- Email: [email protected]
A concrete realization of specific heat-phonon spectrum inversion for YBCO
13 December 1999
Physics Letters A 264 Ž1999. 68–73 www.elsevier.nlrlocaterphysleta
A concrete realization of specific heat-phonon spectrum inversion for YBCO Dai XianXi a
a,b,)
, Tao Wen a , GuiCun Ma a , JiXin Dai
c
Group of Quantum Statistics and Method of Theoretical Physics, Department of Physics, Fudan UniÕersity, Shanghai 200433, PR China b International Center for Theoretical Physics, Trieste, Italy c Department of Chemistry, New York UniÕersity, New York, NY 10003, USA Received 1 September 1999; accepted 18 October 1999 Communicated by L.J. Sham
Abstract In this Letter, a phonon spectrum of YBCO is obtained from its experimental specific heat data by an exact inversion formula with eliminating divergence parameter wDai Xianxi, Xu Xinwen and Dai Jiqiong, Proceedings of Beijing International Conference on High Tc Superconductivity, Sept. 4–8 Beijing, China, Ž1989. 521x, wDai Xianxi, Xu Xinwen and Dai Jiqiong, Phys. Lett. A 147 Ž1990. 445x. The results are comparable to that from neutron inelastic scattering. Some key points of specific heat-phonon spectrum inversion ŽSPI. theory as well as a method of asymptotic behavior control are discussed. An improved unique existence theorem is presented. A universal function set for the numerical calculation in SPI is obtained, which will make the inversion method applicable and convenient in practice. This is the first time to realize the specific heat-phonon spectrum inversion in a concrete system. q 1999 Elsevier Science B.V. All rights reserved. PACS: 74.25.B; 65.40; 02.
1. The specific heat-phonon spectrum inversion problem The specific heat-phonon spectrum inversion ŽwSPIx. has attracted theoretical and application interests for many years w1–6x. In the early 90’s, N.X. Chen proposed an interesting result based on Mobius inversion formula w6x. But up to now, no one has been successful to obtain the phonon spectrum directly from real specific heat data.
Suppose that the renormalized effective Hamiltonian of lattice is: Hˆ s Ý " v q bˆ qq bˆ q
Ž 1.1 .
q
the specific heat can be expressed in the unit system with k B s " s 1 as: `
CÕ Ž T . s
H0
v
ž / T
2
exp Ž vrT . exp Ž vrT . y 1
2
g Ž v . dv
Ž 1.2 .
)
Corresponding author. E-mail: [email protected]
Where g Ž v .,T,k B and " are the density of states of phonon or phonon spectrum, temperature, Boltzmann constant and Planck constant respectively. The spe-
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 6 1 - 6
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
cific heat-phonon spectrum inversion problem is as follow: Instead of knowing the g Ž v . to get the specific heat C Õ ŽT ., if one measures the C Õ ŽT . from experiment first, what is the g Ž v .? In our previous work w1,2x, an exact solution of g Ž v . is obtained with an important parameter s for canceling divergence: gŽ v. 1 s
v =
v
`
Hy`
i kqs
ž / T0
Q˜0 Ž k . dk
Ž ik q s q 1 . G Ž ik q s q 1 . z Ž ik q s q 1 . Ž 1.3 .
where Q˜0 Ž k . is the Fourier transform of Q0 Ž x .: Q0 Ž x . s C Õ Ž T0 e x . eys x
Ž 1.4 .
ŽT0 is a free parameter with temperature dimension.. According to this formula, inputting the C Õ , one can get the phonon spectrum as an output. As pointed out in Refs. w1,2x, the Riemann zeta function z Ž i k q 1. in the denominator is falling in the Riemann strip when s s 0. Riemann hypothesis is an unsolved problem for more than a century. Although it was proved in the Hadamard’s famous paper w7x that for z s i k and z s 1 q i k, z Ž z . has no zeros, the parameter s is still needed for canceling the divergence in the Fourier transforms. Because in general cases,
™`
™ const.; A T
™0
™; A T
lim C Õ Ž T .
T
s1
Ž 1.5 .
which implies the non-existence of Q˜0 Ž k . in the classic function class. Considering the asymptotic behavior of C Õ : lim C Õ Ž T .
T
s2
'A T D.
Ž 1.6 .
where D is the dimension, we introduced the parameter s for canceling divergence: 0 s s1 - s - s 2 s D
Ž 1.7 .
