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Chirality-dependent macroscopic force between chiral molecules and achiral matter
Physics Letters A 373 (2008) 9–12
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Physics Letters A www.elsevier.com/locate/pla
Chirality-dependent macroscopic force between chiral molecules and achiral matter Yonghong Hu, Zhongzhu Liu, Qing Xu, Jun Luo ∗ Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 July 2008 Received in revised form 23 October 2008 Accepted 30 October 2008 Available online 6 November 2008 Communicated by P.R. Holland PACS: 14.80.Mz 11.30.Rd 31.15.B-
a b s t r a c t A non-zero macroscopic chirality-dependent force between achiral matter and homochiral molecules due to the exchange of light particles is shown theoretically. It has the opposite sign for molecules with opposite chirality. As an example, this force between a copper block and a vessel of chiral molecules (methyl phenyl carbinol nitrite) is calculated in the crystal field theory. The magnitude of the force is estimated with the published limits of scalar and pseudo-scalar coupling constants. Its possible influence to the gravitational experiments testing the equivalence principle is discussed as well. © 2008 Published by Elsevier B.V.
Keywords: Macroscopic force Axion Chiral molecule Parity violation
1. Introduction Recently, the force mediated by axions or other particles has become a focus of researches [1–7]. Axion has an abstruse parentage. So, force mediated by it displays complicated forms [1]. It is detected as the force between spins [2]. And it also is detected as a force magnetizing electrons [3–5]. Moreover, the CP asymmetry of the force is emphasized. One proposed to test this property by using precise measurement of gravitational forces [6–9]. Before doing such test, one must know how a chiral force can form between macroscopic objects. In this paper, we shall study the force between achiral matter and homochiral molecules mediated by axions. The force appears between valence electrons in the chiral molecules and nucleons in the achiral matter. Our research shows that the total force is nonzero. Moreover, under a space reflection, a left-hand molecule becomes a right-hand one. Synchronously, the force between chiral molecules and the achiral matter converts its sign. Such chirality-dependent force may be a new factor to influence the test of the equivalence principle. Chirality is a basic configuration feature of material, crystals and organic molecules. These materials and their mirror image enantiomers have the same composition but different geometrical
*
Corresponding author. Tel.: +86 027 87556653; fax: +86 027 87542391. E-mail address: [email protected] (J. Luo).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.10.082
structure, left screw or right screw. They have a series of properties, for example, rotatory power and parity non-conserving energy differences [10–15]. Here we give the potential between achiral matter and a vessel of homochiral molecules. Further we count the potential with the crystal field theory and estimate the magnitude of the force between a copper block and a vessel of homochiral molecules. The possible effects in the torsion pendulum experiments testing the equivalence principle are discussed. 2. Theory and model The chirality-dependent macroscopic force due to the exchange of axion may influence the equivalence principle [7–9]. Pospelov’s results indicate the compositions of material may influence the equivalence principle [6]. Our researches will indicate the configurations of material may influence the test of the equivalence principle. When the chirality of material is changed, the sign of the macroscopic force changed. Such parity-violating force due to the exchange of axion is possible to influence the test of the equivalence principle. Now let’s see the effective spin-dependent potential proposed by Moody and Wilczek [1]
H int = h¯ ( g s g p )
σ · rˆ mϕ 1 + 2 e −mϕ r , 8π me c r r
(1)
where c is the light speed in vacuum, h¯ σ /2 is the electron spin, r is the displacement vector between the nucleon and the elec-
10
Y. Hu et al. / Physics Letters A 373 (2008) 9–12
tron, g s and g p are the scalar and pseudo-scalar coupling constants respectively, me is the mass of the electron and mϕ is the mass of exchanged pseudo-scalar particle. The interaction range is supposed as λ = h¯ /mϕ c. Within the electron system of a chiral molecule, the space distribution of valence electrons displays the chiral character of the molecule. The sum of the potential (1) over all nucleons and molecules will be zero due to the zero total magnetic moment of the molecules. However, the potential may take a non-zero value when the effect of the spin–orbital interaction of valence electrons is involved. The valence electrons and nucleons in our study belong to two macroscopic bodies respectively. So, we can study the motions of valence electrons with the adiabatic approximation [16]. Then the total Hamiltonian of the valence electron system in the homochiral molecules can be written as
ˆ = Hˆ 0 + Hˆ 1 + Hˆ 2 , H ˆ1 H
where
M
is
i =1 ξ(r i )l i
the
potential
ˆ2 = − H
(1),
(2)
M
B (r i ) ·
i =1
σi =
· σi represents the orbit–spin interaction Hamiltonian
ˆ 0 is the Hamiltonian of the of the valence electrons [17] and H valence electron system involving only the Coulomb interaction. Moreover, B (r i ) is the magnetic induction at the position of the ith valence electron, li and σi /2 are the orbital angular momentum
2 and the spin of the ith valence electron. And ξ(r ) = h¯2 2
me c
1 ∂ U (r ) r ∂r
is
the orbit–spin coupling coefficient for the electron in the Coulomb potential U (r ). The energy shift of the valence electron system due to the ˆ 1 and Hˆ 2 can be given with the stationary state Hamiltonian H perturbation theory [18]. A simple calculation in the second order correction of the energy shows that the chirality-dependent ˆ 1 + Hˆ 2 ) is [19–21] potential induced by the Hamiltonian ( H EV = 2
Re{Ψn | H 1 |Ψ0 Ψ0 | H 2 |Ψn } E 0 − En
n=0
(3)
.
