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review derivation: upper limit on velocity in condensed matter

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Notes for this derivation:
https://arxiv.org/pdf/2004.04818.pdf

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 4560648264; locally 1719451:
    \(v = \sqrt{ \frac{K + (4/3) G}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{1466} + \frac{4 pdg_{3033}}{3}}{pdg_{3935}}}\)
no validation is available for declarations 4560648264:
4560648264:
2 declare initial expr
  1. 9376481176; locally 2178289:
    \(K = f \frac{E}{a^3}\)
    \(K = \frac{pdg_{2241} pdg_{6235}}{pdg_{5854}^{3}}\)
no validation is available for declarations 9376481176:
9376481176:
3 drop non-dominant term
  1. 4560648264; locally 1719451:
    \(v = \sqrt{ \frac{K + (4/3) G}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{1466} + \frac{4 pdg_{3033}}{3}}{pdg_{3935}}}\)
  1. 9674924517:
    \(K >> G\)
    \(pdg_{1466} > pdg_{3033}\)
  1. 6504442697; locally 9155336:
    \(v = \sqrt{ \frac{K}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{K}{pdg_{3935}}}\)
no check performed 4560648264:
6504442697:
4560648264:
6504442697:
4 substitute LHS of expr 1 into expr 2
  1. 9376481176; locally 2178289:
    \(K = f \frac{E}{a^3}\)
    \(K = \frac{pdg_{2241} pdg_{6235}}{pdg_{5854}^{3}}\)
  2. 6504442697; locally 9155336:
    \(v = \sqrt{ \frac{K}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{K}{pdg_{3935}}}\)
  1. 8090924099; locally 5077893:
    \(v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{2241} pdg_{6235}}{pdg_{3935} pdg_{5854}^{3}}}\)
valid 9376481176:
6504442697:
8090924099:
9376481176:
6504442697:
8090924099:
5 declare initial expr
  1. 8908736791; locally 2438445:
    \(\rho = \frac{m}{a^3}\)
    \(pdg_{3935} = \frac{pdg_{9863}}{pdg_{5854}^{3}}\)
no validation is available for declarations 8908736791:
8908736791:
6 multiply both sides by
  1. 8908736791; locally 2438445:
    \(\rho = \frac{m}{a^3}\)
    \(pdg_{3935} = \frac{pdg_{9863}}{pdg_{5854}^{3}}\)
  1. 2397692197:
    \(a^3\)
    \(pdg_{5854}^{3}\)
  1. 8688588981; locally 7834577:
    \(a^3 \rho = m\)
    \(pdg_{3935} pdg_{5854}^{3} = pdg_{9863}\)
valid 8908736791:
8688588981:
8908736791:
8688588981:
7 substitute RHS of expr 1 into expr 2
  1. 8688588981; locally 7834577:
    \(a^3 \rho = m\)
    \(pdg_{3935} pdg_{5854}^{3} = pdg_{9863}\)
  2. 8090924099; locally 5077893:
    \(v = \sqrt{ \left( f\frac{E}{a^3} \right) \frac{1}{\rho} }\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{2241} pdg_{6235}}{pdg_{3935} pdg_{5854}^{3}}}\)
  1. 7837519722; locally 5020923:
    \(v = \sqrt{f} \sqrt{\frac{E}{m}}\)
    \(pdg_{2077} = \sqrt{pdg_{6235}} \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}\)
LHS diff is 0 RHS diff is -sqrt(pdg6235)*sqrt(pdg2241/pdg9863) + sqrt(pdg2241*pdg6235/(pdg3935*pdg5854**3)) 8688588981:
8090924099:
7837519722:
8688588981:
8090924099:
7837519722:
8 drop non-dominant term
  1. 7837519722; locally 5020923:
    \(v = \sqrt{f} \sqrt{\frac{E}{m}}\)
    \(pdg_{2077} = \sqrt{pdg_{6235}} \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}\)
  1. 3685779219:
    \(\sqrt{f} \approx 2\)
    \(2 approx \sqrt{pdg_{6235}}\)
  1. 9854442418; locally 4534919:
    \(v = \sqrt{\frac{E}{m}}\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}\)
no check performed 7837519722:
9854442418:
7837519722:
9854442418:
9 declare initial expr
  1. 1556389363; locally 5961293:
    \(E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
    \(pdg_{9838} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}\)
no validation is available for declarations 1556389363:
1556389363:
10 declare assumption