As it was proved w1,2x, for 0 - s - D, the unique existence theorem is guaranteed, and the conclusion is independent with the Riemann hypothesis and Hadamard’s proof. It is needed to be pointed here that the solutions obtained by Montrol w3x, Lifshitz
69
w4x and Chamber w5x are the special case of s s 1 of our formula Ž1.3., but all their solutions can not be applied for low dimensional cases Žas D F 1.. For our result, one can prove that the exact solutions are s-independent as 0 - s - D. It will be shown in this letter that a suitable choice of s is also useful to control the asymptotic behaviors. In addition, based on the formula Ž1.3., the Debye’s spectrum and Einstein’s spectrum has been obtained exactly from the specific heats respectively w1x. We should also emphasize that the inverse Fourier transform may be divergent in previous work due to the limited accuracy and data incompleteness, and make the numerical calculations impossible. Therefore, it is of importance and meaningful to develop the following ‘asymptotic behavior control theory’ and the ‘universal function set method’ so that the concrete specific heat-phone spectrum inversion problem can be realized. As one applies SPI formula to a concrete systems, one will face many practical problems and difficulties. For instance: How to distinguish C p and C Õ from experimental data? How to extract the lattice specific heat? The most important question is how to deal with the instability problem for such an ill-posed integral equation? To make a first attempt, the inversion of high Tc superconductor’s spectrum is chosen as our topic. It is mainly because that there have been many excellent experimental specific heat works in the past several years, e.g. w8,9x. Our criteria for choosing the experimental specific heat data are the suitable accuracy and a wide coverage of temperature regions. Specifically, in the low temperature region, C Õ ; A T s 2 can be seen, while in high temperature region, C Õ turns to be a constant and the Dulong Petit’s law is satisfied. We will discuss later the above conditions are crucial to control the asymptotic behavior. The experimental work of Bessergenev et al w8x on the specific heat of HTS material YBCO is found to be a suitable choice for our purpose. Because in that work, the oxygen components are well controlled in a large temperature region and with higher accuracies. As a reasonable approximation, we neglect the difference between the C p and C Õ for solid, based on the analysis of Landau and Lifshitz w10x. The specific heat of electron can be separated from the total for
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
70
T ) Tc ,Cen s g T. For T - Tc , BCS theory is used to estimate the contribution from electron: Ces s 3.15ty3 r2 ey1 .764r t , Ces s 8.5e
y1 .44r t
t - 0.17 0.17 F t - 0.5
,
Ces s 2.43 q 3.77 Ž t y 1 . ,
0.5 F t - 1.0
where t s TrTc . Then we obtained an analytic expression for phonon specific heat for YBCO.
ž
C Õ Ž T . s a1T a 2 exp y
exp
T q a4
b2
ž / ž /
C Õ Ž T . s b 1T exp y CÕ Ž T . s
a3
T c1 y c 2 T y X0 dx
/
,
T - 16 16 F T - 45
, q c2 ,
T G 45
q1
where a1 s 0.17, a2 s 2.0, a 3 s 117.7, a 4 s 20.0, b 1 s 2.8, b 2 s 50.7, c1 s y1109.4, c 2 s 301.6, X 0 s y110.3, dx s 105.0. We can see that the main contribution is of 2D character and controlled by the third law and Dulong–Petit law. The first class of Fredholm integral equation is known to be a famous ill-posed problem in mathematics. Although the solution does exist uniquely, the small deviation of input may cost a big difference in the output. Much efforts have been made to overcome this instability such as the maximum entropy method, Tikhonov regulation method, et al. But in the problem of SPI, it is studied under some artificial ‘specific heats’. With our understanding, the uniqueness of a solution to the problem is closely related with the function class defined and then with the physical nature. This point is clearly seen in the studies on the singular states in quantum mechanics w11x. Why some singular states are physical and others are not? It depends on the natural laws. These laws, e.g. the orthogonality criteria, select the physical states from the function class. If we consider the phonon spectrum is relatively smooth, and the oscillation with extremely high frequencies can be ignored, then the Hadamard’s instability can be neglected. Another main difficulties in the inversion is the incompleteness of data. In practice, people can not obtain the experimental data in the whole tempera-
ture region 0 F T F `. If there is no information in these limiting region, the solution will be undefined. The important merit of the specific heat-phonon spectrum inversion is these asymptotic behaviors can be controlled from physics: in the ultra low temperature region, it is controlled by the third law and in the high temperature it is governed by the Dulong– Petit law. With the aid of these laws, one can obtain the asymptotic complete data or information. In our case, we can choose s s 1.5 to improve the convergences. At a first glance, one may think the FFT is powerful in our case. But it turns out that FFT does not work in SPI. The main reasons is that after certain k, the F˜0 Ž k . ' Q˜0 Ž k .w G Ž ik q s q 2. z Ž ik q s q 1.xy1 may increase, which means the computing accuracy is inadequate, one should cut off the calculation at certain k; In order to work in a large scale of k, one must increase the computing accuracies of Q˜0 Ž k . and its denominator. However, as k ` the factor G Ž i k q s q 1. does not turn to infinity, but turns to zero exponentially:
™
G Ž z . s AŽ s .
sq 1r2
ž
qi s q
exp ik ln A Ž s .