Where |Ψ is the wave function of valence electron system in the molecules, the subscript “0” represents the initial state and “n” represents final state. And Re{ } denotes the retention of the real part of the products of matrix elements. With the single-electron approximation, the wave functions |Ψ are written as the product of single-electron wave functions [22]. Then, we further factor each single-electron wave function as the product of the orbital wave function |ψ and the spin wave function |m [23]. Employing the relation of (σ · A )(σ · B ) = A · B + i σ · A × B, the sum over spins reduces the expression (3) to the following form, EV = 2
i
j
n=0
1 E i0 − E in
Re ψin |
g s g p h¯ 8π me c
mϕ +
1
ri j
e −ri j mϕ × r |ψ · ψ |ξ( r ) l |ψ . ij i0 i0 i i in 2 ri j
(4)
Where r i j is the displacement vector from the ith electron to the jth nucleon. |ψi0 and |ψin are the initial and final states of the ith valence electron at the energy levels of E i0 and E in respectively. Let |ψ L be a state of left-handed molecule. Under the space reflection P , it becomes a state of the right-handed molecule |ψ R as |ψ R = P |ψ L . The orbital angular momentum li is an axial vector. It doesn’t change under the space reflection. But r i j will change its sign as a polar vector. Therefore the products of matrix elements have the following relation:
ψ Ln | H 1 |ψ L0 ψ L0 | H 2 |ψ Ln = −ψ Rn | H 1 |ψ R0 ψ R0 | H 2 |ψ Rn .
(5)
It means that the energy E V takes the opposite value for the molecules with opposite chirality.
3. Calculation and result Before the calculation of the potential (4), we firstly estimate its magnitude. The energy difference can be taken as ( E 0 − E n ) ≈ 1 eV [24]. The matrix element ψ0 | h2¯ σ · ξ(r )l|ψn is approximated as the energy split due to the spin–orbit interaction [25]. In published theoretical calculations, the value can be read that h2¯ ψ0 |σ · ξ(r )l|ψn ≈ 0.01 eV [26]. So, we get the estimation 1
(E 0 − En )
h¯ ψ0 | σ · ξ(r )l|ψn ≈ 10−2 .
(6)
2
The average distance between nucleons and electrons is supposed to be 0.5 × 10−3 m. And the Compton wavelength of the pseudoscalar particle is supposed to be 10−3 m. Then we get the potential between such a pair of nucleon and electron as
ψn |
g s g p h¯ σ · r
8π me c
mϕ +
1 r
e −rmϕ r2
|ψ0 ≈ 2.8 × 10−8 g s g p .
