and higher fullerenes can confine atoms [8], ions, and even small molecules [9] in their inner cavity. The current experimental strategy involves a set of chemical reactions that close an open-cage fullerene encapsulating chemical species. An analogous scheme might be applied to smaller fullerene structures. Theoretical calculations of the energy levels of atoms [10–12] and small molecules [13] within fullerene cages reveal stable compounds. We have chosen to study ‘prolate-spheroidal’ shaped cages because the Schro¨dinger equation of an electron in the presence of two fixed positive nuclei is separable in the prolate-spheroidal coordinates, defined as, / – the azimuthal angle, and n and g, related to r1 and r2, the distances of the electron from the two nuclei, as, r1 þ r2 r1 r2 ; g¼ ; ð6Þ n¼ R R
ð7Þ
where me denotes the electron mass. W(n, g, /) is a product of three functions, W(n, g, /) = N(n)H(g)U(/), where pffiffiffiffiffiffi Uð/Þ ¼ eim/ = 2p with m being zero or an integer. N and H are solutions of generalized spheroidal wave equations of the form, 0 m2 ð1 n2 ÞN0 þ k2 n2 jn N ¼ 0; ð8aÞ 1 n2 0 m2 H ¼ 0; ð8bÞ ð1 g2 ÞH0 þ k2 g2 1 g2 where 0 designates the derivative operation, j = 2mee2R/ h2, 2me B ¼ h2 , B is the parameter of separation of variables, and e The electronic energy E e is a function of k2 12 mh2e R2 E. three quantum numbers: the magnetic quantum number, m, and two spheroidal quantum numbers nn and ng which determine the number of zeros of the functions N(n) and H(g), respectively. The full inter-nuclear potential, V(R) e plus the inter-nuclear repulsion, of Fig. 1, is E e þ e2 =R. V ðRÞ ¼ E
ð9Þ
This potential denotes the eigenenergies of Eq. (7) under the Born–Oppenheimer approximation. For a free Xþ 2 molecular ion equation (8) is solved numerically for each value of m and R, using the methods of Refs. [14,15]. We now consider placing the Xþ 2 molecule inside prolate-spheroidal shaped fullerene cages. The two smallest fullerene structures C20 [16] and C24 [17,18] containing an H2 molecule are shown schematically in Fig. 2. The average ˚ , respectively. radii of these structures are 2.02 and 2.18 A ˚ [19]. Effectively, the thickness of the fullerene shell is 1 A The different types of boundary conditions that may be imposed on the Xþ 2 system are: (i) hard box confinement, in which the wave function decays to zero at the box boundaries; (ii) soft box confinement, where the walls of the box induce finite confining potentials. Practically, these boundary conditions can be realized by negatively charging the cage, causing the electron to be repelled from the cage’s surface. In order to achieve the strongest confining effect we apply the ‘hard box’ conditions. The soft box case is analyzed in [5]. The prolate-spheroidal surface is obtained by fixing R and n and varying g, in terms of which, the x, y, and z Cartesian coordinates on the surface are given as,
D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
Fig. 2. Small fullerene cages C20 (left) and C24 (right) encapsulating a hydrogen molecule. 1 1 R 2 ðn 1Þ2 ð1 g2 Þ2 cos /; 2 1 1 R 2 y ¼ ðn 1Þ2 ð1 g2 Þ2 sin /; 2 R z ¼ ng. 2
x¼
ð10Þ
Because g can be written as g ¼ cos h, where h is an angle, it is easy to see from Eq. (10) that when R becomes very small, making n2 1, the surface approaches that of a sphere of radius r = Rn/2. The hard-box boundary conditions are now, Wðr ¼ rc Þ ¼ 0;
ð11Þ
where rc = ncR/2. This condition means that the wave function decays to zero on a surface defined in Cartesian coordinates as, x2 y2 z2 þ þ ¼ 1. r2c R2 =4 r2c R2 =4 r2c
ð12Þ
In Fig. 3, we show that the surface given by Eq. (12) becomes independent of R when r2c R2 =4. In our calculations this condition is fulfilled for the R values of interest, where (R/2rc)2 < 0.1.