1
k
/ ž / Ž ½ ž /5 ž / tany1
2
k
='2p exp yktany1 = 1qO
y ik q s q 1 .
sq1
sq1
1 z
(
Ž 1.8 .
2
where AŽ s . s k 2 q Ž s q 1 . and z s ik q s q 1 s AŽ s . exp itany1 Ž krs q 1 . . We found the following expression is useful to get accuracy of 10y9 only keeping the first 5 terms, denoting z s ik q s q n q 1 and n s 9,
G Ž ik q s q 1 . s
eyz z zy1r2 '2p 8
Ł Ž ik q s q j . js1
= 1q
1
1 q
12 z
571 y
2488320 z 4
288 z
139 2
y
q O Ž zy5 .
51840 z 3
Ž 1.9 .
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
71
While calculating z Ž i k q s q 1., instead of using Dirichlet formula: z Ž z . s Ý`ns 11rn z , we apply the Hardy formula Ž1929., which can effectively improve the convergence:
z Ž z. s
`
1 1y2
1yz
1
1
Ý z y z Ž 2 n. ns1 Ž 2 n y 1 . Ž 1.10 .
For example, if s s 1.5 and 1000 terms are taken into account, The accuracies reached by Hardy formula and Dirichlet formula are 10y1 1.5 and 10y3 respectively. One can prove the following ImproÕed Unique Existence Theorem: When the asymptotic behavior of C Õ ŽT . satisfies the following necessary and sufficient condition: < Q˜0 Ž k . < s O z Ž i k q s q 1 . G Ž i k q s q 2 .
½
s O s sq 3r2 exp yk tany1
k
ž /5 sq2
,
Fig. 1. The universal function set Gl Ž v .4.
useful to improve the compactness of the sample points of g Ž v . simultaneously. We choose Hermitian function as our basis, since it is complete and orthonormal set in Žy`,`., and its Fourier Transform can be obtained exactly. Let
Ž 1.11 . the inversion equation Ž1.2. does possess the unique exact solution Ž1.3.. Referring to our previous work w2x and considering the asymptotic behavior of K˜ 0 and the solvable condition of the integral equation, one can prove this theorem. The main improvement here is that the above condition is necessary and sufficient, but it is in the k space. The condition in Ref. w2x is only necessary, and is in the T space directly. Because the oscillating denominator turns to zero exponentially, so the numerator must go to zero faster than the denominator does. But we can not require the limited accuracy of experimental data to guarantee such asymptotic behavior in large k space. This is the reason of increase of F˜0 Ž k . after a certain k in some practical calculations. The best way is to reduce the weight of the F˜0 Ž k . for large k. According to our theorem, one only can search for the solution within the function class, in which C Õ ŽT . satisfies the condition Ž1.11.. After a careful investigation, we find that of the best ways is to expand the C Õ ŽT . or Q˜0 Ž x . by a function set, which is also
1r2
a
ž'
unŽ x . s
n
p 2 n!
/
ey1 r2 a
2
x2
Hn Ž a x .
Ž 1.12 .
Q˜0 Ž x . can be expanded by u l Ž x . as: ny1
Q0 Ž x . s
Ý
Cl u l Ž x .
Ž 1.13 .
Cl u˜ l Ž k .
Ž 1.14 .
ls0
and ny1
Q˜0 Ž k . s
Ý ls0
It is very important to emphasized that the expression Ž1.14. guarantees that the Q˜0 Ž k . satisfies the asymptotic behavior Ž1.11. automatically. The expansion coefficients can be obtained by the orthogonality: `
Cl s
Hy`u Ž x . Q Ž x . dx l
0
Ž 1.15 .
It is interesting to notice, we have the following general solution of the phonon spectrum: ny1
gŽ v. s
Ý ls0
c l Gl Ž v .
Ž 1.16 .
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
72
where G l Ž v .4 are the universal function set, which is system independent and can be calculated in advance:
Gl Ž v . s
1
v
=
`
Hy`
v
i kqs
ž / T0
u˜ l Ž k . dk
G Ž ik q s q 2 . z Ž ik q s q 1 .
Some universal function sets calculated with high accuracies
Ž 1.17 .