(7)
The chirality factor χ is must be introduced to measure the asymmetry of the chiral molecule [24]. In most literatures, the factor takes the value χ = 10−3 . With the factor, the potential between a nucleon and an electron is about 2.8 × 10−13 g s g p . Now, we calculate the matrix element with the crystal field theory. Frequently, this theory is used to determine the correction of d orbit of electrons in molecules [16]. Here, it is used to study a kind of chiral molecules that each molecule is made up of a central atom and several ligands. In the theory, ligands and the central atom are regarded as point-like particles. Valence electrons are regarded to be in hydrogen-like atomic orbits. In this picture, only the Coulomb potential provided by ligands effectively impact valence electrons. Let r, θ , ϕ and R, Θ , Φ be the spherical coordinates of valence electrons and ligands, respectively. Then the Coulomb potential of a valence electron is represented as [16] U (r ) =
l ∞ l=0 m=−l
4π 2l + 1
Y lm (θ, ϕ )
ρ ( R , Θ, Φ)
l r< l+1 r>
∗
Y lm (Θ, Φ)
dτ , (8)
where r> and r< represent the bigger one and the smaller one of r and R respectively. Y lm (θ, ϕ ) is the usual spherical harmonics, and ρ ( R , Θ, Φ) is the charge density of ligands in the molecule. Write the wave function of a electron as the product of the radial component and the angular component |ψ = R C L (r ) · Y LM (θ, ϕ ), where C , L and M are the main quantum number, the angular moment quantum number and the spin magnetic quantum number respectively. These matrix elements in the potential (4) can be given according to quantum mechanics and the coupling law of angular momentums. In Fig. 1, the system of a rectangular copper block and a rectangular vessel filled with homochiral molecules is shown. We suppose the copper block and vessel have the same section, but different thickness. The nucleon number N 1 in the copper block and the molecule number N 2 in the vessel can be determined approximately according to their volumes and densities. The number of valence electrons in a molecule is supposed to be N 3 . We make a coordinate whose origin is placed at the center of the vessel and the z axis points to the mass centre of the copper block. In the coordinate, the kth chiral molecule has a central atom with the position vector rk , its ith valence electron has the position vector r i . The jth nucleon in the copper block has the position vector r j . The displacement vector from the central atom of the kth chiral molecule to the jth nucleon is rkj . Other displacement vector of the ith valence electron with respect to the central atom is rki , and we have the equation r i j = rkj − rki . We note that the vector rkj has a macroscopic scale, but the vector rki has only a microscopic scale. So, we can take |r i j | ≈ |rkj | approximately. The electron wave
Y. Hu et al. / Physics Letters A 373 (2008) 9–12
11
0.08 × 0.01 m3 and 0.08 × 0.08 × 0.002 m3 , respectively. Take the inter-plane distance to be 0.5 mm. The force between the copper block and the chiral molecules in the rectangular vessel is about −2.36 × 1025 g s g p for λ = 10−3 m. The measured limit of scalar and pseudo-scalar coupling constants is g s g p /(¯hc ) < 1.5 × 10−24 for λ = 10−3 m [3]. So the force is less than 1.12 × 10−24 N. And the rate of the force to the gravity for the copper block and the vessel of homochiral molecules is less than 5.4 × 10−15 , which is consistent with Pospelov’s result [6]. Therefore we can say that the chirality of the test mass plays a meaningless role in the torsion pendulum experiment at present and its effect on the test of the equivalence principle is negligible [28]. 4. Conclusions and discussion Fig. 1. The geometric configuration of the copper cube and the cubical vessel filled with homochiral molecules and the scheme of position vectors of the central atom of the kth molecule, the ith valence electron of the molecule in the vessel and the jth nucleon in the copper cube. The origin of the coordinate is put at the mass center of the vessel. Table 1 The charge and spherical coordinates of atoms in methyl phenyl carbinol nitrite. Atom
H0
H1
H2
H3
C1
C2
Charge R
0.28 1.13256 1.9036 5.6589
0.28 2.2104 2.0969 2.0969
0.28 2.2104 2.0969 1.0246
0.28 2.1967 1.3877 1.5708
−0.28 0.00 0.00 0.00
−0.84 0.49 3.1416 0.00
Θ Φ
function and the spin–orbit coupling are both the functions of rki . Therefore, the product of matrix elements in Eq. (4) can be written as
1 e −ri j mϕ ψin | mϕ + r i j |ψi0 · ψi0 |ξ(r i )li |ψin 2 ri j
ri j
1 e −rkj mϕ ψin |rki |ψi0 · ψi0 |ξ(rki )li |ψin . ≈ − mϕ + 2 rkj
rkj
(9)
According to the potential (4), the macroscopic force between the copper block and the vessel of homochiral molecules along the z axis can be expressed as 3 (m r 2 1 − g s g p h¯ ϕ kj + 3mϕ rkj + 3)| z j − zk |e
N
Fz ≈
4π me c
N
N
2 2
k=1 j =1 i =1
× Re
n=0
i =1 n=0
E 0i − E ni
≈ 4 × 10−18 .