243
Eqs. (8a) and (8b) for N(n) and H(g) should be solved simultaneously to provide the common values and k for a given R. The equation for N(n) is solved using a fourthorder finite differences algorithm. The differential equation for H(g) is solved by reformulating it as a recursion formula [5]. Fig. 4 presents the ground state energy (1rg) for a Xþ 2 (X = H, D) ion inside prolate shaped hard boxes with different box radii rc. The general trend observed is an increase of the energy for increasingly small boxes, in agreement with [5–7]. We note that because we restrict r2c R2 =4, the ˚ results are meaningful only for R 6 1.2 A ˚. rc = 1.8 A Two important features are demonstrated in the inset of Fig. 4: (i) the equilibrium distance shifts slightly to the left; (ii) the potential becomes steeper, causing the vibrational frequency to increase. The combined effect of these two features is to reduce Rt, causing the enhancement of nuclear tunneling. Fig. 5 presents the enhancement factors calculated using þ þ Eq. (5) for the Dþ 2 and H2 molecular ions. For a D2 system ˚ the enhancement factor for a cage of rc = 1.5 A is f 1013. ˚ , the enhancement For compressed cages, where rc < 1.5 A factor may be significantly larger than 1015. As the fusion 76 1 þ rates for free Dþ s 2 and H2 molecular ions are 10 74 1 and 10 s , respectively [4], the calculated enhancement factors are not nearly enough in themselves to sustain fusion. Yet, coupled with other effects, such as sending a compression wave through a set of Dþ 2 ions caged in an array of fullerene molecules, this mechanism may increase fusion rates in a more significant way. It is of interest to connect the enhanced tunneling rates observed here with an improved screening of the Coulomb repulsion between the nuclei caused by the increase in electronic charge inside the cage. A simple definition of the effective electronic charge eeff when the system is confined within a cage is the ratio between the normalization of the electronic wave function in the absence and presence of the cage
4 5
r =4 A
r =2 A c
V(R) [eV]
V(R) [eV]
z [A ]
rc =3 A
2
-15.4
0
c
-5
-15.8 -16.2
-10
0.8
1 1.2 R [A]
1.4
-15
0
0
2 y [A ]
4
Fig. 3. The y–z projection of the surfaces defined by Eq. (12) for different ˚ ; dashed: R = 1 A ˚ ; dashedinter-nuclear separations. Full: R = 1.5 A ˚. dotted: R = 0.5 A
0.2
0.4
0.6
0.8
1 R [A]
1.2
1.4
1.6
Fig. 4. Energy of the Xþ 2 molecular ions inside prolate-spheroidal boxes. ˚ (dashed), rc = 2 A ˚ (dotted), rc = 1.8 A ˚ (dashedrc = 1 (full), rc = 2.5 A dotted). The inset zooms on the equilibrium distance.
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D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
The reason for this enhancement is the effective increase in the electronic charge in the inter-nuclear region due to the molecular cage. The effect increases as the cage size decreases. Thus, sending a compression wave in an array of fullerene molecules containing Dþ 2 molecules can greatly enhance the probability of fusion.
15
30
10
10
10
f
10
5
10
20
10
0
1.4
f
10
1.6 1.8 2 2.2 r [A] c
Acknowledgment
10
10
This project was supported by the Feinberg graduate school of the Weizmann Institute.
1
10
References 1
1.5
2 2.5 rc [A]
3
3.5 4
Fig. 5. Tunneling enhancement ratio vs. sphere radius (log–log scale) for þ two isotopes of the hydrogen molecular ion: Dþ 2 (full), H2 (dashed). The ˚ ˚ inset zooms on the 1.5 A < rc < 2 A region.
R eeff ¼ R1 V
2
R1
2
¼ R1nc
drjWj drjWj
1
dn dn
R1 R1 1 1
2
2
2
2
dgðn g ÞN ðnÞH ðgÞ dgðn2 g2 ÞN2 ðnÞH2 ðgÞ
.