G l Ž v .4 have been ŽSee Fig. 1.. Fig. 2. The phonon spectrum obtained of YBCO by specific heatphonon spectrum inversion from the experimental data w8x.
2. Results, discussion and concluding remarks In summary, the procedure for practical calculation of SPI can be outlined as following: 1. Calculate Q0 Ž x . by inputting the experimental data of lattice specific heat C Õ ŽT . into Ž1.4.. 2. Calculate the the coefficient Cl 4 by expanding Q0 Ž x . in the Hermitian function set Ž1.12.. 3. Obtain the phonon spectrum GŽ v . directly from the universal set G l Ž v .4 by Ž1.16.. By inputting the experimental data of specific heat of YBCO w8x and considering the unique existence theorem and asymptotic behavior control, the phonon spectrum of YBCO is obtained and shown in the Fig. 2. The results show that: 1. The phonon spectrum of YBCO obtained by this inversion formula has an acoustic peak, which is in good agreement with that obtained by the neutron inelastic scattering w12x, including the location Žin about 20 meV. and height of the peak. 2. It is well known that the phonon spectrum changes with the oxygen concentrations. Even for the inelastic neutron scattering, the structure of the curves are complicated. What we are expecting is that SPI can obtain the average feature of the spectrum. It is very interesting that our results show a plateau in high frequencies. The plateau covers the region from 40 meV to 80 meV, just likes that observed in the experiments of neutron scattering. This is in qualitatiÕely agreement with the
experiments w12x. It is also interesting that there is a shoulder or hint of small peak at about 50 meV, which is in consistent with the experiments to a certain extent. In fact in the high frequencies, the experiments shows a peak also. It is easy to be understood that in this region, the accuracy of the results are pretty low. Because the integral kernel in Eq. Ž1.3.:
v
ž / T
2
exp Ž vrT . exp Ž vrT . y 1
2
,
goes down exponentially. If one tries to get information in high frequencies, the higher accuracies of the experimental data will be needed. Finally, we need to emphasize that our SPI exact solution formula is of closed form. The Debye spectrum and Einstein spectrum can be exactly obtained from this formula, including the important cutoff factor w1x. Comparing to Chen’s Mobius inversion formula w6x, one can not get cutoff factor, although the v 2 spectrum is obtained. Also the values of Mobius are unknown for large n, because it is hard to determine weather they are prime numbers or not. From practical applications point of view, in our SPI formula, one only need to perform Fourier transform once, while in Mobius inversion formula, an infinite number of Laplace and inverse Laplace transform is
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
needed, which make this method difficult to be applied in numerical calculation. This is the first try to get a practical phonon spectrum from the specific heat. After overcome a series of difficulties, the result from the exact solution formula Ž1.3. or Ž1.16. – Ž1.17. is in good agreement with the experiments by neutron inelastic scattering w12x. Therefore, the theory of the specific heat phonon spectrum inversion can be believed to be a promising theory and method in physics.
Acknowledgements The Authors would like to acknowledge Professors C.N. Yang, N.X. Chen,Yu Lu, W. Evenson and Ma Yongli for their significant discussions and ICTP for their hospitality. This work is in Projects 19834010 and 19575009 and supported by National Natural Science Foundation of China.
73
References w1x Dai Xianxi, Xu Xinwen, Dai Jiqiong, Proc. Beijing International Conference on High Tc Superconductivity, Beijing, China, September 4–8, 1989, pp. 521–524. w2x Dai Xianxi, Xu Xinwen, Dai Jiqiong, Phys. Lett. A. 147 Ž1990. 445. w3x E.W. Montroll, J. Chem. Phys. 10 Ž1942. 218. w4x I.M. Lifshitz, Zh. Eksp. Theor. Fiz 26 Ž1955. 551. w5x R.G. Chambers, Proc. Phys. Soc. 78 Ž1961. 941. w6x N.X. Chen, Phys. Rev. Lett. 64 Ž1990. 1193. w7x J. Hadamard, Bull. Soc. Math. France 24 Ž1896. 199. w8x V.G. Bessergenev, Yu.A. Kovalevskaya, V.N. Naumov, G.I. Frolova, Physica C 245 Ž1995. 36. w9x D.M. Ginsberg, Physical Properties of High Temperature Superconductors I-IV, World Scientific, Singapore et al., 1989–1994. w10x Landau, Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, Beijing et al., 1980. w11x Dai XianXi, JiXin Dai, JiQiong Dai, Phys. Rev. A 55 Ž1997. 2617. w12x B. Renker, F. Gompf, E. Gering, D. Ewert, H. Rietschel, A. Dianoux, Z. Phys. B Condensed Matter 73 Ž1988. 309.