We would like to thank Dr. Chengang Shao for helpful discussions. This study was supported by National Basic Research Program of China (Grant No. 2003CB716300) and the National Natural Science Foundation of China (No. 10121503). References
(10)
The summations over all valence electrons and nucleons can be done through integration over the volumes of the copper block and the vessel. Now we calculate the force for homochiral molecule methyl phenyl carbinol nitrite in the vessel. The efficient charges and coordinates of ions in a methyl phenyl carbinol nitrite molecule (lefthanded or right-handed) are listed in Table 1 [27]. The unit of R is angstrom and the charges are given in electrostatic unit. With these data, the value of the products of matrix elements in Eq. (4) can be given according to the crystal field theory. Their sum over valence electrons is N3 Re{ψni |r i |ψ0i ψ0i |ξ(r i )li |ψni }
Acknowledgements
−rkj mϕ
5 rkj
ψin |rki |ψi0 ψi0 |ξ(rki )li |ψin . E i0 − E in
We have demonstrated that the effective potential describing the interaction mediated by light particles, such as axions, can produce a parity-violating macroscopic force between chiral molecules and achiral matter. This force is equal in magnitude but opposite in sign for molecules with different chirality. This difference may conceivably be detectable experimentally and it may be a possible factor causing the violation of the equivalence principle. We calculated the magnitude of the force between a copper block and the homochiral molecules (methyl phenyl carbinol nitrite) in a rectangular vessel with the crystal field theory. For the interaction range λ = 10−3 m, the calculated force is estimated to be −2.36 × 1025 g s g p . According to the published limits on the scalar and pseudo-scalar coupling constants [3], the force is less than 1.12 × 10−24 N, which is still far beyond the up-to-date detection limit in laboratory [7–9]. The rate of the calculated force to the gravity is less than 5.4 × 10−15 . So the influence of such force to laboratory gravitational experiments may be safely ignored and its influence in the test of the equivalence principle is too weaker to be determined [28].
(11)
For a methyl phenyl carbinol nitrite molecule, E 0i − E ni ≈ 1 eV and N 3 = 48 is taken in Eq. (11). Take the dimensions of the copper block and the rectangular vessel in Fig. 1 to be 0.08 ×
[1] J.E. Moody, F. Wilczek, Phys. Rev. D 30 (1984) 130. [2] T.C.P. Chui, W.-T. Ni, Phys. Rev. Lett. 71 (1993) 3247. [3] G.D. Hammond, C.C. Speake, C. Trenkel, A.P. Paton, Phys. Rev. Lett. 98 (2007) 081101. [4] A.N. Youdin, D. Krause Jr., K. Jagannathan, L.R. Hunter, Phys. Rev. Lett. 77 (1996) 2170. [5] W.T. Ni, S.S. Pan, H.C. Yeh, L.S. Hou, J. Wan, Phys. Rev. Lett. 82 (1999) 2439. [6] M. Pospelov, Phys. Rev. D 58 (1998) 097703. [7] B.R. Heckel, C.E. Cramer, T.S. Cook, E.G. Adelberger, S. Schlamminger, U. Schmidt, Phys. Rev. Lett. 97 (2006) 021603. [8] C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt, H.E. Swanson, Phys. Rev. D 70 (2004) 042004. [9] J.H. Gundlach, G.L. Smith, E.G. Adelberger, B.R. Heckel, H.E. Swanson, Phys. Rev. Lett. 78 (1997) 2523. [10] E.U. Condon, Rev. Mod. Phys. 9 (1937) 532. [11] D. Kang, C. Rosenblatt, F.M. Aliev, Phys. Rev. Lett. 79 (1997) 4826. [12] D.K. Kondepudi, G.W. Nelson, Nature 314 (1985) 438. [13] A.S. Lahamer, S.M. Mahurin, R.N. Compton, D. House, J.K. Laerdahl, M. Lein, P. Schwerdtfeger, Phys. Rev. Lett. 85 (2000) 4470. [14] A. Bakasov, T.K. Ha, M. Quack, J. Chem. Phys. 109 (1998) 7263. [15] J.K. Laerdahl, P. Schwerdtfeger, H.M. Quiney, Phys. Rev. Lett. 84 (2000) 3811. [16] H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry, Wiley, New York, 1944. [17] J.C. Slater, Quantum Theory of Matter, McGraw–Hill, New York, 1968. [18] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965, Chapter X. [19] M.A. Bouchiat, C. Bouchiat, J. Phys. (Paris) 35 (1974) 899.
