Question

What limits the large number of Bravais lattices to only 14? Discuss it with an example​

Answer 1

Answer:

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The Bravais lattices come from unit cells which have an internal symmetry. ... There are again not so many possibilities to have an internal symmetry, so this only makes 14 Bravais lattices out of the 7 crystal systems.

Answer 2

Answer:

ɪɴ ɢᴇᴏᴍᴇᴛʀʏ ᴀɴᴅ ᴄʀʏsᴛᴀʟʟᴏɢʀᴀᴘʜʏ, ᴀ ʙʀᴀᴠᴀɪs ʟᴀᴛᴛɪᴄᴇ, ɴᴀᴍᴇᴅ ᴀғᴛᴇʀ ᴀᴜɢᴜsᴛᴇ ʙʀᴀᴠᴀɪs (),[] ɪs ᴀɴ ɪɴғɪɴɪᴛᴇ ᴀʀʀᴀʏ ᴏғ ᴅɪsᴄʀᴇᴛᴇ ᴘᴏɪɴᴛs ɢᴇɴᴇʀᴀᴛᴇᴅ ʙʏ ᴀ sᴇᴛ ᴏғ ᴅɪsᴄʀᴇᴛᴇ ᴛʀᴀɴsʟᴀᴛɪᴏɴ ᴏᴘᴇʀᴀᴛɪᴏɴs ᴅᴇsᴄʀɪʙᴇᴅ ɪɴ ᴛʜʀᴇᴇ ᴅɪᴍᴇɴsɪᴏɴᴀʟ sᴘᴀᴄᴇ ʙʏ:

{\ᴅɪsᴘʟᴀʏsᴛʏʟᴇ \ᴍᴀᴛʜʙғ {ʀ} =ɴ_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+ɴ_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+ɴ_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}}\ᴍᴀᴛʜʙғ{ʀ} = ɴ_{}\ᴍᴀᴛʜʙғ{ᴀ}_{} + ɴ_{}\ᴍᴀᴛʜʙғ{ᴀ}_{} + ɴ_{}\ᴍᴀᴛʜʙғ{ᴀ}_{}

()

ᴡʜᴇʀᴇ ᴛʜᴇ ɴɪ ᴀʀᴇ ᴀɴʏ ɪɴᴛᴇɢᴇʀs ᴀɴᴅ ᴀɪ ᴀʀᴇ ᴘʀɪᴍɪᴛɪᴠᴇ ᴠᴇᴄᴛᴏʀs ᴡʜɪᴄʜ ʟɪᴇ ɪɴ ᴅɪғғᴇʀᴇɴᴛ ᴅɪʀᴇᴄᴛɪᴏɴs (ɴᴏᴛ ɴᴇᴄᴇssᴀʀɪʟʏ ᴍᴜᴛᴜᴀʟʟʏ ᴘᴇʀᴘᴇɴᴅɪᴄᴜʟᴀʀ) ᴀɴᴅ sᴘᴀɴ ᴛʜᴇ ʟᴀᴛᴛɪᴄᴇ. ᴛʜᴇ ᴄʜᴏɪᴄᴇ ᴏғ ᴘʀɪᴍɪᴛɪᴠᴇ ᴠᴇᴄᴛᴏʀs ғᴏʀ ᴀ ɢɪᴠᴇɴ ʙʀᴀᴠᴀɪs ʟᴀᴛᴛɪᴄᴇ ɪs ɴᴏᴛ ᴜɴɪǫᴜᴇ. ᴀ ғᴜɴᴅᴀᴍᴇɴᴛᴀʟ ᴀsᴘᴇᴄᴛ ᴏғ ᴀɴʏ ʙʀᴀᴠᴀɪs ʟᴀᴛᴛɪᴄᴇ ɪs ᴛʜᴀᴛ, ғᴏʀ ᴀɴʏ ᴄʜᴏɪᴄᴇ ᴏғ ᴅɪʀᴇᴄᴛɪᴏɴ, ᴛʜᴇ ʟᴀᴛᴛɪᴄᴇ ᴡɪʟʟ ᴀᴘᴘᴇᴀʀ ᴇxᴀᴄᴛʟʏ ᴛʜᴇ sᴀᴍᴇ ғʀᴏᴍ ᴇᴀᴄʜ ᴏғ ᴛʜᴇ ᴅɪsᴄʀᴇᴛᴇ ʟᴀᴛᴛɪᴄᴇ ᴘᴏɪɴᴛs ᴡʜᴇɴ ʟᴏᴏᴋɪɴɢ ɪɴ ᴛʜᴀᴛ ᴄʜᴏsᴇɴ ᴅɪʀᴇᴄᴛɪᴏɴ.