Physics Letters A 264 Ž1999. 68–73 www.elsevier.nlrlocaterphysleta
A concrete realization of specific heat-phonon spectrum inversion for YBCO Dai XianXi a
a,b,)
, Tao Wen a , GuiCun Ma a , JiXin Dai
c
Group of Quantum Statistics and Method of Theoretical Physics, Department of Physics, Fudan UniÕersity, Shanghai 200433, PR China b International Center for Theoretical Physics, Trieste, Italy c Department of Chemistry, New York UniÕersity, New York, NY 10003, USA Received 1 September 1999; accepted 18 October 1999 Communicated by L.J. Sham
Abstract In this Letter, a phonon spectrum of YBCO is obtained from its experimental specific heat data by an exact inversion formula with eliminating divergence parameter wDai Xianxi, Xu Xinwen and Dai Jiqiong, Proceedings of Beijing International Conference on High Tc Superconductivity, Sept. 4–8 Beijing, China, Ž1989. 521x, wDai Xianxi, Xu Xinwen and Dai Jiqiong, Phys. Lett. A 147 Ž1990. 445x. The results are comparable to that from neutron inelastic scattering. Some key points of specific heat-phonon spectrum inversion ŽSPI. theory as well as a method of asymptotic behavior control are discussed. An improved unique existence theorem is presented. A universal function set for the numerical calculation in SPI is obtained, which will make the inversion method applicable and convenient in practice. This is the first time to realize the specific heat-phonon spectrum inversion in a concrete system. q 1999 Elsevier Science B.V. All rights reserved. PACS: 74.25.B; 65.40; 02.
1. The specific heat-phonon spectrum inversion problem The specific heat-phonon spectrum inversion ŽwSPIx. has attracted theoretical and application interests for many years w1–6x. In the early 90’s, N.X. Chen proposed an interesting result based on Mobius inversion formula w6x. But up to now, no one has been successful to obtain the phonon spectrum directly from real specific heat data.
Suppose that the renormalized effective Hamiltonian of lattice is: Hˆ s Ý " v q bˆ qq bˆ q
Ž 1.1 .
q
the specific heat can be expressed in the unit system with k B s " s 1 as: `
CÕ Ž T . s
H0
v
ž / T
2
exp Ž vrT . exp Ž vrT . y 1
2
g Ž v . dv
Ž 1.2 .
)
Corresponding author. E-mail: [email protected]
Where g Ž v .,T,k B and " are the density of states of phonon or phonon spectrum, temperature, Boltzmann constant and Planck constant respectively. The spe-
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 6 1 - 6
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
cific heat-phonon spectrum inversion problem is as follow: Instead of knowing the g Ž v . to get the specific heat C Õ ŽT ., if one measures the C Õ ŽT . from experiment first, what is the g Ž v .? In our previous work w1,2x, an exact solution of g Ž v . is obtained with an important parameter s for canceling divergence: gŽ v. 1 s
v =
v
`
Hy`
i kqs
ž / T0
Q˜0 Ž k . dk
Ž ik q s q 1 . G Ž ik q s q 1 . z Ž ik q s q 1 . Ž 1.3 .
where Q˜0 Ž k . is the Fourier transform of Q0 Ž x .: Q0 Ž x . s C Õ Ž T0 e x . eys x
Ž 1.4 .
ŽT0 is a free parameter with temperature dimension.. According to this formula, inputting the C Õ , one can get the phonon spectrum as an output. As pointed out in Refs. w1,2x, the Riemann zeta function z Ž i k q 1. in the denominator is falling in the Riemann strip when s s 0. Riemann hypothesis is an unsolved problem for more than a century. Although it was proved in the Hadamard’s famous paper w7x that for z s i k and z s 1 q i k, z Ž z . has no zeros, the parameter s is still needed for canceling the divergence in the Fourier transforms. Because in general cases,
™`
™ const.; A T
™0
™; A T
lim C Õ Ž T .
T
s1
Ž 1.5 .
which implies the non-existence of Q˜0 Ž k . in the classic function class. Considering the asymptotic behavior of C Õ : lim C Õ Ž T .
T
s2
'A T D.
Ž 1.6 .
where D is the dimension, we introduced the parameter s for canceling divergence: 0 s s1 - s - s 2 s D
Ž 1.7 .