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M.A. Bouchiat, C. Bouchiat, J. Phys. (Paris) 36 (1975) 493. B.Y. Zel’dovich, et al., Zh. Eksp. Teor. Fiz. Pis’ma Red. 25 (1977) 106. S.F. Mason, G.E. Tranter, Mol. Phys. 53 (1984) 1091. R.A. Hegstrom, D.W. Rein, P.G.H. Sardars, J. Chem. Phys. 73 (1980) 2329. R.A. Harris, L. Stodosky, Phys. Lett. B 78 (1978) 313.
[25] [26] [27] [28]
G.L. Tan, M.F. Lemon, D.J. Jones, R.H. French, Phys. Rev. B 72 (2005) 205117. P. Rudra, J. Math. Phys. 6 (1965) 1278. E.U. Condon, W. Altar, H. Eyring, J. Chem. Phys. 5 (1937) 753. S. Schlamminger, K.-Y. Choi, T.A. Wagner, J.H. Gundlach, E.G. Adelberger, Phys. Rev. Lett. 100 (2008) 041101.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Chirality-dependent macroscopic force between chiral molecules and achiral matter Yonghong Hu, Zhongzhu Liu, Qing Xu, Jun Luo ∗ Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 20 July 2008 Received in revised form 23 October 2008 Accepted 30 October 2008 Available online 6 November 2008 Communicated by P.R. Holland PACS: 14.80.Mz 11.30.Rd 31.15.B-
a b s t r a c t A non-zero macroscopic chirality-dependent force between achiral matter and homochiral molecules due to the exchange of light particles is shown theoretically. It has the opposite sign for molecules with opposite chirality. As an example, this force between a copper block and a vessel of chiral molecules (methyl phenyl carbinol nitrite) is calculated in the crystal field theory. The magnitude of the force is estimated with the published limits of scalar and pseudo-scalar coupling constants. Its possible influence to the gravitational experiments testing the equivalence principle is discussed as well. © 2008 Published by Elsevier B.V.
Keywords: Macroscopic force Axion Chiral molecule Parity violation
1. Introduction Recently, the force mediated by axions or other particles has become a focus of researches [1–7]. Axion has an abstruse parentage. So, force mediated by it displays complicated forms [1]. It is detected as the force between spins [2]. And it also is detected as a force magnetizing electrons [3–5]. Moreover, the CP asymmetry of the force is emphasized. One proposed to test this property by using precise measurement of gravitational forces [6–9]. Before doing such test, one must know how a chiral force can form between macroscopic objects. In this paper, we shall study the force between achiral matter and homochiral molecules mediated by axions. The force appears between valence electrons in the chiral molecules and nucleons in the achiral matter. Our research shows that the total force is nonzero. Moreover, under a space reflection, a left-hand molecule becomes a right-hand one. Synchronously, the force between chiral molecules and the achiral matter converts its sign. Such chirality-dependent force may be a new factor to influence the test of the equivalence principle. Chirality is a basic configuration feature of material, crystals and organic molecules. These materials and their mirror image enantiomers have the same composition but different geometrical
*
Corresponding author. Tel.: +86 027 87556653; fax: +86 027 87542391. E-mail address: [email protected] (J. Luo).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.10.082
structure, left screw or right screw. They have a series of properties, for example, rotatory power and parity non-conserving energy differences [10–15]. Here we give the potential between achiral matter and a vessel of homochiral molecules. Further we count the potential with the crystal field theory and estimate the magnitude of the force between a copper block and a vessel of homochiral molecules. The possible effects in the torsion pendulum experiments testing the equivalence principle are discussed. 2. Theory and model The chirality-dependent macroscopic force due to the exchange of axion may influence the equivalence principle [7–9]. Pospelov’s results indicate the compositions of material may influence the equivalence principle [6]. Our researches will indicate the configurations of material may influence the test of the equivalence principle. When the chirality of material is changed, the sign of the macroscopic force changed. Such parity-violating force due to the exchange of axion is possible to influence the test of the equivalence principle. Now let’s see the effective spin-dependent potential proposed by Moody and Wilczek [1]
