## 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
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: failed
8106885760:
5838268428: N/A
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:
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:
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:
2897612567:
7701249282:
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: error for dim with 5789289057
2897612567:
5646314683:
5789289057: N/A
2897612567:
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:
7701249282:
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:
10 declare assumption
1. 4107032818; locally 6901924:
$$E_{\rm Rydberg} = E$$
$$pdg_{9838} = pdg_{2241}$$
no validation is available for declarations 4107032818:
4107032818:
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: failed
3935058307:
9640720571: N/A
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: failed
9640720571: failed
5789289057: error for dim with 5789289057
5838268428: N/A
9640720571: N/A
5789289057: N/A
Physics Derivation Graph: Steps for upper limit on velocity in condensed matter

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
9863 variable m
$$m$$
real
• mass: 1
mass of atom or molecule
8
4567 constant c
$$c$$
['real']
• length: 1
• time: -1
speed of light in vacuum 299792458   meters/second
32
3141 constant \pi
$$\pi$$
['real'] dimensionless pi 3.1415   dimensionless
72
3285 variable A
$$A$$
real
• mass: 1
atomic mass
3
1370 constant \alpha
$$\alpha$$
['real'] dimensionless fine-structure constant 1/137.03599999   dimensionless
5
1999 constant e
$$e$$
['real']
• electric charge: 1
charge of an electron 1.602*10^{-19}   Columb
6
9838 variable E_{\rm Rydberg}
$$E_{\rm Rydberg}$$
real dimensionless Rydberg energy
2
2241 variable E
$$E$$
real dimensionless bonding energy
• str_note
6
1054 constant \hbar
$$\hbar$$
['real']
• length: 2
• mass: 1
• time: -1
Reduced Planck's constant 1.0545718*10^{-34}   meter^2 kilogram second^-1
33
2515 constant m_e
$$m_e$$
real
• mass: 1
mass of electron 9.1093837015E^{-31}   kg
7
2077 variable v
$$v$$
real
• length: 1
• time: -1
longitudinal speed of sound in condensed matter
9
4635 constant v_u
$$v_u$$
real
• length: 1
• time: -1
upper limit on velocity in condensed matter 36100   m/s
1
6235 variable f
$$f$$
real dimensionless proportionality constant
• str_note
5
3033 variable G
$$G$$
real
• length: -1
• mass: 1
• time: -2
shear modulus
2
1466 variable K
$$K$$
real
• length: -1
• mass: 1
• time: -2
bulk modulus
2
3935 variable \rho
$$\rho$$
real
• length: -3
• mass: 1
density
7
5854 variable a
$$a$$
real
• length: 1
atomic separation
• str_note
5
5916 constant m_p
$$m_p$$
real
• mass: 1
mass of proton 1.67262192369E^{-27}   kg
3
7940 constant \epsilon_0
$$\epsilon_0$$
real
• electric charge: 2
• length: -3
• mass: -1
• time: 2
vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum 8.8541878128E-{12}   F/m
14
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 1346919: name 'electric_charge' is not defined
• in step 1452028: name 'electric_charge' is not defined
• in step 1452028: name 'electric_charge' is not defined
• in step 2951905: name 'electric_charge' is not defined
• in step 5023393: name 'electric_charge' is not defined
• in step 5023393: name 'electric_charge' is not defined
• in step 5522705: name 'electric_charge' is not defined
• in step 5522705: name 'electric_charge' is not defined
• in step 7588540: name 'electric_charge' is not defined
• in step 7588540: name 'electric_charge' is not defined
• in step 9573616: name 'electric_charge' is not defined
• in step 9573616: name 'electric_charge' is not defined