As it was proved w1,2x, for 0 - s - D, the unique existence theorem is guaranteed, and the conclusion is independent with the Riemann hypothesis and Hadamard’s proof. It is needed to be pointed here that the solutions obtained by Montrol w3x, Lifshitz
69
w4x and Chamber w5x are the special case of s s 1 of our formula Ž1.3., but all their solutions can not be applied for low dimensional cases Žas D F 1.. For our result, one can prove that the exact solutions are s-independent as 0 - s - D. It will be shown in this letter that a suitable choice of s is also useful to control the asymptotic behaviors. In addition, based on the formula Ž1.3., the Debye’s spectrum and Einstein’s spectrum has been obtained exactly from the specific heats respectively w1x. We should also emphasize that the inverse Fourier transform may be divergent in previous work due to the limited accuracy and data incompleteness, and make the numerical calculations impossible. Therefore, it is of importance and meaningful to develop the following ‘asymptotic behavior control theory’ and the ‘universal function set method’ so that the concrete specific heat-phone spectrum inversion problem can be realized. As one applies SPI formula to a concrete systems, one will face many practical problems and difficulties. For instance: How to distinguish C p and C Õ from experimental data? How to extract the lattice specific heat? The most important question is how to deal with the instability problem for such an ill-posed integral equation? To make a first attempt, the inversion of high Tc superconductor’s spectrum is chosen as our topic. It is mainly because that there have been many excellent experimental specific heat works in the past several years, e.g. w8,9x. Our criteria for choosing the experimental specific heat data are the suitable accuracy and a wide coverage of temperature regions. Specifically, in the low temperature region, C Õ ; A T s 2 can be seen, while in high temperature region, C Õ turns to be a constant and the Dulong Petit’s law is satisfied. We will discuss later the above conditions are crucial to control the asymptotic behavior. The experimental work of Bessergenev et al w8x on the specific heat of HTS material YBCO is found to be a suitable choice for our purpose. Because in that work, the oxygen components are well controlled in a large temperature region and with higher accuracies. As a reasonable approximation, we neglect the difference between the C p and C Õ for solid, based on the analysis of Landau and Lifshitz w10x. The specific heat of electron can be separated from the total for
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
70
T ) Tc ,Cen s g T. For T - Tc , BCS theory is used to estimate the contribution from electron: Ces s 3.15ty3 r2 ey1 .764r t , Ces s 8.5e
y1 .44r t
t - 0.17 0.17 F t - 0.5
,
Ces s 2.43 q 3.77 Ž t y 1 . ,
0.5 F t - 1.0
where t s TrTc . Then we obtained an analytic expression for phonon specific heat for YBCO.
ž
C Õ Ž T . s a1T a 2 exp y
exp
T q a4
b2
ž / ž /
C Õ Ž T . s b 1T exp y CÕ Ž T . s
a3
T c1 y c 2 T y X0 dx
/
,
T - 16 16 F T - 45
, q c2 ,
T G 45
q1
where a1 s 0.17, a2 s 2.0, a 3 s 117.7, a 4 s 20.0, b 1 s 2.8, b 2 s 50.7, c1 s y1109.4, c 2 s 301.6, X 0 s y110.3, dx s 105.0. We can see that the main contribution is of 2D character and controlled by the third law and Dulong–Petit law. The first class of Fredholm integral equation is known to be a famous ill-posed problem in mathematics. Although the solution does exist uniquely, the small deviation of input may cost a big difference in the output. Much efforts have been made to overcome this instability such as the maximum entropy method, Tikhonov regulation method, et al. But in the problem of SPI, it is studied under some artificial ‘specific heats’. With our understanding, the uniqueness of a solution to the problem is closely related with the function class defined and then with the physical nature. This point is clearly seen in the studies on the singular states in quantum mechanics w11x. Why some singular states are physical and others are not? It depends on the natural laws. These laws, e.g. the orthogonality criteria, select the physical states from the function class. If we consider the phonon spectrum is relatively smooth, and the oscillation with extremely high frequencies can be ignored, then the Hadamard’s instability can be neglected. Another main difficulties in the inversion is the incompleteness of data. In practice, people can not obtain the experimental data in the whole tempera-
ture region 0 F T F `. If there is no information in these limiting region, the solution will be undefined. The important merit of the specific heat-phonon spectrum inversion is these asymptotic behaviors can be controlled from physics: in the ultra low temperature region, it is controlled by the third law and in the high temperature it is governed by the Dulong– Petit law. With the aid of these laws, one can obtain the asymptotic complete data or information. In our case, we can choose s s 1.5 to improve the convergences. At a first glance, one may think the FFT is powerful in our case. But it turns out that FFT does not work in SPI. The main reasons is that after certain k, the F˜0 Ž k . ' Q˜0 Ž k .w G Ž ik q s q 2. z Ž ik q s q 1.xy1 may increase, which means the computing accuracy is inadequate, one should cut off the calculation at certain k; In order to work in a large scale of k, one must increase the computing accuracies of Q˜0 Ž k . and its denominator. However, as k ` the factor G Ž i k q s q 1. does not turn to infinity, but turns to zero exponentially:
™
G Ž z . s AŽ s .
sq 1r2
ž
qi s q
exp ik ln A Ž s .