H int = h¯ ( g s g p )
σ · rˆ mϕ 1 + 2 e −mϕ r , 8π me c r r
(1)
where c is the light speed in vacuum, h¯ σ /2 is the electron spin, r is the displacement vector between the nucleon and the elec-
10
Y. Hu et al. / Physics Letters A 373 (2008) 9–12
tron, g s and g p are the scalar and pseudo-scalar coupling constants respectively, me is the mass of the electron and mϕ is the mass of exchanged pseudo-scalar particle. The interaction range is supposed as λ = h¯ /mϕ c. Within the electron system of a chiral molecule, the space distribution of valence electrons displays the chiral character of the molecule. The sum of the potential (1) over all nucleons and molecules will be zero due to the zero total magnetic moment of the molecules. However, the potential may take a non-zero value when the effect of the spin–orbital interaction of valence electrons is involved. The valence electrons and nucleons in our study belong to two macroscopic bodies respectively. So, we can study the motions of valence electrons with the adiabatic approximation [16]. Then the total Hamiltonian of the valence electron system in the homochiral molecules can be written as
ˆ = Hˆ 0 + Hˆ 1 + Hˆ 2 , H ˆ1 H
where
M
is
i =1 ξ(r i )l i
the
potential
ˆ2 = − H
(1),
(2)
M
B (r i ) ·
i =1
σi =
· σi represents the orbit–spin interaction Hamiltonian
ˆ 0 is the Hamiltonian of the of the valence electrons [17] and H valence electron system involving only the Coulomb interaction. Moreover, B (r i ) is the magnetic induction at the position of the ith valence electron, li and σi /2 are the orbital angular momentum
2 and the spin of the ith valence electron. And ξ(r ) = h¯2 2
me c
1 ∂ U (r ) r ∂r
is
the orbit–spin coupling coefficient for the electron in the Coulomb potential U (r ). The energy shift of the valence electron system due to the ˆ 1 and Hˆ 2 can be given with the stationary state Hamiltonian H perturbation theory [18]. A simple calculation in the second order correction of the energy shows that the chirality-dependent ˆ 1 + Hˆ 2 ) is [19–21] potential induced by the Hamiltonian ( H EV = 2
Re{Ψn | H 1 |Ψ0 Ψ0 | H 2 |Ψn } E 0 − En
n=0
(3)
.
Where |Ψ is the wave function of valence electron system in the molecules, the subscript “0” represents the initial state and “n” represents final state. And Re{ } denotes the retention of the real part of the products of matrix elements. With the single-electron approximation, the wave functions |Ψ are written as the product of single-electron wave functions [22]. Then, we further factor each single-electron wave function as the product of the orbital wave function |ψ and the spin wave function |m [23]. Employing the relation of (σ · A )(σ · B ) = A · B + i σ · A × B, the sum over spins reduces the expression (3) to the following form, EV = 2
i
j
n=0
1 E i0 − E in
Re ψin |
g s g p h¯ 8π me c
mϕ +
1
ri j
e −ri j mϕ × r |ψ · ψ |ξ( r ) l |ψ . ij i0 i0 i i in 2 ri j
(4)
Where r i j is the displacement vector from the ith electron to the jth nucleon. |ψi0 and |ψin are the initial and final states of the ith valence electron at the energy levels of E i0 and E in respectively. Let |ψ L be a state of left-handed molecule. Under the space reflection P , it becomes a state of the right-handed molecule |ψ R as |ψ R = P |ψ L . The orbital angular momentum li is an axial vector. It doesn’t change under the space reflection. But r i j will change its sign as a polar vector. Therefore the products of matrix elements have the following relation:
ψ Ln | H 1 |ψ L0 ψ L0 | H 2 |ψ Ln = −ψ Rn | H 1 |ψ R0 ψ R0 | H 2 |ψ Rn .