ð13Þ
Applying this definition to systems with different cage radii, rc = Rnc/2, we find, as expected, that as the volume of the confining sphere decreases, the effective charge becomes larger. For example, for the Hþ 2 ion eeff = 1.005 for ˚ , eeff = 1.05 for rc = 1.8 A ˚ , and eeff = 1.1 for the ulrc = 2 A ˚ . Such small increase in effectra small sphere of rc = 1.3 A tive charges lead to the large enhancement factors observed here. In conclusion, endohedral confinement of the hydrogen molecular ion leads to an enhanced nuclear tunneling rate.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
S. Ichimaru, Rev. Mod. Phys. 65 (1993) 255. L.W. Alvarez et al., Phys. Rev. 105 (1957) 1127. C. DeW Van Siclen, S.E. Jones, J. Phys. G 12 (1986) 213. L. Bracci, G. Fiorentini, G. Mezzorani, J. Phys. G 16 (1990) 83. S. Mateos-Cortes, E. Ley-Koo, S.A. Cruz, Int. J. Quantum Chem. 86 (2002) 376. E. Ley-Koo, S.A. Cruz, J. Chem. Phys. 74 (1981) 4603. J. Gorecki, W. Byers Brown, J. Chem. Phys. 89 (1988) 2138. H. Shinohara, Rep. Prog. Phys. 63 (2000) 843. K. Komatsu, M. Murata, Y. Murata, Science 307 (2005) 238. J.P. Connerade, V.K. Dolmatov, P.A. Lakshmi, S.T. Manson, J. Phys. B 32 (1999) 239. J.P. Connerade, A.G. Lyalin, R. Semaoune, S.K. Semenov, A.V. Solov’yov, J. Phys. B 34 (2001) 2505. A.L. Buchachenko, J. Phys. Chem. B 105 (2001) 5839. J. Cioslowski, J. Am. Chem. Soc. 113 (1991) 4139. E.A. Hylleraas, Z. Phys. 71 (1931) 739. G. Jaffe, Z. Phys. 87 (1934) 535. P.R. Taylor, E. Bylaska, J.H. Weare, R. Kawai, Chem. Phys. Lett. 235 (1995) 558. F. Jensen, H. Koch, J. Chem. Phys. 108 (1998) 3213. B. Paulus, Phys. Chem. Chem. Phys. 5 (2003) 3364. Y.B. Xu, M.Q. Tan, U. Becker, Phys. Rev. Lett. 76 (1996) 3538.
Enhancement of nuclear tunneling through Coulomb-barriers using molecular cages Dvira Segal
a,*
, Tamar Seideman b, Gershon Kurizki a, Moshe Shapiro
a,c,d
a
b
Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3113, USA c Department of Chemistry, University of British Columbia, Vancouver V6T1Z1, Canada d Department of Physics, University of British Columbia, Vancouver V6T1Z1, Canada Received 4 November 2005; in final form 18 December 2005 Available online 24 January 2006
Abstract We present an electrostatic mechanism that gives rise to dramatic enhancements of tunneling through the inter-nuclear Coulomb-barrier. The enhancement is due to an increase in the effective negative electric charge in the mid region between two protons (deutrons) þ belonging to an Hþ 2 ðD2 Þ molecule placed inside a molecular cage, such as a fullerene molecule. This charge increase results in a marked reduction of the outer regions of the Coulomb-barrier, thereby dramatically increasing the nuclear tunneling rates. Coupled with compression of the molecular cage this effect may lead to meaningful enhancements in fusion rates. 2006 Elsevier B.V. All rights reserved.
Fusion of two nuclei, such as that of two deutrons to form a helium nucleus 2Dþ ! He2þ ;
ð1Þ 6
typically requires temperatures in excess of 10 K [1] and/ or high compression. The widespread skepticism regarding the occurrence of fusion at much lower temperatures (‘cold fusion’) stems from the fact that tunneling rates at room temperatures are way too small to give rise to fusion. An exceptional process is a muon catalyzed fusion demonstrated experimentally by Alvarez et al. [2]. In this special case the heavy (negative) muon causes the nuclei to bind closely together, thus enhancing the reaction rate. In general, the estimates of the tunneling rates are based on the following arguments: The rate of the fusion reaction is given by 2
w ¼ CjWðRN Þj ;
ð2Þ
where C is a constant and jW(RN)j2 is the probability–density for two nuclei to be at zero separation, or more pre*
Corresponding author. Fax: +972 8 9344123. E-mail address: [email protected] (D. Segal).