1
k
/ ž / Ž ½ ž /5 ž / tany1
2
k
='2p exp yktany1 = 1qO
y ik q s q 1 .
sq1
sq1
1 z
(
Ž 1.8 .
2
where AŽ s . s k 2 q Ž s q 1 . and z s ik q s q 1 s AŽ s . exp itany1 Ž krs q 1 . . We found the following expression is useful to get accuracy of 10y9 only keeping the first 5 terms, denoting z s ik q s q n q 1 and n s 9,
G Ž ik q s q 1 . s
eyz z zy1r2 '2p 8
Ł Ž ik q s q j . js1
= 1q
1
1 q
12 z
571 y
2488320 z 4
288 z
139 2
y
q O Ž zy5 .
51840 z 3
Ž 1.9 .
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
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While calculating z Ž i k q s q 1., instead of using Dirichlet formula: z Ž z . s Ý`ns 11rn z , we apply the Hardy formula Ž1929., which can effectively improve the convergence:
z Ž z. s
`
1 1y2
1yz
1
1
Ý z y z Ž 2 n. ns1 Ž 2 n y 1 . Ž 1.10 .
For example, if s s 1.5 and 1000 terms are taken into account, The accuracies reached by Hardy formula and Dirichlet formula are 10y1 1.5 and 10y3 respectively. One can prove the following ImproÕed Unique Existence Theorem: When the asymptotic behavior of C Õ ŽT . satisfies the following necessary and sufficient condition: < Q˜0 Ž k . < s O z Ž i k q s q 1 . G Ž i k q s q 2 .
½
s O s sq 3r2 exp yk tany1
k
ž /5 sq2
,
Fig. 1. The universal function set Gl Ž v .4.
useful to improve the compactness of the sample points of g Ž v . simultaneously. We choose Hermitian function as our basis, since it is complete and orthonormal set in Žy`,`., and its Fourier Transform can be obtained exactly. Let
Ž 1.11 . the inversion equation Ž1.2. does possess the unique exact solution Ž1.3.. Referring to our previous work w2x and considering the asymptotic behavior of K˜ 0 and the solvable condition of the integral equation, one can prove this theorem. The main improvement here is that the above condition is necessary and sufficient, but it is in the k space. The condition in Ref. w2x is only necessary, and is in the T space directly. Because the oscillating denominator turns to zero exponentially, so the numerator must go to zero faster than the denominator does. But we can not require the limited accuracy of experimental data to guarantee such asymptotic behavior in large k space. This is the reason of increase of F˜0 Ž k . after a certain k in some practical calculations. The best way is to reduce the weight of the F˜0 Ž k . for large k. According to our theorem, one only can search for the solution within the function class, in which C Õ ŽT . satisfies the condition Ž1.11.. After a careful investigation, we find that of the best ways is to expand the C Õ ŽT . or Q˜0 Ž x . by a function set, which is also
1r2
a
ž'
unŽ x . s
n
p 2 n!
/
ey1 r2 a
2
x2
Hn Ž a x .
Ž 1.12 .
Q˜0 Ž x . can be expanded by u l Ž x . as: ny1
Q0 Ž x . s
Ý
Cl u l Ž x .
Ž 1.13 .
Cl u˜ l Ž k .
Ž 1.14 .
ls0
and ny1
Q˜0 Ž k . s
Ý ls0
It is very important to emphasized that the expression Ž1.14. guarantees that the Q˜0 Ž k . satisfies the asymptotic behavior Ž1.11. automatically. The expansion coefficients can be obtained by the orthogonality: `
Cl s
Hy`u Ž x . Q Ž x . dx l
0
Ž 1.15 .
It is interesting to notice, we have the following general solution of the phonon spectrum: ny1
gŽ v. s
Ý ls0
c l Gl Ž v .
Ž 1.16 .
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
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where G l Ž v .4 are the universal function set, which is system independent and can be calculated in advance:
Gl Ž v . s
1
v
=
`
Hy`
v
i kqs
ž / T0
u˜ l Ž k . dk
G Ž ik q s q 2 . z Ž ik q s q 1 .
Some universal function sets calculated with high accuracies
Ž 1.17 .
G l Ž v .4 have been ŽSee Fig. 1.. Fig. 2. The phonon spectrum obtained of YBCO by specific heatphonon spectrum inversion from the experimental data w8x.