(5)
It means that the energy E V takes the opposite value for the molecules with opposite chirality.
3. Calculation and result Before the calculation of the potential (4), we firstly estimate its magnitude. The energy difference can be taken as ( E 0 − E n ) ≈ 1 eV [24]. The matrix element ψ0 | h2¯ σ · ξ(r )l|ψn is approximated as the energy split due to the spin–orbit interaction [25]. In published theoretical calculations, the value can be read that h2¯ ψ0 |σ · ξ(r )l|ψn ≈ 0.01 eV [26]. So, we get the estimation 1
(E 0 − En )
h¯ ψ0 | σ · ξ(r )l|ψn ≈ 10−2 .
(6)
2
The average distance between nucleons and electrons is supposed to be 0.5 × 10−3 m. And the Compton wavelength of the pseudoscalar particle is supposed to be 10−3 m. Then we get the potential between such a pair of nucleon and electron as
ψn |
g s g p h¯ σ · r
8π me c
mϕ +
1 r
e −rmϕ r2
|ψ0 ≈ 2.8 × 10−8 g s g p .
(7)
The chirality factor χ is must be introduced to measure the asymmetry of the chiral molecule [24]. In most literatures, the factor takes the value χ = 10−3 . With the factor, the potential between a nucleon and an electron is about 2.8 × 10−13 g s g p . Now, we calculate the matrix element with the crystal field theory. Frequently, this theory is used to determine the correction of d orbit of electrons in molecules [16]. Here, it is used to study a kind of chiral molecules that each molecule is made up of a central atom and several ligands. In the theory, ligands and the central atom are regarded as point-like particles. Valence electrons are regarded to be in hydrogen-like atomic orbits. In this picture, only the Coulomb potential provided by ligands effectively impact valence electrons. Let r, θ , ϕ and R, Θ , Φ be the spherical coordinates of valence electrons and ligands, respectively. Then the Coulomb potential of a valence electron is represented as [16] U (r ) =
l ∞ l=0 m=−l
4π 2l + 1
Y lm (θ, ϕ )
ρ ( R , Θ, Φ)
l r< l+1 r>
∗
Y lm (Θ, Φ)
dτ , (8)
where r> and r< represent the bigger one and the smaller one of r and R respectively. Y lm (θ, ϕ ) is the usual spherical harmonics, and ρ ( R , Θ, Φ) is the charge density of ligands in the molecule. Write the wave function of a electron as the product of the radial component and the angular component |ψ = R C L (r ) · Y LM (θ, ϕ ), where C , L and M are the main quantum number, the angular moment quantum number and the spin magnetic quantum number respectively. These matrix elements in the potential (4) can be given according to quantum mechanics and the coupling law of angular momentums. In Fig. 1, the system of a rectangular copper block and a rectangular vessel filled with homochiral molecules is shown. We suppose the copper block and vessel have the same section, but different thickness. The nucleon number N 1 in the copper block and the molecule number N 2 in the vessel can be determined approximately according to their volumes and densities. The number of valence electrons in a molecule is supposed to be N 3 . We make a coordinate whose origin is placed at the center of the vessel and the z axis points to the mass centre of the copper block. In the coordinate, the kth chiral molecule has a central atom with the position vector rk , its ith valence electron has the position vector r i . The jth nucleon in the copper block has the position vector r j . The displacement vector from the central atom of the kth chiral molecule to the jth nucleon is rkj . Other displacement vector of the ith valence electron with respect to the central atom is rki , and we have the equation r i j = rkj − rki . We note that the vector rkj has a macroscopic scale, but the vector rki has only a microscopic scale. So, we can take |r i j | ≈ |rkj | approximately. The electron wave
Y. Hu et al. / Physics Letters A 373 (2008) 9–12
11