ɪɴ ᴄʀʏsᴛᴀʟʟᴏɢʀᴀᴘʜʏ, ᴛʜᴇ ʙʀᴀᴠᴀɪs ʟᴀᴛᴛɪᴄᴇ ᴄᴏɴᴄᴇᴘᴛ ᴏғ ᴀɴ ɪɴғɪɴɪᴛᴇ ᴀʀʀᴀʏ ᴏғ ᴅɪsᴄʀᴇᴛᴇ ᴘᴏɪɴᴛs ɪs ᴇxᴘᴀɴᴅᴇᴅ ᴜsɪɴɢ ᴛʜᴇ ᴄᴏɴᴄᴇᴘᴛ ᴏғ ᴀ ᴜɴɪᴛ ᴄᴇʟʟ ᴡʜɪᴄʜ ɪɴᴄʟᴜᴅᴇs ᴛʜᴇ sᴘᴀᴄᴇ ɪɴ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴅɪsᴄʀᴇᴛᴇ ʟᴀᴛᴛɪᴄᴇ ᴘᴏɪɴᴛs ᴀs ᴡᴇʟʟ ᴀs ᴀɴʏ ᴀᴛᴏᴍs ɪɴ ᴛʜᴀᴛ sᴘᴀᴄᴇ. ᴛʜᴇʀᴇ ᴀʀᴇ ᴛᴡᴏ ᴍᴀɪɴ ᴛʏᴘᴇs ᴏғ ᴜɴɪᴛ ᴄᴇʟʟs: ᴘʀɪᴍɪᴛɪᴠᴇ ᴜɴɪᴛ ᴄᴇʟʟs ᴀɴᴅ ɴᴏɴ-ᴘʀɪᴍɪᴛɪᴠᴇ ᴜɴɪᴛ ᴄᴇʟʟs.

ᴀ ᴘʀɪᴍɪᴛɪᴠᴇ ᴜɴɪᴛ ᴄᴇʟʟ ғᴏʀ ᴀ ɢɪᴠᴇɴ ʙʀᴀᴠᴀɪs ʟᴀᴛᴛɪᴄᴇ ᴄᴀɴ ʙᴇ ᴄʜᴏsᴇɴ ɪɴ ᴍᴏʀᴇ ᴛʜᴀɴ ᴏɴᴇ ᴡᴀʏ (ᴇᴀᴄʜ ᴡᴀʏ ʜᴀᴠɪɴɢ ᴀ ᴅɪғғᴇʀᴇɴᴛ sʜᴀᴘᴇ), ʙᴜᴛ ᴇᴀᴄʜ ᴡᴀʏ ᴡɪʟʟ ʜᴀᴠᴇ ᴛʜᴇ sᴀᴍᴇ ᴠᴏʟᴜᴍᴇ ᴀɴᴅ ᴇᴀᴄʜ ᴡᴀʏ ᴡɪʟʟ ʜᴀᴠᴇ ᴛʜᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴛʜᴀᴛ ᴀ ᴏɴᴇ-ᴛᴏ-ᴏɴᴇ ᴄᴏʀʀᴇsᴘᴏɴᴅᴇɴᴄᴇ ᴄᴀɴ ʙᴇ ᴇsᴛᴀʙʟɪsʜᴇᴅ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴘʀɪᴍɪᴛɪᴠᴇ ᴜɴɪᴛ ᴄᴇʟʟs ᴀɴᴅ ᴛʜᴇ ᴅɪsᴄʀᴇᴛᴇ ʟᴀᴛᴛɪᴄᴇ ᴘᴏɪɴᴛs. ᴛʜᴇ ᴏʙᴠɪᴏᴜs ᴘʀɪᴍɪᴛɪᴠᴇ ᴄᴇʟʟ ᴛᴏ ᴀssᴏᴄɪᴀᴛᴇ ᴡɪᴛʜ ᴀ ᴘᴀʀᴛɪᴄᴜʟᴀʀ ᴄʜᴏɪᴄᴇ ᴏғ ᴘʀɪᴍɪᴛɪᴠᴇ ᴠᴇᴄᴛᴏʀs ɪs ᴛʜᴇ ᴘᴀʀᴀʟʟᴇʟᴇᴘɪᴘᴇᴅ ғᴏʀᴍᴇᴅ ʙʏ ᴛʜᴇᴍ.[] ᴛʜᴀᴛ ɪs, ᴛʜᴇ sᴇᴛ ᴏғ ᴀʟʟ ᴘᴏɪɴᴛs ʀ ᴏғ ᴛʜᴇ ғᴏʀᴍ:

{\ᴅɪsᴘʟᴀʏsᴛʏʟᴇ \ᴍᴀᴛʜʙғ {ʀ} =x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}\ǫǫᴜᴀᴅ {\ᴛᴇxᴛ{ᴡʜᴇʀᴇ }}\ʟᴇǫ x_{ɪ}<}{\ᴅɪsᴘʟᴀʏsᴛʏʟᴇ \ᴍᴀᴛʜʙғ {ʀ} =x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}+x_{}\ᴍᴀᴛʜʙғ {ᴀ} _{}\ǫǫᴜᴀᴅ {\ᴛᴇxᴛ{ᴡʜᴇʀᴇ }}\ʟᴇǫ x_{ɪ}<}

()

ᴜsɪɴɢ ᴛʜᴇ ᴘᴀʀᴀʟʟᴇʟᴇᴘɪᴘᴇᴅ ᴅᴇғɪɴᴇᴅ ʙʏ ᴛʜᴇ ᴘʀɪᴍɪᴛɪᴠᴇ