2. Results, discussion and concluding remarks In summary, the procedure for practical calculation of SPI can be outlined as following: 1. Calculate Q0 Ž x . by inputting the experimental data of lattice specific heat C Õ ŽT . into Ž1.4.. 2. Calculate the the coefficient Cl 4 by expanding Q0 Ž x . in the Hermitian function set Ž1.12.. 3. Obtain the phonon spectrum GŽ v . directly from the universal set G l Ž v .4 by Ž1.16.. By inputting the experimental data of specific heat of YBCO w8x and considering the unique existence theorem and asymptotic behavior control, the phonon spectrum of YBCO is obtained and shown in the Fig. 2. The results show that: 1. The phonon spectrum of YBCO obtained by this inversion formula has an acoustic peak, which is in good agreement with that obtained by the neutron inelastic scattering w12x, including the location Žin about 20 meV. and height of the peak. 2. It is well known that the phonon spectrum changes with the oxygen concentrations. Even for the inelastic neutron scattering, the structure of the curves are complicated. What we are expecting is that SPI can obtain the average feature of the spectrum. It is very interesting that our results show a plateau in high frequencies. The plateau covers the region from 40 meV to 80 meV, just likes that observed in the experiments of neutron scattering. This is in qualitatiÕely agreement with the
experiments w12x. It is also interesting that there is a shoulder or hint of small peak at about 50 meV, which is in consistent with the experiments to a certain extent. In fact in the high frequencies, the experiments shows a peak also. It is easy to be understood that in this region, the accuracy of the results are pretty low. Because the integral kernel in Eq. Ž1.3.:
v
ž / T
2
exp Ž vrT . exp Ž vrT . y 1
2
,
goes down exponentially. If one tries to get information in high frequencies, the higher accuracies of the experimental data will be needed. Finally, we need to emphasize that our SPI exact solution formula is of closed form. The Debye spectrum and Einstein spectrum can be exactly obtained from this formula, including the important cutoff factor w1x. Comparing to Chen’s Mobius inversion formula w6x, one can not get cutoff factor, although the v 2 spectrum is obtained. Also the values of Mobius are unknown for large n, because it is hard to determine weather they are prime numbers or not. From practical applications point of view, in our SPI formula, one only need to perform Fourier transform once, while in Mobius inversion formula, an infinite number of Laplace and inverse Laplace transform is
D. XianXi et al.r Physics Letters A 264 (1999) 68–73
needed, which make this method difficult to be applied in numerical calculation. This is the first try to get a practical phonon spectrum from the specific heat. After overcome a series of difficulties, the result from the exact solution formula Ž1.3. or Ž1.16. – Ž1.17. is in good agreement with the experiments by neutron inelastic scattering w12x. Therefore, the theory of the specific heat phonon spectrum inversion can be believed to be a promising theory and method in physics.
Acknowledgements The Authors would like to acknowledge Professors C.N. Yang, N.X. Chen,Yu Lu, W. Evenson and Ma Yongli for their significant discussions and ICTP for their hospitality. This work is in Projects 19834010 and 19575009 and supported by National Natural Science Foundation of China.
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References w1x Dai Xianxi, Xu Xinwen, Dai Jiqiong, Proc. Beijing International Conference on High Tc Superconductivity, Beijing, China, September 4–8, 1989, pp. 521–524. w2x Dai Xianxi, Xu Xinwen, Dai Jiqiong, Phys. Lett. A. 147 Ž1990. 445. w3x E.W. Montroll, J. Chem. Phys. 10 Ž1942. 218. w4x I.M. Lifshitz, Zh. Eksp. Theor. Fiz 26 Ž1955. 551. w5x R.G. Chambers, Proc. Phys. Soc. 78 Ž1961. 941. w6x N.X. Chen, Phys. Rev. Lett. 64 Ž1990. 1193. w7x J. Hadamard, Bull. Soc. Math. France 24 Ž1896. 199. w8x V.G. Bessergenev, Yu.A. Kovalevskaya, V.N. Naumov, G.I. Frolova, Physica C 245 Ž1995. 36. w9x D.M. Ginsberg, Physical Properties of High Temperature Superconductors I-IV, World Scientific, Singapore et al., 1989–1994. w10x Landau, Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, Beijing et al., 1980. w11x Dai XianXi, JiXin Dai, JiQiong Dai, Phys. Rev. A 55 Ž1997. 2617. w12x B. Renker, F. Gompf, E. Gering, D. Ewert, H. Rietschel, A. Dianoux, Z. Phys. B Condensed Matter 73 Ž1988. 309.