0.08 × 0.01 m3 and 0.08 × 0.08 × 0.002 m3 , respectively. Take the inter-plane distance to be 0.5 mm. The force between the copper block and the chiral molecules in the rectangular vessel is about −2.36 × 1025 g s g p for λ = 10−3 m. The measured limit of scalar and pseudo-scalar coupling constants is g s g p /(¯hc ) < 1.5 × 10−24 for λ = 10−3 m [3]. So the force is less than 1.12 × 10−24 N. And the rate of the force to the gravity for the copper block and the vessel of homochiral molecules is less than 5.4 × 10−15 , which is consistent with Pospelov’s result [6]. Therefore we can say that the chirality of the test mass plays a meaningless role in the torsion pendulum experiment at present and its effect on the test of the equivalence principle is negligible [28]. 4. Conclusions and discussion Fig. 1. The geometric configuration of the copper cube and the cubical vessel filled with homochiral molecules and the scheme of position vectors of the central atom of the kth molecule, the ith valence electron of the molecule in the vessel and the jth nucleon in the copper cube. The origin of the coordinate is put at the mass center of the vessel. Table 1 The charge and spherical coordinates of atoms in methyl phenyl carbinol nitrite. Atom
H0
H1
H2
H3
C1
C2
Charge R
0.28 1.13256 1.9036 5.6589
0.28 2.2104 2.0969 2.0969
0.28 2.2104 2.0969 1.0246
0.28 2.1967 1.3877 1.5708
−0.28 0.00 0.00 0.00
−0.84 0.49 3.1416 0.00
Θ Φ
function and the spin–orbit coupling are both the functions of rki . Therefore, the product of matrix elements in Eq. (4) can be written as
1 e −ri j mϕ ψin | mϕ + r i j |ψi0 · ψi0 |ξ(r i )li |ψin 2 ri j
ri j
1 e −rkj mϕ ψin |rki |ψi0 · ψi0 |ξ(rki )li |ψin . ≈ − mϕ + 2 rkj
rkj
(9)
According to the potential (4), the macroscopic force between the copper block and the vessel of homochiral molecules along the z axis can be expressed as 3 (m r 2 1 − g s g p h¯ ϕ kj + 3mϕ rkj + 3)| z j − zk |e
N
Fz ≈
4π me c
N
N
2 2
k=1 j =1 i =1
× Re
n=0
i =1 n=0
E 0i − E ni
≈ 4 × 10−18 .
We would like to thank Dr. Chengang Shao for helpful discussions. This study was supported by National Basic Research Program of China (Grant No. 2003CB716300) and the National Natural Science Foundation of China (No. 10121503). References
(10)
The summations over all valence electrons and nucleons can be done through integration over the volumes of the copper block and the vessel. Now we calculate the force for homochiral molecule methyl phenyl carbinol nitrite in the vessel. The efficient charges and coordinates of ions in a methyl phenyl carbinol nitrite molecule (lefthanded or right-handed) are listed in Table 1 [27]. The unit of R is angstrom and the charges are given in electrostatic unit. With these data, the value of the products of matrix elements in Eq. (4) can be given according to the crystal field theory. Their sum over valence electrons is N3 Re{ψni |r i |ψ0i ψ0i |ξ(r i )li |ψni }
Acknowledgements
−rkj mϕ
5 rkj
ψin |rki |ψi0 ψi0 |ξ(rki )li |ψin . E i0 − E in
We have demonstrated that the effective potential describing the interaction mediated by light particles, such as axions, can produce a parity-violating macroscopic force between chiral molecules and achiral matter. This force is equal in magnitude but opposite in sign for molecules with different chirality. This difference may conceivably be detectable experimentally and it may be a possible factor causing the violation of the equivalence principle. We calculated the magnitude of the force between a copper block and the homochiral molecules (methyl phenyl carbinol nitrite) in a rectangular vessel with the crystal field theory. For the interaction range λ = 10−3 m, the calculated force is estimated to be −2.36 × 1025 g s g p . According to the published limits on the scalar and pseudo-scalar coupling constants [3], the force is less than 1.12 × 10−24 N, which is still far beyond the up-to-date detection limit in laboratory [7–9]. The rate of the calculated force to the gravity is less than 5.4 × 10−15 . So the influence of such force to laboratory gravitational experiments may be safely ignored and its influence in the test of the equivalence principle is too weaker to be determined [28].
(11)
For a methyl phenyl carbinol nitrite molecule, E 0i − E ni ≈ 1 eV and N 3 = 48 is taken in Eq. (11). Take the dimensions of the copper block and the rectangular vessel in Fig. 1 to be 0.08 ×
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