  1. 4107032818; locally 6901924:
    \(E_{\rm Rydberg} = E\)
    \(pdg_{9838} = pdg_{2241}\)
no validation is available for declarations 4107032818:
4107032818:
11 declare initial expr
  1. 8106885760; locally 9431422:
    \(\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)
    \(pdg_{1370} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{4567} pdg_{7940}}\)
no validation is available for declarations 8106885760:
8106885760:
12 multiply both sides by
  1. 8106885760; locally 9431422:
    \(\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}\)
    \(pdg_{1370} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{4567} pdg_{7940}}\)
  1. 8857931498:
    \(c\)
    \(pdg_{4567}\)
  1. 5838268428; locally 6181437:
    \(\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)
    \(pdg_{1370} pdg_{4567} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{7940}}\)
valid 8106885760:
5838268428:
8106885760:
5838268428:
13 substitute LHS of expr 1 into expr 2
  1. 4107032818; locally 6901924:
    \(E_{\rm Rydberg} = E\)
    \(pdg_{9838} = pdg_{2241}\)
  2. 1556389363; locally 5961293:
    \(E_{\rm Rydberg} = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
    \(pdg_{9838} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}\)
  1. 3291685884; locally 3642765:
    \(E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
    \(pdg_{2241} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}\)
valid 4107032818:
1556389363:
3291685884:
4107032818:
1556389363:
3291685884:
14 substitute LHS of expr 1 into expr 2
  1. 3291685884; locally 3642765:
    \(E = \frac{ m_e e^4 }{ 32 \pi^2 \epsilon_0^2 \hbar^2}\)
    \(pdg_{2241} = \frac{pdg_{1999}^{4} pdg_{2515}}{32 pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2}}\)
  2. 9854442418; locally 4534919:
    \(v = \sqrt{\frac{E}{m}}\)
    \(pdg_{2077} = \sqrt{\frac{pdg_{2241}}{pdg_{9863}}}\)
  1. 3935058307; locally 2063484:
    \(v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)
    \(pdg_{2077} = \frac{\sqrt{2} \sqrt{\frac{pdg_{1999}^{4} pdg_{2515}}{pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2} pdg_{9863}}}}{8}\)
valid 3291685884:
9854442418:
3935058307:
3291685884:
9854442418:
3935058307:
15 simplify
  1. 3935058307; locally 2063484:
    \(v = \sqrt{ \frac{m_e}{m} \frac{e^4}{32 \pi^2 \epsilon_0^2 \hbar^2} }\)
    \(pdg_{2077} = \frac{\sqrt{2} \sqrt{\frac{pdg_{1999}^{4} pdg_{2515}}{pdg_{1054}^{2} pdg_{3141}^{2} pdg_{7940}^{2} pdg_{9863}}}}{8}\)
  1. 9640720571; locally 4586348:
    \(v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\)
    \(pdg_{2077} = \frac{\sqrt{2} pdg_{1999}^{2} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{8 pdg_{1054} pdg_{3141} pdg_{7940}}\)
LHS diff is 0 RHS diff is sqrt(2)*(pdg1054*pdg3141*pdg7940*sqrt(pdg1999**4*pdg2515/(pdg1054**2*pdg3141**2*pdg7940**2*pdg9863)) - pdg1999**2*sqrt(pdg2515/pdg9863))/(8*pdg1054*pdg3141*pdg7940) 3935058307:
9640720571:
3935058307:
9640720571:
16 substitute LHS of expr 1 into expr 2
  1. 5838268428; locally 6181437:
    \(\alpha c = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar}\)
    \(pdg_{1370} pdg_{4567} = \frac{pdg_{1999}^{2}}{4 pdg_{1054} pdg_{3141} pdg_{7940}}\)
  2. 9640720571; locally 4586348:
    \(v = \frac{e^2}{4 \pi \epsilon_0 \hbar} \sqrt{\frac{m_e}{2 m}}\)
    \(pdg_{2077} = \frac{\sqrt{2} pdg_{1999}^{2} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{8 pdg_{1054} pdg_{3141} pdg_{7940}}\)
  1. 5789289057; locally 5883117:
    \(v = \alpha c \sqrt{ \frac{m_e}{2 m} }\)
    \(pdg_{2077} = \frac{\sqrt{2} pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{2}\)
LHS diff is 0 RHS diff is sqrt(2)*sqrt(pdg2515/pdg9863)*(-4*pdg1054*pdg1370*pdg3141*pdg4567*pdg7940 + pdg1999**2)/(8*pdg1054*pdg3141*pdg7940) 5838268428:
9640720571:
5789289057:
5838268428:
9640720571:
5789289057:
17 declare initial expr
  1. 5646314683; locally 6979804:
    \(m = A m_p\)
    \(pdg_{9863} = pdg_{3285} pdg_{5916}\)
no validation is available for declarations 5646314683:
5646314683:
18 substitute LHS of expr 1 into expr 2
  1. 5646314683; locally 6979804:
    \(m = A m_p\)
    \(pdg_{9863} = pdg_{3285} pdg_{5916}\)
  2. 5789289057; locally 5883117:
    \(v = \alpha c \sqrt{ \frac{m_e}{2 m} }\)
    \(pdg_{2077} = \frac{\sqrt{2} pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{9863}}}}{2}\)
  1. 2897612567; locally 8323044:
    \(v = \alpha c \sqrt{ \frac{m_e}{A m_p} }\)
    \(pdg_{2077} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{3285} pdg_{5916}}}\)
LHS diff is 0 RHS diff is pdg1370*pdg4567*sqrt(pdg2515/(pdg3285*pdg5916))*(-2 + sqrt(2))/2 5646314683:
5789289057:
2897612567:
5646314683:
5789289057:
2897612567:
19 maximum of expr
  1. 2897612567; locally 8323044:
    \(v = \alpha c \sqrt{ \frac{m_e}{A m_p} }\)
    \(pdg_{2077} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{3285} pdg_{5916}}}\)
  1. 6259833695:
    \(A\)
    \(pdg_{3285}\)
  1. 7701249282; locally 9568206:
    \(v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }\)
    \(pdg_{4635} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{5916}}}\)
no check performed 2897612567:
7701249282: error for dim with 7701249282
2897612567:
7701249282: N/A
20 declare final expr
  1. 7701249282; locally 9568206:
    \(v_u = \alpha c \sqrt{ \frac{m_e}{m_p} }\)
    \(pdg_{4635} = pdg_{1370} pdg_{4567} \sqrt{\frac{pdg_{2515}}{pdg_{5916}}}\)
no validation is available for declarations 7701249282: error for dim with 7701249282
7701249282: N/A
Physics Derivation Graph: Steps for upper limit on velocity in condensed matter

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
7940 constant \epsilon_0
\(\epsilon_0\)
real
  • length: -3
  • time: 2
  • mass: -1
  • charge: 2
vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum 8.8541878128E-{12}   F/m
14
6235 variable f
\(f\)
real
proportionality constant
  • str_note
5
1370 constant \alpha
\(\alpha\)
['real']
fine-structure constant 1/137.03599999   dimensionless
5
9863 variable m
\(m\)
real
  • mass: 1
mass of atom or molecule 8
2241 variable E
\(E\)
real
bonding energy
  • str_note
6
1466 variable K
\(K\)
real
  • length: -1
  • time: -2
  • mass: 1
bulk modulus 2
5916 constant m_p
\(m_p\)
real
  • mass: 1
mass of proton 1.67262192369E^{-27}   kg
3
1999 constant e
\(e\)
['real']
  • charge: 1
charge of an electron 1.602*10^{-19}   Columb
6
4635 constant v_u
\(v_u\)
real
  • length: 1
  • time: -1
upper limit on velocity in condensed matter 36100   m/s
1
5854 variable a
\(a\)
real
  • length: 1
atomic separation
  • str_note
5
1054 constant \hbar
\(\hbar\)
['real']
  • length: 2
  • time: -1
  • mass: 1
Reduced Planck's constant 1.0545718*10^{-34}   meter^2 kilogram second^-1
33
3033 variable G
\(G\)
real
  • length: -1
  • time: -2
  • mass: 1
shear modulus 2
2515 constant m_e
\(m_e\)
real
  • mass: 1
mass of electron 9.1093837015E^{-31}   kg
7
3285 variable A
\(A\)
real
  • mass: 1
atomic mass 3
2077 variable v
\(v\)
real
  • length: 1
  • time: -1
longitudinal speed of sound in condensed matter 9
4567 constant c
\(c\)
['real']
  • length: 1
  • time: -1
speed of light in vacuum 299792458   meters/second
32
9838 variable E_{\rm Rydberg}
\(E_{\rm Rydberg}\)
real
Rydberg energy 2
3935 variable \rho
\(\rho\)
real
  • length: -3
  • mass: 1
density 7
3141 constant \pi
\(\pi\)
['real']
pi 3.1415   dimensionless
55
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