0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.12.076
˚ , where cisely, at the inter-nuclear distance RN 106 A the strong nuclear forces become operative. In order to reach such a small inter-nuclear separation, the system must tunnel through the entire RN < R < Rt repulsive region of the inter-nuclear potential V(R) of Fig. 1, where Rt, satisfying the E = V(Rt) relation, is the smallest classical turning point at energy E. The nuclear probability–density at RN is given in WKBJ approximation as, ( Z 1=2 ) RN 2M 2 jWðRN Þj A exp 2 ðV ðRÞ EÞ dR ; ð3Þ h2 Rt where A is a normalization constant and M is the reduced mass of the nuclei. Using Eqs. (2) and (3), the fusion rate for the ground vibrational energy E = E0 can be written approximately as [3,4], h i 2ME0 exp 1 pð2Rt =AM Þ1=2 ; wC 2 2 ð4Þ 4p h AM where AM = h2/Me2 is the Bohr radius associated with the reduced nuclear mass M. It is clear from Eq. (4) that even a slight decrease in Rt can result in a vast increase in the tunneling rate. Below, we
242
D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
-15.5 V(R) [eV]
V(R) [eV]
10
with R being the inter-nuclear distance. The range of variation of prolate spheroidal coordinates is 1 6 n 6 1, 1 6 g 6 1, and 0 6 / 6 2p. With the above coordinate definitions, the Schro¨dinger equation for an electron in the presence of two fixed positive nuclei is given as, ( h2 4 o 2 o o o ðn 1Þ þ ð1 g2 Þ 2 2 2 on og og 2me R ðn g Þ on # ) n2 g2 o2 4e2 n e2 þ 2 þ Wðn; g; /Þ 2 2 Rðn g2 Þ R ðn 1Þð1 g2 Þ o/
0
-16
0.6
1 1.4 R [Angstrom]
-10
¼ EWðn; g; /Þ; 0
1
2 3 R [Angstrom]
4
5
Fig. 1. Molecular potential energy V(R) curve and ground state vibrational function (inset) of the Hþ 2 system. The smallest classical turning point Rt is marked at the inset.
show that it is possible to induce such a decrease by confinþ ing the Hþ 2 ðD2 Þ molecular ion within a cage of a much larger molecule, such as C20 [5]. By causing a small increase in the electronic charge between the nuclei, the cage provides an effective screening of the Coulomb repulsion and hence lowers the Coulomb repulsion. The more effective screening lowers Rt and greatly increases w. In a more quantitative manner it follows from Eq. (4), that the lowering of Rt by a factor c < 1, where ð1Þ ð0Þ c ¼ Rt =Rt , increases the tunneling rate by 12 1 ð0Þ 1 c2 . f wð1Þ =wð0Þ ¼ exp p 2Rt =AM ð5Þ ð0Þ
Taking Rt for the lowest vibrational state of Dþ 2 to be ˚ , and AM = 2.9 · 1014 m, we see that a 40% change 1 A in Rt, i.e., taking c = 0.6, results in an enhancement factor w(1)/w(0) of 1025. In order to estimate the values attainable for c in practice we have performed detailed calculations of the electronic ground state of Xþ 2 (X = H, D) inside molecular cages [5–7], e.g., small fullerene molecules. It was verified experimentally that C60 and higher fullerenes can confine atoms [8], ions, and even small molecules [9] in their inner cavity. The current experimental strategy involves a set of chemical reactions that close an open-cage fullerene encapsulating chemical species. An analogous scheme might be applied to smaller fullerene structures. Theoretical calculations of the energy levels of atoms [10–12] and small molecules [13] within fullerene cages reveal stable compounds. We have chosen to study ‘prolate-spheroidal’ shaped cages because the Schro¨dinger equation of an electron in the presence of two fixed positive nuclei is separable in the prolate-spheroidal coordinates, defined as, / – the azimuthal angle, and n and g, related to r1 and r2, the distances of the electron from the two nuclei, as, r1 þ r2 r1 r2 ; g¼ ; ð6Þ n¼ R R
ð7Þ
where me denotes the electron mass. W(n, g, /) is a product of three functions, W(n, g, /) = N(n)H(g)U(/), where pffiffiffiffiffiffi Uð/Þ ¼ eim/ = 2p with m being zero or an integer. N and H are solutions of generalized spheroidal wave equations of the form, 0 m2 ð1 n2 ÞN0 þ k2 n2 jn N ¼ 0; ð8aÞ 1 n2 0 m2 H ¼ 0; ð8bÞ ð1 g2 ÞH0 þ k2 g2 1 g2 where 0 designates the derivative operation, j = 2mee2R/ h2, 2me B ¼ h2 , B is the parameter of separation of variables, and e The electronic energy E e is a function of k2 12 mh2e R2 E. three quantum numbers: the magnetic quantum number, m, and two spheroidal quantum numbers nn and ng which determine the number of zeros of the functions N(n) and H(g), respectively. The full inter-nuclear potential, V(R) e plus the inter-nuclear repulsion, of Fig. 1, is E e þ e2 =R. V ðRÞ ¼ E
ð9Þ
This potential denotes the eigenenergies of Eq. (7) under the Born–Oppenheimer approximation. For a free Xþ 2 molecular ion equation (8) is solved numerically for each value of m and R, using the methods of Refs. [14,15]. We now consider placing the Xþ 2 molecule inside prolate-spheroidal shaped fullerene cages. The two smallest fullerene structures C20 [16] and C24 [17,18] containing an H2 molecule are shown schematically in Fig. 2. The average ˚ , respectively. radii of these structures are 2.02 and 2.18 A ˚ [19]. Effectively, the thickness of the fullerene shell is 1 A The different types of boundary conditions that may be imposed on the Xþ 2 system are: (i) hard box confinement, in which the wave function decays to zero at the box boundaries; (ii) soft box confinement, where the walls of the box induce finite confining potentials. Practically, these boundary conditions can be realized by negatively charging the cage, causing the electron to be repelled from the cage’s surface. In order to achieve the strongest confining effect we apply the ‘hard box’ conditions. The soft box case is analyzed in [5]. The prolate-spheroidal surface is obtained by fixing R and n and varying g, in terms of which, the x, y, and z Cartesian coordinates on the surface are given as,
D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
Fig. 2. Small fullerene cages C20 (left) and C24 (right) encapsulating a hydrogen molecule. 1 1 R 2 ðn 1Þ2 ð1 g2 Þ2 cos /; 2 1 1 R 2 y ¼ ðn 1Þ2 ð1 g2 Þ2 sin /; 2 R z ¼ ng. 2
x¼
ð10Þ
Because g can be written as g ¼ cos h, where h is an angle, it is easy to see from Eq. (10) that when R becomes very small, making n2 1, the surface approaches that of a sphere of radius r = Rn/2. The hard-box boundary conditions are now, Wðr ¼ rc Þ ¼ 0;
ð11Þ
where rc = ncR/2. This condition means that the wave function decays to zero on a surface defined in Cartesian coordinates as, x2 y2 z2 þ þ ¼ 1. r2c R2 =4 r2c R2 =4 r2c
ð12Þ
In Fig. 3, we show that the surface given by Eq. (12) becomes independent of R when r2c R2 =4. In our calculations this condition is fulfilled for the R values of interest, where (R/2rc)2 < 0.1.
243
Eqs. (8a) and (8b) for N(n) and H(g) should be solved simultaneously to provide the common values and k for a given R. The equation for N(n) is solved using a fourthorder finite differences algorithm. The differential equation for H(g) is solved by reformulating it as a recursion formula [5]. Fig. 4 presents the ground state energy (1rg) for a Xþ 2 (X = H, D) ion inside prolate shaped hard boxes with different box radii rc. The general trend observed is an increase of the energy for increasingly small boxes, in agreement with [5–7]. We note that because we restrict r2c R2 =4, the ˚ results are meaningful only for R 6 1.2 A ˚. rc = 1.8 A Two important features are demonstrated in the inset of Fig. 4: (i) the equilibrium distance shifts slightly to the left; (ii) the potential becomes steeper, causing the vibrational frequency to increase. The combined effect of these two features is to reduce Rt, causing the enhancement of nuclear tunneling. Fig. 5 presents the enhancement factors calculated using þ þ Eq. (5) for the Dþ 2 and H2 molecular ions. For a D2 system ˚ the enhancement factor for a cage of rc = 1.5 A is f 1013. ˚ , the enhancement For compressed cages, where rc < 1.5 A factor may be significantly larger than 1015. As the fusion 76 1 þ rates for free Dþ s 2 and H2 molecular ions are 10 74 1 and 10 s , respectively [4], the calculated enhancement factors are not nearly enough in themselves to sustain fusion. Yet, coupled with other effects, such as sending a compression wave through a set of Dþ 2 ions caged in an array of fullerene molecules, this mechanism may increase fusion rates in a more significant way. It is of interest to connect the enhanced tunneling rates observed here with an improved screening of the Coulomb repulsion between the nuclei caused by the increase in electronic charge inside the cage. A simple definition of the effective electronic charge eeff when the system is confined within a cage is the ratio between the normalization of the electronic wave function in the absence and presence of the cage
4 5
r =4 A
r =2 A c
V(R) [eV]
V(R) [eV]
z [A ]
rc =3 A
2
-15.4
0
c
-5
-15.8 -16.2
-10
0.8
1 1.2 R [A]
1.4
-15
0
0
2 y [A ]
4
Fig. 3. The y–z projection of the surfaces defined by Eq. (12) for different ˚ ; dashed: R = 1 A ˚ ; dashedinter-nuclear separations. Full: R = 1.5 A ˚. dotted: R = 0.5 A
0.2
0.4
0.6
0.8
1 R [A]
1.2
1.4
1.6
Fig. 4. Energy of the Xþ 2 molecular ions inside prolate-spheroidal boxes. ˚ (dashed), rc = 2 A ˚ (dotted), rc = 1.8 A ˚ (dashedrc = 1 (full), rc = 2.5 A dotted). The inset zooms on the equilibrium distance.
244
D. Segal et al. / Chemical Physics Letters 420 (2006) 241–244
The reason for this enhancement is the effective increase in the electronic charge in the inter-nuclear region due to the molecular cage. The effect increases as the cage size decreases. Thus, sending a compression wave in an array of fullerene molecules containing Dþ 2 molecules can greatly enhance the probability of fusion.
15
30
10
10
10
f
10
5
10
20
10
0
1.4
f
10
1.6 1.8 2 2.2 r [A] c
Acknowledgment
10
10
This project was supported by the Feinberg graduate school of the Weizmann Institute.
1
10
References 1
1.5
2 2.5 rc [A]
3
3.5 4
Fig. 5. Tunneling enhancement ratio vs. sphere radius (log–log scale) for þ two isotopes of the hydrogen molecular ion: Dþ 2 (full), H2 (dashed). The ˚ ˚ inset zooms on the 1.5 A < rc < 2 A region.
R eeff ¼ R1 V
2
R1
2
¼ R1nc
drjWj drjWj
1
dn dn
R1 R1 1 1
2
2
2
2
dgðn g ÞN ðnÞH ðgÞ dgðn2 g2 ÞN2 ðnÞH2 ðgÞ
.
ð13Þ
Applying this definition to systems with different cage radii, rc = Rnc/2, we find, as expected, that as the volume of the confining sphere decreases, the effective charge becomes larger. For example, for the Hþ 2 ion eeff = 1.005 for ˚ , eeff = 1.05 for rc = 1.8 A ˚ , and eeff = 1.1 for the ulrc = 2 A ˚ . Such small increase in effectra small sphere of rc = 1.3 A tive charges lead to the large enhancement factors observed here. In conclusion, endohedral confinement of the hydrogen molecular ion leads to an enhanced nuclear tunneling rate.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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